Worldviews, Science and Us: Philosophy and Complexity

Worldviews,

Science and Us Philosophy and Complexity

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editors

Carlos Gershenson Brussels Free University, Belgium

Diederik Aerts Brussels Free University, Belgium

Bruce Edmotads Manchester Metropolitan University Business School,

UK

University o f Liverpool, Ul< 11- 14 September 2005

Worldviews,I 4 d

Science and Us 5

Philosophy and Complexity

World Scientific NEW JERSEY * LONDON

SINGAPORE * BElJlNG

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SHANGHAI * HONG KONG

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Published by World Scientific Publishing Co. Re. Ltd. 5 Toh Tuck Link, Singapore 596224 USA ofice: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK ofice: 57 Shelton Street, Covent Garden, London WC2H 9HE

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WORLDVIEWS, SCIENCE AND US Philosophy and Complexity Copyright 0 2007 by World Scientific Publishing Co. Re. Ltd.

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ISBN-13 978-981-270-548-8 ISBN-10 981-270-548-1

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CONTENTS

Introduction Carlos Gershenson, Diederik Aerts and Bruce Edmonds

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Restricted Complexi$y, General Complexity Edgar Morin

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Complexity Science as an Aspect of the Complexity of Science Don C. Mikulecky

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On the Importance of a Certain Slowness Paul Cilliers

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Simplicity is Not Truth-Indicative Bruce Edmonds

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Can the Whole be More than the Computation of the Parts? A Reflection on Emergence Camilo Olaya

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Why Diachronically Emergent Properties Must Also Be Salient Cyrille Imbert

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On the Relativity of Recognising the Products of Emergence and the Nature of the Hierarchy of Physical Matter Kurt A . Richardson

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Truth in Complex Adaptive Systems Models should be Based on Proof by Constructive Verification David Shipworth

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Complexity as an Epistemic Revolution: Considerations on the New Science in the Context of Western Intellectual History Damian Popolo

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Metaphors and Method: Epistemological Considerations on Complexity Science Rodrigo Zeidan and Maria Fonseca

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Some Problems for an Ontology of Complexity Michael McGuire

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How to Love the Bomb - Trying to Solve the Prisoner’s Dilemma with Evolutionary Game Theory Vasco Castela

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Physical Complexity and Cognitive Evolution Peter Jedlicka

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Informational Dynamic Systems: Autonomy, Information, Function Walter Riofrio

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Grasping the Complexity of Living Systems Through Integrative Levels and Hierarchies J. M. Siqueiros and Jon Umerez

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Simulation as Formal and Generative Social Science: The Very Idea Nuno David, Jaime S i m i o Sachman and Helder Coelho

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A Compromise Between Reductionism and Non-Reductionism Eray Ozkural

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The Complexity of Information-Processing Tasks in Vision John Symons

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On the Possible Computational Power of the Human Mind Hector Zenil and Francisco Hernandez- Quiroz

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How Does Complex Mathemical Theory Arise? Phylogenetic and Cultural Origins of Algebra? Helen De Cruz

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INTRODUCTION

CARLOS GERSHENSON~,DIEDERIK AERTS~,AND BRUCE EDMONDS~ Centrum Leo Apostel, Vrije Universiteit Brussel Centre for Policy Modelling, Manchester Metropolitan University

Scientific, technological, and cultural changes have always had an impact upon philosophy. They can force a change in the way we perceive the world, reveal new kinds of phenomena to be understood and provide new ways of understanding phenomena. Complexity science, immersed in a culture of information, is having a diverse but significant impact upon philosophy’. It has highlighted classes of phenomena composed of multiply interacting parts where there may be no simple adequate analytic model, so that new ways of modelling and understanding these phenomena become necessary. In turn these new methods, based upon complex simulation, have brought new phenomena into the domain of science by making possible their formal representation. The result has been a rapid co-development of new domains and new methods that suggests one of the “paradigm” shifts described by Kuhn2. Previous ideas do not necessarily sit comfortably with the new paradigm, resulting in new ideas or new interpretations of old ideas, that is to say the “knock-on” effect is a larger evolution of our worldviews. In this book, researchers from different backgrounds join efforts to update thinking upon philosophical questions with developments in the scientific study of complex systems. The contributions are quite diverse, in the sense that they focus on a wide range of topics. However, they all share the common goal of increasing our understandings and improving our descriptions of our complex world. Of course, the background to this is that there are many that hold worldviews that are essentially untouched by complexity science. To take an example there is the view that our world only appears complex but would, if we could only understand it in the right way, be revealed as essentially simple. Whether the world is simple or complex the fact is that, as humans, there are severe limits to the complexity which we can comprehend. Thus,

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regardless of how complex the world isa, there is an inescapable tension between the relative simplicity of human cognition and the apparent (or real) complexity of the phenomena they are trying to understand. The different philosophies and approaches can, to a large extent, be seen as responses to this tension. Thus some seek to show that this tension is not feasibly soluble, which may have consequences for how we think about or manage the world we inhabit. Some seek to explain productive ways forward-how one might explain and understand the complexity we observe. Others take a step back to examine what consequences the tension itself has upon the scientific method. This volume is the fourth in a series entitled ‘Worldviews, Science and Us’ published by World Scientific. The series collects interdisciplinary articles with the global aim of stimulating new insights about the world and the place of human in this world, and also about the role science plays in the construction of new aspects of worldviews. Many chapters in this volume are derived from presentations given at the Philosophy and Complexity session of the Complexity, Science and Society conference, held in Liverpool, UK, between September llthand 14th 2005. We appreciate the work of the program committee for that session, which was able to select relevant works from all the submissions received. The program committee members were: William Bechtel, Mark Bedau, Jacques Dubucs, Bruce Edmonds, Carlos Gershenson, Francis Heylighen, Alicia Juarrero, Michael Lissack, Chris Lucas, Edgar Morin, Robert Pennock, Kurt Richardson, John Symons, Pedro Sotolongo, Jean Paul Van Bendegem, Franz Wuketits, and Roger Young. We would also like to thank Robert Geyer, Jan Bogg, and Abbie Badcock for their organizational support at the conference. Clem6nt Vidal helped in polishing the translation of Edgar Morinls chapter. Finally the speakers and participants of the session and the quality of their discussions motivated us to compile this volume. After the conference, the participants were invited to prepare a manuscript and more people were invited to contribute to the volume. An internal review process followed, where contributors reviewed other submissions. From these, the actual chapters were selected to be included in the volume. In what follows, summaries of the chapters are given. They are diverse enough to make it difficult to classify in different topic areas, so the order is simply dictated by similarity. aThis is obviously not a question that can itself be settled in a scientific way.

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From a philosophical perspective, Edgar Morin reviews the historical development of complexity, noting that the current accepted notion of complexity is restricted. Morin aims for a generalized complexity, which implies an integrative change in our way of thinking and knowing, in education and science. Don Mikulecky discusses similar issues from a scientific perspective, putting an accent on scientific modeling. Paul Cilliers notes the importance of a certain slowness in modern life, since current practices seem to promote a cult of speed, where faster is better. However, this is far from being the case, as proper adaptation requires certain slowness to maintain the integrity of a system. In a similar spirit, Bruce Edmonds refutes Occam’s razor, arguing that there is no reason to suppose that a simpler theory is more likely to be true. Emergence is discussed by several authors. Camilo Olaya reflects on the use of rule-based modeling and process thought to understand complex systems. Cyrille Imbert develops the notion of salience as a requirement for diachronically emergent properties. Kurt Richardson explores the ontological status of emergent products. David Shipworth proposes the use of proof by constructive verification to determine truth in complex adaptive systems, since a realist truth model of emergent properties is not feasible, thus favoring epistemology over ontology. Damian Popolo discusses the impact of complexity in science as an epistemic revolution, while Rodrigo Zeidan and Maria Fonseca present epistemological considerations on complexity science. Michael McGuire discusses ontological problems of complexity. Vasco Castela uses evolutionary game theory to show the importance of the role of emotions in the evolution of cooperation and social modeling. Peter Jedlicka studies the evolutionary increase of physical complexity and its relationship with cognition. Walter Riofrio proposes ‘Informational Dynamic Systems’ to study the pre-biotic world, explaining naturalistically the physical emergence of functions and information. Jeszis Sigueiros and Jon Umerez present a conceptual overview of different perspectives that were developed during the 20th century to understand biological complexity. Nuno David, Jaime S i m i o Sichman and Helder Coelho propose the use of intentional knowledge to successfully simulate social systems, as opposed to formal and empirical perspectives. Eray Ozkural uses algorithmic information theory to propose a ‘mild’ non-reductionism, with implications for philosophy of mind. John Symons questions the use of the information-processingmodel of the mind, focussing

4 on vision. Hkctor Zenil and Francisco Herna’ndez Quiroz discuss the possibility of using artificial neural network models to characterize the computational power of the human mind. Helen De Cruz argues for an active externalism as a requirement for the emergence of algebra, studying its emergence in different cultural settings.

References 1. Heylighen, F., P. Cilliers, and C. Gershenson (forthcoming 2007). “Complexity and Philosophy”. In Bogg, J. and R. Geyer (eds.) Complexity, Science and Society. Radcliffe Publishing, Oxford. 2. Kuhn, Thomas S. The Structure of Scientific Revolutions. University of Chicago Press, 2nd Ed.

RESTRICTED COMPLEXITY, GENERAL COMPLEXITY*

EDGAR MORIN CNRS Emeritus Director Centre d’Etudes Transdisciplinaires. Sociologie, Anthropologie, Histoire Ecole des Hautes Etudes en Sciences Sociales

Why has the problematic of complexity appeared so late? And why would it be justified? 1. The three principles of the rejection of complexity by

‘classical science’ Classical science rejected complexity in virtue of three fundamental explanatory principles: The principle of universal determinism, illustrated by Laplace’s Daemon, capable, thanks to his intelligence and extremely developed senses, of not only knowing all past events, but also of predicting all events in the future. The principle of reduction, that consists in knowing any composite from only the knowledge of its basic constituting elements. The principle of disjunction, that consists in isolating and separating cognitive difficulties from one another, leading to the separation between disciplines, which have become hermetic from each other. These principles led to extremely brilliant, important, and positive developments of scientific knowledge up to the point where the limits of intelligibility which they constituted became more important than their elucidations. In this scientific conception, the notion of “complexity” is absolutely rejected. On the one hand, it usually means confusion and uncertainty; *Presented at the Colloquium “Intelligence de la complexit6 : Qpist6mologieet pragmatique”, Cerisy-La-Salle, France, June 26th, 2005”. Translated from French by Carlos Gershenson

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the expression “it is complex” in fact expresses the difficulty of giving a definition or explanation. On the other hand, since the truth criterion of classical science is expressed by simple laws and concepts, complexity relates only to appearances that are superficial or illusory. Apparently, phenomena arise in a confused and dubious manner, but the mission of science is to search, behind those appearances, the hidden order that is the authentic reality of the universe. Certainly, western science is not alone in the search of the “true” reality behind appearances; for Hinduism, the world of appearances, the m&y& is illusory; and for Buddhism the samsara, the world of phenomena, is not the ultimate reality. But the true reality, in the Hindu or Buddhist worlds, is inexpressible and in extreme cases unknowable. Whereas, in classical science, behind appearances, there is the impeccable and implacable order of nature. Finally, complexity is invisible in the disciplinary division of the real. In fact, the first meaning of the word comes from the Latin complexus, which means what is woven together. The peculiarity, not of the discipline in itself, but of the discipline as it is conceived, non-communicating with the other disciplines, closed to itself, naturally disintegrates complexity. For all these reasons, it is understood why complexity was invisible or illusory, and why the term was rejected deliberately.

2. Complexity: A first breach: irreversibility However, a first breach is made within the scientific universe during the nineteenth century; complexity would appear from it de facto before starting to be recognized de jure. Complexity would make its appearance de facto with the second law of thermodynamics, which indicates that energy degrades into caloric form: this principle lies within the scope of the irreversibility of time, while until then physical laws were in principle reversible and that even in the conception of life, the fixism of species did not need time. The important point here is not only the irruption of irreversibility, thus time, but it is also the apparition of a disorder since heat is conceived as the agitation of molecules; the disordered movement of each molecule is unpredictable, except at a statistical scale where distribution laws can be determined effectively. The law of the irreversible growth of entropy has given place to multiple speculations, and beyond the study of closed systems, a first reflection about

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the universe, where the second law leads toward dispersion, uniformity, and thus towards death. This conception of the death of the universe, long ago rejected, has appeared recently in cosmology, with the discovery of black energy. This will lead to the dispersion of galaxies and would seem to announce us that the universe tends to a generalized dispersion. As the poet Eliot said: “the universe will die in a whisper” ... Thus, the arrival of disorder, dispersion, disintegration, constituted a fatal attack to the perfect, ordered, and determinist vision. And many efforts will be needed-we are not there precisely because it is against the reigning paradigm-to understand that the principle of dispersion, which appears since the birth of the universe with this incredible deflagration improperly named big bang, is combined with a contrary principle of bonding and organization which is manifested in the creation of nuclei, atoms, galaxies, stars, molecules, and life.

3. Interaction Order/Disorder/Organization

How is it that both phenomena are related? This is what I tried to show in the first volume of La Mkthode (The Method). We will need to associate the antagonist principles of order and disorder, and associate them making another principle emerge that is the one of organization. Here is in fact a complex vision, which one has refused to consider during a very long time, for one cannot conceive that disorder can be compatible with order, and that organization can be related to disorder at all, being antagonist to it. At the same time than that of the universe, the implacable order of life is altered. Lamarck introduces the idea of evolution, Darwin introduces variation and competition as motors of evolution. Post-darwinism, if it has, in certain cases, attenuated the radical character of the conflict, has brought this other antinomy of order: chance, I would say even a vice of chance. Within the neodarwinian conception, to avoid calling “creation” or “invention” the new forms of living organization such as wings, eyesone is very afraid of the word ‘Linventionlland of the word ‘kreation”---one has put chance at the prow. One can understand the rest of the fear of creation because science rejects creationism, i.e. the idea that God is a creator of living forms. But the reject of creationism finished in masking the creativity that manifests itself in the history of life and in the history of humanity. And, from the philosophical point of view, it is rather recently

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that Bergson, and then in another way, Castoriadis, put at the centre of their conception the idea of creation. In addition, in the beginning of the twentieth century, microphysics introduced a fundamental uncertainty in the universe of particles that ceases to obey the conceptions of space and time characteristic of our universe called macro-physic. How thus these two universes, that are the same, but at a different scale, are compatible? One begins today to conceive that one can pass, from the micro-physical universe to ours, since between them a certain number of quantum elements are connected, in virtue of a process called decoherence. But there remains this formidable logical and conceptual hiatus between the two physics. Finally, at a very large scale-mega-physical-Einstein’s theory discovers that space and time are related to one another, with the result that our lived and perceived reality becomes only meso-physical, situated between micro-physic reality and mega-physical reality

4. Chaos

All this made that the dogmas of classical science are reached, but de fucto: although increasingly mummified, they remain. Yet a certain number of strange terms would appear. For example, the term “catastrophe”, suggested by RenC Thom to try to make intelligible the discontinuous changes of form; then the fractalism of Mandelbrot; then the physical theories of chaos, which distinguishes itself from the rest, since today it is thought that the solar system, which seems to obey an absolutely impeccable and measurable order with the most extreme precision, considering its evolution in millions of years, is a chaotic system comprising a dynamic instability modifying for example Earth’s rotation around itself or around the Sun. A chaotic process may obey to deterministic initial states, but these cannot be know exhaustively, and the interactions developed within this process alter any prevision. Negligible variations have considerable consequences over large time scales. The word chaos, in these physics, has a very limited meaning: that of apparent disorder and unpredictability. Determinism is saved in principle, but it is inoperative since one cannot know exhaustively the initial states. We are in fact, since the original deflagration and forever, plunged in a chaotic universe.

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5. The emergence of the notion of complexity However, complexity remained always unknown in physics, in biology, in social sciences. Admittedly, after more than half a century, the word complexity irrupted, but in a domain that also remained impermeable to the human and social sciences, as well as to the natural sciences themselves. It is at the bosom of a sort of nebulous spiral of mathematicians and engineers where it emerged at about the same time, and became connected at once, in the forties and fifties, with Information Theory, Cybernetics, and General Systems Theory. Within this nebula, complexity will appear with Ashby to define the degree of variety in a given system. The word appears, but does not contaminate, since the new thinking remains pretty confined: the contributions of Von Neumann, of Von Foerster will remain completely ignored, and still remain in the disciplinary sciences closed on themselves. One can also say that Chaitin’s definition of randomness as algorithmic incompressibility becomes applicable to complexity. consequently, the terms chance, disorder, complexity tend to overlap one another and sometimes to be confused. There were breaches, but still not an opening. This would come from the Santa Fe Institute (1984) where the word will be essential to define dynamical systems with a very large number of interactions and feedbacks, inside of which processes very difficult to predict and control take pla,ce, as “complex systems”, where the classical conception was unable to be considered. Thus, the dogmas or paradigms of classical science began to be disputed. The notion of emergence appeared. In “Chance and Necessity”, Jacques Monod makes a great state of emergence, i.e. qualities and properties that appear once the organization of a living system is constituted, qualities that evidently do not exist when they are presented in isolation. This notion is taken, here and there, more and more, but as a simple constatation without being really questioned (whereas it is a conceptual bomb). It is like this that it was arrived to the complexity I call “restricted”: the word complexity is introduced in “complex systems theory”; in addition, here and there the idea of “sciences of complexity” was introduced, encompassing the fractalist conception and chaos theory. Restricted complexity spread rather recently, and after a decade in France, many barriers have been jumped. Why? Because more and more a theoretical vacuum was faced, because the ideas of chaos, fractals, disorder, and uncertainty appeared, and it was necessary at this moment that the

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word complexity would encompass them all. Only that this complexity is restricted to systems which can be considered complex because empirically they are presented in a multiplicity of interrelated processes, interdependent and retroactively associated. In fact, complexity is never questioned nor thought epistemologically. Here the epistemological cut between restricted and generalized complexities appears because I think that any system, whatever it might be, is complex by its own nature. Restricted complexity made it possible important advances in formalization, in the possibilities of modeling, which themselves favor interdisciplinary potentialities. But one still remains within the epistemology of classical science. When one searches for the “laws of complexity”, one still attaches complexity as a kind of wagon behind the truth locomotive, that which produces laws. A hybrid was formed between the principles of traditional science and the advances towards its hereafter. Actually, one avoids the fundamental problem of complexity which is epistemological, cognitive, paradigmatic. To some extent, one recognizes complexity, but by decomplexifying it. In this way, the breach is opened, then one tries to clog it: the paradigm of classical science remains, only fissured. 6. Generalized complexity

But then, what is “generalized” complexity? It requires, I repeat, an epistemological rethinking, that is to say, bearing on the organization of knowledge itself. And it is a paradigmatic problem in the sense that I have defined “paradigm”a. Since a paradigm of simplification controls classical science, by imposing a principle of reduction and a principle of disjunction to any knowledge, there should be a paradigm of complexity that would impose a principle of distinction and a principle of conjunction. In opposition to reduction, complexity requires that one tries to comprehend the relations between the whole and the parts. The knowledge of the parts is not enough, the knowledge of the whole as a whole is not enough, if one ignores its parts; one is thus brought to make a come and go in loop to gather the knowledge of the whole and its parts. Thus, the principle of reduction is substituted by a principle that conceives the relation of whole-part mutual implication. =Cf La Me‘thode

4 , Les ide‘es, p.211-238, Le Seuil, 1990

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The principle of disjunction, of separation (between objects, between disciplines, between notions, between subject and object of knowledge), should be substituted by a principle that maintains the distinction, but that tries to establish the relation. The principle of generalized determinism should be substituted by a principle that conceives a relation between order, disorder, and organization. Being of course that order does not mean only laws, but also stabilities, regularities, organizing cycles, and that disorder is not only dispersion, disintegration, it can also be blockage, collisions, irregularities. Let us now take again the word of Weaver, from a text of 1948, to which we often referred, who said: the XIXth century was the century of disorganized complexity and the X X t h century must be that of organized complexity. When he said “disorganized complexity”, he thought of the irruption of the second law of thermodynamics and its consequences. Organized complexity means to our eyes that systems are themselves complex because their organization supposes, comprises, or produces complexity. Consequently, a major problem is the relation, inseparable (shown in La Me‘thode 1) , between disorganized complexity and organized complexity. Let us speak now about the three notions that are present, but to my opinion not really thought of, in restricted complexity: the notions of system, emergence, and organization.

7. System: It should be conceived that “any system is complex” What is a system? It is a relation between parts that can be very different from one another and that constitute a whole at the same time organized, organizing, and organizer. Concerning this, the old formula is known that the whole is more than the sum of its parts, because the addition of qualities or properties of the parts is not enough to know those of the whole: new qualities or properties appear, due to the organization of these parts in a whole, they are emergent. But there is also a substractivity which I want to highlight, noticing that the whole is not only more than the sum of its parts, but it is also less that the sum of it parts. Why? Because a certain number of qualities and properties present in the parts can be inhibited by the organization of the whole. Thus, even when each of

12 our cells contains the totality of our genetic inheritance, only a small part of it is active, the rest being inhibited. In the human relation individualsociety, the possibilities of liberties (delinquent or criminal in the extreme) inherent to each individual, will be inhibited by the organization of the police, the laws, and the social order. Consequently, as Pascal said, we should conceive the circular relation: ‘one cannot know the parts if the whole is not known, but one cannot know the whole if the parts are not known’. Thus, the notion of organization becomes capital, since it is through organization of the parts in a whole that emergent qualities appear and inhibited qualities disappearb. 8. Emergence of the notion of emergence

What is important in emergence is the fact that it is indeductible from the qualities of the parts, and thus irreducible; it appears only parting from the organization of the whole. This complexity is present in any system, starting with H 2 0 , the water molecule which has a certain number of qualities or properties that the hydrogen or oxygen separated do not have, which have qualities that the water molecule does not have. There is a recent number of the Science et Awenir journal devoted to emergence; to relate emergence and organization, one wonders wether it is a hidden force in nature, an intrinsic virtue. From the discovery of the structure of the genetic inheritance in DNA, where it appeared that life was constituted from physicochemical ingredients present in the material world, therefore from the moment that it is clear that there is not a specifically living matter, a specifically living substance, that there is no d a n vital in Bergson’s sense, but only the physicochemical matter that with a certain degree of organizing complexity produces qualities of the living-f which self-reproduction, self-reparation, as well as a certain number of cognitive or informational aptitudes, as from this moment, the vitalism is rejected, the reductionism should be rejected, and it is the notion of emergence that takes a cardinal importance, since a certain type of organizing complexity produces qualities specific of selforganization. The spirit (mens, mente) is an emergence. It is the relation brainbI develop the idea that organization consists of complexity in La Me‘thode 1, La nature de la nature,p.94-151, Le Seuil, 1977. ‘Science and Future, a popular French journal (Translator’s Note)

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culture that produces as emergent psychic, mental qualities, with all that involves language, consciousness, etc. Reductionists are unable to conceive the reality of the spirit and want to explain everything starting from the neurons. The spiritualists, incapable of conceiving the emergence of the spirit starting from the relation brainculture, make from the brain at most a kind of television.

9. The complexity of organization The notion of emergence is a capital notion, but it redirects to the problem of organization, and it is organization which gives consistence to our universe. Why is there organization in the universe? We cannot answer this question, but we can examine the nature of organization. If we think already that there are problems of irreducibility, of indeductibility, of complex relations between parts and whole, and if we think moreover that a system is a unit composed of different parts, one is obliged to unite the notion of unity and that of plurality or at least diversity. Then we realize that it is necessary to arrive at a logical complexity, because we should link concepts which normally repel each other logically, like unity and diversity. And even chance and necessity, disorder and order, need to be combined to conceive the genesis of physical organizations, as on the plausible assumption where the carbon atom necessary to the creation of life was constituted in a star former to our sun, by the meeting exactly a t the same t i m e a b s o l u t e coincident-f three helium nuclei. Thus, in stars where there are billions of interactions and meetings, chance made these nuclei to meet, but when this chance occurs, it is necessary that a carbon atom will be constituted. You are obliged to connect all these disjoined notions in the understanding that was inculcated to us, unfortunately, since childhood, order, disorder, organization. We then manage to conceive what I have called the self-ecc-organization, i.e. the living organization.

10. The self-eco-organization The word self-organization had emerged and had been used as of the end of the 50’s by mathematicians, engineers, cyberneticians, neurologists. Three important conferences had been held on the topic of “selforganizing systems”, but a paradoxical thing, the word had not bored in biology, and was a marginal biologist, Henri Atlan, who retook this idea, in

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a great intellectual isolation within his corporation, in the 70’s. Finally the word emerged in the 8O’s-9O’s in Santa Fe as a new idea, whereas it existed already for nearly half a century. But it is still not imposed in biology. I call self-eco-organization to the living organization, according to the idea that self-organization depends on its environment to draw energy and information: indeed, as it constitutes an organization that works to maintain itself, it degrades energy by its work, therefore it must draw energy from its environment. Moreover, it must seek its food and defend against threats, thus it must comprise a minimum of cognitive capacities. One arrives to what I call logically the complex of autonomydependence. For a living being to be autonomous, it is necessary that it depends on its environment on matter and energy, and also in knowledge and information. The more autonomy will develop, the more multiple dependencies will develop. The more my computer will allow me to have an autonomous thought, the more it will depend on electricity, networks, sociological and material constraints. One arrives then to a new complexity to conceive living organization: the autonomy cannot be conceived without its ecology. Moreover, it is necessary for us to see a self-generating and self-producing process, that is to say, the idea of a recursive loop which obliges us to break our classical ideas of product producer, and of cause + effect. In a self-generating or self-producing or self-poetic or self-organizing process, the products are necessary for their own production. We are the products of a process of reproduction, but this process can continue only if we, individuals, couple to continue the process. Society is the product of interactions between human individuals, but society is constituted with its emergencies, its culture, its language, which retroacts to the individuals and thus produces them as individuals supplying them with language and culture. We are products and producers. Causes produce effects that are necessary for their own causation. Already the loop idea had been released by Norbert Wiener in the idea of feedback, negative as well as positive, finally mainly negative; then it was generalized without really reflecting on the epistemological consequences which it comprised. Even in the most banal example which is that of a thermal system supplied with a boiler which provides the heating of a building, we have this idea of inseparability of the cause and effect: thanks to the thermostat, when 20” is reached, the heating stops; when the temperature is too low, the heating is started. It is a circular system, where the effect itself intervenes in the cause which allows the thermal autonomy of --f

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the whole compared to a cold environment. That is to say that the feedback is a process which complexifies causality. But the consequences of this had not been drawn to the epistemological level. Thus feedback is already a complex concept, even in non-living systems. Negative feedback is what makes it possible to cancel the deviations that unceasingly tend to be formed like the fall in temperature compared to the standard. Positive feedback develops when a regulation system is not able anymore to cancel the deviations; those can then be amplified and go towards a runaway, kind of generalized disintegration, which is often the case in our physical world. But we could see, following an idea advanced more than fifty years ago by Magoroh Maruyama, that the positive feedback, i.e. increasing deviation, is an element that allows transformation in human history. All the great transformation processes started with deviations, such as the monotheist deviation in a polytheist world, the religious deviation of the message of Jesus within the Jewish world, then, deviation in the deviation, its transformation by Paul within the Roman empire; deviation, the message of Mohammed driven out of Mecca, taking refuge in Medina. The birth of capitalism is itself deviating in a feudal world. The birth of modern science is a deviating process from the XVIIth century. Socialism is a deviating idea in the XIXth century. In other words, all the processes start by deviations that, when they are not suffocated, exterminated, are then able to make chain transformations.

11. The relationship between local and global

In logical complexity, you have the relation between the local and the global. One believed to be able to assume the two truths of the global and of the local with axioms of the style: “think globally and act locally”. In reality, one is, I believe, constrained in our global age to think jointly locally and globally and to try to act at the same time locally and globally. Also, which is also complex, local truths can become global errors. For example, when our immune system rejects with the greatest energy the heart that one grafts to him, like a nasty foreigner, this local truth becomes a global error, because the organism dies. But one can also say that global truths can lead to local errors. The truth of the need to fight against terrorism can lead to interventions, which will favor even more the development of terrorism, just look at Irak.

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12. Heraclitus: “live of death, die of life”

In this union of notions logically complex, there is a relationship between life and death. I often quoted the illuminating phrase of Heraclitus, from the VIth century b.C.: “live of death, die of life”. It became recently intelligible, from the moment when we learned that our organism degrades its energy, not only to reconstitute its molecules, but that our cells themselves are degraded and that we produce new cells. We live from the death of our cells. And this process of permanent regeneration, almost of permanent rejuvenilization, is the process of life. What makes it possible to add to the very right formula of Bichat, saying: “life is the ensemble of the functions that fight against death”, this strange complement that presents us a logical complexity: “Integrating death to fight better against death”. What one again knows about this process is extremely interesting: it has been learned rather recently that cells that die are not only old cells; in fact apparently healthy cells receiving different messages from neighboring cells, “decide”, at a given moment, to commit suicide. They commit suicide and phagocytes devour their remains. Like this, the organism determines which cells must die before they have reached senescence. That is to say that the death of cells and their postmortem liquidation are included in the living organization. There is a kind of phenomenon of self-destruction, of apoptosis, since this term has been taken from the vegetal world, indicating the split of the stems operated by trees in autumn so that dead leafs fall. On the one hand, when there is an insufficiency of cellular deaths following different accidents and perturbations, there are a certain number of diseases that are deadly in the long run, like osteoporosis, various types of sclerosis, and certain cancers, where cells refuse to die, becoming immortal, forming tumors and go for a stroll in the form of metastases (It can seem that it is a revolt of cells against their individual death that lead to these forms of death of the organism). On the other hand, the excess of cellular deaths determine AIDS, Parkinson’s, and Alzheimer’s disease. You see at which point this relationship between life and death is complex: it is necessary for cells to die, but not too much! One lives between two catastrophes, the excess or insufficiency of mortality. One finds again the fundamentally epistemological problem of generalized complexity.

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13. On non-trivial machines Living beings are certainly machines, but unlike artificial machines that are trivial deterministic machines (where one knows the outputs when one knows the inputs), these are non-trivial machines (von Foerster) where one can predict innovative behaviors. We are machines, this truth was already in L'homme-machine of La Mettrie. We are physical machines, thermal machines, we function at the temperature of 37'. But we are complex machines. Von Neumann established the difference between living machines and artificial machines produced by technology: the components of the technical machines, having the good quality of being extremely reliable, go towards their degradation, towards their wear, from the very start of their operation. Whereas the living machine, made up mainly by components far from reliable, degrading proteins-and one understands very well that this lack of reliability of proteins makes it possible to reconstitute them non stop-is able to be regenerated and repaired; it also goes towards death, but after a process of development. The key of this difference lies in the capacity of self-repair and self-regeneration. The word regeneration is capital here. One can say that the characteristic of innovations that emerge in the evolution of life (which are determined by environmental changes, or by the irruption of multiple hazards), such as the appearance of the skeleton in vertebrates, wings in insects, birds, or bats, all these creations, are characteristic non-trivial machines. That is to say, it gives a new solution to insurmountable challenges without this solution. All the important figures of human history, on the intellectual, religious, messianic, or politic levels, were non-trivial machines. One can advance that all the History of Humankind, which begins ten thousand years ago-is a non-trivial history, i.e. a history made of unforeseen, of unexpected events, of destructions and creations. The history of life that precedes it is a nontrivial history, and the history of the universe, where the birth of life and then of humankind are included, is a non-trivial history. We are obliged to detrivialize knowledge and our worldview. 14. To complexify the notion of chaos We have seen how the notion of system brings us to complexities of organization which themselves lead us to logical complexities. Let us look now at the notion of chaos, as it appears within chaos theory, and which comprises disorder and impredictibility. The beat of the wings of a butterfly

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in Melbourne can cause by a succession of chain processes a hurricane in Jamaica, for example. Actually, I believe that the word chaos must be considered in its deep sense, its Greek sense. We know that in the Greek worldview, Chaos is at the origin of Cosmos. Chaos is not pure disorder, it carries within itself the indistinctness between the potentialities of order, of disorder, and of organization from which a cosmos will be born, which is an ordered universe. The Greeks saw a bit too much order in the cosmos, which is effectively ordered because the immediate spectacle, the impeccable order of the sky that we see each night with the stars, is always in the same place. And if the planets are mobile they also come to the same place with an impeccable order. However, we know today with the widened conceptions of cosmic time that all this order is at the same time temporary and partial in a universe of movement, collision, transformation. Chaos and Cosmos are associated-I have employed the word Chaosmos-there is also a circular relation between both terms. It is necessary to take the word chaos in a much deeper and more intense sense than that of physical chaos theory. 15. T h e need of contextualization Let us take again the “complexus” term in the sense of “what is woven together”. It is a very important word, which indicates that the breaking up of knowledge prevents from linking.and contextualizing. The knowledge mode characteristic of disciplinary science isolates objects, one from another, and isolates them compared to their environment. One can even say that the principle of scientific experimentation allows to take a physical body in Nature, to isolate it in an artificial and controlled laboratory environment, and then study this object in function of perturbations and variations that one makes it undergo. This indeed makes it possible to know a certain number of its qualities and properties. But one can also say that this principle of decontextualization was ill-fated, as soon as it was ported to the living. The observation since 1960 by Jane Goodall of a tribe of chimpanzees in their natural environment could show the supremacy of observation (in a natural environment) over experimentation (in a laboratory) for knowledged. A lot of patience was necessary dSee “ L e Paradigme Perdu”, pp. 51-54

19 so that Jane Goodall could perceive that chimpanzees had different personalities, with rather complex relations of friendship, of rivalry; a whole psychology, a sociology of chimpanzees, invisible to the studies in a laboratory or in a cage, appeared in their complexity. The idea of knowing the living in their environment became capital in animal ethology. Let us repeat it, the autonomy of the living needs to be known in its environment. From now on, becoming aware of the degradations that our technoeconomic development makes to the biosphere, we realize the vital link with this same biosphere that we believe to have reduced to the rank of manipulable object. If we degrade it, we degrade ourselves, and if we destroy it, we destroy ourselves. The need for contextualization is extremely important. I would even say that it is a principle of knowledge: Anybody who has made a translation in a foreign language will seek an unknown word in the dictionary; but with words being polysemous, it is not immediately known which is the good translation; the sense of the word will be sought in the sense of the sentence in the light of the global sense of the text. Though this play from text to word, and from text to context, and from context to word, a sense will crystalize. In other words, the insertion in the text and in the context is an evident cognitive necessity. Take for example the economy, the most advanced social science from a mathematical point of view, but which is isolated from human, social, historic, and sociologic contexts: its prediction power is extremely weak because the economy does not function in isolation: its forecasts need to be unceasingly revised, which indicates us the disability of a science that is very advanced but too closed. More generally, mutual contextualization is lacking in the whole of social sciences. I have often quoted the case of the Aswan dam because it is revealing and significant: it was built in Nasser’s Egypt because it would make it possible to regulate the course of a capricious river, the Nile, and to produce electric power for a country which had a great need for it. However, after some time, what happened? This dam retained a part of the silts that fertilized the Nile valley, which obliged the farming population to desert the fields and overpopulate large metropolises like Cairo; it retained a part of the fish that the residents ate; moreover today, the accumulation of silts weakens the dam and causes new technical problems. That does not mean that the Aswan dam should not have been built, but that all the decisions taken in a techno-economic context are likely to be disastrous by their consequences.

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It is like the deviation of rivers in Siberia that the Soviet government made and where the perverse consequences are more important than the positive ones. It is thus necessary to recognize the inseparability of the separable, at the historical and social levels, as it has been recognized at the microphysical level. According to quantum physics, confirmed by Aspect’s experiments, two microphysical entities are immediately connected one to the other although they are separated by space and time. Even more, one arrives to the idea that everything that is separated is at the same time inseparable. 16. The hologrammatic and dialogical principles

The hologrammic or hologrammatic principle should also be advanced, according to which not only a part is inside a whole, but also the whole is inside the part; just as the totality of the genetic inheritance is found in each cell of our organism, the society with its culture is inside the spirit of an individual. We return again to the logical core of complexity which we will see, is dialogical: separability-inseparability, whole-parts, effect-cause, productproducer, life-death, homo sapiens-homo demens, etc. It is here that the principle of the excluded middle reveals its limit. The excluded middle states “A cannot be A and not A”, whereas it can be one and the other. For example, Spinoza is Jewish and non-Jewish, he is neither Jewish, nor non-Jewish. It is here that the dialogic is not the response to these paradoxes, but the means of facing them, by considering the complementarity of antagonisms and the productive play, sometimes vital, of complementary antagonisms.

17. For the sciences, a certain number of consequences Regarding sciences, we can see a certain number of consequences. First of all, classical science is somehow complex, even when it produces simplifying knowledge. Why? Because science is a quadruped which walks on the following four legs: the leg of empiricism made of data, experimentation or observation; the leg of rationality, made of logically constituted theories; the leg of verification, always necessary; and the leg of imagination, because great theories are products of a powerful creative imagination. Thus science is complex, produced by a quadruped movement, which prevents it from solidifying.

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The objective knowledge which is its idea, resulted in the need of eliminating subjectivity, i.e. the emotional part inherent to each observer, to each scientist, but it also comprised the elimination of the subject, i.e. the being which conceives and knows. However, any knowledge, including objective, is at the same time a cerebral translation starting from data of the external world and a mental reconstruction, starting from certain organizing potentialities of the spirit. It is certain that the idea of a pure objectivity is utopian. Scientific objectivity is produced by beings who are subjects, within given historical conditions, starting from the rules of the scientific game. The great contribution of Kant was to show that the object of knowledge is co-constructed by our spirit. He indicated us that it is necessary to know knowledge to know its possibilities and limits. The knowledge of knowledge is a requirement of the complex thinking. As Husserl indicated in the ~ O ’ S , in particular in his conference on the crisis of European science, sciences developed extremely sophisticated means to know external objects, but no means to know themselves. There is no science of science, and even the science of science would be insufficient if it does not include epistemological problems. Science is a tumultuous building site, science is a process that could not be programmed in advance, because one can never program what one will find, since the characteristic of a discovery is its unexpectedness. This uncontrolled process has lead today to the development of potentialities of destruction and of manipulation, which must bring the introduction into science of a double conscience: a conscience of itself, and an ethical conscience. Also, I believe that it will be necessary to arrive more and more to a scientific knowledge integrating the knowledge of the human spirit to the knowledge of the object which this spirit seizes and recognizing the inseparability between object and subject.

18. Two scientific revolutions introduced complexity de facto

I already indicated how the concept of complexity emerged in a marginal fashion in a sphere of mathematicians/engineers. It should be indicated now that the X X t h century knew two scientific revolutions which de facto introduced complexity without, however, recognizing this notion that remains implicit. The first revolution, after the thermodynamics of the XIXth century, is that of the microphysics and cosmophysics that introduced indeterminism,

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risk-where determinism reigned-and elaborated suitable methods to deal with the uncertainties met. The second revolution is that which gathers disciplines and restores between them a common fabric. It begins in the second half of the XXth century. Thus in the ~ O ’ S , Earth sciences designed Earth as a complex physical system, which makes it possible today to articulate geology, seismology, vulcanology, meteorology, ecology, etc. At the same time, ecology develops as a scientific knowledge bringing together data and information coming from different physical and biological disciplines in the conception of ecosystems. It makes it possible to conceive how an ecosystem either degrades, develops, or maintains its homeostasis. From the ~ O ’ S , the ecological conception extends to the whole biosphere, necessarily introducing knowledge from the social sciences. Although ecology, at the biosphere level, cannot make rigorous predictions, it can give us vital hypothesis, concerning, for example, global warming, which manifests itself by the melting of glaciers in the Antarctic or the Arctic. Thus ecology, cosmology, and Earth sciences have become poly-disciplinary sciences, even transdisciplinary. Sooner or later, this will arrive in biology, from the moment when the idea of self-organization will be established; this will arrive in the social sciencese, although they are extremely resistant. Finally, the observer, chased by the objectivity postulate, was intrcduced into certain sciences, such as microphysics where the observer perturbs what it observes. In the case of cosmology, even if one does not adhere to what Brandon Carter called the anthropic principle, which holds account of the place of humans in the universe, one is obliged to conceive that this universe, among its perhaps negligible possibilities, had the possibility of human life, perhaps only on this planet Earth, but perhaps also elsewhere. Thus, the common fabric between the human, the living, and the Universe can be restored, which implies a complex conception capable at the same time to distinguish the human from the natural and to integrate it. 19. The insertion of science in History

In addition, there is the problem of the insertion of the Sciences in human

History. You know that there are two conceptions of history of sciences, the internalist conception and the externalist conception. The internalist mode ‘Cf my Humanite‘ de l’humanite‘, La Me‘thode 5, Le Seuil.

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sees the development of sciences in isolation, only in function of their internal logic and their own discoveries. The externalist mode sees them in function of historical and social developments which determine the scientific developments. I think that it is necessary to link both, and this is appropriate for other developments than those of sciences. Thus, some wanted to understand the perversion of the Soviet Union starting from internal factors, such as insufficiencies of the Marxist doctrine, limitations of that of Lenin. Others wanted to impute it to external elements such as the surrounding and hostility of the capitalist powers with regard to the Soviet Union or former elements such as the backwardness of tsarist Russia. Whereas the true cognitive game is to link these two aspects in a dialogical fashion. If one continues to place oneself from the viewpoint of modern Western history of science, one sees how from its marginal and quasi-deviating birth in the XVIIth century, it is developed in the XVIIIth, introduced in universities in the XIXth, then in states and companies in the XXth, and how it becomes central and driving within human history in the form of techno-science, and produces not only all the major elements for a renewed knowledge of the world and beneficial effects for humanity, but also formidable and uncontrolled powers which threaten it. I don’t know if I am right or wrong in retaking an expression of Vico, but it is necessary for us to arrive to the “Scienza Nuova”. Very precisely, Vico inscribed the historical perspective at the heart of the scienza nuova. It is necessary to amplify the idea of scienza nuova by introducing the interaction between the simple and the complex, by conceiving a science that does not suppress disciplines but connects them, and consequently makes them fertile, a science which can at the same time distinguish and connect and where the transdisciplinarity is inseparable from complexity. I repeat it, as much as the compartmentalization of disciplines disintegrates the natural fabric of complexity, as much a transdisciplinary vision is capable of restoring it.

20. The link between science and philosophy

The link between science and philosophy has been broken. Still in the XVIIth century, the great scientists were at the same time great philosophers. Certainly, they did not identify Science and Philosophy. When Pascal made his experiments in Puy de DGme, he did not think about the bet problem. But in the times of Pascal, Gassendi, Leibniz, there was not

24 this cut. This became a frightening ditch. The ditch of ignorance separates the scientific culture from the culture of the humanities. But the current has started to be reversed: the most advanced sciences arrive to fundamental philosophical problems: Why is there a universe out of nothing? How was this universe born from a vacuum which was not at the same time the vacuum? What is reality? Is the essence of the universe veiled or totally cognizable? The problem of life is posed from now on in a complexity that exceeds biology: the singular conditions of its origin, the conditions of emergences of its creative powers. Bergson was mistaken by thinking that there was an e'lan vital, but was right while speaking about creative evolution. He could even have spoken about evolutionary creativity. Today we can foresee the possibility of creating life. From the moment when it is believed that life is a process developed starting only from physicochemical matter under certain conditions, in underwater thermal vents or elsewhere, one can very well consider creating the physical, chemical, thermodynamic conditions which give birth to organisms gifted with qualities that one calls life. We can also foresee the possibility to modify the human being in its biological nature. Therefore, we have to meditate about life, as we never did it. And at the same time we must meditate about our relationship with the biosphere. Thus all the most advanced sciences arrive to fundamental philosophical problems that they thought to have eliminated. They do not only find them, they renew them. If one defines philosophy by the will and capacity of reflection, it is necessary that the reflectivity is also introduced into the sciences, which does not eliminate the relative autonomy of philosophy nor the relative autonomy of scientific procedures compared to philosophical procedures. Finally and especially, any knowledge, including the scientific one, must comprise in itself an epistemological reflection on its foundations, principles, and limits. Still today there is the illusion that complexity is a philosophical problem and not a scientific one. In a certain way, it is true, in a certain way, it is false. It is true when you place yourselves from the point of view of an isolated and separated object: the fact that you isolate and separate the object made the complexity to disappear: thus it is not a scientific problem from the point of view of a closed discipline and a decontextualized object. But, as soon as you start to connect these isolated objects, you are in front of the problem of complexity.

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21. Second epistemological rupture with restricted complexity It is here that a second epistemological rupture with restricted complexity appears. Restricted complexity is interested essentially in dynamical systems called complex. That is to say, it constitutes its own field, within the field of sciences. But generalized complexity not only concerns all fields, but also relates to our knowledge as human beings, individuals, persons, and citizens. Since we have been domesticated by our education which taught us much more to separate than to connect, our aptitude for connecting is underdeveloped and our aptitude for separating is overdeveloped; I repeat that knowing, is at the same time separating and connecting, it is to make analysis and synthesis. Both are inseparable, and our atrophy of the capacity to connect is increasingly serious in a globalized, complexified mode, where it is a matter of generalized interdependence of everything and everyone. The International Ethical, Political and Scientific Collegium has formulated a declaration of interdependence which it would wish to see promulgated by the United Nations. We must think the interdependence in all fields, including the complex relation between the parts and the whole. We need to be able to face uncertainties of life whereas nothing prepares us for it. We need to face complexity, including for action, whereas one opposes the cautionary principle to the risk principle, while Pericles had truly expressed the union of the two antagonistic principles when he said during a speech to the Athenians during the Peloponnesian war: “we Athenians, we are capable of combining prudence and audacity, whereas the others are either timorous or bold”. It is the combination which we need. Also, precaution needs today sometimes much invention. We need to deeply reform all our way of knowing and thinking. 22. The principle of ecology of action

The principle of ecology of action is, in my opinion, central: from the moment an action enters a given environment, it escapes from the will and intention of that which created it, it enters a set of interactions and multiple feedbacks and then it will find itself derived from its finalities, and sometimes to even go in the opposite sense. The ecology of action has a universal value, including for the development of sciences, whose destructive nuclear consequences were absolutely unexpected.

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Think that when Fermi elucidated the structure of the atom in the ~ O ’ S , it was a purely speculative discovery and he had by no means thought that this could allow the fabrication of an atomic bomb. However, a few years later, the same Fermi went to the United States to contribute to the fabrication of the atomic bomb that would be used in Hiroshima and Nagasaki. When Watson and Crick determined the structure of the genetic inheritance in DNA, they thought that it was a great conquest of knowledge without any practical consequences. And hardly ten years after their discovery, the problem of genetic manipulations was posed in the biology community. The ecology of action has a universal value. One can think of examples in our recent French history: a dissolution of the Parliament by President Chirac to have a governmental majority led to a socialist majority; a referendum made to win general support led to its rejection. Gorbachev tried a reform to save the Soviet Union but this one contributed to its disintegration. When one sees that a revolution was made in 1917 to suppress the exploitation of man by his fellow man, to create a new society, founded on the principles of community and liberty, and that this revolution, not only caused immense losses of blood, destruction, and repression by a police system, but, after seventy years, it led to its contrary, i.e. to a capitalism even more fierce and savage than that of the tsarist times, and with a return of religion! Everything that this revolution wanted to destroy resurrected. How not to think about the ecology of action!

23. Creating “Institutes of fundamental culture”

The reform of the spirit seems to me absolutely necessary. Once that I had understood that the reform of thought, deep work that I carried out in La Me‘thode, is a necessity, I accepted the offer of a Minister of Education when he called me for the reform of the content of secondary education. I tried to introduce my ideas of reform of thought into an educational project. I saw its total failure-finally it did not failed, it was not applied!-That pushed me to reflect even more. I wrote a book called La TCte bien fuite (The head well made), then on the initiative of UNESCO I made a book called Les Sept savoirs ne‘cessaires a l’e‘ducation du f u t u r (The seven knowledges necessary in the education of the future). Following a University which will be created on these principles in Mexico, I had the more restricted but maybe more necessary idea of creating “Institutes of fundamental culture”, which would be sheltered in a University or independent, addressing everybody, i.e. before University or during

27 University or after University, students, citizens, members of trade unions, entrepreneurs, everybody. Why the word “fundamental culture”? Because it is that what is missing. In fact, it is the most vital matter to be taught, the most important to face life, and which is ignored by education. (1) Knowledge as a source of error or illusion; nowhere the traps of knowledge are taught, which come owing to the fact that all knowledge is translation and reconstruction. (2) Rationality, as if it were an obvious thing, whereas we know that rationality knows its perversion, its infantile or senile diseases. (3) Scientificity. What is science, its frontiers, its limits, its possibilities, its rules. Moreover, there is an abundant literature, but which has never been consulted by the scientists who are recruited at CNRS for example. Most of the time, they do not know anything about the polemic between Niels Bohr and Einstein, the works of Popper, Lakatos, Kuhn, etc. (4) What is complexity. And also: A teaching on “what is the human identity and condition”, which is not found anywhere. A teaching on the global age, not only today’s globalization, but all its antecedents starting from the conquest of America, the colonization of the world, its current phase, and its future prospects. A teaching on human understanding. A teaching concerning the confrontation of uncertainties in all the fields: sciences, everyday life, history (we have lost the certainty of progress, and the future is completely uncertain and obscure). A teaching on the problems of our civilization. That is for me the fundamental teaching that can aid the reform of the spirit, of thought, of knowledge, of action, of life. 24. I conclude: generalized complexity integrates restricted complexity Unfortunately, restricted complexity rejects generalized complexity, which seems to the former as pure chattering, pure philosophy. It rejects it because restricted complexity did not make the epistemological and paradigmatic

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revolution which complexity obliges. That will be done without a doubt. But in the meantime, we see that the problematic of complexity have invaded all our horizons, and I repeat “problematic”, because it is an error to think that one will find in complexity a method that can be applied automatically to the world and anything. Complexity is a certain number of principles which help the autonomous spirit to know. Whereas a program destroys the autonomy of the one who seeks, the problematic of complexity stimulates an autonomous strategy, obliges in the field of action-once that one knows that the ecology of action can pervert the best intentions-to reconsider our decisions like bets and incites us to develop an adequate strategy to more or less control the action. In other words, in all the fields, I would say “help yourself and the complexity will help you”, which has nothing to do with the mechanical application of a program or a rule. It is a deep reform of our mental functioning, of our being. These ideas now marginal, deviating, begin to constitute a tendency still in minority, or rather tendencies since there are several paths to go towards complexity. These ideas, these deviations, can be developed and become cultural, political, and social forces. The probabilities of a global future are extremely alarming: our spaceship is pulled by four engines without any control: science, technology, economy, and the search for profit-all this under conditions of cham since the techno-civilizational unification of the planet, under the Western push, causes singular cultural resistances and cultural and religious re-closings. The planet is in crisis with all the possibilities, ones regressive and destructive, others stimulant and fertile, such as invention, creation, new solutions.

25. We should even apprehend the possibilities of

metamorphosis We should even apprehend the possibilities of metamorphosis because we have completely astonishing examples of it from the past. The change in certain places where there have been demographic concentrations in the Middle East, in the Indus basin, in China, in Mexico, in Peru, from prehistoric societies of hundreds of men, without cities, without state, without agriculture, without army, without social class, to enormous historical societies with cities, agriculture, army, civilization, religion, philosophy, works

of art ... that constituted a sociological metamorphosis. Perhaps we are going towards a meta-historical metamorphosis suitable for the birth of a society-world at a global scale. I would say that complexity does not put us only in the distress of the uncertain, it allows us to see besides the probable, the possibilities of the improbable, because of those which have been in the past and those that can be found again in the future. We are in an epoch of doubtful and uncertain combat. That makes one think of the Pacific war, after the Japanese had broken into the Pacific Islands and had begun to threaten California, there was a gigantic naval fight over 200 kilometers along the Midways between the Japanese and American fleets: battleships, aircraft carriers, submarines, planes. The global vision was impossible for both of them: there were sunken Japanese ships, sunken American ships, planes that did not find the enemy fleet; in short, total confusion, the battle divided in several fragments. At a given moment, the Japanese Admiral realizing his losses in battleships and planes, thought that they were defeated, thus called for retreat. But the Americans, who had lost as much, were not the first to think that they were defeated; after the Japanese retreat, they were victorious. Well, the outcome of what will happen, we cannot conceive it yet! We can always hope and act in the direction of this hope. The intelligence of complexity, isn’t it to explore the field of possibilities, without restricting it with what is formally probable? Doesn’t it invite us to reform, even to revolutionize?

COMPLEXITY SCIENCE AS AN ASPECT OF THE COMPLEXITY OF SCIENCE

DONALD C. MIKULECKY Senior Fellow, Virginia Commonwealth University’s Center for the Study of Biological Complexity, Richmond, VA

1. INTRODUCTION How can we treat science as an object of scientific inquiry? The central problem arises with that question. Science has tried to rid itself of circularity and in so doing has become a very limited method for examining the complex world it tries to have us understand. Self reference is at the center of so many of the interesting thing we want to understand including and especially life itself. The existence of this self referential character is the essence of what we have come to call “complexity”. The works of Robert Rosen [l, 2, 31 spell this out in great detail. This series of investigations began over a half century ago yet still remains virtually unrecognized by the vast majority of those who call themselves “scientists”. That fact alone can be a springboard to launch a study of science as an object, which is what this study is all about. I have reviewed the technical aspects of Rosen’s work elsewhere [4]and will consider the broader philosophical implications here. Using the ideas Rosen developed, we can begin with the following working definition of complexity:

Complexity is the property of a real world s y s t e m that is manifest in the inability of a n y one formalism being adequate to capture all its properties. I t requires that we find distinctly different ways of interacting with systems. Distinctly diflerent in the sense that when we make successful models, the formal systems needed t o describe each distinct aspect are NOT derivable from each other. Rosen created a dichotomy between complex system and simple systems, or mechanisms. The essence of that dichotomy is summarized in Table 1.

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31 Table 1. Complex systems vs. simple mechanisms. Complex Simple No largest model Largest model Whole more than sum of parts Whole is sum of parts Causal relations rich and intertwined Causal relations distinct Generic Non-generic Analytic Synthetic Analytic = Synthetic Non-fragment able Fragmentable Non-comput able Computable Real world Formal system

This set of properties that distinguish disjoint categories of objects is the subject of Rosen’s life work. Any brief summary will be lacking the content developed to establish them in detail, but can give a sense of what the categorization entails. The categories are disjoint because of a central characteristic of the way we go about practicing science. We create a surrogate world, the world of simple systems, to enable our minds to try to grasp the complexity of the world we wish to comprehend. Hence the real world is complex and must be reduced to something less than its many faceted reality in order to be studied. The surrogate simple world is a world of formal systems or models, that replace the many faceted complex reality with something our minds can work with. Our understanding depends on this process. The process has consequences that can not be escaped. The table of characteristics is a summary of those consequences, and they will be discussed briefly.

1.1. The largest Model The formal system science has traditionally used to model the complex world has a largest model. This model is the crux of scientific description and can be called the “Newtonian Paradigm” using terminology adopted by Rosen. The Newtonian Paradigm is a world constructed in terms of non-linear dynamics in its many forms. The most modern forms all resolve to the largest model formulated as non-linear dynamics. Included are chaos theory, cellular automata, and non-linear systems of partial differential equations or state space descriptions, among others. These formalisms lend themselves well to the use of modern computing methods and therefore computability becomes a hallmark of the largest model used to represent

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simple systems. Real systems go beyond computable properties and can only be approximated by computer simulation. This has been said in many ways, such as “The map is not the territory”, and will often be acknowledged in specific instances, only to be denied vigorously when proposed as a general truth. In the working definition of complexity, this is manifest as the need for distinct formalisms in order to completely represent complex reality. In fact, there is no finite number of such descriptions.

1.2. Why is the whole more than the s u m of its parts? A mantra in complexity research is the notion that any complex whole is more than the sum of its parts. The world of simple systems created by science does not escape this idea. We will give some examples of models that can demonstrate that the same material parts can be assembled in different ways to make very different functioning wholes. The key word being “functional” for it is in function that this property of complex reality is rooted. More interesting is the recognition that function is as much an ontological aspect of complex reality as are the atomic and molecular material building blocks. This must be so if the notion that the whole is more than the sum of its parts is to have any meaning. Reducing a complex whole to smaller units and eventually to atoms and molecules destroys function and destroys the very identity of the complex whole. This thing called “function” that characterizes complex wholes is as real as the material aspects of the system. Simple systems can be reduced to their parts and are called machines or mechanisms for this reason. This leads to an unfortunate ambiguity between different meanings of the word “machine”. An automobile is a man made machine and can seem to be identical with the formal description used to create it. However, the real world automobile is a complex entity only approximated by its formal description even though it can be built from that formal description. It will be important to keep the vernacular meaning of machine, the complex reality, separated from its formal description as a machine or simple mechanism. In the relational systems theory being developed as an approach to complex system description using the ideas introduced by Rosen, the material description is sacrificed in favor of a functional description. The complex whole is reduced to a combination of functional components that can not be mapped to material parts on a one to one basis. These functional components only have definition in the context of the complex whole and are destroyed when it is reduced to material parts. Hence the functional components are the

33 way that the things that are “more than the sum of the material parts” become entities having an ontology of their own. This idea is central to an understanding of complexity. 1.3. Causality and information: Science of method and science of content Science has often been characterized by its methodology. This has been the key feature that has kept science from being “contaminated“ by the logical problems that are inherent in things like circularity and self reference. The limits imposed by this methodology also keep science from dealing adequately with complex reality for this very reason. Within science a very limited definition of 1Linformation77 was developed by Shannon and Weaver to provide a tool in communications theory. The shortcomings of this very sterile definition show up again and again because it does not satisfy the intuitive notion if what the term “information” connotes. This fact is closely linked to the relationship between syntax and semantics. The concept of information they formulated deals only with syntax while what is really needed is a way of also dealing with semantic content. Rosen introduced another kind of information. The difference between the two versions of information is related to the complexity/simple mechanism/machine dichotomy. Science, by dealing with mechanisms, led to questions about iihow” things work. The other aspect to information deals with “why” things are as they are. The question “how does it work?” is a reductionist/mechanist question. The strict methodology of hard reductionist science does not allow “Why?,’ questions because the method is impotent to deal with them. Yet there is much information content in the answer to the question “Why?” It was Aristotle that taught us that. Aristotle gave four answers to the question “Why?” These causes or LLbecauses’’ are the material, efficient, formal, and final causes. They are often illustrated as the answers to the question “Why a house?” Material cause: Bricks, wood, metal, glass, and other materials that make up the house. Efficient cause: The builders that put those materials into the form of a house giving it a function beyond that of the unassembled materials. Formal cause: The plan or blueprint used to give the house its form as a house. The purpose or function of the house, to be a Final cause: dwelling place.

34 In machines these causes can always be separated. In real complex systems they are entwined and not capable of being separated as in a machine. The example to follow, the distinction between an organism and a machine, will establish this in a clear example.

1.4. W h i c h i s generic, physics or biology? The widely accepted myth that biology is special and that physics is generic is totally wrong. In fact, far more can be learned about the material world by a careful study of biology than can ever be learned from physics. Physics is the model of hard reductionist science. Physics envy has done real damage to fields that want to be considered “scientific” such as psychology and sociology among others. Biological and social systems are clearly complex only because they have resisted the attempt to find an acceptable largest model as has been done in physics.

1.5. Analytic us. synthetic models The distinction between analytic and synthetic models is a mathematical one. It is based on the use of direct products and direct sums as the way the models are constructed. The definition of these structures comes from category theory. Rosen developed his own form of category theory but it does not differ that much from other versions. Category theory is that branch of mathematics created to overcome some problems in set theory. It is actually a way of modeling mathematics itself. To summarize the essential difference between analytic and synthetic models is a matter of looking at the way set like entities can be combined. Product sets are based on a set theoretical operation called a Cartesian product. The Cartesian product of a pair of sets A = (al, a2, “3) and B = (bl, b2) is symbolized as A x B and consists of the collection of pairs (al, b l ) , (all b2), (a2, bl), (a2, b2), (a3, b l ) , (a3, b2). The direct sum of these entities is merely the combination of the members of A and B. In simple systems the direct product can always be reduced to the direct sum. They are equivalent. This is simply a mathematical way of describing the reducibility or fragmentability of reductionist models. In the case of complex systems, this is never true. There is more involved because complex systems are modeled in terms of their functional components and these functional components consist of direct products that are not capable of being reduced to parts without the destruction of the functional component.

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1.6. h g m e n t a b i l i t y It follows directly from what has been developed so far that the complex system with its context dependent functional components can not be fragmented into material parts. Simple mechanisms or machines can always be fragmented into material parts.

1.7. Computability This subject can fill a number of books. It is the subject of heated debates and for those who have placed their faith in computers and computable models, the stakes are very high. The sides of this debate can be exemplified on one hand by the hard core proponents of “artificial life” and on the other hand by Robert Rosen and many others who have come to see his understanding of the complex real world as a fundamental breakthrough in world models. The proponents of the reductionist/mechanistic view believe that the Church-Turing thesis is correct. 2. SCIENCE AS A COMPLEX SYSTEM

Science is a complex system. It has many facets and many different ways of being defined, described and analyzed. The most obvious use of the word is as a noun. What does the noun stand for? It may stand for an ideal that we sometimes call ”pure” science. Such ideals are approximated in reality and never really realized completely. Science as a noun also stands for a body of knowledge. Science is also a set of human institutions. Science can also mean those things that the people we recognize as scientists do. When we speak of science in that way we can also speak of an activity and we sometimes refer to doing science so there is a verb form involved. We often describe things as scientific as in a scientific study or a scientific expedition. Then we see that there are adverbs and adjectives as well. There is also the scientific method. There is a philosophy of science. This philosophy is very important because it is the only way we can complete the system. That is, if we were forced to study science using science we would be caught in an impredicative self referential loop. This may be a partial explanation of why scientists seem to avoid philosophy, hoping to escape the need for another approach.. Science as a human institution also has a very important set of political facets. The products of scientific activity are often a source of wealth and power. The doing of science is an expensive enterprise and it must have

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its patrons. Both complexity theory and the philosophy of science run into this political aspect of science in a number of ways. By describing the limits of the reductionist/mechanist paradigm there is a danger of casting doubt on the ability of science to produce what its patrons expect from it. Teaching such things to science students can place them in situations that are difficult. Students must choose projects to satisfy their mentors who in turn must satisfy their patrons. Reflecting these strong shaping forces back on the definition we so roughly outlined, it should be clear that differences between science as an ideal and science as what is practiced by scientists can be very great if the ideal is involved with showing limits rather than convincing patrons that their investment will pay off. As the research tools and equipment grow in expense and sophistication, the tendency to occupy one’s time using the equipment also grows. Thus the activity we call science becomes driven by questions that can be answered by the equipment rather than the desire to know in its “purest” form. The methodology of science is not codified in any clear way. Books have been written about the best way to accept candidates for scientific law and scientific theory. Books have also been written about the failure of other disciplines to satisfy the criteria of the scientific method. Periodically there are so called paradigm shifts claimed. Often these are not universally recognized as such, but what ever name one wants to assign, change does occur in the method and this can be significant in its magnitude. Proponents of complexity theory in its many forms often ask for it to be a candidate for such status. Is there a model for this kind of complexity in systems of human thought and the activities that feed in to such systems? Clearly there is. Robert Rosen chose Number Theory as his example. He reviewed the efforts of the formalists to purge the field of all circularities and self referential loops. To make a very long story very short, the attempt failed miserably. Each attempt to present a finished product was met by demonstrations that important things about numbers were left out. This was because they were insisting that the theory be self consistent and rightly so. It took Kurt Godel to prove that such systems can not be both self consistent and complete. By requiring self consistency they doomed their efforts to produce an incomplete system. The only way to deal with complex systems was to rely on something external to attempt the completion. But then the larger system is subject to the same problem and an infinite regression results. This issue has come up often in discussions among complexity theory proponents. One way some seem to satisfy themselves that they need not heed

37 the problem is by claiming that Number Theory is mathematics and not science and that Godel’s proof has not been shown to apply to physical systems. This seems to be a false solution to the dilemma, especially since all the models science uses are mathematical 3. COMPLEXITY AS AN ATTRIBUTE OF NATURE

The use of the term “complexity” has come to be commonplace both in the scientific literature as well as in the lay press. A complex idea in its own right, complexity presents a challenge to those who would define it or codify its contents. Why is ”What is complexity” a question not so easily answered? Universities and other research institutions have programs in ”complexity research” and journals carry this word in their title. What is complexity? What does it mean to be ”complex”? The dictionary does not help us here. There is more to the idea than what a dictionary definition suggests. We need to look within science, where words are carefully defined and have more precise meanings, since it is here where the answer does lie. The concept of complexity that is of so widespread interest was born within the confines of the scientific community, had to struggle for existence, and finally grew into a substantial force in scientific thinking. The answer lies in the nature of science. Science helps us to understand the world around us by creating a surrogate world that is understandable. The real world is where complexity has its home. The world science gives us has become simple and understandable. 3.1. Hard Science i s built o n Cartesian Reductionism

Hard science can be best-defined in terms of Cartesian Reductionism and the machine metaphor. Descartes gave us the machine metaphor and he did it in a very interesting way. In the case of living systems he saw the body as a biological machine and the mind as something apart from the body. This is called Cartesian Dualism and survives to this day as one approach to the so-called mindlbody problem. What the machine metaphor did was to set the tone for modern science. It is still with us. Descartes really did not know what a machine is, or if he did, he never told anyone. Ironically, not only do we not have a good definition of complexity, but we also lack one for a machine. The duality between these concepts is the key to the resolution of the problem. The importance of the machine metaphor is in the intuitive concept of machine that almost everyone shares. A machine is

38 built up from distinct parts and can be reduced to those parts without losing its machine-like character. We call this idea ”Cartesian Reductionism”. We have seen that complex (real) systems can not be successfully reduced to material parts without the loss of some significant attributes in the process. This led to the axiom that the whole is more than the mere sum of its parts. Adopting this axiom as a truth leads to a conclusion that is inescapable. If the whole is more than the sum of its parts there must be more to things than atoms and molecules. Reducing a complex whole in the real world always destroys something. What is lost is an elusive entity, but is also the central idea in the concept of complexity as applied to the real world. If there is more to things in the real world than atoms and molecules then this something that is there has an existence, an ontology. This is the reason why as a concept complexity is a difficult one. It can not be both ways. Either a real world whole is more than the sum of its parts or it can be reduced to atoms and molecules with nothing being lost and the world is made of machine-like things. Cartesian reductionism does not work for making models of complex systems; it only reduces them to simple mechanisms that may reflect some aspect, but merely as a shadow of the complex whole

3.2. The Newtonian paradigm is the modern manifestation of hard science Newton gave us, among other things, three laws of motion, which were intended to describe the motion of the planets. It turned out that methodology spawned by the application of these laws could be applied in a seemingly perfectly general way. This broader application has been the foundation of the modern scientific method and will be referred to here as the Newtonian Paradigm. In the center of this paradigm is dynamics. Dynamics is the way the laws of motion get applied. The local description of the motion is formulated as a differential equation called an equation of motion. The equation of motion is manipulated by using the calculus (integrated) and results in a trajectory, which is an algebraic equation for calculating a particle’s position as a function of time. Later, this was made somewhat more complicated by quantum mechanics, but the central philosophy has never changed. The paradigm has been generalized from particle motion to all systems if we recognize that quantum mechanics is part of that generalization. The world-view called reductionism has as a central theme that anything in

39 the material world can be reduced to particle motion, that is, the motions and interactions of atoms and molecules. When we look carefully at the subject matter of physics, we see that it is the application of the Newtonian Paradigm to the universe. This application then makes the world into simple mechanisms. That is to say that the subject matter of physics is the study of simple mechanisms. Note that in this context, "simple" means the opposite of complex, not the opposite of complicated. 3.3. Complexity is the result of the failure of the

Newtonian Paradigm to be generic

A careful look at the real world tells us that it is complex. Yet, the success of the Newtonian Paradigm cannot be ignored. Most of modern science and technology is the result of it. For that reason alone it is difficult to suggest that it has limits and to then make that suggestion stick. Not only does the paradigm have limits, but also those limits are what gave rise to a concept like complexity. 3.4. The way science i s done: The modeling relation

How is science done? It is a combination of using our senses to observe the world around us and then to use some mental activity to make sense out of that sensory information. The process is what we will call the modeling relation. If we call the world we are observing and/or trying to understand the Natural System and the events that make it change as we observe causality, then that represents our object of study. What we do in our minds is to encode the natural system into another system that is of our making or choosing which we can call a formal system. Once we have chosen a formal system, we can manipulate it in various ways with the objective of mimicking the causal change in the natural system. These manipulative changes in the formal system we will call implication. Finally, once we think we have an appropriate formal system and have found an implication that corresponds to the causal event in nature, we must decode from the formal system in order to check its success or failure in representing the causal event. Figure 1 represents the modeling relation we have just described. If all the parts of the diagram are in harmony, in other words if 1 = 2 3 4, we say that the diagram commutes and we have a model. A model of the world is the outcome of a successful application of the scientific method, but it can also arise in other, less formal ways. Whenever someone tries to make sense out of the world, they are trying to construct a successful

+

+

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CAUSAL EVENT

DECODING IMPLICATION

SYSTEM

ENCODING Figure 1.

modeling relation, or a model. 3.5. Complex systems and simple systems are disjoint

categories that are related by the modeling relation The world therefore divides naturally into those things that are simple and those things that are complex. The real world is made up of complex things. It is something we encounter constantly and try to make sense of. The way we do that is to encode it into formal systems. Therefore the world of simple mechanisms is a fictitious world created by science or, more specifically, by physics as the hard version of science as a formal system which hopes to model the real world. This is the world of the reductionist. It is modeled by the Newtonian Paradigm and simply needs sufficient experimentation to make it known to us. Those experiments involve reducing the system to its parts and then studying those parts in a context formulated according to dynamics. Therein lays the reason for the restrictions on the scientific method. The method involves a rigid empiricism that feeds data into the formal system. Here is the philosophical rub. The epistemology spills over into an ontology. The modeling relation is forgotten (but it is always there) and the formal system has become reality. Now the definition of complexity is complete. The world, from which we single out some smaller part, the natural system, is converted into a formal system that our mind can manipulate and we have a model. The world is complex. The formal system we chose to try to capture it can only be partially successful. For years we were satisfied with the Newtonian Paradigm as the formal system, forgot about there even being and encoding and decoding, and gradually began to change the ontology so that the Newtonian Paradigm actually replaced or became the real world (at least

41 as seen through the eyes of science). As we began to look more deeply into the world we came up with aspects that the Newtonian Paradigm failed to capture. Then we needed an explanation. Complexity was born! This easily can be formalized. It has very profound meaning.

4. THERMODYNAMIC REASONING AS A

TRANSITION TO COMPLEXITY SCIENCE Rosen had little to say about thermodynamics in his critique of reductionism in the form of the Newtonian paradigm. There is probably a good reason for this. Clifford Truesdell [5] once made a very good case for thermodynamics having a certain “strangeness” as a part of physics, or we could say as part of the reductionist Newtonian paradigm. This strangeness needs to be considered in more detail for it reveals the seeds of the ideas Rosen found to be true about the mechanistic approach to reality. Thermodynamics is probably more poorly understood by mechanistic scientists than any other branch of physics. The reasons for this are deep and revealing. One of the most serious consequences of the inability of those doing either mechanistic science or thermodynamics to see the problem clearly is the resultant gap in the Newtonian largest model. It is possible to see mechanistic reasoning and thermodynamic reason as different models of complex reality in the spirit of the ideas discussed here. Instead, from its beginnings thermodynamics was put under severe pressure to conform to the mechanist’s largest model and thereby suffered a lack of development as its own alternative to mechanistic physics. Thermodynamics came into being for very practical reasons. The boring of cannons, the brewing of beer and the steam engine are but a few of the reasons that the frictionless world of Newton’s paradigm needed to be patched up. Heat as a form of energy also had to be dealt with. Perpetual motion machines had to be dealt with in a rational manner to curtail the squandering of time and energy, sometimes by very bright scientists. As progress was made, there were also immediate problems presented to the those who had been content with the frictionless world where heat was merely another form of energy and nothing more troublesome than that. Since the material world was to be understood in mechanistic terms, a new kind of mechanics called “statistical” mechanics had to be developed to try to make a bridge between thermodynamic reasoning and mechanistic reasoning. Why is this so? The answer is one of the best demonstrations of the

42 42 power of Robert Rosen’s analysis even though he may not have ever seen it himself. There is a fundamental difference between mechanistic reasoning and thermodynamic reasoning and it can not be erased by the limited successes of the use of statistical mechanics to bring them together. Thermodynamics is about those properties of systems that are true independent of their mechanism. This is why there is a fundamental asymmetry in the relationship between mechanistic descriptions of systems and thermodynamic descriptions of systems. From the mechanistic information we can deduce all the thermodynamic properties of that system. However, given only thermodynamic information we can deduce nothing about mechanism. This is in spite of the fact that thermodynamics makes it possible for us to reject classes of models such as perpetual motion machines. (This does not stop such models from appearing in subtle forms in the modern literature.) This asymmetry is poorly understood because thermodynamics is not a tool mechanists see as valuable. Some of this attitude is justified by the structure of thermodynamics as it exists in texts and courses. In some ways the field is a prisoner of its own history. It is useful to examine that history with this problem in mind. 4.1. Classical or LLequilibrium”t h e r m o d y n a m i c s a n d i t s limits

Thermodynamics recognizes distinct systems categories based on the constraints imposed on the system in order to study it systematically. This thought process is very much like the defining of a “universeof discourse” in other fields. A system is defined in terms of the way we allow it to interact with the outside world. Often it is useful to see these interactions as having to occur across some real or imaginary boundary or “skin” surrounding the system. There are only three possible categories based on this mode of reasoning:

(1) The isolated system. Isolated systems are isolated from the rest of the world by a boundary that is impermeable to both energy and matter flow. Such systems define the notion of “equilibrium”, in the thermodynamic sense, as the state the system is in after a sufficient amount of time is allowed to pass. The equilibrium state is characterized by the disappearance of the “transient” flows of matter and energy within the system that brought it to this final state wherein nothing further can happen. That is true in the large. Very locally, fluctuations around this equilibrium point are

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happening as a manifestation of the thermodynamic property called “temperature”. The fluctuations are atomic and molecular motion and increase in magnitude as heat energy is added to the system and decrease in magnitude as it is removed. Isolating the system will result in a system at constant temperature consistent with its heat content. (2) Closed systems are systems with boundaries that will allow the transfer of heat energy through them but these boundaries are impermeable to matter flow. Thus closed systems contain a fixed amount of material but can be warmed or cooled by manipulating their heat content. Such systems are capable of being put in contact with reservoirs of heat energy at different temperature to induce a flow of heat through them and are therefore capable of being maintained in stationary states away from equilibrium. (3) Open systems. These systems can exchange matter and heat energy with their environment and need never reach an equilibrium. These systems can exist in stationary states, periodic states, as well as chaotic states. The laws and ideas of thermodynamics were developed with great care by doing thought experiments on equilibrium systems. Virtual “quasistationary processes were conceived by imagining so called “infinitesimal” changes to take the isolated system from one equilibrium state to another. Sadi Carnot developed a clever method of bringing the system through a cycle so that it always ended in its starting equilibrium state. One central characteristic of equilibrium is the existence of a thermodynamic variable called “entropy”, S, defined as the quotient of the heat energy in the system, Q, and absolute temperature of the system, T:

S = Q/T More useful was the notion of the entropy change associated with an isothermal quasi-stationary process:

dS = dQ/T

(2)

The entropy of a system, to oversimplify somewhat, is a measure of the “quality” of the energy of that same system. This follows from reasoning about heat engines being operated in cycles. In order for a heat engine (e.g. a steam turbine) to produce work that can be used it must have a source of

44 hot steam and that steam must flow through it to a reservoir that is cooler. The First law of thermodynamics is nothing new to physics since it is simply the idea that energy can not be created or destroyed. Hence a stem engine uses some of the heat energy, converts it t o mechanical work, another form of energy, and allows the rest of the heat energy to pass through it with the matter (in this case water and/or water vapor) that is conserved in the process. The Second law of thermodynamics forbids the complete transfer of heat energy to mechanical work as well as forbidding the operation of the engine between a source and sink at the same temperatures. This is why the entropy became a useful and necessary concept. The result of “cooling” the matter in the process of extracting mechanical work, increased the heat/ absolute temperature quotient or the entropy of the system. One way of stating the second law is that there is a requirement that any real process must result in the increase of entropy overall. It is possible to make devices that locally decrease entropy, but only if the global result is an increase. At equilibrium, there is no change in the amount of entropy. dS = 0

(3)

>0

(4)

In any real process,

dS

The fact that so much of what we know about thermodynamics came from reasoning involving equilibrium and isolated systems is ironic when we recall that it was practical matters, the doing of things, the carrying out of processes that motivated the entire development. Nevertheless, by resorting to what now seems to be a rather clumsy thought process involving the carrying out of processes by performing small incremental changes, much was learned. What is problematic is that the world of real processes is what is of interest and equilibrium are the natural endpoints of some of those processes in the situation where the system is totally isolated. 4.2. Dissipation, frzction, and irreversibility

Energy is conserved, but entropy must increase in the carrying out of any process. The first two laws of thermodynamics are imbedded in that sentence. The statement is true of all systems regardless of mechanism. The result of this is that physics has two realizations. The original physics of

45 Newton and his followers was the physics of systems that had dynamics yet were without the creation of entropy. The “ideal” pendulum, for example, can not exist and is the end of a limiting process where friction is diminished to zero. Friction is another manifestation of the second law of thermodynamics. It is the recognition that in any real process some of the energy must be converted to heat energy. This heat energy can not be converted back to the original form of energy by the same process. The process is “irreversible”. One good way to see this is in a famous experiment performed by Joule to measure the factor that must be used when energy in the form of mechanical work is changed to heat energy by any process. The number he obtained was called the mechanical equivalent of heat. The device he used has value as a source of insight far beyond its utility in obtaining that number. The device was a jar of water surrounded by an insulating jacket having the purpose of isolating the system thermally. Thus heat energy could not enter or leave the system through its walls. At the top of the jar was mounted a well insulated crank attached to a paddle wheel inside the jar. Turning the crank allowed mechanical work to be performed on the water in the jar. The system allows a simple mechanistic explanation for how the mechanical work is converted to heat energy. The water is put into motion in a directed manner by the paddle wheels, but looses that directed motion eventually, the motion rather becoming “random” motion characteristic of heat energy. Having a thermometer mounted so that the water temperature can be monitored allows the direct relation between the work done turning the crank and the appearance of heat in the jar, using the heat capacity of water to calculate the amount of added heat energy from the increase in measured temperature. The process is irreversible. There is no way to reverse this randomization of molecular motion to turn the paddle wheel and thereby gain back the mechanical work. The energy has been converted once and for all as long as it is confined to the thermally isolated jar. This is a very revealing example of the nature of friction. Sometimes it is difficult to see a mechanistic picture of how this frictional “dissipation” of energy occurs. Chemical reactions have their own version of friction and are as irreversible as any mechanical process. In electricity the resistance to the flow of electrons results in electrical energy being converted to heat. Knowing this, physics has created two categories for the systems it studies, conservative and non-conservative systems. Newtonian dynamics was developed using conservative, frictionless systems (fictional) and then ex-

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tended to the more difficult non-conservative systems that, as real world processes, involved friction necessarily. These non-conservative systems were irreversible and needed to be open if they were to do anything other than wind down to an equilibrium state and stop.

4.3. Preserving the paradigm involved considering fraction, irreversibility and dissipation It wasn’t until the work of Onsager [6, 71 and others in the 1930’s that a systematic physics of non-equilibrium, irreversible systems was developed. This new physics was thermodynamic rather than mechanistic, but the field of statistical mechanics was later extended to the non-equilibrium domain making the problem of backing up the thermodynamic knowledge with mechanical interpretations in terms of atomic and molecular motion very much more difficult. The increase in difficulty has severely limited the scope of these endeavors. Onsager and the others developed a thermodynamic approach and escaped most of that difficulty. They did so in a way that had its own inbuilt limitations, but that would not be seen for about forty more years when Network Thermodynamics was developed by Oster, Perelson, and Katchalsky and independently by Peusner [8]. The approach Onsager and the others introduced is am “phenomenological” approach. This is very different from the mechanistic physics that was used to describe nonequilibrium dynamic systems up to that time. Onsager’s work was part of what won him a Nobel Prize. Later, Prigogine won a Nobel prize for building on the work of Onsager. One can speculate that Katchalsky may have been there in Prigogine’s place had he not been gunned down by terrorists in 1972. That event changed the history of science and the history of the world. It was Kedem and Katchalsky who introduced non-equilibrium thermodynamics to biology and changed the way we dealt with all the dynamic processes occurring in living systems. Then, along with Oster and Perelson, he explored the new approach that married non-equilibrium thermodynamics to the huge body of theory generated for dynamic electric networks. The impact of all this and the work of others that built on it is largely not understood by many even after complexity theory has grown in its impact on science. The reasons are many and not easily identified without being critical of the way science is practiced. The urge to preserve the mechanistic paradigm and to see thermodynamics as only an appendage has been too strong and in that way too destructive. When quantum theory was born it

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also was shaped by the same urges. The difference is that quantum theory is so much more adaptable to the mechanistic reductionist approach. Its mathematics and the interpretation of the mathematics could be given the very same form as what it was clearly showing to be a failure of Newtonian mechanics. Thermodynamics, on the other hand, uses a different form of mathematics. To most scientists satisfaction quantum mechanics simply helped further the knowledge generated by the Newtonian paradigm and did it little harm. It allowed the reductionist philosophy to appear to have established itself at all levels in the material description of reality. It became possible to see a more or less universal bottoms up approach to science. Yet there are findings from network thermodynamics that raise some very interesting, and possibly troubling questions about this. The lack of interest in these questions parallels a similar lack of interest in questions raised about science by Robert Rosen as he explored the complexity of the world science seemed to have mastered. It is very important that these questions and the findings that have been generated by the use of thermodynamic reasoning that is not constrained by the mechanistic mind set do not get totally ignored and forgotten Yet that may be exactly what is happening as reductionism forges ahead. Some insights into these events of scientific history and the philosophical oversight or lack of same can be had thanks to the contributions of Lakoff [9] who has applied cognitive linguistics to similar situations in politics. His ideas can be married to a concept of memes as “packages” of ideas that get passed along in a manner much analogous to genes.

4.4. Framing the question in science:” Don’t think about

the whole system” As we make models of the world we necessarily use what we have already put together in our minds. The idea that complete objectivity is possible arises when the nature of the modeling relation is misunderstood. A model has many components. It consists of a number of subjective choices. The choice of the formal system we use to encode sensory information in order to think about how the real world is actually operating is a subjective choice. The illusion of objectivity arises from the fact that the question has been framed in a particular way. The idea that there is only one encoding possible in order for the model to be “scientific” forces a choice. As soon as we acknowledge that complex reality can and indeed must be encoded into distinctly different formal systems, the illusion of objectivity is de-

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stroyed. Lakoff uses the sentence “Don’t think of an elephant” to illustrate the impossibility of escaping certain thoughts once such a framing event has occurred. Modern science is framed by reductionist thought and this framing is so effective that the illusion of objectivity can do very well. A large part of Robert Rosen’s work is devoted to helping us see the predicament we are in. It is very interesting how different people react to these ideas. If the concept of complexity being used here is not accepted by a person, there is almost no chance that the person will understand why objectivity is not ever possible. The irony is that that position is a subjective choice. If this is understood, it becomes a simple matter to understand why thermodynamic reasoning is only accepted when it has a mechanistic counterpart. The largest model must be able to encompass everything scientific. The idea that distinctly different formal systems must exist as valid models is unacceptable. Thus the “useful” ideas from thermodynamics are made to be derivable from mechanistic reasoning, namely statistical mechanics, and any other conclusion drawn independent of that mechanical base are suspect.

4.5. Reductionism needs a particular kind of mathematics

t o accomplish its goals The formal systems reductionist science use are mathematical. Yet among the rich formalisms in the total mathematical repertoire, only analytical mathematics is able to fill the needs of the reductionist paradigm. Calculus and differential equations are the essence of the Newtonian approach and were indeed created for that use. Thermodynamics and Rosen’s approach to complexity utilize another kind of mathematics. It is topology that enters as the formalism that provides the needed structure for encoding relationships in the real world. The analytical mathematicians also were forced to utilize topology in the breakthroughs that occurred when the realization that so much of interest had to be encoded using non-linear differential equations as the equations of motion. Topology and differential geometry then were found useful even to reductionists. Out of this the nature of chaotic systems was born and chaos theory had a central role during a brief period of the evolution of complexity theory. It is still not clear to everyone that chaotic dynamics are part of the Newtonian paradigm and do not constitute a distinct formalism different from its largest model. The topological mathematics used to model complex systems and the topological mathematics that must be used to arrive at certain unique new conclusions about

49 the real world using thermodynamic reasoning are very closely related. Rosen, as a student of Rashevsky recognized that topological mathematics was necessary to create the relational biology they both had envisioned as the approach to living systems that was not locked to the severe limits of the reductionist paradigm. The elusive qualities of living things that distinguished them from non-living mechanistic things could only be dealt with by encodings into topological mathematical formalisms. Hence Rosen developed the metabolism-repair M, R system as the formalism he would manipulate to come up with a clear distinction between the classes of things we call “organisms” and the class of things we call “machines”. In doing this he forced his audience to make a hard choice. Either they would accept a formalism that “kept the organization” but left out the “physics” or they were locked into the reductionist’s world of physics which necessarily lost crucial aspects of the complex whole as it was reduced to mere material parts. Thermodynamics in its most powerful form mimics this situation. There are many examples, but only a few need be mentioned to make this point.

4.6. Topological reasoning in thermodynamics leads to

powerful results One of the most interesting examples of the comparison between reductionist methods and relational or topological methods is in the proof of the second law of thermodynamics which was accomplished by Caratheodry after a discussion about the proofs existing up to then with Max Planck [8]. The existing proofs were done using the cyclic quasi-stationary processes developed by Carnot. Plank found these a very unsatisfying way to prove something having the global consequences of the second law. Caratheodry was able to construct a topological proof that rested on one piece of empirical input to confirm it. The second law had its global validity established by Joule’s experiment for it was only necessary to give one instance of irreversibility in the real world to make the model valid. The discovery of a metric structure for thermodynamics had been forsaken until Network Thermodynamics in the hands of Puesner revealed that the network thermodynamic model of a system was canonical. This discovery rests on the topological embedding of the system posited by Onsager, Prigogine and the others in a higher dimensional space[8].

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5. Will science extend to the modeling of complex reality or will it be restricted to the limited domain of the largest model formalism it clings to? This is a question that should not have to be asked. It is the self-imposed limits that scientists have used to frame the answer to the question “What is science?” that make it necessary. The framing of the question in this manner is strengthened by the enormous progress being made in technology. The hunger for more and more practical results makes it very difficult to persuade scientists to spend time and resources pursuing an approach that is just beginning to be recognized by significant minority among them. Should bright students be encouraged to work in this new area? Is it wise to devote any significant portion of the scarce resources given to science to the development of these radically new ideas? It seems to be easier for many scientists to choose an alternative definition for the word “complexity” and to make sure that this definition never allows those using it to step outside the reductionist paradigm. There are a few who will not be restricted in that way. Two examples of recent progress in complexity science are very different in their area of application yet they hold in common their dedication to the new challenge to find distinct formal systems that can not be reduced to the Newtonian paradigm. In their book Into the cool [lo] using earlier work by Schneider and Kay [ll]develop a thesis about the synergy between life on this planet and the conditions that sustain that life based on a broader interpretation of the second law of thermodynamics than used before. They see the second law as demanding that systems respond to imposed gradients by some form of self-organization in order to decrease the gradient. The alternative is to have enormous driving forces stressing the system. Hence the biosphere is believed to be the result of the tendency for the sun’s energy to create stressing gradients. They present evidence for the role that life forms, weather patterns, etc. play in keeping the system stable. Clearly, a concept like this one is not going to be something we prove or disprove. At a time when so much controversy centers around the impact of human activity on planetary conditions, their endeavors do provide a model worth examining further before dismissing it out of hand merely because it forces us to abandon the restricted universe of discourse reductionist science has forced upon us. The second area where some bold new ideas are being offered ins in

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the domain of what we often call the “mind-body” problem. The work of Bach-y-Rita as interpreted by Kercel [12, 131 and others goes tom the heart of many modern controversies. Their interpretation of real experiments involving the sensory behavior of humans is shaking quite a few foundations. At the center of all their models is the central role played by closed loops of causality. These loops are the self-referential heart of complex reality and the human mind seems replete with them. This work and he work of Louie [14], a student of Rosen, showing that Rosen’s {M,R} systems necessarily have non-computable aspects suggests that “artificial intelligence” and “artificial life” are indeed merely machine intelligence and simulations of systems that have some limited life-like qualities. These technological marvels are distinctly different from human intelligence and living systems in many import ways. These new investigations as well as the areas of thermodynamics that have been largely disregarded because of their distinctly non-mechanistic character strongly suggest that science will find a way to include other formalisms and break free from the restrictions it has imposed in itself.

References 1. Rosen, R. Anticipatory Systems, New York, Pergamoii ,1985 2. Rosen, R. LifeIitself, New York, Columbia, 1991 3. Rosen, R. Essays on LifeIitself, New York, Columbia, 2000 4. Mikulecky, D. C. The circle that never ends: Can Complexity be made simple? In Complexity in Chemistry, Biology, and Ecology, D. Bonchev and D. H. Rouvray, eds. New York, Springer, 2005. 5. Truesdell, C. Rational Thermodynamics, New York, McGraw-Hill, 1969 6. Onsager, L. Reciprocal Relations in Irreversible Processes I, Phys. Rev., 1931a, 37, 405-426. 7. Onsager, L. Reciprocal Relations in Irreversible Processes 11, Phys. Rev., 1931b, 38, 2265-2279. 8. Mikulecky, D. C. Applications of Network Thermodynamics to Problems in Biomedical Engineering, New York, New York University Press, 1993. 9. Lakoff, G, Don’t Think of an Elephant! Know Your Values and Frame the Debate White River Junction, Vermont, Chelsea Green Publishing, 2004 10. Schneider, E. D. and Sagan, D. Into the Cool, Chicago, University of Chicago, 2005 11. Schneider, E. D. and J. J. Kay, Life as a manifestation of the second law of thermodynamics, in Modeling Complex Biological Systems, hl. Witten and D. C. Mikulecky, eds, Special Issue of Mathematical and Computer modeling, 1994, 19, 25-48. 12. Kercel, S. W. Journal of Integrative Neuroscience, 2005, 4, 403-406. 13. Kercel, S. W., Reber, A. S. and Manges, W. W. Some Radical Implications

52 of Bach-y-Rita's Discoveries, Journal of Integrative Neuroscience, 2005, 4, 551-566. 14. Louie, A. H. Any Material Realization of the (M,R)-Systems Must Have Noncomputable Models, Journal of Integrative Neuroscience, 2005, 4, 423436.

ON THE IMPORTANCE OF A CERTAIN SLOWNESS Stability, memory and hysteresis in complex systems*

PAUL CILLIERS University of Stellenbosch fpc@sun. ac.za

In philosophy the winner of the race is the one who can run most slowly. Or: the one who gets there last. Wittgenstein (Culture and Value) 1. Introduction

As a result of a whole range of what one could call “pathologies” in contemporary culture, the idea of “slowing down” has of late been mooted in a number of contextsa. A few can be named briefly. The “Slow Food” movement, which started in Italy but has a worldwide following, extols the virtues of decent food made from decent ingredients without compromise. The resistance shown to “junk food” is not only based on aesthetic considerations, but also on ethical (and nutritional) ones. The movement promoting “Slow Cities”, also of Italian origin, fosters an understanding of cities which is more humane. Such cities should encourage walking rather than driving, have small shops with local products rather than shopping malls and, in general, provide opportunities for the community to interact, not to live in isolation. “Slow schooling” is a movement which questions educational processes in a world geared for instant results. It emphasises the contextual nature of knowledge and reminds us that education is a process not a function. On a more personal level, “slow sex” involves attitudes which try to prevent that the values of the marketplace also rule in our intimate relationships. We need to recognize that the journey is *This paper first appeared in Emergence: C o m p l e i t y and Organization. Permission to use the material is gratefully acknowledged. Honor6 (2004) for a discussion of the emergence of several movements which challenge the “cult of speed”.

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more important than the destination, and that takes time. An immediate or perpetual orgasm is really no orgasm at all. There are a number of very important issues at stake in these examples. In what follows, however, the focus will not be on these social movements as such, but on some of the underlying principles which make the debate on slowness an important one. Through an analysis of the temporal nature of complex systems, it will be shown that the cult of speed, and especially the understanding that speed is related to efficiency, is a destructive one. A slower approach is necessary, not only for survival of certain important values or because of romantic ideals, but also because it allows us to cope with the demands of a complex world in a better way. The argument will be made initially by briefly analysing current distortions in our understanding of time. These distortions result, on the one hand, from the rational and instrumental theories we have about a modern world, and, on the other, from the effects of certain technologies, especially communication and computer technologies. In order to show why these are “distortions”, or at least, to show why these distortions are problematic, the temporal nature of complex systems will be discussed. The relationship between memory and anticipation will be central to this discussion, but attention will also be paid to the importance of delay and iteration. These characteristics of complex systems have important implications for our understanding of the formation of identity, both individual identity as well as the identity of groups. In closing, a number of general cultural issues involving the fast and the slow will be looked at. It is important to realise that the argument for slowness is not a conservative one; at least not in the political sense of the word. It is not merely backwards-looking nor a glorification of what has been. Although it emphasises the historical nature of knowledge and memory, the argument for slowness is forward-looking; it is about an engagement with the future as much as with the past. Slowness is in itself a temporal notion, and in many ways the opposite of the notion “static”. In point of fact, it is actually an unreflective fastness which always returns you to the same place. It should also be stated up front that there is no argument against an appropriate fastness. A stew should simmer slowly, but a good steak should be grilled intensely and briefly. The argument is against unreflective speed, speed at all cost, or more precisely, against speed as a virtue in itself; against the alignment of “speed” with notions like efficiency, success, quality and importance. The point is that a system which has carefully accumulated the relevant memories and experiences over time will be in a better position

55 to react quickly than one which is perpetually jumping from one state to the other. Perhaps “slow” and “fast” are not exactly the correct terms to use. Terms like “reflective” and “unreflective”, or “mediated” and “unmediated” may be more accurate. Nevertheless, the debate taking place uses “slow” and “fast”, and the terms do have a certain rhetorical significance. If we stay with their use, it is done in a metonymical way. The whole point of this paper is to give them a richer meaning.

2. Living in the Present

In Tame: The Modern and Postmodern Experience Helga Nowotny (1994) argues for a certain shift in our experience of time. In short, in my paraphrase, and incorporating insights from Bauman (e.g. Bauman 1992), the argument is the following: One of the main aims of the instrumental rationality flowing from the Enlightenment was to create conditions in which we are not controlled by contingency. To achieve these conditions, it is necessary to understand, and preferably control, the future. This demands co-ordinated and goal-oriented action in the present. Modernism becomes a project which demands our total commitment against the forces of irrationality and chaos. The modernist project has two important effects on our understanding of time. In the first place, our actions need to be ceordinated. This can only happen if time is universalised in such a way that we all live in the ‘same’ time. This was achieved mainly through technology, i.e. the construction of accurate clocks, and by regulating time globally. Instead of each person or local community living in their own time, it was necessary to synchronise time in such a way that activities in say New York and Paris could be correlated. The effects of this, however, go much further than merely synchronising time in different parts of the globe. It also means that private time and public time are synchronised. We have to live our lives according to a generalised and controlled understanding of time. A subjective, or should one say phenomenological, experience of time has to be sacrificed in order to generate a universal temporal framework in which we can operate efficiently. The second effect of instrumental rationality on our understanding of time is a result of the desire to control the future; for the future to be made knowable. This would only be possible if the future, in some essential way, resembles the present. We cannot anticipate what we do not know, and

56 therefore we should do everything in our power to create a future which does not disrupt the steady progress we are making towards a better world. This modernist strategy is perhaps exemplified best in Hegel’s dialectic of history which is supposed to converge towards an ultimate solution. The actual result of this ideology is to extend the present into the future, causing us to live in a perpetual “present”. This collapse of the diachronic into the synchronic allows instantaneous interaction between everybody; it creates a world which is fast and efficient. The sacrifice made in order to achieve this, however, is nothing short of sacrificing the very notion of temporality. Nowotny (1994: 16) calls it “the illusion of simultaneity”. The way in which contemporary society lives in an eternal present, or what Eriksen (2001) calls the ‘‘tyranny of the moment”, is made possible, and augmented, by the surge in technology, especially computer and telecommunication technology. We are instantaneously in contact with everybody everywhere. Not only has the distinction between home and the workplace collapsed, but also the distinction between work time and private or leisure time. It is expected that we are available, always and everywhere. This state of affairs may have been less detrimental if it did not also demand instant response. The very reason for mobile phones and e-mail lies in the fact that immediate response is possible. It is in this “immediate” that the main problem lies. There is less and less time for reflection. Reflection involves delay, and in a cult of speed, delay is unacceptable. This move away from reflection to immediate response has profound implications for our understanding of what it is to be human (see Parkins 2004: 376-379), t o which we shall return. The “illusion of simultaneity”, the idea that if we live quickly and efficiently in the present we are somehow closer to reality, is nevertheless exactly that: an illusion. We cannot escape our temporal nature, and our persistence in trying t o do so can only lead t o pathology. The necessity of delay and reflection needs to be re-evaluated. This can be done from a number of perspectives. A Freudian analysis would show that instant gratification is actually a destruction of pleasure. More sublime pleasure can be found only if desire is delayed, anticipated as a memory of something still t o come, yet something which should also in principle be able to surprise us. Derrida calls the illusion of living in the present, of thinking that we have access to an objective understanding of reality if we live “in” it, the “metaphysics of presence” (Derrida 1976: 49). He introduces the notion of dzffkrance specifically to undermine the metaphysics of presence (62). DzffLrance is a notion which intertwines difference (as a spatial notion, one

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could say) and delay (to defer, a temporal notion) as the engines of meaning (Derrida 1982). The present consists only as a combination of memory (of what has been) and anticipation (of what is to come). In his novel Slowness Milan Kundera (1996) uses the metaphor of somebody riding on a motorcycle as being constantly in the present. Speed and the demands of the machine reduces his horizon to something immediate. Someone walking, however, is moving at a pace which allows for a much wider horizon. The stroll unfolds in time in a way which opens up reflection about where we are coming from and where we are going to, as we walk. This theme of both the past and the future being present in a meaningful experience of the present could be pursued in much more detail from both a Freudian and Derridean perspective - and several others too - but the argument for a meaningful temporality, i.e. something slower, will be made here from the perspective of the dynamics of complex systems. 3. Complex Systems, Temporality and Memory

An important aspect of complex systems, one which certainly complicates our understanding and modelling of such systems, is their temporal nature. Complex systems unfold in time, they have a history which co-determines present behaviour and they anticipate the future. Moreover, as we know at least since the work of Prigogine, the behaviour of complex systems are not symmetrical in time. They have a past and a future which are not interchangeable. This being “situated in time” does not always receive adequate attention in our analysis of complexity. The central notion at stake when we talk of time and complexity is that of “memory”. Memory is the persistence of certain states of the system, of carrying something from the past over into the future. It is not merely the remembering of something in the past as if belonging to that past, it is the past being active in the present. We should therefore not think of memory in abstract terms, but of memory as something embodied in the system. In many respects the system is its memory. If one accepts a n understanding of complexity which emphasises the relational nature of the system, it is useful to think of systems as networks where the connections between the nodes are more important than the nodes themselves. The nature of these connections is a result of which states of the network are “retained”, thus the structure of the system is a result of the sedimented history of the system.b bThis argument can also be made using the example of the brain, and links with many

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It is important to remember that memory is not merely a cumulative process. The structure in the network of relationships can only develop if certain states of the network are not maintained. Memory is a result of a process of selection. The states which are significant are repeated more often and therefore form more permanent links in the network. Less significant states will fade away over time. Memory is only possible if the system can also forget‘. What is important to note at this stage is that memory is not an instantaneous thing, it takes time to develop, it is slow. If one characterises memory as the past being carried over into the future, it follows that the future can only be anticipated in terms of the memory of the system. Anticipation is not, or at least, should not be, simply an extrapolation of the present. It is a complex, non-linear process which tries to find some trajectory, some way of “vaulting” from that which has already been experienced to that which has to be coped with. The quality of the anticipation is a function of the quality of the memory. A more varied, richer, deeper and better integrated memory will open up more sophisticated anticipatory capabilities. The obvious question now would be to ask how such a rich memory is formed. This is a complex issue, but for the sake of the argument at stake here, one can say the following: memory is information from the environment which has been filtered, it is that which has been interpreted - by the memory already sedimented in the system - as significant. The identity of the system is, in some sense, its collection of dynamic memories. The implication is that the system cannot reflect, or act upon, everything which is going on in the environment at a given moment. If that were the case, the system would always be merely a reflection of its environment and would have no identity of its own. In order for it to be a system at all, a system which has its own identity, which can react to the environment and not merely mirror it, a certain hysteresis is required. The system must be slower than its environment. The notion of hysteresis is an important oned. It links to the notions of delay and dzffirance discussed above. An event in the environment of the system does not have inherent and immediate significance for the system. Freudian arguments in an interesting way. See Cilliers (1998:4547,92,108)for further discussion. CThisprocess is known as the “use principle” or Hebb’s rule. For more detail see Cilliers

(1998: 17-18,93-94) dHysteresis is the “lagging of effect when cause varies” (Oxford Concise)

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Its significance is established in time as it is reenacted in the system and carried over into the future. In a way, the significance of an event has always already been established (in terms of the memory of the system), but never completely or finally since the significance is always also to be determined by what is still to come. The system has to hang on to some aspects with a certain tenacity; not let go of them too quickly. There is risk involved in this, of course. The system has to invest resources in this process. It cannot maintain everything; it has to select. If too many of the wrong things are carried over it will impair the system’s performance. However, if not enough is carried over, it will also fail. To put it in slightly different terms: the system has to find a way to discriminate between information and noise. If it follows every trend in its environment, it will also be following noise. If it reacts too slowly it will only follow the low-frequency trends, which may also be just noise. The system must be stable enough not to be buffeted around by every fluctuation, and it must be flexible enough to be able to adapt when necessary. Where this optimal point lies is not a question which can be answered from an objective viewpoint. The balance between stability and change is a contingent thing which plays itself out in time. What one can say, though, is that merely to be fast will destroy the system. The argument for a certain slowness should start to take shape now. A viable system has to be able to resist some of the dynamics in its environment. There should be a temporal space in which the past is allowed to play itself out in interaction with present. There must be time for reflection and interpretation. The faster the system becomes, the shallower its resources will be. Ultimately quick behaviour will be no more interesting than Brownian motion. It must be stressed again that the argument for a certain slowness is not a conservative argument. A certain amount of conservation is a prerequisite for a system to maintain itself, of course. The important point, to which we shall return, is that a “slow” strategy is not a backwards-looking one. If a somewhat slower tempo allows a system to develop a richer and more reflective memory, it will allow the system to deal with surprises in its environment in a better way. The argument of slowness is actually an argument for appropriate speed. There is no objective or immediate rule for what that speed is. If anything, it is a matter of experience, and experience (as Aristotle urged) has to be gained, it cannot be “given” in an immediate way. It is experience which determines which piece of meat should be fried quickly and which should simmer slowly in the stew. She who fries

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everything quickly will only have something nice to eat now and then, and then purely by chance.

4. Integrity, Identity and Reflection In his novel The Discovery of Slowness, Sten Nadolny (2003) g'ives us a fictionalised account of the life of John Franklin. Franklin, a l S t h century explorer primarily obsessed with finding the Northwest passage, was slow. His advance (in the Royal Navy) is also slow, mainly because being slow is confused with being stupid. Since he is not stupid he is gradually awarded command, and those working with and under him discover the advantages of being slow. Franklin is persistent, dependable and trustworthy. Even in war, thorough reflection pays dividends not always immediately apparent. His political career, as governor of Van Diemen's Land (now Tasmania), ends badly only because he is disgraced by those out for quick and selfish results. His principles are not negotiable. Franklin is a worthwhile human being because he has integrity. There is a substance to his personality which may seem opaque at first, but eventually shows itself as solidity. The nature of his integrity is directly coupled to his slowness. He assimilates, integrates and reflects before he acts. This is sometimes a ponderous process, and he pays a price for it. Under normal circumstances it is easy not to notice someone like this, or to pass him by, but when there is a crisis, it is him they turn to. He can be trusted, he will come up with something. This is most significant. It is exactly when one would think that being fast is what is required that slowness proves its worth. The link between slowness and integrity is also an issue in J.M. Coetzee's (2005) novel Slow Man. Here we have a character who resists change, despite the cruel demands being made on him. He clings to a set of values important to him, and this gives his personality substance. However, he is too stubborn, and eventually he cannot adapt to new circumstances. One has tremendous sympathy for him, but he turns out to be too slow, and pays the price for it. Even so, it is clear that when there is a choice between the loneliness of the slow and the superficial companionship of the quick, the author sides with the slow. Integrity is more important than a certain kind of success. Despite Coetzee's darker view, there is no reason why slowness should be solitary and sad. Quite the contrary is true. In his novel Slowness, Milan Kundera (1996) shows with great conviction how a certain slowness

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is a pre-requisite for being fully human. What is at stake in this novel is not moral integrity, or a kind of Calvinist dependability, but the sensuality of human interaction, the beauty of a relationship which unfolds in time, the ecstasy of a love which has a history and a future. Being human implies having a body, something with its own rhythms and demands. If we reduce all of this to something merely instrumental, to transactions written in legal terms (not in lyrical prose), if we demand results now, then we will stop being human. Language cannot be reduced to a code; it plays itself out in a certain context. What is more, even if we immerse ourselves in the context we have to wait beyond the last sounds. When all is said, the meaning has not finally arrived yet. It is the anticipation of what it could yet mean which draws us forward. Einmal ist Leinmal. Many may feel that the novel is an outdated art form, something to be replaced with the fast and immediate communication of the digital code. In his book The Moment of Complexity Mark Taylor (2003) seems to lean in this direction. For him, something of a paradigm shift has occurred in the last few decades. We live in a new world with new forms of communication and new forms of learning and human interaction - something he seems quite willing to sell. He resonates with a fast world, something new and exciting. His emphasis on the new goes hand in hand with a nearly interchangeable use of the notions “noise” and “complex”. This problematic conflation is the result of an understanding of complexity primarily informed by chaos theory and of information as entropy. Such an understanding, inspired by the theories of Shannon and Chaitin, will attribute the highest information content to a purely random sequencee. Although these notions are important in the context of computation, they are less useful when talking about complex systems in general. Living systems, including the social systems Taylor explores, are neither random nor chaotic. Despite the fact that they are constituted through non-linear interaction and that they are capable of novel and surprising behaviour, they are well-structured and robust. They persist through time and maintain themselves. When we encounter behaviour that we do not understand, or cannot decode, it often looks like noise, but once it is understood we can see the patterns. These patterns are not merely or only an order imposed by the observer, but also characteristics of the system itself. Complexity may look like noise, but all noise “See Hayles (1999) for a discussion of these issues. Primary sources are Shannon (1949) and Chaitin (1987).

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is not something complex waiting to be decoded. Sometimes noise is just noisef, Taylor’s argument is seductive, but, to my mind, wrong if not harmful. In his fervour to embrace the posthuman he looks at the history of being human with a certain disdain. It seems as if he thinks that complexity is a recent discovery and forgets that being human has always been complex. He embraces the present and wants to deal with it quickly and efficiently. We can be educated instantaneously by electronic means and thus we should make a radical break with old methods. In his excitement he forgets that complex systems, even those in our postmodern world, are constituted historically, that they develop and change and that one of their primary functions is to distinguish between information and noise. This cannot be done at the press of a button. The ideas of the posthuman and the cyborg are of undeniable importanceg, but in our enthusiasm to embrace new modes of being we should be careful not to effect a transformation into something inhumanh. Machines are fast, but they are machines. The present argument is not for an a prior2 rejection of the possibility of machines with human capabilities, or one that denies the intimate relationship humans have always had with technology. Our cultural existence presupposes the use of tools. The difference between using a quill and word-processor may have huge implications on a practical level, but they also share some essential features’. The notion “posthuman” is thus an ambiguous one. If it signifies a tight coupling between the body and technology, we have always been posthuman. If it signifies the obsolescence of the body, perhaps in the sense that a “person” could be downloaded instantaneously as software and run on a machine, it becomes a problematic notion at odds with the idea that human identity is also the result of a certain temporal embodiment. The general argument presented here maintains that any complex system, organic or not, would have to incorporate a certain slowness. The need for slowness, and a warning against the embracing of the

‘A similar, and more detailed argument is made in Cilliers (2005). Wee Badmington 2000 for a collection of philosophical essays on the posthuman. hSee Hayles (1999) for a detailed discussion of cybernetics, the development of the posthuman and the importance of embodiment. See Braidotti (2005) for an affirmative discussion of the posthuman which is neither a euphoric, uncritical acceptance of advanced technology, nor a nostalgic lament for the decline of classical humanism. ‘There would definitely be a lot less drivel to wade through if it was not possible to write so quickly.

63 fast, can perhaps be motivated best from the perspective of philosophy. Philosophy, in its most general form, is essentially the art of reflection. Wendy Parkins (2004) analyses contemporary culture as one moving away from reflection, and argues that what we need is an “ethics of time”. She does not elaborate much on what such a n ethics should look like, but it is something which needs careful attention, not only from a moral perspective, but also from the purely pragmatic perspective of how to live and survive in a fast world. Such an “ethics” will be complex in itself. It will have t o unfold in time and be conscious of its own temporal nature. For now, instead of rushing around like the red queen in a world where change is virtuous merely because it is change, we can start by taking some time out to reflect. At this point the argument for slowness becomes a political one: we should put up some resistance to a culture in which being fast is a virtue in itself. With a little more regularity we should say “no”.

References 1. Badmington, Neil (ed.). 2000. Posthumanism. London: Palgrave. 2. Bauman, Zygmunt. 1992. Intimations of Postmodernity. London: Routledge. 3. Braidotti, Rosi. “Cyberfeminsim with a difference.” http://www.let.uu.nl/womensstudies/rosi/cyberfem.htm (Accessed 8 August 2005) 4. Chaitin, G.L.J. 1987. Algorithmic Information Theory. Cambridge: Cam-

bridge University press. 5. Cilliers, Paul. 1998. Complexity and Postmodernism. Understanding Complex Systems. London: Routledge. 6. Cilliers, Paul. 2005. Complexity, Deconstruction and Relativism. Theory Culture €4 Society, Vol. 22 (5), pp 255-267. 7. Coetzee, JM. 2005. Slow Man. London: Seeker and Warburg. 8. Derrida, Jacques. 1976. Of Grammatology. Baltimore: John Hopkins University Press. 9. Derrida, Jacques. 1982. ”Diffkrance” in Derrida, Jacques. Margins of Philosophy. Chicago: The Harvester Press, pp. 1-27. 10. Eriksen, Thomas Hylland. 2001. Tyranny of the Moment: Fast and Slow Tame in the Information Age. London: Pluto Press. 11. Hayles, N Katherine. 1999. How W e Became Posthuman: Virtual Bodies in Cybernetics, Literature, and Informatics. Chicago: The University of Chicago Press.

12. Honork, Carl. 2004. I n Praise of Slowness: How a Worldwide Movement is Challenging the Cult of Speed. London: Orion. 13. Kundera, Milan. 1996. Slowness. London: Faber and Faber. 14. Nadolny, Sten. 2003. The Discovery of Slowness. Edinburgh: Canongate. 15. Nowotny, Helga. 1994. Time: The Modern and the Postmodern Experience.

Oxford: Polity Press.

64 16. Parkins, Wendy. 2004. “Out of Time: Fast subjects and slow living”. In Time and Society, Vol. 13 No 2, pp. 363-382. 17. Shannon, C.E. 1949. Communication in the presence of noise. Proc. IRE., Vol. 37, pp. 1cb21. 18. Taylor, Mark C. 2003. The Moment of Complexity: Emerging Network Culture. Chicago: The University of Chicago Press.

SIMPLICITY IS N O T TRUTH-INDICATIVE

BRUCE EDMONDS

Centre for Policy Modelling Manchester Metropolitan University http://bruce. edmonds.name

1. Introduction The notion of simplicity as an important property of theories is traditionally ascribed to William of Occam (1180), who extensively used the principle in argument to rebut over-elaborate metaphysical constructions. It has been invoked as part of the explanation why the Copernican account of planetary motion succeeded over the Ptolemaic one whilst the evidence was still equivocal. Newton made it one of his rules of reasoning: “we are t o admit n o more causes of natural things than such as are both true and suficient to explain their appearances f o r Nature is pleased with simplicity and aflects not the pomp of superfluous causes.” [Ref. 1, page 31. Einstein chose the simplest possible system of tensor equations to formalise his theory of General Relativity.2 Phrases like “for the sake of simplicity” are used to justify many modelling decisions (e.g. Ref. 3, page 1). More fundamentally, some go further, claiming (or more often, just assuming) that a simpler theory is somehow more likely to be true (or closer to the truth) than a more complex theory. For some surveys on the philosophy of simplicity see Refs. 4 and 5. In this paper I will argue (along side others, including those in Refs. 6 and 7) that, in general, there is no reason to suppose that the simpler theory is more likely to be true. In other words simplicity does not tell us anything about underlying ‘model bias’.a In this I am saying more than just that simplicity is not necessarily truth-indicative. For although I admit there may be special circumstances where simplicity is truth-indicative, I do not ~~~~

~

~

aModel bias is that effect that the form of a model has upon its efficacy, for example using a series of sinusoid functions to model some data rather than a polynomial of some order.

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66 see that there is any evidence that simplicity and truth are fundamentally related. By analogy, there may be special circumstances where the colour of an object is an indication of its mass (e.g. when someone has gone round calibrating weights and painting them accordingly) but, in general, colour is not at all an indication of mass. To say only that “colour is not necessarily an indication of mass” would be highly misleading, for unless somehow contrived, colour and mass are completely unrelated. I claim that there is no more connection between simplicity and truth than between colour and mass - they are unrelated properties that are only correlated where this is contrived to be so (i.e. by prior assumption or arrangement). Thus, in particular, when t w o theories have equal explanatory power there is no particular reason to prefer the simpler other than convenienceb - it may be that the more complex turns out to more useful in the future, for example by being more amenable to generalisation or explaining new data. Our preference for simpler theories (apart from their obvious practical advantages such as ease of manipulation) can be explained by the facts that, firstly, humans are known to elaborate unsuccessful theories rather than attempt a thorough revision and, secondly, that a fixed set of data can only justify adjusting a certain number of parameters to a given degree of precision. No extra tendency towards simplicity in the natural world is necessary to explain our preference for simpler theories. Thus Occam’s razor eliminates itself (when interpreted in this form). I will start by arguing that a tendency towards elaboration along with the pragmatic advantages are sufficient to explain our preference for simper theories. Then I will briefly look at a couple of a priori arguments that supposedly justify a bias towards simplicity. I follow this by reviewing some evidence as to whether simpler theories are likely to be true taken from the field of Machine Learning, since this is a case that helps distinguish between induction in general and induction by humans. This is followed by a section discussing some special cases where we do have some reason to expect there to be a bias towards simplicity. I briefly consider some of the concepts that have been called “simplicity” in the literature. I conclude with a plea for the abandonment of the use of simplicity as a justification for modelling decisions.

than the peculiar case where one has a limited set of data to fit as discussed in the section on special cases below.

67 2. Elaboration

If one has a theory whose predictions are insufficiently accurate to be acceptable, then it is necessary to change the theory. For human beings it is much easier to elaborate the theory, or otherwise tinker with it, than to undertake a more radical shift (for example, by scrapping the theory and starting again). This elaboration may take many forms, including: adding extra variables or parameters; adding special cases; putting in terms to represent random noise; complicating the model with extra equations or rules; adding meta-rules or models; or using more complicated functions. In Machine Learning terms this might be characterised as a preference for depth-first search over breadth-first search. Classic examples of the elaboration of unsatisfactory theories include increasing the layers of epi-cycles to explain the observations of the orbits of planets in terms of an increasing number of circles and increasing the number of variables and equations in the national economic models in the UK. In the former case the elaboration did increase the accuracy on the existing data because the system of epi-cycles can fitted arbitrarily well to this data, but this is better done with ellipses. Given enough data the system of epi-cycles can be used to successfully predict the orbits to any desired degree of accuracyc, but ellipses will do it with less data and considerably less calculation because the form of ellipses is more suited to describing the true orbits.d Once the a priori bias towards circles is abandoned the system of epi-cycles becomes pointless. In the later case of the macro-economic models the elaboration did not result in the improved prediction of future trends', and in particular these models have failed to predict all the turning points in the UK economy.e Why humans prefer elaboration to more radical theory change is not entirely clear. It may be that it is easier to understand and predict the effect of minor changes to the formulation of a theory in terms its content, so that, if one wants to make a change where one is more certain of improvement, 'One needs t o be a little careful here, unlike ellipses, the system of epi-cycles does not provide of itself any information about the future course of orbits, but given a postulated orbit shape (including ellipses) epi-cycles can be used t o express such orbits and hence used in accurate prediction of the future courses of the planets as well as ellipses. dBeyond a certain point they will do better than an ellipse because they will be able t o include the Eisteinian corrections, however this can still be done easier with corrections t o an ellipse. =Although one was apparently predicted, but this was due t o intervention by the modeller on the basis of his expert knowledge (that destocking that can occur after an oil shock) for details about this see Ref. 8.

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minor changes are a more reliable way of obtaining this. In this case a more certain but marginal improvement may well be preferred to highly uncertain significant improvement. Alternatively, it may be that using a certain model structure biases our view because we get used t o framing our descriptions and observations in this way, using variations of the model as our ‘language’ of representation - akin to Kuhns ‘theoretical spectacle^'.^ In other words, once we have started to think about some phenomena in terms of a particular model, it becomes difficult to think about it in other ways. Finally, it may be due to simple laziness - a wish to ‘fit’ the current data quickly rather than going for longer-term fundamental success ( e g prediction on unseen data). Regardless of the reasons for this tendency towards elaboration, we are well aware of this tendency in our fellows and make use of this knowledge. In particular, we know to distrust a theory (or a story) that shows signs of elaboration - for such elaboration is evidence that the theory might have needed such elaboration in the past - for example because it had a poor record with respect to the evidence. Of course, elaboration is not proof of such a poor record. It may be that the theory was originally formulated in an elaborate form before being tested, but this would be an unusual way for a human to proceed. Thus when presented with alternative theories developed by our fellows, one simpler than the other, we may well guess that the complex one has been elaborated and this would be some (albeit fallible) evidence that it has needed such elaboration. Here it is knowledge of the (human) process that produced the theories that informs us implicitly about their past record against the evidence and past record against evidence is the only guide we have to the future performance of theories. This knowledge, along with an understandable preference for theories that are easily constructible, comprehensible, testable, and communicable provide strong reasons for choosing the simplest adequate theory presented to us. An extra tendency for simplicity to be, of its own, truth-indicative is not needed to explain this preference. In addition to this preference for choosing simpler theories, we also have a bias towards simpler theories in their construction, in that we tend to start our search with something fairly simple and work ‘outwards’ from this point. This process stops when we ‘reach’ an acceptable theory (for our purposes) - in the language of economics we are ‘satisficers’ rather than ‘optimisers’f. fAn ‘optimiser’is someone who searches for the best solution, whilst a ‘satisficer’accepts

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This means that it is almost certain that we will be satisfied with a theory that is simpler than the best theory (if one such exists, alternatively a better theory). This tendency to, on average and in the long term, work from the simpler to the less simple is partly a consequence of the fact that there is a lower bound on the simplicity of our constructions. This lower bound might be represented by single constants in algebra; the empty set in set theory; or a basic non-compound proposition. An alternative approach might be to start from a reasonably complex theory an look at modifications to this (in both simpler and more complex directions). For example in ethology one might start from a description of how an animal appeared to behave and then eliminated irrelevant aspects progressively. In this case, unless one had definite extra evidence to the contrary the sensible bias is towards the original, more complex, account. It is important to constrain our theorising in order that it be effective, but one does not have to use simplicity as this constraint. In" we argue for the use of evidence to determine the starting point for model adaptation rather than simplicity. 3. A Priori Arguments

There have been a number of a priori arguments aimed at justifying a bias towards simplicity - I discuss two of these below. It is impossible to disprove all such arguments in a single chapter so I will confine myself to these two and then make some more general arguments why any such attempt is likely to be mistaken. Kemeny" makes an argument for preferring simpler theories on the presumption that there is a sequence of hypotheses sets of increasing complexity and that a completely correct hypotheses exists - so that once one has reached the set of hypotheses that contains the correct one it is not necessary to search for more complex hypotheses. Thus once the evidence shows one has an acceptable theory one should not look for more complex theories. However this does not show that this is likely to be a better or more efficient search method than starting with complex hypotheses and working from there. The conclusion is merely a reflection of the assumptions he has made. It also does not deal with the case where there is not a single completely 'Lcorrect7'theory but a series of theories of increasing complexity that more precisely fit the data as they get more complex. a solution that meets a given standard.

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Li and Vitbnyi12 show that it is possible to code hypotheses so that the shorter codes correspond to lower measures of complexity so that this coding can be interpreted as prior probabilities (in the Bayesian sense). Thus they suggest that this is a sensible a priori basis for Bayesian prior estimates of hypothesis probability. However, in this case, there is no necessary relation between the complexity of the hypotheses and the length of the codes that is evident before this relation to probabilities is established. Just because such numbers can be interpreted as prior probabilities is not sufficient reason for supposing that it is sensible to do so. As an indication that all arguments that simplicity is a priori likely to be truth-indicative are wrong, consider the following thought experiment. In this experiment there are two lkg masses, A and B, of the same weakly radioactive material, in which atoms currently decay at an average rate of 1 atom per minute. By each mass there is a Geiger counter which detects when an atom in the mass decays and sends a particle towards the counter. The task is to predict which counter will register a particle first after each hour on the clock begins. Now any model which predicts A and B half the time will, in the long run, do equally well. In this case it is abundantly clear that simpler theories are not more likely to be correct - correctness is determined by the proportion of A and B that the theory predicts and nothing else. Now, quite reasonably, one might object that a sensible model concerning radioactive decay is not a directly predictive one but one which specifies the unpredictability of the phenomena and concentrates on ‘second-order’ properties such as the probability distribution. However, this is beside the point - it is a truism to say that those phenomena where our simple theorising succeeds have some simple behaviour and those where such theories do not hold require more complex ones. If the thesis that simplicity is truthindicative is restricted to only those aspects of the natural world where it works, it has some legitimacy but this is merely begging the question - it can not then be invoked to justify the selection of theory about phenomena where it is not known before hand that the right theory is simple. For example, we rightly do not attempt to predict the exact position of each grain of sand with our mathematical models of sand piles but instead concentrate on those aspects of that are amenable to our modelling techniques, such as relation between the frequency and size of a~a1anches.l~ In general we are highly selective about what we attempt to model -we usually concentrate upon that tip of the natural world iceberg which is not too complex for our abilities. In fact there is evidence that we find more complex patterns more

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difficult to per~eive’~, so that we tend to notice the simpler patterns and attribute the more complex ones to meaningless background or randomness. Theoretical results in Machine Learningl5>l6show that, in general, no learning or search algorithm (i.e. method of induction) is better than another. In particular that if a bias towards simplicity is sometimes effective, there must be other domains in which it is counter-productive. To gain any improvement in inductive ability one must apply knowledge about the particular domain one is concerned with. However, these results are extremely abstract and dominated by search spaces that are seemingly random and discontinuous almost everywhere. These theorems do not rule out the possibility that Nature is biased towards producing data that is more amenable simple than these extreme cases. Thus we look to some evidence as to this.

4. Some Evidence from Machine Learning We have two explanations for our preference for simpler theories once the pragmatic advantages are factored out (all evidence being equal): firstly, our knowledge that theories tend to be elaborated when unsuccessful and, secondly, that there is an inherent bias towards simplicity in the natural world. If we were to hold to Occam’s razor (in the form that simplicity is truth-indicative) then we would choose the first because this is sufficient to explain the phenomena - the postulated bias in the natural world is an ‘unnecessary entity’. Since I don’t hold with this form of Occam’s razor I need to look for some evidence to distinguish between the two explanations. Since the tendency towards elaboration is a characteristic of a lot of human theory construction, the critical evidence needs to come from non-human theory construction - situations where theory construction is not based upon a process of elaboration to see if simplicity is truth-indicative there. If a bias towards simplicity was useful apart from our human prelediction for it, this would support its general use. Recently there have been such studies in the field of Machine Learning - where a computer program (rather than a human) attempts the induction. This gives one a test bed, for one can design the induction algorithm to use a simplicity bias or otherwise and compare the results. In one of these studied7 a comprehensive evaluation of all possible theories in a given formal language (to a given depth) were analysed against some real-world data series as follows: firstly as to their effectiveness at fitting some initial portion of the data (the in-sample part of the series), secondly as to their success predicting the continuation of this data (the

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out-of-sample part), and finally, as to the theory’s complexity (measured in this case by the size or depth of the formal expression representing the theory). The theories that were best at fitting the in-sample data were selected. Within this set of ‘best’ theories it was examined whether the simpler theories predicted the out-of-sample data better than the more complex theories. In some cases the simpler hypotheses were not the best predictors of the out-of-sample data. This is evidence that on real world data series and formal models simplicity is not necessarily truth-indicative. In a following study on artificial data generated by an ideal fixed ‘answer’18, it was found that a simplicity bias was useful, but only when the ‘answer’ was also simple. If the answer was complex a bias towards complexity aided the search. This result can be interpreted as exploiting available information about the solution in the search for it - the same lesson that one can draw from the “NO Free Lunch” theorem16. Webblg exhibited an algorithm which systematically extended decision trees (making them more complex) so that they gave the same error rate on the in-sample data, and, on average, gave smaller error rates on the out-ofsample data for several real-life time series. This method was based upon a principle of similarity, which was used to restrict the set of considered hypotheses. A useful survey of results in Machine Learning is Ref. 20 which could be considered a parallel paper to this one, concentrating on the usefulness of a simplicity bias in Machine Learning and only touching upon the wider, philosophical, issue. Thus there is some evidence that, when considering non-human induction, a simplicity bias is not generally helpful or truth-indicative. Rather it seems that it is often used as an ill-defined stand-in for domain knowledge - perhaps this is a reflection of our own ~ 0 g n i t i o n . lThe ~ evidence does not say that a biased towards simplicity is never useful but is sufficient to exclude it being routinely assumed.

5. Special Cases

Although, there is no reason to suppose simplicity is in general truthindicative, there are special circumstances where it might be. These are circumstances where we already have some knowledge that would lead us to expect that the solution might be simple. That is the evidence points to a particular theory (or class of theories) and those happen to be simple. I briefly consider these below. The first is when the phenomena are the result of deliberate human

73 construction. Deliberate human constructions are typically amenable to an almost complete analysis assuming a 'design stance', that is considering the construction from the point of view of it being designed.21 That is, that they are frequently modular, and the result of relatively simple principles iterated many times. If someone asks you to guess the next number is the sequence: 2 ,4,8, 16 you will correctly guess 32, because the nth power of two is the most familiar pattern that describes these five numbers, and you an rely on the fact that the human will have chosen such a pattern and that those patterns tend to be simple (if possibly obscure). It would not be sensible to guess the number 31, despite the fact that there is a rule that would make this the correct answer.g However here we already have some knowledge about what kinds of rule/pattern people are likely to choose and that these are more likely to be simple. The simplicity of these kinds of phenomena is only a hallmark of deliberate, conscious human construction. Products of our unconscious brain or social constructs such as language may be extremely complex for these were not the product of an intentional design process. Thus artists may construct extremely complex artefacts because they do not consciously design every detail of their work but work intuitively a lot of the time with parts and media that are already rich in complexity and meaning. Apart from human construction there are other circumstances where one has good reason to expect simplicity, namely the initial stages of processes that start with the simplest building blocks and work from there. That is the underlying process is known to be one of elaboration. Examples of these might include the construction of progressively higher elements in the early universe, the reactions of bacteria to external stimuli, or, possibly, the first stages in the evolution of life. Evolution started from relatively simple organisms and evolved from there. Obviously when life started the introduction of variety by mutation would be unlikely to result in simplification, since the organisms were about as simple as they could get while still being able to reproduce in their environment. Thus the effective lower bound on complexity of viable organisms means that there is a passive drift towards greater complexity (as opposed to an active drive towards complexity22). However this bias is only significant at the start of the process because the space of possible organisms is so great that once an organism has evolved to be somewhat complex it is almost as likely to evolve to be simpler as ~~~

gFor example, the number of areas that n straight lines, each crossing the perimeter of a circle twice and such that no three lines intersect in a single point, cut that circle into.

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more complex - the lower bound and the ‘inhabited’ part of the possibility space do not impinge upon the possibilities that much so as to significantly bias its evolution towards complexity. Another situation is where one already knows that there is some correct model of some minimum complexity. In this case one heuristic for finding a correct model is to work outwards, searching for increasingly complex models until one comes upon it. There are, of course, other heuristics - the primary reason for starting small are practical; it is far easier and quicker to search through simpler models. In more common situations it might be the case that increasingly complex models may approximate the correct model increasingly, but never completely, well or that no model (however complex) does better than a certain extent. In the first case one is forced into some trade-off between accuracy and convenience. In the second case maybe no model is acceptable, and it is the whole family of models that needs to be changed. Clearly if one has some information about the complexity of the sought after model before the search starts, using that information can make search more efficient, but then this is merely a case of exploiting some knowledge about the solution - it is not an reason for a general bias towards simplicity in other cases. In such circumstances as those above there is some reason to err towards simplicity. However in these circumstance the principle is reducible to a straight forward application of our knowledge about the phenomena that leads us in that direction - principles of simplicity do not give us any ‘extra’ guidance. In these circumstances instead of invoking simplicity as a justification the reason for the expectation can be made explicit. Simplicity as a justification is redundant here. Likewise if one had some evidence that the desired model or theory was complex before the search starts, a bias away from simplicity would be helpful.

6. Versions of “Simplicity”

In order to justify the selection of theories on the basis of simplicity, philosophers have produced many accounts of what simplicity as. These have included almost every possible non-evidential advantage a theory might have, including: number of parameter^^^; extensional p l ~ r a l i t ;y ~ ~ ~ ~ ~ fal~ifiability~~; likelihood26,6; stability27 logical expressive power28; and content2g. These approaches give an uncomfortable feeling of putting the cart before the horse - for instead of deciding what simplicity could meaningfully be and then seeing if (and when) it is useful to be biased towards

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simpler models when trying to explain or predict some observations, they seem to have started with the presumption that a bias towards simplicity is right and then try to find an account of simplicity that is compatible with this idea. In some cases this has almost come full circle. Sober3’ characterises simplicity as informativeness - so that instead of asking whether simplicity is informative he seeks to show that simplicity (as informativeness w.r.t. a specified question) is, in fact, simple! As a result of the prior philosophical need for a definition of “simplicity” to represent a useful bias in model/theory selection, these definitions do not tend to be properties that are the opposite of accounts of complexity. This, in itself, is evidence for the failure of the project to justify a bias away from complexity - it is not the case that a reasonable definition of complexityh is the starting point and then this shown to be counter truth-indicative, but that properties that have nothing to do with a lack of complexity are labelled as “simplicity” in an apparently desperate attempt to justify this assumed bias. If, as I have argued, simplicity is not truth-indicative, this whole enterprise can be abandoned and the misleading label of ‘simplicity’removed from these other properties. This epithet transferral, far from producing insight has produced a fog of differing ‘simplicities’and ‘complexities’which do much to hinder our understanding of the modelling process. Theories can posses a lot of different advantages that are not directly linked to its success at explaining or predicting the evidence, restoring the correct labels for these advantages will help (rather than hinder) the elucidation of their different and interacting roles.

7. An Example - Curve fitting by parameterisation

A particular case of hypothesis selection that has been discussed in the literature is curve fitting. This is the case of deciding which of a variety of hypotheses (in different functional curve ‘forms’) one will select given a set of data (typically given as data points). Typically these ‘forms’ include parameters that are adjusted to fit the data, so that each form corresponds to a family of curves, and the member of that family determined by the choice of values for the parameters. Curve fitting can be a misleading example as it can be difficult to rid oneself of ones intuitions about what sort of curves are useful to posit. One hFor this reason a definition of complexity is not relevant here. For those who are interested in what might be a reasonable definition of complexity see Ref. 31.

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can have strong visual intuitions about the suitability of certain choices which strongly relate to a set of heuristics that are effective in the domains one happens to have experienced. These intuitions might not be helpful in general. In particular, one might happen to know that there is likely to be some random noise in the data, so that choosing a curve that goes through every data point is not likely to result in a line that reflects the case when more data is added. In this case one might choose a smoother curve that might approximate to a local average of the data. A traditional method of smoothing is choosing a polynomial of a lower order or with fewer parameters. This is not, of course, the only choice for smoothing; one might instead use, for example, local regression32 where the fitted curve is a smoothed combination of lines to fit segments of the data. Thus an assumption that a curve with a simpler functional form might be more appropriate for a certain set of data can depend on: firstly, that one has knowledge about the nature of the noise in the data and, secondly, that one chooses the simplicity of the functional form as ones method of smoothing. If, on the other hand, one knew that there was likely to be a sinusoid addition to the underlying data one might seek for such regularities and separate this out. Here a preference for simplicity is merely an expression of a search bias which encodes one’s prior knowledge of the domain. argues that simplicity is justified on the A recent series of grounds that its use can result in greater predictive accuracy on unseen data. This is based on results obtained in Ref. 35. Simplicity in this case is defined as (effectively) the Vapnik-Chervonenkis (VC) dimension36 of the set of curves which in very particular circumstances is equivalent to the number of adjustable parameters in the equation form. The advantages of ‘simplicity’ in this account amount to the prescription not to try and fit more parameters that you have data for, since the larger the set of hypotheses one is selecting from the more likely one is to select a bad hypothesis that ‘fits’ the data purely by chance. The extent of this overfitting can be estimated in particular cases. The argument is that in these cases one can know to choose a form with fewer parameters if one does not have enough data to justify estimating any more, even if such more complex forms may appear to fit the data better. However, this does not effect the general argument - if you have two models whose predictive accuracy, once adjusted f o r its expected overfitting o n limited data, is equal then there would be no reason to choose the family which might be considered simpler to have a simpler form. In circumstances with a fixed amount of data the estimation

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of the extent of overfitting might or might not tip the scales to lead one to select the simpler model. This account gives no support for the thesis that the simplicity of a model gives any indication as to its underlying model bias. Thus, in circumstances where one can always collect more data (so that effectively there is an indefinite amount of data), these arguments provide no reason to select a simpler model, but rather suggest one should collect more data to distinguish which model is better in general. In this case, the decision of when to stop seeking for a model which gives increased predictive accuracy is a practical one: one has to balance the cost of collecting the additional data and using it to search for the most appropriate model against the goodness of the parameterised model. Also the connection between the VC dimension and any recognisable characteristic of simplicity in the family of curves is contingent and tenuous. In the special case where the only way of restricting the VC dimension (or in finite cases, the number of hypotheses) is through the number of adjustable parameters, then it is the case that an equational form with more adjustable parameters will require more data for accurate parameterisation. However there are other ways of restricting the set of hypotheses to reduce the VC dimension; as discussed abovelg successfully used a similarity criterion to do this. Thus one can avoid overfitting by restricting the VC dimension of the set of hypotheses without using any criteria of simplicity or parsimony of adjustable parameters. In a related study3’ examined the connection between the complexity of expressions and (indirectly) their ambiguity and concluded that any measure that restricted the space of models would be equally effective. Of course, one can decide to define simplicity as the VC dimension, but this is begging the question again and one would need to justify this transferred epithet. To summarise this section, there is a limit to the accuracy with which one can adjust a given number of unknown parameters given a certain, jixed amount data - one is only justified in specifying a curve to the extent that one has information on which to base this. Information in terms of a tightly parameterised curve has to come from somewhere. However, in the broader picture where different families of curves are being investigated (for example, by competing teams of scientists continually searching out more data) these considerations give no support to the contention that the simpler family has an advantage in terms of predicting unknown data better.

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8. Concluding Plea It should be clear from the above that, if I am right, model selection ‘for the sake of simplicity’ is either: simply laziness; is really due to practical reasons such as cost or the limitations of the modeller; or is really a relabelling of more sound reasons due to special circumstances or limited data. Thus appeals to it should be recognised as either spurious, dissembling or confusing and hence be abandoned. Indeed rather than assuming that there will always be a simple adequate theory (if only we were clever enough and worked hard enough to find it) we should keep an open mind. On the face of it, it would seem reasonable to expect that complex phenomena will require complex theories for many purposes, and that simple theories will only be adequate for a restricted range of phenomena. However we should always allow for the possibility that occasionally some apparently simple phenomena will require complex theories and apparently complex phenomena allow for simple theories. The point is one can not tell in advance and so it is unwise to make assumptions about this. However, there is a form of Occam’s Razor which represents sound advice as well as perhaps being closer to its Occam’s original formulation (usually rendered as “entities should not be multiplied beyond necessity”), namely: that the elaboration of theory in order to fit a known set of data should be resisted, i.e. that the lack of success of a theory should lead to a more thorough and deeper analysis than we are usually inclined to perform. It is notable that this is a hallmark of genius and perhaps the reason for the success of genius - be strict about theory selection and don’t stop looking until it really works.

Acknowledgements

I would like to thank all those with whom I have discussed or commented upon these ideas (most of whom were not inclined to change their opinions on this matter), including: the participants of the 20(th)Wittgenstein Symposium in 1999; and those at the workshop on the Evolution of Complexity at the VUB in 1997. Also to several anonymous referees who have motivated me to make my position clearer (if more stark). References 1. Newton, I. Philosophia naturalas Princapia Mathematica, (1686).

79 2. Einstein, A., Relativity: the special and the general theory - a popular exposition, London: Methuen, (1920). 3. Gertsbakh, E. and Gertsbakh, I., Measurement Theory for Engineers, Springer, (2003). 4. Hesse, M., Simplicity, in P. Edwards (ed.), The Encyclopaedia of Philosophy, vol. 7., New York: Macmillan, 445-448, (1967). 5. Zellner, A., Keuzenkamp, H. and McAleer, M. (eds.), Simplicity, Inference and Modelling. Cambridge: Cambridge University Press, (2001). 6. Quine, W. V. 0. Simple Theories of a Complex World. In The Ways of Paradox. New York: Random House, 242-246, (1960). 7. Bunge, M., The Myth of Simplicity. Englewood Cliffs, Prentice-Hall, (1963). 8. Moss, S., Artis, M. and Ormerod, P., A Smart Macroeconomic Forecasting System, The Journal of Forecasting 13,299-312, (1994). 9. Kuhn, T. S. The Structure of Scientific Revolutions. Chicago, University of Chicago Press, (1962). 10. Edmonds, B. and Moss, S., From KISS t o KIDS - an anti-simplistic modelling approach. In P. Davidsson et al. (eds.): Multi Agent Based Simulation 2004. Springer, Lecture Notes in Artificial Intelligence, 3415, 130-144, (2005). 11. Kemeny, J. G. Two Measures of Complexity. The Journal of Philosophy, 52, 722-733, (1953). 12. Li, M. and VitBnyi, P. M. B. Philosophical Issues in Kolmogorov Complexity, in Automata, Languages and Programming, 19th International Colloquium, Lecture Notes in Computer Science, 623,1-15, Springer, (1992). 13. Bak, P. How Nature Works: The Science of Self Organized Criticality. Oxford, Oxford University Press, (1997). 14. Charter, N. The Search for Simplicity: A Fundamental Cognitive Principle? The Quarterly Journal of Ezperimental Psychology, 52A,273-302, (1999). 15. Schaffer, C., conservation law for generalization performance. In Proceedings of the llth International conference on Machine Learning, 259-265. New Brunswick, NJ: Morgan Kaufmann, (1994). 16. Wolpert, D. The lack of a priori distinctions between learning algorithms. Neural Computation, 8, 1341-1390, (1996). 17. Murphy, P.M.; Pazzani, M.J. Exploring the Decision Forest: an empirical investigation of Occam’s razor in decision tree induction, Journal of Artificial Intelligence Research, 1, 257-275, (1994). 18. Murphy, P. M. An empirical analysis of the benefit of decision tree size biases as a function of concept distribution. Technical report 95-29, Department of Information and Computer Science, Irvine, (1995). 19. Webb, G . I. Further Evidence against the Utility of Occam’s Razor. Journal of Artaficial Intelligence Research, 4, 397-417, (1996). 20. Domingos, P. Beyond Occam’s Razor: Process-Oriented Evaluation. Machine Learning: ECML 2000, llth European Conference on Machine Learning, Barcelona, Catalonia, Spain, May 31 - June 2, 2000, Proceedings, Lecture Notes in Artificial Intelligence, 1810, (2000). 21. Dennett, D. C., The Intentional Stance. Cambridge, MA: A Bradford Book,

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MIT Press, (1987). 22. McShea, D. Meatzoan Complexity and Evolution: is there a trend? Ewolution, 50, 477-492, (1996) 23. Draper, N. R.; Smith, H. Applied Regression Analysis. New York: John Wiley, (1981). 24. Goodman, N. The Structure of Appearance. Indiapolis: Bobbs-Merrill, (1966). 25. Popper, K. R. Logic of Scientific Discovery. London: Hutchinson, (1968). 26. Rosenkrantz, R. D. Inference, Method and Decision. Boston: Reidel, (1976). 27. Turney, P. The Curve Fitting Problem: A Solution. British Journal for the Philosophy of Science, 41, 509-530, (1990). 28. Osherson, D.N. and Weinstein, S. On Advancing Simple Hypothesis. Philosophy of Science, 57, 266-277, (1990). 29. Good, I. J. Corroboration, Explanation, Evolving Probability and a Sharpened Razor. British Journal f o r the Philosophy of Science, 19, 123-43, (1969). 30. Sober, E. Simplicity. Oxford: Clarendon Press, (1975). 31. Edmonds, B., Syntactic Measures of Complexity. Doctoral Thesis, University of Manchester, Manchester, UK, (1999). (http://bruce.edmonds.name/thesis/) 32. Cleveland W. S., Devlin S. J., Grosse E. Regression By Local Fitting - Methods, Properties, And Computational Algorithms. Journal Of Econometrics, 37, 87-114, (1988). 33. Forster, M. and Sober, E. How to Tell when Simpler, More Unified, or Less Ad Hoc Theories will Provide More Accurate Predictions. British Journal fo. the Philosophy of Science, 45, 1-35, (1994). 34. Forster, M. Model Selection in Science: The Problem of Language Invariance. British Journal for the Philosophy of Science, 50, 83-102, (1999). 35. Akaike, H. Information Theory and an Extension of the Maximum Likelihood Principle, in Petrov, B. N. and Csaki, F. (eds.) 2nd International Symposium on Information Theory, 267-281. Budapest: Akademai Kiado, (1973). 36. Vapnik V. N. and Chervonenkis A. Y. , Necessary And Sufficient Conditions For The Uniform-Convergence Of Means To Their Expectations, Theory Of Probability Applications, 26, 532-553, (1981). 37. Pearl, J. On the Connection Between the Complexity and Credibility of Inferred Models, International Journal of General Systems, 4, 255-264, (1978).

CAN THE WHOLE BE MORE THAN THE COMPUTATION OF THE PARTS? A REFLECTION ON EMERGENCE

CAMILO OLAYA Institute of Management University of St. Gallen (HSG) Dufourstrasse 4Oa, CH-9000,St. Gallen, Switzerland E-mail: [email protected] The term ‘emergence’ is frequently found in Complexity Science; however, it gives the impression that earlier debates on this topic are not taken fully into account. By considering this situation, the article underlines the need of a consistent discussion when addressing emergence if this term is going t o have an added and meaningful value for learning about complex systems. The promising suitability of process thought and the significance of detailed rulebased modeling are derived from this reflection as further paths t o explore.

1. Introduction

Lately everything seems to emerge. Qualities, properties, matter, life, mind, ideas, inventions, computed macro-states, and many other things, now ‘emerge’. In particular, in what is loosely known as Complexity Science (CS),” the term ‘emergence’ is widely invoked as a pivotal notion, although it is used in diverse senses. And it is not the purpose here to offer a new definition; the article presents important distinctions that usually are absent in CS, regardless of been heavily discussed in other areas, in particular in philosophy. The central point is that emergence, taken as a metaphysical inquiry, seems to be overlooked when developing explanations by means of computer simulation. However, as John Holland reminds us in his book ‘Emergence’,’ CS is committed to make models of the world. “In order to frame the subject, somehow, CS refers here t o approaches related t o nonlinear dynamics, agent-based modeling, chaos theory, artificial life, social simulation, complex adaptive systems, system dynamics, and alike, that claim t o address ‘complexity’ using typically computer simulation. Yet this does not exclude indirect connections to related areas that also address emergence like molecular and evolutionary biology, quantum theory, matter theory, and the philosophy of mind.

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a2

W h a t type of explanation regarding emergence can CS really deliver? The paper introduces a reflection on this question. As a result, an ontological shift to process philosophy arises as a natural option so as to address novelty and change, the motivating concerns behind the term ‘emergence’. The suitability of bottom-up computer modeling is particularly underlined, arising from this shift. 2. The Emergent Question

Emergence still remains as one of the hardest notions to demarcate beyond the sentence ‘the whole is more than the sum of the parts’. Usually the word ‘more’ refers to unpredictable novelty. It has produced an intense debate since we have to face the appealing paradox: How can something that arises from a set of parts be more than the sum of those parts? Fundamentally this question is a metaphysical inquiry, or at least it used to be so; the original thesis proposed the generation of new units, new kinds, new entities, new qualities. This task has been historically cumbersome, to say the least. Jones2 recently remarks: Emergence is largely used as an explanation: such and such an object - ranging from atoms through multicellular organisms to consciousness - is an emergent property of some ensemble of parts. But this leaves an inadequate level of understanding The term emergence has a somewhat equivocal press. For some it is an explanation for the appearance of wholes or coherent objects in any system of things or understandings; while for others it is either meaningless or, conversely, an invocation of something mystical (p. 418). The heated dispute may have been started in 1920 with Samuel Alexander for whom emergence necessarily admits no explanation; he claimed that it has to be accepted as a ”brute empirical fact” (as cited by Pap).3 The reply of Stephen Pepper in 1926 posts the warning sharply: either it is the sum of the parts or it is not4 (no paradox here). Looking again at the quote above of Jones, it seems there has not been too much advance since then. What follows is a brief account of some important distinctions through this philosophical debate;b these distinctions (highlighted in italics) are relevant today, on a first basis because many of them seem to be trivialized or overlooked. bFor comprehensive overviews see e.g. [5], [6].

a3 Going back to 1926, Pepper, ignoring the advice of Alexander, addresses the subject. First, he identifies emergence as a kind of change; second, this change, he summarizes, is derived from three propositions: (i) there are different levels of existence in terms of degrees of integration; (ii) there are marks that make these levels distinguishable from one another over and above the degrees of integration; (iii) these marks can not be deduced from marks at other higher or lower levels. He focuses on laws as the means to describe what emerges, though he remarks that "accurately speaking, we must first observe, laws can not emerge. Emergence is supposed to be a cosmic affair and laws are descriptions" (p. 242).4 Pepper challenges the implicit puzzle of the last two premises: On the one hand, (ii) affirms that there is cumulative change; however, he argues that if we work with laws then what take place are shifts at levels of description, where variables can always be distinguished and placed at proper levels. On the other hand, (iii) affirms that change is not predictable; Pepper asserts that either change is predictable (as in "part of the system") or it is epiphenomena1 since "if they represent irreconcilable inconsistencies in the physical system, they are . ~ alternot laws at all, but statements of chance occurrences" ( ~ . 2 4 4 )The native to laws is to a r m that what emerge are qualities (sensory qualities - metaphysical simples); Pepper dismisses this, rejecting such approach as a theory of epiphenomenalism (no added value) and therefore unsatisfactory for metaphysics. Therefore, for him emergence is meaninglessc Soon after in 1929, Watts Cunningham' underlines that novelty should not imply unintelligibility, i.e. if we have dynamic systems whose very nature change with time, there are emergents which, though non-predictable, can be intelligible; he affirms this assertion for both substancial emergence (a constituent or entity appears de novo) and structural emergence (a novel arrangement of old material). In 1939 Ablowitz conceives the emergence of unpredictable and intelligible qualities as due to a new togetherness of the partsg (hence structural emergence). However Malisoff, in the same year, emphasizes the level of description defined by the observer (thus, we can have alternative descriptions depending on the degree of organization that the observer attributes to the system); this eliminates the mystery on the question, he argues, since it is subject to the mode of observation, i.e .relative to particular specifications or postulates." Stace replies that emergence produces novelty, but what is interesting, he underlines, is absoC A full discussion and criticism of the argument of Pepper is developed by Meehl and

~el~ars.'

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lute novelty (anything which had never appeared before) and not relative novelty (relative to a particular situation).l' Gotshalk, in 1942, emphasizes that novel emergents are irreducible and that they bring into prominence individuality and quality; for him emergents are underivative ingredients of the real and not merely causal reductions;12 it is implicitly a criticism to Pepper (emergence now is meaningful) and Malisoff (for whom the question is only epistemological). Also in 1942 Campbell Garnett emphasizes that the failure of the theory of emergence to establish a vera causa (as it is was suggested by Alexander) implies the incompatibility with scientific methods; as a result, he adheres to the organismic interpretation: a single nature within which two irreducibly distinct types of process are functionally integrated in one organic whole;13 hence, he rejects ontological emergence since complete independence is proclaimed between layers. In the same journal, Henle criticizes the prominent role of the observer for addressing predictability (as in "quality A is predictable by person B on evidence C with degree of assurance D"); for him novelty exists in the universe though this does not entail emergence since two conditions have to be met: it must be logically novel relative to all previously exemplified characteristics, and it must not be merely a spatio-temporal characteristic.'* Bahm in 1947 supports organicism: things are organic unities, i.e. they are both one and many, identity and difference, permanence and change, stability and novelty, substance and function; for him, emergence is just a natural consequence of recognizing those levels or dimensions15 (and therefore there is no paradox). Pap in 1952 attacks the notion of absolute unpredictability ("pure metaphysical obscurantism"); he proposes the possibility of having laws that can correlate an a priori unpredictable quality Q with certain causal condition^.^ Berenda, in similar lines, supports the ability for understanding (structured in reason) as different from the ability for learning (developed in time), for him prediction is just a matter of time, therefore he rejects logical unpredictability.16 Somehow summarizing these earlier debates, Kekes in 1966 characterizes emergence with novelty (when a token of a type appears which never existed before) and unpredictability (this occurrence of novelty is unpredictable as based on the theories of lower levels of existence); his type of novelty is in the lines of Berenda, a priori unpredictability, prior to the occurrence of the event but explicable afterwards. Kekes postulates both novelty and unpredictability as necessary and sufficient conditions of emergence. Moreover, for him qualities and laws as well can emerge, and the emergence of any of

85 them would imply the emergence of an entity in general;d finally, he rejects the triviality of having a type of emergence where, in spite of an adequate lower level theory, logical (metaphysical) unpredictability persists.” More recently, Klee proposes an unified treatment of emergence favoring properties over laws as the emergent units within a micro-deterministic view of nature, i.e. parts determine wholes; however, the preference for properties is for practical reasons given the more complicated path, he argues, implied by a discussion around a theory of lawlikeness.18 In 1997 Bedau proposes weak emergence which is based on the necessary condition of deriving dynamically macrostates via computer simulation; here what emerges is a macrostate of a systemlg (though he seems to equate properties with behaviour). Recent significant questions on ontological emergence (and non-reductive physicalism) can be found in the work of Kim; e.g. he criticizes the notion of downward causation and he suggests the conditions of novel causal powers and non-functionalizability for having ontological emergen~e.~ In a needed call for order, Silberstein and McGeever2’ urge to distinguish between ontological and epistemological emergence; they underline that it is common to draw fallacious conclusions on the existence of ontological emergence (a feature of the world) based on epistemological emergence (fundamentally a description, an explanation, an artefact).e Cunningham in 2001 makes a similar clarification suggesting three types of emergence: ontological emergence (based on a duality, close to the proposals of Campbell Garnett of 1942 and Bahm of 1947) and two types of epistemological emergence: multiple realizable properties (determined fully by the organization of the parts) and highly configurational and holistic properties which is roughly similar to the notion of weak emergence of macrostates of Bedau since it is the power of computation that determines the possibility

dHe assumes order in nature which can be described by laws (determinism), then he argues: ”At any rate, emergent laws describe regularities holding in nature which must have been novel at some time. These regularities describe the behaviour of entities which may or may not have emergent qualities; yet the regularities described by emergent laws could not be described by laws formulated to describe the behaviour of entities on a lower level of integration” (p. 366). ‘Van Gulick’l makes a classification using this distinction, finding four types of epistemological emergence and six types of ontological emergence; e.g. Georgiou” favors the epistemological approach because for him emergent properties provide the identity of a system since they allow an observer to say ‘this is a system of such-and-such and not otherwise’ for they act as unifying reference points for the group of interrelated parts which constitute the system” (p. 242).

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of addressing system-level marks (these are properties for Cunningham); then, he places complex dynamic systems studies in general as an example of epistemological emergence.23 And the story goes on. After examining the discussion, a first impression is that, in CS, either emergence is implicitly taken as a basic principle not to be discussed (as the proposal of Alexander and thus rather unintelligible) or it is understood only as an epistemological issue (sometimes with hasty ontological claims). Or even worse, sometimes it reminds a basic premise treated also as a result .~ are various complications. First or conclusion - Petitio P r i n ~ i p i i There of all, regardless of the development of the previous discussions and the diverse proposals available, ambiguity seems more the rule than the exception; e.g. sometimes emergence seems to denote metaphysical assumptions, and sometimes it seems to point at a level of description or explanation; and many times what we find is a sophisticated (and sophistical) muddle; beyond a discussion around words like ‘ontology’ or ‘epistemology’, it is manifest the confusing inconsistent use of the term, e.g. in the lines of the warning made by Silberstein and McGeever, presented ab0ve.g This should not be a problem as long as assumptions are held consistently taking into account the relevance of that distinction (not very frequent in any case). A second complication is that the emergents take a wide range of forms; in the literature we witness the emergence of properties, behaviours, en-

Roughly speaking: ” Why is it emergent? Because it is complex”. ”Why is it complex? Well, because it is emergent.” Apart from the risk of explaining ‘emergence’ in terms of the perhaps more obscure term ‘complexity’, a short example may picture the panorama: On the one hand, ”Complexity is a marvelous and essential property of our universe. The existence of complexity provokes the emergence of behaviors full of richness but hard t o investigate”(p. 33).24 On the other hand, ”If ‘complexity’ is currently the buzzword of choice for our newly minted millennium, as many theorists proclaim, ‘emergence’ seems t o be the explanation of the hour for how complexity has evolved. Complexity, it is said, is an emergent phenomenon” (p. 18).25The problem for a research program like CS is not only the seemingly unavoidable ambiguity of the terms as such (though a modern Tower of Babel is not a desirable situation), the question aims at reflecting on assumptions, meaningfulness, consistency, and relevance. gThis illustrative example, cited in [20] may suffice: ”Chaos brings a new challenge t o the reductionist view that a system can be understood by breaking it down and studying each piece. This view has been prevalent in science in part because there are so many systems for which the behaviour of the whole is indeed the sum of its parts. Chaos demonstrates, however, that a system can have complicated behaviour that emerges as a consequence of simple, non-linear interaction of only a few components. The interaction of components on one scale can lead to complex global behaviour on a larger scale that in general cannot be deduced from knowledge of the individual components” (p. 186).

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tities, laws, structures, events, qualities, patterns, among others, treated in a seemingly careless way; even sometimes these terms are used on an exchange basis. As a further consideration, in order to hold a consistent position (specially if we are going to assume, accept, or address metaphysical emergence), we can consider to challenge also our ontological ground and not only our conceptualization of ‘emergence’, a term that as time goes by becomes too sophisticated, more disperse, and less useful. The presented historical account emphasizes some difficulties behind the real motivation of the endeavour: to address novelty and change. Could an ontological shift be a more suitable way for tackling the emergent question?

3. Explanations of Uniformity Emergence used to be a cosmic affair (as in ”emergence is a feature of the world”); in fact, a number of discussions within CS seem to point at absolute emergence, but what seems to prevail - at the end - is a particular level of description that habitually takes the form of causal accounts and/or universal laws; such view involves a sort of uniformity that makes harder to address novelty and change. This is presented next. Explanations based on causal accounts hold that there is something special about causal interactions that increases our understanding of the world; causality is the crucial relationship which makes information explanatorily relevant.26 An account of ontological emergence is naturally a formidable challenge for such approach, e.g. it has been widely addressed in terms of multiple complex causal networks or sophisticated law-like expressions.h Silberstein and McGeever summarize the popular position:20 Complexity studies is inclusive of chaos theory, connectionism and artificial life, in that it seeks a grand unified theory of the fundamental and universal laws that govern behavior of all complex and non-linear systems. It would be fair to say that many complexity theorists are after a unified theory of ‘emergence’ (see Kauffman and Wolfram for examples). Complexity theorists (often) reject part-whole reductionism, but they hold that there are fundamental laws of the universe which pertain not only to all of the physical, chemical and biological systems, in the universe, but also to all systems (such as the economy and other social systems) that reach certain degree of ‘complexity’. These ‘meta-laws’ or ‘power-laws’ hRather recent examples are [27], [28], [29].

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govern the emergence or evolution of a system’s behavior in the world (p. 190). Such quests seem inappropriate for coping with emergence; for instance, universal laws are opposed to individuality, ignoring the remark of Gotshalk of 1942: ”The most obvious philosophical importance of the principle of emergence, as I see it, is that it brings into intellectual prominence factors in the universe that actually have been present and in their way important all the time, namely, individuality and quality. In modern scientific causal analysis, both of these factors have tended to be lost from view”12 (p. 397, emphases added). Seemingly they are still lost from view since it can be said that many discussions on emergence have taken an uniform nature for granted (or explicitly as e.g. Kekes17, presented above), as it is introduced next. We try to make sense of a higher level ‘property’ (let’s say) called ‘emergence’ that is not only in function of the parts although it is derived from them. This should challenge some basic frameworks.’ Why? Within a causal view, we seem to follow Aristotle where there is a necessity in causal relations coming from the fact that they derive from the essential characteristics of material things.31 A very brief note on Aristotelian properties: ”essential properties of a thing are those on which its identity depends. They are the properties of a thing which it could never have lacked, and which it could not lose without ceasing to exist or to be what it is. Accidental properties of a thing are properties which that thing could either have had or lacked”(p. 374).32 Such a frame imposes strong constraints for having a notion of emergence within a world of properties ascribed to static substances which in turn seem to be assumed as self-identically enduring stuff - see e.g. [33],[34]; hence substancial emergence (as in Watts Cunningham,’ presented earlier) based on essential qualities, becomes difficult to address. This world depicted above is usually expressed with laws: ”The statement that an object follows a law involves the presumption that there is some informant agency that makes an object to behave in a certain way

‘For instance, Humphreys suggests the examination of the common ontological minimalism, a doctrine based on a small set of premises: (i) to hold a small set of fundamental constituents of the world; (ii) the primacy of intrinsic (non-relational) properties; (iii) all non-fundamental individuals and properties are composed from those fundamental entities. These premises are rooted in two doctrines: a pre-eminence of the logical reconstruction of concepts, and a Humean account of causality.30

89 and not to behave in any other way...If there is only one informant agency, for all objects of a kind in all time, we call it a law”(p. 9).35 We can appreciate why individuality and novelty are hard to meet, nature is assumed uniform;j more specifically this use of laws neglects endogenous change, e.g.: ”the events the law describes also do not change endogenously unless an exogenous force is introduced into the system. The model is universally deterministic. Given complete information about the initial and subsidiary conditions, the law allows us to retrodict events precisely on to the past and to predict them to precisely on to the future”(p. Claims of emergence, framed in such assumptions and focused on universal and abstract modeling, are - seemingly unnoticed - reductionist accounts (regardless of regularly affirming the opposite) whenever order is taken as reducible to a single mode of relatedness, but apparently believing, at the same time, in something ‘more than the sum of the parts’; this is indeed a mystic path. Already for Pepper was clear that we can not have both. If there is a function (non-linear usually; it comes to mind the condition of non-functionalizability of Kim), if there is a law (restricted in any form), or if there are causal statements, then to deal with absolute novelty beyond ‘the parts’ is not uncomplicated - see e.g. [37], [38]. The old warning of Gotshalk made in 1932 helps to rephrase the point and the relevance:3g Uniformity, if anything significant, is a relation between a genuine diversity. But the causal theory, as set forth, actually applies to no genuine diversity. We raise no question of the theory’s abstract truth; the theory, as stated, is abstractly true enough; the same cause, under similar conditions, no doubt should produce the same effect, again and again It merely states (in sum) that A is always A, and fails, so far as experience goes, to state a relationship between real diversity. In short, it amounts to a true but trifling proposition (pp. 142-143). In such scenario, prediction is very narrow: on the basis of empirical data”we may have a well-confirmed ‘emergence law’ to the effect that whenever a system instantiates basal condition M it instantiates an emergent, E” (p. 8);5 to label it ‘prediction’ is almost a compliment; and this type JThis is perfectly illustrated in the quote of the physicist Wigner: ”It is ...essential that, given the essential initial conditions, the result will be the same no matter where and when we realize these ...If it were not so ...it might have been impossible for us to discover laws of nature” - as cited by Schlesinger (p. 529).36

90 of inductive predictions is not the interesting matter around the emergent question and unpredictability. What we would like to understand is why A is not always A , and not why A is always A. In short, immutable substances related to a sort of causal forces or lawlike statements for complex systems is a contradiction in terms; it is the search for predictive theories of ‘novelty’. 4. Addressing Novelty

Complexity Science is supposed to address change, at least if the habitual allusion to ‘emergence’ is going to be conceived as a kind of change. There are already accounts of emergence understood as processes of interacting, immutable (unchanging) entities (or as transitions between different entities) where results can be repeated as long as the circumstances are similar. But a static concept of entities is at the center of the puzzle, that is, the neglect of transition within entities. Time is left out. Why? Leclerc expresses it ~ n m i s t a k a b l y : ~ ~ A stuff endures by virtue of its inertness; its endurance is conceived to be the retention of its self-identity... [which]is irrelevant to any temporal passage; the stuff is what it is quite irrespective of any transition ...it means that stuff cannot endure... it is eternal ... For if all substance be eternal, what then is duration? What is time? An ontology of eternity can in the end only explain duration as an appearance and thus as illusory (pp. 227-228, 229). How to include time? One natural option is to include duration in the very character of substances, i.e.”it must be the nature of substance to be in passage, in transition”(p. 230).33 A natural option, then, is to conceive them as processes of activities, e.g. a complex system might be conceived as a process of evolving agents; these agents are thus activities; moreover, they are agents because they act, which in turn can be conceived as the use of rules in interaction processes. The significant term is ‘process’ which takes the center of the stage. The basic shift is from static to changing entities. Our current foundations do not ease such a view given that we are embedded in Western metaphysics which since Aristotle favors things and static s ~ b s t a n c e sYet, . ~ ~ process can become a fundamental notion with which events and experience might be (re)-interpreted;*l Leclerc remarks the shift: ”For Aristotle substance is ‘informed’matter; for Whitehead substance or actual entity is ‘informed’activity” (p. 242).33 Therefore

91 emergents would not be merely things/properties/etc. that surface at some point of time within a particular ‘process’. Emergence would be fundamentally a process in the lines of thought outlined by thinkers like Bergson or Whitehead (and surely back to Heraclitus of Ephesus). An agent i s a process of activities as well. And the process of multiple inter-acting agents, i.e. rule users, that unfolds through time is what we distinguish as emergence. No interactions, no emergence. No process, no emergence. After all emergence is a kind of change. For instance, the earlier dilemma seems less complicated now. Emergence arises from unique interactions unfolding through time, in this sense is constituted by and generated from interacting dynamic agents. And this process only arises as long as we have time and interactions, in this sense is autonomous at the system level (by principle a process is not separable as such, it is a whole). hrthermore, regarding hierarchies, the spatial multilayered scale is confronted with a more important temporal scale where ”higher levels reproduce themselves over longer durations than lower levthis statement suggests a different way to explore the concept e l s ” ( ~9);42 . of ‘different layers’ of existence. In short, the system-level process of interactions unfolding through time as a unit cell-complex is what we would call emergence; the system i s that process whose behaviour we assess as emergent from local interactions. Emergence then is just the process at the system level, nothing more. To name the system is to name ‘what emerges’ so to speak. No need of something ‘ e ~ t r a ’ . ~ The main goal at this point is to consider this exploration for CS. To develop the connection between emergence and process thought is not the purpose of this article.’ But for now suggestive ideas would be: To act i s to exist. Being is becoming. To understand the becoming of a dynamic

k T ~ proposals o of dynamic interactions that would constitute non-magical explanans for ‘emergent’ phenomena are presented in [43]and [44]. ‘A broad view on process thought is introduced in [40]. The philosophy of organism, as proposed by Whitehead, particularly in his major work of 1929 ‘Process and can be a starting point for approaching process thought more formally. Yet, beyond this point, a process philosophy may have diverse interpretations and this article is not necessarily committed with particular assumptions; we can expand it in different ways, e.g. presentationalist/idealist paths closer t o the ideas of Whitehead or re-presentationalist/realist paths closer to the Evolutionary Epistemology of Campbell and Popper or the Evolutionary Realism of Dopfer. Also limitations and specific criticisms can be found, e.g. shortcomings related to the nature of composite entities in this in particular Douglas examines framework have been debated, e.g. [33],[45],[46],[47]; and rejects the possibility of characterizing emergence within process thought.48

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process is to explain it as such. Substances are processes of activities. Agency is constant transition and it is by nature social. Actual occasions are unique entities. There are no essential causes, entities exist in change and are social by nature. Durations, activities, and phases take place identifying an operating unit. Novel togetherness is a principle, not a derivative. But there is no morphological analysis by components of static ‘stuff’ although processes indeed can be characterized. The shift would underline that emergence, altogether with contingency, novelty, and creativity, become fundamental categories of metaphysical ~nderstanding.~’ This does not necessarily resolve the issue; but the debate would be seemingly more intelligibly. Intuitively a kind of change could be better addressed within an ontology of change; after all process philosophy presents an ontology committed to novelty.

5. And Individuality Another central point from the discussion on emergence is individuality. How to approach it? This question is related to the methods and types of explanation that CS aims to provide. The reflection above brings elements to bear in mind. The general concern is still model building which in fact is a major feature of CS: the use of computer simulation models. This particular characteristic is also a current challenge in the realm of explanation, as Berger notices: ”It is reasonable to say that modelling explanations dramatically increase our understanding of the world. But the modelling explanations found in contemporary scientific research show that the interesting claims of causal accounts are untenable...An adequate account of scientific explanation must accommodate modelling explanations, because they are simply too central to ignore”(pp. 329, 330).26 Keeping in mind the previous discussion, simulation models can be characterized as systems of interacting agents using evolving rules. Addressing the evolution of dynamic sets of rules (structure, origination, adoption frequencies, selection criteria, individual and collective retention, etc.) we can explain the becoming of emergents which are constituted by interacting evolving agents. This is particularly relevant for modeling social systems; for instance, the ‘parts’ can be related to agents using rules, e.g. cognitive, behavioural, and blueprint rules, see [49]. Each agent is unique and is changing constantly; they may belong to a generic class but each agent does not necessarily apply in the same way his particular set of rules, and a given agent does not act necessarily in the same way today than yester-

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day, even in similar circumstances; moreover, the set of rules of each agent evolves as well. And in a further level, the agents also evolve as such, that is, the generic class of agents evolves too; perhaps over a longer time scope. In one word, there is variety: ”many individuals of a kind, where each constitutes an actualization of a distinct idea. There are, on the one hand, many individuals that are many matter-energy actualizations, and, on the other hand, many ideas that can be actualized”(p. 14).35Furthermore, this variety changes through time - variation (e.g. mutations in biology). Explicitly, the implication is to question abstract modeling (e.g. ”Kauffman’s style”). A foundation on processes of activities underlines the primacy of acting individuals (agents), variety and variation. The inclination here is for a detailed, qualitative, rule-based, ’expert system’ approach where details do matter.m 6. Looking Ahead

Bearing in mind the emergent question, the last point to comment is that an approach based exclusively on computation has to specify clearly how it is going to meet ontological emergence when such claims are asserted. It is not straightforward to declare non-reductionism using computer simulation. Being closed software systems, it is not evident how something arising from a computation can be (or even just represent) more than essentially that computation (which is a formal deduction, or at least a positive calculation anyway) beyond suggestive metaphors or the association of reductionism with linearity. If we are going to address emergence then the type of epistemological bridge between computer simulation and metaphysical emergent entities has to be built - and carefully - if we are really going to meet it. Can it be done? After all ontological emergence used to be the original question. Already W i m ~ a t noticed: t~~ ”Most scientists regard a system property as emergent relative to properties of its parts if it depends upon their mode of organization - a view consistent with reduction”(p. 269).” He refers to areas such as non-linear dynamics, connectionist modelling, “‘Indeed, Chris Langton, leading researcher on artificial life, has already stated that ”the quest for global rather than local control mechanisms (rules) is the source of the failure of the entire program of modeling complex behaviors up to the present, especially, much of the work on artificial intelligence”(p. 24).50 - see also [51]. “It is also a matter of what is understood by ‘reduction’; Wimsatt clarifies: ”Most scientists see reductionism as a species of explanatory mechanistic materialism which may not even involve laws or theories, as those terms are traditionally understood by philosophers”(p. 292).

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chaos, and artificial life. In principle, epistemological - or ‘weak’ - emergence (close to equating ‘emergent’with ‘computed’) does not exclude with what might be labelled ontological reduction i.e. all higher level entities and processes are nothing more than certain combinations of lower base level entities and processes,52which habitually is what seems to be rejected in the discourse of CS. But there is nothing wrong with recognizing reducti0nism.O What is wrong is whenever scientists claim to have non-reducible accounts of (ontological) emergence on grounds that do not necessarily allow such attempts. Reduction concerns explanation; and it seems that when the term ’emergence’ is used in CS, many times can be successfully understood only as a type of reductive explanation. However, maybe there are real emergent (novel and unpredictable) entities, and maybe they can be suitable of being consciously targeted (or rejected) by CS. An ontological shift to process could be fully explored; even more, perhaps then, the question on emergence might be dissolved. A further step takes place in the roles and functions ascribed to simulation models; detailed modeling does not mean to dismiss the system-level focus of explanations. Quite the contrary. For example, we can follow a path closer to the semantic view where a theory is the whole class of its models (all possible interpretations), such a family of models would be a highly abstract conception, ”carrying no information other than the structure of its parent formal system”(p. 301).58The ‘synoptic vision’, closer to the metaphorical view of explanation, has been also suggested5’. Or we can develop for instance the proposal of Hayek“ with his ‘explanation of the principle’ where we focus on types of patterns to expect; instead of ‘prediction’ we can refer to ‘orientation’ and we can prefer the term ‘cultivation’ over ‘control’. To sum up, process thought arises as a natural option for addressing unpredictable novelty, the traditional concern behind the term ‘emergence’. It does not mean the usual notion of process in uniform nature where is a channel to represent the interaction of unchanging parts in order to label

OIn his stimulating paper, Bedaulg exposes the promising possibilities of working with weak emergence, though he also rejects the meaningfulness of addressing absolute emergence - ”logically possible but uncomfortable like magic; and irrelevant”. In related lines, B e ~ h t e lhas ~ ~to ‘defend’ and to make a case for the compatibility of complex systems and reduction; he is right and a defence should not be necessary; unfortunately it seems it is needed; moreover, the role of mechanism, as a kind of explanation, has been recently developed in this context - see [54], [55], [56];in particular for complex systems see [57].

95 the change of macro-states at the system level as ‘dynamic’; that is, it is not merely the variation between two or more consecutive states; this is misleading since processes of activities are left out, that is, agency (as acting) is left out. W h a t is underlined here is a foundational shift, with a first indication t h a t favors detailed modeling. This may shed a light on understanding novelty; and it may provide an intelligible discourse on emergence. After all, since 1926, the challenge of Pepper still deserves a satisfactory answer within CS. But all of these concerns are related to the questions of change and novelty coherent with a creative and diverse (complex) world; right there is where the discussion should take place.

Acknowledgments T h e author thanks three anonymous reviewers for their helpful criticism and suggestions. Further thanks go t o Peter Douglas, Kristjan Ambroz and Markus Schwaninger, for their valuable comments on earlier drafts of this paper. T h e usual disclaimer applies.

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97 Science, 43 (3), 371-388 (1992). 33. I. Leclerc, Whitehead’s Tkansformation of the Concept of Substance. The Philosophical Quarterly, 3 (12), 225-243 (1953). 34. J. Barnes, The Cambridge Companion to Aristotle. Cambridge, U.K., Cambridge University Press (1995). 35. K. Dopfer, Evolutionary Economics: A Theoretical Framework. In: K. Dopfer (Ed.), The Evolutionary Foundations of Economics, Cambridge, U.K., Cambridge University Press, 3-55 (2005). 36. G. N. Schlesinger , Qualitative Identity and Uniformity. NoZis, 24 (4), 529541 (2000). 37. M. Code, Order and Organism. Albany, State University of New York Press (1985). 38. W. C. Wimsatt, Emergence as Non-Aggregativity and the Biases of Reductionisms. Foundations of Science, 5,269-297 (2000). 39. D. W. Gotshalk, Uniformity and Induction. The Journal of Philosophy, 29 (6), 141-152 (1932). 40. N. Rescher, Process Philosophy. In: E. N. Zalta (Ed.), The Stanford Encyclopedia of Philosophy, Summer 2002 Edition, The Metaphysics Research Lab, Stanford University, (2002). 41. A.N. Whitehead, Process and Reality. Corrected edition by D. R. Griffin and D. W. Sherburne.Cambridge MA., New York, The Free Press (1978). 42. A. Gare, Process Philosophy and the Emergent Theory of Mind: Whitehead, Lloyd Morgan and Schelling. Concrescence, 3, 1-12 (2002). 43. J. Bickle, Understanding Neural Complexity: A Role for Reduction. Minds and Machines, 11, 467-481 (2001). 44. F. M. Kronz and J. T. Tiehen, Emergence and Quantum Mechanics. Philosophy of Science, 69, 324-347 (2002). 45. G. Gentry, Prehension as Explanatory Principle. The Journal of Philosophy, 35 (19), 517-522 (1938). 46. W. H. Leue, Process and Essence. Philosophy and Phenomenological Research, 21 ( l ) ,62-79 (1960). 47. M. Code, On Whitehead’s Almost Comprehensive Naturalism. Process Studies, 31 ( l ) , 3-31 (2002). 48. P. Douglas, Whitehead and the Problem of Compound Individuals. Process Studies, 33 ( l ) , 8&109 (2004). 49. K. Dopfer, The Economic Agent as Rule Maker and Rule User: Homo Sapiens Oeconomicus. Journal of Evolutionary Economics, 14, 177-195 (2004). 50. P. Thompson, Biology. In: W. Newton-Smith (Ed.), A Companion to the Philosophy of Science, Malden/Oxford, Blackwell Publishers, 16-25 (2000). 51. R. M. Burian, Comments on Complexity and Experimentation in Biology. Philosophy of Science. Supplement. Proceedings of the 1996 Biennial Meetings of the Philosophy of Science Association. Part II. Symposia Papers, 64, 279-291 (1997). 52. R. N. Brandon, Grene on Mechanism and Reductionism: More Than Just a Side Issue. PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association, Vol. 1984, Vol. 11: Symposia and Invited Papers, 345-

98 353 (1984). 53. W. Bechtel, The Compatibility of Complex Systems and Reduction: A Case Analysis of Memory Research. Minds and Machines, 11, 483-502 (2001). 54. P. Machamer, L. Darden and C. F. Craver, Thinking About Mechanisms. Philosophy of Science, 67 (l),1-25 (2000). 55. S. S. Glennan, Rethinking Mechanistic Explanation. Philosophy of Science, 69, 342-353 (2002). 56. J. G . Tabery, Synthesizing Activities and Interactions in the Concept of a Mechanism. Philosophy of Science, 71, 1-15 (2004). 57. W. Bechtel and R. C. Richardson, Discovering Complexity: Decomposition and Localization as Strategies in Scientific Research. Princeton, N. J., Princeton University Press (1993). 58. M. Hesse, Models and Analogies. In: W. Newton-Smith (Ed.), A Companion to the Philosophy of Science, Malden/Oxford, Blackwell Publishers, 299-307 (2000). 59. A. L. Plamondon, Whitehead’s Organic Philosophy of Science. Albany, State University of New York Press (1979). 60. F. A. Hayek, Degrees of Explanation. The British Journal of the Philosophy of Science, 6 (23), 20S225 (1955).

WHY DIACHRONICALLY EMERGENT PROPERTIES

MUST ALSO BE SALIENT

CYFULLE IMBERT IHPST / Universit Paris 1 Panthon - Sorbonne In this paper, I criticize Bedau’s definition of ‘diachronically emergent properties’ (DEPs), which says that a property is a DEP if it can only be predicted by a simulation (simulation requirement) and is nominally emergent. I argue at length that this definition is not complete because it fails t o eliminate trivial cases. I discuss the features that an additional criterion should meet in order t o complete the definition and I develop a notion, salience, which together with the simulation requirement can be used t o characterize DEPs. In the second part of the paper, I sketch this notion. Basically, a property is salient when one can find an indicator, namely a descriptive function (DF), that is such that its fitting description shifts from one elementary mathematical object (EMO) to another when the property appears. Finally, I discuss restrictions that must be brought t o what can count as DFs and EMOs if the definition of salience is t o work and be non trivial. I conclude that salience (or a refined version of it) can complete the definition of DEPs.

In the literature about emergent properties, attention has recently been paid to properties that emerge diachronically in complex systems, for example when a tornado ends up appearing in a usually quiet area. The central idea behind diachronic emergence is that these emergent properties can be explained or derived, at least in principle, from the system microdynamics (so they have nothing mysterious), but only in a complex way, e.g. for the apparition of the tornado by numerically solving NavierStokes equations. In the past decade, different definitions have been proposed to characterize more precisely diachronically emergent properties (DEPs). Stephan’ emphasizes that DEPs are highly unpredictable, as demonstrated in the case of deterministic chaos. Bedau2 claims that a property is diachronically emergent (he uses the term ‘weakly emergent’) if it can be derived from the system’s microdynamics and external conditions but only by simulation. I label this latter condition the ‘simulation requirement’ (SR). In this paper, I focus on Bedau’s definition, which I am definitely sup-

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portive of, and I try to refine it further by making it more stringent. I agree that the definition is sufficiently broad to be fruitfully applied to a large variety of systems that exhibit properties that seem to be DEPs. Yet, I argue that the SR is not enough to define DEPs and that the definition, as stated by Bedau, encompasses too many properties. I try in a second step to build a notion, namely salience, that is appropriate to complete the definition and capture only the right set of properties.

1. Preliminaries In order to present the problem, I shall restrict for simplicity to an example belonging to discrete mathematics, namely the case of a one dimensional CA system composed of cells, each colored black or white.a A rule determines the color of each cell at each step. Below is rule 110. It says for example that when a CA and its neighbours are black, the CA turns white at the next step.

Figure 1. Rule 110

The evolution of such a CA system, composed of automata following rule 110, depends on its initial condition. Figures 2 and 3 represent this evolution when the initial condition is a single black cell. Each line represents the system state at a given step and the evolution is downwards. One can see on these figures that in this case, the evolution is neither highly regular nor completely random. We can witness the emergence of patterns such as occasional big triangles or as the vertical column of triangles in the right. For the discussion that follows, it is appropriate that I give pet names to the properties that the discussion is going to revolve upon. ‘Blackcell’refers to the property ‘having CA 10 in a black state’ and it is true at time 20. ‘Bigtriangle’ refers to the property ‘having a big triangle appearing around CA number 20 on the lattice’ and it is true around time 80. ‘Microstate50’ aFor a clear presentation of cellular automata, see for example [3].

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Figure 2. Rule 110, starting from a single black cell, 250 steps. Figure taken from A New Kind of Science, chap. 2, reproduced with permission of Stephen Wolfram, LLC.

refers to the property ‘being in the configuration that the system is in at step 50’ and it is true at time 50. ‘Verticalcolumn’ refers to the property ‘having a vertical column of triangles appearing around CA 60’ and it is true around time 350. These are property names but for linguistic commodity, I shall also use these names to refer to the instantiation of these properties at the mentioned times and also as if they designated emerging things. I also need to introduce scraps of terminology. I shall use ‘microproperty’ to refer to a property of a basic constituent. In our examples, the two basic microproperties of a CA are ‘being black’ or ‘being white’ and you can specify them by including time, e.g. ‘being black at time 3’. Following the uses in statistical physics, I shall call ‘microstates’ conjunctive properties that consist of the ordered enumeration of all the microproperties of the parts of the system S. I shall also call ‘macroproperties’ proper-

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Figure 3. Rule 110,500 steps. Figure from A New Kind of Science, chap. 2, reproduced with permission of Stephen Wolfram, LLC.

ties that can only be possessed by the whole system but not by its part. Such properties are sometimes referred to in the literature as ‘nominally emergent’.4 Structural properties, involving microproperties and a relation between them are a subspecies of nominally emergent proper tie^.^ Examples of macroproperties are Blaekcell, Microstate50 or, for a cup of water, fluidity or transparency, because molecules of water cannot be said to be fluid or transparent. I will not venture here in giving a more detailed general definition of macroproperties because I shall not need that much to define DEPs. Bedau himself acknowledges that “full understanding of nominal emergence would require a general theory of when macro entities have a new kind of property that their constituents cannot have”.4 That should not worry us too much because in our CA system example, we can very easily, if we need to, depart on a purely logical ground what counts as a macroproperty from what does not.

103 2. Statement of the problem (Pb)

I can now phrase precisely why I am not satisfied with Bedau’s definition of DEPs. The consequence of equating the fact of being a DEP with the fact of satisfying the simulation requirement (SR) is that

(1) Blackcell, Microstate 50 and any coined conjunctive property that you feel like building up by the conjunction of microproperties, e.g. ‘having CA 24 black at time 3 and CA 47 white at time 567’- I call these “deceptive properties’; (2) respectable properties such as Verticalcolumn, Bigtriangle, or any emerging pattern - I call these “target properties”; must be treated on a par and considered as DEPs because we need to run a computer simulation to predict them all.

I think this is an unpalatable consequence of the definition, which shows that the SR is not enough to define DEPs. 2.1. Two ways to face (Pb)

I shall now examine two ways to answer my claim that the SR is not sufficient to define DEPs and I shall show why I think these answers are unsatisfactory.

2.1.1. Is nominal emergence enough to solve (Pb)?

I have assumed in my phrasing of the problem that the SR is what matters in the definition of DEPs. Bedau may perhaps argue that I have not faithfully reported his view, because he says: “Assume that P is a nominally emergent property possessed by some locally reducible system S. Then P is weakly emergent if and only if P is derivable from all of S’s micro facts but only by simulation”. So according to Bedau the SR applies only to nominally emergent properties. Thus I can be reproached for having forgotten the condition about P being nominally emergent or else for having misused the notion: Blackcell (‘having CA 20 black’) seems to be a property of the whole system and so to be nominally emergent ... but isn’t it because I used a logical trick to forge Blackcell, instead of simply talking of a less intricate microproperty which applies to CA 20 at time 10, namely ‘being black’? Well, maybe. I leave to the reader’s metaphysical intuitions whether the property ‘having a big wart on her nose’ (which is true of the witch that gives the apple to Snow White) is not a very respectable property too and

104 is really the same as ‘being in a warty state’ (which, for a coarse-grained description of the witch, is true of the descriptive cell corresponding to the witch’s nose). In any case, the conclusion should at least be that if we want to be able to determine which properties are DEPs we really need much more theorizing about nominal emergence to select the right properties. But I strongly doubt that a good theory about nominal emergence will do. A reason is that, from a descriptive point of view, properties such as Bigtriangle or the appearance of a tornado in a fluid, which are paradigms for diachronical emergence, describe local features indeed,b and that these features do not necessarily involve the whole system for being predicated but can be predicated of just a few CAs, even if they are generated by the whole system evolution (just like Blackcell). So from the nominal emergence perspective, the difference between Blackcell and Bigtriangle is very slight, whereas my guess is that Blackcell is not a DEP in our example, whereas the appearance of a tornado is. Another problem is that properties which are pure conjunction of microproperties and involve the whole system, such as Microstate50, will still be nominally emergent, however unremarkable they may be. It is still possible to hope that a strengthening of nominal emergence may make it possible to get rid of them too. Yet it seems to me that with this strategy, we are progressively lead towards a different kind of emergence leaning on a strong distinction between whole and parts. The solution I propose in the second part of this paper, with the use of the notion of salience, does not make use at all of nominal emergence.

2.1.2. Is there more to diachronical emergence than mere computational irreducibility? A second way to face (Pb) is to hold tight like this: “Yes indeed, the SR is what only matters to define DEPs and there is no reason to make further distinctions between properties, because as far as weak emergence is concerned, these properties are really on a par. If you want to predict the apparition of Blackcell or Microstate50, you have to run a simulation and let all these properties diachronically emerge. These properties are so to speak “as diachronically emergent as” Bigtriangle or Verticalcolumn because the same computational effort (modulo time linearity to account for different appearance times) is required to predict them all. This computational effort bSee also note g.

105 corresponds precisely to the running of the dynamics of the whole system and this is where the notion of whole system has a part to play.” I do not think that Bedau would hold tight like this. In [2], the examples are about the emergence of periodic or stable configurations in the Game of Life and about adaptation of mutation rates in a model of evolving life. That is not exactly what one would call trivial or insignificant properties of a system, as Blackcell or Microstate50 can be assessed to be. So if the definition purports to include in its scope such insignificant properties, this should be made clear and the examples are misleading because they point at a specific class of very remarkable DEPs. Besides, if there is nothing more to diachronic emergence than the SR, then it is maybe better not to talk of emergence at all and to speak merely of properties of computationally irreducible systems6 , or of properties that are for a system computationally irreducible: if one views the evolution of a system as a computation, then a system is computationally irreducible if given its initial state, there exists no shorter computation that enables to predict a later state of the system. In other words, there exists no shortcut to predict the properties of the system. Yet, I am quite reluctant to content with an equivalence between diachronic emergence and computational irreducibility. Giving a clear definition of computational irreducibility and finding that there are systems that are computationally irreducible, as in the case of low-dimensional chaos7, is one thing. Then, it could be a brute fact about nature that the properties that cannot be predicted except by a simulation are not really worthy of interest. Take for example a classical gas. Perhaps it is true that there is no way to predict exactly its future microstates (that is to say, to predict the position and velocity for each of the particles), except by a simulation. But who really cares? We can still use statistical physics to (try to) calculate more remarkable properties. So computational irreducibility would not be such a big issue, if it was just met in the case of such anonymous properties. But this is not the case. And a good incentive to talk about diachronically emergent properties is the two-piece fact (S) that, as abundantly illustrated by [6] and as can be seen in figure 3 with Bigtriangle or Verticalcolumn: (Sl) there seems to be computationally irreducible systems and (S2) these systems generate among other things some properties (e.g. periodicity when Verticalcolumn appears) that can be as respectable, remarkable, new, worthy of interest or what you like as the properties we

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106 ordinarily predict using analytical methods. My conclusion is that there is really something wanting in a definition of diachronic emergence that only relies on the SR. Various requirements can be found in the literature’, for emergent properties, in particular novelty and irreducibility.c The SR requirement seems to fulfill the irreducibility requirement. So my claim is that, in the case of diachronic emergence, something like the novelty requirement must also be honored one way or another. 2.2. How to complete the simulation requirement?

It did not go unnoticed that I left (S2) quite vague and used so far unsatisfactory adjectives (‘interesting’, ‘remarkable’, etc.). The purpose was to keep my different claims distinct so that one can agree with my diagnosis (the SR is not enough) and possibly disagree with the way I try to solve the problem (what is missing). What I want to do now is to give a criterion that enables to depart the computationally irreducible properties that seem to have nothing special (deceptive properties) from the ones that seem to have something more (Verticalcolumn, Bigtriangle). But before presenting my solution, I think it is appropriate to list the constraints that weigh on this criterion. 2.2.1. T h e criterion m u s t n o t rely o n intrinsic features of the property but m u s t be contextual (Cl)

As I stated above, properties that are produced by a computationally irreducible system can be the same as the ones we predict in other analytically solvable cases, e.g. periodicity of a configuration. Therefore one should not be after a criterion that would be based on a logical analysis of each property, or on a complexity measure or any other feature characterizing intrinsically the properties. To put it differently, what is striking is not to witness a complex, or ordered or random behavior in a CA system but, for example, to witness the apparition of a complex or random behavior emerging out of an initial simple output or, like in the example of CA 110, of structures emerging out of an almost uniform initial condition.d In other ‘This is in no way is in a definition of or a necessary condition for emergent properties. In particularly, I do not assume that there is a single concept of emergence or that any concept of emergence must meet these requirements. dSee [6, pp.149-1521 for examples.

107 words, what matters is which new features the system dynamics is able to generate out of a context that did not possess these features, and that is why I say the criterion we look for is contextual.

2.2.2. The criterion must be local, that is to say must rely on preceding microstates only (C2) Most of the listed properties (deceptive and target properties) are new in a very weak sense: they are properties that were not possessed before (the

system was never in a state identical to Microstate50 before). Verticalcolumn and Bigtriangle seem to be new in a stronger sense because the system seems to locally undergo a change. In the same time, nothing precludes that similar instances of target properties may have appeared before or may appear again later: tornadoes do re-emerge in the same areas. I conclude that the desired criterion, if it is to characterize novelty, must be local in the sense that it is possible to determine if a property is novel only by looking at the microstates of the system just before the property appears. Local novelty is also an attractive feature because it is a weak requirement in contrast with other kinds of novelty such as absolute novelty (if the supposedly new features were never possessed before). And I am after a weak criterion because I believe that the SR is what matters first in diachronical emergence and I want only to refine it by disqualifying a very specific set of deceptive properties. The requirement of local novelty is also there to do credit to the idea that the apparition of an emergent property corresponds to an observable change in the system state. Emergent properties may involve no new causal power, as this is precisely the case with diachronic emergence, where all causal effects can be traced back to the system’s microdynamics. But one can still require that the emergence of these properties correspond to the apparition of novel features characterizing the system.

2.2.3. The criterion must be contextually absolute (C3) Once a context - that is to say the previous microstates - is given, the novelty of a property must be assessed on the basis of objective properties. For example, there seems to be an objective sense in which one can say that a tornado is a new structure that locally appears, because one can find objective indicators to signal this appearance, such as a sudden local increase of average vorticity. In particular, although I call ‘salience’ the property that emergent properties must possess, I do not want salience to

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be relative to an observer and her cognitive apparatus.

3. Salience In the last section of this paper, I try to develop a concept, salience, that fulfills the previously listed requirements and I claim that DEPs must also be salient. Salience is an independent notion and I believe that it can prove to have a wider application than the question of emergence.e The definition of salience also raises its own set of problems, but since it is not my goal to fully develop this notion here, I shall be quite sketchy, leave aside most of these problems and be content to show how salience could be used to complete the definition of DEPs. The idea I am going to develop is that a property is salient if one can find a descriptive indicator which can be calculated for any state of the system and that undergoes a change when a salient property appears. To make things clearer, I start with two clear-cut examples. In the case of phase transitions, order parameters (when one manages to find one) yield such indicators. For example, in the case of the transition from ice to water, the quantity pice - psystem,where p indicates the density, is zero till the transition and then grows. Another more elaborate example from dynamical system field, which I borrow from Rueger', is the case of the damped non linear oscillator, which is described by van der Pol's equation: x - a(1- x2)x x = 0, where x = d x / d t and a is a damping parameter. For a dynamical system, it is appropriate to study properties of the phase space, which plays for us the role of a descriptive indicator. With a = 0, the oscillator is undamped and the phase space can be portrayed by concentric ellipses, each ellipse representing a trajectory (see figure 4). If the damping is gradually turned on, trajectories are no longer periodic and the phase space is made of spirales converging to a limit circle (see figure 5). So the turning off of the damping makes new salient properties appear (e.g. periodicity) and this can be indicated by the topological change of the phase space, which is described by topologically inequivalent objects (two objects are topologically inequivalent if one cannot smoothly deform one object to transform it into the other). This specific and very well chosen example enables Rueger to provide a very clear criterion of novelty, using the notion of topological inequivalence.

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eFor example for the question of what it is to identify a non trivial property in a data basis obtained by a number crunching simulation or by experiments.

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Figure 4.

Figure 5.

Undamped oscillator, taken from Synthese, 124, 2004

Damped oscillator, taken from Synthese, 124, 2004

Yet it seems to me that he actually shows how novelty can be defined in a very particular case (dynamical system studies) and not in general. I am not sure my criterion of salience will be as clear and elegant, but it aims at being more general.

3.1. Preliminary definitions

I assume that at each time, the system can be suitably represented for the properties that are studied by data originating in n measurements and I call ‘state of the system’ this n-tuple of measurements. Examples of states are a triplet (P,V,T) for a gas at equilibrium, data describing a velocity field

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for a fluid, what I called ‘microstates’ for the above CA system or subparts of them (for example the subpart corresponding to CA number 15 to 25, that is to say where Bigtriangle does appear). I call ‘trajectory’ the sequence of states that one gets by varying a parameter describing the system. This parameter can be time, like in the case of the above CA system, but it need not always be. For example, temperature can be chosen for an ice to water phase transition. What is only required to get a well-defined trajectory along a parameter is that one single state can be ascribed to each new value of the parameter along the trajectory. Suppose for example you study properties of cooling of glasses. Since, for a similar initial condition, the end state of the glass and its properties depend on the cooling rate (see figure 6)’actual trajectories depend on the cooling rate. Therefore temperature cannot be taken as the varying parameter defining trajectories. Specifying exactly the trajectory does matter because salience is a contextual notion (Cl) and because the part of the trajectory before a property appear provides the context against which the property stands out. I call ‘descriptive function’ a single-valued mathematical function that

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Figure 6. Glass cooling modelled by a two well system (the fraction of the systems in the higher well is plotted). Figure taken from Dynamics of Complez systems,by Bar-Yam, reproduced with permission of Addison-Wesley.

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takes as an argument the n-tuple describing the system.f This is in a no way a definition of a new object. This definition only provides a label that is stamped on already existing functions. Here are examples of descriptive function:

0

0

0

0

for a spin lattice, the total magnetization or typical size of clusters of similarly oriented spins; an estimation, using an auto-correlation function, of the correlation between measurements at spatial or temporal distance r or t ; the probability value of an event, for example of having two similar colors following each other on the above CA system; a complexity measure such as Kolmogorov complexity of a state; etc.

Descriptive functions are aimed at giving a common indicator all along the trajectory, so that it be possible to make comparisons and thus determine if the system exhibits a new behavior. In the case of Verticalcolumn, we can for example study the spatial correlations around CA number 60 in the 100 steps before Verticalcolumn is instantiated. On this interval, which plays the role of a context, the system undergoes fluctuations and correlations are low. When Verticalcolumn appears, fluctuations stop and the correlations suddenly get high. To be more precise descriptive functions should themselves be described by other more elementary mathematical object (EMO) which enable to fit them locally. In the case of the oscillator, the phase space was described by different geometrical figures (circles and spirales). The novelty is then signaled by the shift in the description of the descriptive function from one E M 0 to another one. In the case of single-valued functions, EMOs are elementary functions such as z2 and power laws, l o g ( z ) ,e z p ( z ) ,trigonometric functions, etc. 3.2. Towards a definition of salience

I can try now to propose a first definition of salience.

A property P of a system S, which appears at t or on interval I is fThis is actually a restriction that I make for simplicity because one may need more complicate objects than singlevalued function to describe suitably the systems. In Rueger’s example of the oscillator, the system was described for each value of the damping parameter by trajectories within a phase space,

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salient for trajectory T of S iff there exists different EMOs A and B and a descriptive function DF of the states of S’s trajectory such that: 0

0

DF is described by A on an interval C of T (the context) before t or I; the description of DF shifts from A to B at t or on I with the apparition of property P.

This definition, because of the existence condition, does not provide an algorithm to determine if a property is salient. So the burden of the proof is for those who want to claim that a given property is salient, because they need to find an appropriate indicator, which can be far from obvious, even for phase transitions. Further, this definition, stated like this, can easily be found wanting. For example, it would be necessary to have a way to determine what the minimal size of the context needs to be in each case. In the following paragraphs, I shall study a few obvious corrections that should be brought to the definition in order to make it work better and also indicate in which directions some work needs to be done if a definition of this kind is to be given a chance to be well-grounded in the end.

3.3. How to avoid ad hoc descriptive functions? Remember salience aims at eliminating undesirable properties such as Microstate50 or Blackcell. An immediate objection is that it is very easy to find descriptive functions that will exhibit a change in behavior when Microstate50 or Blackcell appear. ‘Blackcell’ refers to the property of having CA 10 in a black state at time 20. If I take the color of CA 10 as a 1-tuple state, and if ‘black’ is encoded by 1 and ‘white’ by 0, and if CA(t,x) denotes the state of CA number x at time t , then the function t Hb(t - 20).6(1- C A ( t ,lo)), where 6 denotes Dirac function, does exhibit a a sudden change in behavior when Blackcell appears. A similar trick can give the same result for Microstate50. That is why it is necessary to require that descriptive functions do not include functions of the parameter defining the trajectory (such as t H6(t - 20)) that have an ad hoc salient feature, which is made to appear at t or I (see definition above). Further, similar variables of the n-tuple should be treated similarly or contribute formally in a similar form to the descriptive function in order to avoid ad hoc treatments. The consequence is that most descriptive functions, like in the above examples, will be sums or integral functions. A cautionary remark needs to be made here. Such restrictions are not

113 aimed at getting rid of properties like Blackcell or Microstate50 in any circumstances. Suppose that CA 10 -happens to be white from step 0 to 19 and then turns black at step 20. Then it is legitimate to say that Blackcell is salient and for that we no longer need an ad hoc function as above. Function CA (t,10), which is tolerated by the definition, will nicely do the job because it will have a zero value till step 19 and then will abruptly take value 1 at step 20.g Similarly, if it happens that Microstate50 is random and appears after a succession of non-random states, then this provides a good ground to call it ‘salient’. 3.4. Requirements about describing EMOs

Notice first that I do not define in the abstract what can count as EMOs. What EMOs are will depend in each case of what kind of mathematical object the descriptive function is. For example, EMOs will be different if you want to study the evolution of a phase space or of an order parameter. A fully assumed consequence of my definition is that a curve can have different salient parts even if you can fit it with one single curve. Take for example a curve that you can fit with function x Hx2 0 . 1 ~ With ~ . my definition, it can be said to have different salient parts. Around x = 0, the curve is correctly described by x2. For large x, it is best described by 0.1x3. A more worrisome problem is this. Take a curve F that always has 0 value. Then you can very easily find two polynomials A and B such that 1) A has value 0 till x = l and then has a more complicated behavior 2) B has value 0 after x = 1 and a complicated behavior before.h A and B are 2 EMOs and the description of F shifts from one to the other. So it seems that it is possible to find salience really everywhere. To guard against that, it is necessary to require that there is no E M 0 that fits F on both intervals where A and B are supposed to fit it and that is in the same time simpler than A and B. Here, the function x H 0 is simpler than A and B. The outcome of this is that, in order to fully develop the notion of salience, I need to be able to estimate the simplicity of mathematical objects. In the previous example, one must be able to explain why x 2 0.1x3

+

+

gNote that in this case, nominal emergence would disqualify Blackcell. hIn short, I need to set restrictions on which EMOs can be used to fit the curves. It probably did not go unnoticed to the reader that the discussion of salience turns out to be reminiscent of discussions about induction, curve-fitting and projectibility, and here in particular of Goodman’s paradox about grue emeralds (see [9,chap. 3-41 and [lo, chap. 51. A complete definition of salience should show how to avoid the goodmanian pitfalls or how to get out of them.

114 is less simple than x2 or 0.1z3. Since the last two expressions are composed of one element and the former of these two elements, the answer may seem obvious. But this seems all too relative to the mathematical framework that is taken as basic. Taking as basic the functions z H g(z) = z2 0.1z3 one gets z2= 0.5(g(z) h ( z ) )and z2seems and z H h ( z )= z2- 0.1z3, now less simple than g(z). As a reviewer points out, this makes salience relative to a privileged descriptive mathematical framework. A way out of the deadlock would perhaps be to argue that there are good objective grounds, in each case of data mining and salient properties detection, to privilege a descriptive mathematical framework and basic EMOs on the basis of which projectable quantities characterizing the system the descriptive function and more generally the mathematical description represent. I do not go any further in the discussion of the concept of salience, since as just shown, this would require a close scrutiny of much debated questions, like simplicity and curve fitting. One more thing still. In this paper, the concept of salience was only aimed at giving a way to single out the properties that are considered as remarkable in the study of a system, e.g. in physics. But determining what patterns or properties are salient for a subject is also a question that is crucial in cognitive sciences.' I make no claim in this paper about the link between the two notions, which may be somewhat related, even if not identical. This latter point can be illustrated as follows. Indeed, a salient property (as defined above) in a data basis can be imperceptible for a subject. That is why data mining requires resort to statistical tests made by computers. In the same time, perceptual abilities for pattern recognition also prove very useful to detect interesting p r o p erties, for example in hydrodynamic simulations, and the existence of the supposedly detected properties can be checked afterwards with statistical indicators. Finally, a system (e.g. a Hopfield network3) could be trained to detect in any circumstances a given pattern of CAs whereas this pattern need not always be salient (in my sense), since the notion is contextual. Besides, the pattern detection system may treat differently inputs of the same kind ( i e corresponding to similar variables of the n-tuple in the mathematical description) and thereby use for detection a function that would not qualify as a descriptive function.

+

'See Ill] for example.

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4. Conclusion We can now check that the concept of salience meets the requirements listed above, or that a well-grounded and refined version of it could be hoped to. It is local, because it depends only on the preceding part of the trajectory. It is contextual because any property can happen t o be salient if it is generated by a context against which some of its features stand out. It is objective (at least provided a descriptive mathematical framework is given) because it relies on the construction of mathematical indicators and not on our epistemic interests, perceptual abilities or practical goals: for example, the position at a given time of a planet, on which one wants t o send a shuttle, is very unlikely t o be salient. Salience, as I built it, is a very weak and purely descriptive notion, which aims a t grasping the idea that the apparition of a non-trivial property is simultaneous with a significant change in some of the values of the observable quantities characterizing the system. Once again, I think that using a concept as weak as possible t o complete the definition of DEPs is appropriate because I believe that the SR is the crucial element in it. But my conclusion is that DEPs satisfy the SR and are salient.

Acknowledgements Special thanks to Anouk Barberousse, Jacques Dubucs and Paul Humphreys for discussing different versions of this paper and also to an anonymous reviewer for very helpful and very apposite critical comments, which helped a lot t o improve the paper. All remaining shortcomings are of course mine.

References 1. A. Stephan, “Phenomenal Emergence”, Networks, vol. 3-4, 91-102 (2004). 2. M. Bedau, “Weak emergence”, Philosophical Perspectives: Mind, Causation and World, 11, Oxford Blackwell Publisher (1997). 3. G. Weisbuch, Dynamique des systbmes complexes, Une introduction a m rkseauz d’automates, InterEditions/Editions du CNRS (1989). 4. M. Bedau, “Downward causation and the autonomy of weak emergence”, Princzpia, vol. 6, 5-50 (2003). 5. T. O’Connor, Hong Yu Wong, “The Metaphysics of Emergence”, N o h , forthcoming.

6. S. Wolfram, A New Kind of Science, Wolfram Media, Inc. (2002). 7. J.P Crutchfield, J.D Farmer, N.H Packard, R.S Shaw, “Chaos”, Scientific American, vol. 255, 46-57 (1986).

116 8. A. Rueger, “Physical Emergence, Diachronic and Synchronic”, Synthese, vol. 124, 297-322 (2000). 9. N. Goodman, Fact, Fiction and Forecast, Harvard University Press (1983). 10. N. Goodman, Language of Arts, Hackett Publishing Company, Inc. (1976). 11. D. Marr, Vision, A computational investigation into the human representation and processing of visual information, New York: Freeman (1982).

ON THE RELATIVITY OF RECOGNISING THE PRODUCTS OF EMERGENCE AND THE NATURE OF THE HIERARCHY OF PHYSICAL MATTER

KURT A. RICHARDSON

1. Introduction On the matter of privileging a so-called ‘scientific method’ over all other methods in the discovery of ‘truth ’ the philosopher Paul Feyerabend said:

It is thus possible to create a tradition that is held together by strict rules, and that is also successful to some extent. But is it it desirable to support such a tradition to the exclusion of everything else? Should we transfer to it the sole rights for dealing in knowledge, so that any result that has been obtained by other methods is at once ruled out of court?’ Like Feyerabend my answer to the question is “ a firm and resounding

NO.” In this chapter I would like to explore my reasons for this answer by exploring the ontological status of emergent products. I am particularly interested in demonstrating the relativity of recognising the products of complexity, i.e., that emergent-product recognition is problematic and not ‘solved’ by the application of one particular method over all others. I would also like to explore the consequences of this in regard to the hierarchy of physical matter and the hierarchy of human explanations. 2. A Theory of Everything My starting point for such an exploration is the assumption that the Universe, at some level, can be well-described as a cellular automaton (CA). Of course by selecting this starting point I am making the assertion that not only does some form of cellular automata represent a plausible theory of everything, but that a theory of everything does indeed exist. This would seem to be a radically reductionist starting point, but I am claiming a lot less than may at first appear. Rather than defend this starting point in

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118 the main part of the chapter, I have added a short appendix that discusses the reasonableness of the CA assumption. One thing is clear, a theory of everything doesn’t even come close to offering explanations for everything so we needn’t get too concerned about the ‘end of science’ or human thinking just because such a theory might exist and might be discoverable. Whether one accepts my starting point or not it should be recognised that this is a philosophical exercise; to explore the ontological and epistemological implications of assuming that the Universe is well-described as a CA at some level. It doesn’t necessarily follow that the Universe actually is and only is a CA. The details of the CA, however, do not change the philosophical conclusions. Richardson15 has explored these issues at length and argued that, whether or not a CA model is a plausible theory of everything, a philosophy of complexity based on this assumption does seem to have the capacity to contain all philosophies, arguing that each of them is a limited case of a general, yet empty, philosophy (akin to the Buddhist ‘emptiness ’. This seems to be a fantastically audacious claim, particularly given the heated debate nowadays over global (intellectual) imperialism. However, one should realise that the resulting general philosophy is quite empty - which I suppose is what we would expect if looking for a philosophy that was valid in all contexts - and explicitly avoids pushing to the fore one perspective over any other. Quoting Feyerabend’ again, “there is only one principle that can be defended under all circumstances and in all stages of human development. It is the principle: anything goes.” In a sense, all philosophies are a special case of nothing!

3. S o , What is Emergence? In the founding issue of the journal Emergence, Goldsteinlo offers the following description of ‘emergence ’: Emergence ... refers to the arising of novel and coherent structures, patterns, and properties during the process of self-organisation in complex systems. Emergent phenomena are conceptualised as occurring on the macro level, in contrast to the micro-level components and processes out of which they arise. For the rest of this paper I would like to consider how patterns and properties emerge in simple CA, and how the macro is distinguished from the micro. I will argue that the recognition of emergent products, which is often how the macro is distinguished from the micro, is method dependent

119 (and therefore relativistic to a degree). The recognition of emergent products requires an abstraction away from what is, and as with all abstractions, information is lost in the process. Of course, without the abstraction process we would have all information available which is no better than having no information at all. "Only a tiny fraction of [the richness of reality] affects our minds. This is a blessing, not a drawback. A superconscious organism would not be superwise, it would be p a r a l y ~ e d ". ~I do not intend to offer a full review of the term 'emergence' here. The interested reader might find Goldsteinlo or Corning3i4 useful as additional sources on the evolution of the term itself.

3.1. Complex cellular automata and patterns Figure l b shows part of the space-time diagram for the simple CA &36 ( n = 20, k = 3). Even though the initial conditions selected for the particular instance of CA were selected randomly, a pattern emerges that is neither totally ordered (an example of which is given in Figure la) nor quasi-chaotic, or quasi-random (Figure lc). CA &36 is an example of a (behaviourally) complex CA as defined by Wuen~che'~.There isn't a lot else we can say about the pattern (although there are a wealth of statistical analyses we might perform to quantify certain aspects of the pattern) that emerges except that maybe it is rather aesthetically pleasing and that its appearance from such a simple set of rules, as well as a random initial configuration, is quite a surprise. So even though it is often said that these patterns emerge from the application of the CA rule, it is not particularly clear as to what is emerging - what are the emergent products here? Is it appropriate to suggest that the space-time pattern is emergent, given that it is a perfect image of the ' micro '-level? What would the macro level be in this example? It is not immediately clear at all. There is no autonomy in this example, as all cells are connected to all others - the pattern is simply the result of applying the CA rule time step after time step. One thing we can say is that the pattern was not wholly predictable from the CA rules, and so in that sense the pattern is novel given our lack of knowledge of it before the system was run forward. But did it really emerge? Or, are our analytical tools just too feeble to characterise the patterns beforehand, in which case the fact that the patterns are regarded as emergent simply reflects our lack of knowledge about this simple system - our theories are simply not powerful enough. Following Bedeau', Goldstein" refers to this aspect of emergence as ostensiveness, i.e., the

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Figure 1. Examples of (a) ordered, (b) complex, and (c) chaotic automatons

products of emergence are only recognised by showing themselvesa. This is also linked to the idea of radical novelty'' , meaning that the occurrence of these patterns was not predictable beforehandb. These patterns are of interest in that they illustrate some aspect of the concept emeTgencec, but they do not tell us much at all about the dynamics of the space-time plot. If we look very closely however we begin to notice that there are patterns within the patterns. The triangles have different sizes and their evolution (shape and movement) 'down' the space-time plot seems to indicate some kind of interaction between local areas of the overall pattern. We might also notice that certain areas of the space-time plot are tiled with a common sub-pattern - a pattern basis. Nansen and Crutchfieldll demonstrated how an emergent dynamics can be uncovered by applying a filter to the space-time plot. Figure 2 shows the result of a ' pattern-basis filter' applied to a particular instance of the Space-time plot of CA rule &36. Once the filter is applied an underlying dynamics is displayed for all to see. What can we say about this alternative representation of the micro-detailed space-time plot? Firstly, we can develop an alternative macro-physics for this apparent 'Even though for the products t o show themselves, an observer must recognise them as products. bThough having run this particular CA on many occasions, the qualitative nature of the pattern was of no surprise to me at all - otherwise I would've found it very dificult indeed to preselect an appropriate case. It would be a very straightforward exercise indeed for me to produce a library of CA patterns from all possible initial conditions though I would be limited by such resources as time and, the related resource, processing power. The point is that I would have to run all the cases fully, there is no shortcut no algorithm - I could implement to speed the task up, i.e., CAs are intrxtable. CIt is still a little dissatisfying to regard these patterns as emergent as they are simply an alternative expression of the CA rule itself, cut-off at an arbitrary time step.

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Figure 2.

CA &36 after pattern basis filtering

dynamical behaviour that is entirely different (from a language perspective at least) from the micro-physics of the CA. If we see these two ‘physics’ side by side (Figure 3) it is plainly clear how different they are. The micrcphysics description uses a microlanguage16 (which is sometimes referred to as the ‘language of design’) comprised only of 0s and Is, whereas the macrolanguage is comprised of an entirely different alphabet: a , p, y+, y-. There is no simple mapping between these two languages/physics, i.e., there is no way (currently) to move between the two languages without some kind of information loss - the translation is lossy (as it is in the translation of human languages). As S t a n d i ~ h ’remarks, ~ “An emergent phenomenon is simply one that is described by atomic concepts available in the macrolanguage, but cannot be so described in the microlanguage”(p. 4). This imperfect relationship comes about because, although the macrophysics is a good theory, it is not a complete description of the micro-physics, it captures some of the details but not all. A qualitative change in language is equivalent to a change in ontology, but what is the ontological status of the emergent products? At the micro-level, only cells (that can only be in states ‘0’ or ‘ 1’) absolutely exist. But can we say with equal confidence that a-particles, for example, exist? Given the filtered version of the space-time diagram showing the apparent evolution of a set of quasi-particles, as well as their compelling macro-physics, it is very tempting indeed to answer in the af-

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a+y.--r y-+a+2y+ y+ + a 2y + a + 'y 111 110 101 100 011 010 001 000 0 0 1 1 0 1 1 0

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R + VI - b y +' + ;

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Y++Y"P . . 7' + a + y- + y + a + y' y' + p + y- + 0 (b)

(a) The micro-physics of level 0 and (b) the macro-physics of level 1

firmative. However, we need to bear in mind how the original patterns were created and how we uncovered the quasi-particle dynamics. The patterns were generated by recursively applying a micro-rule to each cell in the system. Each cell is treated exactly the same as the others - none receive any special or unique treatment. The application of the rule does not change as the system configuration evolves. And, most importantly, each cell in the configuration depends on the states of all the other cells - a particular sequence of 0s and 1s representing the evolution of a particular cell, only occurs that way because all the other cells evolve in their particular way. Although the rule is only applied locally, the dynamics that emerge are a result of global interactions. So a-particles only emerge because P-particles emerge (as well as the other particles) - the emergence of a-particles does not occur independently of the emergence of ,&particles. As a result it would be a mistake to assume that some absolute level of ontological autonomy should be associated with these particles, though of course the macro-physical description provides evidence that making such an assumption is for many intents and purposes useful. However, as with all assumptions, just because making the assumption leads to useful explanations, it does not follow that the assumption is correct. Even the motion of these particles is only apparent, not real; it is simply the result of reapplying the CA rule for each cell at each time step. Whilst discussing Conway's Game-of-Life (a form of two-dimensional CA), philosopher Daniel Dennett6 asks: [Slhould we really say that there is real motion in the Life world, or only apparent motion? The flashing pixels on the computer screen are a paradigm case, after all, of what a psychologist would call apparent motion. Are there really gliders [or, quasi-particles] that move, or are there just patterns of cell state that move? And if we

123 opt for the latter, should be say at least that these moving patterns are real? Choosing ‘0’ and ‘1’ to represent the two cell states perhaps further suggests the illusion of motion and ontological independence. The choice of ‘0 ’ and ‘ 1’ is a little misleading as it can easily be interpreted as being ‘on’ and ‘off’ or ‘something’ and‘nothing’. Whether a cell is in state ‘0 ’ or ‘ 1’, the cell itself still exists even if a‘black square’ is absent. We could easily use ‘ x ’ and ‘ N ’ to represent the two states and the result would be the same. In uncovering the macro-dynamics we had to, however reasonable it may have seemed, choose a pattern-basis to be ‘removed’ or filtered-out. By clearing-out the space-time diagram we allowed ‘defects ’ in the space-time diagram to become more clearly apparent. We also chose to set to ‘0’ those areas of the space-time diagram that were tiled evenly with the pattern-basis, and ‘ 1’ those regions which represented distortions in the pattern-basis; we assumed an even ‘background ’ and then removed it to make visible what was in the ‘foreground’d. Again, it is tempting to interpret the ‘0’-covered (white areas) as ‘nothing’ and the ‘1’-covered (black) areas as ‘something’. This would be incorrect however, as all the cells that comprise the CA network are still there, we have simply chosen a representation that helps us to ‘see’ the particles and ignore the rest. However, it must be noted that though ‘0’ and ‘1’ do not map to ‘nothing’ and ‘something’ at the CA-substrate level, this is exactly the mapping made at the abstracted macro-level. So in any absolute sense, the emergent products do not exist as such. They do however exist in the alternative (although deeply - complexly related) macro-level description of the CA. The change in perspective is equivalent to an ontological shift, where the existence of the emergent products is a ‘feature’ of the new perspective though not of the absolute CA perspective; what existed at the micro-level does not exist at the macrolevel and vice versa. This new perspective is an incomplete description, yet the fact that the new description is substantially complete would suggest the substantial realism of the particles on which the new description is based. For this reason we say that though the particles do not absolutely exist, they do indeed exhibit substantial realism and as such, for many intents and purposes, can be treated as if they were real (for this reason they ~~~

dAt the CA-substrate level the background is not ontologically different from the foreground - any difference is achieved through filtering.

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are often referred to as ‘ quasi-particles’ rather than just ‘particles ’). Quoting Feyerabend again, Abstractions remove the particulars that distinguish an object from another, together with some general properties ... Experiments further remove or try to remove the links that tie every process to its surroundings - they create an artificial and somewhat impoverished environment and explore its pec~liarities.~ The process of abstraction is synonymous with the filtering process described above. Feyerabend goes on to say (p. 5): In both cases, things are being taken away or ‘blocked off’ from the totality that surrounds us. Interestingly enough, the remains

are called ‘real‘, which means they are regarded as more important than the totality itself. (italics added) So, what is considered real is what still remains after the filtering process is complete. What has been removed is often called background ‘noise’, when really what we have accomplished is the removal of ‘details’ not ‘noise’, for the sake of obtaining a representation that we can do ‘science’ with. 4. The Hierarchy of Reality: Nested versus Convoluted

This view of existence changes the popular image regarding the hierarchy of physical matter considerably. The popular image of the hierarchy of matter (illustrated in Figure 4ae) is that starting with atoms (for argument sake - of course more fundamental (or, more real) entities are known to the physical sciences) molecules are formed which are little more than agglomerations of atoms. Here, atoms are regarded as the ‘micro’ and molecules are regarded as the ‘macro’, and the filter applied is some mysterious process known as the ‘scientific method ’ (although any competent and experienced scientist would rightly question the existence of such a universal met hod). From molecules, molecular complexes are formed, from =In considering Figure 4b it is very tempting indeed to assume that the extra dots (in comparison to Figure 4a) are no more than noise. However, it is important to note (but not necessarily easily grasped) that the existence of the seemingly more real pattern (represented in Figure 4a) depends upon the existence of what is easily labelled ‘noise’ and vice verse. Each ‘noisy’ dot is potentially significant in the future evolution of the system as a whole as it is a part of the system, not just the result of some impure measurement.

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which cells are formed, from which multi-cellular organisms are formed, and so on right up to galactic clusters and beyond. In such a neat nested hierarchy every ‘entity’ has a place which distinguishes it from other entities as well as determining which ‘science ’ should be employed to understand it. Furthermore, every ‘thing’ at one particular ‘level’ in the hierarchy is made-up off those ‘things’ said to exist at the ‘level’ below. This would seem to suggest that molecular physics/chemistry, say, should in principle be derivable from atomic physics. However, even though such a trick is supposedly possible (assuming such a neat hierarchical organisation), in reality scientists are very far indeed from pulling off such a stunt. Nowadays the reason offered for this is simply the intractability of emergence which is well known to nonlinear researchers. Rather than derive biology from chemistry, scientists are forced to assume that the ‘entities’ of interest (which are the result of a particular filtering, or coarse-graining) to their particular domain do exist as such. Once such a step is taken, a science can be developed independently of the science that deals with the supposedly ‘simpler ’ ‘ lower-level ’ ‘entities ’. Making such an assumption (which is often made implicitly rather than explicitly) has worked out rather well, and an enormous amount of practical understanding has been accumulated that allows us to manipulate these ‘entities ’ in semi-predictable ways, even if they don’t happen to truly exist! Technology is the obvious example of such success.

4.1. Non-composite secondary structures

As well as the background ‘ pattern basis’ and the fundamental particles listed in Figure 3, CA &36 “supports a large number of secondary structures, some having the domain and particles as building blocks, some not describable at the domain/particle 1evel”ll . Though it has been acknowledged above that a complete ‘physics‘ of one level cannot be derived from the ‘physics’ of the layer below, it is often assumed that the lower-level contains all that is necessary to allow the higher level to emerge. This assumption needs to be explored further, but just because we don’t have access to a perfect mapping from the lower level to the next level up, it doesn’t mean that some day such a bootstrapping method will not be uncovered. In the CA model the fundamental ‘lowest’ level - level 0, say - does indeed contain ‘everything’ there is to know about the CA universe; the space-time diagram is indeed a complete description. However, the move to

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Figure 4. (a) Abstracted ‘reality’ versus (b) reality?

the particle level description - level 1 - is an imperfect ontological shift. As such, the set of quasi-particles describing the particle-level description is not s a c i e n t to describe all the structures that are observed at the lower level (as a result of ’background’ filtering), i.e., one cannot reverse the filtering process and perfectly reconstruct the micro-level. Hansen and Crutchfield’s discovery that secondary structures also exist, indicate that another level of ‘existence’ - level 2 - can be uncovered. if all of these secondary structures were made up of the particles already described ( a ,p, etc.) then we could argue that a ‘composite ’ particle level (2) exists ‘above‘ the quasi-particle level (1). In the natural sciences this would suggest that chemistry is indeed derivable from physics, or that all of chemistry emerges from all of phgszcs. However, the existence of “secondary structures’‘ that are not “describable at the domain/particle level” suggests that there are entities existing at the higher composite level (2) that can’t possibly be explained in terms of the particle level (1) description, even intractability was not an issue, i.e., there exist higher-level entities that are not combinations of a , p, or y quasi-particles. Another way of saying this is that matter is not conserved between these ascending levels. Assuming these observations have merit in the ‘real‘ world, this would suggest for example that chemistry cannot

127 possibly be derived from physics in its entirety. This perhaps is no great surprise; physics is an idealisation and so a chemistry derived directly from such a physics could only represent an idealistic subset of all chemistry even if we could somehow overcome the computational irreducibility of predicting the macro from the micro. This observation indicates a different meaning to the aphorism “the whole is greater than the sum of the parts.“ Traditionally this has only really meant that, because of networked causality, the properties of the whole can not be directly derived from an analysis of the interacting parts; basically, even though our ontology is (assumed to be) correct, our analytical tools aren’t quite up to the challenge of reconstructing the whole from its parts. Novelty in this particular interpretation simply comes from the fact that our ‘ science ’ is unable to predict the macroscopic behaviour from microscopic. The existence of ‘ non-composite secondary structures ’ suggests that, even if we could bootstrap from the micro to the macro, the ‘parts’ do not offer a complete basis from which to derive the ‘whole’f; the ‘whole’ is not greater than the sum of its ‘parts’ because the ‘whole’ is not even constructed from those ‘ parts’ (except in idealised computer models). In this context we would expect to observe chemical phenomena that could never be explained in terms of physics (however complete physics might get - in this view of the Universe physics can never be complete), and that might even contradict predictions made at the level of physics. There are entities that ‘exist’ within a chemical description of ‘reality’ that could never be explained in terms of atoms, say. The fact that each science has facilitated such enormous strides in our ability to rationally affect our environment suggests that, though they may be based on incomplete ontologies, they are in fact very good partial reflections of ‘reality’. The hierarchy of sciences is sustainable and justifiable, but is not an accurate representation of what is. However, the assumption that the physical matter the sciences attempt to explain organises into a nested hierarchy would seem to be questionable. Regarding our hierarchy of ‘explanations ’ as convoluted rather than nested would be more appropriate in light of the CA model.

‘Unless of course we were bootstrapping from level 0 to level 1, as entities existing at level 0 are by definition deemed to absolutely exist.

128 4.2. Does level of abstraction correspond to degree of

realism ? Given that as we move away from the absolute CA reality information is lost, it would be tempting to suggest that the ‘entities’ that ‘exist’ at subsequently higher levels of description, though substantially real, are increasingly less real. If our descriptions at one level were always tied to descriptions at lower levels then this argument would be supportable. But given that particular scientific communities will often operate with little regard to the ontological and epistemological standards of another scientific community, it is quite possible that a macro-explanation contains entities that are more real than those comprising, what seems to be, a microexplanation. If this wasn’t the case and we did indeed tie ourselves to purely quantum physical explanations, say, it is quite possible that, as a result of the multiple abstractions that would be necessary to get from quantum to social reality, we would learn next to nothing about that particular ‘reality‘ (as we would have a partial description of a partial description of a partial description. . . ). The fact that we are comfortable with ignoring quantum descriptions and approaching ‘social reality’ in terms that seem more familiar to us (after all our human perceptions have been honed to give us rather good tools for dealing with this particular level of reality) enables us to make rather more progress in regard to the development of practical understanding for manipulating social reality. In short, a hierarchy of vertical ontologies can be inferred that, though they may actually be inconsistent (or even, incommensurable) with each other, they can still be incredibly useful in providing practical understanding. The hierarchy of physical reality is not a nested hierarchy like the hierarchy of the sciences has been traditionally regarded. 5. Multiple Filters and Horizontal Ontologies Given the fact that the process of filtering-out pattern bases to reveal ‘macroscopic ’ dynamics works so well in the example above, it would be tempting to assume that even though higher-level descriptions are incomplete, there is in fact a ‘ best ’ way to obtain these descriptions. So although the discussion thus far supports the assertion that the recognition of emergent products is tied to our choice of methodology, it also supports (or at least it does not deny) the possibility that one particular methodology is better than all others. I would like to briefly challenge this possibility

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next. This section of the chapter represents a move away from the more scientifically-basedphilosophy presented thus far to a more speculative philosophy. As we move up our constructed hierarchy of explanations, the reality that is ‘explained’ by our explanations seems to be more complex. By complex here I mean that we seem to have increasing difficulty in locating the boundaries, or the ‘quasi-particles’ that describe our systems of interest. This issue is perhaps exemplified at the ‘social level’, and more clearly illustrated by the methodological diversity of the social sciences. Whereas recognising the ‘atoms’ of the physical sciences is seemingly a rather straightforward affair (though still only approximately representative of what is), the recognition of the ‘atoms’ of social science seems rather more problematic. There are many more ways to ‘atomise‘ social reality in an effort to explain a particular phenomenon (of course our choice of ‘atomisation scheme’ determines the phenomena that then takes our interest, and to some extent, vice verse - although there are potentially many different (possibly contradictory) atomisations ’ that may lead to seemingly good explanations for the same phenomena). This is probably why social problems are often described as ‘messy’ problems” ; the articulation of the system of interest itself is problematic, let alone the development of explanations. The fact explanations of social phenomena often have low ‘fitting coefficients‘ whereas explanations of physical phenomenon often correlate very well indeed with perceived (measured) reality, may further suggest the legitimacy of multi-methodological approaches in the social science+. I don’t want to spend much time at all defending the view that multimethodological approaches for the social sciences are a must. For now I am content to point the reader towards the book Multimeth~dology~~ which argues rather well for a pluralist attitude in the social sciences. I think that the failed attempts in the past to develop a ‘Theory of Society’ satisfies me, for now at least, that pluralism is a feature of the social sciences rather than a failure of social scientists to construct such an all-embracing theoryh. gLow ‘ fitting coefficients ’ for explanations of social phenomena is clearly an indication that the explanations proposed are quite incomplete, suggesting the necessary consideration of alternative explanations. Whereas at the level of superstrings we might get very close t o the ‘truth’ with only one model, we might find that ‘truth’ at the social level, say, is at least the sum of all possible models, i.e., a monist notion of truth may work for ‘low-level’ explanations, where as a pluralist notion of truth is essential for explanations of ‘ higher-levels ’. hThe fact there is no complete theory of anything would suggest the importance of pluralism in all realms, but it is clear that some realms of human understanding have

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._ -mE

00 0 0 0 0 0 0 -

O O G O & -

a

Level 0 - CA ‘Reality’: no hierarchy

Figure 5. The convoluted hierarchy of existence. Each ellipse represents the set of ‘entities’ (a theory’s alphabet) that supposedly account for certain phenomena at a certain level ’

.

The different explanations of the same phenomena offered in the social sciences often rely on different ontological assumptions. Given our historical tendency to rank theories in terms of hierarchies of physical matter, we might ask ourselves as to how we might go about ranking social theories. In other words, are the entities used to construct explanation A more real that those entities used to construct explanation B? I very much doubt that a consistent answer exists for this particular question. The reason this issue is raised here is simply to support the suggestion that as we abstract away from the fundamental substrate of existence (level 0) towards the more familiar social level we experience a kind of ontological explosion (see Figure 5). This explosion occurs in explanation-space and represents the theoretical diversity inherent in our attempts to explain phenomenon in our terms rather than the Universe’s terms (whatever they might be). As the explosion grows in intensity we find that there are many more legitimate ways to approach our explanation building. Each approach relies on different ways to recognise the ‘atoms’ of what is and how they interact with each other, i.e., each approach recognises the products of emergence in a different way. Each approach is only capable of explaining certain aspects of the phenomenon of interest, encouraging the increased need for pluralist approaches. The notion of a ‘level’ becomes rather vague, as does the notion of a hierarchy; even a hierarchy of explanations is challenged. Taking this need for pluralism to its extreme leads us into the realms of relativistic philosophies, as opposed to the realist philosophies that tend to developed very well indeed without taking pluralism too seriously. For those fields to continue to develop though I would expect pluralism to become more common. So all my comments above regarding social sciences are true also for physics, just not equally so.

131 reign in the physical sciences. So how do we decide which are the ‘best‘ approaches that lead to the ‘best ’ theories? How do we avoid the choice of ‘filter’ from being completely arbitrary? In the physical sciences consistency with currently accepted theories is often used as an additional filter. Equally important is partial validation through experimentation, and limited empirical observation. Physical scientists are rather fortunate in that many of their systems of interest can be isolated to such a degree that the scientists can actually get away with the assumption that the systems are indeed isolated. This helps a great deal. In the social sciences there is a more ‘try it and see ’ attitude as the application of reductionist methodologies is highly problematic, because of the considerable difficulties in reproducing the same experiment time and time again. Often new situations, or new phenomena, force the social scientist back to square one and a new ‘explanation ’ is constructed from scratch with minimal consideration of what theories went before (although what went before may prove inspirational). The fact that pattern-basis filtering of complex CA leads to alternative representations of the CA reality that account for a good percentage of the systems behaviour would seem to suggest that an ‘anything goes’ relativism in particular contexts is not supportable. The success of physics itself at least supports the notion that some theories are considerably better (in a practical sense at least) than others. However, the fact that these alternative representations are incomplete would suggest that an ‘anything goes ‘ relativism might be valid for all contexts taken together at once. Once a pluralist position is adopted, the key role of critical thinking in taking us from a universal ‘anything ’ to a particular ‘something’ becomes apparent. I do not want to explore any further the epistemological consequences of CA research as this paper is primarily concerned with the ontological status of emergence products (which, as has been indicated, is impossible to explore without concerning oneself with epistemology). If our commitment to a CA model of the Universe is to persist then we need to find CA systems, and the tools for analysing them, that when abstracted multiple times (to ‘higher levels’ of existence) lead to representations that can be usefully ‘atomised’ and analysed in a variety of different ways (analogous to the social sciences).

5.1. A note on intrinsic emergence

Before closing this chapter with some concluding statements I’d like to mention the notion of intrinsic emergence introduced by Crutchfield as it

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offers a computational-based explanation as to why some theories / perspectives / abstractions / etc. are better than others and why we should resist an “anything goes” relativism. According to Crutchfield’ , “The problem is that the ‘newness ’ [which is referred to as ‘novelty ’ herein] in the emergence of pattern is always referred outside the system to some observer that anticipates the structure via a fixed palette of possible regularities.” This leads to the relativistic situation mentioned above. Although there is clearly a relativistic dimension to pattern recognition (and therefore theory development) how might we resist a radical relativism (that would suggest that anything goes in any context all the time)? Crutchfield’ goes on to say, “What is distinctive about intrinsic emergence is that the patterns formed confer additional functionality which supports global information processing... [Tlhe hypothesis ... is that during intrinsic emergence there is an increase in intrinsic computational capability, which can be capitalised on and so can lend additional functionality.” Another way of saying this is that, if we privilege a computational view of the Universe, the structures that emerge support information processing in some way - they increase the computing ability of the Universe. This would suggest that ‘real’ patterns are those patterns which can be linked to some computational task. For example, considering Figure 2 again, if a filter has been chosen that was not based upon the pattern basis then a pattern would still have been uncovered, but it is unlikely that, that pattern would have any relevance to the computational capacity of the system - the odds are the pattern would also appear quite disordered. This of course opens up the possibility that the best ‘filters’ are those that lead to regular patterns, but given that assessments of regularity (or disorder) lie on a spectrum how do we decide which patterns are sufficiently regular to allow us to associate with them some functional importance? Even though certain patterns play a functional role within the Universe (which suggests in absolute terms that not all patterns are ‘real’), it does not follow that we, as participants in the Universe, can unambiguously determine which patterns are significant and which are not. It is not even clear if an ‘outsider’ (God, perhaps) could achieve absolute clarity either. The notion intrinsic emergence shows that the patterns that we might consider ‘real’ allow the system of interest to process information in some way, suggesting that certain patterns are internally meaningful and not arbitrary. However, it does not follow that just because some patterns a more ‘real’ than others we can determine which are which. In a universal and absolute sense, not all filters and patterns are made equal and certain

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filters (and their consequent patterns) are more meaningful than others. In the absence of direct access to absolute and complete descriptions of reality, however, an ‘anything goes ’ relativism (such as Feyerabend’s) cannot be dismissed on rational grounds alone and offers the possibility of a more genuine engagement with the (unseen) ‘ requirements ’ of certain contexts. 6. Summary and Conclusions

In summary I’d like to ‘atomise’ (and therefore abstract in an incomplete way) the assertions made herein concerning the nature of emergents: 0

0

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0

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Emergent products appear as the result of a well-chosen filter rather than the products of emergence ‘showing’ themselves, we learn to ‘see’ them. The distinction between macro and micro is therefore linked to our choice of filter. Emergent products are not real though ”their salience as real things is considerable.. . “ 6 . Following Emmeche et aL7, we say that they are substantially real. In a limited sense, degree of realism is a measure of completeness relative to level 0. Emergent products are novel in terms of the micro-description, i.e., they are newly recognised. There is no such thing as absolute novelty. Even an ‘idea‘ has first to be recognised as such. There will be novel configurations of cell states, but no new real objects will emerge, though new substantially real objects can emerge. In absolute terms, what remains after filtering (the ‘foreground’) is not ontologically different from what was filtered-out (the ‘background ’); what is labelled as ‘not-real ’ is ontologically no different from what is labelled ‘real ’. Product autonomy is an impression resulting from the filtering prccess. The products of emergence, and their intrinsic characteristics, occurring at a particular level of abstraction do not occur independently - rather than individual quasi-particles emerging independently the whole set of quasi-particles emerges together. The hierarchy of explanations correlates only approximately with degree of realism. The emergent entities at level n+l are not derived purely from the entities comprising level n, even if a perfect bootstrapping method that could overcome the problem of intractability was invented. Emergent products are non-real yet substantially real, incomplete

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yet representative; Determining what ‘macro’ is depends upon choice of perspective (which is often driven by purpose) and of course what one considers as ‘micro’ (which is also chosen via the application of a filter). All filters, at the universal level, are equally valid, although certain filters may dominate in particular contexts.

One last remark. Much philosophical debate circles around what the most appropriate filters are for understanding reality. For example, radical scientism argues that the scientific methods are the most appropriate, whereas humanism argues that the filters resulting from culture, personal experience, values, etc. are the most appropriate. An analysis of a CA Universe suggests that it is impossible to justify the privileging of any filter over all others on rational grounds (though there is a way to go before we can say that this conclusion follows naturally from the CA view). Whereas political anarchism is not a particularly effective organising principle for society at large, a complexity informed theory of knowledge does tend towards an epistemological and ontological anarchism, at least as a Universal default position. 7. Appendix A: On the Reasonableness of Assuming that the Universe is, at Some Level, Well-Described as a Cellular Automaton Given that the strength of one’s claims is related (but not necessarily soi) to one’s starting point, I would like to discuss the reasonableness of making the CA-assumption. I would also like to explore briefly what does not follow from assuming the existence of a theory of everything. Firstly, although some form of CA could possibly turn-out to be the theory of everything, I do not believe that inhabitants of the Universe will ever (knowingly) pin-down the details of such a model - the reason for this being that, due to the nature of nonlinearity (assuming the Universe to be inherently nonlinear), there simultaneously exist an infinity of nonlinear (potentially non-overlapping) models that will provide (potentially contradictory) ‘explanations ’ for all observed ‘ facts ’. Laplacian determinism may well operate, but it certainly does not follow that we can observe this exquisite determinism in any complete sense (though clearly we can to a ‘It is quite possible for the conclusions to be correct whilst the reasoning could be fallacious.

135 very limited degree). Secondly, even if we serendipitously stumbled upon THE theory of everything (though we would never know that we h a d ) , just because we have a theory of everything, it by no means follows that we can derive ezplanations for every thing. All we would have is a theory of how the fundamental units of existence interact. Bootstrapping from that primary CA substrate to ’higher levels’ of existence is considerably more challenging than discovering such a fundamental theory in the first place. As Hofstadter12 remarks: a bootstrap from simple molecules to entire cells is almost beyond one6 power to imagine. (p. 548)

I argue herein that not only is information lost as we ‘abstract ’ to more familiar ‘objects’ (which is the very nature of abstraction), but that there is no single foolproof method to select the ‘best ’ abstraction (even if there is such a ‘best’ abstraction). So, claiming that the Universe might be well-described at some level as a CA is not as big a claim as it might first seem. In the argument I develop above, the details of such a CA are not really important for the validity of the conclusions I draw. But why start with a CA, even if the details are not particularly essential. My reason is simply parsimony; a CA model is possibly the simplest construction currently known to us that comes close to having the capacity to contain everything - this possibility arises because of the model’s recursive nature. With his Life game Conway has shown “that the Life universe . . . is not fundamentally less rich than our 0wn.”14. There are a couple of things we can additionally suggest about such a CA model of the Universe. All CA universes are finite universes and as such certain configurations will eventually repeat. When this happens the Universe would fall into a particular attractor in which a sequence of configurations would be repeated for eternity. The idea that our Universe is currently racing around some high-period attractor cycle seems unlikely (given the novelty we observek),or at least at odds with our perspective on j ” ...and

of course the clear and certain truth no man has seen nor will there be anyone who knows about the gods and what I say about all things. For even if, in the best case, one happened to speak just of what has been brought to pass, still he himself would not know.” Xenophanes of Colophon (fragment B34). kNot that the observation of novelty by a participant within the Universe is proof enough that novelty ‘exists’. Plus, the idea of the big bang being following by a big crunch may well suggest that there is indeed some colossally-high period attractor leading cosmic

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things. The phase spaces of CA Universes contain many such attractors, highlighting the potential import of initial conditions'. If our CA Universe is not living some endlessly repetitive nightmare (claimed by big bang / big crunch models), what might be going on instead? Many of the attractors of CA systems exist at the end of fibrous tentacles of configurations that do not occur on any attractor cycle. Figure 6 illustrates this. These transients are the 'gaps' between the system's initial conditions and its final attractor. These transients can have very complex structures and can even support lifelike behaviour. So here is the CA Universe; from a range of initial conditions (all of which could lead to a statistically identical evolution) the Universe would pass through an inordinate number of unique configurations which would never be repeated again, before it finally settled into a fixed pattern of cycling configurations in which no such novelty would ever exist again (unless of course the rules of the game were changed or some external force pushed the system back onto one of the transient branches - which would of course simply result in a partial rerun). We are quasi-entities living on a transient branch of a CA attractor! It would seem that in addition to long transients such a CA would also have to display complex transients, as opposed to ordered or chaotic. Wuenschelg defines complex CAs as those with medium input-entropy and high variance. An example of a CA displaying complex transient dynamics is rule &6cle53a8 (lc-5). The space-time diagram and the phase space for this rule are shown in Figure 7. Though it by no means demonstrates the validity of the CA-hypothesis in itself, an interesting calculation might be to determine the size of CA needed that would have transients at least as long as the time the universe has existed thus far. We'll consider the rule above for this little exercise. Let's assume that one time-step in our CA Universe is equivalent to the small measure of time known as Planck Time"'. Figure 8 plots the maximum transient length for rule &6cle53a8 for increasing CA size (increasing n). If we assume that the Universe is approximately 15 billion years old (or 4.7 x 1017 s) then we require transients at least 8.77 x lo6' steps long. If we approximate the relationship between maximum transient length and evolution. 'Though there are CA models whose phase spaces are characterised by a single attractor - in such circumstances initial conditions are quite independent of the system's end state (cycle). s, is the age of the Universe at which energies were mPlanck time, which is 5.4 x available to enable particle interactions to occur across Planck distances.

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Figure 6. The largest attractor for CA rule &6cle53&, ~ 1 7k.5 , showing complex transient trees of length 72 steps. The period of the attractor cycle is 221 steps

network size shown in Figure 8 by a power law and extrapolate to 8.77 x 1060 (which is a rather extreme extrapolation I know) then we obtain a colossal value of for n, the network size, whereas we get a value for n of 600 if we assume an exponential relationship". This number is either outrageously large or small compared to what we might expect. Chown20 estimated the number of distinct 'regions' that might exist to be loQ3,so we're way off that mark! However, it does serve to illustrate "As a first approximation I found the relationship between network size, TZ, and transient length,

La%, (for rule &6cle53&, k.5)

to be:

n e e x p [ - 11n ( k > ] 0.0126 3.0228 if a power law is assumed, or: n m - -1- - - - - I n ( k ) 0.2368 0.9952 if an exponential law is assumed. ""The only way one region of space can 'know' about the conditions in another is if there has been time for some kind of influence to go back and forth between them-and the maximum speed of any influence is that of light. Hence, the early Universe consisted of los3 regions which were entirely separate, or 'causally disconnected"'2.

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Figure 7. A section of the space-time diagram for CA rule &6cle53a8, -17, k 5 , and the complete attractor field which contains 54 attractors (21 are qualitatively distinct only these are shown)

that CA may be ‘complex’ enough to support the ‘existence’ of complex entities and maybe even ‘Everything‘. A simple 2-state CA of only 600 cells can have transients as long as 1061 steps containing no repeated configurations. This does not confirm that such a system could contain exotic quasi-entities such as humans, but Hansen and Crutchfield (1997) have at least confirmed that simple hierarchies of interacting quasi-particles can be uncovered using appropriate filters. Chou and Fteggia (1997) have even found that constructing CAs that support self-replicating structures is a surprisingly straightforward exercise. It would be interesting to perform the same calculation for CAs with cells that can take on more than 2 states to see if network sizes of log3 can be obtained, although there is no reason to assume this estimate to be correct in the first place. References 1. Bedeau, M. (1997) “Weak emergence,” Philosoph~calperspectives, 11: 375399. 2. Chown, M. (2000) “Beforethe Big Bang,” New Scientist, 166(2241): 24. 3. Corning, P. (2000) “The reemergence of ‘emergence’: A venerable concept in search of a theory,” C o r n p ~ e ~7(6): t ~ , 18-30.

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Figure 8. 25, k 5

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Maximum transient length versus network size for CA rule &6cle53a8, n=5-

4. Corning, P. (2000) “The emergence of ‘emergence ’: Now what?” Emergence, 4(3): 54-71. 5 . Crutchfield, J. P. (1994) “Is anything ever new? Considering emergence,” in G. A. Cowan, D. Pines, and D. Meltzer (eds.), Complexity: Metaphors, Models, and Reality (pp. 515-537). Cambridge, MA: Perseus Books. 6. Dennett, D. C. (1991) “Real patterns,” The Journal of Philosophy, 88: 27-51. 7. Emmeche, C., Koppe, S. and Stjernfelt, F. (2000) “Levels, emergence, and three versions of downward causation,” In P. B. Andersen, C. Emmeche, N. 0. Finnemann, and P. V. Christiansen (eds.), Downward causation (pp. 13-34). Aarhus, Netherlands: Aarhus University Press. 8. Feyerabend, P. (1975) Against Method, London, UK: NLB. 9. Feyerabend, P. (1999) Conquest of Abundance: A Tale of Abstraction Versus the Richness of Being, edited by Bert Terpstra, The University of Chicago Press: Chicago. 10. Goldstein, J. (1999) “Emergence as a construct: History and issues,” Emergence, l(1): 49-72. 11. Hansen, J. E. and Crutchfield, J. P. (1997) “Computational mechanics of cellular automata: An example,” Physica D, 103( 1-4): 169-189. 12. Hofstadter, D. R. (1979) Godel, Escher, Bach: A n Eternal Golden Braid, New York, NY: Basic Books. 13. Mingers, J., and Gill, A. eds. (1997) Multimethdology: The Theory and Practice of Combining Management Science Methodologies, Chichester, UK: John Wiley & Sons. 14. Poundstone, W. (1985) The Recursive Universe: Cosmic Complexity and the Limits of Scientzfic Knowledge, New York, NY: William Morrow. 15. Richardson, K. A. (2004) “The problematisation of existence: Towards a phi-

140 losophy of complexity, Nonlinear Dynamics, Psychology and the Life Sciences, 8(1): 17-40. Ronald, E. M. A., Sipper, A., and Capcarrre, M. S. (1999) “Testing for emergence in artificial life,” in Advances in Artificial Lzfe: 5th European Conference, E C A L 99, volume 1674 of Lecture Notes in Computer Science, Floreano, D., Nicoud, J. -D., and Mondad, F. (eds.), pp. 13-20, Springer: Berlin. Standish, R. K. (2001) “On complexity and emergence,” Complexity International, 9, available at http://journal-ci.csse.monash.edu.au//vo109/standi09/ Vennix, J. A. M. (1996) Group Model Building: Facilitating Team Learning Using Systems Dynamics, Chichester, UK: John Wiley & Sons. Wuensche, A. (1999) “Classifying cellular automata automatically: Finding gliders, filtering, and relating space-time patterns, attractor basins, and the Z parameter,“ Compledty, 4(3): 47-66. ”

16.

17. 18. 19.

TRUTH IN COMPLEX ADAPTIVE SYSTEMS MODELS SHOULD BE BASED ON PROOF BY CONSTRUCTIVE VERIFICATION

DAVID SHIPWORTH Faculty of Science University of Reading It is argued that the truth status of emergent properties of complex adaptive systems models should be based on an epistemology of proof by constructive verification and therefore on the ontological axioms of a non-realist logical system such as constructivism or intuitionism. ‘Emergent’ properties of complex adaptive systems (CAS) models create particular epistemological and ontological challenges. These challenges bear directly on current debates in the philosophy of mathematics and in theoretical computer science. CAS research, with its emphasis on computer simulation, is heavily reliant on models which explore the entailments of Formal Axiomatic Systems (FAS). The incompleteness results of Godel, the incomputability results of Turing, and the Algorithmic Information Theory results of Chaitin, undermine a realist (platonic) truth model of emergent properties. These same findings support the hegemony of epistemology over ontology and point t o alternative truth models such as intuitionism, constructivism and quasi-empiricism.

1. The epistemology of complex adaptive systems This discussion of epistemology starts with a discussion of the nature of formal systems as these are the principle epistemic mechanisms operating within model theory which itself is the dominant epistemology of complex adaptive systems research. In this context ‘model theory’ is used in This discussion the sense of the epistemology of mathematical leads from an analysis of the limitations of formal systems to the choice of ontological logical axioms based on these limitations.

1.1. Formal systems Mathematical modeling involves mapping certain selected observables of the system being modeled onto mathematical systems and in turn onto the formal systems of programming languages (Figure 1 below). The fact

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that the mathematical structures into which the selected observables are encoded are formal systems is of particular importance in light of the known incompleteness of such systems. The mathematician David Hilbert, at the beginning of the 20th century, presented a challenge to the mathematical community. That challenge was to demonstrate the consistency of the axioms of mathematics. This challenge became known as the ‘formalist program’ or f o ~ m a l i s r n . ~ It was well known by the end of the lgth century that mathematical terms had both a syntactic and a semantic content [2] and it was this, the formalists argued, that gave rise to some well documented apparent contradictions within mathematics, such as the Russell para do^.^ The syntactic content of mathematics is that part of mathematics that involves the application of rules to strings of symbols. Certain of these strings of symbols are taken as the axioms of the system; other symbol strings are created through the application of the rules to these axioms to create new symbol strings. This aspect of mathematics is purely logical and the symbol strings mean nothing. It is this aspect of mathematics that was being described by Bertrand Russell when he claimed that ‘pure mathematics is the subject in which we do not know what we are talking about, or whether what we are saying is t r ~ e ’ . ~ J ’ . ~ ~ ~ The semantic component of mathematics arises when meaning is attached to these symbols. This meaning can be as simple as letting these symbols represent number^.^ Without the semantic component of mathematics, mathematical modeling is impossible, as it requires the encoding of the observables of the natural system into the language of mathematics. Likewise, it is impossible to conceive of any applied CAS model without semantic content. It was this ‘meaning’ component of mathematics which Hilbert and the other formalists thought gave rise to mathematical contradict ions. Hilbert could not tolerate any such contradictions and sought to eliminate them through elimination of the semantic content of mathematics. Hilbert argued all semantic functions of terms could be captured by syntactic rules. As Rosen [2, p.71 notes, ‘In the Formalist view, then, Mathematics ... is a game of pattern generation of producing new patterns (theorems) from given ones (axioms) =cording to definite rules (production rules)’. In this view mathematics is not ‘about’ anything, it is simply an exercise in symbol manipulation according to a set of rules. This view of mathematics is highly amenable to computation. Computation is the process of applying predetermined transformation rules to

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a set of inputs to generate a set of outputs. The process is deterministic, predictable, replicable and has of itself no semantic content. To cite Rosen [2], computers, and their theoretical archetypes Turing machines, are ‘...purely syntactic engines’. As such, they have enormous appeal within the formalist program. Indeed Turing was attempting to verify one of the central tenants of formalism, that the truth of all mathematical statements was decidable through a finite number of applications of the rules of the formal system, when he developed the Turing m a ~ h i n e . ~ Unfortunately for Hilbert, Turing, and others working on the formalist program, the logician Kurt Godel, in 1930, proved the entire formalist program was logically impossible. As Rosen [2, p.71 noted ‘Godel’s Incompleteness Theorem devastated the entire Formalist program’. Godel proved that any given formal system was either logically inconsistent, or incomplete. As the former option of inconsistency is anathema to all mathematicians, Godel’s proof is almost universally taken to demonstrate that such formal systems are incomplete. As Casti [6, p.121 notes:

...Godel’s Incompleteness Theorem, ... tells us that no system of deductive inference is capable of answering all questions about numbers that can be stated using the language of the system. In short, every suficiently powerful, consistent logical system is incomplete. Rosen notes that Godel’s Incompleteness Theorem has the following implications. That ‘...Number Theory is already too rich in entailments (or equivalently that machines are too poor in entailments) for there to be any kind of convergence between them. ...’ and that ‘...Mathematics, like natural language itself, cannot be cut off from all referents and remain Mat hemat ~ c s . ’ ~ + ~ This sets profound theoretical limits on the power of mathematical models. It means that no model of a system can, even in theory, ever capture all that can be said about the system that it models. Formal Axiomatic System (FAS) is defined by Chaitin as:

[A Formal Axiomatic System] consists of a finite set of axaoms and of a finite set of rules of inference for deducing the consequences of those ~ x i o m s ~ + ~ Chaitin also notes that:

The sets of axioms that mathematicians normally use are fairly concise, otherwise no one would believe in them. In practice,

144 there’s this vast world of mathematical truth out there-an infinite amount of information-but any given set of axioms only captures a tiny, finite amount of this information. That, in a nutshell, is why Godel incompleteness is natural and inevitable rather than mysterious and complicated.s~p.171

As noted by Chaitin [4, p.111, ‘Theorems are found by writing all the possible grammatical statements in the system and testing them to determine which ones are in accord with the rules of inference and are therefore valid proofs.’ That is, a statement can be grammatically correct and yet still be false. Indeed, only a small proportion of grammatically correct statements of any given formal system are provable as theorems of that system. Hilbert, in developing the mathematics of formal systems as part of the formalist program, devised a means of constructing all provable theorems within any given formal system. An alternate way to view the structure of a Formal Axiomatic System is as a mathematical graph ( GFAS)(a set of ‘vertices’ (points) joined by a set of ‘edges’ (lines)). The vertex set ( V F A S of ) such a graph would correspond to the union of the set of axioms of the FAS, and the set of all syntactically correct statements provably true through application of the FAS’s rules of inference to the set of axioms of the FAS and subsequently derived theorems. The edge set ( EFAS)would be defined by any two vertices connected by application of the FAS’s rules of inference to the set of axioms of the FAS and subsequently derived theorems. Here we follow the logic used by Godel in the incompleteness theorem and equate ‘truth’ with ‘proof’ and note that in the context of a FAS, proof means something that is provable through execution of a properly constituted algorithm on a Universal Tur~ have one connected ing Machine. By definition, this graph G F A would component containing all vertices. It is to be remembered however that FAS are literally meaningless-i.e. they are devoid of semantic content. The use of FAS to model any meaningful system fundamentally changes this graph. Chaitin [9, 41 studied the arithmetic of real numbers (a semantic system) through analysis of real number Diophantine equations. He found that there are an infinite number of essentially random but provably true arithmetic facts that cannot be derived from any other facts through application of the rules of any FAS model of the arithmetic of real numbers. Following from above, we define the graph of the semantic system of arithmetic G s e r n - ~as , the vertex set V s e r n -of ~ all syntactically correct and provably true statements within the semantic system of arithmetic. We

145 define the edge set E s e r n - as ~ the union of the edge sets ( E F A S )of all legitimately constructed FAS models of arithmetic. We know from Chaitin that each individual FAS model of arithmetic defines a connected component on this graph of arithmetic. Further, differently defined formal systems will have, as provably correct theorems of that system, different sets of vertices on the graph G s e r n - ~ That . is, different formal systems will, through defining different edges on the graph, link different vertices of the graph into connected components. However Chaitin [9, 41, through his construction of the universal Diophantine equation (R), has demonstrated that the graph of arithmetic also contains an infinite number of disconnected vertices that are arithmetically provably true, but which are impossible to prove true (i.e. unreachable on G s e r n - ~by ) any FAS model ( GFAS)of arithmetic. Chaitin notes with respect to the Diophantine equations which constitute these disconnected vertices:

Mathematical reasoning is essentially helpless in such a case, since there are no logical interconnections between the Diophantine equations generated in this way. ..Randomness, uncertainty and unpredictability occur even in the elementary branches of number theory that deal with Diophantine equations4 This discovery is central to the epistemology of complex systems in which the permutation spaces defined by the building blocks of the systems and the rules of inference governing their combination into aggregates is approaching the complexity of that defined by number theory. Indeed as Morowitz [lo, p.195-61 notes:

This is one of the core elements of the series of studies designated as complexity-the combinatorics of a set of simple constructs operating jointly produces an array of possibilities too large to deal with, sometimes transcomputational. This is interesting in the light of Chaitin’s [4, p.181 comment:

Although Godel’s original theorem seemed to apply only to unusual mathematical propositions that were not likely to be of interest in practice, algorithmic information theory has shown that incompleteness and randomness are natural and pervasive. The argument is that if a comparatively simple and well-defined system such as Diophantine equations can never be captured by the construction

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of any formal system there is every reason to suppose that the same will apply to more complex systems. This has two implications that are central to this argument. Firstly, as noted above, any mathematical model executed on a digital computer is by definition a formal axiomatic system. This FAS model (syntactic) defines an edge set ( E F A S )on the vertex set of all provably true statements about the semantic system Vs,, it is modelling. The graph created by this vertex set and edge set contains a connected component, (the set of entailments of the model), and an indeterminate number of disconnected vertices. Each of these vertices represents a ‘fact’ about the system being modelled but one that is unprovable by that model. A model with different axioms, and/or rules of inference, defines a different FAS and a different set of truths. Again the graph thus created has a connected component and an indeterminate number of disconnected vertices. Note that while any number of models can be defined, we know from Godel and Chaitin that none of them can connect all the vertices on the graph. This is precisely the point made by Rosen [2, p.8-91 when he notes that ‘The relation of Number Theory to any formalization of it is the relation of the complex to the simple...We shall argue that a formalization constitutes a (simple) model of that which it formalizes’. Rosen [2, p.91 goes on to note that:

For our purposes, the point of doing all this is that we are going to identify this category of simple models of a system of entailment with ‘what we can know’ about it. ... When we do this, all of epistemology gets wrapped up in that category of models. ‘What we can know’ about such a system thus becomes a question of how good we are at navigating in this category of models. Casti reaches the same conclusion. ‘The best we can hope for is to illuminate various facets of natural systems using different - and inequivalent - formal d e s c r i p t i ~ n s ’ . ~ ~ ” ~ ~ ~ ~ ~ p . ~ * ~ It should be noted that the formal results of incompleteness of Godel and Chaitin pertain to the relationship between mathematical systems such as arithmetic, and formal systems models of them. This is illustrated in Figure 1 in the lower half of the diagram. In the quotes above however, both Rosen, and more explicitly Casti, suggest a parallel if less formally defined incompleteness relationship exists between natural systems and mathematical model of them (the relationship on the left hand side of Figure 1).While this is intuitively reasonable, extrapolating from the relationship between a formal system and the mathematical system it models, to a mathematical

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Natural System

f

% 5’

f

Computer Simulation

f$

Formal System

1

Mathematical System

7

~

Figure 1. Modelling relation (synthesising elements from Casti [6] and Traub [ll])

system and the natural system it models is contentious and ultimately an argument from analogy. It is not central to the argument of this paper.

1.2. Mathematical Realism us. Intuitionism Chaitin’s construction of the universal Diophantine equation, raises another issue, and one which bears directly on the ontology of complex adaptive systems models. This point revolves around the question of what constitutes a valid proof. What constitutes a valid proof also determines what we accept as being true for, as Snapper [12, p.541 points out ‘...we prove that something is t r u e ’ . What constitutes a valid proof is in turn, dependent on the logic system subscribed to. For classical mathematics and conventional science this is the classical logic laid down by the Greeks. Classical logic is founded on the three laws of Aristotelian logic: the law of identity, the law of non-contradiction and the law of the excluded middle.3+’.22 These are ontological laws, they are not proved in any sense, they are effectively the axioms of classical logic. Of these ontological propositions, the ‘law of the excluded middle’ is not accepted by mathematical intuitionists. The argument here is that it should not be accepted by complex adaptive systems modellers either. The law of the excluded middle states that any declarative s e n t e n c e that is any grammatically correct sentence in a formal or natural language which makes an assertion about a fact-has to be either true or false. It has to be either true or false because it either does, or does not, correspond to the fact about which it is making an assertion. This correspondence is

148 wholly independent of any verification of the proposition of the declarative sentence, or of the individual making the declaration. As Snapper notes, classical or Platonic truth is ‘verification Subscribing to the law of the excluded middle imposes a Platonic character to truth. That is, the truth or falsity of the sentence exists independently of time and independently of any observer. In the Platonic notion of truth, a sentence is true if it corresponds to the fact about which it is making the assertion. This is therefore also known as the correspondence theory of truth. Mathematical intuitionists, such as Brouwer, Heyting and Dummett, do not subscribe to the law of the excluded middle.13 If, ontologically, one rejects the law of the excluded middle, then it is impossible to say anything about the truth or falsity of a declarative sentence before its truth or falsity has been verified in some epistemologically acceptable manner. Just as classical mathematics is based on an epistemology of classical logic, intuitionistic mathematics is based on its own system of intuitionistic logic formalized by Heyting.14 In intuitionistic mathematics, intuitionistic logic defines what constitutes a valid proof, i.e. it defines its epistemology. Intuitionistic mathematics also gives hegemony to epistemology over ontology. While in classical logic truth (ontology) logically pre-exists, and is independent of, proof (epistemology), in intuitionistic mathematics proof (epistemology) logically pre-exists, and is independent of, truth (ontology). Here proof defines truth rather than simply resolving its status. In intuitionism, prior to a properly constructed verification of a sentence’s assertion, the sentence has no truth value ascribed to it. It is neither true nor false but simply of indeterminate truth status. The ontology of intuitionistic truth is termed ‘constructivist’ and truth is established through a process of ‘constructive verification’. It is strictly inductive in nature, rejecting some classical proofs derived from deductions based on prior ontological assumptions of binary [true; false] truth states. It is a pragmatic theory of truth that is based on the work of the philosophical pragmatists William James, Charles Peirce and John Dewey.12 Chaitin notes (with some irony) both the role Godel and computation have played in strengthening the arguments for such an inductive, constructivist epistemology.

‘...measuring the complexity (algorithmic information content) of m i o m s and showing that Godel incompleteness is natural and ubiquitous, deepens the arguments that forced Godel, in spite of himself, in spite of his deepest instincts about the nature of mathematics, to

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believe in inductive mathematics.

f53P.4

‘So in the end it wasn’t Godel, it wasn’t Turing, and it wasn’t my results that are making mathematics go in an experimental mathematics direction, in a quasi-empirical direction. The reason that mathematicians are changing their working habits is the computer. I think it’s an excellent joke! (It’s also funny that of the three old schools of mathematical philosophy, logicist, formalist, and intuitionist, the most neglected was Brouwer, who had a constructivist attitude years before the computer gave a tremendous impulse to constructivism.) ’16~P.26 There are both pragmatic and philosophical reasons not to subscribe to the law of the excluded middle. Many of the philosophical arguments are presented in the collective works of the mathematical intuitionists and presented in edited volumes on the philosophy of mathematics such as Benacerraf and Putnam [13] and Hart [17]. While these reasons are persuasive, they are not discussed here. Likewise, the arguments of the intuitionists for rejecting the proof mechanisms and class creation mechanisms of classical mathematics are not addressed. The relevant issue is, given what we know of the character of complex adaptive systems, and the means at our disposal to study such systems, whether the truth status of complex adaptive systems models should be based on constructive verification or be verification free. 2. Characteristics of CAS theory and practice There are two dominant characteristics of complex adaptive systems theory and practice which are relevant to this decision, namely our methods of studying them using computer experimentation, and their apparent non-conformity to the dominant scientific reductionist ontology of unidirectional causality. These issues are addressed in turn. 2.1. The epistemology of computer experimentation

Until fairly recently scientists could be catagorised as either experimentalists or theoreticians. Over the past thirty years or so a new type of scientist has emerged who does not fall easily into either of these two categories as he deals with neither real systems nor theory in the strictest sense. He is the computational scientist. The

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computational scientist uses the computer as his main apparatus to perform computer ‘experiments’ or computer simulations. This quote by Jain [18, p.l] introduces one of the central aspects of the epistemology of complex adaptive systems theory, that is, the increasing role played by computer simulation and experimentation as a mechanism for understanding certain classes of systems. Indeed Coveney and Highfield [3, p 2771 note that, in the CAS field of artificial life ‘The most important contemporary Alife system, Tierra, has an evolutionary dynamic of such great complexity that the only way to investigate its behaviour is to perform experiments on it within a computer.. .’. The relationship between complex adaptive systems theory and computer simulation is well expressed by Holland [19, p.951.

Because CAS are so intricate, computer-based models, with their well defined, manipulateable mechanisms, provide a crucial intermediate step in the search for CAS laws. Such models, where they mimic relevant CAS phenomena, put CAS data in a rigorous format, thereby facilitating the descriptions of patterns and laws. Another aspect central to the importance of simulation in the complex adaptive systems field is that simulation allows for the exploration of the behaviour of systems that could, for practical or ethical reasons, never be experimented on. This is particularly so for many ecological or biophysical systems for which experimentation is either impossible or would yield results which are not generalisable. Complex adaptive systems theorists argue that their models can illustrate the mechanisms driving the behaviour of whole classes of such systems, thus allowing for the development of ‘qualitative’ understandings of such s y ~ t e m s . ~ ~ ~ P . ~ ~ Chaos theory has shown that there are classes of systems that are governed by deterministic laws, but which are intrinsically unpredictable. This is the study of deterministic chaos, which has been well documented in the popular scientific press.20 The discovery of the existence of such systems raises profound epistemological questions. If we accept that there are classes of systems that are deterministic but the behaviour of which is intrinsically unpredictable. If we also accept that the identification of emergent properties of classes of deterministic systems such as fitness landscapes or cellular automata is likewise a priori impossible from an analysis of their component parts. Then the fields of chaos

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theory and complex adaptive systems theory to which these observations are central, must adopt an epistemology which accepts as ‘true’ those systems properties which can be constructively demonstrated to exist. That is, a property of a system exists where it can be constructively demonstrated to exist through an appropriately constituted computer experiment, and where its existence can be verified by reference and replication of that experiment. The statement ‘system X exhibits property Y’ is neither true nor false until it has been constructively demonstrated to be either true or false by an appropriately constituted computer experiment. Such a method of proof is termed ‘proof by constructive verification’ and is as has been discussed the hallmark of the intuitionistic philosophy of mathematics.

2.2. The ontological rejection of uni-directional reductionist causality The term reductionism is used in two distinct ways in the description of knowledge systems. The most widely understood use of the term reductionism describes systems of knowledge that only permit objects to be described using a restricted set of descriptors and relations. The arguments outlined above about the incompleteness of formal axiomatic systems shows that complex adaptive systems modeling is reductionist in this sense. A second and more formal definition of reductionism is a descriptor of the ontological position that all properties of a system observable at one hierarchical level are explicable through analysis of the system’s components at a lower hierarchical level. This is the reductionism of Newton and Descartes, that is as dominant an ontology of ‘conventional’ science as adherence to the law of the excluded middle is of ‘conventional’ mathematics. Conventional science assumes that things can always be completely explained by looking at their components. This is reductionist in the sense that the characteristics and behaviour of anything can be reduced to the characteristics and behaviour of its parts. Here causation is uni-directional and always operates from smaller to larger scales. Morowitz [lo] argues that an important aspect of the property of emergence in complex adaptive systems is that it is fundamentally antireductionist. In complex adaptive systems displaying emergent properties, the rules governing the behaviour of elements at one hierarchical level are frequently explained by (emergent) properties only observable at a higher hierarchical level. CAS epistemologists such as Morowitz [lo] and Rosen [2]

152 argue that this, while scientifically unconventional, is still a perfectly valid form of science. As Morowitz [lo, p.1961 points out

Under this mode of operation the successful theories may generate emergent properties not directly derivable from or perhaps even imagined from the simple constructs but b y running the program. The emergence is one of the desirable features of complexity theory. In complex adaptive systems it is argued that it is possible to explain the existence and behaviour of elements of a system in terms of the function that system itself serves with larger systems. Here causation can operate in both directions (smaller H larger) across all hierarchical levels of system scale. Subscription to the concept of bi-directional causality differentiates complex adaptive systems theory from the conventional scientific ontology of uni-directional causality. Consequently, to the extent to which reductionism is defined as subscription to the ontology of uni-directional causality, complex adaptive systems theory is non-reductionist. 3. Conclusion: Truth in complex adaptive systems models should be based on proof by constructive verification

We have seen from the arguments on the use of the computer as an experimental tool that the epistemology of complex adaptive systems accords with a system of proof by constructive verification. Further, if we accept that bi-directional causality across scales is a defining ontological feature of complex adaptive systems, then identification of emergent properties of classes of such systems is a priori impossible from an analysis of their component parts. It therefore follows that the existence of such properties can only be demonstrated through adoption of an epistemology which accepts as ‘true’ those system properties which can be constructively demonstrated to exist. That is, emergent properties of complex systems can only be said to exist if we can construct systems that display those emergent properties. Because, as Morowitz notes, such properties frequently cannot even be imagined, it is not possible to postulate their existence through the formulation of declarative sentences within the language of the formal systems of complex adaptive systems models. Because it is not possible to formulate appropriate declarative sentences which proclaim the existence of such emergent properties, it is not possible to ascribe truth value to them on the basis of their correspondence with ‘facts’ which, likewise, cannot be

153 imagined. Indeed it is even questionable as to whether such qualitatively different system properties operating at higher hierarchical levels can be expressed within the limited vocabulary of the formal systems of CAS models. This is because emergent properties are robust statistical aggregates of what are frequently transcomputational sets of the combinatorics of deterministic relations between elements of the system at lower hierarchical levels. For this reason, the detection of emergent properties through the analysis of individual deterministic relations is not possible. For these reasons, it seems unreasonable, to ascribe truth value to a priori unimaginable and qualitatively different properties of systems that are both structured and limited by our choice of formal axiomatic system. We therefore see that both the epistemological practice of CAS researchers, and one of the defining ontological assumptions of CAS theory, lie in opposition to the dominant deductive ‘realist’ ontology of classical mathematics based on an ontological acceptance of the law of the excluded middle. Rather, CAS ontology and epistemological practice lie in accord with the constructivist epistemology of intuitionism and other inductive logical frameworks. Thus, it is argued that the truth status of emergent properties of complex adaptive systems models should be based on strictly inductive logics and on proof by constructive verification. Accepting this leads to very real implications for building CAS models in practice. It becomes necessary to ask the question whether there are computer hardware or software environments that operate in a way that either uses, or entails, principles of mathematical logic based on adherence to the law of the excluded middle that the strictly constructive proofs of intuitionistic mathematicians would reject? The answer is yes. The most notable example of this is the use of ‘autoepistemic logics’ in multi-agent CAS models. Autoepistemic logics are a class of non-monotonic logics and were developed by Moore [21] and Przymusinski [22]. Autoepistemic logic introduces an additional syntactical element to those of standard propositional logic. This additional element is the modal operator 0 representing knowledge. This is used to indicate knowledge of the truth status of a declarative sentence (S) and is used as follows: US indicates that the truth status of S is known; 04indicates that the truth status of 4 is known; and TOS indicates that the truth status of S is not known. This operator is used, in conjunction with an implicit acceptance of the law of the excluded middle, to support reasoning based on knowledge of

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facts. This supports the reasoning that -4,i.e. S is assumed to be false unless it is known to be true. Thus autoepistemic logics, while not strictly classical, are nevertheless predicated on an acceptance of the law of the excluded middle and contain deductive elements unacceptable t o the strictly constructive proofs of intuitionists. Autoepistemic logics have been used for multi-agent modeling in Prolog and other languages. An example used in the field of organizational modeling is the SDML language developed by Moss and o t h e r ~ . ~T~h?e ~ * logic underpinning this agent based simulation environment enables deductive logic through adherence t o the law of the excluded middle and is thus unacceptable from a strictly inductive, intuitionistic perspective. It is argued therefore, that autoepistemic logics (or other non-monotonic logics based on the closed world assumption) are incompatible with the epistemological practice of CAS researchers and on bi-directional causality, one of the defining ontological assumptions of CAS theory. Emergent properties of CAS models based on such logics should therefore be regarded as philosophically invalid from a coherent complex systems perspective.

References 1. J. Casti, Reality Rules: 1 & 2-Picturing the world in mathematics‘Vol. 1-The Fundamentals’ & ‘Vol. 2-The Frontier’, New York, WileyInterscience, John Wiley & Sons (1992). 2. R. Rosen, ’What can we know?’, in Beyond Belief: Randomness, Prediction and Explanation in Science, eds J. Casti and A. Karlqvist, CRC Press, p.310 (1991). 3. P. Coveney and R. Highfield, h n t i e r s of Complexity: The search for order in a chaotic world, London, Faber and Faber (1995). 4. G. Chaitin, Information Randomness €9’ Incompleteness: Papers on algorithmic information theory, Znd edn, Series in computer science - Vol. 8, Singapore, World Scientific (1990). 5. J. Casti, ‘Chaos, Godel and Truth’, in Beyond Belief: Randomness, Prediction and Explanation i n Science, eds J. Casti and A. Karlqvist, CRC Press (1991). 6. J. Casti, ‘The outer limits: In search of the ‘unknowable’in science’ in Boundaries and Barriers: On the limits to Scientific Knowledge, eds J. Casti and A. Karlqvist, Reading Massachusetts, Addison-Wesley (1996). 7. G. Chaitin, ‘Thoughts on the Riemann Hypothesis.’ The Mathematical Intelligencer, Vo1.26, No.1, pp.4-7 (2004). 8. G. Chaitin, ‘Computers, Paradoxes, and the Foundations of Mathematics’, American Scientists, Vol. 90, pp.164-171, (2002). 9. G. Chaitin, Algorithmic Information Theory, Cambridge, Cambridge University Press (1987).

155 10. H. Morowitz, ‘Complexity and Epistemology’ in Boundaries and Barriers: On the limits to Scientific Knowledge, ed. J. Casti, p.188-199, Reading Massachusetts, Addison-Wesley (1996). 11. J. Traub, ‘On Reality and Models’, in eds J. Casti and A. Karlqvist, Boundaries and Barriers: On the limits to Scientific Knowledge, Reading Massachusetts, Addison-Wesley, p.238-255 (1996). 12. E. Snapper, ‘What do we do when we do mathematics?’, The Mathematical Zntelligencer, Vol. 10, No. 4, pp.53-58 (1988). 13. P. Benacerraf and H. Putnam, (eds) Philosophy of Mathematics: Selected Readings, Znd edn, Cambridge, Cambridge University Press (1983). 14. A. Heyting, Intuitionism, an introduction, 2nd edn. Amsterdam, North Holland (1966). 15. G. Chaitin, ‘Two philosophical applications of algorithmic information theory’, Springer Lecture Notes in Computer Science (Proceedings DMTCS’03), eds C. Calude, M. Dinneen and V. Vajnovszki, Springer, Vol. 2731, pp.1-10 (2003). 16. G. Chaitin, Randomness in arithmetic and the decline and fall of reductionism in pure mathematics, IBM Research Report No. 27, IBM (1992). 17. W. Hart, (ed) The Philosophy of Mathematics, Oxford, Oxford University Press ( 1996). 18. S. Jain, Monte Carlo Simulations of Disordered Systems, Singapore, World Scientific (1992). 19. J. Holland, Hidden Order: How adaptation builds complexity, Reading Massachusetts, Addison-Wesley Publishing company (1995). 20. J. Gleick, Chaos: Making of a new science, New York, Viking (1987). 21. R. Moore, ‘Semantical considerations on nonmonotonic logic.’ Artificial Zntelligence, Vol. 25, pp.75-94 (1985). 22. T. Przymusinski, ‘On the declarative semantics of deductive databases and logic programs’, in Foundations of deductive databases and logic programming, ed J. Minker, San Francisco, Morgan Kaufmann Publishers, pp.193-216 (1988). 23. S. Moss, H. Gaylard, et al. ‘SDML: A Multi-Agent Language for Organizational Modelling’ Computational €9 Mathematical Organization Theory, Vol. 4,NO. 1, pp 43-75 (1998). 24. 0. Teran, B. Edmonds, et al. ‘Mapping the Envelope of Social Simulation Trajectories’, Proceedings of The Second Workshop on Multi Agent Based Simulation, Boston, Springer Verlag (2000).

COMPLEXITY AS AN EPISTEMIC REVOLUTION: CONSIDERATIONS ON THE NEW SCIENCE IN THE CONTEXT OF WESTERN INTELLECTUAL HISTORY

DAMIAN POPOLO Durham University, UK The paper will seek t o present a genesis of complexity in European philosophical thought. Following the works of Gleick and Hobsbawn it is possible to discern the emergence of complexity’s ethos in the ‘age of revolutions’. Gleick, for example, has noted the influence of natural philosophy on the first precursors of chaos theory. Natural philosophy itself was deeply anchored in German romanticism. The paper will thus seek to present, following Foucault, an ‘Archaeology of Complexity’ which considers Foucault’s definition of epistemes as evolving modes of thought. In particular, the paper will seek to use Foucault’s description of the modern episteme in order t o detect the novelty inherent within the ethos of Complexity.

Cause and effect: such a duality probably never exists, - in truth we are confronted by a continuum out of which we isolate a couple of pieces, just as we perceive motion only as isolated points and then infer it without ever actually seeing it. The suddenness with which many effects stand out misleads us; actually, it is sudden only for us. A n intellect that could see cause and effect as a continuum and a flux and not, as we do, in terms of arbitrary division and dismemberment - would repudiate the concept of cause and effect and deny all conditionality. Friedrich Nietzsche, The Gay Science All is flux, nothing stays still. Heraclitus, from Diogenes Laertius, Lives of Eminent Philosophers Greek philosopher (540 B C - 480 BC) 1. Introduction There has been some debate recently surrounding the question of whether Complexity could be seen as a manifestation of ‘post-modern’ science’. Some scholars have sought to underline the similarities between Complexity science and post-structuralist p h i l o s ~ p h y ~ ySome ~ > ~ .others have perceived

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Complexity as being simply the manifestation of what Lakatos would call a ‘progressive paradigm’. In this view, the emergence of notions such as ‘deterministic chaos’ suggest that Complexity is nothing more than a more sophisticated tool for the understanding of patterns that just happen to appear as being completely random - but that remain patterns nonetheless a . Others have suggested that Complexity, in a number of fields, has changed many paradigmatic assumptions to such a point that it can now claim to be a legitimate scientific revolution in the Kunhian sense6. However, any attempt to contextualise and define Complexity should also have a historical backing for its arguments, otherwise, words such as ‘modernity’ and ‘post-modernity’ run the serious risk of being nothing but convenient terms deprived of any meaning. In this context, Dillon has reminded us of the Nietzschean belief according to which only things which do not have a history can be defined?. Complexity, as a set of ‘research principles’, does have a history - a history which enables us to capture its ethos rather than formulating its essentialist definition. This paper argues that Foucault’s enquiry into forms of knowledge ( episternesb) could and should provide a context to capture such ethos. The Foucauldian framework presented here has the advantages of giving a coherent context for the understanding of modernity’s ethos, without necessarily ‘defining’ it, while at the same time allowing us to consider as ‘post-modern’ (strictly in this Foucauldian sense) any construct of knowledge that does not adhere to modernity’s conceptualisation of t i m e , for modernity is conveniently described as the ‘age of (a precise form of linear) History’. According to Foucault, the modern emergence of time is related to the need of placing contingency upon the objects of enquiry’. Complexity Science - understood as a set of knowledge practices which, as in Prigogine’s caseg, postulate the central importance of time irreversibility - could be understood as an epistemic revolution. Moreover, questions surrounding the way in which notions of temporality should be handled in research and philosophical enquiry have been asked throughout the history of Western thought. Thus, Foucault - through a precise analysis of the modern episteme allows us to put some order in the debate by characterising Complexity not aByrne, for instance, considers Complexity exclusively for its potential in quantitative social science, and believes that the new science represents a serious challenge to ‘postmodernism’. See bFoucault describes an episteme as the ensemble of the underlying rules that characterise the general configuration of knowledge within a precise historical context8.

158 simply as that which looks for indeterminist, as opposed to deterministic, scientific laws; not simply as a field of knowledge that calls for interdisciplinary inquiry and recognises the non-linear character and the interconnectedness of all things; but as a set of practices which depart in their quest for knowledge on the ‘post-modern’ assumption that the arrow of time exists, and that time is irreversibly a constitutive element in the formative processes of things and not simply a convenient and contingent parameter. Foucault’s rigorous analysis of the modern episteme allows for a detailed ‘Archaeology of Complexity’ to take place. Through these epistemic lenses, this paper will seek to identify the genesis of Complexity in the history Western thought, before addressing the question of whether Complexity’s ethos is indeed surpassing the rules of the modern episteme. Additionally, this research will allow us to consider to what extent Complexity represents the reemergence of questions and ideas that have emerged in the European intellectual past, ideas which were precisely discarded by the triumph of the modern episteme. This paper will thus commence with a Foucauldian description of the modern episteme. This will give a firm ground to the concept of modernity, which will allow subsequent sections to deploy the word in a more rigorous manner. Second, the paper will make a short enquiry into modern Continental philosophical history in order to develop a genesis of Complexity. Finally, it will conclude with a few examples of how contemporary Complexity thinking is indeed re-deploying arguments present in Europe’s intellectual heritage which - seen through epistemic lenses - can indeed be considered as post-modern. 2. The Modern Episteme

Foucault defines the era of the modern episteme as the ‘age of History’* in the Classical age elements were classified according to their identities and differences (that is, their representations), that is by the spatial position they occupied in the ordered tables constituted by taximonia and mathesis‘; the collapse of representation obliges elements to be classified according to their proximity in the temporal succession of events. We assist therefore to the substitution of the principle of the Classical tables - Order, with the principle inherent in organic structures - History. As Foucault puts it: ‘History gives place to analogical organic structures [the P.217. While

CFora description of taximonia and mathesis consider chapter 3 in

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modern episteme], just as Order opened the way to successive identities and differences [the classical e p i ~ t e m e ] P.219. ’~ This sudden mutation is responsible for a double fragmentation of knowledge: not only modes of enquiry will differ according to how they deal with the concept of temporality, but emerging empirical ‘sciences’ or disciplines will be formed around the perceived ‘history’ of the elements that compose them. This second event leads to the birth of organic structures as fields of knowledge, fields that will be centred on historical transcendental notions. The Classical configuration of knowledge did not require a concept of temporality, as Gutting puts it: ‘...there was no important distinction between analytic (a priori) and synthetic (a posteriori) knowledge. An analysis of representations in terms of identities and differences was at the same time a connection (synthesis) of them in the ordered tables that express their essential reality’”. Nonetheless, with the decline of the Classical episteme, representation cannot be regarded as the sole foundation of knowledge. Thus occurs a crucial schism. Foucault explains that once the representational foundations of thought fade, analytic disciplines are founded on epistemological grounds that fundamentally differ from the basis upon which synthetic disciplines are based. The result is that ‘on the one hand we have a field of a priori sciences, pure formal sciences, deductive sciences based on logic and mathematics, and on the other hand we see the separate formation of a domain of a posteriori sciences, empirical sciences, which deploy the deductive forms only in fragments and in strictly localised r e g i ~ n s ’ ~ P . ~ ~ ~ . Philosophy, especially after Kant, is given the task to resolve the conflict between the emerging transcendentalisms and the world of representation. Representation, in this sense, is ‘essentially the relation between a subject and the object it thinks and experience^'^^*.^^^. Now this is how an embryonic conceptualisation of temporality is introduced within the general frameworks of knowledge. For this relation can be analysed either in terms of the conditions that precede and therefore ground the possibility of any representation or in terms of how the experiencing subject stands in relation to already represented objects (metaphysics). The first solution corresponds to the creation of a transcendental philosophy of the experiencing subject, while the second leads to a transcendental philosophy of the experienced objects (metaphysics). It is this metaphysical solution that characterises modern, pre-Complexity scientific epistemology. Both solutions respectively correlate to the appearance of analytic and synthetic sciences. Finally, both solutions equally assume that it is necessary to con-

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nect representations in terms of which our experience occurs to either a subject or an object that lies outside that experience but grounds its possibility. Philosophical criticism will therefore question representation from the a priori requirement of an experiencing subject, while metaphysics will seek to understand representations in terms of the experienced objects. In both cases, representation is not a causal source of explanation in itself. Foucault shows how the sciences of economics, biology and philology adopt a particular position within the modern organisation of knowledge. These fields clearly enter the category of synthetic sciences that seek to understand the world of representations through transcendental philosophies of objects (metaphysics). As Foucault put it, ‘Labour, life, and language appear as so many ‘transcendentals’which make possible the objective knowledge of living beings, of the laws production, and of the forms of language. In their being, they are outside knowledge, but by that very fact they are This corresponds to the point enunciated conditions of knowledge. . . above which explains that a transcendental philosophy of objects connects representations in terms of which our experience occurs to an object that lies outside our experience but grounds its possibility. Nonetheless, this philosophy necessitates a crucial analytic of finitude and a corresponding notion of linear temporality to operate. Let us take the analysis of the discipline of economics to illustrate the point. Foucault argues that in the Classical age only purely representational sources of value were commonly considered. The value of things was established through the notion of exchange: this was a purely representational system, for the value of an object was always represented in the value of another. Things did not really have a proper value as such. That is why in the Classical age economic ‘history’ did not exist: ‘value is always related to the system of exchange, in which any changes in value are always correlated with changes in other values. Since the system always involves a finite number of correlated values, changes can only be ~ y ~ l i ~ a l ’ ’ ~ ~ ~ ’ ~ Foucault sees Smith as the first step towards modernity. Smith introduced the idea that labour, and not exchange, was the source of value. Nonetheless, Smith considered that this value still necessitated a representational system of exchange to operate. Indeed, Smith’s thesis is that a thing can purchase another object that embodies a relative quantity of labour: the value of labour is determined by exchange. It is Ricardo who operates the decisive break from the Classical era. After Ricardo, labour is considered to be the sole source of value, regardless of how much this labour can represent in the system of exchange. F’rom now on, ‘a theory of

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production must always precede a theory of c i r c ~ l a t i o n ’ ~ pBut . ~ ~the ~ . transcendental notion of labour, the object through which the subject economic ‘man’ will be analysed, necessitates two fundamental notions of finitude. Firstly, the notion of anthropological finitude ensures that the subject ‘man’ is studied through the object labour. While the technocrats analysed how men stood in relation to representations, ‘modern economics deals with factors that causes men to form such representations - for example, with the bareness of nature that forces us to work, with the bodily deficiencies and the external threats that limit our ability to produce. . .modern economics .~~~. is ultimately based. . . on an ‘anthropology’ of human f i n i t ~ d e ’ l ~ p A problem, however, remains. How do we regard ‘man’ (or any other object of metaphysical enquiry) simply as a constituted subject, how can we stop this man from building alternative forms of production, how can we make sure that a transcendental philosophy of the object does not become a transcendental philosophy of the subject? This is precisely the problem that the realm of philosophy must try to resolve, and that is how ‘the study of man, precisely as a constituting subject, is the central concern of modern philosophical reflection.. . ’10P.199. What it is usually referred to in the social sciences as the ‘structure and agency debate’ and in the natural sciences as ‘paradigmatic assumptions’ are issues inherent within the modern episteme. These issues are about setting firm parameters of enquiry. Historical finitude is crucial for the placement of paradigmatic contingency upon the subjects of enquiry. Ricardo demonstrates that there is a historical precedent in all forms of production and capital accumulation that cannot be transcended. The culmination of this historical contingency is expressed in Marx’s maxim - men produce their own history, but not in the conditions of their choosing. Furthermore, precisely because historical finitude is meant to place contingency, the emerging notion of history must necessarily be unidirectional. That is how Foucault can argue that ‘we witness the birth of this grand linear and homogeneous series which is the process of production’ that introduces ‘the possibility of a continuous historical time’ which Ricardo, for instance, thinks ‘in terms of a persistently slower m i s of time, and eventually.. . a total suspension of history’ 8PP.253-63 (emphasis added). As Prigogine showed, time reversibility is what mainly characterised modern scientific enquiry. If a quote could summarise the analytic of finitude within the modern episteme it would have to be the following: ‘What is essential is that at the beginning of the nineteenth century a new arrangement of knowl-

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edge was constituted, which accommodated simultaneously the historicity of economics (in relations to the forms of production), the finitude of human existence (in relation to scarcity and labour), and fulfilment to an end to History.. .History, anthropology, and the suspension of development are all linked together in accordance with a figure that defines one of the major networks of nineteenth century thought.. . Finitude, with its truth, is posited in time; and time is therefore jinite’8pp.262-73 (emphasis added). The developments within the other two fields studied by Foucault, biology and philology, also follow similar patterns of linear historicism. Similarly, Prigogine’s description of modern physics reflects similar phenomena. If we apply Foucault’s definition of the modern organisation of knowledge, arguments such as Popper’s and Kuhn’s make us realise that what we often brand as ‘science’ is centred on presuppositions, or conjectures, that should be regarded as a transcendental philosophy of the object, for the subjects of scientific enquiry are usually examined by how they stand in relation to the basic presuppositions of the paradigm, and not vice-versa. In other words, the paradigm places contingency upon the object of research, and - as Prigogine argues - this contingency is often elaborated in terms of time reversibility. Kuhn demonstrates that those elements that do not comply with paradigmatic assumptions are often d i ~ c a r d e d ~ pThe . ~ ~study . of reality seems to follow from metaphysical assumptions. DuprC, in a book with a very pertinent title ( T h e Disorder of Things), seeks to illustrate the ‘dependence of modern science on metaphysics’ and argues that ‘science itself cannot progress without powerful assumptions about the world it is trying to investigate. . .without.. . a prior metaphysics’ll. While witnessing scientific enquiry in action, Latour also noted the power of paradigmatic assumptions12, both at the level of the analytic of finitude and historical linearity. Although the argument cannot be articulated in any length here, ‘scientific’ theoretical assumptions are not that different from the a posteriori transcendental notions that Foucault argues cause the necessary linearity of history.

3. Continental Philosophy and the Collapse of the Modern Episteme The limits of the modern episteme have however been challenged by philosophers who were keen to conceptualise temporality not as an element of contingency, but as the cause of perpetual and dynamic change. As stated above, Foucault explains that as the era of representation withers, paving

163 the way for the modern episteme, the realm of philosophy finds itself divided into three distinct areas of enquiry: ‘The criticism - positivism metaphysics triangle of the object was constitutive of European thought from the beginning of the nineteenth century to B e r g ~ o n ’ ~ PBergson? ,~~~. Why Bergson? Are we to understand that Henri Bergson was the first thinker to push Western philosophy beyond the modern organisation of knowledge? Henri Bergson is one of the most important figures within the spiritualist tradition, a tradition that advocated the presence of a distinctive philosophical experience. Bergson followed with a lot of interest the developments that were taking place within the philosophy of science. More precisely, PoincarC’s initial doubts concerning the infallible objectivity of science, and his disciple’s (Le Roy) reinforcement of these doubts, must have played an essential role in the constitution of Bergson’s thought. At the core of Bergson’s philosophy of science lies the conviction that the scientific method adopts a cinematographical view of temporality, which implies, as Gutting puts it, ‘that science views reality not as a continuous flux (the duration that in fact is) but as a series of instantaneous ‘snapshots’ extracted from this Science’s cinematographical view of duration is due to the fact that it is primarily concerned with action. As thought that is primarily concerned with practice, science must abstract from that concrete reality that we experience, in which temporality is not simply another form of space, but a ‘wholly qualitative multiplicity, an absolute heterogeneity of elements For Bergson, in the real continuum of which pass into the ~ther’~~J’.~’. duration there are no distinct elements that precede or follow real points in ‘time’. In this context, it becomes meaningless to speak of an a priori or an a posteriori: Bergson envisages a notion of temporality as a ‘continuous flux of novelty in which nothing is ever fixed, complete, or separate. In this flux, anything that we can say exists ‘now’ also incorporates into a qualitative whole everything we can say is ‘past’, a whole that is itself being incorporated into the new synthesis of the ‘future”13. The distinction between the synthetic and the analytic disappears in the flux of time, for it is precisely this continuous temporal vortex that r e p resents the formation of things (their synthesis) and their intrinsic (their essence - in Kantian, the ‘thing in itself’) reality. This is the main postulate of what has been referred to as Bergson’s ‘superior Empiricism’. And again, it is precisely this refusal to deal with transcendentalisms that characterises Bergson’s drive for an immanent reality that can be experienced

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while refusing to be cut into bits and abstracted. Clearly, the emergence of such ontology revolutionises the basis of the modern episteme. And it is precisely by demolishing the modern episteme ’s understanding of all possible paths and conditions for the attainment of knowledge that Bergson issues a challenge to Kant’s Copernican revolution in philosophy and in science: unlike the transcendental procedure of Kant, [Bergson’s philosophy] does not refer to the conditions of all possible experience; rather, it is moving toward ‘the articulations of the real’ in which conditions are neither general and abstract nor are they broader than the conditioned.. . Bergson insists upon the need to provide a genesis of the human intellect”14. Having established the fact that science fails to tackle the issue of real temporality, Bergson argues that philosophy might have been expected to occupy this empty ground. However, this was not to be. Modern philosophy, Bergson argues, has not challenged the view of time as ‘nothing more than a fourth spatial dimension, which could readily be viewed as having no creative efficacy, as merely the vehicle for the automatic unrolling of a nornologically determined s e q ~ e n c e ’ ~ ~ PThis . ~ ’ . modern scientific vision of time, as Prigogine maintains, is all but dead. Now we can start to understand how Bergson goes beyond the modern episteme. His conceptualisation of temporality refuses the a priori / a posteriori distinction upon which the modern organisation of knowledge is based. His refusal of transcendentalism, coupled with his insistence on the realm of immanence, has produced, amongst many, challenges for the modern notions of abstraction, temporality, empiricism, science and freedom. Bergson lies at the heart of Deleuzian philosophy14, whose conceptualisation of time is highly relevant to the ethos of Complexity, as Manuel De Landa has shown3. Prigogine makes of Bergson a cornerstone in his theorizing on Complexity and quantum physics. Another timely reminder of Complexity’s origins can be found in the works of Eric Hobsbawm. Hobsbawm rightly asserts that the principles of Complexity did not ‘appear’ but reappeared under the misleading tag of ‘chaos theory’. The reappearance of such ideas was possible thanks to the increasing calculating powers of computers. According to Hobsbawm the reemergence of ‘Complexity’ has profound implications for the concept of causality. Such an approach does have the potential to undermine several ‘political, economic and social’ assurnption~l~. Crucially, Hobsbawm identifies the genesis of Complexity in the truly European ‘epistemic civil war’, characterised by two contrasting interpretations of the role of rea-

165 son after Enlightenment. Such an account of contrasting ‘modernities’ is also present in Negri’s and Hardt’s Empire“. The following quote is of fundamental importance to critically assess the genesis of Complexity: Developments within ‘chaos theory’ in the 1970s and 1980s are not without commonalities with the emergence, in the beginning of the lgth century, of a ‘romantic scientific school’, of which Germany was the primary focus (Naturalphilosophie); that reacted to the dominant ‘classical’ science, essentially represented by practices in France and in Great Britain. It can be interestingly observed that two eminent pioneers of the new science (Feigenbaum and Libchaber) were in fact inspired by Goethe’s - radically anti-Newtonian - theory of colours, and also by Goethe’s treatise ‘On the Transformation of Plants’, which can be considered as a precursor of antiDarwinian and anti-evolutionist theories17. Hobsbawm retrieves this crucial information on Goethe’s fundamental influence on the two precursors of Complexity in Gleick’s seminal book18. The genesis of Complexity is therefore to be located in a moment of dislocation between two opposing visions of ‘scientific’ knowledge, ultimately based upon two different visions regarding the role of reason after the Enlightenment. What we are referring to in here, in the guise of Complexity, is the re-introduction of German romanticism in modern science. It is tempting to interpret recent, Complexity-related developments within the epistemology of science as the re-introduction of romanticism’s ethos, should we say, ‘through the back door’. As such, the future of Complexity should be seen in terms of the ‘longue duree’ as one of the possible paths to knowledge opened by the European intellectual heritage. It is in fact impossible not to consider the intrinsically German nature of these events. As Hobsbawn puts it: . . . it was in Germany, where classical liberalism was weak, that a rival scientific ideology to the classical was most popular. This was Natural Philosophy.. . it was speculative and intuitive. It sought expressions of the world spirit, or life, as the mysterious organic union of all things with each other, and good many other things which resisted precise quantitative measurement of Cartesian claritylg. More importantly, it was Natural Philosophy that provided the intellectual bedrock to ‘Goethian Physics’, which in turn considerably influenced the thought of thinkers such as Nietzsche, Husserl and Heideggerd. Crucially these thinkers put an alternative conceptualisation of time a t the dConsider, for example 2o

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centre of their attempt to demolish modern metaphysics (consider, for instance, Being in Time). Heidegger related the triumph of mechanistic, metaphysical epistemology to his contemporary political situation by correlating political events to ‘the Forgetfulness of Being’ and to technology: The wilful construction of a world empire to last four millennia shows [historical linearity]. . .a preference for quantity over quality that is alien to genuine creators like the Greeks. Empire-building stems not primarily form dictators and authoritarian states, but from the metaphysical essence of (emphasis added). This ‘verdict’ on Nazism was ‘delivered in the summer of 1941, when Hitler’s power was at its height’21. Similarly, when Europe was on the verge of the Holocaust, Husserl considered that it was appropriate to relate the socio-political environment of his Germany to the ‘Crisis of European Science’22. Nietzsche first individuated this fatal crisis. That great lover of Goethe’s Natural Philosophy, Nietzsche, is the kernel that unites French poststructuralisme and the intrinsically German experience of modern ‘rationalistic thought’. The crisis, as Antimo Negri argues in an important study of Nietzsche’s ideas on the implications of ‘classical’ science and the impact of such ideas on German philosophy as a whole23; is the belief that science’s ‘modern’ purpose is to provide metaphysical certainty. Drawing from this idea, Negri also examines how Heidegger argues that “the fundamental nature of contemporary human existence, which seeks for certainty and security everywhere, is being consolidated under the powers of the convictions professed by the principles of reason”, and ironically, Heidegger states that “Leibniz, who discovered the principles of sufficient reason, is also the inventor of life insurance”. But what kind of certainty can ‘open-ended’ scientific research, always marching towards the - ultimately unreachable - goal of rediscovering the why and how of all things; bring to human existence? Heidegger fears that such impossible research may end up ‘exhausting man’. Here, we are talking about a fear that can only be dissipated if men, including men of science, do not suffer from the exhaustion of starting always again from the beginning. It can only be dissipated if man has the courage to start again there where he though he had finished: Husserl maintains that the future of European culture depends on this??23p.26 (my translation). eThe influence of Nietzsche on Foucault, Derrida and Deleuze is undeniable. See

l3

167 The quest for certainty and securityf is ultimately the biggest danger facing European civilisation. It would be clear, following Foucault’s analysis of the modern episteme, that transcendental philosophies of the object - metaphysics - necessitate two crucial elements in order to function: a concept of analytic finitude and a concept of linear temporality. Heidegger’s conviction according to which the metaphysical nature of his time is what forced a strict philosophy of the limit and of historical linearity to take place is entirely compatible with Foucault’s description of the modern episteme. Metaphysics is necessarily a philosophy of the limit - the contingency which in scientific epistemology translates into paradigmatic thought. The necessary securing of limits is entirely encapsulated in Leibniz’s thought: ‘if one is to establish the elements of human knowledge some fixed point is required, on which we can safely rest and from which we can set out without fear24P.13.It is with this quote in mind that Nietzsche exhorted men of science to build their house on the Vesuvius - only through the acceptance of temporality as the principal creative and destructive force in nature will modern human beings ‘always start again from the beginning’, without fear.

4. Complexity and the Modern Episteme There are at least two ways in which the scientific developments elaborated by Prigogine, a flagship of the loosely defined ‘Complexity Science’, can be broadly understood within a Bergsonian framework. Firstly, Prigogine quickly acknowledges the cinematographical nature of pre-complexity science and secondly, he indicates how this precise view is caused by the notion of linear. Prigogine articulates the idea that contemporary science uses a static notion of time principally by making reference to Einstein and Hawking. Einstein famously declared that ‘time is an illusion’. More recently, “in his Brief History of Time, Hawking introduces ‘imaginary time’ to eliminate any distinction between space and time”gp.58. Thus Classical and Modern physics have maintained a spatialised vision of time that is basically static, and, even today, scientists assume as a ‘matter of faith that as far as the fundamental description of nature is concerned, there is no arrow of time’g. Prigogine argues that ‘Nature involves both time-reversible and time‘For a comprehensive argument, see

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irreversible p r o c e s ~ e s ’ ~ pReversible .~~. processes deny temporality as a constitutive apparatus of the process. Examples of these processes can be found in Newton’s formulation of classical physics and in Schrodinger’s basic equation of quantum mechanics. In both cases, equations are invariant with respect to time inversion. Contrarily, time irreversible processes break time symmetry. In these processes temporality does affect how the general rules of motion will impact the system in a precise temporal context. More importantly, time irreversibility produces entropy. An example of time irreversible processes is the second law of thermodynamics. However, Prigogine argues that time reversibility is produced firstly because we accept to reduce the analysis to an elementary level, and secondly because we abstract: ‘Reversible processes correspond to idealizations: We have to ignore friction to make the pendulum work r e ~ e r s i b l y ’ ~In. Foucault’s words, these metaphysical transcendentalisms adopt reductionism because they ‘must deploy the deductive forms only in fragments and in strictly localised regions’. Once Prigogine dismisses the idea that entropy might be caused by insufficient data or faulty examination, the ideas that follow from his arguments suggest that, if we bear Foucault in mind, time reversibility is a particular cause of a transcendental philosophy of objects, that is, a metaphysical system that ignores elements which do not happen to coexist with the basic premises of a paradigm. This causes the need to discard incompatible elements (precisely such as the second law) on the grounds of humanity’s imperfect observation capacities or on the inadequacy of its instruments (in other words, limits). However, according to Prigogine: ‘The results presented thus far show that the attempts to trivialize thermodynamics.. .are necessarily doomed to failure. T h e arrow of t i m e plays a n essential role in the f o r m a t i o n of structures in both the physical sciences and b a ~ l o g y ’ ~ *(emphasis .~~ added). Time irreversibility becomes undeniable once, on the one had, we adopt a more i m m a n e n t approach to nature, and on the other, we look at populations and not at single elements that compose them. The apparently contradictory pulls towards immanence and connectivity are resolved through the continuum of time and this notion of empiricismg. Prigogine is happy to demonstrate that the results of his own research concord with Bergson’s conceptualisation of temporality: I’m certainly not the first one to have felt that the spatialization of time g For

a more detailed account of Prigogine’s empiricism and its significance for Complexity’s ethos see 25

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is incompatible with both the evolving universe, which we observe around us, and our own human experience. This was the starting point for the French philosopher Henri Bergson, for whom ‘time is invention or nothing at all’. . . I mentioned one of Bergson’s later articles, “The Possible and the Real”. . . where he expressed his feeling that human existence consists of ‘the continual creation of unpredictable novelty’, concluding that time proves that there is indetermination in n a t ~ r e ’ ~ P . ~ ~ . Contemporary cognitive research is equally finding an inspiration in the thought of Bergson. For example, the dynamical hypothesis of experience elaborated by Robbins relies heavily on Bergson’s concept of continuous time. As Robbins explains, ‘the dynamical hypothesis can be placed within the framework created with the theories of Henri Bergson.. . From this emerges a basis for supporting experience, semantics and the fundamental intuition behind representations - representative thought’26. However Robbins goes beyond Bergson, his work has approached Heidegger’s theories of being in time and related these insights to the concept of emergence inherent in Complexity scienceh. Robbins takes note of Heidegger’s thought, which contributed in questioning ‘the distinction between a conscious, reflective, knowing ‘subject’ and a separate ‘ o b j e ~ t ” ~ ~ p . ~ ~ ~ Heidegger does indeed see ‘representations as a derivative phenomenon, which occurs only when there is a breaking down of concernful action. Knowledge lies in the being that situates us in the world, not in reflective representation’26. Moreover, it is equally important to note that Heidegger was inspired by Bergson, and to realise that the essential insight on the role of time is what links these authors and turns the modern episteme on its head: Heidegger was certainly aware of Bergson. Cassirer was straightforward, noting, “it is the lasting achievement of the Bergsonian metaphysic that it reversed the ontological relation assumed between being and time”. The relationship of subject and object in terms of time constitutes the fundamental framework within which ‘situatedness’ truly lies.26Pp.320 In conclusion, Robbins locates the struggle between different theories of experience in a confusion over real and abstract space and abstract time, he can thus affirm that ‘Bergson’s theory.. . can explain the reconstruction of the image of a past experience via an event occurring in the environment; it can at least logically support the phenomenal experience of the memory image26P.337’,and can finally conclude by stating that Bergson hSee references to Gleick in

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170 offers a viable theory of conscious perception, a basis without which any theory of cognition and memory has hitherto only been tentative. It truly acknowledges time. In fact, with respect to the problems of a scale of time, it raised questions 100 years ago that still have not dawned upon representationalism. It relies, integrally, upon dynamics, yet makes clear why dynamics must enlarge its vision of time before it can support a theory of ,ind.26p.333 Thus, he who Foucault describes as the end point of the modern episteme, precisely by virtue of his opposition to the two axis of modernity namely a varying analytic of finitude and linear historicism - has become an inspiration for truly ‘post-modern’ research (in the Foucauldian sense) at the cutting edge of contemporary physics and cognitive science

5. Conclusion

This paper sought to present Foucault’s ‘Archaeology’as an adequate conceptual means for charting the origins of Complexity. Following Foucault, the paper has 1) presented the emergence of linear temporality as an essential feature of the modern episteme, specifically related to birth of transcendental philosophies of the object (modern metaphysics); 2) indicated how figures present in the European philosophical tradition challenged the basic understanding of time inherent in the modern episteme, making special references to Bergson and Heidegger and 3) outlined how these figures, precisely through their alternative conceptualisation of time, are becoming increasingly influential in Complexity-related research. There is no doubt that the question of how time should be conceptualised is becoming increasingly topical in contemporary scientific research. Moreover, such questions directly tackle the issue of how such conceptualisation of time as an irreversible feature of dynamical (as opposed to static) processes should be understood. It appears that an accurate understanding of change - more precisely irreversible change - has become a primary objective in many research agendas. The acceptance of such irreversibility (or, as Prigogine puts it, the ‘End of Certainty’) radically undermines the nature of metaphysical knowledge, and clearly constitutes one of the pillars of Complexity Science. The message inherent to Complexity’s ethos is beautifully encap sulated in the simple concluding remark of Etienne Klein’s exploration of the philosophy of time in contemporary physics: ‘We must learn to love irre~ersibility’~~.

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References 1. G. MorCoI, ”What Is Complexity Science? Postmodernist or Postpositivist?,” Emergence 3, no. 1 (2002). 2. P. Cilliers, Complexity and Postmodernism: Understanding Complex Systems (London ; New York: Routledge, 1998). 3. M. De Landa, Intensive Science and Virtual Philosophy, Transversals. (London ; New York: Continuum, 2002) 4. W. Rasch and C. Wolfe, eds., Observing Complexity: Systems Theory and the Postmodern Episteme (Minneapolis: University of Minnesota press, 2000). 5. D. S. Byme, Complexity Theory and the Social Sciences : A n Introduction (London: Routledge, 1998). 6. Thomas Kuhn, The Structure of Scientific Revolutions (Chicago: University of Chicago Press, 1970). 7. M. Dillon, ” Poststructuralism, Complexity and Poetics,” Theory, Culture €4 Society 17, no. 5 (2000). 8. Michel Foucault, The Order of Things: An Archaeology of the Human Sciences, World of Man (London: Routledge, 1989). 9. Ilya Prigogine and Isabelle Stengers, The End of Certainty: Time, Chaos, and the New Laws of Nature, 1st Free Press ed. (New York: Free Press, 1997). 10. Gary Gutting, Michel Foucault’s Archaeology of Scientific Reason, Modern European Philosophy. (Cambridge England ; New York: Cambridge University Press, 1989) 184. 11. John Dupr6, The Disorder of Things: Metaphysical Foundations of the Disunity of Science (London: Harvard University Press, 1995) 1-2. 12. B. Latour, Science i n Action : How to Follow Scientists and Engineers through Society (Milton Keynes: Open University Press, 1987). 13. Gary Gutting, French Philosophy in the Twentieth Century (Ca,mbridge, U.K. ; New York: Cambridge University Press, 2001) 51. 14. K. A. Pearson, Philosophy and the Adventure of the Virtual. Bergson and the Time of Life (London: Routledge, 2002) 11-13. 15. E. J . Hobsbawm, Age of Extremes : The Short Twentieth Century, 1914-1991 (London: Abacus, 1995). 16. Michael Hardt and Antonio Negri, Empire (Cambridge, Mass. ; London: Harvard University Press, 2000). 17. Hobsbawm, E. L ’age des extremes (Bruxelles: Editions Complexes, 1999), 756 - my translation. 18. J. Gleick, Chaos (London: Vintage, 1998). 19. E. J . Hobsbawm, The Age of Revolution : Europe, 1789-1848 (London: Abacus, 1997) 335. 20. Harold Bloom, The Anxiety of Influence : A Theory of Poetry, 2nd ed. (New York; Oxford: Oxford University Press, 1997) 50. 21. M. Inwood, Heidegger: A Very Short Introduction (Oxford: Oxford University Press, 2000) 6. 22. Edmund Husserl, The Crisis of European Sciences and Transcendental Phenomenology; an Introduction to Phenomenological Philosophy, Northwestern

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23. 24. 25. 26. 27.

University Studies i n Phenomenology B Existential Philosophy (Evanston,: Northwestern University Press, 1970). Antimo Negri, Nietzsche: La Scienza Sul Vesuvio, 1. ed. (Roma: Laterza, 1994). Michael Dillon, Politics of Security : Towards a Political Philosophy of Continental Thought (London ; New York: Routledge, 1996). D. Popolo, "French Philosophy, Complexity, and Scientific Epistemology: Moving Beyond the Modern "Episteme"," Emergence 5, no. 1 (2003). S. Robbins, "Semantics Experience and Time," Cognitive Systems Research 3 (2002): 304. Klein, E. Les Tactiques d e Chronos, Champs : Flammirion 2004.

METAPHORS AND METHOD: EPISTEMOLOGICAL CONSIDERATIONS ON COMPLEXITY SCIENCE

RODRIGO M. Z E I D A N ~AND MARIAG . DERENGOWSKI F O N S E C A ~ ‘Escola de GestGo e Negdcios , UNIGRANRIO Rua da Lapa, 86, Centro, Rio de Janeiro-RJ, CEP: 20021-180 Email: [email protected] Instituto de Economia, Universidade Federal do Rio de Janeiro Av. Pasteur 250, Urca, 22290-240, Rio de Janeiro-RJ Email: [email protected] Any scientific development entails many epistemological considerations. In this paper we consider two stages of scientific development - the metaphorical and empirical levels. When applied to transposition of complexity science t o other disciplines, in special economics, we conclude that complexity is still, in many aspects, at the metaphorical level. This implies that transposed concepts should be scrutinized to verify its validity towards the development of complexity science.

1. Introduction

A presentation at a complexity conference sometime in the early 2000’s: its main purpose is to relate the philosophical implications of the fractal gravitational singularity concept. Fractal gravitational singularity? What does it mean? A gravitational singularity is a precisely defined concept (such as a point of infinite space-time curvature) and a fractal is also a clearly defined concept (a fractal is a geometric object which is rough or irregular on all scales of length)”. However a fractal singularity makes no sense from a scientific point of view, not even as a metaphor. This kind of misinterpretation has been the subject of popular science debate, and it shows how easily the scientific method can be ignored when multidisciplinary conceptual transpositionb occur. The question is then: aMost concepts relating t o complexity theory are rigorously defined in physics or biology bTransposition is here defined as a metaphor for the process of spreading scientific concepts and models from one discipline into another. Carruthers’ calls the recipient of transposing concepts emerging sciences, but this may cause confusion because of the diffusion of the term emergence.

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174 does complexity theory has a sound epistemology? Although the answer should be trivial, when transposing complexity theory to other areas such as economics, or sociology, there are many epistemological implications regarding the usage of the scientific method. Carruthers' summarizes the nature of modern science in three main questions, shown below.

Table 1. The nature of modern science - discussion points2. How is science done? Dynamics of scientific Psychology of scienideas tific creativity. By whom? Consensus as the ref- Science as truth erence frame Where? Advocacy Science as process Who pays? The role of experi- Peer opinion ment Disciplinary The yearning for The 'anxiety of influstructure truth ence' Decision making The emergence of The discipline of dogma quantitative science Science policy New ideas and their Synthesis-its hazards fate The social context The role of personal Big science as the contact emergence of an executive class Big science/little sci- The role of tribalism The institutional conence I I text The tendency of spe- Pressure to conforcialization mity The yearning for gen- Is anybody listening? erality and synthesis The powerful influ- The significance of ence of aesthetics heroes Competition

This paper is related to the dynamics of scientific ideas and is an extension from Zeidan and Fonseca2, where it was argued that, regarding the transposition of complexity into economics, nothing prevents complexity

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studies from trying to incorporate the orthodox economic theory language while developing new models and insights into economics processes. Again economics is used as a case study for more general analysis, in this case as a recipient of transposing concepts from original complex adaptative disciplines such as biology. What we propose here is to analyze the two epistemological levels of scientific evolution of multidisciplinar complexity theory, the first one a metaphorical level and the second an empirical level, both related to the dynamics of scientific ideas and the scientific method.

2. The Metaphorical Level Complexity as a metaphor represents many powerful insights for many disciplines. Carrut hers' groups emergence disciplines as computation, evolution, cognition, cosmology, particle physics and mathematics, biology and the new medicine, economics, political systems, and cultural dynamics. Those areas would be where complexity would thrive. For instance, that is true of economic processes, where the idea of complex adaptative systems to economic interactions is a very persuasive one [e.g. Refs. 3, 4,51. Moreover, there are many studies where biological agent models are transposed to economic agent models [e.g. Refs. 6, 71. Complex agentbased dynamics models the world's behavior through the use of "agents". In this context, an agent is a computer program that acts according to a set of rules, e.g., in a self-interested manner as it deals with numerous other agents inside a specific economic environment - this arrangement can be set-up to mimic almost any interactive system, including consumer theory. These developments are then the result of the fact that economics systems can be analyzed as complex adaptative systems, since there are many non-linear dynamics, feedback processes, hierarchies etc. More important, economics systems are agent-based, and thus regular complex adaptative systems originating in biology are easily transposed into economics complex adaptative systems. The metaphorical level is then the level where ideas and concepts from complex adaptative systems cross over from discipline to discipline. This leads to many new insights, and incorporating complexity concepts leads to many adaptations to fit these concepts into the specific context of the discipline. It seems that there is where the fractal singularity concept emerged, as a new concept for transposing ideas into another area. However, this is a crucial part of the scientific development of crossing-over models, since this

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is where it runs into scientific terminology being used without the slightest knowledge of its real meanings. The metaphorical conceptual nonsense created in transposing complexity into other disciplines has a powerful justification in Kuhn's work - much cited in philosophical complexity works [e.g. Refs. 9, 10, and probably a good percentage of philosophical complexity papers presented at conferences around the world]. The biggest problem with Kuhn's view is that, up to the point of the paradigm shift, proponents of the older and newer theories cannot be speaking the same scientific language, and, in fact, alienate each other from their respective views of the world. This problem permeates complexity studies in economics, where most economists cannot begin to cope with the new theory because of its inherent incommensurability2. However, Kuhn's paradigm shift and incommensurability in no way implies an anything goes relativistic view of the dynamics of science. New paradigmatic science still would have to have consensus as the reference frame, and internal coherence. This would be derived from new concepts and ideas, and new meanings of old ones - but those new meanings would still have to make sense inside a reference frame. What is happening is that to deconstruct what is now referred as the old Cartesian world-viewc internal coherence in many philosophical complexity works have been lost, even in the metaphorical level. The metaphorical level is inherently theoretical and can lead to endemic insights in the process of transposition. For instance, Goldsteing proposes a self-transcending construction of emergence new formalism that is a very powerful insight and is born out of philosophical theory, even if relating some discussion to biology theoretical work. The main problem with the metaphorical level in the process of the dynamics of complexity theory is that its lack of rigor can lead to the hubbubs and nonsensical notions. Moreover, many implication are derived from it without any semblance of a rigorous demonstration - for instance, complexity was used as tool for raising ethical Issues Concerning the Use of Animals in Scientific Experimentation". The metaphorical level is where many epistemological problems can arise. The fact that it is a speculative stage on the dynamics of scientific nature means that many propositions, implications and conjectures

=This view of cartesianism as anathema to complexity is permeating many works in complexity philosophy, even if proper definition of cartesianism seems to be lost to. It seems that cartesianism and newtonian science means only non-complexity science.

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will be raised when transposing concepts and notions to other disciplines. By the very loose nature of this speculation many nonsensical ideas will Aoat around, without the necessary justification and demonstration of its importance. Moreover, without rigorous screening many implications will leak through the next stage, and thus will slow the dynamics of the scientific development of complexity.

3. Empirical Level

- The Method

This is the level where applications of the theory occur. Not only application in terms of objective results, but developments posterior to the consolidation of metaphorical concepts. It involves the yearning for generality and synthesis, and the development, in natural sciences, of quantitative testable results. In social sciences like economics, it involves models that lead to policy prescription and/or better comprehension of economic phenomena. Supposedly at this level the transposition would have implemented consensus as the reference frame and new dogma would emerge. However, at this stage in complexity theory there is still search for the new dogma, although many concepts have spread out as reference frame. But to this day the lack of the institutional context permeates many transpositions of complexity into social science disciplines, and hence Zeidan and Fonseca2 showed how this lack of institutional context leads to developments that disregard almost completely the present orthodoxy in economics. The allure of the transposition process is the fact that in the originating disciplines, like biology, complexity has yielded consistent results, to the point of a paradigmatic shift in evolutionary theory, for instance. The role of experiment is then much more important, and formalization through models has to lead to good empirical results. At this point in the complexity social sciences, however, empirical research has not yielded good results, and thus complexity is stuck in a metaphorical state, when regarding many discipline transposition by this point. To take an example, in consumer theory, the similarities between the agent-based model7 and regular orthodox consumer theory is presented in table 22. The table represents the metaphorical transposition of an agent-based model to answer the same questions as orthodox consumer theory. On the empirical level, however, agent-based models are resolved through simulations. In current state, however, simulations have not yielded consistent results, and thus agent-based models can not offer better results than regular orthodox theory - yet. Santa Fe instate, for instance, is most known

178 Table 2. Metaphor and Method in Complexity Economics. Heterodox Theorv’ Orthodox Theory Two bundles of goods (A and B) Individual agents can be repreare related through preference: sented by a string such as: A = ~ 1 x 2...~xn. 3 x is either 0,1, or * A(B, AiB, or A B (indifferent) (indifferent) For any group of bundles an in- From a number of strings a schemata is developed. difference curve is derived. For any indifference curve there is For any schema H there is a fita utility function f(x) associated ness function f(x) associated with it. with it. There is a probability p associ- There is a probability p associated with the selection of a bun- ated with the selection of a string. dle. The selection is done through The selection is done through satisficing. maximization. ~

for using computer simulations to study complex systems, including environmental, economics, and social systems. Some results are encouraging, but most are still tentative, with no replacement of orthodox theory - again yet. 4. The Role of Metaphor When dividing the epistemology of science in two dividing lines, it should clear the role of each line and its connection to the scietific development process. While in the empirical level results can be adequately verified, and thus it is where science has its best screening process, the metaphorical level is mainly speculative and is not verifiable per si, implying a non-screening process and a loose part of the scientific development process. The role of metaphor resides in the insights and theoretical inquiries that lead to better theories. It was in metaphor that complexity science reached a multitude of scientific areas - in economics, for instance, the idea of complex adaptative systems is very attractive to deal with economic processes. The main problem is translating this inquiries into models and methods for working out sustainable results. Although there is no adequate test for metaphors as in models, the role of metaphors can be analyzed by reverting to its basic core language. In this respect Kuhnian’s ideas of in-

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commensurability are an important part of how the evolution of scientific knowledge takes place in the metaphor stage. In that respect, metaphor occurs as part of the paradigm shift that leads to new emerging theories. In that process the language is fluid to the extent that concepts are not interchangeable between competing theories. This can lead to a relativistic point of view where fractal singularities are acceptable terms for new theories. This interpretation of Kuhn’s point of view is very pervasive in social sciences circles dealing with complexity. Kuhn is often cited as the basis for new concepts as fractal singularities. However, Kuhn’a analysis is not a relativistic analysis and it leads to scientific evolution. Extreme views of Kuhn’s incommensurability concept seems as wrong as fractal singularities. In this way the role of metaphor has to strike a balance between novelty concept and transpositions and scientific background that will eventually lead to model. How to strike this balance depends on the scientific community, and can lead to better thory or even to the burial of valid theories. But the main point is that metaphors should be constrained by its commensurability, not with the theories that it is trying to replace, but with itself. Metaphor should have internal consistency, and it seems that in much of transposition of complexity into other scientific are= this internal consistency is being lost.

5. Final Comments Separating the two levels of the scientific development of complexity can be most useful to verify how transposition of complexity into other disciplines is evolving. Moreover, it leads to better comprehension of how concepts and ideas are being incorporated into other disciplines, with better chance to envision the hubbubs and nonsense being created in the process. Instead of a pure Kuhnian approach, where incommensurability is the norm, by having a metaphorical and an empirical level, some commensurability is possible, especially when regarding inter-disciplinary commensurability - i.e. ensuring that transposed concepts and ideas are compatible with its origin, and also make sense with the recipient discipline, and, more importantly, internal commensurability. Not doing so leads to fractal singularities and even more outrageous concepts.

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References 1. Carruthers, P. (1988). Emerging syntheses in modern science. European Jour-

nal of Physics, 9: 110-116. 2. Zeidan, R. M., Fonseca, M. G. D. (2004) Epistemological considerations on agent-based models in evolutionary consumer choice theory. Emergence: Complexity and Organization. Boston, MA: , v.6, n.3, p.32 - 39. 3. Arthur, W. B. (1995) Complexity in Economic and Financial Markets, Complexity, v.1, n.1, April. 4. Anderson P.W., Arrow K.J., Pines D. eds. (1988) The economy as an evolving complex system; Addison-Wesley Pub.Co, Reading Ma. 5. Cohendet P., Llerena P, Stahn H., Umbauer G. eds. (1998) The Economics of Networks, Interactions and Behaviours Springer, Berlin. 6. Holland, J. H. & Miller J. H. (1991) Artificial Adaptive Agents in Economic Theory, American Economic Review, May, pp. 365-370 7. Goldberg, D. E. (1989) Genetic Algorithms: in Search Optimization and Machine Learning, Reading, MA, Addison-Wesley. 8. Sokal, A. & Briemont, J. (1998) Intellectual Impostors. London: Profile Books. 9. Goldstein, J. (2005). Impetus without Drive or Teleology: The Selftranscending Construction of Emergence. paper presented at the Complexity, Science and Society Conference. University of Liverpool. 10. Waldrop, M. Mitchell. (1992) Complexity: The Emerging Science at the Edge of Order and Chaos. New York: Simon & Schuster. 11. Robinson, F. (2005) The Relevance of Complexity Science to the Ethical Issues Concerning the Use of Animals in Scientific Experimentation: the Matter of Constraints. Paper presented at the CSS Conference - Liverpool.

SOME PROBLEMS FOR AN ONTOLOGY OF COMPLEXITY

MICHAEL McGUIRE Department of Applied Social Sciences London Metropolitan University

Ontology, like the broad philosophical tradition of metaphysics of which it is part, stands ‘beyond physics’ since its propositions are not exclusively subject to empirical refutations. But neither is each discipline fully autonomous from the other. Where possible, ontology ought to avoid direct contradiction with physical evidence. Similarly, many fundamental assumptions and justifications within science have a basis that cannot be established by the experimental method. These notoriously include standard principles of scientific methodology such as the faith in inductive reasoning, indispensability arguments, the value of simplicity, and of course the range of entities to which is ontologically committed. Whilst it is contestable as to whether the range of mathematical techniques and physical observations that are associated with ‘complexity theory’ (henceforth ‘CT’) really form a distinct branch of science in the way say, that quantum mechanics does, they must certainly aspire to the status of scientific theories. And, whether one is willing to engage with this fact or not, such aspirations entail a set of underlying ontological commitments. Complexity theorists can of course proceed to work in an instrumental fashion, without bothering to interrogate their foundational ontological assumptions. But if such assumptions turn out to be incoherent, then C T itself may ultimately inherit those problems. In this paper I intend to press this question and to indicate some potential problems for CT when it is wedded to certain traditional ontologies - in particular those involving the substance/particular and universal/property dichotomies. A highly plausible alternative - an ontology of patterns a also has its own set of problems. If the difficulties of this more radical ontology are also insuperable then one possible conclusion about the apparent lack of theoretical unity to CT is precisely because it rests upon aFor which I have pressed the case in my [l,21

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suspect foundations. In what follows then I will outline some reasons for thinking that standard ontologies cannot meet the muster, before moving on to outline a few of the questions for the alternative that I advocate.

1. Ontology Two (broad) senses of ontology are operative in contemporary analytic philosophy. (i) ‘Classical’or philosophical ontology - what Aristotle called the “science of being qua being’’.b The approach deals with questions about objects, existence, properties] relations, parts, boundaries] measures, causality, possibility and so on. The development of formal techniques in ontology has increasingly enabled such questions to be addressed in more rigorous terms rather than by what D.C. Williams termed ‘speculative cosmology’ [3] (ii) ‘Technical’ Ontology. An approach that has roots in formal ontology but is mostly applicable to developments in computer science, particularly artificial intelligence (AI) research. Here the focus is less upon real world objects, but upon ways of specifying what is required to exist for some conceptualisation to function correctly. In A1 for example knowledge based systems need to be able to represent the range of assumptions they deploy to operate. Specifying an ontology will help the programmer attain this as well providing a shared vocabulary that can operate across other systems in a coherent and consistent manner‘ In this paper it is the former approach that I will draw upon for I intend to ask what is the most general kind of thing that theories of complexity could take to act as the ontic vehicle for the phenomena they purport to identify. Given the discontinuities in content between such theories this will be, by necessity, an exercise in what Strawson calls ‘descriptive’rather than ‘revisionary’ metaphysics [4]. But if the ontological commitments of CT could be given some satisfactory metaphysical grounding, then the radical nature of some of its propositions may in turn suggest that serious revision to the traditional cast of metaphysical candidates ontologists are familiar with may be necessary. bAristotle uses this term in Metaphysics (Books, G, E & K ) =See Tom Gruber’s useful discussion of this kind of ontology at http://wwwksl.stanford.edu/kst/what-is-an-ontology. html

183 2. What should ontology do for complexity theory?

Whilst it cannot directly arbitrate upon empirical questions, a coherent ontology ought to offer a variety of clarifications and conceptual syntheses for CT. Insights into causal relations in complex processes; specifications of the conditions for ‘self-organisation’; a better understanding of part-whole relations, in particular the thorny question of emergence, and so on. Above all, it should offer a taxonomy of the different classes of entities posited by C T for its laws and generalisations to function consistently and effectively. As a branch of metaphysics, ontology is permitted to organise the natural kinds of science into broader more general classes in order to discover otherwise unspecifiable relations between them. Indeed, it can ask not just how many different kinds of entity there are, it can also put this question foundationally - how many basic kinds of entities there are.d Since philosophical predilections, like those in science, are for simplicity it is usually held to be desirable to be sparse with the number of such basic kinds, and to permit only those kinds thought to be explanatorily necessary any kind of status. Of course natural science (or at least that branch practised by physicists) has been marked from its inception by the prejudice that just one set of fundamental ontological kinds - those located at the atomic level - suffice to support its propositions. There has never been a satisfactory rationale provided for this prejudice and with the discovery of subatomic classes of entity, it is one that has had to be seriously qualified. Whilst reductionism remains the unstated prejudice for most scientists, the faith in a unique level of kinds to which all other phenomena reduces exhaustively has become increasingly less plausible. Whilst there may be supemeniencee between explanatory levels, or a nomological “marching in step” many properties in the world such as those involving biological or mental phenomena do not appear to be exhaustively reducible (i.e. explicable) in terms of some unique fundamental level. Philosophers of science such as Nancy Cartwright now argue strongly against the idea of a seamless hierarchy of reducing levels, advocating instead a ‘patchwork’ of laws, of discontinuities that undermine the traditional deference to fixed nomodIt is worth noting that a categorisation of basic metaphysical kinds does not rule out or reduce away large scale, or ‘common sized’ objects. It merely suggests that they might also be subject to unification at deeper levels of abstraction eLoosely, the idea of supervenience holds that all non physical phenomena is realised by physical phenomena without reducing to it. More precisely, a set of properties P1 supervenes on a set of properties P2 if and only if there can be no changes or differences in P1 without there being changes or differences in P2. See [5, 61

184 logical reducibility (see her [7, 81). Such views sit well with many of the underlying assumptions of theories of complexity. But how does this cash out in ontological terms? Are the ideas of holism and emergence, of ‘infinite’ or irreducible ontological kinds it hints at as equally questionable as the commitments of the scientific reductionist? 3. What ought complexity theory to ask of ontology?

Complexity theory contains a number of mysteries that have tended to make more orthodox scientists uncomfortable with the substance of many of its claims. Do chemical reactions such as the Belousov-Zhabotinsky reagent really indicate a kind of self determination aspect at the sub-organic level? Can attractors be regarded as ‘genuine objects’ in the way electrons often are, or are they convenient theoretical posits that do no more than to unify tendencies within dynamical systems? Is the apparent ubiquity in nature of power laws like Zipf’s law something that is ontologically significant, or is it no more than a ragbag of mathematical coincidences and pseudo-continuities? Indeed is there even really any such thing as a ‘complexity science’ or must we accept, as some critics argue that “there is no such theory, and it even may be that there are no such structures as they are implied to exist by Prigogine, Haken, and their collaborators.”f Of the many questions of this kind surely one of the most mysterious for any putative ontology of complexity is how it ought to represent the relationship between order and chaos. C T would seem to have an urgent requirement for an adequate set of conceptions here, not least because of one of its governing assumptions - that many important structures arise at or near some kind of ‘boundary’ between order and chaos. But not only is there more than one conception of order which is used, such concepts tend to be formulated it in negative terms - as that which is non-random or foregrounded from the ambiently chaotic. A variety of further mysteries then arise; how do ‘ordered’ objects differ from ‘random’ objects? Can there be a random object? Can there be absolute randomness? Why should order be so fundamental? And so on. This central dichotomy between ordered and ‘disordered’ phenomena appears to require some urgent conceptual refinements. Ontology ought to provide us with at least some kind of insights here. But when we come to examine how order/disorder distinctions might be handled within traditional ontologies, we find all of them wanting in f[9, p. 4471

185 some respect. 4. Some ‘ontologies’ of complexity

Since the ontology of complexity has not been widely discussed the metaphysical preferences of complexity theorists are not immediately obvious. But it does not take much reflection to see where the predominant sympathies lie. Thus, we hear of; “generic low-energy properties determined by a higher organizing principle and nothing else” [lo]; or how “the validity of a kinetic equation” can imply “ergodic properties” [ll, p. 431; or of “collection(s) of molecules which have the property that the last step in the formation of each molecule is catalyzed by some molecule in the system” [12, p. 2851. Tacit appeals by complexity scientists to the category of properties, together with their close association with the idea of emergence, immediately suggest that the traditional philosophical distinction between particulars and properties (or its close cousin substances and universals) is at work here. So what more precisely is a property and how does it relate to the category of particular? Properties have been conceptualised in various ways. One idea has them as sets - specifically sets of particulars, though this idea quickly runs into technical problems which are not relevant here [13, p.211. A more natural view sees them in terms of things which are ‘had’ by a particular, features or characteristics such as ‘redness’ ‘roundness’ and so on. But there is a further, important ingredient of the distinction between properties and particulars which relates to how these objects can be distributed in space and time. Properties can be repeated, particulars cannot. Thus, one approach to an analysis of the distinction might focus upon location and repeatability as follows:

A particular P is an object which has a unique location. That is, it can exist at only one place in space-time

By contrast:

A property K is an object which can have more than one location. That is, it can exist at multiple places in space-time Thus a single hydrogen atom might count as particular since it will be uniquely located, whilst various characteristics it has such as ‘mass’, or ‘charge.’’ will count as properties since other atoms may have them at the same time. A more elaborate version of the particular/property distinction

186 is the distinction between substances and universals. A substance counts as a particular since it will be uniquely located. But it involves further ideas such as Locke’s idea of them as a ‘substratum’ or property bearer [14, pp. 39-50]. Aristotle offered a still more subtle and complex view of substances. In the Metaphysics he defined substances as “impredicables” objects which cannot be predicated of other objects. Substances here serve as summum genera - paradigm subjects of scientific investigation. Like properties, universals have the capacity for multiple location but as with the particular/substance continuity they too carry slightly more metaphysical baggage than their more sparse cousin. Plat0 and Aristotle famously disagreed about universals and so set off an argument that has been running in philosophy ever since. On Plato’s view, true universals were ‘forms’existing in Platonic heaven that could only be physically manifested in imperfect ways. For Aristotle, all universals were ‘immanent’ - located materially - a further basis for the greater palatability of an Aristotelian ontology to natural scientists than a Platonic 0ne.g But whilst properties can be contrasted with particulars in terms of their locative character they need not be taken to really exist in the way their more tangible partners usually are. Instead, as Quine has repeatedly argued (see for example his [16, p.1221) their role may be purely semantic - indicative of a range of linguistic proclivities amongst certain language users. Nonetheless the presence in language of predicates which indicate recurrence - (“is red”, “is complex”’), has been taken by many to imply some deep ontological fact - the existence of properties or their more robust metaphysical correlate, universals. Properties are often taken to be a more neutral basis for the concept of a universal - they may even be seen as a way of “solving the problem of universals” [13, p. 751. Where particulars and properties are taken to exist in conjunction we have a ‘two’ category ontology. Whether either is held to exist in isolation a more economical ‘one’category ontology emerges. Some of the basic possibilities & associated terminology are set out in Table 1. The substance/particular, property/universal ontology appears to enjoy the widest acceptance amongst scientists in general as well as with complexity scientists, so it is here where I will focus my main objections. However there are of course other kinds of ontologies a complexity theorist might appeal to. A Whiteheadian ontology of processes or events for example is gThis is the medieval distinction between universalia ante rem and universalia in rebus. See [15]for more on this

187 Table 1. Particulars One Category Ontology

Two Category Ontology

jects part icularised of properties things with particularised features All objects are combinations of particulars and properties

an obvious metaphysical resource for complexity science [17]. The appeal of this lies in the attractions of the dynamical - a ‘Heraclitean’ universe of flux. But note that processes can be analysed in terms of particulars and their properties - for example, processes might be construed as particulars that have temporal stages or parts.. Objects are thereby seen as four, rather than three dimensional - extended in time as what are sometimes referred to as continuants (objects with temporal parts) as opposed to occurrents, objects which are fully present a t each moment. We shall return to this issue shortly. A further potential ontological category for complexity science that might be considered is the idea of Facts or States of Affairs. Both were famously invoked by Wittgenstein, though they have been less popular within philosophy than ontologies based on the particular/property dichotomy. A fact is a specific kind of states of affairs - those which exist in actuality and which can be the referents of true propositions. Facts have a range of their own problems which I will not discuss here, but as a species of states of affairs one can assume that any problem had by the latter will affect the former.h A state of affairs is normally defined as the complex that is made up of the conjunction of a property and particular and is ontologically implied by the use of propositions of the logical form P ( z ) . But therein lies the problem for the putative category. Either it reduces t o particulars and properties - in which case it must inherit their problems, or it is an irreducible complex. But if it does have an existence ‘over and above’ the particular and property that makes it up then the failure t o understand such problem is technical - the so-called ‘slingshot’ argument which, it is held, implies t h a t any ontology of facts would ultimately be committed t o a very unlikely entity - the one ‘great fact’. See [18].

188 how it is distinct makes it a rather mysterious entity. Alternatively, as a complex simpliciter, it may just turn out to be a less well developed version of the ontological category I am about to discuss.’ Similar reflections apply to a further move that might be made within this particular species of ontological candidates - that of taking information itself as a kind of ontological category. For not only are there problems arising from the different interpretations of information available. There is also the fact that it too appears to be largely subsumed by the candidate I shall argue for shortly. 5. Nominalism and Complexity Advocates of a one category involving only particulars - nominalists - have included amongst their illustrious ranks thinkers as diverse as Leibniz, Hume and Quine. It is not only the desire for ontological simplicity that drives the interest in this approach. There is also the requirement to respect empirical constraints and to assume the existence only of that which is observable. By occupying distinct (and finite) portions of space-time particulars fit these requirements in a way which multiply located properties seem not to. But nominalism, whilst appealing in the abstract, works less well in the context of scientific practice - not least because of the commitment of scientists t o mathematical properties and entities. In spite of elegant attempts to explain away the role of mathematical claims in science‘ the majority verdict appears to be that pure nominalism is an extremely difficult position t o maintain. The same reflections would appear to hold where complexity theories are interpreted in nominalist terms. Here, a purely nominalist ontology would appear to fail for two fairly clear reasons. First, realist interpretations of complexity would appear to involve commitments t o many kinds of phenomena that require multiple location for example attractors, scale phenomena, or dissipative chemical reactions. The dynamical systems approach surprises us precisely because, as Andy Clark argues it uses the “same basic approach to tackle many superficially very different kinds of real world phenomena” [2l, p.1151. This ‘same a p proach’ assumes ontological continuities which underpin the behavioural modelling of distinct systems - not the distinct behaviours in distinct systems that the nominalist must advocate if they are to be consistent. Second, and following from this, nominalists must deny the reality of relations since ‘See [19] for a discussion of states of affairs and a more developed account of whether universals and particulars from irreducible complexes JSeefor example [20] which attempts to develop an approach to ‘science without numbers’

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these are in effect ‘dyadic’ properties (properties that obtain in virtue of a connection between more than entity. For example the relation x > y can be thought of as a dyadic property of x, or of y). But most complexity sciences have instances of phenomena which appear to imply the very opposite - an intimately relational universe, where discrete elements appear to suddenly operate in close harmony and ceoperation to produce higher order phenomena. The more developed idea of particulars - as substances not only falls foul of these problems of multiple location but has some problems of its own. For example, a common distinction here - between primary & secondary substances does not appear to be a comfortable one within CT. A primary substance has traditionally been identified with specific, or individual objects, whilst a secondary substance is read as a more general type of the object concerned - a kind. Thus, Napoleon would count as an instance of a primary substance and ‘man’ as a secondary substance. But the kind of modularity within complex systems that Bechtel and Richardson identify [22] would appear to make room for graduated classes of complex objects. That is, a continuum of overlapping objects of varying degrees of complexity - not objects which are forced to accommodate absolute distinctions between general and primary kinds. Worse, as we saw earlier, primary substance in the Aristotelian sense is understood as an ‘impredicable’ - something which can be predicated of anything else ( e g one cannot say “John is a George Bush” since GB is a primary substance). But this seems to be an ontological commitment C T ought to avoid for it seems plausible that it should allow for complexity to go ’all the way down’. That is, C T ought to allow for infinite nesting of objects within objects, not metaphysical full stops. And it appears to be a further prerequisite for CT, not only that its basic objects can generate new basic objects, but that basic objects can pass out of existence altogether. This again is not easily made consistent with a substance ontology. A recent nominalist compromise has been to appeal to the category of a trope. Tropes avoid the problems to do with the recurrence of features across multiple locations by ‘particularising’ properties. Thus, trope theorists can say that any objects properties are made up of bundles of tropes TI . . . T, where each T is a particularised property - e.g. the tropes Redl . . . Red, or Shape1 . . . Shape,. Tropes have found some support amongst analytic ontologists, perhaps because they permit one to have ones nominalist cake and eat it. But there are many more philosophers who have found tropes

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to be much too mysterious an entity to fully commit to.k And an interpretation of complex objects and processes in terms of tropes does not seem very intuitively promising. Every complex object or process would have to consist of infinite numbers of tropes which would need to ‘come into being’ and then simply disappear as new complex objects emerge. And since there can be no trope heaven, there could be no explanation of the hows, whens or wheres of any of these manifestations. We would simply have to accept that tropes appear in the location they do and then disappear. No general laws could be invoked that allow for recurring phenomena to underpin this. For C T then, a commitment to tropes would seem to be more of an act of philosophical faith than a clarification of its underlying ontology.

6. Problems with the alternatives to nominalism Alternatives to a nominalist ontology face their own kinds of objection. There is not space here to run through every kind of possible ontological solution or each of their specific problems. But as an illustration we can pick out two classes of problems which appear to pose serious problems for a seamless application of standard ontologies to CT. Whilst these are not conclusive they seem to me to require so much ‘fixing’ that what results lacks in elegance. All the more so if, as I will suggest, there is an ontological candidate which not only avoids these problems, but which accommodates certain requirements of complexity theories far more neatly. The first class of problems relates to problems to do with explanation. The second class relates to tensions between fundamental ontological requirements of complexity theories and the suitability of the available candidates. As an example of the first class of problems we might consider explanations of certain kinds of change and how these are dealt with in CT. It is the kinds of explanation of these changes offered by complexity theories which contributes a great deal to the idea that they have a distinct scientific character from more orthodox approaches. For example one claim that is frequently made is that, as result of increases in the magnitude in control parameter for some system such as temperature, ‘spontaneous’ manifestations of completely new kinds of behaviours or characteristics can occur. These so called ‘phase transitions’ do not appear to be straightforward kinds of changes which are gradual and long term. Rather these are sudden, and dramatic and frequently require new kinds of laws or explanations. kSee [23] for a defence of tropes. See [24] for reasons to question them

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Finding ways to accommodate explanations of change and generation in terms of universals has been a widely discussed problem in philosophy. A prima facie problem of contradiction appears to emerge when the predicates that refer to properties relate to properties had by an object at different times. Since it appears natural to think of any object as being wholly present at different temporal points, we appear to be committed to the belief that an object can be both p and not p . For example a piece of paper may be white, but later it may have been painted black. But then we seem to be committed to saying that the paper, which is wholly present throughout its existence both has whiteness and does not have whiteness. Such a contradiction is clearly untenable with our common sense view of the world. One obvious response to what goes wrong here [25, p.2041 - the so-called ‘perdurantist’ view - is to say that objects are not wholly present at individual times - that is, they do not endure, they rather perdure. This entails a view of objects as four dimensional with a succession of temporal ‘parts’ or slices that constitute their worm-like extensions through time. But for this to work one must not only commit oneself to the counterintuitive view that nothing is ever wholly present, but to the vagaries of time indexed properties - ‘the property of being P at t’. The idea of temporal parts and time indexed properties has many critics, but in any case it does not appear to deal with what would be required for CT. For temporal parts may succeed each other as one time indexed property is replaced by another. But to explain phase change in C T by saying that one property is replaced by another seems to distort the force of the scientific explanation here - which is to say that the properties undergo a kind of orderly mutation into different properties. Not only that, there are transparently identifiable control parameters that make this possible. Since the same universal or property couldn’t span the mutation (different properties by definition are involved), and since the idea of simple replacement of one property by another is not a very accurate illustration of what is happening here, properties seem to be weak ontological vehicles for what is, after all, one of the most important forms of explanation offered by complexity theories. The category of process is a direct response to these kinds of problems. In fact Heraclitus’s claim that ‘everything flows’ was a specific rebuttal to the atomists perception of elements of reality where objects do not change but merely exchange properties. Processes do better than properties here - though it is less clear how well they deal with the idea of spontaneous phase change. Since processes just are four dimensional entities, change is fundamental to them. But C T makes specific claims about certain kinds of changes.

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Unless there is a further ontological distinction amongst processes to allow for this, one change is much the same as another. And if there a further ontological distinction is required it looks like any economy that might have been gained by the appeal to processes over properties has been lost. Since processes are a direct response to the problem of change it is not very surprising that they do better than properties. However it is less clear how well they deal with the second of the problems raised above. 7. Problems with the Order/Disorder Distinction Amongst the most serious kinds of objection faced by standard ontologies when it comes to accommodating the research programme of CT must be those centring on their weaknesses in accounting for the key distinction between order & randomness. Universals standardly depend upon tacit concepts of order in lots of ways - not least the fact that they must be conceived of as occurring in orderly complexes - i.e. ‘attached’ in regular ways to one or more particulars. As Wittgenstein pointed out, the orderly conjunctions of universals and particulars indicates something significant “. . . universals and particulars may appear to be mutually independent but this independence is a form of connection with states of affairs - a form of dependence”. Indeed, there is even a suggestion of metaphysical priority for: “it is essential to objects that they should be possible constituents of states of affairs” [26, 2.011 & 2.0121. But the question of how they do this has never received any satisfactory answer. Indeed, the role of ordering (processes) is usually ignored or brushed to one side. One motivation for this wilful denial to address something so fundamental may lie in the disaster that results from one famous attempt to explain the orderly connection between particulars and their properties. On this (Platonic) line an additional universal is introduced as a way of providing the regular link between particulars and properties. But this only serves to lead, inexorably, to a logical and metaphysical regress (for what then ‘binds’ the binding universal, other than a further one? And so on).’ Aristotelian style solutions avoid regress by simply positing the binding between a universal and particular as something primitive and inexplicable. The problem then is any concept of how this orderly connection holds must remain to unexplained. As its stands then accounts of order on this line appear to be missing, incoherent or to lead to inconsistency. ‘The so-called ‘Third M a n Regress’ [27]

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If the tactic of ignoring order or providing a metaphysical basis for it fails then what of attempts to analyse it away in other terms? In general specific attempts to analyse order in terms of properties seem to lead to further failings. For example, one way this might be essayed is in terms of a second order property - a property of a property which holds in virtue of other things being the case. On this line order is conceptualised in terms of a property that exists only where other properties do. But not only have second order properties tended to be regarded as an unnecessary or mysterious kind of commitment,”’. it is not clear whether they could offer the kind of conceptions of order that is consistent with the claims of complexity scientists. CT appears to permit different degrees of order - scalar conceptions rather than the discrete ‘pockets’ which second order properties (or third order, and so on) would seem to represent. So second order properties fail twice here. First in the failure to illuminate notions of “degrees” of order. Second, because a commitment to second order properties as an analysis of order would further commit CT to a hierarchy involving properties of properties, of properties and so on. Again the threat of a (vicious) infinite regress raises its head. Perhaps then it would be better to deal with order by treating it as a first order property simpliciter. But this too has some undesirable outcomes. For presumably randomness could be a lack of the property of ‘order’. Random phenomena plays a key explanatory and causal role in CT - for example in discussions of entropy production. To allow causal efficacy to arise as a result of ‘lack’ of another thing sits uneasily with our normal understanding of causation. So if order could be a first order property, so too would randomness need to be - if only for the reason that this would facilitate the distinction of random things from orderly things. But then suppose there were such a property of “randomness”. This would imply the existence of some method of being able to distinguish a random property from an orderly property. But then we would be saying that there is potentially decidable method for deciding something is random. Yet we know from [31] that there is no such method - indicating that a binary string has the property of randomness for example is not a thing that be definitively demonstrated. To do so would require a program of less complexity than the random string it outputs. But then the string would have a minimal program and would have some pattern. To claim “‘For some objections to second order properties see [28]. For some arguments in favour see (291. (301 discusses the idea of a second order logic

194 that the randomness of a binary string is demonstrable leads inevitably to the contradiction of saying it is not random. At this point it becomes clearer why our other candidate ontological category - that of processes, runs into problems. Whilst processes appear to deal with change more naturally than the’contortions we are forced into with properties, similar problems seem to arise when it comes to applying the distinction between order and randomness to them. For we similarly ask - what distinguishes a random from a non random process? Either we get the unilluminating response “because one process is orderly and one is not”. Or, by attempting to definitively identify a process as ‘random’ we run the risk of encountering the same kind of contradictory outcomes as before. 8. Further, More General Objections

In each case where a property is used to explain order there seems to be something extra required - order (however it be explained) and the entity that is ordered. Further more general kinds of objection to this picture that arise suggest a further kind of strategy might be worth considering. (i) Objections relating to economy. If ‘order’ appears to be something that is an additional requirement for properties, either in ontological or analytic terms, might it not be more economical to drop the commitment to properties, processes or whatever and to investigate whether order itself could be made to do the ontological legwork? (ii) Objections relating to reduction. It appears that there may be reasons for suggesting that traditional categories can in any case be themselves subsumed into this putative category of order. These reasons revolve around arguments that appear to suggest that the very conceivability of properties, particulars and so on depends upon a prior conception of order.

An indication of this potential conceptual reducibility can be seen if we examine again the way considered earlier in which universals, particulars or indeed processes are distinguished. As we saw, one powerful way in which this can be done is in terms of location and more particularly, the way that the identity conditions for properties, particulars and the like are fixed in terms of differing characteristics of location. An extended definition incorporating identity might go as follows: A particular P is the same P if it is identifiable as the same P

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across unique locations at different times

A property Py is the same Py if it is identifiable as the same Py across multiple locations at the same time

A Process Po is the same Po if it is identifiable as the same Po across multiple locations at different times In other words, we obtain the outlines of what Kant called a ‘transcendental’ argument for a priority of order, or pattern over these Categories.” That is, an argument which demonstrates the plausibility of accepting the existence of something by showing that it is a necessary condition for positing the existence of other things. Here it relates to how certain kind of prior order needs to be in position if we are to invoke the existence of properties or particulars. And there are other ways too in which this apparent conceptual priority of order might be illustrated. For example, set theory - a foundational tool amongst analytic philosophers for the conceptualisation of phenomena in both the physical and the mathematic contexts can be shown to be lacking when it comes to analyse order itself. [33] indicated how the ‘Wiener-Kuratowski’ procedure, which attempts to analyse ordered pairs in purely set-theoretic terms requires a further element - that of choice, one which in effect, masks its own implicit commitment to the priority of 0rder.O There is not enough space here to consider many other similar examples, but the implications are clear enough - the apparent metaphysical priority of order over more developed ontological categories indicate that, for CT, this may be the best direction for developing its own ontology. 9. Patterns

If venerable and long serving range of metaphysical candidates such as these are found wanting as the referents of the language of complexity, then a neg“See (321 for a discussion of this form of argumentation OThe Wiener-Kuratowski procedure states that an ordered pair < x,y > can be construed in terms of a class such that < q y >=def. {{x}, {x,y}}. Any pair < x,y > can be introduced if the following condition holds: < x,y >=< w, z > iff (z= w&y = z). Since {{a},{a,b}} and {{a,b}, {a}} denote the same class it would seem that ordered pairs (and hence order) has been analysed in terms of an unordered entity - the class. But as Hochberg points out if one “uses sets like {{a},{a,b}} and {{b}, {b, a}} one chooses which of { { a } , { a , b } } and { { b } , { b , a } } is to represent < a , b > and which represents < b, a >. This shows that in the construal of < a,b > as {{a}, {a,b}} one implicitly takes a and b in and ordering for one takes the element in the unit set as the first element”. Russell dismissed the whole procedure as a trick.

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ative conclusion might be that is the language of complexity itself which is wanting. But another, more positive conclusion seems possible, one that draws upon these suggestions of some kind of priority of order. Such priority suggests that there might be a further powerful and plausible alternative ontological candidate in waiting - a reification of order in all its different forms itself. This category of ‘pattern’ (to use a more convenient term) not only sits very neatly with fundamental conceptions in C T - specifically a definition of complexity itself, it also offers a certain kind of objectivity in apprehending the notoriously subjective divide between what can be considered to be ‘ordered’ and what is not. For there is a definition of pattern that both allows for a degree of observer relativity - patterns that exist only in the eyes of beholders (faces in Rorschach inkblots for example) as well as permitting only a certain class of patterns, the ‘real’ patterns, to have scientific relevance. The definition, taken from the idea of computational complexity - states that a pattern in some binary sequence can be defined if it has a compression - a program shorter in length than the sequence itself that can be used to generate the sequence. And the real pattern in a sequence is only identified where a ‘minimal program’ for it has been identified - the shortest possible sequence which can be inputted to a Turing Machine so as to produce that sequence as an 0utput.P There are many kinds of lhring machine, but since the definition relativises all definitions of length to the Universal Turing machine - that Turing machine which can simulate any member of the class of Turing Machines there is no ambiguity in what counts as the shortest program for produce the sequence and hence defining the pattern there. There are too many qualifications and developments of this basic idea to go into it in detail hereq but it proved tempting enough for Daniel Dennett to make use of it in his [36] as a way of better defending his views about mental representation and the idea of

PSee[34] and [35] for the earliest discussions of the compression criterion for pattern and the idea of algorithmic complexity qIn my (forthcoming) I consider a way in which computational complexity conceptions of pattern can be augmented by a complementary, more naturalised conception of them. This relates to the idea of symmetry and the deeper notion that goes with this - the notion of invariance (through transformations of various kinds). This idea provides for a continuity between overtly mathematical or geometrical patterns and the regularities in physical objects. On the one hand there is invariance and stability in the face of transformation through mathematical operations, on the other stability in the face of transformations through physical processes such as time, causal effect and change. A plausible bi-conditional holds, one that relates the computational complexity conception of pattern t o this symmetry, or invariance notion

197 an intentional state. Such states are, in a nutshell, grounded in real patterns. But Dennett showed little interest in further developing this brief foray into ontology and the idea of pattern as an ontological category of interest has not been developed much further within analytic philosophy. Aside from an eccentric paper from the 1940’s and some discussions within the philosophy of mathematics, little else has been said.’ This seems odd, not least because patterns appear to play a central theoretical role in many other ways that simply grounding intentional, or mental states. We can see oblique references throughout philosophical history to them in the form of causal regularities; as dispositions and capacities; in the appeal to laws, the salience of structures and structural thinking in the philosophy of science, and so on. In some cases key forms of scientific reasoning such as induction, or the best way to connect statistical observations into hypotheses (the socalled ‘curve-fitting’ problem) can be shown to have specific derivations from complexity definitionss 10. Patterns and Complexity

This widespread (but tacit) use of patterns in philosophy and science indicates that, at the very least, there is something here worthy of further investigation. And for CT patterns may not just be an interesting potential metaphysical resource - they may be indispensable to its practice. The very foundation of CT lies in the task of explaining how elements as diverse as molecules, plants, or economic actors act collectively to to produce complex patterns of behaviours. To this can be added a range of other desirable features which indicate their usefulness for complexity science. For example: (i) Definitions and identity criteria for other ontological candidates such as properties are frequently unclear. Patterns by contrast can draw upon resources for definition and identification from algorithmic and invariance notions, resources which offer impressive levels of clarity. ‘The discussion is 1371. 1381 argued that mathematical objects can be dispensed with in terms of structures or patterns such as ( N >) where each number functions as no more than ‘position’ in this pattern. It is not a view he has developed in any great ontological detail “Solomonoff 139) develops a model of induction which involves parallel conceptions to Chaitin & Kolmogorov’s ideas about compressibility. I401 indicates a way in which Solomonoff’s idea can be extended into the “MinimumDescription Length Principle” as a way of solving the curve-fitting problem

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(ii) As we saw, there is a case for arguing that patterns subsume many traditional metaphysical categories. The necessity for depending upon some form of regularity or continuity in order to identify and reidentify anything at all provides both justification for priority as well as suggesting something very basic underpin this need for ordering. (iii) Patterns serve as precisely the kind of entities that can provide a grounding for the apparently universal capacity within science, and especially complexity science to distinguish between more or less ordered phenomena. (iv) Even better, a multiplicity of patterns allows complexity science to justify its rejection of reductionism and so permit a plurality of explanatory levels and processes. (v) Patterns provide a way of grounding the apparent ubiquity of regularity in the world given its extremely low probability. Continuities in object identity, regular causal connectivity or the tendency of complex systems to resist the pull of entropy all acquire an ontological grounding they would otherwise lack. (vi) By extension, patterns not only offer important grounding concep tions into the concepts of ‘Laws of Nature’ and of theory change in science (the development of increasing sphiosticated compressors of the world. As Ray Solomonoff’s work indicated [39] they may illuminate some of the mechanisms for inductive reasoning and its successes (the search for minimal programs in the identification of natural phenomena). (vii) Patterns seem also to offer insights into numerous other mysterious, but desirable principles of scientific reasoning - in particular the effectiveness of Occams razor - the appeal to simplicity (or shortest compressor).

11. The problems with patterns In addition to the difficulties faced by other metaphysical candidates, the variety of such a list seems to me to be a further reason why CT should take the category of pattern highly seriously. But of course, life is never simple. For whilst patterns certainly seem to have some good credentials to be the referents of claims within the sciences of complexity, this does not mean that there are not also number of problems which would need to be addressed if they are to function in effective ontological terms. I will list a

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few of the more significant ones these before concluding. (i) There appear to be a range of problems involving the nature of the relationship between a pattern and the elements which go to provide its ontological foundation. Prominent here is the issue of foundational elements - especially should a one category ontology which involves only patterns be thought desirable. For without particulars, or other traditional ‘substantial’ entities to play the role, it must be the case that patterns serve as the constituents of patterns. For many philosophers this would lead to an undesirable outcome - the lack of a categorical base. And since patterns seem always to be patterns of other things we would seem to end up lacking in any ‘material’ like stuff in world. Instead we seem only to have a chain of dependencies - something like a series of brackets within brackets, within brackets - a series that never reaches any substantial kind of thing. (ii) There are also problems of extending the computational complexity line to patterns in, or of physical things. Since the algorithmic definition of patterns requires clearly definable binary sequences for it to work properly, using it to define patterns in physical objects requires that binary sequences can be shown to adequately model or simulate such objects. But this then invokes the problem of what a ‘truthful’ or even ‘adequate’ representation really is. And this remains one of the most controversial and difficult problems in philosophy. (iii) The problem of pattern identity. In the formal context pattern identity is fairly clear. P=P’ if both have the same minimal program. Or, focussing upon patterns elements, one might say that P=P’ if there is an isomorphism or 1:l mapping between the elements. In less formal contexts, especially where there is certain ambiguity in saying what the elements of some pattern actually are, identity conditions are less clear. For which ‘elements’ would we select for the mapping? And how 1:l could it really be? Yet without a proper criterion for identity there could be no proper ontology of patterns. (iv) The risk of tautology. Patterns may be a desirable way of accounting for the ubiquity of order. But as orderly things there is the very real risk that we end up saying no more than ‘order is understood by way of order’. (v) The problem of accounting for causality. It certainly seems plau-

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sible, as we saw earlier, t o say that the regularities seen in causal relations have a pattern t o them. But if causality is also understood in purely physicalist terms as some kind of “bumping” or “impacting” effect (cf. [41, p. 181) then the question remains as to what could be doing the bumping and impacting

I do not think these problems are fatal, though they are certainly difficult problems that need to be addressed. My (forthcoming) addresses some of them and it is to be hoped that additional lines of research can deal with others. But if, in the end, an ontology of patterns turns out t o be as unworkable for complexity science as traditional ontologies seem t o be, then perhaps we would need t o look again at how scientifically robust the patchwork of claims and observations that make it up can really be taken to be. References 1. McGuire M.R. 1999 On a General Theory of Pattern Doctoral Dissertation, University of London 2. McGuire M.R (forthcoming) Patterns Kluwer Academic Press 3. Williams, D.C. 1953 “On the Elements of Being”, Rev. Metaphysics 7. 3-18, 171-192 4. Strawson, P. 1959 Individuals: A n Essay in Descriptive Metaphysics, Rout-

ledge 5. Davidson, D. 1970, “Mental Events”, in D. Davidson, Essays on Actions and Events, Oxford, Clarendon Press., pp. 207-227 6. Kim, J. 1993 Supervenience and Mind: Selected Philosophical Essays ,Cambridge: Cambridge University Press 7. Cartwright, N.1983 How the Laws of Physics Lie, Oxford: Clarendon Press 8. Cartwright, N 1989 Natures Capacities and their Measurement, Oxford: Clarendon Press 9. Anderson, P & Stein, D 1987 ”Broken Symmetry, Emergent Properties, Dissipative Structures, Life: Are They Related, in F. Eugene Yates (ed.), SelfOrganizing Systems: The Emergence of Order (NY: Plenum Press, 1987), p. 445-457 10. Laughlin, R.B. and Pines, D.2000 “The Theory of Everything,” Proceedings of the National Academy of Sciences 97 (l),28-31 11. Prigogine, I. (et al) 1976 “Kinetic theory and ergodic properties” Proc. Nati. Acad. Sci. USA Vol. 73, No. 6, pp. 1802-1805 12. Kaufmann, S. 1993 The Origins of Order: Self-organization and Selection in Evolution, Oxford, OUP. 13. Oliver, A. 1996 “The Metaphysics of Properties”, Mind, 105, (pp. 1-75) 14. Ayers, M. 1991 Locke, Vol 2, Part I “Substance and mode”, Routledge, London.

20 1 15. Armstrong, D.M. 1978 Nominalism and Realism: Universals €9 Scientific Realism, Vol I, Cambridge, Cambridge University Press 16. Quine, W.V.O. 1960 Word and Object, Cambridge, Mass: MIT Press 17. Whitehead, A.N. 1929 Process and Reality, New York: Macmillan 18. Neale, S. 1995, “The philosophical significance of Godel’s Slingshot.” Mind Vol. 104, NO. 416, pp. 761-825. 19. Armstrong, D.M. 1997 A World of States of Aflairs, Cambridge, CUP 20. Field, H. 1980 Science Without Numbers: A Defence of Nominalism Princeton, NJ: Princeton University Press 21. Clark, A. 1997 “Emergence and Explanation” from Being There: Putting Brain Body and World Together Again, Cambridge MA: MIT Press, p.103128 22. Bechtel, W., Richardson, R. C. (1992). Emergent Phenomena and Complex Systems. In A. Beckermann, H. Flohr, J. Kim (Eds.) Emergence or Reduction? Essays on the Prospects of Nonreductive Physicalism. (257-288). Berlin: Walter de Gruyter 23. Simons, P. 1994 “Three trope theories of substance”, Philosophy and Phenomenological Research, 54, pp. 553-575 24. Denkel, A. 1997 “On the compresence of tropes”, Philosophy and Phenomenological Research, 57, pp. 599-606 25. Lewis, David. 1986. O n the Plurality of Worlds, Oxford: Blackwell. 26. Wittgenstein, L. 2001 Tractatus Logico-Philosophicus, (trans. David Pears and Brian McGuinness), Routledge 27. Vlastos, G. 1954 “The Third Man Argument in the Parmenides” Philosophical Review, 63, 319-349. 28. Kim, J . 1998 Mind in a Physical World: A n Essay on the Mind-Body problem and Mental Causation. Cambridge (Mass.): A Bradford Book, The MIT Press 29. Gozzano, S. 2003 “Second Order Properties: Why Kim’s Reduction Does Not W o r k , Logic and Philosophy of Science, Vol. 1 No. 1 30. Boolos, G. 1975 “On Second-Order Logic”, Journal of Philosophy, 72 pp. 509-527; 31. Chaitin, G 1975 “A theory of program size formally identical to information theory”, Journal for the Association of Computing Machinery, 21, pp. 403424 32. Cassam,Q 1987, “Transcendental Arguments, Transcendental Synthesis and Transcendental Idealism” Philosophical Quarterly 33. Hochberg, H. 1981 “The Wiener-Kuratowski Procedure and the Analysis of Order”, Analysis, 41, pp.161-163. 34. Chaitin, G. 1966 “On the length of programs for computing finite binary sequences’ Journal for the Association of Computing Machinery pp. 547-569 35. Kolmogorov, A. 1963 “On tables of random numbers”, Sankhya, A, 25, pp. 26-376 36. Dennett, D. 1991 “Real Patterns” Journal of Philosophy 88, pp. 27-51. 37. Bush W.T. 1940 “Concerning the concept of pattern” Journal of Philosophy, 37, pp.113-134

202 38. Resnik, M. 1997 Mathematics as a Science of Patterns, Oxford: Clarendon Press 39. Solomonoff, R. 1964 “A formal theory of inductive inference”, Infomation & Control, 7, pp 1-22 (Pt I) & pp. 224-254 (Pt 11) 40. Rissanen, J. 1978 “Modelling by the shortest data description”, Automatica, 14, pp. 465-71 41. Crane, T . 1995 “The mental causation debate” Proceedings of the Aristotelian Society Supplementary Volume L X I X

HOW TO LOVE THE BOMB Trying to solve the prisoner’s dilemma with evolutionary game theory

VASCO CASTELA University of Manchester School of Social Sciences - Philosophy Economists traditionally see altruistic acts as irrational. However, in the Prisoner’s Dilemma, a rational player can do worse than a moral player. The rules of the game imply that one cannot defend one’s best interest if one tries to. Game theory has struggled t o explain how an agent could have access t o the strategically best outcome without behaving irrationally, but with little success. Can a complex systems approach do better? Peter Danielson, using Evolutionary Game Theory, has avoided some of the assumptions of Game Theory by using a complexity approach to reframe the problem, and offers a solution of sorts. According t o Danielson, the foundations of altruism are ‘mechanisms of deterrence that rely on credible threat - we are nice for fear of retaliation. He is both right and wrong. It will be argued that utilitarian, consequentialist principles must have been at work t o create the conditions for altruistic acts t o be performed. It is wrong to expect, however, that the same reasons are the reasons for action. In order for a model of genuine altruism to be possible, an extra cog must be inserted in the mechanism of causality in order t o distance moral action from its strategic advantages. If emotions fulfill this role, we can tell a story in which it is rational to act on altruistic motivations and materially advantageous to hold such motivations. Moral sentiments can be seen as a tool designed by evolution t o help optimize cooperation in a social environment. The proposed account integrates the Humean theory of motivation with Robert Frank’s commitment model and Aristotle’s views on moral education, keeping an adequate story of how it can be in our material interest to be moral without having to renounce t o the existence of genuine acts of altruism.

1. How to love the bomb

- the MAD strategy

In most situations of strategic tension, the outbreak of actual conflict does not imply the end of diplomacy. A war ordinarily involves a number of battles and plenty of time and opportunity for negotiations, such as temporary ceasefires and conditional or unconditional surrenders. This was not the case in the Cold War, as the highly destructive power of nuclear

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weapons meant there was a decisive advantage in being the one to strike first. After an initial attack, in which the enemy’s defenses would be rendered inoperative, victory was sure. This of course made giving the order of striking first both very tempting and quite final, as once an attack had been launched and the bombs were in the air, there was no turning back. The instability of this situation led both American and Soviet governments to adopt a doctrine of nuclear deterrence known as Mutual Assured Destruction (MAD). The idea was for each to eliminate the opponent’s strategic advantage of striking first by stockpiling strike-back missiles that could be launched within minutes of the initial attack or that would survive the blasts by being placed in bunkers. If each state could provide evidence of adequate retaliatory capability, this would give the other state a good reason not to make a first move, as it was to expect heavy casualties in case of war, regardless of whether it was the first to attack. The situation seemed under control until an influential game theorist claimed MAD was not entirely safe, as mutual destruction could not be assured - which implied it could not be a reliable deterrent. After an initial strike from the Russians, for instance, why would the US retaliate, if this no longer served any rational purpose? If we consider a situation in which Soviet missiles had been launched, this would mean American bombs had already failed to act as a deterrent. Why use nuclear weapons, at that point, for killing millions of Russian civilians, with no possible strategic gain to be had? If retaliation was irrational, there would be no assurance of mutual destruction, and the strategic advantage of striking first would once again be in place. The solution the game theorist devised to this dangerous situation was a doomsday device, a machine that would be programmed to automatically retaliate in case of attack. The existence of such a machine would bring the situation back to a MAD condition of equilibrium. MAD was the basis for the plot in Stanley Kubrick’s hilarious dark comedy, Dr. Strangelove or How I Learned To Stop Worrying And Love The Bomb. In the film, the President of the US discovers, following the launch of a nuclear strike by a rogue American general, that the Russians had built a doomsday device to ensure the effectiveness of nuclear deterrence, only had kept it a secret as they were saving the announcement for the upcoming elect ions.

205 2. The problem

MAD may not seem such a good idea now, but it was the best solution that could be found to a puzzling problem: rational action does not always bring about the most materially favourable state of While acting in the interest of national safety, nations were paradoxically putting themselves in increasingly great danger. Note that the notion of rationality we are discussing is that of self-interest theory, underlying game theory and mainstream economics (and cold war politics), according to which an action can only be rational so long as it is in the interest of the person who performs it. No form of giving can then ever be rational, unless it happens as a part of a trade. Or, to use the terms of game theory, an agent's action is said to be rational if it maximises his utility, according to a set of personal preferences. While this theory of rationality may seem to make rather pessimistic assumptions regarding human nature, it does have the merit of acknowledging the trivial fact that systems (states, companies, cells, humans, and so on) must conform to the laws of physics. Systems must conserve their resources and ensure their security if they are to survive. In a situation where conflict seems likely, it is perhaps inevitable that states will care about their national security above that of their opponents, if they are to remain a state. The same is true regarding the actions of individuals. We could explain altruistic behaviour by saying, like Hume, that when we act unselfishly we are merely satisfying a natural instinct of benevolence, which would enable us to claim that altruism is compatible with self-interest.17 If we think like Hume, altruistic behaviour no longer seems to be counter-intuitive. Such psychological explanations, probably of great importance in a complete ethical theory, do not deal, however, with the very real problem that the range of psychological reasons must be restricted by material demands, if we want to have a system that survives. Regardless of whether we prefer to count altruistic interests as rational or irrational, we must provide an explanation for how they arise and remain viable. The strategic advantage of striking first, during the cold war, before MAD was put in place, was not simply a mistake of the analysts. It was quite real. While both nations would be better off by not going to war, striking first was the only safe way for each to avoid total annihilation. Diplomatic negotiations, interestingly, could not help, as no promise could "An interesting analysis of this and other interesting commitment issues involving the

MAD strategy can be found in 141.

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be trusted in that situation. What could the penalty for lying be after the trusting party had been destroyed? One of the most popular and systematic approaches to try to solve the problem of trying to make peace between rationality and cooperation has been to study the Prisoner’s Dilemma, a strategic puzzle which closely mirrors the nature of the cold war situation.

3. The Prisoner’s Dilemma

The limitations of rational action were popularly studied in game theory in the form of the Prisoner’s Dilemma (PD). The dilemma arises out of the following situation. Consider that two individuals, partners in crime, are arrested and each is interrogated in private. Each of them is told that all charges against him will be dropped if he rats on his partner, who will as a consequence get a longer prison sentence. Game Theory predicts that, as each will realise that, regardless of what the other does, he will always get a lower prison sentence by ratting on his partner, they will both do it, and both will get iong prison sentences. In game theory terms, mutual defection will be the dominating strategy. If they had both chosen to cooperate and remained silent, however, they would both have been better off, each with a relatively short sentence. The paradox is that, similarly to the cold war situation, as each rationally chooses to do what is in his best interest (in game theory terms, what awards the highest payoff), they both stand to lose. As Skyrms4 notes, however, from the standpoint of rational choice theory, the subjects’ utility function is up to them. Game theory does not say humans are selfish, it rather describes the behaviour of self-interested agents. The distinction makes sense because game theory imposes no restrictions on what an agent’s interests can be. So if a person has a preference for the happiness of others, game theory would recognize it is in his selfinterest to seek it. If we imagine a version of the PD in which the prisoners are partners not in crime but in kindness, and the payoffs are cheques of different amounts to the prisoners’ favourite charities, the results will be exactly the same. Each kind player, by trying to maximise the amount of the cheque to his charity, which he thinks will bring the greater good, will act rationally by not cooperating. Acting so as to maximise their nonselfish interests would still bring about the same suboptimal outcome and they will both come out with lower cheques. So we can see that the origin of the problem as defined by game theory is not that players are selfish, but that they care to maximise their inter-

207 ests, regardless of whether these are selfish interests or not. This suggests

the obvious solution: what if each would care about the other’s interests? According to Peter Kollock, game theory’s assumption that agents make ~ Kollock, a subjective decisions based on fixed payoffs is m i ~ t a k e n .For sense of moral righteousness could affect them, making cooperation seem more attractive. As an example, imagine that in the PD the agents would prefer to help their friend rather than try to avoid prison time at all cost. The payoff for cooperation would then be higher and cooperation would be rational. However, as we have seen above regarding Hume, this solution would solve the problem at the expense of taking the explanation of why they would adopt this preference for granted. In practice, economists label altruistic actions as irrational.’ This is a fair assumption, or at least a good starting point, as it deals with the problem of physical viability of a system. It makes sense to expect not only that the agent will satisfy his interests, but also that these interests will be self-serving. Any other claim will contradict the Darwinian model of evolution. Evolution does not care about subjective moral principles. A genetically inherited trait only gets selected if it directly contributes (in a material sense) to the survival of the individual and, to quote Robert Frank (1998:54), “We can’t eat moral sentiments”. So, in order for altruistic behaviour to emerge and spread, it seems it must be advantageous in a material way. How could this work? There have been a few interesting attempts to try to explain how altruistic behaviour could have emerged (See [3] or [l],for a detailed survey), but, as we will see, they all have their limit at ions.

4. Naturalistic accounts of the emergence of altruism The idea of altruism arising by group selection was first mentioned by Darwin, who claimed that a group in which individuals would self-sacrifice for the common good would be superior to other groups in the fight for survival. Altruism could then emerge by the normal process of natural selection. The problem with this argument is that every element inside the group would have an advantage in saving the energy costs of cooperation by being a free-rider, exploiting the altruist. This implies that a mutant, born with no instinct to cooperate, would have better survivability and so would in time leave more offspring than a cooperator, therefore soon dominating the population. Altruistic behaviour would not be, to use the term coined by evolutionary biologist Maynard Smith, an “evolutionarily stable

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strategy”

.’

Kin selection was proposed by Hamilton and can be understood as a variety of the group selection argument, but with much better support.6 Hamilton claims it would make sense for someone to sacrifice himself for his family as the genes he shares with his kin would live on. This is an argument easily understood today if we think in terms of Richard Dawkins’ notion of the selfish gene.1° The phenotype (the body, a physical implementation of the genotype) may be sacrificed in order to save the genotype (the DNA). An instinct for sacrificing oneself for the sake of our genotype would be an Evolutionarily Stable Strategy (ESS). Kin selection, however, does not explain why some of us are willing to risk our lives to save a stranger and surely, in today’s social life, most of our interactions are not with kin. It could be argued that this is a recent phenomenon in the history of human evolution, and that perhaps our genetic programming has not had time to adjust to our new lifestyles. Robert Frank, however, correctly argues that genetically a second cousin is almost no different from a stranger, and so even if humans used to live in small groups in recent natural history, kin selection could not by itself explain the evolution of altruism.’ For Robert Trivers, reciprocity is the basis of altruistic behaviour.” If a set of players was to play the PD a number of times, a failure to cooperate could be punished in the next interaction.b The success of the Tit for Tat strategy in Axelrod and Hamilton’s study of the PD lent considerable empirical support to this claim.12 Reciprocity can also work indirectly through r e p ~ t a t i o n In . ~ real-world situations, behaviour that could appear altruistic at prima facie could contribute to our establishing a reputation for being honest or kind, which could give us an advantage in future interactions. In many cases, we have an interest in keeping a promise because we would be punished if we did not. However, we all know situations where we could stand to gain by behaving selfishly but behave altruistically, not merely cooperating with

bTo be precise, the number of plays would have to be unknown to the players for cooperation to be possible. If they both knew they would play 10 times, for instance, it would be rational for both to defect on the 10th game, as there would be no possibility of retaliation. This would mean there would be no possibility of retaliation on the 9th game either, as both would know they would defect on the last game anyway. This would then be true of all games, and so they would never cooperate. The versions of the puzzle that are the hardest t o solve (and the most interesting for the purposes of this paper) are either the single-shot PD or a variant in which the number.

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others for mutual advantage but going to some considerable trouble to help a stranger, for instance, with no hope of material gain. Reciprocity’s major drawback is that it fails to explain cooperation when there is no possibility of retaliation. Even when humans play a PD-style game a single time (also known as the single-shot PD), when not only defection is the only rational choice but also it cannot entail retaliation, results do not follow the predicted result of game theory. Many of us give tips in restaurants when we are on holidays away from home. Why would anyone pay for good service after he already got it, if he could simply leave and save some money? Some of us would risk our lives to save someone we do not even know. Reciprocity does not seem to be able to solve our problem. Our aim should then be to explain how altruism is possible in the single-shot PD, which reciprocity fails, not least because thinking of altruism as being based purely on conscious seeking of material gain surely contradicts our common sense understanding of what genuine altruism is. For some of those not satisfied with the standard accounts of how altruistic behaviour emerges and who do not want to abandon game theory’s concept of rationality (which has the merit of dealing with the issue of physical viability of ethical systems), a complexity approach seemed an interesting route. Skyrms, Danielson and Axelrod, among others, wanted to see how adding an evolutionary dimension to traditional models would change the nature of the PD, making cooperation a rational strategy.* 21 12

According to Jason Alexander, there are two main approaches to evolutionary game theory: one in which strategies reproduce by natural selection and another in which the mechanisms of selection of strategies are included in the model.8 We will take no more than a brief look at an example of the former type, which we will call type 1, as it takes for granted the mechanisms we want to understand. We will then discuss in more depth an example of the latter type, which we will call type 2. We will see what this approach achieves and how it fails to escape some of the rationalistic pitfalls of traditional game theory.

5. Complex Systems and Evolutionary Game Theory

Even in the 21st. century, weather forecasting seems to be a fiddly bit of science. Forecasts are famous for being all too often disastrously wrong. Only short term prediction is reliable, and this is still on a probability basis. The reason why it is hard to predict the weather is not so much that we do

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not understand the mechanics involved in it. The main problem does not lie in a lack of knowledge regarding intra-molecule interaction, but on the fact that the behaviour of every single molecule in the atmosphere has a role in the behaviour of the system as whole. As the climatic system cannot be decomposed into large chunks of functional parts as, say, the engine of a car, it resists traditional methods of analysis. The behaviour of the climate is a result of the rich interactions between its elements - it is a complex system. Human society, like the weather, is composed of a high number of functional elements. Interactions between humans are complex and it certainly seems that any model that ignores them will be overly reductive. In weather forecasting, thousands of measurements of the temperature and pressure of the atmosphere at different altitudes are obtained by satellite and weather balloons and are inputted into computer simulations. In the simulation, time can then be fast forwarded to yield predictions. The more detailed the model is, the more accurate will the predictions be. Much as even a perfect understanding of the interaction between two molecules would not yield good weather forecasting, game theoretic analysis of the interaction between two rational agents will not yield adequate models of the mechanisms of altruism. Evolutionary Game Theory (EGT) introduces the concept of population in game theory, eliminating a perhaps excessive reduction of standard game theory that may be throwing away the baby with the bath water. When playing the PD in a game theory environment, the agent who cooperates when the other defects will have the lowest payoff of all, as we have seen. He will get the so-called “sucker’s payoff’. It is only mutual cooperation that is beneficial for both, which is the reason why defection is such an alluring strategy. EGT models have been instrumental in showing that a cooperator will not necessarily do worse than a defector even when playing with other defectors. Consider a typical type 1 model: there is a virtual space, a grid in which a number of strategies, some “Cooperate” and others “Defect”, placed at random, interact. The game theoretic concept of utility is replaced with that of fitness (something similar to health points which reflect the ability to reproduce more effectively), and strategies play the one-shot PD repeatedly for fitness points. Note that repetition does not change the fact that the game keeps the characteristics of the one-shot PD. As long as the rules of the system make sure the players have no possibility of retaliating (for instance by not giving agents ways of identifying and remembering other players) they play each game as if it was the first and only time they played.

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Let us run the model. We will notice that the strategy of Defect, freeriding on Cooperate (taking advantage of its kindness) and not losing too many fitness points against other Defect strategies, will do quite well. Defect will do better on average than Cooperate. So far this confirms the predictions of game theory. Now consider we introduce an extra element of complexity in the model. When strategies are very healthy, having enough fitness points, we allow them to produce offspring. This is where EGT starts yielding considerably different results from game theory. If we program the rules of the model so that the offspring remain within a short distance from the parents, as happens in many biological systems, we will notice that local pockets of Cooperate will start to grow, as Cooperate will have a good probability of playing against other Cooperate strategies and get more fitness points on average. In this scenario, and depending on the variables regulating spatial distribution and the fitness points attribution matrix, Defect may be driven into extinction, or keep only a small population, surviving in relative isolation on the borders of populations of Cooperate they can exploit. A moral player does better that a rational player, and so cooperation can become an Evolutionarily Stable Strategy. (See [4]for a detailed explanation of the mechanics of standard EGT models). In order to achieve the results described above, however, careful tweaking of the variables is essential. We do get the results we want, but at the expense of rigging the experience, making sure the model is set-up so that nothing else can happen. This does not detract from the fact that it is good news that such a set-up is possible at all. There are however many more possible scenarios in which cooperation cannot evolve, due to the natural advantages of free-riding. The model described is also abstracting away quite a few important things. The characters of the plot are excessively flat, for instance. These agents always cooperate or always defect, regardless of context. Calling them strategies is more of a metaphor, as they are equivalent to their strategies, which means there is no process of decision. It is useful to have discovered that the spatial distribution of cooperators and defectors can be a deciding factor in cooperation dominating the population, but surely we do not always or mostly interact with our kin. The argument that was valid against kin selection can be again brought in here. There are, however, other more sophisticated applications of EGT that deal to this issue, such as Paul Danielson’s, which we will now examine.

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6. A solution to the Prisoner's Dilemma?

The EGT model we will look at is inspired in David Gauthier's naturalistic account of For Gauthier, morality is grounded on a principle of rational choice: we cooperate, respecting a contract that limits our freedom, because it is in our interest to do so. For instance, we accept that we are forbidden to steal as that agreement protects us from others who could steal from us. We trade freedom for security - an argument famously associated with Thomas Hobbes." Reason alone, however, does not seem to be enough to solve the one-shot PD, as we have seen. So how can we use the merits of rational choice theory to solve the problem? Gauthier's solution is that agents should apply rational choice not only to the maximisation of preference but also to the selection of behavioural disposition^.'^ They can then avoid the sub-optimal mutual defection outcome by acquiring a disposition to behave altruistically simply because it is rational to do so. After these behavioural dispositions have been acquired, the agent is merely serving his own interest in being altruistic. He now needs nothing but reason in order to solve the PD. Gauthier's account is apparently compatible with the demand based on common-sense that the label of genuine altruism can only be applied to behaviour that is disinterested. As the agent has a genuine interest in the welfare of his partner in the PD, he is not motivated by a material/strategic personal gain, even if he will in fact stand to gain from his generosity. The agent's behaviour can be self-interested without being selfish. Gauthier's account seems then to roughly match our demands of a reasonable account of altruism. Gauthier has been attacked over the years, however, for failing to explain how these dispositions would work. Even with a disposition to cooperate, the rational player knows that it is in his best interest to defect, and this is what he will still do. This is where EGT comes in. Peter Danielson used EGT models in an attempt to improve on Gauthier's work on altruism.21 For Danielson, the problem of mutual defection in the one-shot PD comes from the fact that, within game theory, the strategy of the rational agent always involves straightforward maximisation. This means the only rational course of action is for the agent to maximise utility at each point at which there is a decision to be made. The straightforward maximiser cannot but choose defection in the one-shot PD as, at the moment of decision, defection maximises utility. Danielson was interested in producing an agent that would not maximise utility at each decision point but one that would maximise average utility,

213 as only this agent could solve the PD. In order to produce it, Danielson asks

us to consider not the PD, but a slightly modified version of it that shares most of its structure. In what Danielson calls the Extended Prisoner’s Dilemma (XPD), one of the agents makes his move first. According to game theory, this does not change the payoff matrix or the sub-optimal result of mutual defection. The first player will still be better-off defecting, regardless of what the second player does. The second player will also still be better-off defecting, again regardless of what the first player has decided. However, now it would be rational for the first player to cooperate if he could somehow be sure that the second player would respond to cooperation and only to cooperation with cooperation. It would also be rational for the second player to use this strategy if proof that he will use it is the only thing that can convince the first player to cooperate. Danielson creates a constrained maximiser that is able to show his moral engine to his opponent as evidence of his intention to respond to cooperation with cooperation (and to defection with defection). The agent, a computer program, gives access to his own code to the other agent, which can run it to test it. As Danielson runs an EGT model in which there are such moral agents, these will soon dominate the population, even if they start off with just a small proportion of elements. Again, a moral player does better than a rational player, and this time no careful tweaking of initial conditions is necessary. We have seen before that no form of discussion or promise-making could result in a commitment in the game theory analysis of the PD. That is because talk is cheap. Regardless of what promises the agent would make to cooperate, he would have no reason to keep them because he knows that, as he will not play again against the same opponent, he cannot suffer retaliation for lying. Knowing this, his opponent would have no reason to take his promises seriously, and would be better off ignoring them altogether. Both will defect. In Danielson’s XPD, however, commitment becomes possible because there is no need for promises. There is in fact no real commitment, in the psychological sense. If I say I behave like a human being because I am a human being and present as evidence my healthy human body, I am not just promising to behaving like a human being, I am proving I will behave like one, by necessity. Similarly, the agent that plays first knows that the other player will respond to cooperation with cooperation in virtue of its moral make-up. It will be irrational for the first player to defect, as the retaliation is inevitable and automatic - the equivalent to the cold war’s doomsday device. It seems we have got to love the bomb.

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7. The role of morality According to LaCasse and Ross, Danielson’s work promises to provide the answer to what they believe is an essential question: “What is the role of morality?”22 In their paper, Morality’s Last Chance, they start from a position of moral scepticism, doubting that any role can be found, and argue that Danielson fails to satisfy, actually giving them tools to strengthen the case against the existence of morality. An analysis of LaCasse and Ross’s critique of Danielson will provide us with interesting insights on the advantages and limitations of the uses of EGT in research on altruism. For LaCasse and Ross, morality is certainly not something that exists independently from human beings, as in Plato. They do not find Rawlsian arguments, inspired in Kant’s claim that moral behaviour is an inescapable consequence of being rational, particularly compelling. For them, if there is such as thing as morality at all, it requires a kind of justification that could only be found in contractarianism, in the tradition of the work of David Gauthier. The central argument in Danielson’s argument, which is in Gauthier’s tradition, is that a moral player does better (materially speaking) than a rational player. LaCasse and Ross believe that, if the claim turns out to be true, there is then a role for morality. In order to test Danielson’s claim, they proceed to analyse Danielson’s XPD from a conventional game theory approach. For them, according to game theory, as player 2 reveals his strategy before player 1 makes his move, player 2 should actually be represented as playing first. Player 1’s actions are indeed fully determined by player 2’s strategy. If we look at it this way, a straightforward maximiser does not do worse than a moral agent. Indeed they do exactly the same. It seems, then, claim LaCasse and Ross, that once we change the PD into the XPD, give agents the ability to apply rational choice in choosing their behavioural dispositions (strategies) and reveal their strategies to others, there is no further role for morality, as reason alone can solve the dilemma. 8. Discussion

If LaCasse and Ross are right, Danielson’s work is a complete waste of time. But let us consider what kind of argument would convince them they were wrong. What kind of argument would they accept? Hume said that “ought” cannot be derived from “is”, to mean that normative statements (what is morally good) cannot be derived from prudential considerations (what is advantageous). LaCasse and Ross seem to agree with Hume, when

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they refuse to accept Danielson’s explanation for how altruism can exist. His “is” arguments cannot bridge the gap t o “ought”. However, they seem to demand such impossible bridging arguments, since they consider that the failure t o provide them can help them to prove morality has no role. LaCasse and Ross seem to expect a normative account of morality to be based on prudential explanations. This is indeed an impossible mission. Nevertheless, failing it has no implications for the possibility of a naturalistic account of altruism. If Hume was right in saying that reason is but a slave of passions, such an account cannot be grounded on rational reasons. According to Hume’s account of action, reason only job is to find the means to ends defined by passions (desires). Emotions, then, are more likely acceptable candidates for the causes behind altruistic action, when we ask for a psychological explanation. And when we ask for the foundations of such psychological explanations, we will have to look a t what evolutionary pressures produced such emotions, so as to complement Hume’s account of motivation. Economist Robert Frank offers us an account of altruism that deals with both the psychological and the evolutionary aspects of the problem. For Frank, emotions, like Danielson’s or Gauthier’s behavioural dispositions, act as enforcers of commitment. This needs not be seen as a deviation from rationality if we accept Hume’s minimal notion of reason, according t o which it is rational t o do whatever we desire. We have expressed some reservations regarding the fact that Hume’s account in incomplete, but Frank’s, however, includes the essential explanation for why moral sentiments do not compromise the organisms’ viability. Holding moral sentiments, for Frank, may be an evolutionary advantage, as this can make the agent a preferred target for cooperation. The agent’s sense of justice implies that he may seek revenge at any cost, when he considers he has been treated unfairly. This will act as a deterrent t o potential offenders. These mechanisms can only work if it is somehow visible to others that the agent is a moral agent (perhaps because he sounds and looks honest). It must also be true that mimicking the appearance of a moral agent is quite costly, from an evolutionary point of view, or we would all pretend to be honest without being it, enjoying the advantages without incurring the costs. While it is not easy to provide empirical proof to support these claims, Frank’s story succeeds in providing an account for situations in which one is generous even when one knows the other may have no chance of returning one’s generosity. Holding moral sentiments may mean that, a t times, one

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will engage in costly self-sacrifice for someone else’s sake. However, the theory will remain compatible with Darwinist evolution as long as holding moral sentiments is advantageous on average. To be fair to Hume, we should note that he does include a crucial component of a kind of non-intentional consequentialism in his account of how desires are formed. The story goes as follows. We desire t o help some rather than others because we have come to love them. But for Hume we come t o love those who have something that interest us: a quality we admire, fortune, influential friends. This is not because we are planning to take advantage of such people. The love is genuine. It is simply in our nature to love those we admire. After this genuine love is formed, it will suit our purposes to help such people, as they will be in a position to return more than we have gave them - only this advantage is now accidental. Interestingly, Hume’s account of how we come to love can help t o solve a major problem in Frank’s theory, as we will see. Robert Frank goes one step too far when he argues that the selfish person who, after reading his book, now knows that being an altruist is advantageous, will start acting altruistically. This is the mistake Gauthier has made and Danielson was trying to avoid. Acts of genuine altruism do involve sacrifices, and if we accept Frank’s account, it is only advantageous to be an altruist on average. Each individual act of altruism remains something to be avoided by a selfish person, regardless of whether he agrees with Frank’s account. So Frank still leaves us in need of an explanation for how we can come t o be altruists. Evolution has made it possible for us to feel the emotions that make genuine altruism possible, but we are still not born natural altruists. How can these emotions be used in the right way? How can they serve our interest? Hume’s account of how we come to love others can be part of it. We love those who have something to offer us. But here we have an interesting case in which genuine sacrifice leads to material advantage, and that cannot always be the case. Genuine altruism can involve straightforward loss, like the sacrifice of one’s own life for the sake of someone else’s welfare. A r i ~ t o t l eoffers ~ ~ an interesting account of moral education, according to which we come t o know what is right, as children, by doing the right thing. We simply go through the motions, mechanically at first, and then end up enjoying it. This is indeed how we come to like music, or sports: by listening to music, by doing sports, not by theorising about it. It could be the way how we come to enjoy moral action. So where does this leave Danielson’s work on altruism? His model shows

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that while cooperation may seem irrational at a certain level of analysis, it may actually become rational at another, once we understand the details of the mechanism that produces it. It is true that Danielson did cheat. He did change the rules of the game: the XPD is simply not the PD. In the standard PD, the players cannot commit, by an impossibility defined by the rules. EGT also corrupts the essential nature of the one-shot PD. It is true that in the models described, players do not have a memory of what is happening and therefore cannot retaliate, but as the system will hold such a memory under the form of physical location and respective proportions of cooperators and defectors, punishment can come under the form of a defector being eventually surrounded by defectors as it reproduces. What Danielson did was to reframe the question, transforming the problem. Perhaps expressing the problem of altruism by looking for a traditional game theoretic solution to the prisoner’s dilemma was like asking what the evolutionary advantage of having half an eyeball could be (half an eyeball that will fall off its socket), on the route to a full eye. These are not problems evolution had to deal with. It could be fine, then, to rephrase a question that made little sense. Danielson’s approach is promising in the sense that it can reveal that some problems that apparently can only be solved by genuine altruism can actually be shown by EGT analysis to be solvable by some form of reciprocal altruism. Davidson has not dealt satisfactorily, however, with the issue of genuine altruism. Peter Danielson does not include an explanation for how us, humans, could show our lines of code to others, in order to prove our intentions to cooperate. Is it the case that only computers can behave morally? It would be possible to improve the model by including some extra details, perhaps giving the agents a neural network and some kind of simulated emotions. This would make it possible to study scenarios that could explain why humans are rarely very good liars, for instance. Perhaps evolution never had to solve commitment problems. It is not hard to conceive of an evolutionary story in which bad liars were produced as an accident, as a compromise between the advantages of being able to lie convincingly, pretending to be a cooperator, and the advantages of communicating one’s emotions effectively and automatically, as infants. This could mean the efforts of EGT must be combined with those of developmental psychology and neuroscience.

218 9. Conclusions

One important lesson to learn is that models of complex systems are not at all immune to rationalistic assumptions. These can indeed dangerously creep up unseen, covered up by the apparent realism of this methodological approach. This seemingly down to earth, pragmatic nature of Evolutionary Game Theory cannot make us forget its models are still heavily theory laden. Danielson’s story is a story of how to love the bomb, as it claims altruistic actions can be made rational if they can rely on a mechanism of deterrence. We have seen that being moral cannot compromise a system’s physical viability. The merit of the Prisoner’s Dilemma puzzle is that it highlights this issue, a problem the Humean account of motivation does not deal with directly. Acting altruistically therefore needs to be materially advantageous, at least on average. In search for the elusive connection between morality and rationality, Danielson has followed Gauthier’s approach of reducing altruism to rational action. Danielson demands a direct connection between physical viability and action, one which makes genuine altruism impossible. If adequate evidence that genuine acts of altruism are an illusion and can be explained in terms of pure rationality could be provided, we should be ready to accept it. It seems, however, that those who are happy about reducing altruism to rationality do not do it because they have found such evidence, but rather because they are confused by the fact that altruistic acts paradoxically seem to serve the individual who performs them. Although their intuition might be telling them the same story those who defend in genuine altruism believe in, they reject it possibly on the grounds that we may be fooling ourselves. In fact, we often do. Perhaps we like to think we are being kind when we are actually always trying to further our interests. But why would we be any happier for thinking we are morally good if we are not? If we accept this happiness is a result of purely cultural reasons, we are still left with no explanation for why culture would favour such self-delusion. Keeping genuine altruism and explaining why it is possible offers us a much more satisfactory story of human nature. An extra cog is needed in the mechanisms of causality; one that sometimes seems to be turning the wrong way, and emotions are a good candidate for this. Danielson’s other interesting work in applied ethics has done much to establish EGT as a powerful tool for understanding and predicting human behaviour. However, EGT, in its present stage, is unable to offer an adequate account of altruistic behaviour. Danielson may be right in ex-

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pecting deterrence to be the reason (foundation) why altruism exists, but he should not expect it to be the reason (motivation) for why we act altruistically. Perhaps it is simply the case that some of our interests are in fact the welfare of others, and that it is rational to act on them. Hume’s psychological story can now become plausible and is confirmed and completed by Frank’s and Aristotle’s account. It is not possible to solve the single-shot Prisoner’s Dilemma with game theory or evolutionary game theory. Only genuine altruism can solve it. Deterrence may have shaped emotions, the mechanisms that allow commitment, but there are surely cases in which it is not out of fear or considerations of possible material advantages that we are generous or kind. This is not being sentimental. It is important t o realise that although the moral person performs altruistic acts for moral reasons, not self-interest, he is not a fool. He will profit from cooperation and will refuse t o be exploited. Attempts t o free-ride will be punished. This means there are no exceptions to the advantages of being a moral person, as not only it serves the person who performs the altruistic acts when he deals with other moral people, but also with those whose behaviour is more often governed by self-interest. Mechanisms of reciprocity such as fear of retaliation are but one of the reasons why we cooperate, and moral sentiments are an essential complement t o such mechanisms. There is space for altruism within rationality. We should then love the bomb, only not because it motivates us t o cooperate, but rather because the strategic virtues of deterrence are part of what makes us want to act morally.

Acknowledgments

I conducted the research presented in this paper with a PhD scholarship from the Portuguese Foundation for Science and Technology (FCT). I would also like to thank Peter Goldie, Federica Frabetti and Marina Hasselberg, for a number of interesting discussions on the topic of altruism References 1. R. Frank Passions within Reason: the strategic role of emotions. Norton (1988). 2. C. Darwin The Descent of Man and Selection in Relation to Sex. New York: Appleton (1871) 3. F. Heylighen, Evolution, Selfishness and Cooperation. In: Journal of Ideas Vol 2, NO 4, pp 70-84 (1992)

220 4. B. Skyrms (Ed.) Evolution of the Social Contract Cambridge University Press (1996) 5. P. Kollock, Transforming Social Dilemmas: Group Identity and cooperation. In: B. Skyrms (Ed.) Evolution of the Social Contract Cambridge University Press (1996) 6. W. D. Hamilton, The Genetical Evolution of Social Behaviour I and 11. In: Journal of Theoretical Biology, 7: 1-16, 17-32 (1964) 7. J. Maynard Smith, Group Selection and Kin Selection. In: Nature, 201: 11451147 (1964) 8. J. M. Alexander, Evolutionary Game Theory. In: Edward N. Zalta(ed.), /textitThe Stanford Encyclopedia of Philosophy (Summer 2003 Edition) URL = ~http://plato.stanford.edu/archives/sum2OO3/entries/g~e-evolutionary/~. 9. M. Nowak and K. Sigmund, Evolution of Indirect Reciprocity by Image Scoring / The Dynamics of Indirect Reciprocity. In: Nature 437:1291 (2005) 10. R. Dawkins The Selfish Gene. Oxford University Press (1976) 11. R.L. "rivers, The Evolution of Reciprocal Altruism. In: /textitQuarterly Review of Biology, 46: 35-57 (1971) 12. R. Axelrod and W. D. Hamilton, The Evolution of Cooperation. In: Science, 211: 1390-96 (1981) 13. S. Okasha, Biological Altruism. In: Edward N. Zalta(ed.) The Stanford Encyclopedia of Philosophy (Summer 2005 Edition) (2005) URL = ~http://plato.stanford.edu/archives/sum2OO5/entries/altruism-biological/~. 14. M. Osborne, An Introduction to Game Theory, Oxford University Press (2003) 15. D. Gauthier, Morals by Agreement, Oxford University Press (1986) 16. T. Hobbes, Leviathan, Oxford World's Classics (1998) 17. D. Hume, A Treatise of Human Nature Oxford University Press (1978) 18. D. Hume, Enquiries Concerning Human Understanding and Concerning the Principles of Morals Oxford University Press (1975) 19. D. Gauthier and R. Sugden (eds.), Rationality, Justice and the Social Contract, Harvester Wheatsheaf (1993) 20. P. Danielson (ed.) Modelling Rationality, Morality and Evolution, Routledge (1998) 21. P. Danielson, Artificial Morality, Routledge (1992) 22. C. LaCasse and D. Ross, Morality's last chance. In: P. Danielson (ed) Modelling Rationality, Morality and Evolution, Routledge (1998) 23. J. Rawls, A theory of justice, Harvard University Press (1971) 24. Crisp, Roger (ed.), Nichomachean Ethics, Cambridge University Press (2000)

PHYSICAL COMPLEXITY AND COGNITIVE EVOLUTION*

PETER JEDLICKA Institute of Clinical Neuroanatomy J. W. Goethe University fiankfurt/Main, Germany E-mail: [email protected]

Our intuition tells us that there is a general trend in the evolution of nature, a trend towards greater complexity. However, there are several definitions of complexity and hence it is difficult to argue for or against the validity of this intuition. Christoph Adami has recently introduced a novel measure called physical complexity that assigns low complexity to both ordered and random systems and high complexity to those in between. Physical complexity measures the amount of information that an organism stores in its genome about the environment in which it evolves. The theory of physical complexity predicts that evolution increases the amount of ‘knowledge’ an organism accumulates about its niche. It might be fruitful t o generalize Adami’s concept of complexity to the entire evolution (including the evolution of man). Physical complexity fits nicely into the philosophical framework of cognitive biology which considers biological evolution as a progressing process of accumulation of knowledge (as a gradual increase of epistemic complexity). According t o this paradigm, evolution is a cognitive ‘ratchet’ that pushes the organisms unidirectionally towards higher complexity. Dynamic environment continually creates problems t o be solved. To survive in the environment means t o solve the problem, and the solution is an embodied knowledge. Cognitive biology (as well as the theory of physical complexity) uses the concepts of information and entropy and views the evolution from both the information-theoretical and thermodynamical perspective. Concerning humans as conscious beings, it seems necessary to postulate an emergence of a new kind of knowledge - a self-aware and self-referential knowledge. Appearence of selfreflection in evolution indicates that the human brain reached a new qualitative level in the epistemic complexity.

1. Introduction Our intuition suggests that there is a general trend in the evolution of nature, a trend towards greater complexity. According t o this view evolution is a progressive process with Homo sapiens emerging a t the top of the life’s *This work is supported by Graduiertenkolleg: Neuronale Plastizitat - Molekule, Strukturen, Funktionen (J. W. Goethe University)

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222 hierarchy. An alternative opinion is that there is no trend in evolution and we are just one of many leaves on the evolutionary tree. There are several definitions of complexity and hence it is difficult to argue in a quantitative way for or against the validity of each of these views. In my article, I will be focusing on a novel measure of biological complexity (physical complexity) that has been proposed by Christoph Adami. 2. Kolmogorov - Chaitin complexity and biology

One well-known definition of complexity is the Kolmogorov-Chaitin complexity (introduced independently by Kolmogorov ( l ) , Chaitin (2) and Solomonoff (3)). It represents an algorithmic measure of system’s randomness (4, 5). It defines the complexity of an object (or a process) by the size of the smallest program for calculating it. For example, a long sequence of digits with a regular pattern (e.g. 01010101...) can be compressed to a much shorter description/program (“repetition of 01”) and hence it has a small amount of complexity. In contrast, an infinite random sequence of digits “19255324193625168147...” has no intrinsic structure and cannot be compressed at all. The only way to express the sequence is to enumerate all the digits that it consists of. Thus, Kolmogorov - Chaitin’s definition implies that maximum complexity is ascribed to a completely random process that is algorithmically incompressible (Fig. 1, c.f. Ref. 6).

complexity

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Figure 1

However, although useful for the theory of computation, KolmogorovChaitin complexity does not satisfy our expectations inspired by biology that most complex organisms are neither completely regular nor utterly

223 random but lie between the two extremes of order and randomness (7). What makes living systems complex is the interplay between order and randomness (8). This principle has been supported by recent research on biological networks (9). Complex networks of cellular biochemical pathways have neither regular nor random connectivity. They have a so called scale-free structure (10). A scale-free network contains a small number of hubs - major nodes with a very high number of links whereas most nodes in the network have just a few links. “Scale-free” means that there is no welldefined average number of connections to nodes of the network. In case of a biochemical network, nodes and links represent molecules and their chemical reactions, respectively. In such a molecular network, hubs are important molecules that participate in a large number of interactions (e.g. CAMP, H20) in comparison to other molecules that take part in a few biochemical signaling paths (11). An important consequence of the scale-free architecture is robustness against accidental failures of minor nodes/links and vulnerability to disfunctions of major hubs. (Note that this may bring interesting insights into the molecular pathophysiology of various diseases.) Surprisingly, recent mathematical analysis of various scale-free networks (including biochemical signaling networks) has revealed self-similarity (fractal pattern) in their structure (12). Self-similarity is a typical property of systems that are on the edge of order and chaos. Such a critical state (with a tendency to phase transitions) might be useful for optimizing the performance of the system (13). To sum up, we need a measure which would capture the complexity of dynamical systems that operate between order and randomness.

3. Physical complexity Christoph Adami (14) has recently introduced a novel measure called physical complexity that assigns low complexity to both ordered and random systems and high complexity to those in between (Fig. 2). Physical complexity measures the amount of information that an organism stores in its genome about the environment in which it evolves. This information can be used to make predictions about the environment. In technical terms, physical complexity is a shared (mutual) Kolmogorov complexity between a sequence and a n environment (for mathematical equations see Ref. 14). Information is not stored within a (genetic) sequence but rather in the correlations between the sequence and what it describes. By contrast, KolmogorovChaitin complexity measures regularity/randomness within a sequence and

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therefore fails to address its meaning in the environment.a Information is a relative quantity - always about something, in case of the genome it is about the niche that an organism lives in (16, 17). The theory of physical complexity uses two central concepts of Shannon’s theory of information: entropy and information. Entropy is a measure of potential information, it determines how much information a sequence could hold. Entropy of a sequence can be compared to the length of a tape, and information to the length of a tape portion containing recordings (14). Physical complexity has the advantage that it is a practical measure, because entropy of an ensemble of genetic sequences can be measured by estimating the probabilities of finding the genetic sequences in the environment.

maximum I

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Figure 2 .

Evolution is a slow process and therefore obtaining exact data is difficult. How could we test the hypothesis of the evolutionary complexity increase? An interesting option is to study digital evolution (18). Digital organisms (digitalia) are self-replicating computer programs (sequences of instructions) that mutate and compete for space and computer time (19). Thus, digital evolution occurs since the three conditions of evolution are met: replication, variation (mutation) and competition/selection (due to differential fitness/replication rate). Digital organisms have a much shorter generation time (seconds) and their physical complexity can be measured exactly. In this case, physical complexity is the information in a digitalia aInterestingly, Chaitin has proposed his own definition of life based on Shannon’s concept of mutual information. See Ref. 15.

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program that is vital to organism’s survival (replication). Of course, digital evolution is a highly simplified model of biological evolution and this creates the possibility of artifact conclusions or generalizations. On the other hand, simplification is at the same time the greatest strength of simulated evolution experiments since it allows us to find and see the forest (general principles of evolution), not just the trees. Interestingly, Adami’s experiments on digital organisms revealed a steady trend toward increased physical complexity in their evolution within a fixed environment (16). According to his theory and simulation data, evolution increases the amount of ‘knowledge’ an organism (or a population of organisms) accumulates about its niche. Since in this context entropy is a measure of potential information, biological evolution leads to a decrease of entropy. Natural selection turns an e m p t y tape into a filled tape: entropy into information. The information-filled part of the tape is important for the survival of an organism in the environment. If the selective advantage fixes a beneficial mutation within the population, the amount of information (physical complexity) increases. Adami views natural selection as a unidirectional filter (see below the discussion of ratchetting) that lets information enter the genome, but prevents it from flowing out. Adami’s simulations were done in a fixed environment. He discusses that a rapidly changing environment as well as several other factors that were not included in his experiments (high mutation rates, sexual recombination, co-evolution between species occupying different niches) may lead to complexity declines. However, these factors are ambivalent - sometimes they help, rather than hinder, the evolution of complexity (14). Therefore he argues that there are good reasons to expect that the theory of physical complexity will reveal an overall trend towards higher complexity in biological evolution.

4. Physical complexity and the concept of cognitive biology It might be fruitful to generalize Adami’s concept of complexity (which has been primarily thought to describe the evolution of genome) to the entire evolution including the evolution of man (Fig. 3). Physical complexity fits nicely into the philosophical framework of cognitive biology which considers biological evolution as a progressing process of accumulation and application of knowledge, i.e. as a gradual increase of epistemic complexi t y (20, 21). Cognitive biology provides a broader philosophical frame for Adami’s approach since the central idea of his theory of physical complexity

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is relating complexity to the system’s ‘knowledge’ (information) about its environment. Physical complexity can be viewed as a special case of epistemic complexzty. This becomes clear if we look at many common features of Adami’s ideas and cognitive biological ideas. According to the paradigm of cognitive biology, evolution as a whole is a cognitive ‘ratchet’ that pushes the organisms unidirectionally towards higher complexity. Epistemic ‘ratchetting’ operates at all hierarchical levels, from molecules to societies (20). Dynamic environment continually creates problems to be solved (i.e. each niche is a solution problem). To survive in the environment means to solve the problem, and the solution is an embodied knowledge.

The Concept of Physical Complexity w.krl.caltech.edu/-adami

1

generalization to the entire evolution including the evolution of man

The Paradigm of Cognitive Biology w.fns.uniba.sk/-kbi/kovlablprincip.htrn

Figure 3.

‘Ratchetting’ is a general phenomenon that has been usually described in the context of thermodynamics. Cognitive biology acknowledges that progress in evolution has thermodynamical reasons (20). Both cognitive biology and the theory of physical complexity use the concepts of infomnat i o n and entropy and strongly emphasize that there is a close connection between thermodynamical thinking and information-theoretical approaches. By thermodynamical reasoning we can identify a ‘differentiation of a system from environment’ as a dissipative ‘movement from thermodynamic equilibrium’ (22). Biological evolution creates organisms with an ever increasing amount of embodied ‘knowledge’ and with an ever farther distance from thermodynamical equilibrium (20). Living systems are far away from the equilibrium because of the information stored in their genomes (17). John

227 Polkinghorne (23) predicts that “by the end of the twenty first century, information will have taken its place alongside energy as an indispensable category for the understanding of nature.” The paradigm of cognitive biology points t o the same direction.

5. Self-referential cognition - a ‘Big Bang’ of complexity in cognitive evolution? Concerning humans as conscious beings, it seems necessary t o postulate an emergence of a new kind of knowledge - a self-aware and self-referential knowledge. We not only know, we know that we know. We not only have informations but we possess informations about informations (metainformations) as well. Cognitive biology itself is an example of the selfreferential knowledge. It is a theory (knowledge) about accumulation of knowledge. Appearence of selfreflection in evolution indicates that the human brain reached a new qualitative level in the epistemic complexity. One may speak of a cognitive ‘Big Bang’. The expression of this cognitive ‘explosion’ that occured in human species may be found in the development of science, art, culture and religion. Many writers have noticed this interesting phenomenon. Gilbert Keith Chesterton wrote that “man is not merely an evolution but rather a revolution.” Since the most primitive men drew pictures and the most intelligent monkeys don’t, “art is the signature of man” (24). What are the mechanisms of self-awareness? Can we describe the emergence of self-referential knowledge in the mathematical language of empirical science? Is it possible to extend the theory of physical complexity and formalize the meta-knowledge which is characteristic for our species? Interestingly, Godel’s theorem shows a basic limitation of formal methods due to self-reference (25). His results demonstrate that it is impossible to achieve a complete knowledge of a (nontrivial) formal system with the means available within the system. To fully understand the formal system one must go outside the system (otherwise one falls into inconsistencies). It is interesting to apply this principle to human cognition. According t o the computationalism, our mind basically functions as a certain (unknown) computational algorithm. However, a Godelian line of thinking would suggest that we are not able to find this out because we cannot step out of the algorithm. If this reasoning is valid then there are two possibilities: either we are no mechanical machines or we are machines but we cannot know it because of the limitation imposed by computational selfreference. It is

228 the limitation of an observer in “Cartesian prison” (26). In a paraphrase of Stephen Hawking’s words (27): we don’t see the mind from the outside. Instead, we and our models, are both part of the mind we are describing. Thus, a theory of human cognition is self-referencing, like in Godel’s theorem. One might therefore expect it to be either inconsistent, or incomplete. So it seems to be improbable that a complete and consistent formalization of human epistemic complexity is possible (28). Based on Godelian and other arguments, some authors argue that the evolutionary leap from ‘pure’ cognition to self-referential cognition might have been governed by some novel noncomputational principles. (It should be noted that Godelian arguments are highly controversial and the interesting debate about their validity continues. See e.g. Ref. 29, 30, 31.) To account for the emergence of consciousness, new physical principles (32) or known physical (quantum) principles operating in the brain in a special way have been suggested. An interesting quantum brain proposal was published by Jeffrey Satinover (33) that combines findings in computational neuroscience, nonlinear dynamics and quantum physics. Since indeterminism observed in quantum events is sometimes interpreted as a fundamental time asymmetry (34), it is tempting to speculate that if quantum brain hypotheses contain some truth then a deeper link might connect thermodynamic, cosmological and epistemic ‘ratchetting’ processes (for introduction to quantum ‘ratchetting’ see e.g. Ref. 35). If it is not possible to entirely reduce emergent human consciousness to neuronal computation (28), then an interesting philosophical question arises, namely what kind of emergence is responsible for it. Mark Bedau distinguishes nominal, weak and strong emergence (36, see also Ref. 37). If a new kind of causation powers (so called ‘downward causation’) has been brought about by the emergence of selfreferent cognition, then we can call it a strong emergence, according to Bedau’s definitions. The existence of downward causation belongs to intensely discussed topics in the philosophy of science. I a recent article, George Ellis (38) describes a hierarchical view of the complexity of the universe with the autonomously effective mind at the top of it: “the higher levels in the hierarchy of complexity have autonomous causal powers that are functionally independent of lower-level processes. Top-down causation takes place as well as bottom-up action, with higher-level contexts determining the outcome of lower-level functioning ...” These words resemble the Satinover’s hypothesis (33) of a hierarchy of nested networks (similar to Hofstadter’s “tangled hierarchy”, Ref. 39). An attractive speculation is that the strong emergence might be related to

229 quantum phenomena since quantum theory postulates that quantum events cannot be reduced t o some form of bottom-level microscopic law governing their outcomes (41, 40). In addition to information, the downward causation (or - in Aristotelian terms - the inside-out causation, Ref. 42) seems t o be a very interesting topic for discussion and research in complexity science in the next years.

Acknowledgements T h e author thanks Dr. Stephan W. Schwarzacher for valuable discussions and for reading the manuscript

References 1. A.N. Kolmogorov, Three approaches to the quantitative definition of information. Problems Information Transmission 1:l-7 (1965) 2. G.J. Chaitin, On the length ofprograms for computingfinite binary sequences. J Assoc Comput Mach 13:547-569, (1966) 3. R.J. Solomonoff, A formal theory of inductive inference, part 1 and 2. Inform Contr, pp. 1-22, 224-254 (1964) 4. G.J. Chaitin, Computers, paradoxes and the foundations of mathematics. Sci Am 90:164-171 (2002a) 5. G.J. Chaitin, On the intelligibility of the universe and the notions of simplicity, complexity and irreducibility. In: Grenzen und Grenzberschreitungen, XIX. Deutscher Kongress fur Philosophie (Hogrebe W, Bromand J, ed) Akademie Verlag, Berlin, pp. 517-534 (2002b) 6. J.P. Crutchfield, When Evolution is Revolution Origins of Innovation. In: Evolutionary Dynamics Exploring the Interplay of Selection, Neutrality, Accident, and Function (Crutchfield JP, Schuster P, ed) Santa Fe Institute Series in the Science of Complexity, Oxford University Press, New York (2002b) 7. J.P. Crutchfield, What lies between order and chaos? In: Art and Complexity (Casti J, ed) Oxford University Press (2002a) 8. P. Coveney, R. Highfield, Frontiers of Complexity: The Search for Order in a Chaotic World, Ballantine Books (1996) 9. A.L. Barabasi, Z.N. Oltvai, Network biology: understanding the cell’s functional organization. Nat Rev Genet 5:lOl-113 (2004), www.nd.edu/ networks 10. A.L. Barabasi, E. Bonabeau, Scale-free networks. Sci Am 288:6(r69 (2002) 11. D. Bray, Molecular networks: the top-down view. Science 301:1864-1865 (2003) 12. C. Song, S. Havlin, H.A. Makse, Self-similarity of complex networks. Nature 4331392-395 (2005) 13. S.H. Strogatz, Complex systems: Romanesque networks. Nature 433:365-366 (2005) 14. C. Adami, What is complexity? Bioessays 24:1085-1094 (2002)

230 15. G.J. Chaitin, Toward a mathematical definition of “life”. In: The M m i m u m Entropy Formalism (Levine RD and Tribus M, ed), pp. 477-498, MIT Press (1979) 16. C. Adami, C. Ofria, T.C. Collier, Evolution of biological complezity. Proc Natl Acad Sci U S A 97:4463-4468 (2000) 17. C. Adami, Information theory in molecular biology. arxiv.org/abs/qbio.BM/0405004 (2004) 18. The Digital Life Laboratory at the California Institute of Technology: http://dllab.caltech.edu 19. C.O. Wilke, C. Adami, The biology of digital organisms. Trends Ecol Evol 17~528-532 (2002) 20. L. Kovac, Fundamental principles of cognitive biology. Evol and Cognition 6~51-69 (2002) 21. H. Kuhn, Origin of life and physics: Diversified macrostructure - inducement to form information-carrying and knowledge-accumulating systems. J Res Develop 32: 37-46 (1988) 22. 1. Prigogine, The End of Certainty. Free Press (1997) 23. 3. Polkinghorne, The person, the soul, and genetic engineering. J Med Ethics 30~593-597 (2004) 24. G.K. Chesterton, The everlasting man. Ignatius Press (1993, orig. 1925) 25. K. Godel, Uber formal unentscheidbare Satze der Principia Mathematica und venuandter Systeme I. Monatshefte fur Mathematik und Physik 38:173-198 (1935) 26. K. Svozil, On self-reference and self-description. In: Functional Models of Cognition, (Carsetti A, ed) Kluwer, Dordrecht, p. 18S197 (2000) 27. S. Hawking, Godel and the end of physics. Lecture given on Texas A&M’s College Station campus. www.damtp.cam.ac.uk/strtst/dirac/hawking/ 28. P. Jedlicka, Neuroethics, reductionism and dualism. Trends Cogn Sci 9:172 (2005) 29. B. Giolito, Monographica: John Lucas against mechanism. Etica & Politica, (5) (2003) http://www.univ.trieste.it/ etica/2003-l/index.html 30. M. Redhead, Mathematics and the mind. Brit J Phil Sci 55:731-737 (2004) 31. S. Bringsjord, A. Shilliday, J. Taylor, B. Bram van Heuveln, J. Baumes, K. ROSS, Y. Yang, A new Giidelian argument for hypercomputing minds based on the busy beaver problem. J Comput Appl Math (2005) http: //kryten.mm.rpi.edu/laihyper 11.pdf 32. R. Penrose, Shadows of the Mind: A Search for the Missing Science of Consciousness, Oxford University Press (1994) 33. J. Satinover, Quantum Brain, John Wiley & Sons (2001) 34. D. Zeh, The Physical Basis of The Direction of Time. -.time-direction.de (2004) 35. M. Brooks, Quantum clockwork. New Scientist 165:28 (2000) 36. M. Bedau, Downward causation and the autonomy of weak emergence. Principia 6:5-50 (2003) 37. P. Humphreys, How Properties Emerge. Philosophy of Science 64:l-17 (1997) 38. G.F. Ellis, Physics, complexity and causality. Nature 435:743 (2005)

231 39. D.R. Hofstadter, Godel, Escher, Bach: A n Eternal Golden Braid, Basic Books (1979) 40. F.M. Kronz, J.T. Tiehen, Emergence and quantum mechanics. Philosophy of Science 69:324-347 (2002) 41. K. Svozil, Computational universes. In: Space Time Physics and Ractality (Weibel P, Ord G, Rossler OE, ed) Springer Verlag, Wien, New York, p.

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INFORMATIONAL DYNAMIC SYSTEMS: AUTONOMY, INFORMATION, FUNCTION

WALTER RIOFRIO Neuroscience and Behawiour Division Universidad Peruana Cayetano Heredia. The main purpose of this paper is the conceptual exploration into the pre-biotic world, a world that we consider t o be made up of a relatively continuous sequence of systems directed toward the origin of the first life forms. In a tentative way, we propose the ‘Informational Dynamic System’. This type of system would constitute the ancestral system that definitively opened the door t o the pre-biotic world. Also, it explains, in a naturalistic way, the physical emergence of functions and information.

1. Introduction

The so-called self-organization phenomena feature multiple types of dynamic systems, and we find them in different sectors of our reality. For example, we have the BCnard cells and the Belousov-Zhabotinskii reactions (or Brusselator when this type of chemical reaction is computer simulated). It was Ilya Prigogine who, working on these chemical processes, coined the term “dissipative structures”, allowing us to understand the emergence of ‘novelties’ in the systems that are far from thermodynamic equilibrium state. From the thermodynamic point of view, these systems are open systems (they exchange energy and/or matter with their environment), and the self-organization phenomena are produced due to the continuous changes in the local interactions of the components that form the dynamic system 111. The Belouzov-Zhabotinskii (B-Z) reaction is an oscillating chemical reaction that is autocatalytic. In other words, products of the reaction increase the rate of their own production. In this process, different steadystates and periodic orbits can be observed. The Cerium oxides, Ce(II1) and Ce(IV), are the chemical compounds in solution that produce an oscillation dynamic in the reaction. The periodic increase in the concentration of one of the chemical compounds and the subsequent decrease of the other is seen

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at the macroscopic level as periodic changes and rhythms of color [2]. What this type of self-organized system has in common with others of the same type is that the continuous entropy production in the system is actively exported (dissipated) to the outside of the system. These types of self-organized dynamic systems reveal to us certain generic properties that can be applied to the rest of the self-organizational phenomena [3-5). (1)The appearance of a global order starting from local interactions. (2) The control of the organization is distributed throughout the whole system. (3) They have a relative “insensitivity” to environmental changes; they are robust. (4) Self-organized phenomena are essentially non-linear. (5) Selforganization contains feedback processes - positive and negative. (6) The degree of coordination or correlation between the system components is associated with the degree of organization in the self-organized dynamic system. (7) Self-organization is fundamentally caused by a dynamic inside the system; because of that, the system possesses “organizational closure”. (8) There are additional properties that begin to emerge in self-organized systems as the internal dynamic becomes more complex. The interest of our work will be centered on the emergence of certain properties in determined, self-organized systems: those that we find in the time before the appearance of the first life forms. We will propose an initial working hypothesis with the purpose of contributing to the discussion and improving the understanding or clarification of what more than likely happened in the distant past, in the transition from being inanimate to being animate. Therefore, we will be occupied with exploring the nature of the local interactions that made it possible for the appearance of the initial forms of dynamic systems that had the global property of being an autonomous agent [6-81. Our strategy will be to approach the problem from a bottom-up perspective in naturalist terms. Consequently, we will make explicit references in our arguments to the ways in which a dynamic organization might be produced in systems that use matter and energy components governed by the laws of physics, chemistry, and physical chemistry. With this in mind, we will put forward the possibility of the emergence of other properties that will, gradually, provide the dynamic systems we are talking about with a character that is more similar to living phenomena. These systems will continue to be governed by the abovementioned laws; however, these new properties will begin to participate actively in its dy-

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namic organization such that they would allow the emergence of the dawn of an open-ended evolution (biological evolution). Our conceptual proposal furthermore has two additional ends: the construction of arguments that support the possibility and the necessity of proposing one definition of information and one definition of function in naturalist terms. In other words, we defend that both notions are not purely thought exercises, whether in abstract, formal, or epiphenomena1 terms, or that for their definition they need some type of adscription imposed from the human world or from that of the human-generated devices. In this work, we put forward a type of self-organized dynamic system that we believe to be the system that opened the doors of the pre-biotic world, and because of that, will begin the step from the world of the inanimate to the world of living systems: the Informational Dynamic System.

2. Some preliminary reflections

We consider it to be reasonable to postulate theoretic constructs that can provide us insight into the nature of the dynamic organization that those preparatory systems must have possessed for the future emergence of living systems. We also believe that the origin of pre-biotic systems already carry with them a certain level of complexity in their dynamic organization. Put in different terms, to understand the conditions that make the appearance of living forms possible is to understand that their most basic properties were present at the origin of the pre-biotic world. So, Informational Dynamic Systems already have in their organization those minimum capacities that are present in whatever living system and are expressed at levels of increasing complexity as we advance in evolution. We can state that the self-organized type of system that is related to living phenomena is one of a chemical nature. Hence, the normativity that encompasses chemical reactions and the physical chemical requirements will occupy a special place in our discussion. In addition, it is feasible to envision the pre-biotic world made up of a series of systems that are related in more or less successive stages. In each step, we would find systems with certain capacities that they had incorporated from previous systems. In turn, they would explore different strategies for improving what they already had and would have the chance of producing the emergence of new capacities that together with the existing ones would prepare for the emergence of the first living systems. Before the genesis of pre-biotic systems, we find different, self-organized

235 systems that should gradually carry us to this new form of existence. It must be pointed out that even though the purpose of our work is not to discuss this prior time, it is possible for us to make certain clarifications about it. These very simple, inorganic, chemically composed, self-organized systems were found in extremely specific regions of the planet. The Earth’s planetary conditions during those epochs make it possible to hypothesize the existence of the first forms of chemical self-organization on the surface of certain minerals [9-131 or in regions very close to volcanoes [14-181. What must be stated clearly is the definite impossibility of trying to find a pre-biotic system as we have visualized in our proposal. In the first place, after so long a time since their disappearance, all traces of them would have vanished as well. In the second, their “building blocks” must have been metabolized by other systems in the pre-biotic world as well as by the first living ones. We also want to explain that when referring to the origin of the prebiotic universe, it is illogical to speak in terms of a Darwinian evolution since neither DNA nor RNA had appeared, and there was nothing at all like “genetic material” in existence. These emerged in the far distant future of the history of the pre-biotic world. Therefore, we cannot speak of fitness and other concepts connected with biological evolution. There is just openended evolution - a type of evolution whose components are related to the idea of evolution by natural selection. Nevertheless, it is possible for us to refer to the capacities that permit the gradual building of the structure and dynamic organization of those systems in terms of their internal properties. Therefore, at some moment, future generations of the different populations of these systems will possibly be ‘seen’ by natural selection [19-211. If we want to suggest some reference to the succession of changes in the dynamic systems, those that are strongly linked to the emergence of life, we could speak of a certain proto-evolution. An example of such could be a variant of the evolvability notion applied to the pre-biotic world [22-261.

3. The Informational Dynamic Systems

We place our theoretical reflections in the distant past, and this era tell us that the types of chemical constituents we might find available on the Earth would be incredibly simple chemical compounds, mainly inorganic, yet perhaps a very small amount of organic ones. We might even consider a

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possible abiogenic synthesis of such compounds like the thioester [27]. The thioester compounds are those that are formed when sulfhydryl, an organic group that contains a Sulfur-Hydrogen bond (R-SH), joins with another compound called carboxylic acid (R’-COOH). During the chemical reaction, a water molecule is released, forming this way the thioester compound (RS-CO-R’). Taking into account the current understanding about what the conditions of primitive Earth could have been like and meteor impacts during this time period, it is possible to theorize the feasibility of the existence of amino acids and carboxylic acids. Furthermore, the fact of massive volcanic activity makes it possible to theorize the existence of sulfhydryl [28-321. For the purposes of the work, we set a line of demarcation around the origin of pre-biotic systems: it would be a type of system that has managed a certain degree of autonomy by its own, self-maintained organization dynamic. Therefore, it is a type of dynamic system much more robust than the self-organized system we mentioned at the beginning, which we will call a “strict sense’’ self-organized system. This opens up space for the possibility of returning to the most robust and complex dynamic organization in the continuous succession of a type of pre-biotic system that will end up producing the first life forms. In that sense, we consider that our hypothetical informational dynamic system provides an explanation for the emergence of that organizational dynamic, and with it, allows us to understand the emergence of the global property known as autonomy. Generally speaking, self-organized systems are systems far from thermcdynamic equilibrium. BCnard cells, Belousov-Zhabotinskii reactions, living systems, and pre-biotic systems are ones that are far from thermodynamic equilibrium. One difference that is crucial to our argument is that the first two examples are systems far from equilibrium because of external causes: the matter and energy requirements that their processes needs invariably depend upon the environmental conditions. On the contrary, the other two cases maintain themselves in nonequilibrium conditions for reasons that are systemically intrinsic. With this statement, let us begin our study. The correlation among processes is an expected phenomenon when a system is in far from equilibrium conditions [33, 341. One important step to the pre-biotic world is the appearance of a correlation between endergonic and exergonic processes. The former needs energy requirements, whereas

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the latter produces energy. The existence of compounds with “high energy” bonds significantly favors this correlation between the abovementioned processes, forming a micro-cycle that is capable of generating work (chemical work). The inorganic pyrophosphate and/or a simple thioester could have been the ancestral “energy currency” molecules of these types of chemical bonds. In current cells, we find that the thioesters are bound up with the esters formation, including those we find in complex lipids. We also find them actively participating in the synthesis of peptides, fatty acids, sterols, terpenes, porphyrins, and other compounds. With this last reference, we have just established the feasibility of maintaining an adequate energy interchange between the exergonic-endergonic cycle and the production of compounds (accomplishment of a chemical work) that have much to do with the formation (and reparation) of a protomembrane (Amphiphilic polymers) [35, 361. In this way, the informational dynamic systems manage to keep themselves in nonequilibrium conditions because there are two fundamental requirements within its dynamic organization. Another way of putting it is that in the first place, we would have the exergonic-endergonic cycle made possible by a high energy bond carrier compound that as a group is a process with the capability of producing work. In the second place, we would have the protoplasmic membrane related in some way to the compound acting like the “energy currency” molecule. As we are visualizing the possible interaction of these types of processes, later on, we will see how it is that they allow certain initial degrees of autonomy with respect to the ever changing environmental conditions. Another important topic to cover is what we refer to as the possibility of forming oligomeric compounds by noenzymatic condensation. Much has been said concerning abiotic chemistry in regards to this theme and until recently the results were very limited for abiotic polymerization reactions starting from their monomers - whether amino acids or nucleotides. Nevertheless, a group of researchers has recently proven, experimentally, that ‘carbonyl sulfide (COS)’ - a very simple gas formed from volcanic activity and doubtlessly present in the pre-biotic era - intervenes in shortchain peptide condensation. In short, the molecular reaction involves the bond of the COS to the amino acid by the amino group, producing afterwards a cyclical compound called amino acid N-carboxyanhydride (NCA). The NCA will bond with another amino acid and a dipeptide will form, freeing a COz molecule [37].

238 The results of scientific research demonstrate to us the complete possibility of abiogenic short-chain peptide synthesis in very similar conditions to those existing in the pre-biotic times. With these results, it is possible to propose in a rational way that short-chain peptides could have managed to attain a sequence of interdependent reactions: the beginnings of protometabolism. The three groups of processes that we have presented bring us directly to the dynamic organization that constitutes the informational dynamic system.

3.1. Organization of the Informational Dynamic System In order to have the capacity of being more robust than the “strict” selforganized systems and to be able to have the necessary and sufficient conditions to begin the slow but sure road to improving their levels of complexity, the informational dynamic system has to be made up of at least three different classes of process groups. The first class of processes is fundamental for providing the system a certain degree of independence with respect to its environment since it generates the conditions for being in nonequilibrium state. This is the endergonic-exergoniccycle capable of producing - in each cycle - quantities of free energy to the system that are appropriate to generate work (of a type fundamentally chemical). This cycle is, in its own right, the one providing the system with the far from equilibrium characteristic since it forms a component part of its dynamic organization. To be a system in the nonequilibrium state is a priority intrinsic to the system; because of that, it is the most basic fundamental of informational dynamic systems. The second process group would form another important part of the dynamic organization of these systems. This is the one that makes up the protoplasmic membrane. First, this component permits a separation, a physical barrier between the system and its environment, causing from there a different distribution of compounds, a different dynamic, and a different interaction among the processes. It also permits certain properties of chemistry to have a focal point of action to generate possible problematic circumstances which in turn produce the conditioning situations for exploring different solution strategies for the system in the future. Some examples we can cite are colligative properties, cinetic chemistry, electrochemical gradients, and others.

239 It is in the protoplasmic membrane that we will objectively be able to observe the emergence of the autonomous capacity. It is the part of the system that regulates the interaction with its environment as well as providing the necessary constituents (matter and energy) so that the internal system processes continue producing their mechanisms in conditions compatible to their survival. In this way, system cohesiona is not disturbed [38, 391. This part of the system organization is one that is a complete agent since it allows for the resolution of the osmotic problems, the adequate concentration of components in the internal system processes, the management of electrochemical gradients, the reduction of chances for the entrance of foreign elements into its organization, etc. We can state at the same time that the system contains two different types of constriction (40,411. The first one we call “basic constriction”, which directly maintains the f a r from equilibrium state. It is a cycle between the endergonic processes linked with the exergonic ones and with the added capacity of generating work (a necessary and sufficient condition to produce the organizational dynamic phenomena). The second permits a dynamic and agent interaction with its external and internal environment: a semi-selective membrane (protoplasmic membrane) that contains the compounds which will generate processes to give the system the capacity of managing both external and internal conditions. Both types of constriction are so interrelated that a very basic, direct communication exists between them. In addition, we will find other types of processes (that make up, along with the last two, the organizational dynamic identity of the system). The third group is a network of reactions that would perform the organizational dynamic’s reparation, maintenance, and reproduction processes of the informational dynamic system and is in one way or another connected to the constrictions. The informational dynamic system is organized by the interrelation and interdependence of these three classes of processes. The expression of an autonomous and agent behavior implies certain capacities that are made real in the way that the protoplasmic membrane performs and coordinates its internal processes. For these reasons, the detailed analysis of this part of the informational dynamic system is valuable. We must remember that the informational dynamic system is the dy-

aThe idea of cohesion is applied to dynamic systems and their properties that are bonded by an internal dynamic of relationships. This allows the establishment of a causal substrate for delimiting the dynamic system identity [39]

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namic organization that constitutes it. So, the system’s dynamic organization is in fact a collection of processes, interrelated and interdependent, with the additional element that this interconnection among the processes maintains a determined form of concretely connecting themselves with the constriction that permits the system to remain in the far from thermodynamic equilibrium state. An agent behavior brings us to the understanding that a certain degree of managing their internal and external conditions would exist. This management (even though very minimal at the beginning) directs our attention to those processes that would be responsible for interacting and performing recognition, selection, and active or passive transport processes. And this management is done because it is “dictated by” (there is a strong correlation with) the tendency to persist as such - to maintain itself in the state that fundamentally characterizes it; the most primary aspect of its nature is constituted by being a far from thermodynamic equilibrium system. It is possible to conjecture that the basic constriction causes the change of the system’s free energy (AG,,,) to have a tendency towards negative values. Taken as a totality of the processes and its interconnection, this causes it to be immersed in continuous and successive dynamics of increasing order. Therefore, to stop being a far from thermodynamic equilibrium system is the same as saying that the system no longer exists. To strongly influence the conditions for remaining in that state implies seriously placing its survival at risk. This central aspect of its nature is now maintained by the system itself through the incorporation of the basic constriction. This allows the system to maintain itself in the far from equilibrium state. As this constitutes the most original aspect of these systems, it seems to us that we can assume that it governs the multiplicity of transformations of behaviors, components, capacities, and characteristics of the future system. This way, our proposal rests upon the naturalist supposition that informational dynamic systems develop their difference from “strict sense” selforganized systems because they have been able to develop and maintain a capacity that is elemental for their existence: maintaining or increasing the far from equilibrium state. As the constriction maintaining them far from equilibrium is an intrinsic part of their dynamic organization, there are strategies they can develop that manage to keep this state in conditions compatible with the laws im~

241 posed on them by the material world. The strategies we are referring to will be two new characteristics that emerge in the system - in the local interactions - and are directed a t maintaining the far from equilibrium state by a direct or indirect connection with the basic constriction: information and function. Briefly, we think that both information and function emerge at the same time, in unison, in the informational dynamic systems and that the physical emergence of both notions happens in an interrelational way: information-function. Moreover, we postulate that both information and function are relational concepts. For our intentions, understanding the basic difference that exists between a system in an equilibrium state and one in a far from equilibrium state is very important. This important difference permits us to construct a type of norm that is found in certain laws of nature without any adscription coming from the human world: the laws of thermodynamics. Therefore, our proposal of the notion of information-function - as a characteristic emerging in the informational dynamic systems - is a relational concept that is strongly governed (ruled) by the far from thermodynamic equilibrium state. The information-function characteristic has two aspects: the informational function aspect and the functional information aspect. When we observe the transmission of some information in informational dynamic systems, this happens because some type of function is produced. Similarly, when we observe the presence of some function, it happens because some information has been transmitted.

3.2. I n f o r m a t i o n and Function Emergence

Both ideas of information and function are directed towards the network of interrelations among the processes. It is in the processes and the relationships among them that both capacities emerge for the first time in the physical world in such a way that through the function pole (“informational function”), we observe that the contribution of the processes among each other for their appropriate performance in the integral logic of the system’s organization causes the system to maintain itself far from thermodynamic equilibrium. Through the information pole ( “functional information”), we observe that the meaning for the system of some signal or sign (that is the physical means of transporting some matter-energy variation) will be expressed in

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agreement with maintaining the system in the far from equilibrium state. However, it can also be that it was in opposition to maintaining the far from equilibrium state. In the first situation, the meaning of the matter-energy variation that comes to the system will cause a positive response by the system. In the second case, the meaning will generate a negative response by the system. We can see that both information and function are strongly connected by their definition with the idea of far from thermodynamic equilibrium. And this is precisely what we propose. Because of this, it is possible to consider, through a naturalist perspective, the far from thermodynamic equilibrium state as a basic norm that the system imposes on itself for continuing to be the system that it is. This way, information and function will have to be defined starting from what we physically have right now, where there is no place for human adscription, and hence, the possible epiphenomena1interpretation of some of them is avoided from the start. The differences that exist between a system in a thermodynamic equilibrium state and a system in a far from thermodynamic equilibrium state is something we possess firmly in our minds and have been well defined through experimentation. The group of causes and contour conditions to differentiate them can be used in the most explicit way possible. Therefore, for our defense of a class of system with the characteristics we mentioned, we would have in the realm of physical laws a naturalist principle for building up the essential aspects in a normativity. Effectively, a normative principle points out precisely those circumstances when the norm is fulfilled and the difference in those situations when the norm is not fulfilled. For our work, those circumstances in agreement with the far from equilibrium state are fulfilling the norm, and those circumstances not in agreement with that state are not fulfilling the norm. What is interesting about the case is that the norm we just explained essentially permits the way these systems exist. Therefore, this normativity is derived naturally from the physical world. Moreover, the normativity comes from and is dictated by the system itself. In its origins, what we would have might be a normative definition of the information-function notion. It is from this initial appearance of both ideas that we can postulate later on that a kind of symmetry break would arise in the primitive idea so we would find both characteristics acting independently (although on occasion, in some way, interacting as they did at the beginning) and with the possibility of increasing the levels

243 of complexity and sophistication in one of them or in both. Everything taken together brings us to the thesis that starting the process from inanimate to animate could have been produced by the appearance of a type of dynamic system whose organization is an informational and finctional dynamic Organization.

The normative idea of functions (we understand the functions to be “contributing to what”) is founded precisely on this internal normativity that is produced in the informational dynamic system by the components and processes that sustain and constitute it. And all of this creates a tendency to maintain the far from equilibrium thermodynamic state. We consider the function to be located in the action that generates a determined process and what we can understand as “what contributes” (cooperates, favors, supplies) to the interrelation and interdependence among the processes with a view to maintaining the far from equilibrium state [42]. On the inside of the process, each component will have the mechanism that is proper to its chemical constitution, but that chemical action (or group of actions) will have its raison d’gtre because of the process. Then, each process that carries out a determined and particular action (made up by the causal nexuses among the component mechanisms that constitute them) will be assimilated into the dynamic organization only if this particular action is maintaining, improving, or achieving the far from equilibrium. And this action is precisely its function in this organizational logic of the informational dynamic systems. Thus, a function is something that contributes or facilitates to maintaining or increasing the far from equilibrium state. A dysfunction, on the other hand, is something that does not contribute or facilitate to maintaining the far from equilibrium state. Concerning information, any type of signal or sign can be a carrier of what might be information (or what we call ‘potential information’). We consider a signal or sign to be any matter-energy variation (sound wave, electromagnetism, concentration of a chemical compound, change in pH, etc.). The signal or sign that is a ‘potential information’ carrier must be in the surroundings, on the inside, or have transmitted the information to some system component. The important thing for our idea is that the information, properly said - The Information - “has a meaning” (very basic semantic) that is created on the inside of the system. As Menat would say, “It is meaningful information.” [43]. In a naturalist perspective, any signal, sign, or information must always exist with respect to something else and not imposed by some observer

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(we can find other approaches to Biosemantics in [44-481, among others). Therefore, we do not accept the existence of things like signs, signals, or information by themselves. In this way, information will always be information for the system. How can we understand this information for this system? First off, that potential information that carries the signal must have been incorporated to some process. It is in that particular process where the information might be transmitted. Secondly, the ‘potential information’ becomes Information ( “infomation with meaning” for the system) since it has the capacity to produce something (an effect) in the process that incorporated it, in some other process that is directly or indirectly connected to the initial process that incorporated it, or with some aspect of the constrictions that are effectively maintaining the system far from thermodynamic equilibrium. Third, the effect has a repercussion in the system, influencing its own dynamic organization. For the purposes of our work, we will use Collier’s notion of cohesion [38, 391. Cohesion is the idea that would provide the informational dynamic system its identity, in all its transformations in time. Such cohesion is constituted by the group of relationship and interdependencies that exist among the processes; it is a relational and dynamic definition that encompasses the nature of the system organization. The effect of the information that has meaning for the system can be in the maintenance or the increase of the system cohesion. As well, the effect could produce some level of interference in the system cohesion, possibly interrupting one or more processes. It is clear that meaningful information can be caused by some signal (that carries potential information) coming from the environment like a signal that is generated in the internal dynamic of the system. In all cases, whether an effect in favor of or in contrast to cohesion, the system will develop some type of response that will be correlated to that meaningful information and the process or processes enveloped by the effect. Let’s imagine an ideal situation in which all the system processes perform each and every one of its actions in a perfectly balanced and interdependent way. In this situation, the informational dynamic system is normally far from thermodynamic equilibrium (there is a sustained correlation among the processes such that the tendency towards irreversibility may increase). In this ideal case, we can see what is happening in the entire system, what happens among the processes, and inside each one of them as well.

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In this case, each process performs its determined function (contribution) in which this integrated network of relationships is directed towards the cohesion of a far from equilibrium system. At the same time, the information generated in each process and in the interaction among this integrated network of processes will be considered by the system as a type of system checkup to see if the cohesion has not been disturbed. In that sense, the state of being far from thermodynamic equilibrium behaves like the most fundamental norm that the system imposes on itself, for which every strategy elaborated by the system (component, process, relationship, or matter-energy management) for maintaining or increasing this norm will be incorporated as part of its dynamic structure. In order to finish our argument, we present some examples that we consider supports our working hypothesis to a certain degree. First, it is known that the fibrodendritic projections coincide with the isofrequency organization planes (high, medium, and low) in the Central Nucleus of the Inferior Colliculus (CNIC). Tonotopic organization is a fundamental property of the auditory system (491, and it is now well established that the morphological substrate for such tonotopy in the CNIC is its laminar organization [50]. The inhibitory action of GABA and Gly is what shapes the temp+ ral and spectral response of Inferior Colliculus (IC) neurons. Due to the relevance of inhibitory processes in the physiologic responses of the IC neurons, it is realistic to think that we might obtain some explanations of the acoustic information flow dynamics from the neural populations and the histochemical and cytoarchitectural organization of this nucleus. Working with a colleague, who is a neuroscientist, on his acoustic system research [51], we have found a certain correlation between the distribution of neurons in the Central Nucleus of the Inferior Colliculus and the way in which the flow of acoustic information travels through it. It is observed through densitometric quantitative methods that different populations of GABAergic neurons (which are inhibitor neurons) exist in each isofrequency plane that are oriented in transversal planes to each isofrequency plane. Thus, a topological relationship of neurons exists with respect to the laminar organization. This topological dynamic organization in each isofrequency plane correlates clearly with GABA gradient concentration such that it would generate a determined rhythm to the temporary excitatory flow. If only one type of neuron population would exist in each isofrequency plane, then this hypothetical fact could be considered by us as a counterexample to our working hypothesis.

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Second, erythrocytes mature in bone marrow by means of the hormone, erythropoietin, and then are released into the blood stream in where usually are destroyed 120 days after. Recently, levels of anemia have been discovered in returning astronauts as a result of the destruction of erythrocytes. What is surprising in this case is that in place of destroying the oldest cells (which is the expected) the focus of the destruction is the youngest cells. This is a phenomenon known as “neocytolysis” [52]. This leads to the question: Could certain signals exist that would carry a determined type of information for the organism as a whole system such that it would make “functional” the massive destruction of the youngest erythrocytes before the oldest? It is important to point out that erythrocytes do not have nuclei. The topic of viruses seems to us a very revealing case as well. These are entities that are not autonomous systems; nevertheless, in determined conditions, they develop processes that come out to be informational and functional. The processes for entering a cell begin with “recognition” of the target cell receptors, whose activation by the virus will cause notable changes in the plasmatic membrane, and end up with the virus entering the cytoplasm [53]. Here, we can visualize certain “functional information” for the virus that is “dysfunctional information” for the host cell. Its molecular components are such that they can perfectly interact with the organizational logic of the target cell and only because there is that organizational dynamic (informative and functional in the target cell); it is that its components, upon interacting, will carry determined information that makes the functional actions concrete since they contribute to its reproduction and survival. The relational character is revealed in that these same processes are dysfunctional for the cell and the information that certain components of the virus transmit are counterproductive for the cell (to such a degree that the virus’s use of the synthesis and replication machinery of the cell will end up destroying the cell). These are interesting lines of research, attempting to find out which signals permit “blocking” the mechanisms and processes which monitor the internal states so that the virus ‘takes possession” of a large part of the cellular metabolism.

References 1. Nicolis G & Prigogine I, Self-Organization in Nonequilibrium Systems:Fbom Dissipative Structures to Order through Fluctuations. Wiley, New York (1977).

247 2. Strogatz SH, Nonlinear Dynamics and Chaos. Westview Press (1994). 3. Haken H, Synergetics: an introduction. Springer, Berlin (1978). 4. Bak P, How Nature Works: The Science of Self-organized Criticality. Springer, Berlin (1996). 5. Heylighen F, Bollen J & Riegler A (Eds.), The Evolution of Complexity. Kluwer Academic, Dordrecht (1999). 6. Maes P (Ed.), Designing Autonomous Agents: Theory and Practice From Biology to Engineering and Back. MIT Press (1991). 7. Kauffman S, The Origins of Order: Self-organization and Selection in Euolution. Oxford University Press, New York (1993). 8. Bar-Yam Y,Dynamics of Complex Systems. Addison-Wesley (1997). 9. Wachtershauser G, Before enzymes and templates: Theory of surface metabolism. Microbiological Reviews,52: 452-484(1988). 10. Hazen RM, Life’s rocky start. Scientific American, 284 (4): 63-71 (2001). 11. Ferris JP, Montmorillonite catalysis of 30-50mer oligonucleotides: laboratory demonstration of potential steps in the origin of the RNA world. Origins of Life and Evolution of Biospheres,32:311-332(2002). 12. Cairns-Smith AG, Fine-tuning in living systems: early evolution and the unity of biochemistry. International Journal of Astrobiology, 2 (2): 87-90 (2003). 13. Ricardo A, Carrigan MA, Olcott AN & Benner SA, Borate minerals stabilize ribose. Science, 303:196 (2004). 14. Fisk MR, Giovanonni SJ & Thorseth IH, Alteration of oceanic volcanic glass: Textural evidence of microbial activity. Science, 281: 978-980 (1998). 15. Schopf J W , Kudryavtsev AB, Agresti DG, Wdowiak T U & Czaja AD, LaserRaman imagery of Earth’s earliest fossils. Nature,416: 73-76 (2002). 16. Van Zuilen MA, Lepland A & Arrhenius G, Reassessing the evidence for the earliest traces of life. Nature,418: 627-630 (2002). 17. Furnes H, Banerjee NR, Muehlenbachs K, Staudigel H & de Wit M, Early life recorded in Archean pillow lavas. Science,304: 578-581 (2004). 18. Foustoukos DI & Seyfried J r WE, Hydrocarbons in hydrothermal vent fluids: the role of chromium-bearing catalysts. Science, 304:1002-1005 (2004). 19. Lewontin RC, The units of selection. Annual Review of Ecology and Systematics, 1: 1-18 (1970). 20. Sober E, The Nature of Selection: Euolutionary Theory in Philosophical Focus. MIT Press (1984). 21. Brandon RN, Adaptation and environment. Princeton University Press ( 1990). 22. Kirschner M & Gerhart J , Evolvability. Proceedings of the National Academy of Sciences, 95: 8420-8427 (1998). 23, Radrnan M, Matic I & Taddei F, Evolution of Evolvability. Annals of the New York Academy of Sciences, 870(1): 146 - 155 (1999). 24. Csete ME & Doyle JC, Reverse Engineering of Biological Complexity. Science, 295(5560): 1664 - 1669 (2002). 25. Poole A, Phillips M & Penny D, Prokaryote and eukaryote evolvability. Biosystems, 69: 163-185 (2003).

248 26. Tanay A, Regev A & Shamir R, Conservation and evolvability in regulatory networks: The evolution of ribosomal regulation in yeast. Proceedings of the National Academy of Sciences, 102(20): 7203-7208 (2005). 27. de Duve C, The Beginnings of Life on Earth. American Scientist, 83: 428-437 (1995). 28. Bada JL, Miller SL & Zhao M, The stability of amino acids at submarine hydrothermal vent temperatures. Origins of Life and Evolution of Biospheres, 25: 111-118 (1995). 29. Cooper G, Kimish N, Belisle W , Sarinana J, Brabham K & Garrel L, Carbonaceous meteorites as a source of sugar-related organic compounds for the early Earth. Nature, 414: 879-883 (2001). 30. Bernstein MP, Dworkin JP, Sandford SA, Cooper GW & Allamandola LJ, Racemic amino acids from the ultraviolet photolysis of interstellar ice analogues. Nature, 416: 401-403 (2002). 31. Monnard PA, Ape1 CL, Kanavarioti A & Deamer DW, Influence of ionic inorganic solutes on self-assembly and polymerization processes related t o early forms of life-implications for a prebiotic aqueous medium. Astrobiology, 2: 139-152 (2002). 32. Rees DC & Howard JB, The interface between biological and inorganic world iron-sulfur metalloclusters. Science, 300: 929-931 (2003). 33. Kosztin I & Schulten K, Fluctuation-driven molecular transport through an asymmetric membrane channel. Physical Review Letters, 93: 238102 (2004). 34. Levine RD, Molecular Reaction Dynamics. Cambridge University Press (2005). 35. Schopf J W , Microfossils of the early Archaean apex chert: new evidence of the antiquity of life. Science, 260: 640-646 (1993). 36. Keefe AD, Newton GL & Miller SL, A possible prebiotic synthesis of pantetheine, a precursor of coenzyme A. Nature, 373: 683-685 (1995). 37. Leman L, Orgel L & Ghadiri MR, Carbonyl Sulfide-Mediated Prebiotic Formation of Peptides. Science, 306 (5694): 283 - 286 (2004). 38. Collier J, Entropy in evolution. Biology and Philosophy, 1: 5-24 (1986). 39. Collier J , Self-organization, Individuation and Identity. Revue Zntemationale de Philosophie, 59: 151-172 (2004). 40. Patee HH, Laws and constraints, symbols and languages. In Towards a Theoretical Biology 4, Essays, Waddington CH (Ed.). Edinburgh University Press, pp. 248-258 (1972). 41. Kauffman S, Investigations. Oxford University Press, Oxford (2000). 42. Riofrio W , La vida en sus origenes: Las propiedades b&icas. 20th World Congress of Philosophy, Boston, Ma (1998). 43. Menant C, Information and Meaning. Entropy, 5: 193-204. (2003). 44. Millikan RG, Language, Thought and Other Biological Categories. Cambridge MA: MIT (1984). 45. Millikan RG, Biosemantics. The Journal of Philosophy, 86 (6): 281-297 (1989). 46. Millikan RG, Biofunctions: Two Paradigms. In Functions: New Readings i n the Philosophy of Psychology and Biology, Cummins R, Ariew A & Perlman

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M (Eds.). Oxford: Oxford University Press, pp. 113-143 (2002). 47. Emmeche C, A-life, Organism and Body: the semiotics of emergent levels. In Bedeau M, Husbands P, Hutton T, Kumar S & Suzuki H (Eds.), Workshop and Tutorial Proceedings. Ninth International Conference on the Simulation and Synthesis of Living Systems (Alife ZX), pp. 117-124 (2004). 48. Kull K, A brief history of Biosemiotics. Journal of Biosemiotics, 1: 1-34 (2005). 49. von Bitkhsy G, Experiments in hearing. McGraw-Hill, New York (1960). 50. Oliver DL & Morest DK, The central nucleus of the inferior colliculus in the cat. The Journal of Comparative Neurology, 222: 237-264 (1984). 51. MerchLn M, Aguilar LA, L6pez-Poveda EA & Malmierca MS, Immunocytochemical and semiquantitative study on Gamma-aminobutiric acid and glycine in the Inferior Colliculus of rat. Neuroscience, 136 (3): 907-925 (2005). 52. Rice L & Alfrey CP, Modulation of red cell mass by neocytolysis in space and on Earth. European Journal of Physiology, 441(2-3 Suppl):R91-4 (2000). 53. Doming0 E, Webster R & Holland J (Eds.), Origin and Evolution of viruses. Academic Press (2000).

GRASPING THE COMPLEXITY OF LIVING SYSTEMS THROUGH INTEGRATIVE LEVELS AND HIERARCHIES

J , M. SIQUEIROS & JON UMEREZ Department of Logic and Philosophy of Science University of the Basque Country (UPV/EHU) Avda. Tolosa 70, 20018, Donostia-San Sebastian, Spain E-mails: {jsiqueiros, jumerez} @sf. ehu. es http://www. ehu. es/ias-research In this paper we intend to address the conceptual challenge posed by complexity by focusing on living systems. Biological complexity has traits that seem t o be more amenable t o scientific scrutiny and description, without having t o turn t o higher order capacities as sociality or human intelligence. Notwithstanding, biological complexity keeps the conceptual depth and suggestive power that appear lacking in more formal or physical expressions of the idea. The paper presents a conceptual overview of the different perspectives that along the 20th century were developed with the intention to grasp biological complexity (i.e., Organicism, Theory of Levels of Integration, etc.). Concepts that nowadays are so common t o the different approaches on complexity such as emergence, hierarchy, downward causation, wholeness, etc.- arose for the first time in these theoretical perspectives during the early 20th century and in the 60's and 70's. The goal of this paper is t o point out that these accounts still have deep insights and potentialities that are worthy t o recast in order t o advance in the understanding and exploration of biological complexity and complexity in a wider sense, specially the ideas of autonomous hierarchies and the notion of constraints as enabling not only as restrictive material structures.

1. Introduction. Complexity: an elusive idea Along with many of the technological achievements during the last decades, the re-emergence of the idea of complexity and its incorporation into many and diverse scientific disciplines have contributed greatly to the change on how science is done nowadays. On the one hand, complexity has proved to be a good methodological frame for modeling and, on the other, concepts like emergence or systematicity, that once were only peripheral and marginalized, now are almost part of everyday language, since they are tools that allow to approach problems from a new perspective. Moreover, the impact of the idea of complexity has not been only experi-

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enced in the natural sciences but it has reached and influenced a wide span of disciplines, from physics to the humanities and, of course, to what Simon began to call “the sciences of the artificial.” Even so, it is remarkable how difficult is to define the concept of complexity. Complexity is a very elusive idea, due, on the one hand, to the wide range of areas of applicability and, on the other, to the multiplicity of proposed definitions and measures. Nevertheless, it is not our intention to offer a huge list of definitions of what other people think what complexity is.a On the contrary, we will just very shortly describe which general traits epitomize complex systems and then we will characterize what kind of systems are the living ones and which are their peculiarities (as organized complexity). Therefore, in this paper we intend to address, at least partially, the conceptual challenge posed by complexity (i.e., apparent contradiction of simultaneous vagueness and usefulness) by focusing on living systems. It is generally taken for granted that living beings are complex, even characteristically so, and this attribution does not seem especially controversial. Besides, the traits that make biological organization complex seem to be (at least to certain degree) more amenable to scientific scrutiny and descrip tion, without turning to higher order capacities such as sociality or human intelligence. Notwithstanding, biological complexity keeps at the same time some conceptual depth and suggestive power that appear lacking in more formal or physical expressions of the notion of complexity. Turning then to biological complexity, if we review the philosophical panorama of the past century regarding the conceptual tools that have been used to understand and explain living systems, we may easily find (among other equally interesting turning points) that the fate of a cluster of ideas revolving around aspects of complexity, such as hierarchy, levels, emergence, organization, wholes and systems, etc., has been peculiar, ranging from proposal and more or less easy coexistence and debate with other opposing views in the first part of the century to total disappearance from the larger part of that period, and to a renewed interest in its last decade. Witness to this resurgence are, to mention two approaches relevant to this work, the very recent mainstream appeal to build a systemic biology or systems biology (see Ref. 1) and the not as recent constitution of the field of theories or sciences of complexity, including Artificial Life, which we might date back

Lloyd, from MIT, has been compiling for years a “non-exhaustive list of measures of complexity” that can be accessed at the web: http://web.mit .edu/esd.83/www/notebook/Complexity.PDF

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to the mid 80’s (see Refs. 2, 3, 4).b We think, therefore, that it would be worthy to recast some of those perspectives from the first third of the 20th century (emergent evolution, organicism, integrative levels,. . .), as well as those fighting to get some audience during the 60’s and the 70’s (theoretical biology, hierarchy theory, systems science,. . . ), not only in their common generality of approach but paying special attention to their particularities in order to highlight their specific potentialities to advance in our understanding and explanation of biological features and the very phenomenon of life. We pose it that way because, even when some kind of acceptance of the general perspective is granted, problems are continually presented. Some are related to the presumed vagueness of the ideas, other to their contested instrumental fertility, and yet other to their physical implementation or their ontological nature. To answer this kind of objections, besides a general philosophical justification of the approach, it is also convenient to provide more specific and operative ways to apply it to precise cases or areas of research. Within this context we propose in this paper to recuperate two (among many) of those specific views, each belonging to one of the two periods mentioned before. The first is the notion of integrative levels defended by Needham and other organicists during the 30’s (and later, for instance, by Novikoff or Feibleman). The second is the notion of hierarchy developed by Pattee based on the concept of constraint during the late 60’s and the 70’s (and later by others with analogous approaches, for instance, Salthe). The idea of levels of integration has a sound grounding in the work of organicist embryologists and theorists (Needham, BertalanEy, Woodger, Weiss, Waddington, etc.) and it offers theoretical and empirical tools to understand the organization of a living system as it develops. The idea of hierarchy elaborated by Pattee seeks to offer operational tools as the concept of constraint (in the sense of Ref. 5) and the distinction between holonomic and non holonomic ones, in order to explain how levels can emerge and relate to one another in different ways, giving rise to specific hierarchical arrangements, as the one showed by biological organization. The working out of those instruments will allow us to develop an idea of biological hierbHowever, it is not the purpose of this short paper to deal with the logically subsequent issue of trying to assess to what degree do these recent approaches innovate with respect to paths previously open or just develop them further (or even if they aware of them). Therefore, we are not going to discuss here current efforts to approach the complexity of living systems along similar lines to those we will describe in the paper.

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archy as a relation among levels with different and specific dynamics which gives way to emergent processes both upwards and downwards. We claim that this kind of work is a very appropriate way to advance in disclosing what complexity means. Recalling Weaver’s distinction between problems of “simplicity,” “disorganized complexity” and “organized complexity,” we intend to contribute to characterize as precisely as possible a kind of systems, the living ones, which are paradigmatic of that organized complexity whose challenge Weaver thought would be accessible within the next fifty years since his paper was published in 1948.6

2. Sciences of objects and systems, some simple, some complex

It is possible to say that a system is formed by things and the relations among them.7 According to this, systems can be defined in terms of their thinghood or in terms of their systemhood.8 To define a system in terms of its thinghood is to identify it by those traits that are not part of its systemic character. These very general criteria make possible to recognize different systems by the things that form them. In a broad sense we could say, for instance, that there are physical systems, biological systems, social systems, and artificial systems that, even if they are not separate kinds, represent different phenomenological and knowledge realms. To define a system in terms of its systemhood is to identify it by the relations among things. Relations are what connect and join entities giving rise to a system. So, different things can establish similar relationships. The other way around is also possible, the same things can establish very different relations. From this perspective, the very subject of a systemic approach is the analogies between the kinds of relations that things can establish. Complex systems are mostly understood in terms of their systemhood instead of their thinghood. Even though, at this moment, it is almost impossible to offer a general definition of complexity due to the variety and diversity of phenomena that can be labeled as such (See note 1). Moreover, the concept of complexity is related to a field that nowadays is in full development. Still, a general outline or a summary of complex systems according to some of their properties is possible. Beginning at a physical and chemical level we might find the incapability to predict global behaviors from local component behavior -nowlinear systems (see Ref. 9). As well, an immense

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variety of behaviors are possible due to an extreme sensibility to the initial conditions -chaotic systems.lO,ll We have already become familiar with dissipative phenomena and the emergence of order in far from thermodynamic equilibrium conditions -dissipative system^.^^^^^ Somehow a generalization of the previous case is the emergence of robust and stable phenomena that appear form unstable situations due to self-organizing properties -self-organizing systems (see Ref. 14). If we turn to biological (and psycho-social) systems, many of the properties just mentioned for physical and chemical complex systems get intensified through the appearance of functional, regulative, and organizational features and, especially, through informational and symbolic attributes. Finally, artificial systems also can show complex behaviors, as is the case of cellular automata, Boolean networks, or neural networks, among others (see Ref. 15). Apart from chaotic phenomena, most of the cases mentioned above imply a high dependency on a singular relationship between the behavior of individual components and the global or collective one, a situation that unavoidably brings to memory the classic problem of the relationship between parts and wholes, emergent behavior, and the problem of hierarchies. Besides, in the biological realm, this characteristic is autonomously generated, from within the very system, and not instructed from without by any external agent.16 This is the double reason why we are proposing to focus our attention in “grasping living systems complexity through integrative levels and hierarchies.” But first, turning back to Weaver’s 1948 paper, we might recall that it explains how the complexity of a problem (or system) is related to the number of variables involved. In order to conclude this broad characterization of complexity, we will shortly render Weaver’s classification. He recognized three different kinds of problems: a. Problems of simplicity are those that imply two variables (maximum 3 or 4) as in the classical mechanics before 1900.

b. Problems of disorganized complexity are faced when, after 1900, a large amount of variables become tractable through probability theory and statistical mechanics: “It is a problem in which the number of variables is very large, and one in which each of the many variables has a behavior which is individually erratic, or perhaps totally unknown. However, in spite of this helter-skelter, or unknown, behavior of all the individual variables, the system as a whole possesses certain orderly and analyzable average

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properties.” (Ref. 6:538). c. Problems of organized complexity appear when we deal with an intermediate number of variables that cannot be averaged and it is only nowadays that we are becoming able to address them: “The really important characteristic of the problems of this middle region (. . . ) lies in the fact that these problems, as contrasted with the disorganized situations with which statistics can cope, show the essential feature of organization. (. . . ) They are all problems which involve dealing simultaneously with a sizable number of factors which are interrelated into a n organic whole.” (Ref. 6:539).

2.1. Some methodological and conceptual conclusions

We might try to extract some methodological and conceptual conclusions on the development of the idea of complexity. To begin with, simple systems are those that perfectly fit to a Newtonian and Laplacian classical epistem01ogy.l~In order to explain them all that is required is to apply the laws of motion, that is, no alternative descriptions of the system are needed. A few variables are enough, linearity holds and there is a proportional relation between causes and their effects. System behavior can be deduced or predicted by knowing the initial conditions and the individual behavior of the components. If we move on, one of the biggest problems that Laplace already faced was the incapacity of human minds to define all the initial conditions of all bodies in the universe, this was a situation way beyond the two variable classical epistemology of “his” physics; this difficulty is the first instance of complexity. Moreover, in the case of full-fledged complex systems there is no proportional relation between causes and effects, because the system behavior cannot be reduced to the sum of the behavior of its components. Due to this property, besides a description grounded in the laws of motion, an alternative and complementary description may be required; this is to say, it is not enough to describe the local behavior in order to predict the global behavior. Since there is a difference between local and global behavior, two levels of description are required, the first one is usually called “micro” and makes reference to the behavior of local or individual components; the second one is the “macro” because it makes reference to the global or collective behavior of the system. Moreover, this global behavior will be qualitatively different between systems showing disorganized complexity and those displaying organized

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complexity. In the first case, the behavior of the components of the system can be described as Brownian and the global outcome can be described as an average; examples would range from gases to more interesting ones such as some dissipative structures or chaotic systems. The second case would include biological and some social systems, where the behavior of components is locally controlled by the level of the whole system (or specific modules). In general terms, complex systems show a global emergent behavior that is not predictable with the classical mechanics toolkit, due to their non-linear character. Complex systems description always imply, at least, two levels lower and upper levels- and, in order to explain their behavior, a full account of the relations between them is required. The levels at the bottom give rise to global patterns (upward causation) meanwhile those global patterns harness the behavior of the levels at the bottom (downward causation).18 In the case of disorganized complexity, the level above may impose what is known as boundary conditions; in organized complexity, the level above may impose functional constraints on the level below. It is this challenge that Simon, to cite another classical paper, seeks to answer by claiming that the architecture of complexity is necessarily hierarchi~a1.l~ 3. Biological complexity: hierarchies

Aside from the variety of scientific objects that are recognized as complex, the epistemology, the methodologies, and the concepts behind each of the cases above can be historically traced back to, at least, three theoretical frameworks: a) Cybernetics and Systems Science$ b) the work on emergence at the end of the l g t h century and the first half of the 20th; and c) the organicist theories in biology. All these different perspectives shared a set of common interests and intuitions, which basically were to overcome the limits of reductionism and determinism (curiously enough they also share some participants and spatio-temporal coordinates). In order to develop a scientific approach to a more complex phenomenology, their effort was focused, for instance, on a wider notion of causality, an understanding of the relationship between levels, an appreciation of contextual aspects, etc. Of the entire conceptual “Even if they had a different origin and development, somehow they have come t o meet in a common corpus (see Refs. 20, 21, 22, 23).

257 universe around complexity, we are interested in hierarchies and how they are expressed and studied in biological cases. Hierarchical approaches to what nowadays are considered, in its most general sense, complex systems phenomena have been developed all along ~ ~may, , ~ ~ for instance, recall the end of the lgth century to the p r e ~ e n t .We the suggestions of Mill or Lewes in the second half of the lgth century. Mill defined what he called heteropatic laws in contrast to compositional ones and Lewes distinguished among resultant and emergent properties. During the 20th century the most important approaches were that of Emergent Evolution developed primarily by Lloyd Morgan or Wheeler,27i28the Theory of Integrative Levels exemplified in the works by Needham and N o v i k ~ f f , ~ ~ ~ ~ and the Materialist Emergentism well represented in the collection of papers edited by Sellars et al.31116 Simultaneously, Woodger was claiming that the most important feature of living beings was their hierarchical organization (we may notice that in the 30’s BertalanfFy was closely connected to Needham and W ~ o d g e r ) In . ~ the ~ years between the forties and the sixties and almost the seventies, hierachical approaches to natural or artificial system seemed to disappear from mainstream science. However, resurgence began as the work of Herbert Simon on the “Architecture of Complexity” in 1962 or Polanyi’s 1968 paper on the idea of constraint were p ~ b 1 i s h e d . By l ~ ~the ~ 1970’s the situation changed and we can highlight Howard Pattee’s work, among others, as well as that of T u r ~ h i nin ; ~the ~ 1980’s we will underline the contribution by Salthe,34 and today there is the work developed by Heylighen on metasystem and evolutionary transition^.^^,^^

3.1. Organicism Organicist theories in biology developed mainly during the interval between the two world wars by practitioners of theoretical biology generally linked to e m b r y ~ l o g y . Organicism ~~>~~ was a research framework that developed an epistemology addressed to cope with the complexity of life. It might be among the first ones in doing so in the 20th century since it dates back to the early days of the 1910’s with the work of Ross Harrison. Among those who were involved on the new perspective were the just mentioned Harrison, Bertalanfi, Needham, Waddington, Weiss, and Woodger among others. Organicism in a wider scope was a response to a set of general claims and debates developed all along the lgth century between vitalism and materialism (e.g., some of these were on naturalism vs. supranaturalism, determinism us. freedom, mechanicism vs. vitalism). A very specific

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one was related to how biological form should be explained, a problem that defied every mechanisic or vitalist explanation since the 18th century (and still does to some extent), due the emergence of order apparently “for free.” So, in a much more narrow sense organicism offered an alternative program regarding the emergence of embryological order. The main difference between this alternative view with reference to vitalism and materialism was the search for an emergentist account that would be consistent with physics. All the efforts of organicism were addressed to understand the forms that arise and proliferate in the organic world and the comprehension of the laws that generate them. To explain form it was necessary to understand what was responsible for its origins, maintainance and transformation. This was a twofold situation, on one side, it was necessary to approach the evolutionary aspects of form; on the other, its developmental features. A particular challenge posed by organic forms to any evolutionary or developmental account of it was that organic forms are progressively defined in the process of growing and not piecemeal built. In order to address these problems, organicism developed a set of ideas and concepts that can be classified, following Bertalanfi and Morton Beckner, in three groups: Organization and Wholeness, Directiveness, and Historicity or dynamic^.^^^^^ Organization and Wholeness. The concept of organization made reference to the hierarchical constitution of organisms. Each level was formed by parts which, due to the relations among them, displayed at that level rules and properties that were absent at the levels below. Within this frame some organicists like Harrison and Needham, developed their notions on the connection between structure and function. Akin to the concept of organization was the notion of wholeness. Organicists always emphasized the importance of the appreciation of wholeness. This notion was about regulation, that is, about the top-down perspective that allows to see how the rules at the level of the system exert their power over the relations among its constitutive parts. In other words, the existence of a whole stands on the subordination of parts to the rules or laws that characterizes the systems as such. To make explicit the link between form, wholeness and organization, for the organicists, form would be the expression of the whole. In this sense, their main concern was to understand how, along development and evolution, the relations among parts and the relations among levels give rise to new levels of complexity; and at each step of these processes, properties such as symmetry, polarity, and patterns are the means through which form

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expresses the wholeness of the system i.e. organism. Directiveness. Directiveness is another notion shared by the organicists. As Donna Haraway noticed, all different expressions of organicism agreed on the Aristotelian idea of the necessity of a teleological explanation that would be complementary to a materialist explanation (see Ref. 41:34). Teleology was present in the organicist way of thinking as a property of the whole; according with their view, this property is displayed at several organismal instances, that is, at the developmental process towards specific forms, at reproduction, and certainly in the case of behavior. Organicists highlighted that directedness at any of the instances was not an external imposition, but generated by the very organization of the system. It is interesting to notice that, on this subject, Bertalanffy suggested to replace the word teleology by that of organismic. The word organismic would express the idea that what is characteristic of organisms is that physicochemical processes are organized in such a way that “they are directed to the maintenance, production, or restoration of the wholeness of the organism.” (Ref. 39%). Historicity OT Dynamics. Organicism appreciated that organisms have a developmental and evolutionary history. The historical and dynamical character of organisms were explicitely recognized, specially in their notion of form as a dynamical property and in the rejection of any static notion of it. By this token there were emphasized at least three ideas: one, that development is a process of transformations that depends to some degree on previous states of the organic system. Second, that development is a process that involves novelty. Finally, the historical character is also linked to the idea that the developmental and evolutionary processes are irreversible.

3.2. Theory of Integrative Levels In the 1930’s and 40’s a lot of theoretical work on hierarchy theory was done by some researchers with an organicist background. Most of the researchers that participated on the development of the hierarchy theory proposed that nature is structured in emergent levels that are qualitatively different amongst each other. James Feibleman, Joseph Needham, and Alex Novikoff are some of these theorists that found inspiration in emergentist theories as well as in organicist approaches and developed what was known as Theory of Integrative Levels. Like organicism, this was not a reductionist, mechanicist or vitalist approach, but an emergentist approach grounded on a materialist epistemology.

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The Theory of Integrative Levels stated that the relation among the components of a system is what gives rise to new emergent levels and that new properties are the outcome of the organizational and integrative principles instead of the component properties. The purpose of taking into account the integration of the components in the system’s organization was to include as fundamental the properties of the level considered at the bottom and those of the levels considered at the top, i.e., the so-called emergent. They considered that there are some general levels that can be identified as physical, chemical, biological and social, and that each one has its very own laws as well as its own mechanisms through which changes or in between levels emergence occur. One of the most representative features of the Theory of Integrative Levels is their stand against the reduction of biology to physics and chemistry; Novikoff, for example, was clear about the fact that living organisms are not just machines made of many physico-chemical units that could be taken apart and whose global behavior could be deduced through analysis (see Ref. 30:210). For this theory the main biological levels were the cells, tissues, organs, organisms, and populations. As methodological and ontological aspects, we may consider the following rephrased from Feibleman:42 The analysis of a system must be done considering its lowest and its highest levels in order to achieve the most complete explanation. An organization always belongs to its highest level. At the end, every organization must be explained in terms of its very own level. It is impossible to give a full account of an organization if it is explained only in terms of its lowest or its highest level (e.g., an organism cannot be explained just in terms of genes or ecosystems). 3.3. Pattee and recent work (Pattee on hierarchies €d the concept of constraint)

Since the end of the 60’s, Howard Pattee developed a theory of hierarchies that was very i n f l ~ e n t i a lBetween .~~ 1969 and 1973 he wrote a compact series of papers that are mainly dedicated to the exposition of his hierarchical view (see Refs. 44, 45, 46, 47, 48). This work includes the edited volume entitled precisely Hierarchy Theory. The Challenge of Complex Systems,49 which included papers by J. Bonner, C. Grobstein, R. Levins, H. Simon and H. Pattee himself, besides a long Postscript on “Unsolved Problems

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and Potential Applications of Hierarchy Theory.”48 There is nearly no treatment of the subject since the 70’s that does not turn to, at least, his 1973 edited collection (see Ref. 34 for the best treatment one decade later, and Ref. 50 for a recent review of the state of the art). In a simplified way, it was his previous concern with the nature of molecular control together with the need to find operative ways to deal with complexity, what led Pattee to undertake the task of devising a more comprehensive hierarchical perspective. Following Simon’s path (see Refs. 19, 51), Pattee related hierarchical organization to a best likelihood of evolution, but he went further in order to explore the conditions of possibility for unlimited open-ended evolution of a von Neumannian ancestry.43 Therefore, he complemented Simon’s, mostly structural, kind of hierarchical order with the elucidation of a functional kind, which he needed to give account of control attributes. In order to give an account of this functional kind, he developed Polanyi’s idea of the dual control of laws and constraint^.^ Polanyi’s view of boundary conditions (or constraints) as harnessing the laws of nature and, therefore, neither disobeying them nor being reducible to them, will allow Pattee to develop a biological theory grounded on an illuminating understanding of the role of constraint. Pattee made a distinction between holonomic and non-holonomic constraints: a. Holonomic constraints: These are constraints that freeze or simply eliminate permanently some degrees of freedom making easier to de-

scribe the system and its behavior; e.g. the walls of a Petri dish; boundary conditions, and the surface of a solid. In the cme of dynamical systems, holonomic constraints are rate-dependent to the flux of matter and energy of the system so they arise when certain conditions are present. Holonomic constraints allow the establishment of a statistical and global order. b. Non-holonomic constraints: They are material structures that are rate-independent with respect to the inherent dynamics of the system, though they act locally on it. By the same token, these are non-permanent constraints, they temporally generate relations of dependency among some degrees of freedom of a system. It can be said that this kind of constraint is a material structure that embodies a rule. An example would be a switch, which is independent of the flow of electrons, though the switch x t s on this flow when the rule

262 conditions are satisfied. Due to its dynamical decoupling, it would be more proper to describe it as another (higher) level. Consequently, Pattee insisted on two somehow paradoxical but crucial features of hierarchies: that very organized systems as hierarchies experiment a simplification of their behavior, and that the limitations exerted upon the degrees of freedom at one level result in an increase at the subsequent one. Those operations are performed through material devices in the form of constraints of different kinds, mainly holonomic for structural hierarchies and non-holonomic for functional ones. The very specific control relation is introduced as selection of alternatives. Together with the basic distinction between laws and rules and the hierarchical perspective distinguishing structural and functional, the concept of physical constraint makes room for the natural acceptance of a fully material kind of downward causation. 4. Conclusion

The work of all these researchers on biological complexity offers a toolkit to build an operative approach to phenomena whose epistemological implications go beyond biology and which is interesting for the study of complex systems in general. Beginning with the necessity of a hierarchical arrangement, there is further interplay among structural and functional hierarchies. In this context, the emergence of autonomous hierarchies implies having systems with the capacity to self-impose holonomic and non-holonomic constraints. Those systems and their increasingly complex behaviors appear naturally in biological cases (and, obviously, in socio-cultural ones), but are also quite interesting for artificial systems. Within this kind of full-fledged complex systems, the relations among levels are described as upward and downward causation and are made operative through the concept of constraint. But constraints understood as enabling mechanisms and not only as restrictive material structures. Finally, this scenario sets the conditions to allow the open-ended evolution of complexity.

Aknowledgements Funding for this research was provided by grant HUM2005-02449 from the Ministerio de Educacin y Ciencia of Spain and the FEDER program of the E.C., and grant 9/UPV 00003.230-15840/2004 from the University of the Basque Country. Jes6s M. Siqueiros is holding a predoctoral scholarship

263 from t h e Consejo Nacional de Ciencia y Tecnologia (CONACyT) of Mexico. Jon Umerez holds a Ram6n y Cajal senior researcher position (R y C program is funded by t h e MEC a n d t h e UPV/EHU).

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264 21. L. von Bertalanffy, General Systems Theory. Foundations, Development, Applications. New York, George Braziller (1968). 22. L. von Bertalanffy, Perspectives o n General Systems T h e o y . ScientificPhilosophical Studies. New York, Georges Braziller (1975). 23. N. Wiener, Cybernetics or Control and Communication in the Animal and the Machine. Cambridge, MIT Press (1948 [1961]). 24. J. Umerez, and A. Moreno, Origin of life as the First MST. Control hierarchies and interlevel relation. World Futures, 45, 139-154 (1995). 25. J. S. Mill, A System of Logic, Ratiocinative, and Inductive. London, Longan and Green (1843). 26. G. H. Lewes, Problems of Life and Mind. London, Trench and Turber & Co. Kegan Paul (1875). 27. C. L. Morgan, Emergent Evolution. New York, Henry Holt and Cn. (1923). 28. W. Wheeler, Emergent Evolution and the Development of Societies. New York, Norton (1928). 29. J. Needham, Integrative Levels: A Revaluation of the Idea of Progress. Oxford, Clarendon Press (1937). 30. A. Novikoff, The concept of integrative levels and biology. Science, 101, 209215 (1945). 31. R. W. Sellars, V. J. McGill, and M. Farber (Eds.), Philosophy for the Future. The quest of modern materialism. New York, Macmillan (1949). 32. J. H.Woodger, Biological Principles: A Critical Study. London, Routledge and Keegan Paul (1929). 33. V. Turchin, The Phenomenon of Science. New York, Columbia University Press (1977). 34. S. Salthe, Evolving Hierarchical Systems. Their Structure and Representation. New York, Columbia University Press (1985). 35. F. Heylighen, The growth of structural and functional complexity during evolution. In: F. Heylighen, J. Bollen, A. Riegler (Eds.), The Evolution of Complezity. Dordrecht, Kluwer Academic (1999). 36. F. Heylighen, Evolutionary transitions: how do levels of complexity emerge? Complezity, 6 (l),53-57 (2000). 37. K. Ruiz-Mirazo, A. Etxeberria, A. Moreno, J. Ibbnez, Organisms and their Place in Biology. Theory in Biosciences, 119 (3-4), 209-233 (2000). 38. A. Etxeberria and J. Umerez, Organism0 y Organizaci6n en la Biologia Te6rica Vuelta a1 organicismo 50 aiios despuhs? In: A. Ambrogi (Ed.) Medio siglo de doble hdice. Edicions Universitat de les Illes Balears (in press) (2006). 39. L. von Bertalanffy, Modern Theories of Development. A n Introduction t o Theoretical Biology. London, Oxford University Press (1933). 40. M. Beckner, The Biological Way of Thought. Berkeley, University of California Press (1968). 41. D. J. Haraway, Crystals, Fabrics, and Fields. New Haven, University of Yale Press (1976). 42. J. K. Feibleman, Theory of Integrative Levels. The British Journal f o r the Philosophy of Science, V (17), 59-66 (1954). 43. J. Umerez, Howard Pattee’s theoretical biology. A radical epistemological

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SIMULATION AS FORMAL AND GENERATIVE SOCIAL SCIENCE: THE VERY IDEA*

N U N 0 DAVID1, JAIME SIMAO SICHMAN2 & HELDER COELH03 ‘Department of Information Science and Technology, I S C T E Av. das ForGas Armadas, 1649-026 Lisboa, Portugal E-mail: Nuno. DavidOiscte.pt Computer Engineering Department, University of Sao Paulo Av. Prof.Lucian0 Gualberto 158 tv.3, 05508-900 Sao Paulo SP Brazil E-mail: [email protected] 3Faculty of Sciences, University of Lisbon Bloco C5, Piso 1, Camp0 Grande, 1749-016 Lisboa, Portugal E-mail: hcoelhoOdi. fc.ul.pt

The formal and empirical-generative perspectives of computation are demonstrated to be inadequate to secure the goals of simulation in the social sciences. Simulation does not resemble formal demonstrations or generative mechanisms that deductively explain how certain models are sufficient t o generate emergent macrostructures of interest. The description of scientific practice implies additional epistemic conceptions of scientific knowledge. Three kinds of knowledge that account for a comprehensive description of the discipline were identified: formal, empirical and intentional knowledge. The use of formal conceptions of computation for describing simulation is refuted; the roles of programming languages according to intentional accounts of computation are identified; and the roles of iconographic programming languages and aesthetic machines in simulation axe characterized. The roles that simulation and intentional decision making may be able to play in a participative information society are also discussed.

1. Introduction The analysis of complex systems is being pursued with increasingly more sophisticated information technologies. In particular, the area of computer simulation has acquired a decisive role for analysing societies as complex systems, leaving behind the history of simulation as a secondary methodology in the social sciences. The sources of analogy between agent-based *The title of this paper is inspired by James Fetzer’s article “Formal Verification: The Very Idea”. See Ref. 1.

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267 technologies and social scientific models fomented an unprecedented interdisciplinary effort, which has been creating countless interfaces of research, across the computer and social sciences. Several reasons exist for conducting a philosophical analysis of this scientific domain. In general, social science simulation has contributed to an inter- and multi-disciplinary scientific praxis,a thereby establishing new alternatives to traditional scientific methodologies. This should lead to the elaboration of new philosophical perspectives about the rules of the game as they are played in ~ i m u l a t i o n . ~ Within the scientific community, the existence of methodological aspects that deserve better analysis is recognized. For some, the use of formal models, resulting from the computational nature of simulation, has been considered not only an addition to the established methods but the basis for the emergence of “proper social sciences”.‘ For others, the classical theory of computation does not support an adequate model of reality for simulation in the social sciences, and therefore the formal perspective of computation is not enough.d At any rate, the difficulties in constructing methodological perspectives on simulation raises interesting questions about the kind of scientific knowledge that simulation is providing. Once the philosophy of simulation is analysed, it becomes clear that most essays do not take into account methodological and philosophical aspects of computer science, but are grounded mostly on aspects of social science. Among the diversity of perspectives that may be adopted, most should lead to recognizing additional ways to understand the concepts of computation and programming languages. However, few if any philosophical analyses in this field considered theoretical and practical limits of computation, as well as its new approaching challenges, well understood. Specifically, among the questions confronted by social science simulation is the extent to which formal and empirical methodologies are sufficient to describe the goals and methods of the discipline. In this paper we claim that they are not. As we demonstrated in David et al. (2005)’, simulation reveals new conceptions about the kind of scientific knowledge that computers provide. aOn the interdisciplinary structure of the scientific community in agent-based social simulation see Ref. 2.

bSee the introduction to the issue of JASSS on epistemological perspectives on simulation, Ref. 3. See also Ref. 4. ‘See specifically Ref. 5. See also the debate in Ref. 6. dSee Ref. 8.

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Meanwhile, the meaning of social simulation is strongly connected with two traditional epistemic conceptions of scientific knowledge. The first conception is the formal view of mathematical knowledge, reminiscent of the computer science formal tradition and congruent with the idea of Simulation as numerical calculation. The second refers to the experimental character of simulation, insofar as scientists run their programs in computers like an experimental set-up. Programs, in both senses, are viewed as descriptions of social theories or phenomena that, unlike most theories in the social sciences, are viewed as formal mathematical models. The conflation of the formal and the experimental perspectives of simulation lead many scientists to pose simulation as a way of testing representations of social theories or phenomena that could, in principle, be deduced from general principles. This is based on three tacit methodological assumptions that draw on the formal tradition of classical computational theory, which we will challenge throughout this article:

Assumption 1 . The process of executing a program in a computer can be formally modelled, and thus understood as an automatic process of formal inference. This position was explicitly advocated by Epstein ( 1999)l0, drawing directly on the Turing-Church thesis.e Assumption 2. The sensitivity of complex models to initial conditions does not permit deducing its simulated results from general principles, although it would be possible to do it in principle. Casti, for instance, described this conception in a very explicit way: “In principle, one could trace all the statements in the program representing the simulation and discover exactly why things unfolded the way they did (. . . ) it just cant be done in practice.” (Casti, 2001, p.14, our emphasis)” Assumption 3. The experimental character of simulation in the social sciences can draw on the experimental character of simulation in the natural sciences.f Insofar as the sensitivity of general complex models explains the inability to deduce its simulated results from general principles, the sensitivity of social-scientific models explains similarly the inability to deduce its simulated results from general principles. Casti, for instance, tacitly conflated the present and the previous assumptions as follows:

CForan introduction to classic computer theory, also known as the Church-Turing thesis, see e.g. Ref. 9. ‘For a comprehensive critic see Ref. 8; see also Ref. 12.

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"For the first time in history we are in a position to do bona fide laboratory experiments on these kind of complex systems. (. . . ) We can use these surrogates as laboratories for carrying out the experiments needed to be able to construct viable theories of complex physical, social, biological and behavioural processes." (Casti, 1997, p.35, our emphasis)13 Recently, a number of essays debated epistemological perspectives on social science simu1ation.g Either by viewing simulation as a process of imitation12, stylized facts 14, or intentional adequacy between programs and theory', the tendency is to emphasize the interpretative character of social science in simulation, notwithstanding its application as a useful methodology for approaching complexity. The essay presented by David et al.', in particular, advocates the perspective of intentional computation as the approach able to comprehensibly reflect the multiparadigmatic character of social science in terms of agent-based computational social science. Conversely, essays presented by Moss and Edmonds15 or Boero and Squazonni" argue that simulation provides the social sciences with a powerful instrument to generate empirical evidence, and thereby, contribute to better social sciences. Whereas these two trends may not be contradictory, the formal character of computer science pervades the scientific culture of simulation, as well as most of its methodological arguments. However, the formal perspective does not seem to be compatible with any such views. In this chapter the classical account of computation is demonstrated to be inadequate to secure the goals of simulation in the social sciences. Simulation does not resemble formal deductive demonstrations or generative mechanisms that explain how certain agent-based models are sufficient to generate emergent macrostructures of interest. The justification of results implies additional epistemic conceptions of scientific knowledge, which are tacitly being used in the discipline. Three kinds of knowledge that account for a comprehensive description of the discipline from an epistemological point of view are identified: formal, empirical and intentional knowledge. Intentional knowledge should be considered an outcome of an experimental exercise, albeit not empirical, acquired within a context of limited consensus. The structure of our argument in this chapter is composed of two parts. In the first part the formal and generative perspectives of computation in gSee the introduction to the issue of JASSS on epistemological perspectives on simulation, Ref. 3.

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literature are recalled, proceeding afterwards to refuting the use of such conceptions for describing the scientific practice of simulation. In the second part the role of programming languages in simulation, according to intentional accounts of computation, is discussed. Two types of programs and programming languages in simulation are identified: Programs as text and programs as icons; and languages as abstract machines and languages as aesthetic machines. The use of abstract languages confirms that the method of simulation incorporates formal and empirical methodologies. The use of aesthetic languages demonstrated that it depends fundamentally on intentional methodologies. The roles that intentional decision making may play in a participative information society are also discussed. 2. An Ontological Confusion Conventional methodologies of computer science model the mechanism of executing a program in a computer like a process of formal inference. In complexity sciences, the process of running a program was described not only as an automatic inference procedure but as a formal deductive procedure itself. The computer is seen as a mechanism of formal calculus, where programs represent mathematical functions that map inputs into outputs. The calculus can be modelled in several ways, one of which ascribes the computer the capacity to prove first-order theorems according to a fixed vocabulary. Considerations of brevity and simplicity lead us to call this tradition the FDE argument, as per ‘Formal Deduction through Execution’. Notwithstanding, our goal is quite the opposite, namely to demonstrate that simulation shall not be legitimized under the presumption of resulting from a calculus of formal inference. Additional conceptions of knowledge are needed.

2.1. Against Genemtive Suficiency of Growing Artificial Societies From the Bottom Up The relationship of simulation with the formal perspective of computation was rigorously advocated by Epstein (1999, p.44)lo in his account of “generative social science”, where he writes:h . . . if one accepts the Church-Turing thesis then every computation - including every agent-based computation - can be executed by a suitable register machine. It is then a theorem of logic “

hThe concept of generative social science was also adopted in Ref. 17.

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and computability that every program can be simulated by a firstorder language.” The point is the following: . . . for every computation there is a corresponding logical deduction, and this holds even when the computation involves ‘stochastic’ features since, on a computer, these are produced by deterministic pseudo-random number generation. Even if one conducts a statistical analysis over some distribution of runs, each run is itself a deduction.” (Epstein, 1999, p.44)lo “

The methodological point of social science simulation would be to generate observed social phenomena in computers and thus deductively explain the social phenomena, insofar as there is an intellectual tradition upon which “we deduce propositions expressing observations from other more general propositions” . i Accordingly, scientists “seek to explain macroscopic social phenomena by generating it in an agent-based computational model”. Moreover, “in that event, we can claim that they [the explanations] are strictly deductive.” (p.43, our emphasis and brackets)” We may state Epsteins argument of generative sufficiency as follows:

The Generative Sufcciency Argument. Agent-based models provide formal demonstrations that a given microspecification is a suficient condition, albeit not a necessary condition, to generate a macrostructure of interest. And this leads Epstein to conclude: ‘‘J?rom an epistemological stand point, generative social science, while empirical, is not inductive, at least as that term is typically used in the social sciences.” (1999, p.43)lo 2.2. ‘Generative’ from a Philosophy of Computer Science Perspective The arguments of Epstein can be analyzed from the perspective of the philosophy of computer science. The term ‘formal’ is ubiquitous in computer science. Smithlg, for instance, has acknowledged the ambiguity of the term: ‘Epstein‘s account of scientific explanation seems to be inspired by the work of classical empiricists, such as in Ref. 18.

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“People may believe that developing an idea means formalizing it, and that programming languages are formal languages, and that theorem provers operate on formal axioms - but few write ‘formal’ in formal axioms or daily equations. Moreover, a raft of different meanings and connotations lie just below the surface. Far from hurting, this apparent ambiguity has helped to cement popular consensus. Freed of the need to be strictly defined, formality has been able to serve as a lightning rod for a cluster of ontological assumptions, methodological commitments, and social and historical biases. Because it is tacit, goes deep, has historical roots, and permeates practice, formality has been an ideal foil, over the years, with which to investigate computation. Almost a dozen different readings of ‘formal’ can be gleaned from informal usage: precise, abstract, mathematical, a-contextual, digital, explicit, syntactic, non-semantic, etc.” In a Church-Turing theoretic account, the meaning of ‘formal’ appears connected to the antisemantical reading mentioned above. That is, the idea that a symbolic structure is formal just in case it is manipulated independently of its semantics, in which it is assumed that a theorem is derived by an automatic inference regimen. To some extent, the pervasive use of the term ‘formal’ arises from conflating the terms ‘program computation’ and ‘program execution’ into one single meaning, conveying the same ontological status to two fundamentally distinct p r o c e ~ s e s The . ~ concept of ‘abstract machine’ in classical computation is conflated with the concept of ‘physical machine’ in software engineering. The observed behaviour of a program executing in a computer, which should be the subject of research in the first place, gives way to a computation that is, in essence, a formal model or a theory itself. As Epstein himself states (1999, p.44)lo, “each run is itself a deduction” the actual execution of a program in a physical machine is a theory or a computation in itself. It would be reasonable to question whether the idea of Epstein is not to illustrate anything more than an ideal perspective of computation in real computers. Yet, his efforts are not limited to illustrating that simulation implies deductive and generative conceptions of scientific research but that simulation implies a new kind of scientific empirical research. The point of simulation, in such a sense, is to provide agent-based specifications for which the corresponding program execution should generate patterns

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that match empirical data - for instance, to implement in a computer an agent-based archaeological model that succeeds in generating behaviours that match empirical data and thus explain why the Kayenta Anasazi population of Long House ValleyJ vanished at some point from the valley (1999, p.44)’O. The role of computer science, therefore] is one of formal inference, whereas the role of social science simulation is to use computers to generate behaviours that should match social scientific data. Apart from the arguments of Epstein, there is wide evidence confirming a tacit association of simulation with formal deduction, for instance, by suggesting the high levels of control and objectivity that scientists attribute to simulation results. The claim of Castill that it is possible to trace all statements in the simulation is obvious evidence. Prietula et a1.” advocate that: “computational models are generally less noisy, easier to control, more flexible, more objective”. Sawyer explains: “the contributions of artificial societies to sociological theory will be primarily to theories that are characterized by logical rigor (. . . ) ] which allow precise and logical deductions from abstract principles to empirical hypotheses (. . . ) Artificial societies resemble axiomatic theory in the sense that their propositions are explicitly stated in the form of algorithms or program code and valid derivations may be drawn systematically by running the program” (2003, p.332)’l. For Axelrod, one of the advantages of simulation is that “there are no messy problems of missing data or uncontrolled variables as there are in experimental or observational studies” (1997, ~ . 2 7 ) ’ ~ . The most cited sentence of Axelrod, describing simulation as a bridge between induction and deduction, is actually not far from the idea. Whereas it is claimed that simulated data must be analysed inductively, it is suggested that the data are a necessarily valid consequence of the rules specified: “Simulation is a third way of doing science. Like deduction, it starts with a set of explicit assumptions. But unlike deduction, it does not prove theorems. Instead, a simulation generates data that can be analyzed inductively. Unlike typical induction, however, the simulated data comes from a rigorously specified set of rules rather than direct measurement of the real world. While induction can be used to find patterns in data, and deduction can be used to find consequences of assumptions, simulation modelling can be used as J A small region in northeastern Arizona.

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an aid to intuition.” (Axelrod, 1997, ~ . 2 5 ) ~ ~ However, insofar as the data can be analysed inductively, it seems difficult to conceive how it was generated in the first place without “direct measurement of the real world”. Computers in this sense do not belong to the “real” world. That is to say, the tacit methodological assumption of social science simulation would propose the same ontological status to both a program execution in a physical machine and a formal computation in an abstract machine. We shall call this assumption the FDE argument, as per Formal Deduction through Execution:

The Argument of Formal Deduction through Execution. It is based on three methodological beliefs, which we will show to be contradictory with one another: A) The mapping of classical computer theory into the logic of the method of social science simulation. B) The conflation of the terms ‘computation’ and ‘execution’. C ) The experimental character of simulation, upon which an unexpected result can be a reflection of a mistake in the implementation (bug) or a surprising consequence of the program itself (see e.g. Axelrod, 1997, ~ . 2 7 ) ~ ’ .

2.3. Refutation of Formal Deduction through Execution

In this research, it is claimed that the role of computation in social simulation is not one of formal deduction. The goal is to refute the FDE argument. From a scientific-philosophical perspective our goal may seem somehow trivial. However, we demonstrated that the formal perspective of computer science is recalcitrant within social science simulation. Moreover, the objection to the FDE argument suggests yet another objection to current philosophical thinking in the literature: The objection to characterizing simulation as a basic and alternative epistemic conception to deduction or induction. The objection to the FDE argument parallels an old, but still omnipresent, debate in computer science, centered around the merits of the so-called “formal verification of programs”. The merits of formal methods for verifying programs became particularly controversial by the end of the Eighties and in the beginning of the Nineties after James Fetzer published an eloquent article in the ‘Communications of the ACM’’>23.On one side of the debate were gathered those who considered that computer program-

275 ming was reducible to mathematics - inspired by Hoare or Dijkstrak and the other side was composed of those who saw it as applied mathematics or empirical science. The claim of Fetzer was that the formal verification project carried misleading ideas. The seminal work of Hoare is an obvious example: “Computer programming is an exact science in that all the p r o p erties of a program and all of the consequences of executing it in any given environment can, in principle, be found out from the text of the program itself by means of purely deductive reasoning.” (Hoare, 1969, p.576, our emphasis)25 These ideas often turned out to be misleading. For instance, it was often claimed that the intended behaviours of computers could be completely specified and verified before the corresponding programs were executed on specific computers, by means of purely formal methods. Computer science would be viewed as pure mathematics rather than as applied mathematics or empirical science. Fetzer‘s philosophical refutation of the formal verification project consisted of distinguishing two kinds of programs: those that are and those that are not in a suitable form to be compiled and executed in a machine. The difference between the programs is that the former is verified by reference to abstract machines, whereas the latter require the existence of compilers, interpreters and target machines. Compilers, interpreters and processors are properly characterized according to specific target physical machines. Insofar as a program in an abstract machine does not possess any significance for the performance of a target machine, the performance of that program can be verified formally and conclusively. Conversely, to the extent that the performance of a program possesses significance for the performance of a target machine, that program cannot be conclusively verified a priori. The program must be verified empirically by means of program testing. Hence, the formal verification project is not viable.’ Program testing should be the crucial technique to ascertain the proper behaviour of programs and computers, notwithstanding the use of formal methods during the stages of analysis and design. Our goal is to informally reduce the FDE argument in social science simulation to the formal verification project in computer science. Consider the kSee e.g. Ref. 24. ‘For more details see Ref. 1 or 23.

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formal verification project according to its most radical terms. The intent would be to create formal methodologies that could guarantee that a given specification would correspond to the behaviour of a program executing in a computer. That is, to find deductive procedures to verify conclusively the correctness of a program P in relation to a specification F : I 4 0, in order to guarantee that the execution of P with inputs I would result exactly into the specified outputs 0. The following argument reduces FDE to the formal verification project.

The Argument of FDE Refutation. Consider a specification F1 : I1 -+ 01 and a program P1 as text that can be read, edited, printed. The computation of P1 with inputs I1 is denoted by Pl(I1) = 01 and the execution of P1 after implementation is denoted by Pl'(I1) M 01. Suppose that Pl(I1) = 01, according to a proof of partial correctness.m Suppose, however, that Pl'(I1) M 0 2 , that is the computation and the execution of P1 leads to different results, compatible with assumptions B and C in the FDE argument. However, according to assumptions A and B in the FDE argument, there is a specification F2 : I1 + 0 2 and some program P2 such that the computation with input I1 leads to 0 2 , i.e. P2(11)= 0 2 . So, a specification F2 : I1 -+ 0 2 exists such that the execution of P1 and the computation of P2 satisfy F2. The formal verification project is thus possible: the behaviour of P1 execution (as well as P2 computation) necessarily corresponds to the specification F2 : I1 -+ 0 2 . In short, from a methodological point of view, the FDE argument cannot be sustained. At the very least it is misleading. Even though it does not suggest that both the computation and the execution of P1 necessarily give the same outputs, which would prima-facie instantiate the formal verification project, it may suggest that the execution of P1 necessarily corresponds to a formal computation of some program P2, which is a methodological absurdity. The formal perspective of computation is not enough to support an adequate model of reality for simulation in the social sciences. The logic of simulation implies distinct types of program verifications that reflect epistemological distinctions in the kind of knowledge one can have about programs. One obvious type of knowledge is empirical knowledge obtained through program testing. Another one is intentional knowledge,8 which is discussed in subsequent sections. mOn proofs of partial correctness see e.g. Ref. 25.

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A final comment emerges regarding the perspective of Axelrod22 that poses simulation as a contrast to both induction and deduction, which is after all not far from the formal perspective of Espstein. Whereas Axelrod defines induction as the “discovery of patterns in empirical data”, deduction is understood as the specification of “a set of axioms and proving consequences that can be derived from those assumptions” (1997, ~ . 2 4 ) Yet, ~~. there is no reason for not viewing deduction as a kind of empirical enquiry. Popper26,as many other ‘deductivists’ in the philosophy of science, would say that there is no such thing as induction. Inference through Em piric a1 E n q u i r y deductive@ ar inductively or abductive ly ...

I

I

i

1

de duc tive or hductive@ar abductively ...

I I

! Natural Process Figure 1. A theory square.

Inference through Empirical Enquiry deductive@ or inductively or abduct& ly ...

I I I

I I

I

I

de duc tivc ly or hductivelyor hbductively ...

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Program Execution

Figure 2.

A second theory square.

Consider Figure 1. The theory square represents a traditional epistemic question in the philosophy of science with respect to empirical science: Are correlations between natural properties enquired inductively or deductively? Likewise, it is equally fair to use a theory square in relation to the behaviour of a program executing in a computer, just like in Figure 2. Indeed, whether the enquiry may be inductive, deductive, or even abductive depends on the relevant methodological conception of scientific enquiry, which is by no means a specific dilemma of simulation. To define the epis-

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temic specificities of simulation based on such contrast does not seem to be significantly informative to the point in question.

3. The Role of Programming Languages An informative way to analyse the epistemic status of simulation is to examine how simulations are programmed. Program implementation plays a fundamental part in constructing a simulation. The role of intentional methodologies becomes crucial once it is realised that, rather than one kind of program and programming language, two kinds of program and programming language are used in the implementation of a simulation. Simulations are constructed by means of successive language translators, from program to program, by using simulation platforms, compilers and interpreters. In computer science we usually view a programming language as a n abstract machine and a program as a textual and static entity, which may be read, edited, printed. However, in social simulation we have identified yet another type of programming language and kind of program, which involve the use of intentional knowledge. At least two kinds of programming languages are used in the process of implementing simulations, iconographic and textual programming languages.

Iconographic Programming Languages. Iconographic programming languages consist of a set of subjective rules that model the behaviour of an aesthetic machine. A subjective model, eventually associated with a specific domain composed of organisational or aesthetic abstractions, such as groups of agents, grids, movement, constraints, roles, levels, messages, societies, or specific behaviours, ‘segregation rules’, ‘sexual rules’ and ‘genetic crossing’, ‘culture’, ‘race’, ‘influence’, ‘friendship’, ‘innovation’, ‘state nations’ or ‘political actors’. For instance, the CORMAS platform,” as well as the Swarm simulation system,O support aesthetic human-machine interfaces to specify interactions between individuals and groups sharing renewable natural resources. The commands in the programming languages are icons represented on the screen rather than structures in textual languages. A program is a set of selected icons with no definitive explicit order. The user selects icons with the mouse, after which an automatic code generator transforms icons into sections of code in high-level programming languages, such as SmallTalk or Objective C. However, the first order logics of classic nCornmon-pool Resources and Multi-Agent Systems, see http://cormas.cirad.fr. Osee http://www.swarm.org.

279 computer theory can hardly describe these transformations. Moreover, the sections of code are later linked in some arbitrary way by the user. The specific mapping from icon-level language to high-level language is subjective, domain specific and validated according to a limited level of consensus. In the case of participative-based simulation, stakeholders may be involved in the specification of iconographic programs, but hardly in the semantic process of mapping iconographic programs to high-level programs.

Textual Programming Languages. Textual programming languages include the usual high-level and low-level languages. A high-level language, as defined by Fetzer2?, is a set of abstract rules that model the behaviour of an abstract machine. High-level languages have clear formal semantics, and contrary to icon-level languages, the meaning of their commands must not be subjective. The advantage of programming with high-level languages, such as Java or Objective C, is that there is a one-to-many relationship between the commands that can be written in a high-level language and the counterpart operations that are performed by a machine executing them, on the basis of their translation into machine language. The function of interpreters and compilers is to create a causal mechanism so that programs written in high-level languages may be mapped to low-level languages and later executed by target machines whose operations are causally affected by machine code, which usually consists of sequences of zeros and ones.27 A low-level language is a set of abstract rules that model the behaviour of abstract or target machines. Typically, the lowest-level language programmers use is Assembly language, where there is more or less a one-to-one correspondence between commands and operations. Low-level programming languages therefore play two roles: First, that of an abstract machine, in a way analogous to high-level languages but where, second, unlike high-level languages, there is a one-to-one causal relationship between the commands that occur within a programming language and the operations performed by a target machine. The programming language stands for a virtual machine that may be understood as an abstract entity, which may or may not be causally connected with a target machine. The distinction between programs as text and programs as icons, as well as between abstract and aesthetic machines, reveals the intentional character of social simulation methodologies. Whereas the consistency between two different abstract machines can be specified formally and verified empirically, the consistency between abstract and aesthetic machines must be

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verified intentionally. Indeed, the modelling of target machines by means of abstract machines can be defined according to formal logic, drawing on classic computer theory. Hence, the relative consistency between the highlevel and low-level abstract machines can be demonstrated formally, as well as tested empirically against the observed behaviour of the program. Moreover, insofar as most high-level programming languages stand for abstract machines rather than physical machines, we can say that the process of implementing a high-level program involves the construction of a sequence of embedded models that are causally connected to a target machine. In contrast, whereas abstract machines are specified with well-defined formal semantics, the meaning of iconographic languages and aesthetic machines is negotiated intentionally by the members of the team implementing the simulation, as well as by the stakeholders involved. Since iconographic languages must be mapped into high-level, and ultimately into low-level languages, it becomes clear that the process of implementing icon-level programs involves the construction of a sequence of embedded models connected intentionally to a target machine. The target machine is modelled intentionally according to a limited level of consensus and tested experimentally, albeit not empirically, against the observed behaviour of the icon-level program. From this point of view, the implementation of a program can be viewed as the action of embedding models within models, where the notion of embedding may be envisioned as an intentional, causal or logical relation. In Figure 3, Fetzer’s diagram of language embedding27 is expanded with iconlevel programs and aesthetic machines. The thin arrows represent a possible relation between programs and machines represented by programming languages. The thick arrow represents an actual relation between a low-level program and a target machine. The series of three black dots stands for the possible existence of automatic code generators, compilers and interpreters that effect some causal connection between programs/machines at different levels of embedding. In addition, the series of unfilled dots stands for the existence of implementation teams and stakeholders that exercise intentional connections between abstract and aesthetic programs/machines at different levels, according to a limited level of consensus. The use of iconographic programming languages demonstrates that the logic of simulation incorporates formal and empirical methods, but largely surpasses the use of formal and empirical methodologies. The results of a simulation are outcomes of experimental set-ups, but the results of the experiments can hardly be represented by material conditions of necessity

281 PROGRAMS

MACHINES

HIGH LEVEL

LOW LEVEL

Figure 3. Programs and languages 89 models - an extension t o the diagram of Fetzer with icon-level programs and aesthetic machines.

relating facts about the objective behaviours of the program. The results are appropriately characterized by conditions of intentionality that relate aesthetic components in the program, negotiated according to a limited level of consensus. 4. A Different Idea of Simulation We shall recall the traditional assumptions of simulation in the social sciences, presented in the introduction, which we have been challenging throughout this article. Firstly, whereas it is true that the process of executing a program can be modelled according to a mechanism of formal inference, it does not seem to be significantly relevant to the methodology of social science simulation. Secondly, whereas sensitivity to initial conditions is an important issue in the social sciences, it is not the primary reason explaining the inability to deduce simulation results from general principles, at least from a technical point of view. A tremendous semantic gap exists between the formal-empiricaland the intentional perspective of computation, both of which are used to interpret the behaviour of simulations. The misleading assumption of generative social science is that both formal and intentional representations can be legitimised deductively, insofar as the process of program execution can be understood as a formal deductive mechanism. This reasoning is unsustainable. Firstly, the vocabularies of the low-level abstract machine (e.g. memory registers, bit logical operations), as well as the vocabularies of the

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high-level machine (e.g. complex data structures, objects, graphics), are not identical to the vocabularies of the icon-level machine (e.g. agents, grid, movement, culture, segregation rules). The parts that those vocabularies designate in the world are not the same; from a strict formal point of view the consistency between machines is incommensurable. Secondly, although the consistency between low-level and high-level abstract machines can be verified empirically, the consistency between abstract and aesthetic machines must be verified intentionally. But unlike the process of empirical adequacy, no formal computational theory is available to justify the process of intentional adequacy.P The characterization of simulation as a research practice implies additional epistemic conceptions of scientific knowledge that are being used tacitly in the discipline. There are at least three different kinds of knowledge that can be acquired from computer simulation: formal, empirical and intentional knowledge. Intentional knowledge should be considered an outcome of an experimental exercise, albeit not empirical, acquired within a context of limited consensus. Social science simulation, like the social sciences, is interpretative and diverse theoretically and methodologically. To imagine that simulation could integrate the archipelago of the social sciences, at least as far as that may depend on the establishing of wide consensuses, like those found in the natural sciences, would be a mistake. The conditions for the acceptance of a simulation depend on the particular theoretical-methodological context of the social scientist, can be interpretative and subjective, and may depend on the socioeconomic and sociocultural context. The perspective of intentional computation seems to be the one able to reflect the multiparadigmatic character of social science into social science simulation. Finally, the observation that social science simulation is multiparadigmatic demonstrates the need for participative contexts. As a growing information technology, simulation is being used to assess concrete socioeconomic and environmental/ecological problems. Simulation is a useful methodology to approximate scientists and stakeholders. However, only within a specific interpretative context can a specification or a program be considered as a set of sufficient conditions to explain the observed behaviour

PMoreover, as we mentioned before, even if we ought to stay on the empirical plan (which is not the case), whether the epistemic conception of inquiring the behaviour of computers could be an inductivist or a deductivist one that would be by no means a specific dilemma of simulation.

283 of a simulation. Indeed, from a sociological perspective, simulation should help us pose the process of science as critical thinking in a democratic

context. Acknowledgements Jaime S i m k Sichman is partially supported by CNPq, Brazil, grants 482019/2004-2 and 304605/2004-2.

References 1. James Fetzer, Program Verification: The Very Idea, Communications of the

ACM, 31, 1048-1063(1988). 2. Nuno David, Maria Marietto, Jaime S. Sichman, Helder Coelho, The Structure and Logic of Interdisciplinary Research in Agent-Based Social Simulation, Journal of Artificial Societies and Social Simulation, 7(3), ihttp://www.soc.surrey.ac.uk/JASSS/7/3/4.htmli,(2004). 3. Ulrich Frank and Klaus Troitzsch, Epistemological Perspectives on Simulation, Journal of Artificial Societies and Social Simulation, 8(4), ihttp://jasss.soc.surrey.a~.uk/8/4/7.html~ (2005). 4. Carlos Gershenson, Philosophical Ideas on the Simulation of Social Behaviour, Journal of Artificial Societies and Social Simulation, 5(3), ihttp://jasss.soc.surrey.ac.uk/5/3/8.htmli,(2002). 5. J. Kluver, C. Stoica and J. Schmidt, Formal Models, Social Theory and Computer Simulations: Some Methodological Reflections, Journal of Artificial Societies and Social Simulation, 6(2), ihttp://jasss.soc.surrey.ac.uk/6/2/8.html~(2003). 6. Rosaria Conte, Bruce Edmonds, Scott Moss, Keith Sawyer, Sociology and Social Theory in Agent Based Social Simulation: A Symposium, Computational and Mathematical Organization Theory, 7(3), 183-205(2001). 7. Nuno David, Empirical and Intentional Verification of Computer Programs in Agent-based Social Simulation, Ph.D., University of Lisbon (in Portuguese) (2005). 8. Nuno David, Jaime S. Sichman, Helder Coelho, The Logic of the Method of Agent-Based Simulation in the Social Sciences: Empirical and Intentional Adequacy of Computer Programs, Journal of Artificial Societies and Social Simulation, 8(4), ihttp://jasss.soc.surrey.ac.uk/8/4/2.htmli,(2005). 9. C. Papadimitriou, Computational Complexity, Addison-Wesley (1994). 10. Joshua Epstein, Agent-Based Computational Models And Generative Social Science, Complezity, 4(5), John Wiley & Sons, 41-59 (1999). 11. John Casti, Would-Be Business Worlds. Complexity, 6(2), John Wiley & Sons, 13-15 (2001). 12. Gunter Kuppers and Johannes Lenhard, Validation of Simulation: Patterns in the Social and Natural Sciences, Journal of Artificial Societies and Social Simulation, 8(4), ihttp://jasss.soc.surrey.ac.uk/8/4/3.htmli,(2005).

284 13. John Casti, Would-be Worlds: how simulation is changing the frontiers of science, John Wiley & Sons (1997). 14. Bernd-0. Heine, Matthias Meyer and Oliver Strangfeld, Stylised Facts and the Contribution of Simulation to the Economic Analysis of Budgeting, Journal of Artificial Societies and Social Simulation, 8(4), ihttp://jasss.soc.surrey.ac.uk/8/4/4.htmli (2005). 15. Scott Moss and Bruce Edmonds, Towards Good Social Science, Journal of Artificial Societies and Social Simulation, 8(4), ihttp://jasss.soc.surrey.ac.uk/8/4/13.htmli(2005). 16. Riccardo Boero and Flaminio Squazzoni, Does Empirical Embeddedness Matter? Methodological Issues on Agent Based Models for Analytical Social Science, Journal of Artificial Societies and Social Simulation, 8(4), ihttp://jasss.soc.surrey.ac.uk/8/4/6.htrnlj,(2005). 17. Joshua Epstein, Robert Axtell, Growing Artificial Societies: Social Science from the Bottom Up, MIT press (1996). 18. C. G.Hempel and P. Oppenheim, Studies in the Logic of Explanation, Philosophy of Science, 15, 567-579 (1948). 19. Brian Cantwell Smith, The Foundations of Computing, Smiths introduction to a series of books that report his study of computing in the books The Age of Significance: Volumes IVI. Available at ihttp://www.ageosig.org/people/bcsmith/papersi(1996). 20. M. J. Prietula, K. M. Carley and L. Gasser, Simulating Organizations, MIT Press (1998). 21. Keith Sawyer, Multiagent Systems and the Micro-Macro Link in Sociological Theory, Sociological Methods and Research, 21(3), 37-75 (2003). 22. Robert Axelrod, Advancing the Art of Simulation in the Social Sciences, Simulating Social Phenomena, Springer Verlag, 21-40 (1997). 23. James Fetzer, Computers And Cognition: Why Minds are not Machines, Studies in Cognitive Systems, 25, Kluwer Academic Publishers (2001). 24. Edsger Dijkstra, A Discipline of Programming, Prentice-Hall (1976). 25. C. A. R. Hoare, An Axiomatic Basis for Computer Programming, Communications of the ACM, 12,576-580 (1969). 26. Karl Popper, The Logic of Scientific Discovery, Routledge Classics (1935/1992). 27. James Fetzer, The Role of Models in Computer Science, The Monist, 82(1), La Salle, Illinois 61301,20-36 (1999).

A COMPROMISE BETWEEN REDUCTIONISM AND NON-REDUCTIONISM

ERAYOZKURAL Department of Computer Engineering Bilkent University Bilkent 06800 Ankara, Turkey E-mail: [email protected] This paper investigates the seeming incompatibility of reductionism and nonreductionism in the context of complexity sciences. I review algorithmic information theory for this purpose. I offer two physical metaphors t o form a better understanding of algorithmic complexity, and I briefly discuss its advantages, shortcomings and applications. Then, I revisit the non-reductionist approaches in philosophy of mind which are often arguments from ignorance t o counter physicalism. A new approach called mild non-reductionism is proposed which reconciliates the necessities of acknowledging irreducibility found in complex systems, and maintaining physicalism.

1. Introduction

New mathematical techniques have enabled scientists to gain a better understanding of non-linear dynamical systems, while similar advances have occurred in computer science for constructing models of natural phenomena as information processing systems. A particularly interesting research direction in complexity sciences is the study of conditions under which selfComplexity theorists pay special attention to the organization occurs. concept of emergence. Gershenson and Heylighen give a typical definition of emergence:

'

Emergent properties characterize a system that cannot be reduced to the sum of its parts. The authors then give the most striking example that an animate cell is composed of parts that are inanimate. Thus, to be animate must be an emergent property since the system cannot be explained in terms of its parts. In the same paper, the authors show analysis/reductionism as a shortcoming of classical science, which is indeed the case.

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286 However, the identification of non-reductionism with this particular notion of “emergent property” leaves something to be desired. In this paper, I will argue that the mechanical view of nature is compatible with Heylighen’s definition of complexity: complexity of a system increases with the number of distinct components, the number of connections between them, the complexities of the components, and the complexities of the connections. And I will suggest that complex phenomena like emergence may be understood through the mechanical view. The rest of the paper is organized as follows. In the next section, I will review the relevance of the information theory approach to quantification of complexity. In Section 3, I will highlight the relation of dualist and nonreductionist theories in philosophy of mind, suggesting that extreme nonreductionism leads to extra-physical notions. In Section 4, I will outline mild non-reductionism as a physicalist approach for resolving the tension between reductionism and non-reductionism. 2. Information theory and complexity 2.1. Basic definitions

Algorithmic information theory (AIT) gives a precise mathematical definition of complexity. Algorithmic complexity (or entropy) H ( z ) of a message z is the length of the shortest program that generates it. a This definition captures the essential difference between the complexities of

0101010101010101010101010101010101010101010101010101010101010101 1011100001110100110110110000010011011010100010111000101111000110 While the first sequence can be generated by a simple program consisting of a loop, it takes a longer program to create the second random sequence. In contrast to classical information theory, this definition has also the advantage that it is an absolute quantity per message, rather than characterizing a random source. The algorithmic definition is deemed more fundamental than classical information theory for classical information can be derived from algorithmic information. This definition has not been well received in the philosophy community due to some basic misunderstandings of the theory. Obviously, for there aFormally, H ( z ) = min(lp1 : U ( p ) = z} where U is a prefix-free universal computer, U ( p ) is the output of program p and Ipl is its length. 3,4

287 to be a program, a programming language has to be chosen. Due to the invariance theorem, this choice makes only a constant difference in complexity. However, people have been confused over whether this means that the definition is not absolute as it is claimed. An achievement of AIT is the proof that other mathematical definitions of randomness and incompressibility are identical. That is to say, one cannot find a (much) shorter program for a random sequence, with the consequence that minimal programs are random. This discovery has been confusing as well, causing people to claim that they can denote with a single symbol a message of any length (in which case nothing would be random). While this can be done, such a decoder would be useful for decoding only one message, while a universal computer can handle any message. If one “hacks” a universal computer to achieve this, one will have to encode the additional information inside the computer, and thus algorithmic information is not lost. 2.2. Physical complexity

We offer the following purely physical metaphors to make it easier for the reader to form an intuition about the physical relevance of the information theory point of view. 0

0

physical message complexity: volume of the smallest machine that can output a message in any fashion. The condition is that the output message is encoded as is, it should be written symbol for symbol. Like the AIT definition, with increasing complexity, we would need a larger machine to achieve this. constructive complexity: volume of the smallest machine that can precisely build a desired object (up to physical limits, or required accuracy).

These metaphors are given to furnish the understanding of algorithmic complexity as a physical concept. Every bit in a sequence of bits extends in space, and must exist as a distinguishable physical property. That is to say, length may be conceived as the physical length. Considering physical message complexity, the volume of this machine is similar to the length of a minimal program in a programming language. However, the physical definition is literally universal. The smallest such bPlease note that a formal argument is beyond the scope of this paper.

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volume does not depend on the choice of a computer; since it effectively uses our physical universe as the universal computer in the definition of algorithmic complexity. Likewise, we would not think that we could design a storage medium that is more compact beyond a certain physical limit. More naturally, one may consider algorithmic complexity in relation to thermodynamic entropy. A discussion of various approaches to physical complexity may be found in Bennett’s work. 2.3. Adequacy of algorithmic complexity

Algorithmic complexity beautifully captures our basic intuitions about complex systems. Recalling Heylighen’s definition, we can view a complex system as a graph (or network) where vertices are components and edges are interactions. According to subadditivity of algorithmic information, the complexity of a message is the sum of the complexities of its components, with the addition of a constant term. That is, H ( z ,y) = H ( z ) H(y/z) 0(1) where H(y/z) is conditional (or relative) algorithmic complexity, H ( z ,y) denotes the complexity of concatenation of messages z and y, and 0(1)is a constant. Conditional algorithmic complexity H(y/z) is the length of the shortest program that generates y given the shortest program for z. The complexity of each component therefore does add to overall complexity. However, it is not a simple sum of the complexities of parts. If the components are identical, then there will be little increase. If the components are all very different, each will contribute a large amount of algorithmic complexity. The algorithmic complexity of the graph of the system will also increase with the number of vertices, the number of edges, and the randomness of the topology of the edges.

+

+

2.4. Shortcomings of algorithmic complexity

The definition of algorithmic complexity neglects the temporal dimension which is indispensable in the study of dynamical systems. Without time, we could not formulate such concepts as trajectory and stability. Algorithmic complexity does not distinguish between passive stores of information and active users of information, thus it has to be supplanted by descriptions of a dynamical system in the study of many natural complex systems. However, there are theorems connecting algorithmic information theory and dynamical systems. Also of interest is the relation of algorithmic complexity =A regular topology takes fewer bits to specify than an irregular one.

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and cellular automata dynamics. lo On the other hand, time aspect can be put into new definitions of complexity. The “logical depth” definition of Bennett incorporates time. l1 There are significant features of biological systems that algorithmic complexity does not model. For instance, algorithmic complexity disregards redundancy in the data, while redundancy and fault-tolerance are fine features of biological systems. Biological systems also seem to achieve optimizations other than reducing program-size, e.g., time and energy optimizations. Critics have also argued that algorithmic complexity does not model the interestingness or usefulness of a message. In classical information theory, the notion of “relevant information” has addressed the problem of interestingness. l2 Hutter has pursued in detail an approach based on algorithmic probability including agent models that maximize an arbitrary utility function. l3 Another shortcoming is the fact that algorithmic complexity is uncomputable. Therefore we always have to stick to much simpler approximations if we need to explicitly calculate the complexity of a message. This is not a shortcoming in theory, only a shortcoming in practice.

2.5. Applications and related work Mutual algorithmic information H ( s : y) between two messages quantifies how much algorithmic complexity is shared. In the visual descriptions of a car and a plane, “wheel” could be part of shared information. The joint information H ( a , y ) and mutual information are related via H ( s : y) = H ( s ) H(y) - H ( s ,y) O(1). Mutual information measures how much easier it is to compress two strings together, than separately. Mutual information can be used to quantify the amount of knowledge a biological agent has about the world by the mutual information between the agent and the world. While this a good characterization of the amount of knowledge, it says nothing about the dynamical aspects. The operational facets of intelligence could be incorporated by modelling which events it could predict, an idea that may be important since some theoretical neuroscientists propose that it is one of the main functions of the brain. l4 It is also worth mentioning that a minimal program can be used to make predictions; Solomonoff presents a particularly accessible introduction to applications of algorithmic complexity in A1 using inductive methods. l5 Some philosophical applications of AIT may be found in a paper by Chaitin including a thesis about perception. Another interesting work relating

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290 the information theoretic point of view to complexity sciences is due to Bialek et al. 17, suggesting that predictive information has a close relation to the measures of complexity in dynamical systems theory. 3. Non-reductionism and physicalism

Proponents of non-reductionism often bring the human mind as an example of an irreducible system. In philosophy of mind, the tension between reductionism and non-reductionism has been a major driving theme of contemporary research. A motivating question is to find a theory in which physicalism becomes compatible with non-reductionism. Physicalism is the view that mental events are ultimately physical, i.e., material. I will use the term extra-physical for those concepts which do not befit a physical description. Complexity sciences may be seen as embracing a physicalist methodology, for the tools of a complexity scientist are usually physical: trajectories, dynamics, interactions, entropy, etc. This attitude does not carry over to philosophy of complexity. In philosophical discussions we often see a drift towards some old concepts in philosophy such as dualism.

3.1. Non-reductionism in philosophy of mind A fundamental problem in philosophy of mind is that it does not seem there is a place for qualia such as color, or pain in a human organ. I can see green, but when I open up a brain I do not find any green, only neural tissue. On the other hand, there seems to be a strong relation between experience and the physical state of brain. For instance, we can invoke certain sensations by stimulating the brain with electrodes. In philosophy of mind, we describe this relation as: mental events supervene on physical events. We can speak of a general principle of supervenience in philosophy of mind: there are no mental differences when there are no physical differences. However, it is difficult, if not impossible, to fathom a relation between this complex, impenetrable organ and our basic perceptions. Thus, the philosophers have been creating new ideas for centuries, imagining that the mind must be something else than the brain. This has traditionally had other reasons such as the belief that we are special. Descartes thought that while the animals were automata, we were more than mere machinery. One reason for this was the apparent inadequacy of a mechanism. A machine could do only one kind of a thing, but the human mind seemed to be capable of anything. Thus, the human mind could not be a machine. Of course,

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Descartes would not have known the concept of a universal machine which was discovered by Alan Turing, a machine so flexible that it can mimic any other machine. l8 Descartes proposed that the immortal soul that he imagined interacted with the human body through the pineal gland. This was not a good idea as examinations of the pineal gland showed. In general, the view that the stuff of mind is distinct from matter is known as substance dualism. Descartes subscribed to this view; he thought that mind did not extend in space like matter. There are two other kinds of ontological dualism. P r o p erty dualism suggests that mental properties are fundamentally different than material properties. The weakest statement of dualism is predicate dualism 19, which states that mental (or psychological) predicates cannot be reduced to material predicates, although supervenience holds. A primary non-reductionist argument against physicalism is the knowledge argument. 2o Imagine a neuroscientist who has been born and raised in a black and white room. The neuroscientist knows every physical fact about the brain. However, she has never seen red; she does not know what red is like. The argument then quickly concludes that, the knowledge of red cannot be reduced to physical facts about the brain, so it must be extra-physical.

3.2. Arguments

from ignorance

An extreme non-reductionist attitude may result in dualism. Both the nonreductionist predicate dualism and knowledge argument views purport to show that ordinary physicalism is insufficient to explain the mind. The proponents of these arguments seem to arrive at their conclusions through reinforcing our ignorance in cognitive science. 21 Predicate dualism relies on the fact that there are still many things unknown about the physical processes in the brain. The claim is made that there are no strict bridge laws connecting mental events and physical events. If there were such laws, it would be possible to fully reduce psychological states to physical descriptions. Assume that this is not possible. Then, clearly, there is something more to psychological states than can be explained by a physicist. It is also quite difficult to see how the part of the psychological state that cannot be reduced to a physical description could legitimately exist as part of our causal world. 22 While it is possible to defend that a sufficient form of supervenience would hold in this case, l9 this view has the flavor of an idea that depends on our lack of progress

292 in neuroscience. We do not have a comprehensive neuroscientific theory of how consciousness comes about in the brain. This lack of knowledge is termed the “explanatory gap” and much has been said of it. 23 Thus, the predicate dualist has a lot of leeway to persuade the readers about the non-existence of bridge laws, since neuroscientists have not shown them yet. However, this skepticism does not seem too different from invoking supernatural forces in any phenomenon that science cannot explain currently. It also seems to contradict the expected uniformity of natural law. In particular, why should we think that such bridge laws seem present everywhere but the brain? In the case of the knowledge argument, we are forced to believe that the ultimate neuroscientist can not know what “red” is. However, even today’s neuroscientists know a great deal about the human vision system. Furthermore, if experience of red is a particular physical event as is usually thought by neuroscientists, then it is acceptable that reading a book might not result in the same brain activity. Given appropriate apparatus, however, the neuroscientist might be able to invoke any color sensation. Today, artificial vision is a reality. As science advances, arguments like the knowledge argument lose their ontological appeal, although they may be useful in illustrating the difference between subjective experience and objective knowledge. 24

4. Mild versus extreme non-reductionism The complexity view encourages us to leave behind certain tendencies of classical science. One such tendency may be called extreme reductionism. We often see observations of complex systems such as human societies being fitted to linear or quadratic equations, disregarding complex phenomena like phase transitions. An extreme inductive bias towards simplicity is not uncommon among scientists. Classically, everything must have a clear-cut, simple and perfect explanation, such as the law of motion F = ma. Today, we see that even in physics, the facts are not so simple. The world is imperfect, complex, dynamic and not easily predictable. However, the alternative to extreme reductionism must not be extreme non-reductionism, which may bring about dualism via either of the nonreductionist arguments in Section 3.1. In fact, dualism is one of the classical tendencies that we must abandon. There are still scientists who believe in an inexplicable soul. On the other hand, many other scientists have broken this psychological barrier and see themselves as part of an evolving, complex

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nature. Nevertheless, we still need to clarify in what sense our minds are irreducible.

4.1. Degrees of irreducibility In Section 2, I have reviewed the information theoretic definition of complexity. Algorithmic complexity of a system is a matter of degree. However, the system is fully reducible to small physical interactions. This is the view of mild non-reductionism. Indeed, viewing a complex system as consisting of local interactions was a notable achievement of complexity sciences. In the sense of supervenience physicalism, the properties of such a complex system supervene on the local interactions, and of course the structure of these interactions and the components as discussed in Section 2.3, all of which are physical. An advantage of mild non-reductionism is that it is fully compatible with the mechanical view of nature. Algorithmic complexity clarifies our intuitions about reducibility. Ordinarily, we view reducibility as an all or none issue. However, reducibility does not make sense without certain important variables. Reduction to what, by what amount, and in many cases, by which method? A physicalist would think that everything, including mental states, are fully reducible to physical states. However, in the light of Section 2.2 the physicalist would also agree that we can reduce only so much. If we would like to store all the information about complex entities such as one's psychology, or a living cell, then we cannot hope to compress all this information beyond an optimally small volume of physical space. 4.2. Consequences of irreducibility

If we were to take away only a single piece of information from this optimal description, the reproduced system may degrade in an important way or stop functioning altogether, because there is no fault-tolerance in an optimal description. The minimal program for anything is irreducible. When we look at it, it is a random arrangement of symbols. Yet, it holds crucial information. Note that, if something is random, there can be no more analysis of it. Thus, all regularities in the data have been found in a minimal description. If of course we knew all the regularities in the data, this also means, intuitively, that we can realize the best prediction. The analysis has occurred to the maximum limit in the process of compression if we could have achieved maximal compression. Unfortunately, in the real-world, we can never guarantee that we have

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found a minimal program, or we can never know how close we are to a minimal program. Both of these have turned out to be uncomputable, and we finite beings have to be satisfied with approximations. Beyond the uncomputability of algorithmic complexity there are incompleteness results that place serious limits to analysis which have been rigorously investigated by Chaitin. However, the incompleteness results in mathematics do not show that it is impossible for scientific inquiry to be successful in general. Rather, they deal in what a machine can/cannot infer from given axioms. On the other hand, it has been possible for scientists to discover new postulates such as that the speed of light is constant. As a consequence, we can never be sure whether we have the most elegant theory in any domain. Intuitively, we can imagine that the more the complexity, the more effort it will take to compresslanalyze the data. Then, we need to diminish complexity.

4.3. The need for abstract talk

We can not demand that all sciences are reduced to physics before we begin research. The possibility of reduction to physics is not an issue, the feasibility is. We have to talk at a higher level of abstraction that makes the problem smaller, because we often cannot face it in all its glory. Let us turn our attention to the problem of piecewise analysis and analysis at different levels of description. Often, all the details about a system are not known, or are irrelevant to our purpose. Then, we must have a way of dealing with the relevant aspects of the system. For instance, one might investigate the data transfer rate of the brain, if we are so certain that it is a computer. We then need a reliable way to choose a level of abstraction in which it becomes possible to talk meaningfully about the proposition. That is to say, it should be neither a lower nor higher level of abstraction than necessary. In the case of the data transfer question, we might like to retain only the information about data transmission capabilities of individual links and the network topology. We do not need to concern ourselves with philosophical puzzles to work at this level. In general, we would only like to deal with information that allows us to predict a variable of our choice. There is another reason for abstract talk. In a mechanical universe, we may expect the probability of a rule to be inversely related to its complexity. That is to say, we do not expect arbitrarily complex rules to pop up. In theory some explanations have been given as to why Occam’s razor holds; l5 our natural way of beginning with simple theories and progressing towards

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more complex theories seems to have a sound mathematical basis. This would also explain why random events are not common. If randomness were the common case, then our brains could make little use of all the information about the past they are storing. Pure randomness means precisely that one cannot use previous observations to predict the future. However, this is not the case, we routinely predict and control events around us. This can happen only because their algorithmic complexities are small. In other words, randomness is bounded. Mountains do not appear at once, and weather does not reverse every second. While in dynamical systems, a view is that butterfly effects do occur, in the cosmic scale everything is pretty much uniform. It is a fortunate fact that this predictability also occurs to a healthy extent at the level of human beings. That is, mild non-reductionism also leads to mild indeterminacy. It is not productive to imagine that the macro-scale world is as unpredictable as the most random quantum events. For sufficient prediction, a simple abstract theory might be found by induction. Another reason for abstract talk comes from the question: reduction to what? When we say reduction, we do not mean reduction to an alien language that we do not understand. Instead, we mean a satisfactory explanation in familiar terms. That is to say, if we know x about the world, we would like the new explanation y to be relative to this knowledge, i.e., that it is a short explanation in terms of x. We demand this, because otherwise we cannot integrate y into our body of knowledge. This constraint is captured well by the notion of conditional algorithmic complexity. It is possible that an alien creature may have developed a quite different body of knowledge, and we will have to spend a lot of effort to decypher its musings, i.e., the meaning would have large conditional complexity relative to our culture. Another relevant idea in algorithmic complexity is the notion of algorithmic independence (lack of mutual information). If through observations, a brain can detect “islands” of information, then it can exploit this p r o p erty. l6 Such a capacity for abstraction from sensory information seems to be present in our brains, however we can make few speculations on its properties at the present.

4.4. Algorithmic complexity and emergence

We are now in a better position to review the idea that animate-ness of a cell is more than the sum of its inanimate parts. Or similarly, that a

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brain is more than the sum of its neurons, I have already argued that reduction to a physical description is necessary for the scientifically minded complexity researcher who rejects ontological dualism. First, let us have a look at the meaning of such predicates like animate or conscious that denote complex entities. These have been most likely invented for the purpose of avoiding excessive expcplanations, i.e., to be able to talk at a correct level of abstraction. When we use these terms, we do not feel the need to explain them further, however we can all recognize their presence through their outward appearances. Furthermore, in many common sense problems, we need no more information about such words except for the basic function of distinguishing a dead animal from a living animal, etc. As countless works in philosophy show, it takes a great effort to do otherwise and try to give a complete description. On the other hand, the animate-ness of a cell supervenes on continuing success of a number of complex subsystems that are tightly interconnected. Should one of these crucial systems fail, the cell is no more animate. While like any other machine, the description of the cell is compositional, the topology and nature of interconnections are non-trivial. The outward appearances of complex phenomena are usually not satisfactory for a scientist. Complexity community was not satisfied with the definition of “animate” that it meant being able to reproduce or having genetic material. They rightly sought generic properties of the systems that we call animate. For instance, that they are stable, yet have dynamics at the edge of chaos, that they are made up of parts that are being replaced over time. Attempts have been made to characterize not only earth biology, but any animate matter, and further efforts have been made to characterize evolution. For instance, Chaitin has given a rough mathematical definition of life 25 where he suggests that the interacting parts of such a system (similar to components of a system in Section 2.3) are self-similar, i.e., they have high mutual algorithmic complexity. He also suggests the famous halting probability R as an abstract model of evolution; the approximation procedure to calculate it is presented as a mathematical model of entire evolution. On the other hand, researchers have already successfully reconstructed evolutionary trees from gene sequences using an information theoretic method. 26 The very existence of these generic explanations in complexity sciences suggests that the concepts of animate-ness and evolution may be given abstract descriptions that are independent of particular instances. It may be possible to extract the essential commonality in all living systems into

297 a clear description by choosing an affordable level of explanation. These descriptions may turn out to be much simpler than the dynamics of a cell which is a complex machine. The architecture of the cell may be impossible to infer from the history of evolution prior to the cell, that is to say it contains novel algorithmic information relative to that history. Given the “inanimate” components of a cell, we can construct a large number of systems, however, only a few of them will result in proper function. The algorithmic complexity of the fit organization relative to the set of components may be interpreted as corresponding to new “valuable” or “emergent” bits invented by evolution. The information theoretic view of emergence is similar to Bedau’s notion of “weak emergence” as he distinguishes between nominal, weak, and strong emergence. 27 However, in the ongoing discussion, emergence is independent from the distinction between micro and macro levels.

5. Conclusion

The physical versions of algorithmic complexity help us consider it in a purely physical way, and free of problems in the abstract version. I have shown that algorithmic complexity corresponds to our intuition about complexity, and further clarifies it with theorems. In Section 3 some dangers of extreme non-reductionism have been shown by examining the knowledge argument and predicate dualism which rest on epistemological and absolute irreducibility. These arguments have been criticized as arguments from ignorance. With the same attitude, we would have hardly advanced science to this point. In Section 4,I have proposed using precise and quantitative conceptions of complexity as they clarify our intuitions. Irreducibility of a system comes in degrees, and it may be understood physically. Randomness means the absence of further reducibility, and we cannot reason about random events. The reason for abstract talk is examined as a matter of processing feasibility, purpose, inductive reasoning and opportunity. Finally, instead of stating emergency of a property X as “ X is more than sum of its parts”, I suggest that we can use the precise language of information theory. In particular, “emergent property” can be given a satisfactory meaning in AIT, corresponding to algorithmic information in a system that does not reside in its components or prior history, thus is impossible to predict using the information therein. Mild non-reductionism cannot be the final word on this controversial subject at least for the reason that algorithmic complexity has obvious

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shortcomings. While irreducibility is so important, our intuitions may not suffice. Algorithmic complexity helps us sharpen our methodology with mathematical reasoning, and sheds new light on concepts such as irreducibility, limits to analysis, predictability and emergence.

Acknowledgments

I a m grateful to Cristian S. Calude for technical corrections and suggesting references. Thanks t o anonymous reviewers, Peter Jedlicka, Michael Olea, Gregory Chaitin, and William Modlin for their comments which helped improve this paper.

References 1. Carlos Gershenson and Francis Heylighen. When can we call a system selforganizing? In W Banzhaf, T. Christaller, P. Dittrich, J. T. Kim, and J. Ziegler, editors, Advances i n Artaficial Life, 7th European Conference, ECAL 2003 LNAI 2801, pages 606414. Springer-Verlag, 2003. 2. Carlos Gershenson and Francis Heylighen. How can we think the complex? In Kurt Richardson, editor, Managing Organizational Compledty: Philosophy, Theory and Application, chapter 3. Information Age Publishing, 2005. 3. Information and Randomness - A n Algorithmic Perspective. EATCS Monographs in Theoretical Computer Science 1. Springer-Verlag, 1994. 4. Gregory J. Chaitin. Algorithmic Information Theory. Cambridge University Press, 1987. 5. Thomas M. Cover and Joy A. Thomas. Elements of Information Theory, chapter 1. Wiley Series in Telecommunications. Wiley-Interscience, 1991. 6. W. H. Zurek. Algorithmic randomness, physical entropy, measurements, and the demon of choice. In A. J. G. Hey, editor, Feynman and computation: exploring the limits of computers. Perseus Books, 1998. 7. Charles H. Bennett. How to define complexity in physics, and why. In W. H. Zurek, editor, Complexity, Entropy, and the Physics of Information, volume VIII, pages 137-148, 1980. 8. A.A. Brudno. Entropy and the complexity of the trajectories of a dynamical system. IPrans. Moscow Math. SOC.,44:127, 1983. 9. Fabio Cecconi, Massimo Falcioni, and Angelo Vulpiani. Complexity characterization of dynamical systems through predictability. Acta Physica Polonica, 34, 2003. 10. Jean-Christophe Dubacq, Bruno Durand, and Enrico Formenti. Kolmogorov complexity and cellular automata classification. Theor. Comput. Sci., 259( 12):271-285, 2001. 11. Charles H. Bennett. Logical depth and physical complexity. In Rolf Herkin, editor, The Universal Turing Machine: A Half-Century Survey, pages 227257. Oxford University Press, Oxford, 1988.

299 12. N. Tishby FC Pereira and W Bialek. The information bottleneck method. In Proceedings of the 37th Annual Allerton Conference on Communication, Control and Computing. 13. Marcus Hutter. Universal Artificial Intelligence: Sequential Decisions Based on Algorithmic Probability. Springer-Verlag, 2005. 14. William Bialek. Thinking about the brain. In H Flyvbjerg, F. Jlicher, P. Ormos, and F. David, editors, Physics of Biomolecules and Cells: Les Houches Session L X X V , pages 485-577. EDP Sciences, Les Ulis; Springer-Verlag, Berlin, 2002. 15. Ray Solomonoff. The application of algorithmic probability t o problems in artificial intelligence. In M. Kochen and H. M. Hastings, editors, Advances in Cognitive Science, A A A S Selected Symposia Series, pages 210-227. 1988. 16. Gregory J . Chaitin. Two philosophical applications of algorithmic information theory. In Proceedings DMTCS’03 Springer Lecture Notes in Computer Science, volume 2731, pages 1-10, 2003. 17. William Bialek, Ilya Nemenman, and Naftali Tishby. Predictability, complexity and learning. Neural Comp, 13:2409-2463, 2001. 18. Marvin Minsky. The Emotion Machine, chapter 8. Draft. http://web.media.mit.edu/-minsky/EB/ebB.html. 19. Donald Davidson. Mental events. In L. Foster and J. W. Swanson, editors, Experience and Theory. 1971. 20. Frank Jackson. Epiphenomena1 qualia. Philosophical Quarterly, 1982. 21. Paul S. Churchland. The hornswoggle problem. Journal of Consciousness Studies, 1996. 22. Dan Hutto. Davidson’s identity crisis. Dialectica, 52(1), 1998. 23. Ned Block and Robert Stalnaker. Conceptual analysis, dualism, and the explanatory gap. Philosophical Review, 108:l-46, 1999. 24. Tim Crane. Subjective facts. In H. Lillehammer and G. Rodriguez-Pereyra, editors, Real Metaphysics, pages 68-83. Routledge, London, 2003. 25. Gregory J. Chaitin. Toward a mathematical definition of life. In R.D.Levine and M. Tribus, editors, The Maximum Entropy Formalism. MIT Press, 1979. 26. Rudi Cilibrasi and Paul M. B. Vitnyi. Clustering by compression. ZEEE Transactions on Information Theory, 51(4):1523-1545, 2005. 27. M. A. Bedau. Downward causation and autonomy in weak emergence. Principia, 6:5-50, 2003.

THE COMPLEXITY OF INFORMATION-PROCESSING TASKS IN VISION

JOHN SYMONS University of Texas, El Paso

The basic presupposition of cognitive science is that mental life is more complicated (or complex) than it appears; there is more to memory, attention, perception and the like, than meets the inner eye. In most mainstream cognitive science, the hidden workings of the mind are treated as information processing systems. In defense of their approach, cognitive scientists often point to the importance of articulating the complexity of the information processing processes underlying cognition and perception. The use of complexity in defense of mainstream cognitive science deserves detailed examination, because of its importance and because it is more philosophically problematic than has been recognized to date. This paper focuses primarily on the use of complexity in the science of vision. The information-processing model gained supremacy, in large part, because of its success in the science of vision. Visual perception was one of the preeminent areas of investigation and served as the testing ground for the most important conceptual presuppositions of the discipline. By the mid 1980s, it was commonly assumed that a satisfying explanation of vision should permit us to program a computer to see. From the computationalist perspective, a satisfying explanation of some aspect of mental life is one that allows us to understand the phenomenon in question as a series of steps articulated in ways that can easily be presented to our colleagues without the appearance of question-begging. However, the attempt to articulate the mind’s informational complexity raises a series of fundamental questions. In the cognitive science of vision, we can ask, for instance: (1) What is the relationship between the information-processing complexity that we hypothesize and the actual embodied act of seeing? (2) What predictive power do we gain via an articulation of the information-processing complexity of the visual system?

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30 1 (3) How can we correctly capture the information processing complexity of a particular system? (4) Is it always necessary to provide an information-processing account of a system in order to provide a proper explanation? One can easily imagine explanatory contexts where the informationprocessing complexity that cognitive scientists hope to understand is simply not relevant. After all, most scientific explanations do not take an information-processing form and for the most part, it is not obvious why one should opt for an explanation of this kind. There might be some justification for an information-processing style analysis in engineering contexts, but even here, it is not clear that engineers would really need to know the information-processing properties of the solutions they develop. In addition to their emphasis on the importance of non-enthymatic, proof-like processes in explanations, the computationalist commitment to an informationprocessing accounts of psychological and perceptual processes results from a representationalist conception of mind. As we shall see, the notion of information-processing complexity serves as the intersection for a tangled set of basic questions in the philosophy of cognitive science. This paper examines some of the arguments in support of the information processing approach to the complexity of visual perception. David M a d s defense of the approach serves as the focus of the analysis. Marr’s posthumously published book Vision: A Computational Investigation into the Human Representation and Processing of Visual Information (1982) presents the most prominent computational theory of visual perception. Vision provides an overview of work conducted by Marr and his colleagues at the Artificial Intelligence Laboratory at the Massachusetts Institute of Technology from about 1973 to the late 1970s. Written under the pressure of serious illness and completed after his death by friends and colleagues, it is widely regarded as the most important book in cognitive science. Vision outlines a set of methodological and theoretical principles that have helped to consolidate the self-image of cognitive science as a discipline. Marr’s computational theory of the visual system is seen as providing a model for theoretically sophisticated investigation into the biological mechanisms responsible for mental life. So, for example, Steven Kosslyn claims that Marr’s view provided “. . . an enormous advance over much of the previous work in neuroscience, which tended to minimize theory, and over most of the previous work in artificial intelligence, which tended not to rest on careful analyses of a problem.” (Kosslyn 1994,35) For the purposes of this

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paper, M a d s text provides is an influential example of the reasoning that shaped the prevailing view of complexity in the visual system. 1. Recognizing Complexity

Accounts of the visual system from the computational perspective almost invariably begin by contrasting their approach with earlier attempts to provide “direct” or “na’ive realist” theories of perception. Central to this contrast is the notion of complexity. Computationalists see their work as uncovering the complexity of the information-processing task involved in visual perception that previous theorists had neglected. When scientists are introducing and defending their approaches to vision, they regularly begin by taking a stand on the notion of complexity. So, for example, Alan Yuille and Shimon Ullman begin their introduction to computational theories of low-level vision as follows: Vision appears to be an immediate, effortless event. To see the surrounding environment, it seems we need only open our eyes and look around. However this subjective feeling disguises the immense sophistication of the human (or animal) visual system and the great complexity of the information processing tasks it is able to perform (Yuille and Ullman 1990, 5) Computationalists understand their predecessors as having ignored the complex process underlying visual experience. Nicholas Wade and Michael Swanston also echo this emphasis on the underlying process when they write that: “The world around us seems concrete and immediate, and our ability to perceive it is easily taken for granted. . . Clearly, there must be some process that gives rise to visual experience.” (Wade and Swanston 1991, 1) By contrast with their predecessors’ neglect, they contend that the computational approach to perception as offers the best route to recognizing and untangling the information processing complexity of the visual system. In his introduction to cognitive science, Paul Thagard explicitly connects our recognition of complexity to the problem of crafting a computer program that sees:

For people with normal vision, seeing things seems automatic and easy. You look at a room and immediately pick out the furniture and people in it. The complexity of vision becomes apparent, however, when you try to get a computer to do the same thing.” (Thagard, 1996, 96)

303 Introductory discussions of the science of vision show a striking level of uniformity in their theoretical presuppositions with respect to complexity. They echo a long tradition in cognitive science and cognitive psychology which takes a two pronged approach to defending its approach to vision and to mental life more generally. First, cognitive scientists criticize what they claims is the naivety of their predecessors, who fail to see the complexity of the processes under consideration. Second, they argue that the correct approach to depicting and understanding the complexity of visual perception involves the use of an information-processing model. As early as Ulric Neisser’s 1967 introduction to cognitive psychology we read the following: Even psychologists who ought to know better have acted as if they believed [in naive realism] [. . . ] The information reaching the eye is subjected to complex processes of analysis, extended in time. Visual experience results from some of these processes. [. . .] Perception is not a passive taking-in of stimuli, but an active process of synthesizing or constructing a visual figure. Such a complex constructive act must take a certain amount of time. (1967, 16) Neisser and others argued that the time taken to complete a process is an indication of the level of complexity involved. It is important to note that the complexity which interests cognitive scientists is said to exist at the level of information-processing, rather than at the biological or behavioral levels. Given the emphasis on the informational level analysis, sources of evidence at the biological or even at the behavioral level tend to be marginal to their choice of explanation. Given the nature of our access to the putative underlying information processing, the evidential constraints governing articulations of informational complexity become somewhat problematic. The time that experimental subjects take to perform visually-based tasks serves as the primary empirical basis for the construction of theories of information-processing complexity. However, the problem of understanding what the informational complexity of a process amounts to and how it should be studied is underdetermined by timing evidence alone. Timing measurements alone leave room for a variety of explanations that are compatible with the evidence. Consequently, other, non-empirical constraints must play a role in one’s decision to favor one articulation of information processing complexity over another. Philosophical presuppositions, play an especially prominent role in this kind of decision, as we shall see.

304 2. How Marr saw vision

Marr describes vision as a computational process wherein the brain constructs a three dimensional representation of the distal scene based solely on the illumination of the surface of the retina. According to Marr, the visual system consists of three (or four, if one counts the initial effect of light on the retina) hierarchically organized stages. At each stage a new aspect of the representation is constructed from information originally contained in the retinal image. For Marr, the visual system functions somewhat like an assembly line whose final product is a useful, finished model of the visual scene suitable for the organism’s manipulation. Along the way, each stage in the visual system is explained in terms of the transformation of one representational structure into another according to relatively fixed rules. No new material or content is added to the representations as they undergo the sequence of computational transformations. Instead, increasingly specific features are extracted from information derived from the original retinal image, until the visual system generates a fully elaborated model of the visual scene. Models of this kind represent a data-driven approach to perception. Data-driven models of perception are generally contrasted with theories that view belief and other higher-level phenomena as playing a role in the construction of what it is that the organism perceives. A gestalt theory of perception would be an example of a top-down as opposed to a purely data-driven theory. The specific claims involved in his description of how the visual system actually accomplishes these stages have proven less important to cognitive science than the methodological framework articulated in the opening and closing chapters of Vision. Following Marr’s methodological example, traditional cognitive science has assumed that the first step in the investigation of mental life is the construction of an information-processing model. This construction involves determining a set of well-defined informationprocessing problems that the researcher believes the visual system must solve. Marr recommended that these problems should, at least initially, be studied in-themselves, apart from consideration of the underlying neuroscience. Marr’s suggested methodological tripartition in Vision; his distinction between computational, algorithmic and implementational levels of analysis in cognitive science is consonant with the philosophical functionalist intuition that, when it comes to matters of the mind, neuroscience can be safely ignored, since mental states can be approached at the computational

305 (or functional) level. The notion of information-processing complexity plays a central role in the approach and as we shall see, the claim that these systems are complex is motivated by the assumption that the visual system traffics in information bearing representations. By insulating the information-processing approach from the constraints of biological or behavioral evidence, cognitive scientists were free to craft a system of rules and representations that solved the problems of perception that they believed the visual system evolved to solve. However, because they are crafted in response to computational characterizations of the problems in question, such solutions do not necessarily help us to understand the mechanisms in actual biological vision systems.

3. Information processors vs. resonators

As mentioned above, the computational or information-processing view is almost always introduced by contrast with secalled direct theories of perception. Marr follows this practice when, early in Vision, he distinguishes his approach from J. J. Gibson’s. Gibson, who is widely regarded as the most important modern proponent of the direct theory of perception, denied that perception is mediated by sense data or representations and sought instead to provide an ecological theory of vision that accounted for the dynamical and embodied characteristics of the senses. Gibson recognized that visual perception is mediated by highly complex biological processes. His denial of what Susan Hurley called the classical sandwich - the distinction between perception, cognition and action- is generally overlooked in computationalist characterizations of so-called direct theories of perception. (Hurley 1998) Rather than tackling his criticism of the functional-level amount, critics assume that the ecological theorists share their analysis of the complexity of vision at the functional level and then that they simply fail to explain that complexity properly. This is what motivates the computationalist emphasis on Gibson’s sketchy and highly simplified accounts of the internal processes that subserve the senses. So, for example, Marr takes Gibson to task for ‘seriously underestimat[ing] the complexity of the information-processing problems involved in vision and the consequent subtlety that is necessary in approaching them.’ (1982, 29) M a d s criticism of Gibson focused specifically on the idea of the senses as organizing behavior in relation to ecological invariants. According to Marr, Gibson’s approach was the product of his failure to understand in-

306 formation processing (1982, 29) and he wrote: Although one can criticize certain shortcomings in the quality of Gibson’s analysis, its major and in my, fatal shortcoming lies at a deeper level and results from a failure to realize two things. First the detection of physical invariants, like surface images, is exactly and precisely an information processing problem, in modern terminology. And second, he vastly underrated the sheer difficulty of such detection. (1982, 30) Gibson would object to the idea that the brain does anything like computing the solutions to problems. For example in a passage cited by Marr, he writes “the function of the brain, when looped with its perceptual organs, is not to decode signals, nor to interpret messages, nor to accept images, nor to organize the sensory input or to process the data, in modern terminology.” (as cited in Marr 1982, 29) Consequently Marr and Gibson differed fundamentally on the nature of brain function. For Gibson, the problem with an account like Marr’s is that it relies on what he called “the muddle of representation” (1988, 279) and that it takes what Gibson called a “molecular” rather than a “molar” approach to perceptual systems. (1966, 52). The purpose of Gibsonian explanations, by contrast, is to show how “the receptive units combine and their inputs covary in systems and subsystems” (1966,52) Both Gibson and Marr are interested in understanding the function of vision. However, their views of what constitutes an explanation of this functional level are different. This functional level difference is not debated, and the computationalists, who assume that cognitive science is in the business of providing complex information-processing accounts, see Gibson as simply failing to understand the problem. In Marr’s words: “Although some aspects of his thinking were on the right lines, he did not understand properly what information processing was, which led him to seriously underestimate the complexity of the information-processing problems involved in vision and the consequent subtlety that is necessary in approaching them.” (1982,29) From Marr’s perspective, the important lacuna in Gibson’s approach, was his failure to articulate the process involved in extracting information about the environment from the flowing array of ambient energy. For Marr, it is not enough to say, as Gibson does, that the organism simply ‘Lresonates”with certain invariant features of its visual environment. Marr believed that he could explain, in information processing terms, how the organism extracted these invariant features in the first place. In Gibson’s

307 terms, Marr can be understood as attempting to explain how an organism comes to LLresonate” with some aspects of the environment rather than others. At its core, Marr’s project is an attempt to explain how we recognize things. For Gibson, by contrast, recognition happens via a set of biological and ecological mechanisms. These mechanisms can be described, but beyond that no further explanation is really necessary. The purpose of the functional level analysis according to Gibson is simply to show the interaction and covariation of what he called the “receptive units”. Marr viewed the appeal to mechanisms that resonate with certain features of their environment as missing precisely the information-processingstrategies by which we recognize, to use his example, certain cat-like shapes as cats. Denying representationalism, as Gibson did, does not entail a denial of the possibility of an explanation of recognition per se. Gibson’s criticism of representationalism is unequivocal, and yet it leaves space for explanations of how we might come to judge, for instance that some object that we directly perceive is a cat. He writes for instance: Direct perception is what one gets from seeing Niagara Falls, say, as distinguished from seeing a picture of it. The latter kind of perception is mediated So when I assert that perception of the environment is direct, I mean that it is not mediated by retinal pictures, neural pictures, or mental pictures. Direct perception is the activity of getting information from the ambient array of light. I call this a process of information pickup that involves the exploratory activity of looking around, getting around, and looking at things. This is quite different from the supposed activity of getting information from the inputs of the optic nerves, whatever they may prove to be. (1979, 147) Gibson was also interested in providing an explanation for the capacity to recognize distinct objects as instances of a particular kind. However, Gibsonian accounts of how such recognition takes place, do not employ the representationalist strategies used by the computationalists. Rather than processing information, for Gibson, the organism is simply reacting to patterns of information in its environment. A more appropriate target for Marr’s frustrations and criticism would have been the handful of philoscphers who argue that it is simply impossible to explain how we recognize cats as cats, tables as tables, etc. Marr’s desire to explain how we can, for example, see a cat as a cat or a table as a table, provides the basic impetus for his idea that we should treat vision as a set of information-processing

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problems. Marr actually cites John Austin’s discussion of what it means to say that we recognize or fail to recognize the real shape of an object. (1982, 29) Marr criticizes Austin’s claim that there is no such thing as the real shape of a cat. Austin writes : What is the real shape of . . . a cat? Does its real shape change whenever it moves? If not, in what posture is its real shape on display? Furthermore, is its real shape such as to be fairly smooth outlines or must it be finely enough serrated to take account of each hair? It is pretty obvious that there is no answer to these questionsno rule according to which, no procedure by which, answers are to be determined. (Austin 1966, 67 as cited in Marr 1982, 31) Contrary to the skepticism that Austin promotes, Marr shows how each of Austin’s questions can be given an answer, perhaps not the kind of answer that would satisfy the philosophical skeptic, but certainly the kind that would satisfy ordinary scientific curiosity. Marr sees himself as providing precisely these answers in the fifth chapter of Vision where he provides an account of how the visual system generates representations of shape. The purpose of M a d s account in that chapter is to show how the visual system might provide some content for a judgment that some represented shape can be identified as an instance of some concept- - -how, for example, some represented shape is the shape of a cat. To conclude this section, we have seen that M a d s criticism of direct theories of perception is misdirected. Gibson was not suggesting that Marr’s explanatory goals could not be achieved. Philosophers like Austin and perhaps occasionally Wittgenstein may have held such positions, however, as we have seen, where Gibson and Marr really differed was with respect to the representational approach to perception. The significant difference centers on the notion that the complexity of the information-processing task must be articulated in representational terms. Gibson certainly believed in some sort of mediating process that supported perception. What Gibson denied was the need for a system of representations to serve as mediators between perception, cognition and action. By contrast, the notion that the brain generates a series of representations is central to M a d s view. 4. Representation and Information Processing Complexity The idea that brains traffic in representations was encouraged by re-

309 newed interest in the processes underlying behavior in psychology. While cognitive psychology as a discrete field of study only emerged in the early 1960’s, the idea that recourse to representations and psychological processes is necessary to explain behavior was already in the air thanks to Tolman’s cognitive mapping experiments in the 1930’s. Theories of brain function that posited internal manipulation of symbolic structures seemed to solve a number of important problems for experimental psychology. They provided a model of the kind of internal processing that Tolman’s cognitive mapping experiments were thought to require and, through the physical symbol system hypothesis, they offered a neat solution to the problem of the relationship between psychological and physical properties. Representations stored in a physical form in a computer could be understood as having both semantic and causal properties. Relying on what had, by then, become a standard line of thought in cognitive science; Marr claims that we must have representations in the brain, because in order for an organism to know something about an object, it must be able to represent the object to itself. (1982, 3) Marr offers a clear definition of what he means by ‘representation’. He writes: ” A representation is a formal system for making explicit certain entities or types of information, together with a specification of how the system does this.” (1982, 20). Marr’s line of thought with respect to representation runs roughly as follows: in order to know we must have representations in the brain, and these representations are (or are part of) a formal system that makes certain kinds of information explicit along with a set of instructions or rules for interpreting these representations. In this paper, I will not provide an analysis of Marr’s view of representation per se, instead I will focus on his unique and extremely influential computational account of the relationship between representations. This is where the notion of information-processing complexity plays an important role. According to Marr’s view, cognitive scientists should articulate information-processing problems into their component sub-problems and present an explanation that consists of a string of computable solutions to each of these problems. However, one unresolved issue faced by these explanations involves the criteria for judging whether they fit the mechanism under consideration. For instance, in biological systems a relatively simple neural mechanism such as the edge detecting cells in the visual system may solve what computationalists treat as a complex informationprocessing task. In such cases, the neural mechanism can be interpreted as computing a range of alternative algorithms for the information-processing

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task. The challenge for the computationalist is to decide which of the set of possible solutions to chose. In his defense of the dynamical systems approach to perception and cognition, Tim Van Gelder made the valuable point that simple systems can accomplish feats that can also be construed as incredibly complex information-processing tasks. (Van Gelder 1995;1991, Van Gelder, T. J., & Port, R. 1995) His example of such a system is Watt’s governor. In the late 1700’s James Watt invented a device for controlling the relationship between the speed of a steam engine and the pressure of the steam driving the machine. This device itself is simple, however, given computationalist inclinations, we could interpret it as solving a complex information-processing task. Careful analysis of the problem could, for example, lead us to write a computer program attached to a system of sensors (input) and valves (output) which could calculate the adjustments that are necessary to maintain a constant relationship between the pressure, speed, load etc. While nothing stops us from producing such a system, (let’s call it Turing’s governor) Watt’s governor was a far simpler and more elegant solution.a How much would an articulation of this task in terms of a computer program - the Turing Governor - add to our understanding of what it is that the Watt Governor accomplishes? Practically speaking, a project to develop Turing’s Governor would almost certainly have been an impediment to the creation of a technological solution. It is possible that such a solution may have been of some theoretical interest, but the chances of such theoretical advancement would hardly be sufficient to abandon Watt’s solution Theorists of direct perception, like Gibson and others, saw the senses along the lines that Watt took to the governor. The senses were to be understood in terms of the dynamic interaction between the organism and the environment. Sensory systems were collections of feedback loops that tended towards a kind of equilibrium between organism and environment. In the case of the visual system, these visuomotor loops obviously involved aThe Watt governor is a spindle with two hinged arms. Connected t o the engine’s flywheel, the spindle turns at a rate that varies according t o the demands on the engine (speed and load). As the spindle turns, the hinged arms are driven towards a horizontal position by centrifugal force. The arms are connected t o valves that control the release of steam such that as the flywheel spins faster, the arms of the spindle move upwards, slowing the release of steam and thereby slowing the engine. If the flywheel slows, one gets the opposite effect, the arms descend and the valve opens again, thereby increasing engine speed. This simple device thereby establishes equilibrium between a variety of forces without any recourse t o the complex information-processing techniques that would have been involved in our imaginary Turing governor.

311 some kind of mediating biological process, but these processes were thought to be more like Watt’s governor than my imaginary Turing Governor. For direct theorists of perception, a detailed presentation of the interaction of the organism and the environment was a sufficient explanation of the senses. Computationalist criticisms of direct theories of perception attack this basic idea that the senses can be understood solely in terms of the interaction of their component parts. The study of cells, for example, was seen as failing to explain how an information-processingtask, like recognizing a cat as a cat, could be accomplished. To compensate for this apparent deficiency, an additional level of explanation was introduced wherein the dynamic relationship between organisms and their environment could be analyzed into components that had no direct relationship to physical objects or to observable behavior. This level, the level of information-processing complexity, permits the development of theories of perception that incorporate the allimportant notion of representation. This complexity bears no direct relationship to the philosophical problem of computational complexity and has relatively little to do with Shannon and Weaver’s formal definition of the notion of information. Instead, the standard strategy in cognitive science is to recommend that perception be broken down into a series of stages consisting of transformations over information bearing states or representations. As we have seen, articulations of this complexity may often have no correlate in physical reality. Consequently, it is often very difficult to know what it is that we are being told to recognize as complex.

5. Conclusion

The problem with Marr’s argument for the importance of informationprocessing complexity is its implicit assumption that we can understand what the visual system does or what its function is, without knowing its physiological or anatomical form. Marr’s basic assumption, understood in its strongest form, detaches accounts of this complexity from biological or other sources of evidence and precludes the possibility that any new discoveries regarding the structure or physiology of the brain or mechanism in question could change the way we understand its computational or functional properties. This weakness is due to the lack of evidential constraints on the characterization of information processing complexity. Marr qua scientist was a not dogmatic adherent to the autonomy of the functional level and would probably have been sufficiently skeptical in

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scientific practice not to stick to the strongest version of his methodological principle come what may. Nevertheless, Marr’s methodology does recommend that scientific practices be governed by a principle that is equivalent to the belief that a suitably attuned philosopher or psychologist can determine psychological functions without concern for messy details of biology or ecology. Historically, the bias in favor of the functional or computational level of investigation over the evidence of neuroscience did not originate with Marr. It dates back at least to the classic early works in cognitive psychology and is clearly articulated in Neisser’s Cognitive Psychology. (1967) Again, the dismissal of biological evidence rested, almost from the start on the analogy between minds and computer programs, as Neisser writes: The task of the psychologist trying to understand human cognition is analogous to that of a man trying to understand how a computer has been programmed. In particular, if the program seems to store and reuse information, he would like to know by what “routines” or LLprocedures”this is done. Given this purpose, he will not care much whether his computer stores information in magnetic cores or in thin films; he wants to understand the program not the “hardware.” By the same token, it would not help the psychologist to know that memory is carried by RNA as opposed to some other medium. He wants to understand its utilization, not its incarnation. (1967, 6) This radical denial of biology was not tempered by developments in neuroscience. For example, it is common to read more recent works in cognitive science arguing that: Information processes can and must be studied without reference to the physics or biology of the system carrying them out.. . the algorithm for multiplication can be studied and understood in interesting ways without any knowledge of how it is implemented by neurons in the human brain.. . In general an important clarity of understanding is attained by separately considering an information process as a formal symbol manipulation process and as a physical or biological process. (Stilling et al. 1987 4-5) Even in the congressionally declared decade of the brain one could still read cognitive scientists arguing along similar lines. In his (1996) Paul Thagard lists ‘Neurobiological Plausibility’ as the fourth of five criteria for

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evaluating theories in cognitive science. (1996, 13) In his judgment, only the ‘Practical Applicability’ of a theory to educational techniques and design ranks as a less import3nt criterion. The only real theoretical constraints on computational explanations are the limits of computability. This is a virtue insofar as it seems to allow the investigator to abstract from the mental life of particular cognizers to ”cognition as such,” but it is a vice when it comes to offering the kinds of explanations that most of us seek from the brain and behavioral sciences. Given an appropriate interpretation, we can imagine an information-processing account capturing the essence of the behavior that interests us, but we can only make this claim once we imagine putting the kind of biological or robotic mechanisms of affection and transduction in place that will allow us to imagine our interpretation of the algorithm being enacted. The characterization of those mechanisms was precisely the goal of direct theorists of perception like Gibson. While Marr’s work and influence continue to be celebrated in both philcsophical and scientific circles, the science of the visual system has begun to follow a very different path from the one Marr’s theoretical framework prescribed. For instance, it is now clear to most workers in the field that detaching work at the functional or computational level from considerations of neural anatomy and physiology is a less fertile strategy than taking neural architectures seriously as a source of insight and evidence. Even computationally inclined investigators of vision have parted ways with Marr. For example, the work of computationalists like Steve Grossberg begins by paying attention to the patterns of connectivity and the laminar structure of visual cortex. Rather than believing that computational and biological levels can be held separate in any scientifically useful sense, Grossberg claims to identify the neural architecture with the algorithm. Marrls view was, in Patricia Churchland’s phrase ‘brain-shy’ whereas much of what has happened since the early-80’s has been brain-friendly, and highly successful. Revisiting Marr’s account of complexity is important if we are to understand how the norms governing explanation in the computational functionalist framework have fared in the practical business of scientific investigation.

Acknowledgements

I am grateful to Carlos Gersherson for his patience and assistance and to two anonymous referees for some very helpful comments.

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References 1. Austin, J. 1967. How to do things with words. Cambridge: Harvard University Press. 2. Gibson 1966 The Senses Considered as Perceptual Systems Boston: Houghton Mifflin. 3. Gibson, J.J. 1979. The Ecological Approach to Visual Perception Systems. Boston: Houghton Mifflin. 4. Hurley, S.-1998. Consciousness in Action. Cambridge: Harvard University Press. 5. Kosslyn, S. 1994. Image and bmin: The resolution of the imagery debate. Cambridge: MIT Press. 6. Marr, D. 1982. Vision: A Computational Investigation into the Human Representation and Processing of Visual Information. San Francisco: W.H. F’reeman. 7. Neisser, U. (1967) Cognitive Psychology. Englewood Cliffs, N.J.:PrenticeHall. 8. Stillings, N.A. et al. (1987) Cognitive Science: A n Introduction. Cambridge, Mass.: MIT Press. 9. Thagard, P. 1997. Mind: A n Introduction to Cognitive Science. Cambridge: MIT Press. 10. Van Gelder, T . J. 1995 What might cognition be, if not computation? Journal of Philosophy, 91, 345-381. 11. Van Gelder, T. J., & Port, R. 1995. It’s About Time: An Overview of the Dynamical Approach t o Cognition. In R. Port & T. van Gelder ed., Mind as Motion: Eqlomtions in the Dynamics of Cognition. Cambridge MA: MIT Press. 12. Van Gelder, T . 1998. The Dynamical Hypothesis in Cognitive Science. Behavioral and Brain Sciences. 21, 1-14 13. Wade, N. and M. Swanston, Visual perception: an introduction. London: Routledge 14. Yulle, A.L.& S. Ullman, (1990). ‘Computational theories of low-level vision’. In: D. N. Osherson, Kosslyn, S. M. &. J.M Hollerbach (Eds), Visual Cognition and Action, Volume 2, pages 5-39, MIT Press.

ON THE POSSIBLE COMPUTATIONAL POWER OF THE HUMAN MIND

HECTOR ZENIL* & FRANCISCO HERNANDEZ-QUIROZ+ Math Department, Faculty of Science, National University of Mexico (UNAM) -Forthcoming in: Carlos Gershenson, Diederik Aerts, and Bruce Edmonds (eds.) Philosophy a n d Complexity: Essays o n Epistemology, Evolution, a n d Emergence, World Scientific, 2006-

The aim of this paper is t o address the question: Can an artificial neural network (ANN) model be used as a possible characterization of the power of the human mind? We will discuss what might be the relationship between such a model and its natural counterpart. A possible characterization of the different power capabilities of the mind is suggested in terms of the information contained (in its computational complexity) or achievable by it. Such characterization takes advantage of recent results based on natural neural networks (NNN) and the computational power of arbitrary artificial neural networks (ANN). The possible acceptance of neural networks as the model of the human mind’s operation makes the aforementioned quite relevant.

1. Introduction

Much interest has been focused on the comparison between the brain and computers. A variety of obvious analogies exist. Based on several thoughts, some authors from very diverse research areas (Philosophy, Physics, Computer Science, etc.) claim that the human mind could be more powerful than Turing Nevertheless, they do not agree on what these “super-Turing” capabilities mean. Consequently, there is no universally accepted characterization of this extra power and how it could be related to the human mind, even though there is strong defense of these authors’ theories based on whether or not humans are “super-minds” capable of processing information not Turing-computable. *[email protected] t [email protected]

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Nowadays, it is accepted that the nervous system, including the spinal cord and the neocortex, control behavior. In a simplified picture, the nervous system could be viewed as a device which receives input from various senses and produces output in the form of some action. From a computational point of view, the word “produces” could be harmlessly substituted by “computes”, in the sense that the mind computes in the way a Turing machine does. One of the most interesting and challenging tasks in science is to assume this as a possibility in order to understand how these computations are performed. Thus, it is not surprising that the field of Neuroscience attracts many researchers not only from the fields of Biology and Computer Science, but also from the area of Physics. Contributions to the mind’s neural computation require a strong effort towards interdisciplinary collaboration. As Teuscher and S i ~ p e pointed r~~ out, there are very few problems stemming from neural computation on which a theoretical computer scientist can commence work without further interaction or collaboration with neuroscientists. Since the artificial model was inspired in the natural, in this paper we will explore the bridge between them on the basis of recent work. Since artificial neural networks have been inspired from their biological counterparts, it is natural to ask about the feedback from the artificial to the biological. We attempt build such a bridge by taking the standard model of ANNs and the way in which ARNN admits arbitrary weights, in order to characterize its computational power by using the complexity of the weights or the timing of the pulses. Most of what happens in our brain does so without our being aware of it, so that many things that seem easy to us involve highly complex neural circuits and operations. Most practical applications of artificial neural networks are based on a computational model involving the propagation of continuous variables from one processing unit to the next. Inspired by a simplistic vision of how messages are transferred between neurons, computer scientists invented the artificial computational approach to target a wide range of problems in many application areas. Biological neurons are connected by synapses, which are the links that carry messages between them. Using synapses neurons can carry pulses to activate each other with different threshold values. Neurons are the computational part of the network while links connect different neurons and enable messages to flow among them. Each link is a connection with a weight that affects the carried message in a certain way. In fact each link receives a value from an input neuron, multiplies it by a given weight, and then passes it to another

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neuron. Between this process several sources of computational power can be identified. This means that the source of the computational power of an NN might have at least three possible sources: (a) one carried by the operations in the neuron itself; (b) the message between neurons involving pulses, action potentials (APs) and timing and (c) the complexity of the neural network weights in terms of their capability to encode information which implies (a) and (b). As biologically-inspired devices Artificial Neural Networks (ANNs) have been suggested as a computational theory of mental activity. An ANN is mathematically defined as a weighted directed graph in which each vertex represent a neuron and each arrow a connection. Definition 1 . A n Analog Recurrent Neural Network or ARNN" is

a finite network of nodes (called neurons) and connections, wherein the synaptic weight associated with each connection (determining the coupling strength of the two end-nodes) is a real (analog) value. An ARNN is basically an artificial neural network allowing loops (recurrent) and irrational (even non-Turing computable) numbers as weights (analog). We consider the question about the possible computational power of the human mind in this paper as a problem of computability in general and tractability or practical computability out of its scope then making only emphasis on the computability power defined by the Turing degrees and beyond. The key arguments which form the bases of our position are: (1) ARNNs are a plausible model of the human brain because they define a wide range of different levels of computational power. (2) At least some characteristics of the human mind arise from the human brain. (3) ARNNs are therefore a plausible model of the human mind. (4) It has been argued that the human mind could be depicted or a The

A R N N dynamics i s given by a mapping:

F : W ~ X ( 0 , l ) M+ R N an which each neuron xi in t i m e ( t + l ) behaves as described by the following non-linear equation: aijzj(t) j=1

+

M j=1

b i j u j ( t ) + ci

(2)

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fully simulated by Turing machines, and others, as it has a “superTuring” computational power. (5) Therefore, prima facie, it seems reasonable to explore claims on the human mind’s supposed Turing or “super-Turing” capacities through an analysis of ARNNs in relation to recent work on the brain’s own neural network. ( 6 ) That an analysis of ARNNs supposed Turing and “super-Turing” capabilities (and other more well established findings) gives rise to a taxonomy of computational capabilities. (7) The taxonomy of computational capabilities gives rise to prima facie reasonable hypotheses concerning the human mind. The contribution of this paper to our knowledge is to build up this argument and generate hypotheses binding artificial models to the natural. With reference to argument number 1, the ARNN model even allows to consider a situation in which the weights and the ARNN become equivalent in power to automaton of lower power (including the Turing machine). However, the ARNN (and a wider generalization proposed in this paper, to be fully explored in another, yet to be published) allows us to consider simultaneously, all possible scenarios. Claim number 2 is widely accepted in different degrees of engagement. It is the case of the mind/body problem and our claim is based on the consideration of the mind as the brain operation. Hence, no distinction is made between brain and mind. However, it is our assumption that the mind does not subtract any computational power and fully inherits the whole computational power of the brain. If mind adds some computational power to the overall system our arguments still apply as lower bound. By contrast, if the mind subtract computational power, our claims should be reformulated in terms of it and our arguments will lay as upper bound. Our intent is to discuss the mathematical features which a model should possess, if it is to aspire to certain computable power. Our choice of an ARNN model rests on two considerations: (a) it has been proposed as a model with (potentially) non computable capabilities (provided some requirements are met; see later); (b) as a neural network model, neuroscientists might find it easier to relate it to their own empirical work -most of their research is based on (pseudo)analog values- and therefore they can put forward and test concrete hypotheses to confirm or refute the model. Additionally, the ARNN represents a refinement on what had been typically regarded as the mathematical definition of an ANN, which did not make

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any distinction between the complexity of the weights in terms of the computational power beyond the seminal work of Minsky. Nevertheless, neural networks which run on digital computers operate on a Turing-computable subset of irrational numbers, a strong restriction that determines a priori its computational power. Hence, some enthusiasm generated by important experimental and theoretical results cannot be extended easily to applications because there is no straightforward way to make real numbers available even assuming its possible existence. Digital hardware implementation uses a finite number of bits for weight storage and rational restraint values for firings rates, weights and operations and remain limited to a computational power. Even analog implementations, which are often cited for their ability to implement real numbers easily (such as analog quantities), are limited in their precision by issues such as dynamic range, noise, VLSIb area and power dissipation problems. Thus, in theory and in practice, most implementations use rational numbers, or at most, a subset of irrational numbers, the Turing-computable ones. The classical approach of computability theory is to consider models operating on finite strings of symbols from a finite alphabet. Such strings may represent various discrete objects such as integers or algebraic expressions, but cannot represent general real or complex numbers, even though most mathematical models are based on real numbers. The Thing machineG7and its generalization in the form of the Universal Turing machine (UTM) is the accepted model for computation, while under the Church-Turing thesis, it is considered the authoritative model for effective computation. However, some researchers have proposed other models in which real numbers play the main role in effective computations. Machines with “super-Turing” capabilities were first introduced by Alan TuringGg,which investigated mathematical systems in which an oracle was available to compute a characteristic function of a (possibly) noncomputable set. The idea of oracles was to set up a scheme for investigating relative computation. Oracles are Turing machines with an additional tape, called the oracle tape, which contains the answer to some non-computable characteristic functions. An oracle machine is an abstract machine used to study decision problems. It can be visualized as a Turing machine with a black box, called an oracle, which is able to determine certain decision

bVery Large Scale Integration are systems of transistor-based circuits into integrated circuits on a single chip. For example, the microprocessor of a computer is a VLSI device.

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problems in a single step. However, Turing gave no indication on how such an oracle might be implemented. The Turing machine can write an input on its own tape, then tell the oracle to execute it. In a single step, the oracle computes its function, erases its input, and writes its output to the tape. It is possible to posit the existence of an oracle, which computes a non-computable function, such as the answer to the halting problem or some equivalent. Interestingly, the halting problem still applies to such machines; that is, although they can determine whether particular Turing machines will halt on particular inputs, they cannot determine whether machines with equivalent halting oracles will themselves halt. This fact creates a hierarchy of machines according to their Turing degree, each one with a more powerful halting oracle and an even more difficult halting problem. With such a method, an infinite hierarchy of computational power can easily be constructed by positing the existence of oracles that perform progressively more complex computations which cannot be performed by machines that incorporate oracles of lower power. Since a conventional Turing machine cannot solve the halting problem, a Turing machine with a Halting Problem Oracle is evidently more powerful than a conventional Turing machine because the oracle can answer the halting question. It is straightforward to define an unsolvable halting problem for the augmented machine with the same method applied to simpler halting problems that lead to the definition of a more capable oracle to solve that problem. This construction can be continued indefinitely, yielding an infinite set of conceptual machines that are progressively more powerful than a Turing machine. This build the hierarchy of Turing degrees: Definition 2. The Turing degree of a subset A of natural numbers is the equivalence class consisting of all subsets of B equivalent to A under Turing reducibility. The degree of a set A is denoted by deg(A). The least element in the partial order is denoted b y 0 and is the degree of all recursive sets (computable sets). In other words, two sets of natural numbers have the same Turing degree when the question of whether a natural number belongs to one can be decided by a Turing machine having an oracle that can answer the question of whether a number belongs to the other, and vice versa. So the Turing degree measures precisely the computability or incomputability degree of a subset X . Definition 3. A Turing reduction from a A to B is a reduction which easily computes A assuming B , i.e. A is computable by a Turing machine with an oracle for B .

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Because a language can be always codified by a subset of natural numbers, in terms of computability theory, sets and languages are equivalent. Definition 4 . Formally, a language L is recognized b y a Turing ma-

chine with an oracle A if f o r every string s the machine using A as a n oracle halts o n input s if S E L . A language B is Turing-reducible to a language A if a Turing machine equipped with an oracle f o r A can decide B . Models of hyper-computation tend to be of two general types: One uses oracles or oracles in disguise, and the other uses infinite computation in finite time. Whether or not such machines are physically constructible -most experts believe they are not- studying them improves our understanding of the recursion theory. On the other hand neural networks have been studied as computational devices. In 1956, Kleene showedz7 how to simulate finite automata using McCulloch and Pitts neurons41 and proved that when the weights of the networks are constrained to be integers, the languages accepted by them are exactly the regular languages. On the other hand, M i n ~ k yshowed ~ ~ that neural networks with boolean neurons can simulate any finite automaton. More recently, Hava Siegelmann5’ presented a computation model based on analog recurrent neural networks (ARNN). All this work establish a very strong connection between artificial neural networks and automata theory. Both automaton (including the Turing machine) and neural networks are characterized by the languages they accept and hold the same hierarchy. Siegelmann offered a proof5g in which the set of languages accepted by networks with rational weights are exactly the recursively enumerable languages=. Siegelmann showed that ARNNs are strictly more powerful than the Turing machine model in that they can perform computations provably uncomputable by a universal Turing machine. Disregarding the fact that it seems unrealistic for most experts that those ARNN could be someday constructible it is not easy to discard if such devices are already present in nature taking advantage on the physical properties of the universe or, at least, the way in which they can perform computations over its physical environmentd. According to G. Kreisel himself it is an open question whether ‘Davislg rightly pointed out that even if a subset of non rational numbers is used, namely the set of Turing-computable irrationals, the class of languages recognized by neural networks remains the same, as Siegelmann’s proof on the power of networks with rational weights readily extends to nets with computable irrational weights (as Turing already did with his machines). dAnother interesting question raises by its own right: if there exists a natural device with such capabilities how might we be restricted to take advantage of the same physical

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there is any ”natural phenomenon” leading to an uncomputable numberg. These neural networks satisfy some classical constraints of computational theory: the input, output, and transitions, are discrete and finite. But the overall system is not really finite since it uses arbitrary real numbers, and it is known that it is powerful enough to encode all non-recursively enumerable languages. Consequently, a connection between the complexity of the networks in terms of their information complexity and their computational power had been developed, spanning a hierarchy of computation from the Turing model to a “super-Turing” mathematical model. 2. Modeling the Brain

A clear difference between the brain and the computer is that a digital computer operates by performing sequential instructions from an input program, whereas there is no clear analogy of such a performance in the human brain. However, since any Turing machine working on several tapes is finally equivalent to a single tape Turing machine, the difference may be functional rather than fundamental. The brain as a fully parallel device, could be reduced to a sequential input under some very significant restrictions (for example being discrete or finite). Nowadays, such questions are the subject of scientific and philosophical debate since we have the computational resources to develop live experimentse. However, Digital properties in order t o build an artificial equivalent device? Much of the defense of the work mentioned above have precisely centered on questions such as if we are taking advantage of the resources we have in nature. eThere are at least three important projects currently running: A Linux cluster running the MPI NeoCortical Simulator (NCS), capable of simulating networks of thousands of spiking neurons and many millions of synapses, was launched by Phil Goodman at the University of Nevada. Blue Brain, a 4,000 processor IBM BlueGene cluster, was used t o simulate a brain in a project started in May 2005 in the Laboratory of Neural Microcircuitry of the Brain Mind Institute at the EPFL in Lausanne, Switzerland, in collaboration with lab director Henry Markram. It has as its initial goal, the simulation of groups of neocortical columns which can be considered the smallest functional units of the neocortex. Also running is the NCS, t o be combined with Michael Hines’ NEURON software. The simulation will not consist of a mere artificial neural network, but will involve much more biologically realistic models of neurons. Additionally, CCortex, a project developed by a private company Artificial Development, planned t o be a complete 20-billion neuron simulation of the Human Cortex and peripheral systems, on a cluster of 500 computers: the largest neural network created t o date. Different versions of the simulation have been running since June 2003. CCortex aims t o mimic the structure of the human brain with a layered distribution of neural nets and detailed interconnections, and is planned t o closely emulate specialized regions of the Human Cortex, Corpus Callosum, Anterior Commissure, Amygdala and Hippocampus.

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computers were not designed to be models of the brain even when they are running neural networks to simulate its behavior within their own computational restrictions. Most fundamental questions are however related to its computational power, in both senses: time/space complexity and degree of solvability. Most computational brain models programmed to date are in fact strictly speaking, less powerful than a UTM. Researchers such as Stannett63 have speculated that “if biological systems really do implement analog or quantum computation, or perhaps some mixture of the two, it is highly likely that they are provably more powerful computationally, than Turing machines. This statement implies that true human intelligence cannot be implemented or supported by Turing machines, an opinion shared by Roger P e n r ~ s e who ~ ~ , believes mechanical intelligence is impossible since purely physical processes are non-computable. A position strongly criticized by many researchers in the field since its authors first propagated the idea. However, assuming some kind of relation between the mind and the brain’s physical actions, neural networks may be accepted as a model of the human mind operation. Since such a mindlbrain relation is widely accepted in different ways and levels, we concern ourselves with the computational power of these devices and the features that such networks must possess. We will address which from our point of view, goes to the crux of the matter when the question of the computational power of the brain is raised, that is, its solvability degree. This means that we do not concern ourselves with what could be the recipe in which a simulation could run, since if we restrict ourselves to the discussion of artificial neural networks running on actual digital computers, we will be restricted to the lowest computational degree of solvability. From this, it can be easily deduced that if there is a fundamental difference between the architecture of the brain and digital computers, then the efforts of the artificial neural networks to fully simulate the brain either for the purpose of study or reproduction, are destined to have fundamentally different degrees of power. Based on certain references5 as well as our own research, we have identified at least five mathematical descriptions in which “super-Turing” capabilities have been formally captured: super- task^^^^* and accelerated Turing machines, Weyl machines or Zeus machines7, Trial and Error machines50, Non-Standard Quantum Computation, and the Analog Recurrent Neural Networks59. We have also identified other proposals concerning Turing machines with some kind of interaction between them or the e n ~ i r o n m e n t ’ ~those ~ ~ ~models ~ ~ ~ ,provide a basis for the following claims:

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(1) Minds are not computers, because (most) thought processes are not. (a) It can “violate” Godel’s theorem, therefore is not a computing machine, a claim made most famously by Lucas2’ in 1961. (b) Mind can “solve” the “Entscheidungsproblem” , therefore is not a computing machine. (2) Minds are computing devices but not of the same power of Turing machines (maybe Godel himselff). (a) There are special operations that occur in the brain which are not Turing computable, a claim made most famously by Penr~se~~. (b) The mind could be a machine but with access to a certain oracle (from an external source or from a previously coded internal source). All hyper-computational models presented to date are purely theoretical, and we may well ask whether they can actually be implemented in the sense that the universal Turing machine is implemented or pseudoimplemented in a common digital computer. A personal hyper-computer would be no more an implementation of a hyper-computer model, than a personal computer is of a UTM and nobody has physically implemented any kind of hyper-computer. Most models seems to take advantage of (a) a type of oracle device or (b) an infinite number of steps in a finite amount of time. Jack Copeland has pointed out13 an interesting fact concerning the way Turing machines work. He stated that Turing machines are closed systems that do not accept input while operating, whereas the brain continually receives input from the environment. Based on this observation, Copeland has proposed the coupled Turing machine which is connected to the environment via one or more input channels13. However, as Christof Teuscher and Moshe Sipper have pointed any coupled machine with a finite ‘In his 1951 Gibbs lecture” Godel attempts to use incompleteness t o reason about human intelligence. Godel uses the incompleteness theorem to arrive at the following disjunction: (a) Either mathematics is incompleteable in this sense, that its evident axi o m can never be comprised in a finite rule, that is t o say, the human mind (even within the realm of pure mathematics) infinitely surpasses the powers of any finite machine, or (b) or else there exist absolutely unsolvable diophantine problems (or absolute undecidable propositions) for which it cannot decide whether solutions exist. Godel finds (b) not plausible, and thus he seems have believed that the human mind was not equivalent t o a finite machine, i.e., its power exceeded that of any finite machine, the term used originally for Turing machines

325 input stream can be simulated by a UTM since the data can be written on the machine’s tape before it begins operation. From dynamic systems we often decide almost in an arbitrary way, when a system will be closed in order t o handle it. However, the chain of such external systems, potentially infinite (even just by loops) can create a non-linear system which could truly be more complex and maybe more powerful. Some other claims and critics have been made in this regard. Some other authors claim that the “super-mentalistic” perspective is not a scientific one, as it implies the possibility of assigning non-reducible phenomena t o some sort of information processing. However, we believe that this fact does not preclude a study on what formal properties can be required from non-Turing models of the human mind. A clarification of these properties would help us understand to what extent “super-Turing” models of the mind can or cannot be considered in a scientific way. 3. A computational model for the human mind 3.1. Determining the power of a neural network by the complexity of its weights

Neural Networks are able t o encode information by several ways. If brain’s neural network computes equal or less than the Turing degree of Turing machines without oracles, their weights should code only whole or rational numbers. However, if brain’s NN is more powerful it is clear that it is possible to use an oracle to build a neural network that simulates a Turing machine with that oracle in order to recognize any arbitrary language5’. This can be done by encoding the oracle into a real number r which the neural network “consults” in order to know if a certain input (or a transformation of it) does belong to the language encoded by r. Because we are interested in the set of Turing degrees of the weights of a neural network, and since not always Turing degrees are comparable, it is necessarily t o use the notion of a maximal element. Definition 4 . Let be a partially ordered set (A,<). Then an element r c A is said to be maximalg if, for all aEA,r$a. We are going to take as “information” to the repository of languages. In Computer Science and Linguistics a language is a set of symbols from gNote that the definition for a maximal element is true for any two elements of a partially ordered set that are comparable. However, it may be the case that two elements of a given partial ordering are not comparable as in the case of Turing degrees as Post proved

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an alphabet. As it is well known, every string on an arbitrary alphabet can be encoded into a string on the binary alphabet {O,l}. In the same way, a real number with infinite expansion can also be represented by an infinite binary string. Putting together these facts, a language can be easily encoded into a real number r in the interval [0, 1) taking the n digit of the expansion of r as the value of the characteristic function of the language applied to the n string in {0,1}* ordered lexicographically. Additionally, this encoding is unique. Then given a language L , rL is the real number that encodes it. Definition 5. A real number r L is computable i f and only i f it is the limit of a n effectively converging computable sequence of rational numbers. T h e Turing degree of a real number r L i s defined as the Turing degree of its binary expansionh. Example 1. This definition is quite natural and robust. For instance, let P be the set of natural numbers defined by P={nEN(the digit n of the binary expansion of (A-3)is 1). P is evidently computable. Let be R N a neural net with weights A and A-n. Since A is Thing-computable and A-n too then R N will compute only those languages coded by A and 8-71, Example 2. However if h is used as a weight, where h is the Chaitin constant 0 defined as the probability of halting of a Turing machine Mi for an input xi R N is going to compute definable languages but non-Turing computable. It is evident that R N computes h and those in its same complexity degree. If the set of maximals of R N has only h then the Turing degree of R N is going to be the Turing degree of h, that is 0'. Then if the human mind is capable to hold x,h or any other value as a connection between neurons it will determine its computational power by taking the set of maximals of the weight's Turing degrees. 3.2. Extracting information from the weights

A more interesting question is how to verify what type of languages a weight has coded in order to determine the computational power of a neural network. That means what language a real number encodes. Since the function of the encoding is bijective it is just needed to analyze one by one the digits of the binary expansion of the real number in order to reconstruct the encoded language. Of course, languages encoded in this way hit can be seen as a subset of natural numbers (or in fact the whole set of natural numbers encoded in a single real number concatening all). It is evident that not all real numbers are computable (they are also identified as random in Chaitin theory).

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go well beyond recursively enumerable languages and most of them will go beyond any Turing computable procedure. It is clear that if a weight really encodes a non rational number it is going to be impossible to extract all digits in a finite time and space for most of them. However, if all weights in the brain neural network are finite it will represent an argument for a computationalism position on the human mind (and depending upon the accepted relation between the brain and mind). If a computable irrational number is taken, for example, T , the language it encodes is clearly recursively enumerable but infinite. However this procedure obscures the rules (of a suitable Turing machine) that generate the language and it is very difficult to see how such rules could be derived from the number itself and ) the first n digits of then impossible to determine if the substring n ( ~for K is really going to be K at the end or any other number sharing the first n digits. Even further, it seems to be impossible to distinguish between a computable or non-computable number by this way. Relativization of arbitrary neural networks by using the computational power of their weights by any procedure could allow us to classify them into the well known traditional hierarchies. For this purpose, we will use an oracle to build a neural network that simulates a Turing machine with that oracle to recognize a language. Let M be the neural network we will build. First, we encode the input w into a real number r , as we saw in the last section, then we code the oracle given in the same way. Let o be the real number that encodes the oracle. For each entry i coded in r , the oracle coded in o is consulted, when the answer is yes, the digit i is 1, when it is not, the answer is 0. Then M has two parts, one a sub-network N will simulate a Turing machine, and the other, named 0 will simule the oracle machine5g. Let w ( i ) be the oracle answer for the input i, we denote by c the concatenation c = w(O)v(l) ...w ( i ) ...w(n),and by d(c) a Cantor encoding5g of T . The Cantor encoding can be avoided since we are not concerned in this paper in timelspace complexity. According to Davis’’ it is enough a straight encoding based on length and lexicographical order’ Then, the network M can be described59 as a composition of two subnetworks: the first, N , for which T, is a weight, receives the input w, and ‘~-1, a-2,

b-3, aa-4, ab-6, ba+6, bb-+7,aaa-8, aab-+9,aba-10,

. . . For example

if L is defined by all the strings that begins with an “a” followed by an arbitrary number of “b”sL=(a, ab, abb, abbb, ...}, then the set S will be (2, 5, 11, 23, ...} and the real number encoding L will be L,=0.0100100000100000000000100... (in fact it is a non rational number in this case).

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after a fixed computation N submit yw t o 0, where y comes from w after the computation of the Turing machine simulation, the sub-network which performs the oracle machine and has d ( c ) = o as weight. The output of the sub-network 0 will depend on a binary activation functionJ . The input arrives on two binary input lines. The first is a data line, which is used to carry a binary input signal, when no signal is present, the output is zero. The second is a validation line, which indicates when the data line is active. It takes the value one while the input is present and zero when not. There are also two output lines, that take the same roles. These conventions allow using all external signals to be binary. This gives rise t o a taxonomy of computational capacities summarized in the table of the following section. If any arbitrary neural network N is equivalent to a network N' with two sub-networks M and 0 that simulate the Turing machine and the oracle component respectively, then the computational power of the whole network N will be determined by the sub-network 0. Therefore, it makes sense to classify a neural network by means of the Turing degree of its oracle component. The Turing degree of an oracle in a neural network is the set of maximals of the Turing degrees of the weights of the network. The Turing degree of each weight depends on its encoding capacity.

3.3. Other sources of computational power: spikes and operations In recent years, data from neurobiological experiments has made it increasingly clear that biological neural networks, which communicate through pulses, use the timing of these pulses to transmit information and to perform computation. This realization has stimulated a significant growth of research activity in the area of Spiking (or Pulsed) neural networks (SNNs or PNNs), theoretical analysis, as well as the development of new ANN paradigm^^^,^^. From the perspective of our main concern in this paper, an important question is: What type of information in terms of complexity, might be carried by an action potential? Each action potential could represent a single bit of information, similar to a serial digital communication channel without error checking. However, the information per action potential may differ according to the function of the neural network. In jA sigmoidal type function called the signal function defined as: signal(z) = 0 if xS0 and 1 in other case.

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traditional neural network models the timing of computation steps is usually “trivialized” . Clearly, precisely timing of spikes would allow neurons t o communicate much more information than with essentially random spikes. Otherwise, such temporally-coded information is lost as a source of possible computational power in those traditional models. It has been shown that the use of pulse correlations in addition to pulse rates can significantly increase the computational power of a neural network. These timing encodings finally contribute to the overall power of such networks. Biologically-inspired computation paradigms take different levels of abstraction when modeling neural dynamics. The production of action potentials or spikes has been ab- stracted away in many rate-based neurodynamic models, but recently this feature has gained renewed interest. In biological neural networks, information is transmitted by the conduction of action potentials along axons, and information processing takes place at the synapses, dendrites, and soma of neurons28. It has been shown that firing correlations play a significant computational role in many biological neural systems. Models of biological neural networks in terms of dynamic systems have been studied and formalized by H ~ d g k i n - H u x l e yand ~ ~ FitzHugh-Nagumo equation^^^^^^ and others. Recently experimental evidence has accumulated during the last few years, which indicates that many biological neural systems use the timing of single action potentials (or “spikes”) to encode information1~3~6~51~24~21~70~53,55~56@. Experiments have shown that in vitro biological neurons fire with slightly varying delays in response to repetitions of the same current injection’. Even when this behavior may be explained by noisy versions of neural networks, the source for some kind of possible codification in such processes remains. However, noise certainly affects the computational power of networks of spiking neurons for analog i n p ~ t ~ l ? ~ ~ . Nevertheless, empirical experiments in which complex fluctuations exist, as seen for example in EEGk signals that are generally taken as noise, may be indicative of complex dynamics for processing information similar t o that of pulse propagation networks or spiking neurons, maybe continuously over time. A central theme in this paper is the coding capabilities of neural net-

kElectroencephalographyis the neurophysiologic measurement of the electrical activity of the brain by recording from electrodes placed on the scalp or, in special cases, on the cortex. The resulting traces are known as an electroencephalogram (EEG) and represent so-called brain-waves.

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works in which its computational power lies. Most accepted descriptions of the human neural network involve finite and discrete quantities, but some of them remain as continuous variables from one processing. Computer scientists have been studying ANNs for many years. Although ANNs were inspired by real biological networks, typical ANNs do not model a number of aspects of biology that may prove to be important. Real neurons, as we have seen for example, communicate by sending out little spikes of voltage called action potentials (APs). ANNs, however, do not model the timing of these individual APs. Instead, ANNs typically assume that APs are repetitive, and they model only the rate of that repetition. However some of the computational power of a biological neural network could be derived from the precise timing of the individual APs or other properties inherent to the biological neural network that it is not fully right-modeled. Regular ANNs could never model such a possibility defined as their currently are. Additionally, all ANNs running over digital machines are incapable of simulating any possible analogical signal or full irrational encodings if they were present in human brain’s architecture. Methods for estimating the computational power of neural circuits and their relation to artificial neural networks models have been established. Maass30,32733134 and Markram3’ have recently argued that “in contrast to Turing machines, generic computations by neural circuits are not digital, and are not carried out on static inputs, but rather on functions of time”. These kinds of experiments could ultimately provide a definitive answer to the critical issues which are the concern of this paper. These models are also interesting because even if a neural network is constrained to simple weights, namely whole or rational numbers, it is possible to achieve an extra power from the timing process, encoding what is not possible to encode in the weights. Since they are similar in power to analog signal processing, they can be compared to the traditional hierarchies we have already explored. Even when a mathematically rigorous analysis of the computational power of networks of spiking neurons has so far been missing, they are equivalent to the levels of such hierarchies, simply by replacing weights encodings with pulse timing encodings. In other words, the computational power is transferred from weights to spikes, and in the presence of a mixed model with both weights and spikes encodings, its final computational power will be the power of the most powerful. Both pulse frequencies and correlations which are computationally relevant can be seen as operations involving potentially any possible real numbers in principle equivalent to a weight encoding. Hence, the overall power of

331 these PNNs or SNNs is determined by the maximal Turing degree of the union of the Turing degrees of both weights and time encodings. It has also been shown that SNNs are, with regard to the number of neurons that are needed, computationally more powerful than other neural network models. These models are obviously Turing reducible when variables (the input and all internal operations) are restricted to Turing-computable values if such model is close under its operations. However when arbitrary values are ~~ allowed then they can compute possible non-Turing functions. M a a . 9 ~has shown that, with regard to temporal coding of real-valued variables any continuous function can be approximated arbitrarily closely. In this case, the model depends on the continuity of time. On the other hand, since it is possible to compute fifrom simple operations between whole numbers (as the hypotenuse of a Pythagorean triangle with unitary sides) and even to make proofs of irrationality without calculating their decimal expansion, it might be possible to prove if brain’s neural network can hold non-computable numbers if it is able to perform some special operators where the discrete operations, such as primitive recursion or bounded sums and products, are replaced by operations on continuous ~ . sets of those functions, such as integration or linear i n t e g r a t i ~ n ~The real functionss4 can be definable by a general-purpose analog computer” or GPAC, which is a general computing model evolving in continuos time’. Then even if weights and spikes are restricted to Turing-computable functions, the brain neural network might be not closed if at least an operation is not recursive possibly (see and 16) taking the involved values to nonTuring degree at some point of the computation. Traditional functional operations like Sequential Composition, Parallel Composition or definition by Cases, Primitive Recursion or simply Recursion, Bounded Recursion, Partial Projection, Cut-off Subtraction and the order functions like the Minimization or Projection and Bounded minimization take computable functions into computable functions. By contrast, It is well known that limits take computable functions beyond the class defined by the power of Turing machines. For a more detailed description of such operations see a. The power of human brain operations remains unknown, but even if 162,16 The class of R-recursive functions is very large. It contains many traditionally non-computable functions, such as the characteristic functions of sets of the arithmetical h i e r a r ~ h y ~ *Experimental %~~. proofs beyond this level could be more difficult if not impossible since the construction of a sequence of real numbers which can not be computably diagonalized is used to prove that there are continuous functions without a Turing degree.

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non-traditional operations were possible both theoretical and empirical evidence seem to be hard or impossible since the verification of non-Turing operators require non-Turing inputs and outputs which seems to be undistinguishable from those Turing-computable. Nevertheless neurological tests could be designed in order to achieve some advance in this direction. One could suppose that given the evolution of neural networks models, several better approaches to the biological neural network model could be expected. Other possible sources could be unrepresented in current artificial models. Experimental and theoretical research in the field should continue to relate it more to the biological model. ma as^^^ suggests SNNs as the third generation of the artificial neural network model. Each one seems to offer better approaches to the experimental evidence of the brain’s own neural network. Now, we can build a simplified hierarchy of computational power regarding the two possible sources explored in this paper (weights and spikes, which are equivalent in terms that they can be replaced one for the other preserving the whole complexity of the NN):

Neural Network Architecture R Non-computable numbers

Q Rational numbers Z Whole numbers

Computational Power Turing machine with an oracle depending on the maximal Turing degree of of the weights: w1,w2, . . .’ n E R and pulses pl,p2, . . .’ rn E R, not always in the Arithmetical Hierarchy at most Turing machine at most bounded automata

At the bottom level, we have neural networks computing Turingcomputable languages if the oracle is empty. On the other hand, weight’s and spikes’ complexity could be determined -assuming that the neural network is closed under isomorphism- then they can be decoded into an oracle which will determine its computational power. If the neural network perform not closed operations under isomorphism, the neural network will compute in a range determined by the bounds of such operations, possibly determining an interval within this simplified hierarchy.

333 4. Discussion and Conclusion

Even when the answer concerning the computational power of the human mind should come from neurophysiological and interdisciplinary research, including the correspondence between the natural and the artificial neural network models, the question remains of mathematical and philosophical interest as it is possible to explore all possible scenarios inside the computability theory. We add that if we were able to determine some key properties of the physical universe and the mind we could determine the computational power of the mind, among them: 0 0 0

0 0

if our physical universe is discrete or continuous, and in which case if it can hold non-computable numbers; if it can hold non-computable operations; if the brain inherits those properties; if the mind can take advantage of them, which depends on the precise relationship between the mind and the physical operation of the brain.

As it can be seen, these are not small requirements. However once we can ascertain some fundamental properties of the mind it could be possible to know to which computational level it belongs by its maximal Turing degree. The value of this paper lies in its effort to restore some of the claims which have been made and proven for artificial neural networks to the natural model and the problem of determining the mind’s computational power. Then, it is not just about the way in which a neural network can achieve one or more computational powers, but also which features a brain should have to potentially take its power to one level or another: either less, equal or more powerful than Turing machines. Then, if every language with a given alphabet can be encoded and potentially extracted into a real number (from weights or spikes), this real number can be used as the main component of a neural network that simulates an oracle Turing machine whose oracle is the characteristic function of the language. Therefore by determining the neural network encoding capability and knowing the type of operations permitted in it, its computational power could be determined. We would like to conclude, by remarking that a full model of the mind has a very difficult task ahead. We hope that this paper has succeeded in shedding light on the current research and ways to build certain bridges from the artificial to the natural model, as a possible way to eventually

334 determine the computational power of the human mind taking in consideration all possible sources of power.

Acknowledgments The authors would like to thank the editors and referees for very helpful comments during the preparation of this paper.

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335 tional systems beyond the Turing limit, European Association for Theoretical Computer Science Bulletin, 85:181-189, 2005. 17. M. Davis, Computability and Unsolvability, Dover Publications, 1958. 18. M. Davis, The Undecidable, Dover Publications, 2004. 19. M. Davis, The Myth of Hypercomputation, Alan Turing: Life and Legacy of a Great Thinker, Springer, 2004. 20. R. FitzHugh, Impulses and physiological states in models of nerve membrane, Biophysical Journal, 1:445466, 1961. 21. W. Gerstner, R. Kempter, J. van Hemmen and H. Wagner, A neuronal learni n g rule for sub-millisecond temporal coding, Nature, 383, 76-78, 1996. 22. K. Godel, Some Basic Theorems on the Foundations of Mathematics and Their Implications, in Feferman, et al., 1995, 304-323, 1951. 23. A.L. Hodgkin, A.F. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve, J. Physiol. 117 500544, 1952. 24. J.J. Hopfield, Pattern recognition computation using action potential timing f o r stimulus representation, Nature, 376, 33-36, 1995. 25. J.J. Hopfield, Neural networks and physical systems with emergent collective computational abilities, Proc. Natl. Acad. Sci. USA 79 25542558, 1982. 26. K. Judd, K. Aihara, Pulse Propagation Networks: A Neural Network Model That Uses Temporal Coding by Action Potentials, Neural Networks, Vol. 6, pp. 203-215, 1993 27. S.C. Kleene, Representation of events in nerve nets and finite automata. In C.E. Shannon and J. McCarthy, editors, Automata Studies, pages 3-42. Princeton University Press, Princeton, NJ, 1956. 28. C. Koch, Biophysics of Computation: Information Processing in Single Neurons, Oxford University Press, 1999. 29. JR. Lucas, Minds, Machines and Gdel. Philosophy, 36:112-127, 1961. 30. W. Maass, R. A. Legenstein, and N. Bertschinger. Methods for estimating the computational power and generalization capability of neural microcircuits. In L. K. Saul, Y. Weiss, and L. Bottou, editors, Advances in Neural Information Processing Systems, volume 17, pages 865-872. MIT Press, 2005. 31. W. Maass, Lower bounds for the computational power of spiking neurons. Neural Computation, 8:140, 1996. 32. W. Maass. O n the computational power of neural microcircuit models: Pointers to the literature. In Jos R. Dorronsoro, editor, Proc. of the International Conference on Artificial Neural Networks - ICANN 2002, volume 2415 of Lecture Notes in Computer Science, pages 254-256. Springer, 2002. 33. W. Maass. Neural computation: a research topic for theoretical computer science? Some thoughts and pointers. In Rozenberg G., Salomaa A., and Paun G., editors, Current Trends in Theoretical Computer Science, Entering the 21th Century, pages 680-690. World Scientific Publishing, 2001. 34. W. Maass. Neural computation: a research topic for theoretical computer science? Some thoughts and pointers. In Bulletin of the European Association for Theoretical Computer Science (EATCS), volume 72, pages 149-158, 2000. 35. W. Maass and C. Bishop (Eds), Pulsed Neural Networks, MIT Press 1999.

336 36. W. Maass Networks of Spiking Neurons: The Third Generation of Neural Network Models, Neural Networks, Vol. 10, No. 9, pp. 1659-1671, 1997. 37. T. Natschia, W. Maass, Dynamics of information and emergent computation in generic neural microcircuit models, Neural Networks 18, 13011308, 2005. 38. W. Maass, T. Natschlager, and H. Markram, Real-time computing without stable states: A new framework for neural computation based on perturbations, In Neural Computation, volume 14, pages 25312560, 2002. 39. W. Maass, and P. Orponen, On the effect of analog noise on discrete-time analog computations, Advances in Neural Information Processing 11 Systems 9, 1997, 218224; journal version: Neural Computation 10, 1998. detailed version see http://www.math.jyu.fi/?orponen/papers/noisyac.ps . 40. J . Nagumo, S. Arimoto, and S. Yoshizawa, An active pulse transmission line simulating nerve axon. Proceedings of the IRE, 50:20612070, 1962. 41. W.S. McCulloch and W. Pitts. A logical calculus of the ideas immanent in nervous activity’, Bulletin of Mathematical Biophysics, 5:115-133, 1943. 42. C. Moore, Recursion theory on real-time language recognition by analog computers. Theoretical Computer Science, 162:23-44, 1996. 43. Jerzy Mycka. mu-Recursion and infinite limits, Theoretical Computer Science, 302:123-133, 2003. 44. M. Minsky, Computation: Finite and Infinite Machines, Prentice Hall, New york, London, Toronto, 1967. 45. P. Odifreddi, Classical Recursion theory. Elsevier, 1989. 46. R. Penrose, The Emperor’s New Mind Concerning Computers, Minds, and the Laws of Physics, Oxford: Oxford University Press, 1989. 47. R. Penrose, Shadows of the Mind: A Search for the Missing Science of Consciousness, Oxford: Oxford University Press, 1994. 48. E. Post, “Recursively enumerable sets of positive integers and their decision problems”, bulletin of the American Mathematical Society, 50:284-316. 49. H. Putnam, Renewing Philosophy, Cambridge, MA: Harvard University Press, 1992. 50. H. Putnam, W a l and EWOT Predicates and the Solution of a Problem of Mostowski, Journal of Symbolic Logic 30, pp. 4957, 1965. 51. F. Rieke, D. Warland, R.R. von Steveninck, and W. Bialek, Spikes: Exploring the Neural Code, MIT Press, Cambridge, USA, 1997. 52. J.R. Searle, Minds, Brains, and Programs, Behavioral and Brain Sciences, 3, 417-457, 1980. 53. T.J. Sejnowski. Neural pulse coding. In W. Maass and C.M. Bishop, editors, Pulsed Neural Networks, The MIT Press, 1999. 54. Daniel Silva Graa. Some recent developments on Shannon’s General Purpose Analog Computer, Mathematical Logic Quarterly, 50(4-5):473-485, 2004. 55. W. Singer, Neuronal synchrony: A versatile code for the definition of relations, Neuron, 24, 49-65, 1999. 56. W. Softky, Simple codes versus eficient codes, Cum. Opin. Neurobiol., 5, 239-247, 1995. 57. C. Shannon, Mathematical theory of the differential analyser. J. Math. Phys. MIT, 20:337-354, 1941.

337 58. H.T. Siegelmann, Computation Beyond the T u r i n g Limit, Science, 268:545548, April 28 1995. 59. H.T. Siegelmann, Neural Networks and Analog Computation, beyond the Turing Limit, Birkhauser, 1999. 60. H.T. Siegelmann and E.D. Sontag, Analog computation via neural networks, Theoretical Computer Science, 131:331-360, 1994. 61. H.T. Siegelmann and E.D. Sontag, O n computational power of neural networks, Journal of Computer and Systems Sciences, 50( 1):132.150, 1995. 62. Daniel Silva Graa and Jose Felix Costa, Analog computers and recursive functions over the reals, Journal of Complexity, 19(5):644-664, 2003. 63. M. Stannett. X-machines and the halting problem: Building a super Turing machine, Formal Aspects of Computing 2, 4, 331-341, 1990. 64. W. Rall, Theoretical significance of dendritic tree for input-output relation, in: R.F. Reiss (Ed.), Neural Theory and Modeling, Standford University Press, Stanford, pp. 73-97, 1964. 65. Christof Teuscher and Moshe Sipper, Communications of the ACM, Vol. 45, No. 8, 2002. 66. S. Thorpe, F. Fize, and C. Marlot, Speed ofprocessing in the human visual system, Nature, 381, 520-522, 1996. 67. A.M. Turing, O n computable numbers, with an application to the Entscheidungsproblem, Proc. London Math. SOC.42 (2) (1936) 230-265. A correction in 43 (1937) 544-546 (reprinted in: M. Davis, (Ed.). The Undecidable: Basic Papers on Undecidable Propositions. Unsolvable Problems and Computable Functions, Raven, New York, 1965, pp. 115-154) Page numbers refer to the 1965 edition. 68. J. Thomson, 1954-55, Tasks and Super-Tasks, Analysis, XV, pp. 1-13; reprinted in Salmon, 1970. 69. A.M. Turing, Systems of logic based o n ordinals, Proc. London Math. SOC. 45 (2) (1939) 161-228 (reprinted in: M. Davis (Ed.). The Undecidable: Basic Papers on Undecidable Propositions. Unsolvable Problems and Computable Functions, Raven, New York, pp. 154-222, 1995. 70. K. Wagner and T.M. Slagle. Optical competitive learning with V L S I liquidcrystal winner-take-all modulators. Applied Optics, 32(8):14081435, 1993. 71. P. Wegner and D. Goldin, Interaction, Computability, and Church’s Thesis, University of Massachusetts at Boston, 1999. 72. J. Wiedermann and J. van Leeuwen, Emergence of a Super-Turing Computational Potential in Artificial Living Systems, J. Kelemen and P. Sosik (Eds.): ECAL 2001, LNAI 2159, pp. 5565, Springer-Verlag Berlin Heidelberg, 2001.

HOW DOES COMPLEX MATHEMATICAL THEORY ARISE? PHYLOGENETIC AND CULTURAL ORIGINS OF ALGEBRA

HELEN DE CRUZ Centre for Logic and Philosophy of Science Vrije Universiteit Brussel Pleinlaan 2, 1050 Brussels, Belgium E-mail: [email protected] Algebra has emergent properties that are neither found in the cultural context in which mathematicians work, nor in the evolved cognitive abilities for m a t h e matical thought that enable it. In this paper, I argue that an externalization of mathematical operations in a consistent symbolic notation system is a prerequisite for these emergent properties. In particular, externalism allows mathematicians t o perform operations that would be impossible in the mind alone. By comparing the development of algebra in three distinct historical cultural settings-China, the medieval Islamic world and early modern Europe-I demonstrate that such an active externalism requires specific cultural conditions, including a metaphysical view of the world compatible with science, a notation system that enables the symbolic notation of operations, and the ontological viewpoint that mathematics is a human endeavour. I discuss how extending mathematical operations from the brain into the world gives algebra a degree of autonomy that is impossible to achieve were it performed in the mind alone.

1. What Would Mathematics without Culture Look Like?

Even a concept as seemingly simple as the number two is a highly abstract representation. There is no obvious perceptual similarity between two cows and two bicycles, and yet both collections share the concept of ‘twoness’. A large number of experimental studies nevertheless indicate that both nonhuman animals and human infants have a robust understanding of number (see Ref. 1 for an overview). For example, when presented with two different numerosities of food-items, red-backed salamanders go for the larger quantity 2 . Human infants, likewise, can discriminate between small and larger collections (e.g. two versus three, eight versus sixteen)-even when non-numerical variables such as cumulative surface area are controlled for They can solve simple addition and subtraction problems involving both

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small and larger quantities Recently, single cell recordings in rhesus monkeys have identified number-sensitive neurons-individual neurons that respond only to changes in number, while remaining insensitive to changes in shape or size. Each neuron is tuned to a preferred quantity: a neuron preferentially firing at, say two, will fire a bit less at one or three, and even less when observing higher quantities. These neural tuning curves become broader as quantities increase. Numerosities are thus not represented as a linear mental number line, but more as a logarithmic ruler, in which the psychological distance between one and two is considerably greater than between fifty and one hundred. Yet, mathematical systems without exact number representations can only capture the most rudimentary of numerical relationships. Evidence for this claim comes from two Amazonian cultures, the Pirahii and the Munduruk6 7, where exact number words do not exist, such that the word for ‘one’ can mean ‘two’ and vice versa. People in these cultures cannot even discriminate a box with three fish from a box with four fish painted on it. They do not possess a counting routine or any other cultural tool which helps them to construct positive integer values. Conversely, despite extensive training, no non-human animal has yet been able to learn complex mathematical operations, such as exact positive integer representations. One long-term training programme extending over twenty years involved teaching Ai, a female chimpanzee, to understand and produce Arabic digits Ai never managed to count more than nine items, and never generalized to the counting procedure that children master with ease. These lines of evidence combined suggest that although our mathematical abilities build upon an evolved number sense which we share with other animals, they are clearly more than that. It is as yet unclear how this cognitive adaptation can account for the vast proliferation and complexity of cultural mathematical concepts. In this paper, I will argue that complex mathematical theory can emerge if humans extend their minds into a symbolic notation system. Using algebra as a case-study, I will demonstrate how humans can overcome their cognitive limitations by externalizing operations that are difficult or impossible to perform in the mind alone. I begin by outlining which cognitive processes are involved when people learn and perform algebra. I show how humans combine several evolved specialized neural circuits to solve equations, but that learning algebra still requires synaptic rewiring. I then go on to explain how cultural factors, including religion, symbolic notation and philosophy influence the development and level of abstraction in alge-

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braic systems in China, the Islamic world and Europe. I argue that this development critically depends upon the elaboration of a symbolic notation system, and the assumption that mathematics is a human endeavour, which can be improved by individual mathematicians. Finally, I discuss how mathematics can have emergent properties as a result of this externalism. Mathematics may be a hybrid system of knowledge, in that it contains operations that are performable in the head, and operations that can only exist as symbols.

2. Cognitive Processes Involved in Learning and Performing Algebra Classical algebra is a set of procedures and rules for solving equations with one or more unknowns. How does one acquire this procedural knowledge? It is important to note that the transmission of a concept such as ‘democracy’ does not entail that this concept is downloaded from one brain to another: rather, when I hear about democracy I reconstruct it in my own mind, and it is quite likely that internal representations of democracy are not identical ’. Some concepts are more easily transmitted than others because they require minimal cultural input-in such cases the rest of the concept is ’filled in’ by the recipient’s tacit assumptions. For example, someone who learns that a zombie does not move according to his own free will tacitly assumes that such a being does obey the laws of physics. Thus, only the counterintuitive part of this concept-a violation of intuitions about agency-needs to be transmitted lo. Given the minimal transmission requirements for such religious concepts, it is not surprising that they should have evolved many times independently, occurring throughout a wide array of physical and cultural settings. Neither their birth nor persistence seems to require exceptional cultural conditions l l . In stark contrast to religious concepts, algebra is non-intuitive: it requires a co-optation of evolved cognitive strategies which are normally not deployed together. This results in a new system with emergent properties not present in any of the subsystems. We can observe this in other domains as well: hunter-gatherers for instance, track down animals using a combination of several domains of knowledge outside of biology, including physics (like examining spoor and damage the animal caused to plants) and psychology (imaging what one would do if one were in the position of the prey) 12. As a result, the emergent properties of tracking enable the hunter to work out solutions which would have been impossible in each of

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the subsystems. The drawback for these enhanced cognitive abilities is that such skills are typically hard to learn. Male hunter-gatherers start learning to track when they are adolescents, but only gain expertise-as measured by their return-rates-when they are well in their thirties 13. Likewise, fMRI studies (e.g. Ref. 14) found that people use different brain circuits when they solve equations (Fig. 1):

i I

--.----

mantenor cingulatecortex bilateral horizontalsegment of the intraparietal sulcu! bilateralposteriorsuperior parietallobe

Figure 1. Brain areas involved in solving equations (only left hemisphere shown)

the anterior cingulate cortex, which is otherwise active when people reflect on other people's mental states l5 e the intraparietal sulci, which have been implicated in brain imaging studies involving number l6 e the posterior parietal cortex, which is normally activated in visuospatial tasks, including spatial working memory and attention orienting l'. e

One study investigating the difference between mathematical strategies in normal and in precocious teenagers found that the latter deployed spatial skills to solve algebraic problems by diagramming important relationships apparent in the problem 18. Thus, solving equations depends on a successful co-optation of several brain areas which are normally involved in ecologically relevant tasks such as detecting numerosities and assessing spatial relationships. Perhaps the most surprising finding is that after a learning period adolescents exhibited alterations in their cerebral blood flow when solving equations 19. In other words, the brain rewires its synaptic connections in response to the task-demands. Thus, we cannot explain cultural

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mathematical concepts solely as a result of the combination of specialized neural circuits, since learning these produces permanent synaptic changes in a few well-restricted areas of the brain.

3. Cultural Transmission of Algebra in Context

Like other domains of human culture, algebra has emergent properties which cannot be traced back t o the innate cognitive architecture of our brains. Human culture is cumulative, unlike social traditions in non-human species. Our cultural systems build and elaborate upon the inventions of previous generations, which has resulted in a vast proliferation of artefact types, customs, religious beliefs and social systems that vary across cultures. Why only humans can benefit from this ratchet effect of cultural evolution is not entirely clear. One proposal, the extended mind thesis, formulated by Clark and Chalmers 2o is that humans extend thought processes into the external world. External memory devices such as books or electronically stored documents, instruments such as computers or nautical slide rulers, or simply pen and paper, all serve to extend computations beyond the brain. Clearly, the use of external media enables us t o accumulate information beyond the scope of the individual memory. Moreover, working with external symbolic media may constitute an epistemic act in itself. Some actions performed by use of external media solve problems more easily and reliably than if they had been solved in the mind alone. Take the computer game Tetris: physically rotating the two-dimensional blocks by means of a keyboard in order to fit them in the slots proves far more efficient than mental rotation 21. Note that, in all these cases, cognition only takes place within the human brain 22. The use of external media makes computational solutions possible that could not have been reached without them. In mathematics, a positional system such as the Hindu numerals renders multiplications with large numbers easy and transparent, while multiplication in non-positional systems such as the Roman numerals is a far more daunting task. As I shall demonstrate, the absence or presence of symbols that represent operations influences the degree of abstraction within specific mathematical systems. Only in early modern Europe did various cultural conditions shape the possibility of a consistent externalism of mathematical ideas.

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3.1. Chinese Algebra Chinese algebra first bloomed around the beginning of the Common Era with the anonymous Nine Chapters on the Mathematical Arts (Chiu Chang Suan Shu). It was organized around practical problems of public services, such as taxation and digging canals 23. During the T’ang dynasty (618906), China exhibited an unusual openness to foreign influences, welcoming Arab and Indian scholars. Thanks to this influx, by the end of the Sung dynasty (900-1279), Chinese algebra had attained a level that would only be surpassed in Europe in the course of the eighteenth century. During the Ming dynasty (1368-1644), as a reaction to the Mongol invasions, indigenous culture and values were restored. This led to a stagnation and decline in all sciences. Finally, during the Qing dynasty (1644-1911), there was an increase in influence from European mathematics, which led to the demise of Chinese algebra and other mathematical systems 24. Why did Chinese algebra come to a standstill, and why was it eventually abandoned in favour of Western algebra? A possible factor is the notation system, shown in Fig. 2. To represent a system of equations, counting rods were arranged in such a way that each column was assigned to each equation, and each row corresponded to the coefficients of each unknown. This in turn promoted the invention of matrix methods to solve simultaneous linear equations and higher-order equations. Figure 2, for example, represents the following equations:

{

+

2~ - 3y 82 = 32 - 6 ~- 2y - z = 6 2 3x 21y - 3 z = 0

+

However, these rods were less useful to express general abstract rules other than actual calculations, which preserved the concreteness of Chinese mathematics 23. Consequently, Chinese algebra textbooks never attempted to give an abstract formulation of a general rule, but presented examples that served as paradigms to solve similar problems 2 5 . Next to this, the venerable status of the Nine Chapters on the Mathematical Arts impeded further progress. Numerous mathematicians wrote commentaries on it, and as time went by commentaries on those commentaries, and even those who produced original work felt obliged to extensively refer to it. This attitude stems from the Confucian point of view that only the wise sages of the past have attained true wisdom; it was the duty of aspiring scholars to emulate their mental state 26. The high status of mathematicians gradually eroded as mathematics came to be perceived as a diligent and unquestioning appli-

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111 II

1

coefficients

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I ?quations Figure 2. Representing simultaneous equations with counting rods. Red rods (here shown in grey) indicate positive coefficients, black rods negative coefficients. Redrawn, with permission, from Ref. 23, p. 146, Fig. 4.6.

cation of ancient wisdom, rather than an ongoing creative process 23. Chinese mathematics never became an autonomous discipline: mathematicians remained technical experts dealing with chronology, finances, taxation, architecture, and the military 24.

3.2. Islamic Algebra Islamic algebra developed as a combination of and elaboration upon cultural inventions borrowed from other traditions, notably Greek geometry, Indian numerals and arithmetic, and Nestorian astronomy. This was possible because of the patronage of the Abbasid caliphate (758-1258), centred in Baghdad. It commissioned translations of major works by Aristotle, Galen, and many Persian and Indian scholars into Arabic. This knowledge became appropriated and reconstructed in Islamic mathematics 27. It built upon a combination of Greek geometric visualization and ‘proofs’, and Indian and Chinese arithmetic rules to solve equations quickly and efficiently. Through this eclectic approach, Arabian mathematicians could formulate solutions that they could not find through arithmetic alone. When, for example, Omar Khayyhrn (1048-1123) failed to find an arithmetic solution to cubic

345 equations, he resorted to a geometric method of intersecting cones to solve them 23. After the fourteenth century, Islamic mathematics and other sciences apparently stagnated and declined. A likely cause for this dwindling was the growing incompatibility between Islamic metaphysics and science. According to the Qu’ran, the world is governed by Allah’s uninterrupted control of all events. This line of thought inhibits the possibility of a world governed by rational, knowable and coherent laws of nature, being a principle of Greek philosophy. Nevertheless, between the eighth and eleventh century, Islam did adopt scientific thought in the form of philosophy, astronomy, medicine and mathematics. However, influential theologians such as al-Ash’ari (873-935) and al-Ghazali (1058-1111) questioned the validity of Greek philosophy which they saw as a menace to religious orthodoxy. They became committed to Islamic occasionalism. According to this doctrine, the basic building blocks of nature are individual indivisible ‘atoms’, each of which lasts only a moment and then disappears. Allah creates the world anew each moment of time, thereby giving it pattern and persistence. The world by itself cannot exist without this continuous creation and is therefore not ruled by laws of nature. Such a worldview is deeply incompatible with the basic assumptions of science that the world is governed by rational, coherent and orderly laws 28. Mathematics and its application in astronomy and other branches of science did not escape this devastating attack. The fourteenth century theologian al-Iji wrote that “we can disregard mathematical entities as they are more tenuous than a spider’s web”. From the tenth century onwards, occasionalism became the dominant orthodoxyincidentally, it still is. Hampered by religious orthodoxy, mathematics and other sciences could no longer flourish, with the exception of some applied mathematics that had highly political or religious importance, such as the construction of a round cupola on the square basis of mausoleums for political rulers, or the calculation of the qibla, the relative position of Mecca 27. Like in China, mathematics never became an autonomous discipline, but remained ancillary to engineering, cartography, optics and geography.

3.3. European Algebra

European algebra starts in the late thirteenth century commercial cities of the Catalonian, Provencal and northern Italian regions among mathematicians with a background in teaching commercial calculation methods. However, while showing substantial influence from Islamic mathematics, this algebra went beyond it in its ambition to solve irreducible higher-degree

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problems, starting with Jacopo da Firenze’s Tractatus algorismi (1307) 29. Since the seventeenth century, mathematicians began to work at increasingly abstract problems, including algebraic series and the calculus, culminating in collective conflicts among synthesizing schools with rival programs on the foundations and nature of mathematics. A disparate number of cultural factors have jointly contributed to this evolution. In contrast to Islamic metaphysics, Christian metaphysics endorsed the notions of natural laws and causality. The Scholastics saw evidence of God’s harmonious creation, fully in line with Plato’s system of nature, as a system of causal necessity. The metaphor of this machina mundi is ubiquitous in their writings. The arrival of Arabic translations of Aristotle in the thirteenth century gave this enthusiasm for naturalistic enquiry another powerful boost. Although modern science has traditionally been seen as a child of the Enlightenment, this development has actually been more gradual. In particular, the emergence of Nominalism within Scholasticism in the second half of the thirteenth century, was an important factor in the development of science. The Nominalists introduced empirical investigation, arguing that we can only discover the world by observation 30. Medieval universities institutionalized philosophical enquiry, making it acceptable that knowledge acquired through scientific investigation, became a valid source of knowledge next to the Scriptures 28. Thus, seventeenth century science can be seen as a second phase in an intellectual movement that began in the thirteenth century. Nevertheless, it was not until the seventeenth century that a fundamental shift in the perception of science enabled turnovers such as the transformation from Ptolemaic to Keplerian astronomy. Medieval mathematicians believed that mathematics had deep roots in classical Greece: there always remained an unchanging body of knowledge which could at best be preserved and at worst corrupted by contemporary scholars. Gradually, however, they came to realize that mathematicians could elaborate and even improve upon this body of knowledge. In this newly emerged paradigm, science was no longer perceived as ancient revealed knowledge, but as a human endeavour influenced by culture and circumstance. For example, John Wallis, a seventeenth century historian of mathematics, not only wrote that the Muslims had developed and improved classical texts, but proudly stated the further achievements of his contemporaries 31. European mathematicians thus became increasingly aware that they themselves created the levels of abstraction in mathematics. This realization led to many innovations, such as the introduction of imaginary numbers. Take i, which symbolizes an operation that cannot

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actually be performed, namely extracting a square root from a negative number. By denoting this operation it becomes possible to integrate roots from negative numbers in operations, thus vastly extending the scope of mathematics. Only European mathematics has consistently aimed at representing such opaque operations by a constant set of symbols. Indian mathematicians, for example, used the symbol 0 for both zero and the unknown in equations 23. Such ambiguities, or in the case of Chinese algebra, a lack of external symbols for operations, inevitably limit the degree of abstraction in mathematics.

4. The Extended Mind and the Autonomy of Mathematical

Thought External symbols can provide anchors for thoughts that are difficult to understand or represent. Without them, such thoughts would not survive long in the competition for attention and for cognitive resources that characterizes cultural evolution. Indeed, theoretical models that examine cultural representations from an epidemiological perspective predict that concepts that are hard to learn and hard to represent are quickly outcompeted in favour of intuitive or attention-grabbing ideas. These predictions have been experimentally confirmed in a variety of cultural settings. For example, story recall experiments illustrate that both Hindu 32 and Christian 33 college students do not intuitively think about their respective gods as their theologies require. Christians have difficulties representing God as an omniscient, omnipresent being and distort stories about him to fit intuitive expectations they have about normal people, like that he can only attend to one person or one event at the same time. External representations can anchor non-intuitive concepts on a more permanent basis. Seeing mathematical symbols facilitates the recall of previously stored knowledge. Once mathematical concepts are nested outside of the brain, their evolution and cultural transmission is less vulnerable to corruption by individual mathematicians or to competition from ideas that are easier to learn, that speak more to the imagination, or that pose less computational demands. They gain a degree of autonomy that would be impossible to attain were they represented in the mind alone. In such distributed cognitive systems 34 both the interaction of human minds with external notation systems, and the interaction of many mathematicians’ minds enhance the cognitive capacities of the brain. The consistent striving of European mathematicians to externalize op-

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erations has given mathematics a unique representational power. This ‘unreasonable’ efficiency of mathematics as an epistemic tool is, I argue, an emergent property of its hybrid structure. Since mathematicians routinely express operations that can never be performed in the mind alone (e.g. a negative number, which results from the subtraction of a larger number from a smaller one), mathematics can convey a range of ideas that are actually opaque. Let us examine this distinction between transparent and opaque more closely. A transparent concept is one to which we have semantic access; we intuitively grasp its meaning. In mathematics, Arabic numerals which denote positive integers (1, 2, 3, . . .) are transparent concepts. Studies of brain activation in adults and even five-year-olds have shown that a number comparison task with Arabic numerals (say 5 and 3) activates the same brain areas as one that involves arrays of dots (say five dots and three dots)--even the speed of computation is identical 35. Thus, the brain immediately translates a positive integer into a mental representation of its quantity. In contrast, when confronted with i or -3, no such translation takes place, because these operations cannot actually be performed in the mind (it is impossible to imagine a negative quantity). The only way to represent them is through an external set of symbols, the meaning of which remains semantically inaccessible to us. Through this externalization, we expand our representational abilities by delegating operations that are impossible to perform in the brain to our external environment, by denoting them through a symbolic system. Mathematical concepts, in themselves a hybrid structure composed of transparent and non-transparent symbolic notations, can in its turn become an epistemic tool. We denote processes in nature through mathematical formulations, because the latter have no clear semantic content. Therefore, identical functions and equations can be used in disparate contexts, including physics, biology, and anthropology.

5. Conclusion

In this paper, I have argued that algebra has emergent properties, resulting from the co-optation of evolved mental abilities and the externalisation of operations in a symbolic system. Such a system needs extensive cultural support, both from the metaphysics of the culture it belongs to and the ontological status of mathematics. It is crucial for mathematics and other sciences that a metaphysical view is endorsed in which the world is governed by causal laws, and in which humans are capable of acquiring reliable knowledge of the world. Once mathematicians realized their practice is a

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human endeavour, and that they can improve it, they could create higher levels of abstraction by symbolically representing operations. Early modern European mathematics was not superior t o its Chinese and Arabian contemporaries-rather, it could evolve because cultural conditions were more favourable.

Acknowledgements I gratefully acknowledge helpful comments by Johan De Smedt and Jean Paul Van Bendegem on a n earlier draft. This research is supported by grant OZR916BOF from the Free University of Brussels.

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