Towards a Dynamic Methodology of Science Author(s): Aharon Kantorovich Source: Erkenntnis (1975-), Vol. 14, No. 3 (Nov.,...

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Kantorovich Aharon

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Towards a Dynamic Methodology of Science Author(s): Aharon Kantorovich Source: Erkenntnis (1975-), Vol. 14, No. 3 (Nov., 1979), pp. 251-273 Published by: Springer Stable URL: http://www.jstor.org/stable/20010666 . Accessed: 20/01/2011 06:34 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at . http://www.jstor.org/action/showPublisher?publisherCode=springer. . Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected].

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KANTOROVICH

AHARON

A DYNAMIC

TOWARDS

1.

OF SCIENCE

METHODOLOGY

INTRODUCTION

of science treat theories mainly as static entities. philosophies deal with the degree of support or con? Justificationist methodologies firmation granted to a theory by empirical data, where the methodological - a set of unit which stands vis-a-vis empirical data is a static theory

Orthodox

still assesses a theory as a The Popperian methodology propositions. static unit, granting it a degree of falsifiability and empirical content or a degree of corroboration. Popper makes, however, a big step towards a dynamic approach by stressing the process of theoretical change and growth, in which one theory is refuted and replaced by a better one, i.e. in corroborated by a theory which is richer in content and, in particular, content. Since his theories are static entities, there is only one category of change whereby one theory vanishes while another appears on the scientific scene. Thus, there is no room in the Popperian methodology

theoretical

Indeed, circular

possibilities where theories are modified but not refuted. if for example the first version of the Bohr atom model with electronic orbits is regarded just as a timeless set of propositions,

then we

should

for intermediate

orbits.

elliptic

say that itwas refuted and replaced by the new theory with But it would be a distortion of the facts to exhaust the

of such a theoretical significance change by the term "falsification". a as a initial the version of is often treated intentionally model Firstly, "naive" or a crude version open to improvements and changes. Secondly, even

to be a genuine description, scientists would be in the basic picture which remains invariant under the consecutive changes, and they would regard it more fruitful to view the process as a modification of their basic theory rather than as a replacement of one if it is considered

interested

frozen version by another. According to the Popperian approach, there is no category difference between a theoretical change such a& the above mentioned change in atomic theory and a radical change such as the replace? ment

of classical

found under

physics by relativistic and quantum theories; the same methodological roof of falsification.

Erkenntnis 14 (1979) 251-273. Copyright

?

1979 by D. Reidel

0165-0106/79/0143-0251 Publishing

Co.,

Dordrecht,

both

are

$02.30

Holland,

and Boston,

U.S.A.

252

KANTOROVICH

AHARON

Popper, who was inspired by the theoretical changes of large magnitude which took place in early twentieth century physics, requires every theo? retical change to be as daring as possible. But this does not accord with - in actual scientific practice. We know from the history of science particu? lar physical science that there are periods when the scientific community adopts a conservative policy rather than a radical policy. In such periods a radical departure to an from an established theory is tantamount or sheer metaphysics. unfounded methodology Popper's speculation or a theories held in these low to the scientific grants grade periods, of them as metaphysical. Thus, sticking to the framework categorizes at the end of the nine? classical physics and the mechanistic conceptions teenth century is regarded by Popper as a metaphysical behavior since the theories became at that time unfalsifiable. nowadays Physicists agree with this. But what about the eighteenth century? Would that even then physicists should have made attempts to Popper maintain depart from Newton's theory, for example? Probably not, but there is no which will tell us when to make a daring clear criterion in his methodology classical

would

with our present theoretical frame? not account for the widely held does Thus, Popper's methodology physicist's belief that it would not have been possible at all to arrive at the theoretical

leap and when

to continue

work.

theoretical scientific

the of early twentieth century physics without of the two preceding centuries. And, in general, those the way to a big theoretical patterns of science in which

discoveries endeavor

evolutionary is paved by a preceding

change are

not

accounted

conservative

era of scientific

investigation

for.

science into focus by introducing brought the non-revolutionary he treats the science and of normal paradigm. However, concepts a deliberate as theoretical activity which takes place in normal science Kuhn

his

effort to keep the paradigm intact, and does not give methodological criteria for appraising it as a theoretical change. Lakatos, who carries some such an into the Popperian scheme, makes aspects of Kuhn's message new methodological entity appraisal explicitly. He creates for this purpose a a scientific research program, which replaces the static one research pro? from theory as the basic unit for appraisal. The passage revolution, whereas the gram to another is roughly parallel to a Kuhnian the to within one another from sequence which constitutes a passage theory

which

he calls

research program might

be viewed

as a moderate

or "normal"

theoretical

DYNAMIC

change.

OF SCIENCE

METHODOLOGY

In the latter case a hard

253

is parallel to a Kuhnian pictures the normal scientist as

core, which

paradigm, remains invariant. While Kuhn interest is to defend the paradigm merely a puzzle solver, whose main with takes a Popperian clashes Lakatos attitude empirical data, against in an according to which the scientist is willing to learn from experience the framework active way, namely to change his theories, within of a research program or in a normal science setting. Thus, Lakatos admits in distinct types of theoretical his scheme two methodologically change: or research pro? radical changes in which major theories are overthrown grams are replaced, and paradigm-preserving changes. Both Kuhn's and Lakatos' approaches reflect the Duhem-Quine which says that in a theoretical system any element can be protected are made

in other elements

refutation, provided adjustments The theoretical paradigm or hard core are such elements.

thesis from

of the system. But according

to Lakatos, the presence of a hard core does not ensure continuity in the research program. Hence, he introduces the positive heuristic, which has so to speak, for developing the task of supplying the building materials, the program

and adjusting

2.

McMullin

it to empirical

mcmullin's

data.

dynamic

takes one step further theory in a methodological

approach

toward

the concept of a employing In dynamic [5]. doing this he departs from the falsificationist tradition, with which Lakatos's method? radically ology was tied, and suggests that a research program be treated in an holistic manner as a unified and continuously developing body of knowl? context

edge, rather than as a series of successive theories each of which is falsified in its turn, giving way to its successor which is also doomed to be falsified eventually. He naturally suggests then to use the term "theory" instead of "research program", where theory is interpreted not in the logicist sense of a timeless set of propositions but as a dynamical-historical entity. Thus, the falsificationist is eliminated while Popper's idea of the terminology scientific knowledge is retained. The concept of a research a or now has a right to existence of its own; it program dynamic theory is not merely recruited for the sake of overcoming the difficulties which the falsificationist those which stem from the faces, notably approach growth

of

Duhem-Quine

thesis. However,

McMullin

questions

Lakatos'

concepts

of

254

AHARON

KANTOROVICH

hard core and positive heuristic ment of novelty or the criterion

and suggests that one replace the require? of progressiveness which Lakatos employs of a theory. Let us turn to his objections and

by the criterion of fertility suggestions one by one. Lakatos assigns to the hard core those parts of the theory under evalua? tion which are protected from falsification, the protective whereas belt which undergoes changes includes the auxiliary assumptions or peripheral

around the core of the theory. If so, components which are constructed asks McMullin, in what sense is it the hard core or the main part of the in the process of testing and growth? Thus, he theory which participates suggests, first of all, that the idea of a sharp distinction between "hard" and "soft" parts be abandoned since there is a whole spectrum of "hardness" or "softness" degrees. But the hard side of the spectrum includes not the main

of the theory but the established background theories such as and measurement theories (e.g. optics and radio-optics in or theories mechanics and astrophysics) general (quantum special relativity in contemporary nuclear and particle physics), whereas the atomic, body observation

when a theory is developing theory belongs to the softer edge. Namely, tested and evaluated, only the measurement and observational theories, the highly confirmed general theories and some specific auxiliary theoreti? are tentatively kept unquestionable. The fact that certain of the main theory remain nevertheless unchanged throughout the process might testify for their adequacy, but it does not necessarily mean

cal elements elements

that a decision was made

at the start of the process to protect them from refutation. Indeed, in a model for the growth of knowledge in a dynamic was we at which the methodo? elsewhere arrived theory developed [3], logical

conclusion

that a "good" is distin? theory or research program or theoretical growth, change, relative to empirical means a given amount of new it that for speaking

guished

by a slow

growth;

roughly data the

remains larger is the portion of the theory which or nearly invariant, the more the dynamic theory is progressive. shall later turn to this model and its implications for the appraisal of a

empirical invariant We

dynamic theory. The second problematic

concept Lakatos introduces in his scheme is the is assigned the task of integrating the research

heuristic, which program and assuring continuity about the origin of his positive positive

is not clear in its growth. But Lakatos heuristic. For one thing, it cannot be

DYNAMIC

METHODOLOGY

OF SCIENCE

255

the hard core or from the original theory-version Tx of the some as of Lakatos' formulations; research program, may be implied from

derived

from

of the program the development the case, claims McMullin, us to the logicist back be entirely predetermined by Tl9 bringing a "if the stand of appraising 'positive heuristic' single static theory, since a deductive analysis is entirely contained within the original hypothesis, if this were

would

test of the hypothesis and a systematic observational of the consequences other the On is needed" is all that them of hand, says (op. cit., p. 414). if the positive heuristic is only a "partially articulated" plan for McMullin, the research program, as Lakatos puts it in another place, constructing then the question is inwhat sense the program is an integrated whole which can be appraised as one unit?

3. POSITIVE HEURISTIC THEORETICAL

AND NON-RADICAL CHANGES

is making, let us introduce here the points McMullin in a research the following notation. A dynamic theory which is developed program will be symbolized by a function T which assumes the "value" In order

to discuss

theory) at time tt. That is, T refers to a in such a way that at certain points t{ dynamic entity which evolves more or sets less the it of definite time axis propositions produces along

Ti

(the

/th version

of

the

case of a research program or a dynamic In the prototype theory, is The first theory-version behind T there stands a model. 7\ which generated by the research program may be seen as a simplified or a crude version of the model, with some elements of the model remaining un? utilized explicitly. These elements constitute the "neutral analogy" - to use and Mary Hesse's terminology [2] which serves as a guide for modifying further developing T. Thus, the model generates a succession of theory Tl9 T29 ..., where at each step, more of the neutral analogy turns into positive or negative analogy. The neutral analogy serves as a source and as a guide for developing T, and therefore it may be regarded as a positive heuristic which is "contained within the original hypothesis"

versions

is the nature of this original hypothesis ? Is it just a well defined set of propositions re? which predetermines the evolution of the whole

But what

search program?

It seems that due to the presence

of the neutral

analogy

256

AHARON

KANTOROVICH

it cannot be regarded as such, since only further empirical data, on the one which is inspired by the original hand, plus the imagination of scientists but not always in an unequivocal and explicit way - on the hypothesis other hand, will determine the fate of the neutral analogy in subsequent of the system. We may identify our dynamic theory T with development what Mary Hesse calls (op. cit., p. 9) "modeli" (the subscript should not be confused with that which appears in Ti), namely the model minus the known

in addition to the positive includes, negative analogy, which the original hypothesis may be analogy, also the neutral analogy. Hence, looked at as an initial state of a dynamic entity capable of growth. The of growth come not only from the model potentialities standing in isola? is tion, but also from the background knowledge within which the model The background embedded. besides the includes, knowledge accepted theories and empirical data, partially articulated beliefs which are shared of the scientific community who produce the research by the members Thus it and metaphors related to the program. yields various associations cannot these be analyzed by the logicist's tools in the way a model, but well defined set of propositions can, e.g. by deducing testable predictions from

it at the outset.

We

take therefore

the theory-versions Tt to be the successive versions of each of which includes some neutral analogy - which

the "modeli", gradually diminishes along the sequence. The question which arises now is whether appraisal should be directed towards the model in conjunction or with the background the towards model. McMullin's knowledge only of a dynamic theory refers to the resources of the model, but these cannot be detached from the background and knowledge beliefs. Nevertheless, he claims that appraisal is directed towards the notion

of fertility

evolving theory which since the background

changes, namely towards the "modeli", does not participate in the dynamic knowledge of T's evolution (McMullin, op. cit., p. 422). In the scheme which undergoes

process was developed in [3] the dynamic theory T includes not only the main - but also the one which is subject to evaluation and modification theory elements of the background knowledge which are relevant to the main e.g. general theories which apply to the theoretical entities in the model, and auxiliary theories which apply to the system under investigation and enable the main theory to make observational predictions. However, theory:

since the background

knowledge

remains

invariant throughout

the process,

DYNAMIC

METHODOLOGY

and since appraisal is conditionalized upon is assessed evaluation which under theory

257

OF SCIENCE

its veracity, it is the particular - relative to the background

knowledge. A non-radical

theoretical change, in which a dynamic theory ismodified in the face of a new problematic empirical datum, is an inductive transition is the it is not only the which restricted by positive heuristic. Namely, data, but also the positive heuristic which restricts the number of possibili? ties for modifying T. Symbolically, the transition can be represented by or AT (where the A symbol denotes the formula: E&T^>T'9 E^> change). The fact that the premise for the inductive 'inference' includes not only the empirical datum but also the last version of T, indicates it is not just an induction which the continuity in the transition; is directed from empirical data to a theory, but from data to a change AT in the theoretical system - a system which endures in time. The formula E^+ AT encompasses, therefore, an important category of inductive inferences or theoretical changes. The complement category would include the case of radical theory-innovation which does not involve a modification in an existing explicit theory. In scientifically interesting cases the positive heuristic (e.g. the neutral relevant leaves more parts of the background analogy plus knowledge) than one possibility for modifying T. Thus, even if we have a well defined model, we have to decide what to do with the neutral analogy, since the data do not dictate a unique way to proceed. The inclusion of the criterion of simplicity in our positive heuristic will not help in this respect since it is not always

clear what is the simplest modification. We must conclude, even that if the heuristic is included in the initial theory therefore, positive version Tx it will not be a case of predetermination.

4.

ON

A PROBABILISTIC

MODEL

FOR

IN RESEARCH

THE

GROWTH

OF

KNOWLEDGE

PROGRAMS

In [3] a model was developed which dealt with the dynamic methodological facets of a research program or an evolving theory. In the framework of this scheme we will be able to formulate the issues raised by McMullin and suggest alternative answers to those he provides - especially to his notion of fertility. Let us therefore summarize the main features of the model.

258

AHARON

KANTOROVICH

a research program, as a dynamic system, is divided a two into theoretical components: methodologically component T as was specified above and an empirical or an observational component E which consists of the observational data accumulating in the system. At time ti9 In this scheme

statement E{. The dynamic theory T E is represented by the observational is ideally adjusted to E such that at each step E% is accounted for deductively by Ti9 i.e. T{ -> E{. The evolution of the system is described by two variables which measure

the prior probabilities of the two varying components. At time ti9 the variables acquire the values P(Tt) and P(E?). The probabilities are taken to be representative degrees of rational belief in T{ and E{ as at time t0 - before the program starts to evolve. At time t? the measured first theory-version data Ex Tx is set up to account for the observational contain elements which cannot be explained at the state of knowledge at t0. Thus P(E?) is inversely related to the degree of unexpectedness or novelty - of E1 relative to t0. P(TX) represents the degree of belief in Tx at t0. Hence, P?TJ and P{E?) are not the actual degrees of belief or at the degree of nearness of Tx and Ex time tx but measure expectedness to the state of knowledge at t0 and the same is true for every i. It is of the pre? assumed that the functions P?T) and P(E) are representative

which

vailing

beliefs

of the scientific

community

that is developing

the research -

or at least metaphysical background It is further assumed that within the representative of its leading members. content of framework of a given research program, when the ontological T does not change significantly - i.e. when the passage from Tx to Ti + 1 or a radical change - P(T) varies in an does not involve a revolutionary ~ 1? Ct(T). opposite direction to the empirical content Ct(T)9 e.g. P(T) The same assuption ismade for P(E). Thus P(E) decreases in time if there program

and

share

common

data. Again, of observational is an accumulation, but no withdrawal, at t0. are measured relative to the state of knowledge both contents in the the growth of knowledge Hence the two pairs of variables measure research

program.

context is, however, important variable in a methodological of T which is taken to be the actual probability the degree of confirmation the probability to the fundamental Bayesian assumption of T. According at t0; of Ti at time t{ equals the conditional probability P(Ti\Ex) measured The most

that is, at t{ the probability

of Et equals one and all probabilities

are obtained

DYNAMIC

by

conditionalization

METHODOLOGY

upon

it. Bayes'

259

OF SCIENCE theorem

relates

the posterior

= probability to the prior probabilities: P(Ti\Ei) P^T^?T^iE,). our ideal model, inwhich Tj-^jE'i for every/and the degree of confirmation of T as a function P(T)/P(E).

Now

the methodological

content

1, Tt) consequentlyP(??| of time is simply: C(T) = of the dynamic theory T is

appraisal the behavior of C(T)

by inspecting accomplished The case which interests us most

In =

as a function

of time.

is when

both T and E grow in empirical functions decrease. P(E) decreases

and thus both probability of the growth of observational data. P(T) decreases when in the of to T E the process adjusting theory-version T? is replaced by Ti + 1which has higher empirical content (e.g. when Ti + 1 is more precise than T? or applies to a wider range of phenomena). There is a limit case - when Ei + 1 can be accounted for by Tf and there is no need to change T. In that case P(E) decreases while P(T) remains because

constant, and consequently C(T) increases. The general case of progressive ness is just a generalization of the above "static" case; namely we identify a progressive step by the condition AC(T) > 0. In the case when both or both empirical contents increase, this condition decrease, probabilities for a given rate of empirical growth the rate of theoretical does not exceed a certain upper bound. That is, as can be simply

is achieved when growth

calculated from the formula C(T) = P(T)/P(E) (see [3]), AC > 0 if and < Ci, where Cx is the initial degree of confirmation. only if AP(T)/AP(E) In the limit case of static confirmation of a theory by its successful predic? tions the rate of growth of T is zero and the above condition is obeyed for a every Cx (since always Cx > 0). Even here T generally undergoes a but this is deductive transformation. transformation, content-preserving A deductive change is therefore a limit case of an inductive change. By an inductive change it is meant that new observational data "bring about" a theoretical change with 0. ^ ACt?T) It was shown in [3] that when one uses an additive measure for content one can interpret the above condition for as saying that the progressiveness rate of theoretical growth is smaller than the rate of empirical growth. Hence we might there is a "gain" in say that in a progressive move information. This gain is made possible due to the initial empirical theoretical discovery, i.e. the invention of Ti, which constitutes a big a theoretical in a which adds amount of jump, Popperian style, large

260

AHARON

KANTOROVICH

(untested) empirical content to T. The big initial jump results in a low can increase if initial value of C(T). Later on the degree of confirmation the program is progressive. sum up, our model provides an explication of Lakatos' concept of or non-ad hocness and connects it in certain circumstances progressiveness To

with

the notion

or the Popperian

of confirmation, scheme.

5. THE DYNAMICIST FERTILITY

a notion which

APPRAISAL

is foreign to the Lakatosian

OF THEORIES:

VS PROGRESSIVENESS

scheme it is evident that in contrary to what can be from McMullin's view we can handle the dynamic case by using implied this reflects the difference between his scheme and formal tools. Indeed,

From

the above

ours.

In the logicist approach it by measuring appraises

one picks up a proposition (a static theory) and or probability. But its degree of confirmation, by the logicist's focus on the goal of discovering

is dictated this approach true propositions truth and producing by science. Thus, the static theory of the logicist is an abstraction and is not empirically given; theories under evaluation are in a fluid state and very often no official version of a theory is made since truth can be attributed only is laid down. This abstraction and not to a dynamic process or an historical to a timeless proposition

entity. on the other hand, focuses his attention on additional The dynamicist, also in goals and norms of science ;he is interested not only in truth, but truth. According the way science approaches more and more comprehensive in these norms give science its special characteristics, to the dynamicist, of notions The from other disciplines which strive for truth. distinction and fertility are related to such possible norms. The AC progressiveness - is notion of progressiveness is our modified > 0 criterion - which related to the requirement of economy or non-ad hocness for theoretical norms of "good" dynamic science. But growth, which may be regarded as such a requirement can be applied only to dynamic entities such as theories or research programs which, similarly to static theories, are abstractions. it is therefore not a simple task to determine the boundaries Practically,

DYNAMIC

METHODOLOGY

OF SCIENCE

261

and scope of both dynamic and static theories and there is no preference in this respect of the logicist notion of a theory over the dynamicist's. The fact that a dynamic theory is not a linguistic entity but produces a time-ordered parameter

such entities, leads to the introduction of the time into our formalism. The formalism we use for explicating the set of

includes logicist tools for analyzing each proposi? dynamicist's approach i.e. deducing predictions from each Ti9 comparing tion in the sequence, with empirical data and calculating the degree of confirmation through theorem. On this is superimposed the time parameter. Hence, Bayes' instead of dealing with static logicist relations between statements such as E and T, we compare E(r) and T(f ) as functions of time. We are not only in the probability of T being true given E at a given moment of but also in the process of arriving at a probable theoretical statement. time, We can arrive at true or highly probable propositions about the world by our and to it.We can, theories waiting passively recording data, adjusting interested

on the other hand, and guide us where

invent a daring theory which will anticipate new data to look. In both ways we can arrive at highly probable but the latter way is generally regarded as more scientific. Thus

theories, the criterion or the norm which

requires theory-guided growth of knowledge or norm which will distinguish, e.g., between or phenomenological theories in physics and New?

in this sense is the criterion

epicycle astronomy or structural tonian mechanics McMullin's

criterion

theories of microphysics. of fertility seems to make the above

distinction. a "good" (fertile) theory as that which "successfully guides research over an extended period of time, not only in the sense of correct predictions providing (an ad hoc theory could do this) but also in the sense of imaginatively in the suggesting plausible modifications

McMullin

defines

that allowed predictions to be made original theory/model, modifications which would not have been directly deducible from the original hypothe? sis" (op. cit., p. 401). It seems, however, that this formulation of fertility excludes the methodological situation of a static theory - which is appraised - since it does not suggest or guide modifica? by its successful predictions tions and growth of T. In another place, McMullin suggests the use of instead of the criterion of novelty; fertility thus, an ideal static theory which

successful novel predictions would not be categorized produces as a "good" (fertile) theory. But, as we saw above, the pure deductive case is just a limit case, so that there is no need to replace the criterion of

262

AHARON

KANTOROVICH

- as we do in our scheme, where the AC > 0 novelty but to extend it condition applies also to the static case in which a growing body of data in T. is explained by only deductive manipulations of fertility, McMullin attempts also to clarify the in Lakatos' formulations with respect to the manner by which ambiguity the modifications in T are "guided" or "suggested". He introduces for With

notion

his

a good metaphor "allows the the concept of metaphor-, now the to work and guides it in certain directions". But imagination a we to when certain find criteria for shifted have is and deciding problem in T is "guided" or "suggested" modification by "the theory", or by the this purpose

in a metaphorical way and when the modification original hypothesis, done in an arbitrary manner. The above formulation also shares in common with Lakatos' notion a

to account

is of the

for shortcoming by not being able course T can its of In the situation. evolution, following methodological have some new predictions which agree with the data, but at the expense of an inflatory theoretical growth. That is, a given transition Tt-^ Ti + 1, progressiveness

may produce some successful new predictions, whereas Ti + i has a great content which is in principle testable (i.e. additional deal of additional for but untestable; example cosmological practically empirical content) or astrophysical be conclusively

theories which

have

some empirical

but can

predictions or making

tested only by performing experiments tions in the vicinity of a distant galaxy. Or an hypothesis has a prediction that can be tested only by endangering

observa?

inmedicine human

which

life. Or a

in particle physics which can be effectively tested only at extremely (or high energies which cannot be reached with the available technology Incases social and economic conditions!). indeed at the present political theory

like these the research program may be regarded as progressive or fertile even if the rate of theoretical if it leads to successful new predictions, growth exceeds the upper bound dictated by the rate of empirical growth such that C(T) decreases. Thus the criteria of fertility and progressiveness are too weak in this respect; they do not rule out the so-called "specula? do not keep track with empirical data. The rejection of of this sort is in accordance with the general practice speculative - in in physics periods of normal science. or On the other hand, the theory would not be regarded as progressive in T, fertile in the case when at each step of its evolution the modification

tive" theories which

theories

DYNAMIC

METHODOLOGY

OF SCIENCE

263

growth, follows the observation of new data or empirical in our modified growth. But such a development may be quite progressive sense if a small theoretical growth or a minor adjustment in T, can account for a large increase in the amount of data. Thus C(T) may increase even or the theoretical

if the change in T was made after the data was acquired. Here the criteria which demand the anticipation of some of fertility and progressiveness, new data, are too restrictive.

6. STABILITY

OF A THEORETICAL

SYSTEM

criterion of progressiveness saw, our quantitative implies that a research program or a dynamic theoretical system T, which accounts for, or explains, a growing body of data E, is progressive if the rate of theoreti?

As we

cal growth has an upper bound which is imposed by the rate of empirical a "good" methodo? growth. Thus, not as the logicists would maintain, occurs not situation when itself Tx only logical provides successful predic? tions, but, dynamic data. The

in general, when it constitutes system which accounts for an of a stable

notion

senses of non-ad

hocness:

sense. Maximal

covers two important and continuity - in their generalized in the pure deductive case of a theory

theoretical

novelty is attained

the basis for a relatively stable increasing amount of empirical system

stability are (T\) which remains with the same logical content, while its predictions in accord with new empirical data; this is the ideal case of non-ad hocness or - since the content of the novelty, and, of course, continuity theory remains is a generalization of this unchanged. The general case of progressiveness ideal case, where the invention of Tx initiates a phase in the evolution of the system in which a large amount of new empirical data is explained while T remains relatively stable. So the first sense of non-ad hocness which

is accounted

for is (generalized) non-ad hoc novelty. Secondly, a or some is which heuristic change change obeys positive retains continuity in the growth of T. Stability, indeed, implies continuity. Thus, we have here three facets of a "good" dynamic theory: an economy an economic of theoretical and consequently growth, explanation, theoretical

theoretical stability and fertility. The fertility metaphor can be used here in the following sense: If fertility which includes novelty), is attributed to the dynamic that it provides an explanation to a large amount of new

for a stable theory (in the wide sense, system T, it means observational data.

264

AHARON

KANTOROVICH

But for the system to retain its main features - such that it can be treated as a unified and continuous some entity, or an entity which possesses itmust be identity and not just a succession of loosely connected theories ?n a stable

state and not be "dragged" by the data, i.e. not undergo too large changes. Hence, only when T remains stable in the face of a growing E, we can attribute to it the property of fertility. the evolution of a dynamic theory, the dynamicist In describing views the sequence {Tf} as successive states of one dynamic system which under? goes changes but which endures through a certain period of time. Thus the unit, can also be described system, which is appraised as a methodological an or as an adapting system system, in terms of information-processing notions borrowed from general systems theory. We can view a fertile or theoretical system as a stable system, which growing relatively small changes. processes empirical information while undergoing Thus the desideratum of anchoring our growing factual knowledge in a stable an economically

body of theoretical knowledge can be regarded as one of the goals of science, criteria of increasing which confirmation, implies the methodological hocness. non-ad and theoretical economical However, stability growth must

follow or theoretical

a

of "spontaneous" phase which adds potential empirical

non-stable

discovery

inductive content

jump to the

system.

view of science was developed by E. Laszlo [4]. systems-theoretical an is to science his system, information-processing approach, According to an organismic system, which adapts to "nature". We are analogous in smaller units - dynamic theories which are sub? interested, however, systems of science. Thus, in this view, science is a network of interacting A

which adapt to nature, rather than one monolithic system. a information which comes not only Therefore, dynamic theory processes of the science directly from "nature", but also from other components subsystems

system adapts to its natural system. The process by which a theoretical in can be schematized and scientific environment by the flow diagram is inter? from "nature", which Fig. 1. The detector absorbs information theories includes observation preted and processed by the filter which from filter and the from data The measurement theories. and empirical the enters of science are stored in E. This information other components of the theoretical compo? comparator where it ismatched with predictions nent T. If a prediction of T is not in accord with E, or when no prediction

OF SCIENCE

METHODOLOGY

DYNAMIC

other components

265

of science

comparator

'nature"

effector

control

i Fig.

1.

of T can account which

The

formation, perform detector

diagram

of a dynamic

system

of

science.

for an element

sends a command

T cannot

flow

be confirmed

of E, a signal is emitted to the control to change T. If, on the other hand, a prediction of by E, since E does not include some relevant in?

or to a direction is sent to the effector to make an observation an experiment - the results of which are conveyed through the to E. Thus, empirical information is not passively absorbed from

"nature", but is a product of theory-guided experiments. It is assumed that the lifetime of the system it develops relatively autono? during dynamic when interactions other with mously subsystems of science are relatively weak. Thus, our unit, the theoretical system which learns from experience and which is put to test, is not the whole scientific knowledge system but a more restricted system which possesses an identity of its own and exhibits an holistic

behavior of during a limited period of time. The division or a as scientific discipline such science, particular physics, at a given in the scientific community period into subsystems has its counterpart which is divided into respective "invisible colleges", or specialized groups, who

develop such a group

or research programs; communication inside between quicker and effective than communication

these theories is much

or general conception groups. The partially articulated model, or metaphysical which the of a dynamic picture, inspires development - whether it be a concrete model or an abstract setting - is the theory

different

unifying glue for such a group. In order to describe the evolution

of the system, we have

to choose

266

AHARON

KANTOROVICH

P(E)

region of stability

- -

. P?T\ Fig.

2.

The

phase

space

of

the

and a typical system evolution.

trajectory

which

describes

its

appropriate system's variables. One choice which comes up in a methodo? logical context is the pair of content measures: Ct?T) and Ct(E)9 or the or The the space spun by pair prior probabilities. system's variables is the space of the system. Let us choose the (P(T), P(E)) plane as the phase space of our system. In this plane the system's evolution can be described to the successive states of the system by a set of points corresponding case The deductive limit will be (see [3]). represented by points which lie on

phase

a straight line parallel to the P(E) axis. When the system's evolution adheres to the prescriptions of our ideal model, then P(T?) < P(E?) and the system will move only within the corresponding half square (Fig. 2). The degree of confirmation of T at a given point (state) equals the slope of the line connecting the point to the origin; the smaller the angle is between the line and the diagonal, the higher is C(T) at this point. C(T) varies with the slope: from zero on the P(E) axis to one on the diagonal. In this way,

one can easily visualize the static and dynamic methodological of the system's movement in the phase space, namely the

implications degree of confirmation

at each point and its direction and rate of change. in the case where both Ct(T) and Ct(E) increase in time after the initial state where T? was set up to explain El9 namely that case when T becomes richer in content while accounting for the growing We

are interested

data. This represents for example the evolution of a body of observational is generated by a model, where more and more dynamic theory which neutral analogy is exploited to become positive analogy or when the model is extended. Also when

ad hoc corrections

are added

to T? and T becomes

DYNAMIC more

METHODOLOGY

267

OF SCIENCE

and more

complicated, Ct?T) may increase. Sometimes, are added to the theory and Ct?T) decreases parameters accurate in its predictions, theory later becomes more

however, but when

free the

Ct?T) finally in the the region of non-progressive development = as line and the the the C bounded Cx phase space triangle P(E) axis, by since C(T) decreases when the system moves from its initial state (P?Tx), increases. We

define

PiEx)) to any state represented by a point located in this region. The region in which C(T) line of pure deduction increases includes the vertical is enclosed between this line and the (static T) and the triangle which C = 1 and C = Ci lines; this region will be defined, therefore, as the region of progressive development. we want to characterize a stable behavior of the system, we must a vertical movement with static T as maximally stable under all regard definitions and to set a lower limit for stability. The choice of the C = Cx line as the demarcating line between stable and non-stable growth seems When

natural.

to the methodological It corresponds and progressiveness non-progressiveness. Using

demarcation

between

our system's variables it is bounded. The above definition

implies that the rate of growth of Ct?T) is in accordance with the notion of "region of stability" used in general systems theory; a region of stability is a set of states such that once the the system's system has entered into it it can never leave it namely, behavior under external disturbances are is such that its movements confined to that region. That is, the disturbance exerted on T by the inflow of empirical information E is still controlled by T which is not "dragged along" by E and which keeps its identity. A by-product of a stable evolu? tion under this definition is, therefore, an increasing credibility of T. Thus, if we conceive a dynamic explanation - i.e. the explanation of a growing E - as by a growing T implying a relative stability of the theoretical system as defined above, then it follows that a necessary condition for a dynamic is an increasing credibility. But this is one of the obvious explanation intuitive methodological i.e. that when a new unexpected requirements, datum is explained by a theory, then the theory's degree of credibility should rise. Thus, our characterization of dynamic explanation is consist? ent with

the two requirements: that a dynamic explanation should imply theoretical > and that it also 0. Hence the AC dynamic stability implies demarcation line between progressiveness and non-progressiveness is also a demarcation line between explanation and non-explanation.

268

AHARON 7.

RADICAL

VS.

KANTOROVICH

NON-RADICAL

CHANGES

AND

STRATIFIED

STABILITY The necessary condition for a dynamic explanation, that AC > 0, implies = 1 cannot explain a new un? that a theory T which has reached C(T) in has an infinite datum. any principle theory-version (Although expected number of predictions of which only a finite number can be tested - whence 1 - in practice, when the background knowledge C(T) can never approach includes

established

theoretical

which

the personal probabilities to make only a few observations

statements

and various

are conditionalized, before the degree

upon assumptions it may be enough of confirmation be?

comes close to 1.A more

detailed discussion of this point can be found in can see from Fig. 2 that when the system's trajectory is approach? = 1 smaller line, the rate of growth of T cannot be much ing the C than the rate of growth of the data. This is a state where the system has been [3].)We

fully exploited and it cannot account for unexpected data in a progressive a new theoretical jump which will cause C(T) to way without making to decrease again. The system before reaching such a state had managed of a given order of magnitude, but retain stability under disturbances of higher magnitude; that is, it cannot stay stable under disturbances cannot absorb high increments of empirical input Ct(E) which may be brought about by a radical advance in the degree of accuracy and sophisti? or by exposing a and technology, cation of the experimental methods new of type empirical phenomenon. radically of a dynamic theory indicates that it Thus, a progressive development was initiated by a theoretical discovery Tx which hit close to a stable in its natural environment. state in the evolution of the scientific knowledge in general, Stability is, however, not absolute; of the disturbances to the order of magnitude

it is defined with which

reference

are exerted

on the

system. Take the example of a species which is in a state of stability but may lose it under radical ecological changes. But to make the analogy in which the the case of science is similar to the phenomenon complete, of the changes were partly brought about by the evolution ecological a or a theoretical of science evolution of That itself. is, progressive species of new experimental methods subsystem may bring about the development and techniques which produce in a stable manner. At handle

empirical data that the system cannot this stage, the system should make a

DYNAMIC

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269

to a new state of stability, or a new system should be construc? ted on the basis of the old one. This reminds us, of course, of Bronowski's idea for of stratified stability [1]. Laszlo invokes Bronowski's conception transition

to him, as a result of the evolution of science [4]. According describing the science-system environment its from information ("nature"), input on its way to a states of series intermediary develops and grows through states are more Some intermediary equilibrium with the environment. for a from the environment stable than others, i.e. they resist disturbances the data of enables inflow The of time. empirical relatively long period the inherent in the system system to grow, such that the potentials - are actualized. The we can is whether "hidden strata of stability" question adopt this picture in describing the evolution of subsystems of science. It we may be seems that if we take a major subsystem such as microphysics terms strata of in of its successive evolutionary able to describe phases a major subsystem has an important role in determin? stability. Moreover, ing the strata of stability of the whole system. But what about a dynamic theory which is more restricted in scope, such as kinetic theory; it is more that the strata of stability of the whole system are superimposed conceivable on a subsystem such as this. However, every progressive dynamic theory to the evolution of the whole system; it creates local stability contributes in the framework of the stratum in which it is embedded. And since a state system is not a static state and there may be in degrees of stability, this local stability is part of the evolu? tion of the whole system; many local stabilities raise the degree of stability

of stability fluctuations

of the whole

subsystems system, whereas many non-progressive an approaching termination of the stable phase. and an epistemological The above picture has an ontological of the whole

indicate

import. a is of "natural of all, if the scientific knowledge system product i.e. if it is the fittest, its selection" (e.g. Popper in Objective Knowledge), should reflect some aspects of the structure of the environment. potentials

First

is the counterpart of McMullin's that a fertile This assumption assumption "is one which has a 'good fit' with the structure of the real". We just

model

replace "fertile" by "stable", or "locally stable", and "model" by "system" - since the model generates a theoretical system, determines its scope and our its evolution. with We, however, explicitly rely inspires assumption on a general evolutionary theory. Our second conclusion, which is predominantly states that the strata of stability are not exposed at once; epistemological,

270

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KANTOROVICH

as the observational

advance the rate of inflow and experimental methods of new empirical data increases and the levels of stability are exposed one patterns which have by one. Such a picture may well fit the evolutionary eras and quantum been exhibited by modern science; the relativistic come

to this conception, into being, according before in the of the classical middle in period physics physics the state of stability came to an end. When the system climbs up the

not

could

have

Newtonian before

nor

more it "becomes resistant to distur? stability, progressively from nature", says Laszlo. This is due to the fact that each evolu? tionary phase is initiated by a new theory which is set up to explain the in the previous phase, as well as to explain the empirical data explained strata of

bances

new

data.

our model, we can find a possible pattern evolutionary Employing which an ideal dynamic theoretical system can follow. At first, there is a of data E1 which is major problem or an anomaly, namely a collection at the state of knowledge t0 (we denote the state of unexpected at time t{ by 'V) and then Tx is invented to explain it. The knowledge theoretical jump must result in a decrease in C(T) since at t0 the theoretical

highly

statements and E0 included of established theoretical system consisted statements explained by T0, so that the state t0 is located at observational the point (1, 1) of the system's phase space, and the state tl9 for which Cx < 1, is located somewhere within the upper triangle (see Fig. 2). is when Tx does The only possibility of staying near the C = 1 diagonal or has only a few, but not have any new empirical prediction besides El9 then

it would

not be called a theoretical

or an inductive -

not be regarded as "explained" by Tx (or by T) This criterion of dynamic explanation. methodological to the development of "phenomenological" corresponds would

"jump", according possible

and Ex to our situation

theories, which cases precedes a big theoretical innovation. These phenomeno? phase, or to a logical theories may belong to the previous evolutionary a in research program, where non-progressive they account previous there. and anomalies for arising way problems in many

step comes with the big theoretical discovery. Thus the - a a disturbance in the system data cause, eventually, problematic - which at the start has the effect of con? theoretical decreasing jump firmation. If the disturbance carries the system close to a stable state, C(T) increases again. So that in a successful adaptation process a state of The

crucial

DYNAMIC

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theoretical growth (theoretical jump) is followed by a state of stable growth. The system's trajectory is confined then to the inside of the the closer the the C = 1 line. However, region of stability and approaches = can to 1, the less it enough develop progressively system is to C new unexpected data. This can be seen in Fig. 2 where any accommodate

maximal

data must shift the trajectory to be nearly input of unexpected = 1 diagonal, namely the rate of growth of Ct?T) cannot parallel to the C the smaller than that of C/(E). This is the stage when be significantly new to starts and should radically system degenerate phenomena bring further

innovation. Thus an unstable stage of preliminary or with minimal theoretical development phenomenological empirical a if successful leads to a is followed theoretical which by jump growth stage of stable growth which ends up again with instability and a similar

about

a new theoretical

cycle may start again. What are the methodological scenario ? The first is related

conclusions

one can draw from the above

to the distinction

between

radical and non

a dynamic at the start. When theoretical changes we mentioned a we enters to to to stable have stick make the state, it, namely theory minimal possible changes in the theory, and not to look for a radically radical

theory till T starts to be unstable. (Of course, the judgement if T is stable or not may not be simple.) This recommendation which is widely a in actual in science and in particular obeyed period of normal science in the exact sciences, follows from the assumption of gradual climb

different

the system hits a stable state we have along the strata of stability. When to find first the limits of stability of the system and to exhaust maximally its potential before we try to hit the next stratum. This we should do for two reasons. Firstly, if it is still possible to confront T with new data in a stable manner, itmeans that T can still help us in learning from experience; i.e. we still "gain" more empirical information than we invest in T, and it still functions as an explanatory device (in our dynamic sense). Secondly, since

the n +

1th big theoretical discovery should explain all the data at the nth all the data which could be progres? gathered phase namely - it would be at that to arrive at the n + 1th easier sively explained phase theoretical discovery after most of these data have been accumulated. a big theoretical innovation The Popperian of should be tried after stage this stage. It is outside

the scope of the present

article to enter into the question

of

272

AHARON

KANTOROVICH

transition. We can only pose the question of what of such a transition and may even is the systems-theoretical counterpart answer. If a this sort of transition exists then possible suggest a hint for

an incommensurable

in the new phase the system is better-adapting case corresponds to a rearrange? an such evolutionary Perhaps use Laszlo's to of + ment of the whole closed system nature}, {science new open system terminology, which creates new boundaries between the i.e. the new "nature". Thus it may of science and the new environment, be impossible to compare the quality of adaptation of the two systems to This problem applies to the whole science system different environments.

we cannot

assess whether

to "nature".

and not to any particular

subsystem.

8. CONCLUDING

We

treat dynamic

theories

in a black-box

REMARKS

manner

through

the information

the question of their generic characterization. they process, disregarding of time as at certain moments are manifested Whatever they are, they chains with be cannot but as i.e. identified they statements, theory-versions, to a is our of In this respect of theory-versions. analogous concept theory succes? that of an organism. Thus, we cannot reduce a human being to the in space. manifestations or a a basic metaphor, a of The identity dynamic theory is grounded on not do We a its or model, held by try to developers. general conception, rather we explicate these notions or to capture them by formal tools; rates of information attempt to analyze the external attributes of a theory: sion of his material

flow,

stability

etc. The

directions

in which

to the identity as well or anthropological similar to raises problems

the solution

problem may be sought for may be sociological as formal. Thus the study of scientific theories in psychology. We those related to behaviorism

that believe, however, of scientific character some methodological aspects implied by the dynamic - as has been manner theories may be treated in a behavioristic-like in our scheme. exemplified

The Institute for theHistory Tel-Aviv University

and Philosophy

of Science

DYNAMIC

OF SCIENCE

METHODOLOGY

273

REFERENCES in the Evolution of Complexity: Stratified

[1] Bronowski, J.: 1970, 'New Concepts and Unbounded

Stability [2] Hesse,

M.:

1966, Models A.: 1978,

[3] Kantorovich, Research Programs', E.: [4] Laszlo, 1972, Scientia

107,

[5] McMullin, Science', Dordrecht),

Planes', Zygon and Analogies 'An

Ideal

System's

version

Indiana). (Notre Dame, the Growth of Knowledge 250-272.

for 45, Model

of

the

Evolution

of

in

Science',

379-395.

E.:

1976, 'The Fertility of Theory and the Unit

in: Cohen, R. S. et al. (eds.), pp. 395-432.

Manuscript submitted 7 November Final

Model

of Science

Philosophy 'A General

5, 18-35. in Science

received

7 August

1978

1977

Essays

inMemory

for Appraisal

of Imre Lakatos

(Reidel,

in

Springer is collaborating with JSTOR to digitize, preserve and extend access to Erkenntnis (1975-).

http://www.jstor.org

KANTOROVICH

AHARON

A DYNAMIC

TOWARDS

1.

OF SCIENCE

METHODOLOGY

INTRODUCTION

of science treat theories mainly as static entities. philosophies deal with the degree of support or con? Justificationist methodologies firmation granted to a theory by empirical data, where the methodological - a set of unit which stands vis-a-vis empirical data is a static theory

Orthodox

still assesses a theory as a The Popperian methodology propositions. static unit, granting it a degree of falsifiability and empirical content or a degree of corroboration. Popper makes, however, a big step towards a dynamic approach by stressing the process of theoretical change and growth, in which one theory is refuted and replaced by a better one, i.e. in corroborated by a theory which is richer in content and, in particular, content. Since his theories are static entities, there is only one category of change whereby one theory vanishes while another appears on the scientific scene. Thus, there is no room in the Popperian methodology

theoretical

Indeed, circular

possibilities where theories are modified but not refuted. if for example the first version of the Bohr atom model with electronic orbits is regarded just as a timeless set of propositions,

then we

should

for intermediate

orbits.

elliptic

say that itwas refuted and replaced by the new theory with But it would be a distortion of the facts to exhaust the

of such a theoretical significance change by the term "falsification". a as a initial the version of is often treated intentionally model Firstly, "naive" or a crude version open to improvements and changes. Secondly, even

to be a genuine description, scientists would be in the basic picture which remains invariant under the consecutive changes, and they would regard it more fruitful to view the process as a modification of their basic theory rather than as a replacement of one if it is considered

interested

frozen version by another. According to the Popperian approach, there is no category difference between a theoretical change such a& the above mentioned change in atomic theory and a radical change such as the replace? ment

of classical

found under

physics by relativistic and quantum theories; the same methodological roof of falsification.

Erkenntnis 14 (1979) 251-273. Copyright

?

1979 by D. Reidel

0165-0106/79/0143-0251 Publishing

Co.,

Dordrecht,

both

are

$02.30

Holland,

and Boston,

U.S.A.

252

KANTOROVICH

AHARON

Popper, who was inspired by the theoretical changes of large magnitude which took place in early twentieth century physics, requires every theo? retical change to be as daring as possible. But this does not accord with - in actual scientific practice. We know from the history of science particu? lar physical science that there are periods when the scientific community adopts a conservative policy rather than a radical policy. In such periods a radical departure to an from an established theory is tantamount or sheer metaphysics. unfounded methodology Popper's speculation or a theories held in these low to the scientific grants grade periods, of them as metaphysical. Thus, sticking to the framework categorizes at the end of the nine? classical physics and the mechanistic conceptions teenth century is regarded by Popper as a metaphysical behavior since the theories became at that time unfalsifiable. nowadays Physicists agree with this. But what about the eighteenth century? Would that even then physicists should have made attempts to Popper maintain depart from Newton's theory, for example? Probably not, but there is no which will tell us when to make a daring clear criterion in his methodology classical

would

with our present theoretical frame? not account for the widely held does Thus, Popper's methodology physicist's belief that it would not have been possible at all to arrive at the theoretical

leap and when

to continue

work.

theoretical scientific

the of early twentieth century physics without of the two preceding centuries. And, in general, those the way to a big theoretical patterns of science in which

discoveries endeavor

evolutionary is paved by a preceding

change are

not

accounted

conservative

era of scientific

investigation

for.

science into focus by introducing brought the non-revolutionary he treats the science and of normal paradigm. However, concepts a deliberate as theoretical activity which takes place in normal science Kuhn

his

effort to keep the paradigm intact, and does not give methodological criteria for appraising it as a theoretical change. Lakatos, who carries some such an into the Popperian scheme, makes aspects of Kuhn's message new methodological entity appraisal explicitly. He creates for this purpose a a scientific research program, which replaces the static one research pro? from theory as the basic unit for appraisal. The passage revolution, whereas the gram to another is roughly parallel to a Kuhnian the to within one another from sequence which constitutes a passage theory

which

he calls

research program might

be viewed

as a moderate

or "normal"

theoretical

DYNAMIC

change.

OF SCIENCE

METHODOLOGY

In the latter case a hard

253

is parallel to a Kuhnian pictures the normal scientist as

core, which

paradigm, remains invariant. While Kuhn interest is to defend the paradigm merely a puzzle solver, whose main with takes a Popperian clashes Lakatos attitude empirical data, against in an according to which the scientist is willing to learn from experience the framework active way, namely to change his theories, within of a research program or in a normal science setting. Thus, Lakatos admits in distinct types of theoretical his scheme two methodologically change: or research pro? radical changes in which major theories are overthrown grams are replaced, and paradigm-preserving changes. Both Kuhn's and Lakatos' approaches reflect the Duhem-Quine which says that in a theoretical system any element can be protected are made

in other elements

refutation, provided adjustments The theoretical paradigm or hard core are such elements.

thesis from

of the system. But according

to Lakatos, the presence of a hard core does not ensure continuity in the research program. Hence, he introduces the positive heuristic, which has so to speak, for developing the task of supplying the building materials, the program

and adjusting

2.

McMullin

it to empirical

mcmullin's

data.

dynamic

takes one step further theory in a methodological

approach

toward

the concept of a employing In dynamic [5]. doing this he departs from the falsificationist tradition, with which Lakatos's method? radically ology was tied, and suggests that a research program be treated in an holistic manner as a unified and continuously developing body of knowl? context

edge, rather than as a series of successive theories each of which is falsified in its turn, giving way to its successor which is also doomed to be falsified eventually. He naturally suggests then to use the term "theory" instead of "research program", where theory is interpreted not in the logicist sense of a timeless set of propositions but as a dynamical-historical entity. Thus, the falsificationist is eliminated while Popper's idea of the terminology scientific knowledge is retained. The concept of a research a or now has a right to existence of its own; it program dynamic theory is not merely recruited for the sake of overcoming the difficulties which the falsificationist those which stem from the faces, notably approach growth

of

Duhem-Quine

thesis. However,

McMullin

questions

Lakatos'

concepts

of

254

AHARON

KANTOROVICH

hard core and positive heuristic ment of novelty or the criterion

and suggests that one replace the require? of progressiveness which Lakatos employs of a theory. Let us turn to his objections and

by the criterion of fertility suggestions one by one. Lakatos assigns to the hard core those parts of the theory under evalua? tion which are protected from falsification, the protective whereas belt which undergoes changes includes the auxiliary assumptions or peripheral

around the core of the theory. If so, components which are constructed asks McMullin, in what sense is it the hard core or the main part of the in the process of testing and growth? Thus, he theory which participates suggests, first of all, that the idea of a sharp distinction between "hard" and "soft" parts be abandoned since there is a whole spectrum of "hardness" or "softness" degrees. But the hard side of the spectrum includes not the main

of the theory but the established background theories such as and measurement theories (e.g. optics and radio-optics in or theories mechanics and astrophysics) general (quantum special relativity in contemporary nuclear and particle physics), whereas the atomic, body observation

when a theory is developing theory belongs to the softer edge. Namely, tested and evaluated, only the measurement and observational theories, the highly confirmed general theories and some specific auxiliary theoreti? are tentatively kept unquestionable. The fact that certain of the main theory remain nevertheless unchanged throughout the process might testify for their adequacy, but it does not necessarily mean

cal elements elements

that a decision was made

at the start of the process to protect them from refutation. Indeed, in a model for the growth of knowledge in a dynamic was we at which the methodo? elsewhere arrived theory developed [3], logical

conclusion

that a "good" is distin? theory or research program or theoretical growth, change, relative to empirical means a given amount of new it that for speaking

guished

by a slow

growth;

roughly data the

remains larger is the portion of the theory which or nearly invariant, the more the dynamic theory is progressive. shall later turn to this model and its implications for the appraisal of a

empirical invariant We

dynamic theory. The second problematic

concept Lakatos introduces in his scheme is the is assigned the task of integrating the research

heuristic, which program and assuring continuity about the origin of his positive positive

is not clear in its growth. But Lakatos heuristic. For one thing, it cannot be

DYNAMIC

METHODOLOGY

OF SCIENCE

255

the hard core or from the original theory-version Tx of the some as of Lakatos' formulations; research program, may be implied from

derived

from

of the program the development the case, claims McMullin, us to the logicist back be entirely predetermined by Tl9 bringing a "if the stand of appraising 'positive heuristic' single static theory, since a deductive analysis is entirely contained within the original hypothesis, if this were

would

test of the hypothesis and a systematic observational of the consequences other the On is needed" is all that them of hand, says (op. cit., p. 414). if the positive heuristic is only a "partially articulated" plan for McMullin, the research program, as Lakatos puts it in another place, constructing then the question is inwhat sense the program is an integrated whole which can be appraised as one unit?

3. POSITIVE HEURISTIC THEORETICAL

AND NON-RADICAL CHANGES

is making, let us introduce here the points McMullin in a research the following notation. A dynamic theory which is developed program will be symbolized by a function T which assumes the "value" In order

to discuss

theory) at time tt. That is, T refers to a in such a way that at certain points t{ dynamic entity which evolves more or sets less the it of definite time axis propositions produces along

Ti

(the

/th version

of

the

case of a research program or a dynamic In the prototype theory, is The first theory-version behind T there stands a model. 7\ which generated by the research program may be seen as a simplified or a crude version of the model, with some elements of the model remaining un? utilized explicitly. These elements constitute the "neutral analogy" - to use and Mary Hesse's terminology [2] which serves as a guide for modifying further developing T. Thus, the model generates a succession of theory Tl9 T29 ..., where at each step, more of the neutral analogy turns into positive or negative analogy. The neutral analogy serves as a source and as a guide for developing T, and therefore it may be regarded as a positive heuristic which is "contained within the original hypothesis"

versions

is the nature of this original hypothesis ? Is it just a well defined set of propositions re? which predetermines the evolution of the whole

But what

search program?

It seems that due to the presence

of the neutral

analogy

256

AHARON

KANTOROVICH

it cannot be regarded as such, since only further empirical data, on the one which is inspired by the original hand, plus the imagination of scientists but not always in an unequivocal and explicit way - on the hypothesis other hand, will determine the fate of the neutral analogy in subsequent of the system. We may identify our dynamic theory T with development what Mary Hesse calls (op. cit., p. 9) "modeli" (the subscript should not be confused with that which appears in Ti), namely the model minus the known

in addition to the positive includes, negative analogy, which the original hypothesis may be analogy, also the neutral analogy. Hence, looked at as an initial state of a dynamic entity capable of growth. The of growth come not only from the model potentialities standing in isola? is tion, but also from the background knowledge within which the model The background embedded. besides the includes, knowledge accepted theories and empirical data, partially articulated beliefs which are shared of the scientific community who produce the research by the members Thus it and metaphors related to the program. yields various associations cannot these be analyzed by the logicist's tools in the way a model, but well defined set of propositions can, e.g. by deducing testable predictions from

it at the outset.

We

take therefore

the theory-versions Tt to be the successive versions of each of which includes some neutral analogy - which

the "modeli", gradually diminishes along the sequence. The question which arises now is whether appraisal should be directed towards the model in conjunction or with the background the towards model. McMullin's knowledge only of a dynamic theory refers to the resources of the model, but these cannot be detached from the background and knowledge beliefs. Nevertheless, he claims that appraisal is directed towards the notion

of fertility

evolving theory which since the background

changes, namely towards the "modeli", does not participate in the dynamic knowledge of T's evolution (McMullin, op. cit., p. 422). In the scheme which undergoes

process was developed in [3] the dynamic theory T includes not only the main - but also the one which is subject to evaluation and modification theory elements of the background knowledge which are relevant to the main e.g. general theories which apply to the theoretical entities in the model, and auxiliary theories which apply to the system under investigation and enable the main theory to make observational predictions. However, theory:

since the background

knowledge

remains

invariant throughout

the process,

DYNAMIC

METHODOLOGY

and since appraisal is conditionalized upon is assessed evaluation which under theory

257

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its veracity, it is the particular - relative to the background

knowledge. A non-radical

theoretical change, in which a dynamic theory ismodified in the face of a new problematic empirical datum, is an inductive transition is the it is not only the which restricted by positive heuristic. Namely, data, but also the positive heuristic which restricts the number of possibili? ties for modifying T. Symbolically, the transition can be represented by or AT (where the A symbol denotes the formula: E&T^>T'9 E^> change). The fact that the premise for the inductive 'inference' includes not only the empirical datum but also the last version of T, indicates it is not just an induction which the continuity in the transition; is directed from empirical data to a theory, but from data to a change AT in the theoretical system - a system which endures in time. The formula E^+ AT encompasses, therefore, an important category of inductive inferences or theoretical changes. The complement category would include the case of radical theory-innovation which does not involve a modification in an existing explicit theory. In scientifically interesting cases the positive heuristic (e.g. the neutral relevant leaves more parts of the background analogy plus knowledge) than one possibility for modifying T. Thus, even if we have a well defined model, we have to decide what to do with the neutral analogy, since the data do not dictate a unique way to proceed. The inclusion of the criterion of simplicity in our positive heuristic will not help in this respect since it is not always

clear what is the simplest modification. We must conclude, even that if the heuristic is included in the initial theory therefore, positive version Tx it will not be a case of predetermination.

4.

ON

A PROBABILISTIC

MODEL

FOR

IN RESEARCH

THE

GROWTH

OF

KNOWLEDGE

PROGRAMS

In [3] a model was developed which dealt with the dynamic methodological facets of a research program or an evolving theory. In the framework of this scheme we will be able to formulate the issues raised by McMullin and suggest alternative answers to those he provides - especially to his notion of fertility. Let us therefore summarize the main features of the model.

258

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a research program, as a dynamic system, is divided a two into theoretical components: methodologically component T as was specified above and an empirical or an observational component E which consists of the observational data accumulating in the system. At time ti9 In this scheme

statement E{. The dynamic theory T E is represented by the observational is ideally adjusted to E such that at each step E% is accounted for deductively by Ti9 i.e. T{ -> E{. The evolution of the system is described by two variables which measure

the prior probabilities of the two varying components. At time ti9 the variables acquire the values P(Tt) and P(E?). The probabilities are taken to be representative degrees of rational belief in T{ and E{ as at time t0 - before the program starts to evolve. At time t? the measured first theory-version data Ex Tx is set up to account for the observational contain elements which cannot be explained at the state of knowledge at t0. Thus P(E?) is inversely related to the degree of unexpectedness or novelty - of E1 relative to t0. P(TX) represents the degree of belief in Tx at t0. Hence, P?TJ and P{E?) are not the actual degrees of belief or at the degree of nearness of Tx and Ex time tx but measure expectedness to the state of knowledge at t0 and the same is true for every i. It is of the pre? assumed that the functions P?T) and P(E) are representative

which

vailing

beliefs

of the scientific

community

that is developing

the research -

or at least metaphysical background It is further assumed that within the representative of its leading members. content of framework of a given research program, when the ontological T does not change significantly - i.e. when the passage from Tx to Ti + 1 or a radical change - P(T) varies in an does not involve a revolutionary ~ 1? Ct(T). opposite direction to the empirical content Ct(T)9 e.g. P(T) The same assuption ismade for P(E). Thus P(E) decreases in time if there program

and

share

common

data. Again, of observational is an accumulation, but no withdrawal, at t0. are measured relative to the state of knowledge both contents in the the growth of knowledge Hence the two pairs of variables measure research

program.

context is, however, important variable in a methodological of T which is taken to be the actual probability the degree of confirmation the probability to the fundamental Bayesian assumption of T. According at t0; of Ti at time t{ equals the conditional probability P(Ti\Ex) measured The most

that is, at t{ the probability

of Et equals one and all probabilities

are obtained

DYNAMIC

by

conditionalization

METHODOLOGY

upon

it. Bayes'

259

OF SCIENCE theorem

relates

the posterior

= probability to the prior probabilities: P(Ti\Ei) P^T^?T^iE,). our ideal model, inwhich Tj-^jE'i for every/and the degree of confirmation of T as a function P(T)/P(E).

Now

the methodological

content

1, Tt) consequentlyP(??| of time is simply: C(T) = of the dynamic theory T is

appraisal the behavior of C(T)

by inspecting accomplished The case which interests us most

In =

as a function

of time.

is when

both T and E grow in empirical functions decrease. P(E) decreases

and thus both probability of the growth of observational data. P(T) decreases when in the of to T E the process adjusting theory-version T? is replaced by Ti + 1which has higher empirical content (e.g. when Ti + 1 is more precise than T? or applies to a wider range of phenomena). There is a limit case - when Ei + 1 can be accounted for by Tf and there is no need to change T. In that case P(E) decreases while P(T) remains because

constant, and consequently C(T) increases. The general case of progressive ness is just a generalization of the above "static" case; namely we identify a progressive step by the condition AC(T) > 0. In the case when both or both empirical contents increase, this condition decrease, probabilities for a given rate of empirical growth the rate of theoretical does not exceed a certain upper bound. That is, as can be simply

is achieved when growth

calculated from the formula C(T) = P(T)/P(E) (see [3]), AC > 0 if and < Ci, where Cx is the initial degree of confirmation. only if AP(T)/AP(E) In the limit case of static confirmation of a theory by its successful predic? tions the rate of growth of T is zero and the above condition is obeyed for a every Cx (since always Cx > 0). Even here T generally undergoes a but this is deductive transformation. transformation, content-preserving A deductive change is therefore a limit case of an inductive change. By an inductive change it is meant that new observational data "bring about" a theoretical change with 0. ^ ACt?T) It was shown in [3] that when one uses an additive measure for content one can interpret the above condition for as saying that the progressiveness rate of theoretical growth is smaller than the rate of empirical growth. Hence we might there is a "gain" in say that in a progressive move information. This gain is made possible due to the initial empirical theoretical discovery, i.e. the invention of Ti, which constitutes a big a theoretical in a which adds amount of jump, Popperian style, large

260

AHARON

KANTOROVICH

(untested) empirical content to T. The big initial jump results in a low can increase if initial value of C(T). Later on the degree of confirmation the program is progressive. sum up, our model provides an explication of Lakatos' concept of or non-ad hocness and connects it in certain circumstances progressiveness To

with

the notion

or the Popperian

of confirmation, scheme.

5. THE DYNAMICIST FERTILITY

a notion which

APPRAISAL

is foreign to the Lakatosian

OF THEORIES:

VS PROGRESSIVENESS

scheme it is evident that in contrary to what can be from McMullin's view we can handle the dynamic case by using implied this reflects the difference between his scheme and formal tools. Indeed,

From

the above

ours.

In the logicist approach it by measuring appraises

one picks up a proposition (a static theory) and or probability. But its degree of confirmation, by the logicist's focus on the goal of discovering

is dictated this approach true propositions truth and producing by science. Thus, the static theory of the logicist is an abstraction and is not empirically given; theories under evaluation are in a fluid state and very often no official version of a theory is made since truth can be attributed only is laid down. This abstraction and not to a dynamic process or an historical to a timeless proposition

entity. on the other hand, focuses his attention on additional The dynamicist, also in goals and norms of science ;he is interested not only in truth, but truth. According the way science approaches more and more comprehensive in these norms give science its special characteristics, to the dynamicist, of notions The from other disciplines which strive for truth. distinction and fertility are related to such possible norms. The AC progressiveness - is notion of progressiveness is our modified > 0 criterion - which related to the requirement of economy or non-ad hocness for theoretical norms of "good" dynamic science. But growth, which may be regarded as such a requirement can be applied only to dynamic entities such as theories or research programs which, similarly to static theories, are abstractions. it is therefore not a simple task to determine the boundaries Practically,

DYNAMIC

METHODOLOGY

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261

and scope of both dynamic and static theories and there is no preference in this respect of the logicist notion of a theory over the dynamicist's. The fact that a dynamic theory is not a linguistic entity but produces a time-ordered parameter

such entities, leads to the introduction of the time into our formalism. The formalism we use for explicating the set of

includes logicist tools for analyzing each proposi? dynamicist's approach i.e. deducing predictions from each Ti9 comparing tion in the sequence, with empirical data and calculating the degree of confirmation through theorem. On this is superimposed the time parameter. Hence, Bayes' instead of dealing with static logicist relations between statements such as E and T, we compare E(r) and T(f ) as functions of time. We are not only in the probability of T being true given E at a given moment of but also in the process of arriving at a probable theoretical statement. time, We can arrive at true or highly probable propositions about the world by our and to it.We can, theories waiting passively recording data, adjusting interested

on the other hand, and guide us where

invent a daring theory which will anticipate new data to look. In both ways we can arrive at highly probable but the latter way is generally regarded as more scientific. Thus

theories, the criterion or the norm which

requires theory-guided growth of knowledge or norm which will distinguish, e.g., between or phenomenological theories in physics and New?

in this sense is the criterion

epicycle astronomy or structural tonian mechanics McMullin's

criterion

theories of microphysics. of fertility seems to make the above

distinction. a "good" (fertile) theory as that which "successfully guides research over an extended period of time, not only in the sense of correct predictions providing (an ad hoc theory could do this) but also in the sense of imaginatively in the suggesting plausible modifications

McMullin

defines

that allowed predictions to be made original theory/model, modifications which would not have been directly deducible from the original hypothe? sis" (op. cit., p. 401). It seems, however, that this formulation of fertility excludes the methodological situation of a static theory - which is appraised - since it does not suggest or guide modifica? by its successful predictions tions and growth of T. In another place, McMullin suggests the use of instead of the criterion of novelty; fertility thus, an ideal static theory which

successful novel predictions would not be categorized produces as a "good" (fertile) theory. But, as we saw above, the pure deductive case is just a limit case, so that there is no need to replace the criterion of

262

AHARON

KANTOROVICH

- as we do in our scheme, where the AC > 0 novelty but to extend it condition applies also to the static case in which a growing body of data in T. is explained by only deductive manipulations of fertility, McMullin attempts also to clarify the in Lakatos' formulations with respect to the manner by which ambiguity the modifications in T are "guided" or "suggested". He introduces for With

notion

his

a good metaphor "allows the the concept of metaphor-, now the to work and guides it in certain directions". But imagination a we to when certain find criteria for shifted have is and deciding problem in T is "guided" or "suggested" modification by "the theory", or by the this purpose

in a metaphorical way and when the modification original hypothesis, done in an arbitrary manner. The above formulation also shares in common with Lakatos' notion a

to account

is of the

for shortcoming by not being able course T can its of In the situation. evolution, following methodological have some new predictions which agree with the data, but at the expense of an inflatory theoretical growth. That is, a given transition Tt-^ Ti + 1, progressiveness

may produce some successful new predictions, whereas Ti + i has a great content which is in principle testable (i.e. additional deal of additional for but untestable; example cosmological practically empirical content) or astrophysical be conclusively

theories which

have

some empirical

but can

predictions or making

tested only by performing experiments tions in the vicinity of a distant galaxy. Or an hypothesis has a prediction that can be tested only by endangering

observa?

inmedicine human

which

life. Or a

in particle physics which can be effectively tested only at extremely (or high energies which cannot be reached with the available technology Incases social and economic conditions!). indeed at the present political theory

like these the research program may be regarded as progressive or fertile even if the rate of theoretical if it leads to successful new predictions, growth exceeds the upper bound dictated by the rate of empirical growth such that C(T) decreases. Thus the criteria of fertility and progressiveness are too weak in this respect; they do not rule out the so-called "specula? do not keep track with empirical data. The rejection of of this sort is in accordance with the general practice speculative - in in physics periods of normal science. or On the other hand, the theory would not be regarded as progressive in T, fertile in the case when at each step of its evolution the modification

tive" theories which

theories

DYNAMIC

METHODOLOGY

OF SCIENCE

263

growth, follows the observation of new data or empirical in our modified growth. But such a development may be quite progressive sense if a small theoretical growth or a minor adjustment in T, can account for a large increase in the amount of data. Thus C(T) may increase even or the theoretical

if the change in T was made after the data was acquired. Here the criteria which demand the anticipation of some of fertility and progressiveness, new data, are too restrictive.

6. STABILITY

OF A THEORETICAL

SYSTEM

criterion of progressiveness saw, our quantitative implies that a research program or a dynamic theoretical system T, which accounts for, or explains, a growing body of data E, is progressive if the rate of theoreti?

As we

cal growth has an upper bound which is imposed by the rate of empirical a "good" methodo? growth. Thus, not as the logicists would maintain, occurs not situation when itself Tx only logical provides successful predic? tions, but, dynamic data. The

in general, when it constitutes system which accounts for an of a stable

notion

senses of non-ad

hocness:

sense. Maximal

covers two important and continuity - in their generalized in the pure deductive case of a theory

theoretical

novelty is attained

the basis for a relatively stable increasing amount of empirical system

stability are (T\) which remains with the same logical content, while its predictions in accord with new empirical data; this is the ideal case of non-ad hocness or - since the content of the novelty, and, of course, continuity theory remains is a generalization of this unchanged. The general case of progressiveness ideal case, where the invention of Tx initiates a phase in the evolution of the system in which a large amount of new empirical data is explained while T remains relatively stable. So the first sense of non-ad hocness which

is accounted

for is (generalized) non-ad hoc novelty. Secondly, a or some is which heuristic change change obeys positive retains continuity in the growth of T. Stability, indeed, implies continuity. Thus, we have here three facets of a "good" dynamic theory: an economy an economic of theoretical and consequently growth, explanation, theoretical

theoretical stability and fertility. The fertility metaphor can be used here in the following sense: If fertility which includes novelty), is attributed to the dynamic that it provides an explanation to a large amount of new

for a stable theory (in the wide sense, system T, it means observational data.

264

AHARON

KANTOROVICH

But for the system to retain its main features - such that it can be treated as a unified and continuous some entity, or an entity which possesses itmust be identity and not just a succession of loosely connected theories ?n a stable

state and not be "dragged" by the data, i.e. not undergo too large changes. Hence, only when T remains stable in the face of a growing E, we can attribute to it the property of fertility. the evolution of a dynamic theory, the dynamicist In describing views the sequence {Tf} as successive states of one dynamic system which under? goes changes but which endures through a certain period of time. Thus the unit, can also be described system, which is appraised as a methodological an or as an adapting system system, in terms of information-processing notions borrowed from general systems theory. We can view a fertile or theoretical system as a stable system, which growing relatively small changes. processes empirical information while undergoing Thus the desideratum of anchoring our growing factual knowledge in a stable an economically

body of theoretical knowledge can be regarded as one of the goals of science, criteria of increasing which confirmation, implies the methodological hocness. non-ad and theoretical economical However, stability growth must

follow or theoretical

a

of "spontaneous" phase which adds potential empirical

non-stable

discovery

inductive content

jump to the

system.

view of science was developed by E. Laszlo [4]. systems-theoretical an is to science his system, information-processing approach, According to an organismic system, which adapts to "nature". We are analogous in smaller units - dynamic theories which are sub? interested, however, systems of science. Thus, in this view, science is a network of interacting A

which adapt to nature, rather than one monolithic system. a information which comes not only Therefore, dynamic theory processes of the science directly from "nature", but also from other components subsystems

system adapts to its natural system. The process by which a theoretical in can be schematized and scientific environment by the flow diagram is inter? from "nature", which Fig. 1. The detector absorbs information theories includes observation preted and processed by the filter which from filter and the from data The measurement theories. and empirical the enters of science are stored in E. This information other components of the theoretical compo? comparator where it ismatched with predictions nent T. If a prediction of T is not in accord with E, or when no prediction

OF SCIENCE

METHODOLOGY

DYNAMIC

other components

265

of science

comparator

'nature"

effector

control

i Fig.

1.

of T can account which

The

formation, perform detector

diagram

of a dynamic

system

of

science.

for an element

sends a command

T cannot

flow

be confirmed

of E, a signal is emitted to the control to change T. If, on the other hand, a prediction of by E, since E does not include some relevant in?

or to a direction is sent to the effector to make an observation an experiment - the results of which are conveyed through the to E. Thus, empirical information is not passively absorbed from

"nature", but is a product of theory-guided experiments. It is assumed that the lifetime of the system it develops relatively autono? during dynamic when interactions other with mously subsystems of science are relatively weak. Thus, our unit, the theoretical system which learns from experience and which is put to test, is not the whole scientific knowledge system but a more restricted system which possesses an identity of its own and exhibits an holistic

behavior of during a limited period of time. The division or a as scientific discipline such science, particular physics, at a given in the scientific community period into subsystems has its counterpart which is divided into respective "invisible colleges", or specialized groups, who

develop such a group

or research programs; communication inside between quicker and effective than communication

these theories is much

or general conception groups. The partially articulated model, or metaphysical which the of a dynamic picture, inspires development - whether it be a concrete model or an abstract setting - is the theory

different

unifying glue for such a group. In order to describe the evolution

of the system, we have

to choose

266

AHARON

KANTOROVICH

P(E)

region of stability

- -

. P?T\ Fig.

2.

The

phase

space

of

the

and a typical system evolution.

trajectory

which

describes

its

appropriate system's variables. One choice which comes up in a methodo? logical context is the pair of content measures: Ct?T) and Ct(E)9 or the or The the space spun by pair prior probabilities. system's variables is the space of the system. Let us choose the (P(T), P(E)) plane as the phase space of our system. In this plane the system's evolution can be described to the successive states of the system by a set of points corresponding case The deductive limit will be (see [3]). represented by points which lie on

phase

a straight line parallel to the P(E) axis. When the system's evolution adheres to the prescriptions of our ideal model, then P(T?) < P(E?) and the system will move only within the corresponding half square (Fig. 2). The degree of confirmation of T at a given point (state) equals the slope of the line connecting the point to the origin; the smaller the angle is between the line and the diagonal, the higher is C(T) at this point. C(T) varies with the slope: from zero on the P(E) axis to one on the diagonal. In this way,

one can easily visualize the static and dynamic methodological of the system's movement in the phase space, namely the

implications degree of confirmation

at each point and its direction and rate of change. in the case where both Ct(T) and Ct(E) increase in time after the initial state where T? was set up to explain El9 namely that case when T becomes richer in content while accounting for the growing We

are interested

data. This represents for example the evolution of a body of observational is generated by a model, where more and more dynamic theory which neutral analogy is exploited to become positive analogy or when the model is extended. Also when

ad hoc corrections

are added

to T? and T becomes

DYNAMIC more

METHODOLOGY

267

OF SCIENCE

and more

complicated, Ct?T) may increase. Sometimes, are added to the theory and Ct?T) decreases parameters accurate in its predictions, theory later becomes more

however, but when

free the

Ct?T) finally in the the region of non-progressive development = as line and the the the C bounded Cx phase space triangle P(E) axis, by since C(T) decreases when the system moves from its initial state (P?Tx), increases. We

define

PiEx)) to any state represented by a point located in this region. The region in which C(T) line of pure deduction increases includes the vertical is enclosed between this line and the (static T) and the triangle which C = 1 and C = Ci lines; this region will be defined, therefore, as the region of progressive development. we want to characterize a stable behavior of the system, we must a vertical movement with static T as maximally stable under all regard definitions and to set a lower limit for stability. The choice of the C = Cx line as the demarcating line between stable and non-stable growth seems When

natural.

to the methodological It corresponds and progressiveness non-progressiveness. Using

demarcation

between

our system's variables it is bounded. The above definition

implies that the rate of growth of Ct?T) is in accordance with the notion of "region of stability" used in general systems theory; a region of stability is a set of states such that once the the system's system has entered into it it can never leave it namely, behavior under external disturbances are is such that its movements confined to that region. That is, the disturbance exerted on T by the inflow of empirical information E is still controlled by T which is not "dragged along" by E and which keeps its identity. A by-product of a stable evolu? tion under this definition is, therefore, an increasing credibility of T. Thus, if we conceive a dynamic explanation - i.e. the explanation of a growing E - as by a growing T implying a relative stability of the theoretical system as defined above, then it follows that a necessary condition for a dynamic is an increasing credibility. But this is one of the obvious explanation intuitive methodological i.e. that when a new unexpected requirements, datum is explained by a theory, then the theory's degree of credibility should rise. Thus, our characterization of dynamic explanation is consist? ent with

the two requirements: that a dynamic explanation should imply theoretical > and that it also 0. Hence the AC dynamic stability implies demarcation line between progressiveness and non-progressiveness is also a demarcation line between explanation and non-explanation.

268

AHARON 7.

RADICAL

VS.

KANTOROVICH

NON-RADICAL

CHANGES

AND

STRATIFIED

STABILITY The necessary condition for a dynamic explanation, that AC > 0, implies = 1 cannot explain a new un? that a theory T which has reached C(T) in has an infinite datum. any principle theory-version (Although expected number of predictions of which only a finite number can be tested - whence 1 - in practice, when the background knowledge C(T) can never approach includes

established

theoretical

which

the personal probabilities to make only a few observations

statements

and various

are conditionalized, before the degree

upon assumptions it may be enough of confirmation be?

comes close to 1.A more

detailed discussion of this point can be found in can see from Fig. 2 that when the system's trajectory is approach? = 1 smaller line, the rate of growth of T cannot be much ing the C than the rate of growth of the data. This is a state where the system has been [3].)We

fully exploited and it cannot account for unexpected data in a progressive a new theoretical jump which will cause C(T) to way without making to decrease again. The system before reaching such a state had managed of a given order of magnitude, but retain stability under disturbances of higher magnitude; that is, it cannot stay stable under disturbances cannot absorb high increments of empirical input Ct(E) which may be brought about by a radical advance in the degree of accuracy and sophisti? or by exposing a and technology, cation of the experimental methods new of type empirical phenomenon. radically of a dynamic theory indicates that it Thus, a progressive development was initiated by a theoretical discovery Tx which hit close to a stable in its natural environment. state in the evolution of the scientific knowledge in general, Stability is, however, not absolute; of the disturbances to the order of magnitude

it is defined with which

reference

are exerted

on the

system. Take the example of a species which is in a state of stability but may lose it under radical ecological changes. But to make the analogy in which the the case of science is similar to the phenomenon complete, of the changes were partly brought about by the evolution ecological a or a theoretical of science evolution of That itself. is, progressive species of new experimental methods subsystem may bring about the development and techniques which produce in a stable manner. At handle

empirical data that the system cannot this stage, the system should make a

DYNAMIC

METHODOLOGY

OF SCIENCE

269

to a new state of stability, or a new system should be construc? ted on the basis of the old one. This reminds us, of course, of Bronowski's idea for of stratified stability [1]. Laszlo invokes Bronowski's conception transition

to him, as a result of the evolution of science [4]. According describing the science-system environment its from information ("nature"), input on its way to a states of series intermediary develops and grows through states are more Some intermediary equilibrium with the environment. for a from the environment stable than others, i.e. they resist disturbances the data of enables inflow The of time. empirical relatively long period the inherent in the system system to grow, such that the potentials - are actualized. The we can is whether "hidden strata of stability" question adopt this picture in describing the evolution of subsystems of science. It we may be seems that if we take a major subsystem such as microphysics terms strata of in of its successive evolutionary able to describe phases a major subsystem has an important role in determin? stability. Moreover, ing the strata of stability of the whole system. But what about a dynamic theory which is more restricted in scope, such as kinetic theory; it is more that the strata of stability of the whole system are superimposed conceivable on a subsystem such as this. However, every progressive dynamic theory to the evolution of the whole system; it creates local stability contributes in the framework of the stratum in which it is embedded. And since a state system is not a static state and there may be in degrees of stability, this local stability is part of the evolu? tion of the whole system; many local stabilities raise the degree of stability

of stability fluctuations

of the whole

subsystems system, whereas many non-progressive an approaching termination of the stable phase. and an epistemological The above picture has an ontological of the whole

indicate

import. a is of "natural of all, if the scientific knowledge system product i.e. if it is the fittest, its selection" (e.g. Popper in Objective Knowledge), should reflect some aspects of the structure of the environment. potentials

First

is the counterpart of McMullin's that a fertile This assumption assumption "is one which has a 'good fit' with the structure of the real". We just

model

replace "fertile" by "stable", or "locally stable", and "model" by "system" - since the model generates a theoretical system, determines its scope and our its evolution. with We, however, explicitly rely inspires assumption on a general evolutionary theory. Our second conclusion, which is predominantly states that the strata of stability are not exposed at once; epistemological,

270

AHARON

KANTOROVICH

as the observational

advance the rate of inflow and experimental methods of new empirical data increases and the levels of stability are exposed one patterns which have by one. Such a picture may well fit the evolutionary eras and quantum been exhibited by modern science; the relativistic come

to this conception, into being, according before in the of the classical middle in period physics physics the state of stability came to an end. When the system climbs up the

not

could

have

Newtonian before

nor

more it "becomes resistant to distur? stability, progressively from nature", says Laszlo. This is due to the fact that each evolu? tionary phase is initiated by a new theory which is set up to explain the in the previous phase, as well as to explain the empirical data explained strata of

bances

new

data.

our model, we can find a possible pattern evolutionary Employing which an ideal dynamic theoretical system can follow. At first, there is a of data E1 which is major problem or an anomaly, namely a collection at the state of knowledge t0 (we denote the state of unexpected at time t{ by 'V) and then Tx is invented to explain it. The knowledge theoretical jump must result in a decrease in C(T) since at t0 the theoretical

highly

statements and E0 included of established theoretical system consisted statements explained by T0, so that the state t0 is located at observational the point (1, 1) of the system's phase space, and the state tl9 for which Cx < 1, is located somewhere within the upper triangle (see Fig. 2). is when Tx does The only possibility of staying near the C = 1 diagonal or has only a few, but not have any new empirical prediction besides El9 then

it would

not be called a theoretical

or an inductive -

not be regarded as "explained" by Tx (or by T) This criterion of dynamic explanation. methodological to the development of "phenomenological" corresponds would

"jump", according possible

and Ex to our situation

theories, which cases precedes a big theoretical innovation. These phenomeno? phase, or to a logical theories may belong to the previous evolutionary a in research program, where non-progressive they account previous there. and anomalies for arising way problems in many

step comes with the big theoretical discovery. Thus the - a a disturbance in the system data cause, eventually, problematic - which at the start has the effect of con? theoretical decreasing jump firmation. If the disturbance carries the system close to a stable state, C(T) increases again. So that in a successful adaptation process a state of The

crucial

DYNAMIC

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271

theoretical growth (theoretical jump) is followed by a state of stable growth. The system's trajectory is confined then to the inside of the the closer the the C = 1 line. However, region of stability and approaches = can to 1, the less it enough develop progressively system is to C new unexpected data. This can be seen in Fig. 2 where any accommodate

maximal

data must shift the trajectory to be nearly input of unexpected = 1 diagonal, namely the rate of growth of Ct?T) cannot parallel to the C the smaller than that of C/(E). This is the stage when be significantly new to starts and should radically system degenerate phenomena bring further

innovation. Thus an unstable stage of preliminary or with minimal theoretical development phenomenological empirical a if successful leads to a is followed theoretical which by jump growth stage of stable growth which ends up again with instability and a similar

about

a new theoretical

cycle may start again. What are the methodological scenario ? The first is related

conclusions

one can draw from the above

to the distinction

between

radical and non

a dynamic at the start. When theoretical changes we mentioned a we enters to to to stable have stick make the state, it, namely theory minimal possible changes in the theory, and not to look for a radically radical

theory till T starts to be unstable. (Of course, the judgement if T is stable or not may not be simple.) This recommendation which is widely a in actual in science and in particular obeyed period of normal science in the exact sciences, follows from the assumption of gradual climb

different

the system hits a stable state we have along the strata of stability. When to find first the limits of stability of the system and to exhaust maximally its potential before we try to hit the next stratum. This we should do for two reasons. Firstly, if it is still possible to confront T with new data in a stable manner, itmeans that T can still help us in learning from experience; i.e. we still "gain" more empirical information than we invest in T, and it still functions as an explanatory device (in our dynamic sense). Secondly, since

the n +

1th big theoretical discovery should explain all the data at the nth all the data which could be progres? gathered phase namely - it would be at that to arrive at the n + 1th easier sively explained phase theoretical discovery after most of these data have been accumulated. a big theoretical innovation The Popperian of should be tried after stage this stage. It is outside

the scope of the present

article to enter into the question

of

272

AHARON

KANTOROVICH

transition. We can only pose the question of what of such a transition and may even is the systems-theoretical counterpart answer. If a this sort of transition exists then possible suggest a hint for

an incommensurable

in the new phase the system is better-adapting case corresponds to a rearrange? an such evolutionary Perhaps use Laszlo's to of + ment of the whole closed system nature}, {science new open system terminology, which creates new boundaries between the i.e. the new "nature". Thus it may of science and the new environment, be impossible to compare the quality of adaptation of the two systems to This problem applies to the whole science system different environments.

we cannot

assess whether

to "nature".

and not to any particular

subsystem.

8. CONCLUDING

We

treat dynamic

theories

in a black-box

REMARKS

manner

through

the information

the question of their generic characterization. they process, disregarding of time as at certain moments are manifested Whatever they are, they chains with be cannot but as i.e. identified they statements, theory-versions, to a is our of In this respect of theory-versions. analogous concept theory succes? that of an organism. Thus, we cannot reduce a human being to the in space. manifestations or a a basic metaphor, a of The identity dynamic theory is grounded on not do We a its or model, held by try to developers. general conception, rather we explicate these notions or to capture them by formal tools; rates of information attempt to analyze the external attributes of a theory: sion of his material

flow,

stability

etc. The

directions

in which

to the identity as well or anthropological similar to raises problems

the solution

problem may be sought for may be sociological as formal. Thus the study of scientific theories in psychology. We those related to behaviorism

that believe, however, of scientific character some methodological aspects implied by the dynamic - as has been manner theories may be treated in a behavioristic-like in our scheme. exemplified

The Institute for theHistory Tel-Aviv University

and Philosophy

of Science

DYNAMIC

OF SCIENCE

METHODOLOGY

273

REFERENCES in the Evolution of Complexity: Stratified

[1] Bronowski, J.: 1970, 'New Concepts and Unbounded

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[3] Kantorovich, Research Programs', E.: [4] Laszlo, 1972, Scientia

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[5] McMullin, Science', Dordrecht),

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E.:

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Manuscript submitted 7 November Final

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received

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1978

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(Reidel,

in

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