Fabric of Knowledge

THE IFABRIG OF KNOWLEDGE a study of the relations between ideas

'Contemplate the formative principles of things bare of their coverings'

The Meditations of Marcus Aurelius book I 2 paragraph 8

First published in 1973 by Gerdd Duckworth & Co. Ltd., 43 Gloucester Grescent, London NWL.

All rights reserved. No part of this publication rnay be reproduced, stored in a retrieval system, or transmitted, in any form or by any rneans, electronic, mechanicd hotocopying, reeordlng 'P or otherwise, without prior permission of the copyright o m e r . ISBN O 7156 0714 6

Typesetting by Specidised Offset Services, Liverpool. Printed by kinwin Brothers Limited, Old Woking, Surrey

F O R E W CP R D by PREFACE

C.W. Kilmister

1

T H E A R R A N G E M E N T O F I D E A S . Standpoint - A history of an expepiment - The general pattern of ideas - Perception classes - Chains of notions Integrative levels - Formative grades - Artefacts Notation for levels and grades - Scmantic types Introduction of relations - Names m d notation for the types - Full code sequences - The holotheme: a provisional summary.

2

T H E D E V E L O P M E N T O F A L H E O R U . properties of

relaLions - Relation codes - Operations - Numbers - Mathematical structures - h t o the domain of physics - Relation codes and the holotheme - Rules for placing nolions - The pattern in brief.

40

3 B A C K G R O U N D A N D G O M M E N T . Some other views - l a w s of the levels - The arrival of new levels - Aggregates of matter - Photons - Bther r e c e n t w o r k - Earlier work - The Greeks Mediaevd to modern - Co-ordinate indexing - Data fields - Applications - Conclusion.

65

APPENDIXES

A . A Data Field and its Contents B . Examples of Placement

6 . Previous Notes on the Levels D . Summary of the Relation Code GENERAL INDEX PLACEMENT INDEX

The classification of the elernents of knowledge is largely the province of the librarian and the specialist in documentation, with help from the teacher and the linguist. Such people have sometimes developed a tendency to avoid one of the subjects they must classify, namely arithmetic and the more advanced mathematics built upon it. This attitude is not unreasonable in the light of the forbidding methods of teaching the subject to which they may have been exposed at school. Fortunately, matters are improving fast; but memwhile to present a classification based on a simple and easily rnemorised pattern is safe only so long as n o one hints that the pattern may be related to mathematics and its feIIow studies. Uet at bottom the classifier and the mathernatician are doing the sarne thing, finding and manipulating patterns. Mlhen I saw this book in draft my reaction was one of great pleasure at the pattern it put forward, and this was increased because the approach to my own subject came, as it were, from the side of the arts. It was a way to arrange ideas which arose from the notions used in everyday life, essayed the development of readily-followed rules, and almost as a by-product revealed a scheme for classifying the concepts of mathematics and using them as a means of ordering the other sciences. I hope that one day Mr Jolley wil1 turn his argument round, present it in the fonn of a hierarchy derived from mathematics and then test its applicability to the rest of knowledge. It has become clear that a systematic application of such a hierarchy is not only a vduable classification scheme but serves as a genuine research tool and an effective teaching aid. The advantages to research are gained from the way that a regular pattem reveals gaps where, if the pattern is consistent, something new ought to be found. The teaching advantages spring from the way a suitable framework makes things fit together, so that connections between ideas are easy to mernonse. One way of judging the value of a book is to see how much

furfier work it suggests. There is a great deal to be done in the present field of study; bui Mr Jolley has made the main lines clear. His range includes the social sciences, economics, politics and management, but perhaps I rnay be forgiven if I deal for a moment with my own subject. The non-mathematician rnay skip the next few sentences, in the knowledge that his or her special study is equally recondite, in its own way, though it wil1 seem easy t o those familiar with it. The order and system that MrJolley brings to the modem world of mathematics (specially algebra and analysis) is quite stnking. I t would not be true to say that there was previously chaos here, but the most avowedly systernatic account, say by the Bourbaki school, is of an unsatisfacto-). nature. A starting point for Bourbaki is what is known as a set, a co3lectPon of things of any sort whatever. Bourbaki defines structures within and between sets in such a way that the result is mathematics as we know it. But why does he choose to define these structures and not others? There seems to be no inner motivation: the object is sirnply to impose a pattern that wil1 produce the desired result. By contrast, an arrangement such as that here presented supplies a motive: an order of definition appears, based on the observed properties of the structures tBiemselves, and implies that these are the structures out of which the subject we know exfoliates. A good pattern, like that of the periodic table, must make its own way in &e world. It invites attempts to prove it mistaken, to question its assumptions, to show that its gaps are blemishes and not merely undiscovered country. No doubt, faced with the considerable novelty in these pages, the reader wil1 be tenipted (fairly, I hope) to test their claims. There is always the question of how much reserve should be accorded to a view when it may be mistaken but, equally, rnay just be unfmiliar. There is much here which is novel, not least the belief that a universal classification rnay at last be with us, whose rules are independent of the habits of the dassifier. The pattem presents a considerable challenge: may &e book inspire others to begin the large task that it delineates.

C.W. ~ i l m i s t e r Professor of Mathematics King9sCollege, London.

The first two chapters of this book describe an inquiry into whether the elements of human knowledge may be arranged in an order which is not determined by personal opinion and which is capable of being venfied independently by different people. I t coneludes that this is possible and describes a theory which rnay serve the purpose. This part of the book is based on a paper presented at the First Ottawa Conference on the Conceptual Basis of the Ciassification of Knowledge, which was held in October 1971. 1 wrotc the paper during the summer of that year, using the results of work I had b e e n about twelve years earlier. Almost at the last moment I found myself unable to present it in person, and this task was undertaken at short notice by Robert Fairtllorne, t o whom I owe a considerable debt of gratitude. The third chapter is a commentary, intended to supply a background to the theory, to cornpare my present views with those of others, and to suggest applications of the work to practica1 problems. I should like to thank ASLIB Proceedings and Classqication Society Bulletin, in whose pages some of the material here printed first appeared, and the organisers of the Ottawa Conference, vvithout whose invitation much of this book would still be in the form of disorganised notes in trays on rny study floor.

I The Arrangement of Ideas have been trying for fifteen years to find out how people think. I do not mem how nervous impulses travel about their brains, or how they reason, or whether they visudize, though d l these are part of the problem - that part which is to do with methods and processes. My o m concern has been with a different aspect of the matter: with the materids they use. These materids are ideas, mental images or representations of what may (or at times may not) exist in the world about us. The questions I asked were, what sort of ideas exist, how c m we classify Lhem, and how can we be sure that none have een overlooked? I also wanted to know how simple ideas were assembled into the more complicated sort. I needed this wledge for a highly practica1 puvose: I was concerned a special type of information retrieval which relied for ffectiveness on the assembly of simple notions to form licated descriptions. Systems of this sort demand that ers can put their hands on ideas when they want thern constmction work they have In mind. Pn consequente sential to find a helpful order for the notions &ey Sometirnes the dphabet wil1 do; but this has its ns. To start with, it arranges words, each with one or eas tagging after it, and the ideas take up a random ement which makes it impossible to find them if their ed words are unknown or forgotten. Gonsequently,

other patterns are sought. Most other patterns are equdly arbitrary, based on the rnere decision of an expert or a comrnittee; but even so they are rernarkably helpful, once they have been learned in outline. Any pattern is better than none. However, if it is possible to find an arrangement which is inherent in tihe ideas themselves, this should offer advantages far in excess of those provided by patterns imposed by any personal opinion. It seems clear that the development of a universal ordering of ideas must be based on a study of the notions we believe correspond to the contents of the real world about us. This inquiry must supply rules whereby the notions may be placed in positions which they cannot help but hold. So far as is possible, personal opinion must be ruled out as a reason for placement: the only help must be that which comes from a careful exarnination of the structure of the ideas themselves. Such a scheme of notions wil1 bear directly on linguistics, since it wil1 be tke image which speech must represent, and indeed may even turn out to be the so-called Ueep structure of language' - the external pattem our notions imitate and which must in its turn be symbolized by speech. It will also be relevant to other disciplines, which it TNill support much as the periodic table underlies chemistry. No doubt its gaps wil1 dways be more impofiant than its areas of completeness, for they wil1 correspond to those places where research is still to be done. It wijl have especial value for those whose work is to catdogue m d to classify what we cunenely know. The documents they index and file deal with topics which have a place in it, and the retrieval systems they create wil1 be the more efficient for being more closely moulded to it. I suspect that a most-exact model of this type wil1 also display great economy and coherente of pattern, so that it may 'be of considerable use in the field of education. It is iiikely to set the major subjects of the school and college curriculum in an obvious, understandable, sensible relationship to each other, so that the bewildered scholar may the

The Arrangement of Ideas

13

more readily see where his studies fit in tke vast field of knowledge offered t o him. This is consistent vvith the recent trend towards ernphasizing the unity of knowledge as opposed to the differences between the traditiond main subjects of school curricula. So much for rny standpoint, except for this: exploration and adgustment of such a pattem may never be completed, but I think we now know enough to get the outline right. It is to this matter that this book is directed.

A history of an expe.piment My concern with this problem started in 1959. Bt was a spare-time affair, rnuch intenupted, and it comprised three stages. During the first, which llasted for several years, I examined ideas taken from large numbers of classifications and word Ests which I encountered in my daily Ilfe as an indexing consultant. My aim was to find what major varieties of idea could be distinguished, starting with as few opinions on the matter as possible, and consequently treating a'U other witings on the subject with reserve. P Look no steps t o avoid other peoples9 opinions when they came rny way, but I did not carry out a jiterature search or sit at the k e t of any teacher. Indeed, it was some time beforc I realized I had ernbarked on any special voyage of discovery; when I did so, well on rny journey. When the true position dawned on t seemed better to go forward than to return to find charts of the unknown seas. I think this choice was . Most of the arrangements of knowledge which I met in Iife were n0 more relevant to rny rieeds than the of signatures is to the classification of plants. I had as innocendy as I could, and to compare results with ose of other workers when results were to hand. I decided ould in any case be a wortk-while experiment to go where subject took me, and then, at a suitable moment, "c see wel1 my expenence ageed with that of others. Later, I

14

The Fabric of Kno wledge

read extensively in the field of the classlfication of ideas, and found I was part of a great stream of effort and speculation whose headwaters were lost in the past. Fortunately, I did not then find anything which made me fee1 my results had been reached before. This was indeed good luck: I would very seldom recommend that a study should begin like mine, without a full survey of existing views: the chance of spending years doing work whlch is already done is far too great. The o u t c o m of this period was a general pattem which appeared to be complete, repetitive and interndly consistent; but it was based entirely on observation and cried out for a f o m a l theory, for mathematical underpinning. The second stage of niy inquiry began wïth a search for this. Quite suddedy, in 1965, I redized that set theory provided the pattem I sought. I fastened on this and worked upon it, referring back and forth between the textbooks and rny obsemations, trying to develop a sirnple, ianderstandable terminology with which to talk of what I was about. As rny confidence grew, I began at last to read the history of the matter, and to relate other people9s views to rny own experienee. By 1967 I was satisfied Mrith the formalism, and the present stage could start. This is concerned with consolidation, drawing conclusions, finding applications and developing rules of the sort h i c h may one day grace a textbook on holothemics, by which is meant the study of the whole set of notions we may Ionn. Typical rules are concerned Mrith finding the position occupied by an idea ~ t h i nthe general pattern. Such rules can be reduced to a series of choices between alternatives which come in pairs and are mutually exclusive. Consequently they can generate series of b i n a v digits, which may fit wel1 in the memo-. of a computer. This property may have a special appeal for those who are concerned with handling infomation by electronic rneans. Rules of this sort place ideas according t o five main

The Arrangement of ldeas

15

categones. These are concemed with each notion's perception class, integrative level, formative rank and gade, and sernantic type. Glass, level, rank, gade and type rnay now be described - first briefly, as more or less bald assertion, and then in greater detail.

The general pattern of ideas The concept of a perception class provides the first and most general distinction between the various types of notion we may form. It is based on the difference between the plain ordinary certainties of the world, which are accepted quite generally, and ideas which are held by at least one large group of sane people to conespond to no reality, os whose conespondenee with reality is unproven or agreed to be non-existent. These are special ideas, which may be known to be fantasy, or rnay be hypotheses awaiting proof, or rnay be unprovable though many people assert thern to be true. This sort of speciality has nofiing to do with abstraction. Nurnbers are abstract, but they are universauy held to exist, none the less. They are not doubted or denied as one rnight doubt or deny the existence of the Great God Thor. By contrast, religieus beliefs are special. Believers may be sure they are true, but there are also unbelievers. The proper way to accommodate both is to give a special status to faith, rnaking it quite distinct frorn hypothesis and fiction, yet none îhe less to do with how we interpret the world, how we two classes of notions may be distinpuished: the e, which is lower, and the special, which is higher, ng ideas which are often achieved by our imagination upon and seeking insight into the lower. The main of ideas with which this book is concerned is the r. I t is this lower, mundme, class which most readily ories rnentioned above: those of level, , grade and type. These categones are to do with

constmction, w i t h intemal state or structure. For example, an i n t e p t i v e level m a y b e defined as a consecutive sequence o f sixteen degrees o f complexity. B y contrast w i t h perceplion classes, o f which ( o n t h e definition above) tkere are only t w o , integrative levels are fairly numerous. Eight c m b e defined, each being well k n o m as t h e province o f one or more major sciences. A s a result, three choices are needed i n order t o place an idea i n its level. Fortunately, these generally appear i n t h e guise o f a single choice between eight dternatives. T h u s there is a level concerned Mrith atoms and molecules, and n o laborieus decision procedure is needed t o conclude that t h e concept o f an acid radicd appears there. T h e forrnative grades are t h e sixteen degrees o f complexity within each level. T h e y are divided i n t o t w o ranks, i n each o f which eight o f t h e grades are found. T h u s a level consists o f eight grades whicl-i f o r m its lower rank, and eight which f o r m its higher. A single choice is needed i n order t o determine rank; three are needed t o decide u p o n g a d e ; i n b o t h cases &e declsions are based o n familiar properties o f relations, which are dealt with later i n this account. It is n o t surprising that t h e properties o f relations are relevant i n this context, for i f ideas are t o b e $ven places i n some m a y b y rneans which are n o t arbitrary t h e n t h e y must be placed according t o their nature, which arises f r o m their internal constmction. T h e relations between their parts are therefore o f central is w h y mathernatics, as t h e study o f sact structure, m a y b e expected t o have something y about t h e f o m a l i v e ranks and grades. g a d e contains examples o f eight different , which are t h e semanfic types whose languages t o develop t h e different parts o f verbs, adjectlves. An idea's t y p e , Iike its ious, b u t i n difficult cases a routine o f ons m a y b e called for, and i n these cases r decision m a y b e felt t o b e very like those

FIGURE 1: A hierarchy showing the pattern of ideas as described opposite with numerals attached as explained later

I -1

DIRECTION OF INCREASING COMPLEXITY COMPLETE RANGE OF IDEAS

.. . . . . . . . . . . . . . . . . .. .. .. .. .. .. . ... .... ... . . . . . . . . . . . . . . . . . . .;L~ssES':;:::~,::',:;::,,,~:::::::: j,: : l: : .................................................

I

I

..nvo ~ E

O : MUNDANE

R ~ ~ P ; . ~ ( ~ ~

I

I : SPECIAL

helium . zinc - carbon - tin - iron

...

.l 1

18

The Fabric of Knowledge

employed in grammar t o determine a word's type according to the part it plays in expressing a train of thought. These, then, are the categories of notion round which my inquiry suggested a structure of knowledge might be built.

Perception classes To take things in due order, a more detailed treatment of the categories of notion must begin with a note on the perception classes as such, although the main emphasis of the work must be on the levels, grades and types which appear in them. Utopia, mallom trees, Mr Bumble, the coming of the eoquecigmes, phlogiston, Osiris, vital spirits, Ragnarok, these are examples of notions whose home is in the upper perception class. They are now generally accepted as flction, even those which were once matters of faith, like the existence of Osiris, or of firmly held hypothesis, like the existence of vital spirits. Othcrs, more important in this class, are the notions which are alive in the great world religions: redemption, the hereafter, reincarnation, the Church Triumphmt. These, the concern of theology, I laid aside during my inquiry - at first unconsciously, and later in the belief that they nicht repeat, at a higher position, the patterns I encountered in more mundane affairs. In practice, I found myself beginning my task by contemplating notions of a sort I was later to cal1 'objects'. These were passive rmndane entities with boundaries. Nations, plants and people are examples, and it is clear that notions of this sort appear in the special perception class also: Mr Bumble is modelled on a person, Utopia on a nation, mallom trees on trees of our world of everyday. As T grew to be aware of what I was doing, the distinction between special and mundane grew important to me, and it seemed useful to symbolize it. I made use of a binary notation, simply because there were only two choices. T allotted the numeral O to the mundane, and 1 to the special,

Th e Arrangement of ldeas

19

w i t h t h e result that t h e special appeared i n t h e numericdly later or higher Position. Clearly, any further notation produced as a result o f 'further uiquirjr, could remaln w i t k i n this framework. I thought o f t h e entire set o f notions w e rnight f o n n - t h e holotheme - as potentially suited t o continued division o f this sort, d t h o u g h I &d n o t expect this division t o b e simple. I looked for great complexity, and indeed began b y using an alphabetic notation o n t h e ground that a mere t e n n u m e r d s were unlikely t o b e enough. T h e persistent d u d i t y o f things surprised m e w h e n I encountered it. This binary e f f e c t established itself dunng rny examination o f t h e passive entities o f ordinary life, t o which i t is time t o turn.

Ghains of notions T h e essential process i n t h e study o f integrative levels is that o f rnaking chains o f ideas such that each earlier notion is a constnicrive part o f each later concept, just as bricks m a y b e part o f a house. This is work which is fairly easy t o d o , i f bounded rnaterial bodies - objects - are considered. I was fortunate t o have selected these as t h e first sort o f n o t i o n I would exarnine. T h e follouving is a chain o f objects: a proton an atomic nucleus a carbon a t o m a methyl g o u p a phospholipid molecule a lipoprotein membrane a mitochondrion a rnuscle cel1 a muscle fibre a muscle a heart

a cardiovascular system a human being a degreasing section a painting department a production line a factory a neighbourhood a town a county a nation an alliance the United Nations It is interesting to form other chains and to compare them, setting sirnilar notions together. For example, an electron rnight occupy the same line as a proton, and a plant cel! might appear on the sarne line as an animal cell. M e n this is done, some chains d l 1 be found to have gaps in thern; others wil1 seem t o have a superabundance of ideas. Here is a cornparison of chains: a carbon atom a methyl group a phospholipid molecule a lipoproteh membrane a mitochondnon a muscle cel1

a magnesium atorn a chlorophyll molecule a chloroplast a chloroplast layer a leaf cel1

It is clear that each chain calls for an expansion of the other. When enough chains of this sort have been made, regularities appear. At intervals, the notions foming the basis of important disciplines occur: sub-atornic particles; atoms; molecules; organelles; cells; organs of the body; entire plants or animals; organized human groups; nations. This list is not complete, but it offers a starting point. At an early stage in my study, I named ideas of this sort "units', and I wondered

The Arrangement of Ideas

21

whether there was a general property by which they could always be recognized. The answer, it seemed, was provided by the concept of homeostasis, as first applied in the field of biology. AI1 the units possessed this property to a high degree: @ven favour&le circumstances, they remained in balance with their surroundings. If not too severely damaged, they mended themselives. Even atoms, lacking electrons, trapped them if they could; and if a trade union Post its general secretary it appointed another. Units demanded completeness, and completeness appeared to be judged by degree of ability to survive. However, frorn this point of view some units fared better than others. Molecules were more independent than atoms: they &d not rush into partnership as readily as did atoms, although they too could interact. Gells were more independent than the organelles they contained, which mostly relied on the cells for survival. Leaves and flowers rnight die but the plant coulid still continue. An industrial company rnight close d o m a department, a state might extinguish old divisions, but the company and the state could both live on.

Integrative levels

I remember my s u p n s e when I saw that the more and the less dependent units dternated dong the entire length of any chain Mnthin which no units were omitted. This is the sort of reguj2arity whicfi entourages myone looking for a pattern, because it appears Mrithout being forced, and justifies the procedure of simply gazing at one's problem in a questioning way. My o m first list of units ran as fouovvs: photons PARTICLES

( ~ t rest h mass)

atom MOLECULES

The Fabric of Knowledge organelles CELLS

QrganS PLANTS AND ANIMALS

departments GOMMUNITIES

locd governrnents NATIONS

I noticed with interest that it was notably organic. It did not deal with artefacts, which I w u l d have to examine later. Also, I observed a second pattern in the chain. Every fourth unit seemed in some way to be enninent, to be a culrnination of the ideas which went before. The physical sciences aspired to molecules and molecular substances; the life sciences rose to the study of living beings; the social sciences culminated in the politic-d, social and econoniic behaviour of nations and of their unions and alliances. Such effects as this made it seem very rnuch as if the underlying pattem was binary. If this was so, then it was hard to resist the view that a list containing twelve units was incomplete. There ought to be sixteen. The positions held by the remaining four might, of course, come after that held by nations; but this view had littPe to recommend it counpared with the alternative, that they came before photons and were, in fact, to do with rnathematics. I remernbered reading specuIations that physicists might one day have to make matter out of space, and I guessed that two units concerned with geometnes rnight precede the photons and particles of sub-atomic physics, while two units of even greater simplicity occupied the two positions at the very start of the chain. I therefore added the following t o rny list: members of sets FULL SETS

points LINES, LINEAR SPACES

Two domains of the INANIMATE KINGDOM

Two domains of the ANIMATE KINGDOM

The Arrangement of ideas

25

purpose. I began by considering those ideas which came immediately after units in my chains. There was an immediate result: my colliection of ihese contained a marked preponderance of what I cdled %ssemblies9:groups of units in which no precedence could be found. The two electrons in the innemost orbit of an atom afford an example; so do the members of a leaderless group of people involved in a general discussion; so do a pair of eyes, the atoms in a hydrogen molecule; the sides of a hexagon, a pair of cuffllnks, a swarrn of midges, the voters in an election when each has one vote only . The notions which followed assemblies also had something in common, Sequence, precedence, order appeared in them. Examples are afforded by a production line in a facto-., the digestive system of an animal, the rainwater system of a house. I found the name 'system9 was so frequently used in connection G t h them that for a time I adopted it; but in due course it began to seem better to kmphasise the seriality of the collections of units involved, so I began to use the word 'series' for notions of this type. When units, assemblies and series had been removed from my chains of ideas very little remained. That little, however, was of considerable interest. Pt consisted of reciprocal or interactive series (I thought of them as assemblies of series), and it introduced an effect of feedback. Examples are: the afferent and efferent nervous systems taken together (and generally, with the brain and the spinal cord, called the central nervous sy stem) ; the transmission, ignition, braking and other systems, taken LogeSier, of a motor vehicle without its bodywork (I have often seen those driven past my tvlndow on their way from the engineer to the coachbuilder); the two series of unit parts which between them make two-way conversation possible by telephone. At first, after some hesitation, I called these reciprocal series 'combines9; but when unidirectional sequences had been named 'series' the word Csystem9became free and I applied it to assemblies

26

The Fabric of Knowledge

of series which displayed this internal balancing tendency. It chimed in wel1 with cument practice: general systems theory is concerned with interacting series. Notions of this sort are central to the study of cybemetics. I t is hard t o overrate their

importante. When sufficient series are brought together in this way, the resulting complex system is sufficiently interactive to produce a new unit, ensuring its homeostasis by the co-operation of its many trains of elements. This method of examining ideas reveals four steps, including a lower unit, assembly, series and system, before a higher unit is reached, and since I had found two units, a major and a nunor, in my integrative levels P concluded that each level contained eight steps of this sort. Of these, the first four led up frorn the lower unit to the unit which, being major, I thought of as central t o the level; the second four led on to the lower unit of the next level upward. I found it helpful to think of these as formative stages, stages in the fomation of higher things from lower. Soon each of these eight was to be divided in two, to form sixteen in all. Meanwhile I added a little to my teminology, calling a unit of a lower rank a subunit, an assembly of a lower rank a subassembly, and so on. With this convention, an atom becarne a molecular subunit and the alimentary canal of a human being becarne a subseries in the level of plants and anirnals. To proceed, a new distinction must be called into play. As an immedate example, consider the differente between an atom of copper and copper as a substance. 6)ne is an object; the other is a large collection of such things, an idea vvithout bounds, copper atoms on and on til1 we stop bothering to think about them, copper atoms to infinity. TFbe Same situation is found at other levels: electricity is a boundless collection of electrons, space can be thought of as a boundless collection of points, mist is a boundless collection o1 droplets, water a boundless collection of molecules of that

The Arrangement of ldeas

27

substance, yeast a b o u n d e s s collection o f yeast plants. Further, n o t al1 substances are simple i n t h e sense o f being made o f units o f one sort only. Others are mixtures: air is an example. U e t others have an even more complex internal structure - consider plywood. In t h e case o f plywood t h e substance is bounded i n o n e dimension buk n o specific bounds are set i n t h e other t w o . There is a significant comparison t o b e d r a m between units and assemblies, o n t h e one side, and simples and mixtures, o n t h e other. Units and assemblies are concerned urith objects; simples and mixtures are concemed w i t h substances. There is n o necessary order about tlie internal stmcture o f simples or rnixares; b u t i f sheedike or larriinar substances are considered t h e n an order can b e made: i n t h e case o f &ree-ply plywood &is m a y be: first ply - rniddle ply - last ply. Suppose that sheets are equated t o series, accordingly: can w e t h e n find something which m a y b e equated t o systems? Sheets are bounded i n one dimension: perhaps materials which are bounded i n t w o dimensions might fill this higher r o k . Examples are afforded b y hosepiping, rope, cinematograph film. These are o f indeterrninate l e n g h , b u t breadth and thickness are assumed. A f t e r trying various names for t h e m , I ended b y called t h e m 'stretchesp. T o take this t o a conclusion: Sie n e x t m o v e should b e t o consider substances bounded i n three dimensions. These, however, apped t o u s ceirectly as objects. T h e y are clearly units, as indeed might b e expected, since at this point o n t h e upward path m o t h e r rank ( m o t h e r lower or upper half-level) becomes available. This line o f a r g m e n t led m e t o divide each formative stage i n t o t w o , o n e part serving t o accornmodate objects and t h e other t o accommodate substances. These parts I named k a d e s ' , and it is f r o m this division that t h e sixteen grades o f each integrative level are derived: eight grades i n each rank o f t h e level.

FIGURE 3:The sequence of formative grades in any level, showing its binary pattern, with octal and binary notation as described later first appearance in level of

I

I

I

boundlessness, infinity (after one grade)

subsimpie

reciprocity, symmetry (after two grades)

u d

0.4

2

subseries

0.1 00

0.5

subsheet

0.101 I

0.6

I

subsystem

0.110 l

l

1 1 1 1

0.7

1

order (after four grades)

I

I

U

0.11'1

I substretch

1.000

unit

1.001

simple

1 1

completion (of main unit) (after eight grades)

In this table, reciprocity is shown as a property which first appears in subassemblies: it is to be thought of as a relationship between the things concerned (as one is to the other, so the other is to the one). In this sense, 'symmetry' is another name for it. Note that the g a d e names are chosen with respect to objects and substances: other grade names may be more helpful for other types of idea.

The Alrangement of ldeas

29

Considera~onof rnaterials - vvire, rubber and the like - led me to become concemed vvith the placing of artefacts in my eihain of integrative levels. Books, cups and saucers, chairs, pianos, roofing tiles, bottles of weedkiller, the Venus de Milo, transistors, Jet engines, f o o t b d boots, electric slravers and battleships al1 required a home. It was a land of pitfals. I h e w , for example, that an ìdea was not more advanced than another idea simply beeause it physically contained that other idea: tortoises are contained in shePls, but they are more advanced than the shells whieh contain them. Again, I knew that physical size was nothing to go by: îhe sun, for al1 its irnmerisity, is not as advanced as a bol1 weevil. I suspected that a hinge, iricorporating the facility of movement, might be higher than the door m d door frame which it served to connect, however complex these might be, so long as they were rigid in themselves. Such things as hinges, scissors, locks, latches and the Iike T caliied "djustable devices'. In the end I set up the hypothesis that the two levels between that of rnolecu8es m d that of communities were occupied, on the mechanic slde, by two pairs of units (vvith their appropriate assemblies, series and systerns) as foIIows: single-piece parts ADJUSTABEE DEVICES engines and other organs of machines MACHINES At the level of communities the two branches of the chain came together: a farm, for instance, consists not o d y of people, other animds and plants, but also of buildings, tractors and harvesting machinery, After experimenting wit21 more chains of ideas I concluded that this was satisfactory, thocagh it seemed that unitary completeness could not be judged In the light of homeostasis on the mechanic side. lhnstead, Pt was something to do vvith

fitness for purpose. None the less, T believed I could recognize a unit artefact when I saw one. My lavvnmower ceased to be one when its handle broke away. M i l e I was looking at this part of the problem, I was met several times by the question Why, if tools and machines are inanimate, should they occupy levels which are mainly concemed with life?' There certainly seems to be a good common-sense argument for placing implements and machinery of al1 kinds at some inanimate physical level, moving off sideways, so to speak, from the main stern of the patterii as it rises through the grades. The argument which decided the matter for me may have been somewhat childlike: tools are extensions of the animal. A bicycle is an improved pair of legs, a pair of pliers is a hand with a stronger
Notation for levels and p d e s To proceed, the levels must be named and numbered. The follouing names and numbers, the latter in both octal and linary, may suit: 000 : level 0 : sei-theoretic : rnembers of sets, full sets 001 : level 1 : spatial : points, lines and linear spaces 010 : leve1 2 : subatomjc : photons, rnassy particles such as elec trons 01 1 : level 3 : molecular : atorns, molecules 100 : leve1 4 : cytomechanic : organelles, cells; and also singlepiece parts, adjustable devices 101 : level 5 : biornorphic : organs, plants and anirnals: and also engines and organs of machines, machines 110 : level 6 : cornrnunal : departrnents, organisations 111 : level 7 : national : local governments, nations

This notation may be added to that for ahe perception

The Amangement of Idem

31

classes, placing the class first and interposing a point between the class and the level. Then, in binary, Utopia, Ruritania and other imaginary nations occur in a position Mihose code sequence (class and level) is 1.11 1. Canada, France and other real nations occur at position 0.1 11. Gopper atoms are at 0.011. Ottawa University is at 0.110. In octal, these sequences are 1.7, 0.7, 0.3 and 0.6 respectively. A code sequence for the formative ranks and grades may also be made, starting with a zero for the lower rank of a level and a unit for the upper. If a point is used to separate the level code sequence from the g a d e code sequence, then Canada and France, being main units at nationd level, wil1 gain the sequence 0.1 1 1.1, signifying mundane, national level, upper rank. Cooper atoms, being subunits at molecular level, wil1 gain sequence 0.011.0, meaning mundane, molecular level, lower rank. Within each rank, further notation may then be applied, as follows: units simples assemblies mixtures series sheets systems stretches

: 000 : 001

: 010 : 01 1

: 100 : 101 : 110 : 11 1

(relevant to objects) (relevant to substances) (reIevant to objects) (relevant to substances) (relevant to objects) (relevant to substances) (relevant to objects) (relevant to substances)

In this arrangement, the right-hand position of the code sequences holds a zero if the notion to which it refers concerns an object and a unit if it concerns a substance. In the rnidde position, the zero indicates a sort of slngleness and the unit shows the condition of assembly or mixture - a system being an assembly of single series and a stretch being f o m e d by a rnsxture or assembly of sheets foming an envelope. The way this may be interpreted at the higher levels of integration is still a matter for inquiry. The left-hand position carries a zero if order does not matter arid a unit when order makes an appearance.

32

Ilze Fabric of Kno wledge

The code sequences for Canada, France, copper atoms and the like rnay now be extended further, cornpleting the notation for fomative degsees by adding triads to show the grades Mrithin the ranks. Again a point may be used, this time between rank and gade. Thus the two nations gain the sequence 0.1 11.1.000, the gade triad showing that they are units and the rank monad that they are in the upper part of a level. Atoms gain sequence 0.01 1.0.000, being units also, but in their level's Power rank. As a further example, the digestive tract is a biornovhic subseries and carries the code sequence 0.101.0.100, rneaning niundme, biomorphic, (sub)series as reqilired. Again an octal expression is possible, each binary nurnber, monad or triad, being converted to a numeral from 0 to 7. ?'he nations then obtain sequence 0.7.1.0, which may be shortened to 0710 Mrithout Xoss of infomation. Atorns are numbered 0300; the digestive tract is aIlotted sequence 0504. These four-fipre references give class, level, rank and g a d e in compact fom.

Semantic types The study of semantic types is not unlike that of integative levels. Major varieties of notion are identified and set aside frorn a collection of ideas, type by type, until none remain, For this part of rny inquiry li took words at random from a dictionary, laid aside al1 those which I recognized as referring to objects or substances, and looked carefuHy at the rest. It was almost too obvious that objects and substances, which I called collectively 'ttbings9, were both passivc and cntitive. Since the passive contrasts ~ 4 t h the active, whilst the entitive rnay be set against the attributive, this suggested immediately that three other types of nolion rriight be expected to exist. These would be respectively passive and attributive, active and entitive, and active and attribeitive. Here the use of contrasting pairs of properties offers yet mother binary pattern.

The Arrangement of Ideas

33

There is Iittle difficulty in applying this analysis. Passive attributive notions are easy to find: they are qualities, such as 'blue', "riangular', 'cheap' and korthem'. Active entities are also much in evidente, being phenornena 1ike rain, vvlnd, the battPe of Salamis, a g a n d dance, a surprise. Active attributes describe phenomena Just as those of the passive sort describe things. Since rny daily work at the time I examined this problem was concemed with indexing such occurrences as sales, breakdowns, accidents and incidents, rny files were full of exarnples. They dealt with when and how events occurred. The problern was not ~o find sucb notions but to declde upon what to cal1 them. T could find no cornmonly accepted name for thern. In the end I settled for the word 'mode'. An active entitive notion describes the mode of occurrence of a phenomenon. 1 cornpared modes vvith qualities, and this led me to look again at m n y adjectives I had thought were to do with the properties of things. For exarnple, in the phrase 'that factory is productive', the adjective 6pproductive9is not strictly descrlptive of the factory, but of the phenomenon of manufacturing. Thus I karnt that the part of a word may play in speech is no sure p i d e to the type of notion it represents. It is only a general indieation of this, though often a good one. At this stage, almost al1 the words remaining in my list were verbs. I t kook me a long time to redize what H am sure others could have seen in a flash, namely tbat verbs represent a sort of idea which connects other ideas, that they are to do witli relations. My readings in logic had made me familiar with the distinction between a term (adjeetival or substantive) and a relation. Hcrc I had a third dichotomy.

Introduction ofreladions I'hus at last I found mysePf concerned with the study of relations. Relations give stmcture t o terms, whether the tërms be modes, qualities, things or phenornena. Relations

34

The Fabric of Kno wledge

between sirnpler notions become relations within more complex notions. Each t e m is provided with its own variety of relation: passive entitive relations (like being-bent-round, being-contained-in and being-beside) inform things. Passive attributive relations give form to qualities. Active attributive relations are found in modes. Active entitive relations are embo&ed in phenomena: thus exploding, faging, running, encouniering are the relations we meet in explosions, turnbles, races, meetings. When I had set aside all the words in rny collection which referred to one or other of the four types of term, or of the four types of relation, very little remained. This little comprised two intejections, the indefinite article, and a word of indeterrninate meaning which occurs in a work by Shakespeare: honorificabilitudinitatibus. I decided that my set of semantic types was complete. However, names were still needed for those which were relations. I &d not find m c h help in the literature on the subject. The most widely agreed word in English was 'operations', a name which was applied to any relation of the active sort. In many textbooks on mathematics, passive relations were known simply as kelations9, the passivity being implied. Por these I adopted the tide 'conditions'. Then, after several trials with words whose other meanings often led me astray, I concluded that entitive conditions could wel1 be called %tructures9, whilst those of the attributive sort could be named 'states'. For entitive operations I used the word 'actions' and for attributive operations the title 'activities'. Since binary code sequences can be allotted to classes, levels and prades, it is natura1 to consider allotting them to sernantic types also. The three dichotomies provide a basis for this, the onPy problern being that of deciding upon an order of precedence for thern. On the ground that al1 ideas embody relations, which must therefore come first so as to be available, I synnbolized relations by means of a zero in the right-hand position of the code sequence, whilst terms were

The Arrangement of Ideas

35

by unity. Sirriilar arguments gave attributes over entities, passive ideas precedence over active, ttributive-entitive contrast precedence over that static and the changing.

Names and notation for the types The resulting set of semantic types, with their names and code sequences, was as follows : 000 : passive attributive relations (states) 001 : passive attributive terms (qualities) 0 10 : passive entitive relations (structures) 01 1 : passive entitive terms (things) 100 : active attributive relations (actions) 101 : active attributive terms ( m o d e s ) 110 : active entitive relations (activities) 111 : active entitive terms ( p h e n o m e n a )

The question arises: Do qualities, actions, phenomena, modes and the other semantic types d1 behave in a similar way, so that what we know about objects and substances may be applied to them? It seems that they do. The distinction between object and substance has its analogues: phenomena may be divided into events and processes, qualities into positionals and extensives, modes into datals and duratives. Thus an object suffers an event, a substance undergoes a process. Events occur at instants, processes continue over periods. An object may be found at a given place - say the North Pole - while a process extends over a region. As examples, in the realm of phenomena, consider the process we cal1 trade and the event we cal1 making an exchange, the process we cal1 fire and the event we cal1 the ation of an atom, the process we cal1 traffic and the nt we cal1 the passage of a vehicle, the process we cal1 war the event we cd1 the firing of a gun. In the realm of ties, c o q a r e orientation with angle, temperature with tity of heat.

Further, gades conresponding to units, sirnples, assernblies and the rest are found in respect of d semantic types, not o d y in respect of things. Pn the redm of modes, for instance, we are familiar vvith such rneasurables as miles per hour and centimetres per second. These are complex ideas made out of slrnpler notions. They conespond to series: order is important in thern. To see this, note that miles per hour is not the sarne as hours per mik. Specialists in work study examine trains of events leading to desirable ends and develop rnethods of reaching the sarne ends with less effort. Here we encounter typieal series of phenomena in the upper biornorphie rank. Another exmple can be found when a cheinicd works goes on stream and substances move from vessel to vessel whBlst undergolng a sequence of reactions. I should remark that this oecurs at molecular level even though Sie factory in which ih takes place is at the level of communities. If substances (for example, sulphunc acid) are subscquently recovcred for recycling then the whoPe cycle of ghenomena achieves the status of a systern. The Ksebs cycPe in a living ceU is another case of a molecular systern urithin an object of a higher level. Relakions display pattenis akin to those of Lhe terms in which they are embodied. The coinmand structure of an arrny, for example, is a pattern of relations, and so are large nurribers of politicd theories. At the spatial level and below, connplex arrangements of relations are f o m d in mathematical forrnulae, with place-holders to show cvliere the terrns (usually numbers) may be inserted. These group the terms into assernblies, series and systems of many sorhs. Simple examples are the formula for an arithmetic mean, the pattern made by the elements of a rectanelar matrix, and the hypergeometric series. Full code sequences

The code sequences for semantic types may be added, after

The Arrangement of Ideas

37

yet another point, t o the sequences already given. This adds tbe triad 011 t o the sequences for atoms, nations, and any other objects or substances, so that a full code sequence for Canada is 0.111.1.000.011, or 07103 in octal G t h o u t the points. H f we work entirely u.ithin the world o f mundane notions, the firsh zero in this number is superfluous, so that nations may be @ven the sequence 7 103 - seventh level, upper rank, unitary things. Atoms become placed at position 3003, being third level, lower rank unitary things. This way o f speaking demands that a zero level be allowed, below the first level, much as a gound Roor lies below the first Roor o f a building. This, o f course, is the level o f set Sieory. For example, addition is an assernbly action upon Ule nurnbers which fit in the lower rank o f that level; since numbers are sirnple attnbutes, addition is attributive. It gains code sequence 0024 in o c t d , standing for ground level, lower rank, assembly action. In the longer binary notation this is 000.0.010.100. In both o f these sequences a preliminary zero ( t o indicate rnundane ideas) is orriitted. Generics and collectives M i l e I studied semantic types I found many taxonyrns, words narning ideas which group other ideas into classes. I &stinpished two varieties o f these, the generics and the coIlectives. T o give instances hmediately: in respect o f a poodle, the notion o f a dog is generic and that o f a pet is collective. Iodine is generically a halogen but rnay be collected with other substances as an antiseptic or as a staining agent. Although generics and collectives o f d l semantic types appear possible, most o f those for which we have names are concerned with objects or substances: that is, with things. Generics are always closely concemed with the construction o f tbe notions they bring Logether. Iodine BS a halogen because o f the n u h e r o f electrons in the outer orbit o f each

38

The Fabric of fiowledge

of its atoms. Other elements with the sarne structure in this respect are also halogens. Iodine is a thing, and so is an electron, and indeed it seems that generic ideas always group ideas of the same semantic type as themselves. By contrast, a collective idea groups notions which are of a different semantic type from itself. An example is tbe concept of a weapon. Weapons are things grouped according to a phenomnon, namely their use. Antiques, ironmongery, roofing matenals, stock-in-trade, paint, tiles, professors and public relations offacers: these, too, are collectives. We live in a world of thern. Generics are easy to place in the pattern of integative levels. They occupy the position held by the notions they bring together. Collectives, however, tel1 mother story. A rock may be used as a weapon bui the concept of a weapon is not fully developed at the molecular Pevel where the rock is found. The proper home for a collective is the lowest grade at which its definition is complete. For example, the notion of a weapon may need an idea Erom the upper biomorphic rank as one of its constituents. If Ihis is the most advanced idea required, then this is the concept which determines the place which the notion of a weapon must hold.

The holotheme: a provisional summary It may now be useful to sumrnarize the view I had formed of my subject by the time 1 had finished this part of rny inquiry. Wuman knowledge, I concluded, is concerned with notions which fall into one or other of two perception classes, the lower of which contains eight clearly defined integralive levels each of two ranks each of eight fomative grades each containing eight semantic types each of which may be generic or collective to a greater or less degree. These rnundane ideas conespond to daily reality. The remaining notions may do so but await acceptante, or may be k n o m to be fiction, or may of their nature be unprovable. The complete pattern of al1

The Arrangemen t of ldeas

39

ideas may be cdled the holotheme since other words which might be appropriate are pre-empted for use in representing other meanings by other disciplines. Binary code sequences can be formed in such a m y that they act as descriptors of positions in the holotheme, and these may be shortened by transfomation to an octal form. The pattern they fix can be found by myone tvho makes constmctive chains of notions and examines thern. Within the class of mundane ideas the pattern is regular, to such a degree that it may even be used to validate notions which seek admittance to it. For example, the four elements of mediaevd cosrnology are earth and water, air and fire. These do not fit into the pattern at molecular level as neatly as do atoms, and anyone who tries to use them as subunits, units in the lower rank, of this level wil1 find that the result is an intemption of the major design. It is therefore in order to allot them to the special class, which in the proposed notation foUows that of the cornmon or mundane notions. The holoSieme is a pattern of ideas, not of statements; it is no more than a sequentia1 arrangement of the elements of knowledge, based on a design which seerns to be inherent in sorne of these. It has a place for lions, and a place for bravery, and even a place for the bravery of lions, but no place for the statement that lions are brave. Problems about statements, or about'how we know what we know, are not its concern. But it is time to turn to the mathematics.

The Development of a Theory &operties of relations Students of set theory become familiar with various properties of relations. Since the holotheme seem to be most neatly arranged if account is taken of how notions are constmcted, and since this is to do Mrith the relations embodied in the notions, the properties of relations cannot help but interest those tvkio e x m i n e it. "he relations dedt with in set theory hold between subsets of a main set of terms which Is called the full set. The full set is taken to be one of its o m subsets, md the empty set (the set with no terms in it) is also taken to be a subset of the full set. Every term in a full set is said to be a member of that full set, and it is also a member of many of the subsets. Some of the relations between these subsets of tenns are passive, being condltions; others are active, being operations. Examples of conditions are overlap, absence, presence, containing, rnernbership, identicdlty. Union, ántersection and complementation are examples of operations. These relations may be exarnined by means of a binary code sequence based on &e properties they possess. It is convenient to deal urith conditions first. Four properties are relevant, of which the first is refiexiveness. To place this property first is not an exercise of mere personal choke: reflexiveness can be defincd by the use of one subset only, and since no relation can &>e defined by the use of no

The Development of a Theory

41

this makes reflexiveness the first definable property. A ion is reflexive if it relates a subset (or, more generally, nn whatever) to that Same subset (or, more generally, me term). Identicality is reflexive because any subset is tical with itself. Absence is not reflexive because no of which we we thinking is at that time absent from our thought. Whatever we may symbolize by the letter A, the absence of A is not A. Pf in fact the letter A stands for a subset, and if the Petter R stands for a relation, we may say that reflexiveness is the condition in which A R A. The second property is symmetry, which calls for two subsets if it is to be defined. An example is overlap: if A and B are two subsets which have a cornrnon member then they are said to overlap each other: A overlaps B and B overlaps A, a condition which we rnay represent by saying that A R B The third property is transitivity, which requires three subsets for its definition. The relation of containing is transitive: if A, B and G are three subsets and if A contains B while B contains G then A contains C. We rnay say that we have a transitive relation when A R B and B R C and A R C. The fourth property differs from the others in that it does not hold between subsets of a fuU set but between entirely separate full sets - full sets of different varieties, vyithout any common mmber. So far as I h o w , this between-set has no standard name. I have found it convenient to ransversiveness'. The condition of membership is ve, being between totaljly different full sets. For a man may be a rnember of the subset of men with s. Watever this subset may be, it is certainly not a of the subsets of the full set of men is a mm. The set of subsets has no mernber in common with the full set en. Even the subsets consisting of one member only are thernselves m n . This effect wil1 be familiar to those who w how an index may be represented by a full set of items gs indexed) crossed by a full set of features (character-

istics which, the items may possess). Such a network is called a 'data field9, and in it the two full sets - the items and the features - are visibly transverse t o each other. Another example of different varieties of full set may be taken from geometry. Though all triangles are polygons, n0 triangle is a rectangle. At the degree of specificity at which number of sides matters, triangles and reetangles are members of absolutely different full sets. Relations between these sets are transversive.

Relation codes These properties of conditions may now be employed as a means of placing the conditions Siemselves in order. This is done by forming binary code sequences in which the first or right-hand position is used to show the property defined by one term only - reflexiveness. If a condition is reflexive, this position holds the character 1; otherwise O is witten. The second position deals ~ t the h property defined by two terms - symmetry. A symrnetric condition is denoted by the character 2 ; if a condition is not symrnetric, O is used. The third position is concerned with transitivity, the property defined by means of three terrns. Again the possession of the property is s h o w by 1 and its absence is s h o w by O. The fourth, the left-hand, position holds the character 1 if the condition obtains between full sets, and O if it is to be found within them - that is, between subsets of a full set. This is a natura1 sequence for the proper~es,being clearly taken from a numeric attribute possessed by the properties themselves - a defining attribute. W e n the sequence is put t o use the results are remarkably neat. To show the method, the condition of containing rnay be employed. This relation is taken t0 be not reflexive: it is defined in such a way that things are not permitted t 0 contain themselves. This being so, containing is readily seen t0 be not transversive, transitive, not symmetrie and not reflexive. The pattern of occunences

The Development of a Theory

43

word k o t 9 in this descnption is the pattern of zeros in ation's code sequence: 0100. following is a list of con&tions found in set theory, om absence (the simplest) to recigrocal membership, which el1 into the realm of transversive relations. A point is eed after the symbol which indicates whether the relation oncemed is transversive or oherwise. This helps to separate the three well-knom properties of renexiveness, symmetry and transitivity from the less Miidely treated transversive property, and thus to emphasize whether a condition is found within a fuPl set or between two os more full sets. Here are the conditions: 0.000 : absence 0.001 : presence 0.010 : disjunction 0.0 11 : overlap 0.1 00 : containing 0.1 0 1 : inclusion 0.1 10 : exclusion 0.1 11 : identicality 1.000 : mernbership 1.O01 : non-membership 1.O10 : reciprocal membership

A few comments on the meanings of the narnes of these conditions may be helpful. Disjunction is a relation between two subsets which have no common member: it is not transitive because although (in the case of three subsets) the first and second may be disjoint, and the second and third may be disjoint, nevedheless it does not follow that the flrst and third are disjoint: they rnay overlap. Inclusion is a forrn of containing in which a subset is allowed to contain itself. Exclusion, more commonly called m t u a l exclusion, is a dvanced type of disjunction: here, if there are three we can be sure that if the first and second are and the second and third are disjoint, then the first rd are dlsjoint also. Membership is a transversive

relation between a set of members of subsets and Sie set of subsets of which Lhey are rnernbers. It is an exarnple of a one-way relation between one fuE set and mother. Reciprocal mernbership is an example of a two-way relationship between full sets. For exarnple, in a data field, the items are members of the features and the features are mernbers of the items. From the v i e q o i n t of holothemics, this progession of relation code sequences has two properties of outstanding importante. First, it forms a definition series. Every concept it contains can be defined by the use of notions &ich occur e d i e r , except for absence, & e notion with which it starts. Thus presence is the absence of absence; disjunction is the relation between a present and an absent subset; overlap is the absence of digunction, contalning is the relation between two disjoint subsets and either of thern; Inclusion holds between a present subset and its overlap with another, exclusion is found amonpt many present subsets when these do not overlap; identicdity is the absolute absence of exclusion. These brief definitions need expansion and explanation, but they give the Ravour of the matter. The principle of definition by means of earlier notions continues into the ttransversive series. Secondy, the sixteen code sequences for types of relation conespond exactly vvith h e sixteen sequences representing the ranks and grades of each Pntegrative Ievel. They correspond not only in binary pattem but also in meankg. Thus the e x a ~ n a t i o nof notions occurring at higher levels led to the detection of assemblies, mixhires, series and other sorts of Pdea, which were then observed to exist in the lowest integrative level as wel1 as in the higher levels where they were first detected. These assemblies, mixtures and the like were to do ulith Siin-; Lhings are held together by their Qnternd relations; and now an examination of relations has produced exactly the Same binary pattern, even to the extent of producing a mixture condition (overlap) at Sle g a d e

where mixtures are found,and a serial condition (containing) tvliere series occur.These conditions of the Powest level seem also to be suited to the higher levels where the pattem was origindly found. As an example of this,consider the condition of being a subsidiav, which ocmrs between industrial companies.The companies are red: we are concerned with rnundane ideas at the cornmund level. The class m d level code is therefore 0.110 (or 06 in octal). The condition applies between companies,not wjthin them,so it is in the upper rank. lt. is readiIy seen to be transitive, not symmetric, and not reflexive. Consequcntly it carrles code sequence 1.100 (or 14) for its rank and gade. Findly, being a passive entitive relation, a structure (in this case a structure found in industry) it is of sernantic type 010:that is, type 2. Putting al1 these parts of its code sequence together,the referente 0.110.1.PO0.010is achieved, reduced in octal to 06142. This is the holotheinic position for the rclation,and thc way in which the properties of relations are used in order to work it out bas been clearly show. The relation code sequences throw considerable light on the holothemic pattern. For example, tbe property of renexiveness can be seen to correspond to the property of being a substance:the relation code sequences for the grades of simple,rnixture,sheet and stretch end dth 4, the reflexive symbol. This rnay seem hard to grasp,but it rnay help if we recall that substances have been defined as unbounded in at least one direction. In rnathematics,these is a well-knom eonneetion between the unbounded (the infinite) and the reflexive. An infinite set can be put into one-to-one(reflexive) comespondenee wit161 a proper part of itself.Thus there is an infinite set of natural numbers,and there is an infinite set of even naturd numbers,and yet the even numbers are only a of the whole,For odd numbers are part of the whole as

46

The Fabric of Knowledge

reflexive grade code sequence, 01 1, is the sequence for a mixture. This feels right intuitively: what is a mixture but the overlap of two simples, a region in which both of them are found? The code sequence of overlap incorporates the symmetry symbol, a unit in the middle of the grade triad. Again this is suitable for mixtures: if one substance is mixed with another, that other is mixed with the first. Assemblies also have this sort of symmetry, and their triad - 010, that of disjunction - reminds us that their component parts are disjoint: they do not penetrate each other as the constituents of mixtures do. This is apparent from the zero in the left-hand position, showing the absence of reflexiveness. The connection between h-ansitivity and the upper four grades in any rank may need less comment. Transitivity is the essence of order, and order is apparent in each of these grades.

Opera tions It wil1 now be helpful to consider operations. Operations are active relations and the properties which they possess are not those of reflexiveness, symmetry and transitivity, but of idempotency, commutation and dissociation. Like conditions, operations may be transversive. At the level of set theory this means that they act on whole sets instead of on subsets. An idempotent operation can be perfomed on a single term and wil1 leave that term unchanged. If an operation has this property, then its grade triad has a unit in its right-hand position. Otherurise a zero appears there. Using an arrow to stand for 'leads to' or 'results in', the letter R (as before) for a relation, and the letters A, B and G for subsets, we may say thatc an idempotent operation is one in which R A-A. Sometimes this is written A R A+A, which is helpful if the operation is normally performed on two subsets. An exarnple

The Development of a Theory

47

is the operation called intersection: this consists in forming the subset consisting of members which two subsets have in common. Some people are children, some are male, and it is possible to possess both features at once, so if we intersect the subset of children with the subset of males there will be a resultant subset, Ihe boys. Intersection of the children with the cliildren produces the chlldren as a resultant subset. The operation is idempotent accordingly. This property can be s h o w by rneans of a single t e m (in this case a subset),and thus coresponds to renexiveness. If we possess two subsets on which to operate, then the property of commutation may be shown. A comrnutative operation is such that the order in which the subsets enter into the operation does not matter. Intersection is commutative as well as idempotent: selecting the males who are children produces the same collection of people as does choosing the children who are males. If a double-headed arrow is used to signify 'has the same result as' then the commutative property may be symbolized byA RB-BRA. If an operation can be applied to two subsets, then it can be applied to three if we select a pair out of the three and carry out the operation on these, applying Sle result, by means of a second instance of the operation, t o the remaining subset. Here the order in which the pairs of subsets are taken may affect the final answer. If it does, the operation may be called 'dissociative'. This is not standard terminology. In mathemtics, Jack of dissociation has been taken as the positive property: it is named 'association' and a dissociative operation is called hot-associative9. However, if the pattern of occurrences of the word h o t 9 is to be used as an aid in f o ~ n g relation code sequences (as it is in the case of ditions) then dissociation must be used as the property eh is positive. If a pair of brackets is used, to show which ts are operated upon first, and if a crossed doubleed arrow is employed to mean 'leads t o a different result

f r o m 9 , t h e n dissociation rnay b e syrnbolized b y ( A R B ) R C++=-A R ( B R C). T h e reason for t k e name o f the prope&y rnay n o w b e c l e x : t h e result is affected according t o which o f its associates is first associated w i t h t h e central term. T t is easy t o check that intersection is n o t dissociative; co it m a y b e described as not transversive, n o l dissoeialive, commutative and idernpotent. T h u s it has code sequence 0.001 for its rank and gade. Pf this is compared ~ 6 t ht h e code sequence for t h e condition o f overlap ( n o t transversive, n o t transitive, symrnetric and reflexive - since a subset completely ovedaps itself) t h e t w o wil1 be found t o b e the sarne. In fact, it seerns that conditions are closely followed b y operations which becorne possible w h e n the conditions are available. This rule is catered for b y $he code sequences for semantic types, which place conditions before operations passive before active. T h e full o c t d sequences for overlap and for intersection are 00032 and 00036 sespectively. 1 t h o u g h m u c h has been w i t t e n about t h e operations o f set t h e o v these have n o t 21 been provided w i t h standard names. T h e following list places o n e o s more such operations against t h e code s e v e n c e for each g a d e o f t h e Bower rank o f t h e set-theoretic level, and for this purpose Ik makes use o f o n e or two names vvhic1-i are n o t i n comrnon use. 0.000 : complementahion 0.001 : conservalion 0.010 : adjunction 0.01 1 : union, intersection 0.100 : relative complementation 0.101 : restriction 0.110 : nand, nor 0.1 11 : counter nand, counter nor

Of these, complementation, intersection and union are dealt w i t h i n al1 works o n set theory. Intersection (choosing the mernbers c o m m o n t o b o t h o f t w o subsets) has been mentioned above. U n i o n consists i n choosing t h e members o f one, t h e other, or b o t h o f t w o subsets. Complementation

The Development of a Theory

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consists i n choosing t h e rnembers o f a fuP1 set which are n o t members o f t h e subset which Is complemented. Consenation is t h e stay-the-same operation, and as such it is generally ignored. Works which treat set theory in some depth are likely t o mention relative complementation - choosing t h e members o f one given subset M-ilich are n o t members o f anotlner. I f only one subset is available, it is rejected. Adjunction is less PikePy t o appear: It is t h e f o m ~ a t i o no f t h e subset consisting o f rnernbers o f t w o narned subsets which d o n o t have c o r n m n members: it appears, that is t o say, before overlap is available. Restrietion is n o t a standard name: here it refers t o t h e operation choosing a narned subset b u t ornitting frorn it any members which are d s o rnembers o f another narned subset. I f only one subset is available, it is T h e nand and nor operations are widely used i n t h e llogic o f electronic circuits. T h e first, t h e hot-and' operation, consists i n f o m i n g t h e subset o f dl members o f a full set M-hich are n o t m m b e r s o f b o t b o f t w o named subsets. T h u s i t is t h e complement o f intersection. It is sometimes k n o w n as rejection, or as t h e S c h e f f e r operation, after t h e Lheoretician w h o paid special attention t o it. I n t h e Same w a y , t h e nor operation, not-or, is t h e complement o f union. T h e counter operations t o these, at t h e highest g a d e i n t h e rank, are f o m e d b y means o f an operation o f complementation acting o n nand or o n nor. This has t h e e f f e c t o f making nand nd n o r idempotent ( f o r example, n o t neither A nor A is viously A). T h e names "counter nand' and "counter nor' are

and subsets are basic entities. Their qualities are ers, which are basic attributes. Specificdly, t h e qualities pnate t o present subsets are gositive natural numbers. r o is appropnate t o t h e e m p t y subset. Negative natura]

50

The Fabric of Knowledge

nurnbers, so to speak, suit absent subsets. The negative and positive natural nurnbers, with zero, form the integers. M e n a full set has been attained it becornes possible to relate subsets to it; these relations are accompanied by relations between the natural numbers or the integers exhibited by the subset and the full set. Thus a new type of nurnber, the rational nurnber, appears. Rational numbers occur in the upper rank of the set-theoretic level, where they take part in a pattern which is similar to that of the natural nurnbers in the lower rank. Rationals whose value is less than unity correspond to negative naturals, and those whose value is greater than unity correspond to those of the positive sort. Unity itself, the so-called identify element of rnultiplication, corresponds to zero, the identity element of addition; and as niight be expected these two operations occur in their appropriate ranks - multiplication in the upper, being an operation between the numbers appropriate to full sets, and addition in the lower, being an operation between disjoint subsets. The following is a selection of nurneric conditions and operations found in the set-theoretic level, together with their rank and grade code sequences: 0.000 : being negative 0.001 : being positive 0.010 : 0.011 : 0.1 00 : being greater than 0.1 01 : being greater than or equal t o 0.1 10 : 0.1 11 : equality 1.O00 : being fractional 1.O01 : being whole 1.0 10 : being relatively prime 1.0 11 : having common factors 1.1 00 : being properly divisible 1.101 : 1.1 10 : 1.l 11 : congruence

succession; inversion about zero addition cyclic addition subtraction cyclic subtraction subtraction of a sum ( - (p + q ) ) inversion about unity multiplication forming the highest common factor division division by a multiple ( 1/pq ) averaging

'

The Development of a I;heory

51

This patten? is rich in relationships. A simple example is afforded by the operations whose code sequences end in zero. These may be compared, rank against rank, as follows: (gade) (lower rank) 000 : inversion about zero 010 : addition 100 : subtraction 110 : subtraction of a sum

(upper rank) inversion about unity multiplication division division by a multiple

h o t h e r example is obtained by comparing both entitive and attributive relations, taking these from just one rank and gade, as follows: rank 0, g a d e 100, semntic type

000 : being greater than 010 : containing 100 : subtraction 110 : relative complementation

Here the great similarity between relations of the s m e rank and grade is obvious. If one subset contains mother, its naturd number (the nurnber of its members) is greater than that of the other. If we f o m a relative complement, consisting of members of the larger subset which are not mernbers of the smaller, then at the same time we subtract the naturd nurnber of the smaller subset frorn that of the larger.

Mathema tical structures M e n I have a problem it pushes itself into my thoughts in all the spare moments of the day. During meals H produce numberless jottings on the backs of envelopes, and I end train journeys with diagrams and statements and questions and bits of explanatory paragraphs on sheets of paper pushed hastily into my briefcase as my destination arrives. Such was my exploration of the relation code sequences of the settheoretic level. I was fascinated by the interlinked patterns I found there. Each new discovery added to rny feeling that this was the genuine one and only foundation for the

arrangement o f notions In t h e higher levels o f t h e holotheme. T h e m o s t powerful reinforcement o f this view, however, was rny sudden redization &at h e relation code sequence embodied a complete series o f mathematicd stmctures in exact and proper order. This series, i n order o f increasing complexity, is: semigroup, g o u p , ring, field, vector space. Briefiy, a sernigroup is an m a n g e m e n t i n which addition is possible; a group p e m i t s b o t h addition and subtraction; i n a ring addition, subtraction and multiplication can b e carried o u t ; i n a field, addltion, subtraction, multiplication and division are d1 available; in a vector space, more advanced operatiom can b e employed: an example is involution. I f t h e rank and g a d e code seqenences for uie operations u p o n numbers are e x a ~ n e din succession, it becomes cliear that t h e first four provide al1 that is necessary for t w o sernlgroups - one o f t h e positive naturd numbers, and o n e o f t h e negative naturd nurnbers, acted o n b y addition. T h e n e x t four add what is needed for a group, bnnging t h e semigoups together t o f o r m t h e set o f integers, i n which subtraction is always possible. It m a y be noted &at an integer is o f t e n defined as a set o f ordered pairs of naturd numbers, any one o f Mrhich pairs m a y represent i t ; and t h e introduction o f order is t y p i c d o f t h e transitive and dissociative upper parts o f an integrative rank. These elght grades o f t h e lower rank, containing khe ingedients o f what is called a group under addition, are followed b y a further eight which, i n t h e higher rank, hoPd t h e matenal for a group under multiplication. T h e first four o f these upper eight contain t w o semigroups as before, one dealing w i t h fractions and t h e other w i t h whole numbers: this time t h e senrllgoup operation is multliplication, and division is introduced i n t h e second four t o f o r m t h e complete group. Again a n e w t y p e of number is created: fractions and vvhole numbers f o r m rational numbers, and these are o f t e n defined as ordered pairs o f simpler nurnbers, any one o f vvhlch pairs m a y represent its appropriate rational.

The Development of a 2"heory

53

The additive group Mrlth the two higher semigoups form a ring; when the entire level is taken all together, its two ranks, each with its goup, form a field. Very neatly indeed, the set-theoretic level is a mathematical field, pure and simple. It is to be infemed that every subsequent level is, in its mathematics,' a field also. This leaves vector spaces unaccounted for. A vector space is a collectlon of elements wkch form a g o u p for a commutative operation, and which are subject to d l the operations of a field: for example, there is a vector space of pojynornials. To exhibit an instance: the polynomial 5x4 + 3x3 + 9x2 c m be multiplied (this is the field operation) by the rational number 6%. The commutative operation of the group of t e m s of the polynomial is the sort of addition syrnbolized by the plus s i p in the example; the field operaLion happens .in &is case to be multiplication, but it could as easily have been addition, division or subtraction. The t e m of polynoniials are the units of the next integative level, the level of spaces. In this level, polynornials play the part of integers. The lower rank is the rank of the red numbers, whicli have polynomial representations - for example, a decimal representation. Their vector space lies in one dimension Just as, one level bePow, the proto-space of set may be represented as one dimension in a In the upper rank, vector spaces of many appear, and in this rank the typicd nurnber is rnplex nunibers may be represented by vectors in ace of two dimensions, and also as two-part ing of a r e d number linked t o an imaginay is a real number multiplied by .\/ - 1 ). 'Ihey o be s h o w as a particular type of rational fraction, a whose numerator and denominator are polynornids. they show Liieir character as space-level brothers of nal numbers, which at set-theoretic level may be s h o w ratios of integers, and which (like complex numbers) cupy an upper rank.

FIGURE 4: Some of the structures found in mathematics, related to some of the notions encountered in the set-theoretic level

VECTOR SPACE in next integrative level

Al1 the notions given above are to do with numbers and only a few of those available are given. Cyclic addition is idempotent because p + p (mod p) is p: it has affinities with the set-theoretic operations of union and intersection. The numeric counterpart of the intersection of two different overlapping subsets is addition rnodulo p where p is the number of members in the union of the two. 'Proper' divisibility excludes the operation of dividing a nurnber by itself.

The Development of a Theory

55

Complex numbers may also be represented as a special sort of matrix, and matrices are sets of interlinked vectors. For example, a square matrix of ordes two is a set of four numbers, displayed as two rows of two numbers each. Each row is a vector; each column is a vector. It is remarkably similar in its looks to a data field relating one full set to another. Matrices are found in the upper rank of the spatial level, and may be used to represent the displacements, rotations, reflections, shears and other transformations of geometric shapes. The entities of the spatial level are indeed shapes of this type, and of course their component parts, d o m to the simplest part, a point. The algebraic representation of geometry, which gave so powerful an impetus t o the development of mathematics when it was recognized, arises because at the spatial level geometric figures are accompanied by appropnate numbers just as sets are accornpanied by numbers at the set-theoretic level. Ir may be of interest bere to display the level, rank and grade code sequences of a few notions in the spatial level. Thls level contains many more ideas than does uie level below it, and in consequente the following are the merest handhl comlpared with the n u d e r which are available for study. 001.0.001 : being a power of. Powers are powers of rational (or higher) numbers; the reflexiveness arises because any such number is a power - the k s t - of itself. Multiplied by suitable coefficients, using the contents of the set-theoretic level for this purpose, the powers become the terms of polypomials, and indeed may be thought of as the simplest form of polynomial in thek o m right. 001.0.100 : extracting the root. Many roots are irrational: here is the heart of the matter so far as a large class of real nurnbers is concerned. The grade triad is identical with that of division one rank below and with that of subtraction two ranks below. Just as root extraction (involution) has its connections with real numbers, so division is connected with rationals and subtraction with integers. The relation code sequences reflect this.

56

The Fabm'c of Kno wledge

4

001.0.1 11 : forniing a g e o m e t i c m e a n ( x y ). Compare t h e g a d e triad w i t h that for averaging (forming an arithmetic mean). I n each case t h e operation consists o f assehibling a number o f terms b y means o f a n operation o f g-rade 010 and t h e n applying the.result t o an operation o f grade 100 which is attached t o a nurnber which is t h e n u d e r o f terms entering i n t o t h e first operation. 001.1.000 : being i m a g i n a y ( o f a n u m b e r ) ; forrning an inner product o f vectors; matrix conformation ( t h a t is, matrix rnultiplication). 001.1.010 : t h e operation d x 2 iy 2 . This gives t h e length o f a vector i n terms o f its components along t w o axes at right angles t o each other. It also gives t h e argument o f a c o m p i e x n u m b e r e x ressed i n polar co-ordinates. In t h e f o r m r= i- y P it is f h e cquafion o f a circle. I n passing, t h e condition o f being-at-right-angles also has code sequence 001.1.010.

4:'

This rnay be sufficient to show how the code sequences continue into and Lh~oughthe higher level of the domain of mathematies. Before leaving this (vvith an apology for concentrating so rnuch upon nurnbers) it may be of interest to return to the very beginning and to look at the so-called Peano axiom for the fomation of the naturd numbers on which dl higher mathematics Is built. These axiorns begin with the existence of zero, and zero has code sequence 0.000.0.000.0011 (mundane, ground level, lower rank, lowest grade, quality). In passing, zero ernbodies tlie condition of nullity - 0.000.0.000.000. The axiorns proceed by using the operalion of succession, which (like zero) is at grade 000. Its full code sequence is 0.000.0.000.100. The result is to produce the nurnber one, unity, as the successor of zero. This process is then repeated to produce the later natural nurnbers. It is satisfactory to note that this well-know means of obtaining the natural numbers begins where it shodd, at the beginning. Into the domain ofphysics Sub-atomic physics gains powerful support from rnath-

Ille Development of a Theory

57

ematics. It is interesting t o see how the relation code pattem may continue from the higher rank of the spatial level into the lower sub-atomic rank. For Siis purpose the basic measurables which loom so large in physics and chemistry may be exarnined: mass, length, time and energy. Length appears in the lower spatial rank, with the arsival of the red nurnbers and of continuity, Time occurs in the upper rank of the same level, where the spacetime continuurn is developed. If the lower subatomic rank begins with photons and other particles without rest-mass, then this is where energy appears. Mass then follows, coming into its own in the higher rank of the subatornic level where it is manifest in the electron, the muon, the proton, the neutron and other massy particles. Given length, time and energy, a definition of mass is possible: mass is cdculated by first multiplying energy by time-squared, and then dividing the result by length-squared. Of course, these are not ordinary multiplications and divisions: we are in the presence of new operations, which are not too unlike the ordinary multiplication and division of integers or rational numbers. It is not common practice to define mass in ternis of energy, length and time: the usud process is to define energy, which is done by means of length, time and mass. In the present case, however, this is not possible: mass comes last: if we seek a definition sequence based on a relalion code then the notions to hand are the Mass is not the only physical measurable which can be two or more of the trio, energy, time any others c m be co defined: spin, action, surface tension, gravity and the like. Since on properties suited t o substances rather than exiample, the time is a duration, the length is r than a position - their g a d e triads end in a selection of these measurables, each gade, and with an indication of order

58

The Fabric of Knowledge 010.0.011 : 010.0.101 :

E T : angular momenturn, spin, action E : rnornent of inertia E / L : force

E / L2 : surface tension E / L3 : stress, gravity

E / T : power 010.0.111:

E T / L : nomenturn E / L :~mass

Here, of course, E stands for energy, L stands for length and T refers to time. The s i ~ i l a r i t ybetween the fomula for mass and those for an arithmetic and a geometrie mean is worth noting. Again we have a commutative but not dissociative operation combined with a dissociative operation which is not commutative to produce an operation which is both commutative and dissociative. It is also noteworthy how near mass is to the next rank, the rank in which it is embodied in massy particles. M e n I first encountered this part of the pattern, I was impressed also by the obvious interpretation of gravity as three-dimensional surface tension, a sort of effect of space stretching out, trying to be spacious, and thus bringing bodies together much as I have seen two needes, gently Roating on the surface of very still water, come together by two-dimensional surface tension effects.

Relation codes and the holotheme As I explored the pattem of relation code sequences in the domain of mathematies, it seemed that I was doing three things at once, or perhaps one thing in three dispises. I was trying to work out a mathematicd model to provide an underlying theory for my arrangement of everyday notions taken from higher positions in the holotheme. I was placing ideas in the first two integrative levels just as I would place them in higher levels, using the same mles and the same collection of sernantic types, formative grades, higher and lower ranks. I was ascovering a classification for the

The Development o f a Theory

59

discipline of mathematics itself. The theory or model I soucght, and the actual arrangement of ideas to be found in the lowest levels of dl, were one and the Same thing. I remember Iooking at the symboIs for relations between terms, scouring the pages of textbooks of dl degrees of difficdty, and fin&ng none that were not one or other of rny four semantic types of relation. Nor did I find any mathematical t e m vvhlch was not a passive or active entity or a passive or active attribute. As 1 followed the series of relation codes upwards through the level of space and into the subatomic level where energy and matter appeared, I b e g a to fee1 I was doing no more than unroll, with great labour, a chart of an ocean which was already weP1-knom to everyone but myself. At the same time I knew that other workers at the trade of organising knowledge had reached opinions at variance with mine. Most of these provided a pattem which was far less regular than my set of relation code sequences, a pattern whose very complexity seemed more in keeping with the bewrldering ramifications of knowledge than did my rigid binary scheme. Certainly a binary frarnework is suspiciously neat. I can understand the view that, perhaps unwittingly, I started with it and forced a large number of notions into the pigeonholes it provides. E-Powever, I do not think this was so. I knew nothing of set theory when I began, and the notation I used in order to handle entities and attributes in the early days of my inquiry was based on the alphabet. The pattern of noughts and ones came later, and this helped to persuade me of its validit y. Wh& impressed me most, however, was the sense of surprise I so often felt as bits of the pattern fel1 into place. Such unexpected pleasures came, for example, when 1 realized that the relation code sequences put mathematical structures in order from semigoup to vector space, when I found that they put relations into a definition series and when I saw how tbey could be equated with the binary code

6O

The Fabric of Knowledge

sequences I had derived as a result of looking at ideas taken from levels m c h higher than that of set theory and arithmetic. Another impressive aspect of the pattern was that it seemed to have predictive properties. To choose an instance from Sie cytomechanic level: I knew nothing about cellular biology, but the repeating pattern of units and subunits implied &at I ought to find, inside the living cell, something made out of molecules by the stages of assembling, forming series, and assembling series int5 systems. I looked at articles on the subject and found much about polyribosomes, nucleoli and the endoplasmie reticulunn. I also found the mitochondrion, an organelle complete with its o m enclosing membrane assembled out of standard protein and phospholipid molecules. The mitochondrion contained assemblies which formed series of specialized parts, and these transferred electrons from place to place in the course of building up molecules which actcd as energy stores. These operations were reversible: there was a means of releasing energy as wel1 as of capluring it. I was in the presence of the typicd bdancing effect of a system. I wrote the phospholipids and the mitochondrion into one of my chains of notions, as I have dready shown.

Rules for placing notions Thus I decided the relation code pattem was that of the holotheme itself. I might, I knew, be mistaken; but as my Inquiry continued I came more and more to fee1 that the mistakes were likely to be in the detail rather than in the broad outline of the amangement. It was, I concluded, time to develop mles for placing ideas according to the code. This was not difficult: it was merely a formalization of the way I habitudly found Ivhat I thought was the right code sequence for a notion I sought to place. However, though it was easy, it was important. It was a step towards finding how hard it

The Development of a Theory

6P

rnight be t o teach the method t o others. Here is a typical series of placement mles: First, write down your definition of the rneanlng of the word you wish to place. Next, consider the notion to which it refers or any representation of the notions it groups together. In respect of thls, ask: . . . Is it generally accepted as mundane, irrespective of faith, school or persuasion (0)or is it a matter of belief, fiction or hypothesis (I)? Frorn your answer, write down the code numeral for the significance class. Next, ask: . . . What is the highest unit which it contains or to which it refers or of which it is a representative? Tn which integrative level is this unit? . . . Is this unit in the lower (0) or the upper ( I ) rank of the level? Frorn your answers, write down the code sequence for the integrative level and rank. Next, ask: . . . lis it, or does Pt embody, a relation which is: intransitive or associative (0) or otherwise (l)? non-symmetrie or non-cornrnutative (0)or otherwise ( I ) ? not reflexive or not idempotent (0) or otherwise (I)? From your answers, write down the code sequence for the formative gade. Next, ask: . . . Is it passive (0) or active (I)? . . . Is it attributive (O) or entitive (l)?

62

The Fabric of Knowledge

. . . Is it a relation

(0) or a term (l)?

From your answers, write down the code sequence for the semantic type. These answers . in this order wil1 produce a code sequence for class, level, g a d e and type. Gonvert it to any shorter notation which may be in use, if this is necessary. Last, ask: . . . IS this sufficient, or is further specificity needed?

If the answer is 'yes', apply these placement rules again or use such other mles as may have been chosen so as to achieve the additional specificity required. This is a long series of choices, but experience in making them soon leads to speed. The.class, level and rank are often seen at a dance; the formative stage and the semantic type are supplied in the definition if it is based constmctively on suitable units. The mles are only called upon in difficult cases, and many of these arise only because the definition is poor. Irnprovements in this lead to immediate placement of the notion concerned.

The pattern in brief Again it is time to summarize. The pattern of notions derived from a study of ideas arranged in constmctive order includes, quite naturally, a portion which applies to the simplest, the most fundamental and in a sense the most abstract notions. These are the concepts used in arithmetic and set theory. It transpires that this part of the pattern can be derived by means of a very easy rule from the properties of the relations between terms - natural numbers and subsets - and that the result is a very vast neat arrangement of the notions in use at the set-theoretic level, the ground level of the holotheme.

The Development of a Theory

63

This caters for all semantic types at each grade in each rank of the level, and can be extended directly to cater for the next level, the level of spaces. Gonsequently it covers the whole of mathematics, and places the group, ring, field and vector space framework of the subject in the order it is already known to possess. It continues from thence into the FIGURE 5:The nesting property in the structure of binary relation code sequences, which corresponds to the many nesting and repetitive patterns in the holetheme. The stops or points which break up the code sequences as shown in the text are here omitted

Background and Comment me other views

To start with, it

ndamentd position for the science of sets and numbers ernained large. I concluded that much effort in the field of eneral classification was nullified by unrecognized preonceived opinions, lack of certain types of knowledge, and tion upon words instead of upon the notions these There was general agreement that something like levels existed; the problems arose when attempts to identify them. died the history I was impressed by the extent to e major sciences had had to develop before it had me possible to display any'Lhinglike a complete sequence egrees of complexity in the world around us. In the past, domains of mathematics and of the physicd sciences had n separated by unknown land, while another wide stretch

of ignormce had Lain between the physical sciences and biolocy. Mathematical set and number theory was not available as a unifying subject, and anyone who detected a patten? in part of the holotheme could follow it but a very little way. The criticism which can be made of work done afteï these gaps were closed is quite unfair if levied on earlier schemes for organizing knowledge. As I read, i t seemed t o me there was a great divide between the late nineteenth and the mid twentieth century. Before this divide, praise was due for the avoidance of pitfdls, but it was hard to blame those who fel1 into them. After it, different rules applied. The frontiers of the great domains of knowledge had been pushed back til1 they met; the countryside was mapped, at least in outline. Pitfalls were visible: no praise was now due for avoiding them. On the other hand, it was fair to regard falling into one as calling for a littli censure. There are now several widely-held views which, in the light of this development of the sciences, seem to me to be mistaken. There is, for example, a wish to base an integrative level on the mind and its creations. This is a continuance of a tendency which has been with us for many centuries. It takes several forms. The notions of mathematics may be given a home above the level of whole plants and animals, on the ground that a mind is needed in order that they may be f o m e d or comprehended. A psychological level may be said to exist, above that of biology and below that of society, with mind as its prime inhabitant. These views may occur together. Attempts rnay be made to form an integrative level out of mentefacts: creations of the mind, much as artefacts are (but are they?) creations of the hand. These views may be called instances of a psychocentric fallacy. So far as the relation code sequences are concerned, adoption of this opinion leads t o a break in the continuity of the structurd pattern. The pattern demands that more complicated entities in this advmced domain should be made out of simpler ones; but the mind is not constructed out of hurnan beings, or

Background and Commend

67

nable only by using the notion of a complete animal of sort. IIt is, I suppose, possible that a break of this type y occurs; buit I have not been able t o bring myself to leve so. It seems far simpler to treat the mind as a enomenon, the brain in action, amazingly skilful, Rexible d creative, yet easily sent awry, as are the other functions f the body. Speaking fos myself, my feelings, memories, sensations, emotions, habits, impulses seem to be inside me, d I find it straightforward t o cal1 my mind a phenomenon of the lower biornorphic ievel in consequence. Another example of this kind of mistake is the thinking which provides different levels for plants, thc l o w e animals, and mankind. This has a long history. From early times a minerd level has also been used, placed under that of the vegetables. The whole series Sien runs from mineral, vegetable and animal to abstract - &e abstract providing the mental element which is taken as typical of human beings. Like many other arrangements which do not tally with the relation codes, Siis is none the less a powerful means of locating notions. Glearly no criticism can be levelled against it from a holothemic viewpoint if its object is not to model reality as wel1 as may be, but simply to offer an effective classifying device. Other fallacies are of the dimensional type. Of these, a good example is the belief that to be later in time is to be more advanced in complexity. This is t m e as a general trend over long penods of tirne: the occupants of the biornorphic level have certainly appeared on earth much later than those of Sie molecular, and a theory of continuous creation is needed if we are to argue that all levels are found here or there in the universe at the same time, whatever &at may mem. So far, so good; but there are forces at work which reduce the higher to the lower, and these have led to the postulation of the existente of disintepative levels which hold the broken bits of higher things. Setting aside the question of what may be meant by a broken bit of a mode or

68

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a quality, the attempt to set up disintegrative levels makes it impossible to place - for example - an oxygen atom. Is it disintegrated water, or integrated subatomie particles? The better course is not t o trouble about origins, but only about internal structure. Origins, from this viewpoint, are irrelevant. The dimensions of space also contribilte to confusion. There is a tendency to fee1 "Lalat larger aggregates occupy higher levels than do smaller amounts. This leads to a suggestion that aggregative, as opposed to integrative, levels exist. The problem here is that there is nowhere to stop. It is simpler to return to set theory and t o recall that a larger set is simply a larger set, and that is all.

Laws of the levels Nluch work on Pntegrative levels has been carried out by J.K. Feibleman. In F o c m o n Information and Communication (ASLIB, London, 1965) he lists the following major levels: the physical; the chemieal; the biological; the psychological; the anthropologicd. We mentions a level of geometry below physics, and his anthropologcal level includes human institutions and cultures, thus extending the pattem up t o the higher rank of the national level. He comments: "he picture seems unfinished because of the dead ends and also because of the asymmetry of the whole.' As I write, Feibleman's account Is already seven years old, but this does not take it back beyond the point at which al1 the evidence for a more consistent pattern was available. If rny argument is correct, then Feibleman's work leaves out Parge parts of the dornains of mathematics and of the social sciences, while inserting a mental level in the best traditions of the psychocentric fallacy. Rernovlng the intmsive level, and applying relation code techniques t o the remainder, we go far t o produce the more coherent arrangement which he sought. It is fair t o add that his article ends with the question, 'Are there infrageometrical and supra-cultural levels which have thus far

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69

ed our sensibilities and our hstruments?' As to the -geometrical, it is clear I would answer 'Yes9. The -cuEtural is a different matter. If the cultural level rnay quated with the national, then anything supra-cultural , in the language of these pages, be in the higher erception class. Perhaps that is right; but again, perhaps it is t a matter of how we define the rneanings of our words. In the s m e work, which is a revision of an earlier article in e British Joumal for the Ptiilosophy of Science (May, 954), Feiblernan presents a considerable number of Paws relating tbe levels to each other. The following are examples:

. . . the higher the level, the greater the complexity of entities. . . .in any entity, the higher level depends. upon the lower. . . . the later the date, the higher the level of organization. . . . in any entity the lower level is directed by the higher. The concept of laws of the levels is very attractive, but for my part I should like t o see laws of a more specific type tban these. For example, is it true to say that qualities of higher levels arise from the phenomena found below them? Golour is a case in point: it may be a quality at molecular level, but it sterns frorn the wavelength of light, and vibration of photons is a phenomenon of the subatomic variety. Does length at the spatial Ievel arise frorn a change in numeric rneasure at the set-theoretic? Is the honesty of a community a result of the frequency with which its individual rnembers perform honest actions? This looks Iike a reasonable law, which may give insight into the constmction of ideas at many levels. But are there other sources of qildities? And what about modes? Modes are the qualities of phenomena, which are things in course of alteration; if qualities arise from lower phenomena, do modes arise from lower things? Acidity rnay

7O

The Fabric of Knowledge

be a mode at molecular level, but it is measured by finding the concentration of hydrogen ions in the solution in question, and hydrogen ions are protons, so that the measure arises in the subatomic level, and is to do with Siings, as suggested. If laws of this type can be found, they wil1 @e the student of the holotheme something to get his or her teeth into.

The awival of new levels This may be a suitable point at which to discuss a few problems, chosen more or less at random, concerned with the way the higher grades and levels develop from the lower. These are further examples of the matter from which reliable Paws of the levels may one day be derived. The first of these problems concerns the case of the fealher and the bird. The question is: How does the holotheme unroll in such a way that feathers need not be developed before birds appear, even though feaSiers are constmctive parts of birds? It is of course possible t o argue Ratly that the question arises from the temporal version of the dimensional fdlacy, and therefore should not be asked. It deserves more respect, however. We are familiar with one-member sets in mathematics, and with one-man companies, single-cel1 animals, and one-city nations like Monaco and Singapore. The atoms of inert gases can be ihought of as single-atom molecules in view of their Jack of vdency. In al1 these cases the Same entity seems t o be an inhabitant of two levels at once, depending on how it is regarded. If we look at set-theory we can see in a general way what occurs. We start with the one-member sets. Full sets of this type can show no internal stmcture even though, as full sets, they occupy the start of the upper rank of their level. However, at this position there must be at least some inhabitants with internal stmcture, since otherwise the lower rank would be

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71

tured. Fortunately we do not have far to look. Sets few as two members display structure: in this case the uch a two-member framework is simple, however. To w how simple, we need only consider Uie condition of rlap - rank and grade code 0.01 1. A two-mernber full set not show this, since it cannot be divided into subsets ich have a common member and a different mernber. Only 1 set of three or more mernbers can provide this facility. rt, if we have to build a set of at least three members much internal structure c m be displayed, it is clex igher units may at times hold too few lower units to al1 tlie intemening degrees of complexity. This is what ires to be the case in practice in higher ranks and IS is why birds and their feathers may arrive simulaneously: the birds bring their feathers with them, just as a sufficiently large set brings alll its subsets with

Aggregates o f matter The series unit - simple - assembly - mixture - series sheet - system - stretch has already been described. It is the series of grades on the passive entitive side of the holotheme. It suggests that the most inclusive course tlnrough the avallable notions of things may run alternately from the bounded t o the boundless and then back to the bounded, and then to the boundless again, and so on. An exmple is the progression from a carbon atorn, a unit, through the sirnple carbon to an assembly, a diamond. At the upper rank of the molecuIar level this progession may be examined at g a d e 101, where boundless expanses of molecular materials may be found. Sheets of this type need not be infinite even though they are boundless: for example, like the air or the sea or the upper mantle of the earth they

72

irhe Fabric of Kno wledge

may curve round to form the atmosphere, the hydrosphere and the Iithosphere; and it wil1 be noted that outer shells include inner shells. The relation of inclusion, we note from set theory, carries the same grade code as that of a sheet n m e l y , 101. This is intriguing. The earth is a finite but unbounded laminate of sheets of tbe upper molecular rank. This remark disregards our planet's coat of life, but is quite satisfactory from the viewpoint of astrophysics. Two further gades are available in the holotheme after this, before the domain of the physical sciences is complete. The solar system may perhaps be an inhabitmt of one of these. It is not rnerely Zarger than the earth: it is more cornplex. It is a system in the meaning here adopted for the word: the sun and its planets are held together by counterbalancing forces. If soliar systems are in the penultimate g a d e of the molecular level, as this may indicate, what occupies the highest gade, before the cytomechanic level begins? It may be &at the physical universe as a whole is to be placed in this position. It certainly seems to be unbounded, which is in keeping with the grade code sequence 111. Ml this is highly speculative, but shows how the relation code can be used in the business of placing elements of the cosmos in order. This approach may be contrasted with other attacks oai the problem: for example, the suggestion that aggregatlve levels exist, to cater for collections of gross materia1 bodies. This has already been mentioned, as an instance of a dimensional fallacy.

The lowest rank of the domain of physical science, like the highest, is cunently the site of intense and costly effort, both intellectually and by way of a great investment in equiprnent. Somewhere between the photon and the electron there lie - if the relation code is right - assemblies, series and systems of objects whlch are yet to be found, or at any rate

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73

ified among what is already known. 'Here be quarks' Es sornewhere on the map of this misty territory. It is ting that sine waves appear to possess the highest n code sequence in the spatial level, and that the s which irnlnediately follow thern are described by ans of two such waves, an electric and a magnetic ponent, at right angles to each other along a cornmon is and out of phase in such a way that the locus of the of intersection is a helix. Of course, to visualize a point n this way it is helpful t o think of the waves lanes. The vision of these little intersection themselves along at incredible speed, perhaps lacement of one set of waves along the axis in other set, is one of the means I have tried to of quanturn theory. If only they uld rol1 themselves up into closed circuits they might act e tiny gyroscopes and produce effects of inertia leading to ass. Such imagining is dangerous: one gets laughed at by the xperts. Perhaps it may be best to keep quiet, after noting at if quarks exist and are found - as predicted - to build into subatornic particles with rest rnass, they rnay wel1 n out to be what the relation code knows as systems. separate from the unit in which rhaps that is why quarks are so elusive. blems may be resolved, these notes the cosrnos, the feather and the bird, how the relation code sequence rnay attitude, out of the very attempt to ance Mth it. These are ihe sorts of to yield specific laws of conranks, grades, and semantic types in a consistent way.

Other recent work As I w i t e , the most recent work h i c h is to hand on these

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topics is the Library Association9s Research Pamphlet Number One: 'C6lassification and Information Control' (London, 1969). This contains a series of papers representing the work of the Classification Research Group from 1960 to the year of publication. Since these display the stages of an inquiry which is still in progress, they are not always consistent with each other, and show several alternative views of their subject. They are a most convenient source of examples for any study of the problerns encountered during the search for integrative levels and for a classification of notions based upon tbern. The pnncipal paper is Derek Austin's "he Theory of Entegrative Levels Reconsidered as a Basis for a General Classification9. If niy present views are accepted, then several of the engaging sidetracks have been taken here, including Siat which leads to the dimensional fallacy, with its requirement for aggregalive levels, and that which leads to a viewpoint from which the problem of the bird and its feathers appears forbidding. Oliver k. Reiser9s Tke Integration o f fiman Knowledge (Boston, 1958) presents a philosopher's approach to the study of integrative levels. Et offers no specifically patterned approach, such as can be obtained by way of relation code sequences, but presents much relevant matenal (for example, in the region between mathematics and tbe upper level of the physical domain) and comments on many of the effetts arising from the existente of the levels. Reiser remarks, for example, on the curious property of the square root of minus one, which is a component of complex numbers in one of their forms, noting that it sends matters "ff at right angles' to their previous direction. This he takes to be a typical sign of the arrival of a new stage of integration, and his opinion seems to be confirmed by the way the condition of being off at right angles is found again and again in the relation code pattern. It first appears in a pimilive fonn when two full sets are related to each other in the upper rank of the set theoretic Icvel by means of a data field diagram of rows

Backqound and Comment

75

cted by columns. It also resides in the upper rank of atial level, where complex numbers appear in essenthe way Reiser mentions. Pt is encountered in the ns between the semanlic types, which can be s h o w by of a cube diagram tvhose relation-term axis is at right to the entity-attribute axis d i l e both are orthogond passive-active axis. On Sie active side the systems of ntial equations used by Bertalanffy in %n Outline of eral Systern Theory9 (British Journal for the Philosophy cience 1950, vol. 1) display the same effect. These are the behavlour of biologicd systems, and rectangular pattem of rows and columns. is not to explore the pattern of integrative it is to set up a general Lheory of the ents of systems d e r e v e r they rnay occur. In to study Sie whole of the active side of the in the end ~ t every h notion whose unit in rhe first position of the final ary triad, signifying an active sernantic type. Using the bol x as a place-holder, &is mems aU notions A t h a ary level, rank, grade and type code sequence of the form

a desire to find and understand the pattern of the ut US is built into us, one might say genetically, as a sunrival. To see the possibilities in a situation, even seize a stick for defence, is a fundamental use of imagination. To seek a general understanding of se is made seems no more than a development of seeking a specific understanding of how in to catch a fish'or attain a fruit which is out of reach. Before the arrival of vvntten records we have no means of knovving tvhal speculation and what insight were achieved

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d o m their views of the worId when they had learnt t o spell. From the earliest times we have records of commercial transactions, lists of possessions, codes of laws, letters of command, annals, rituals, royal and religieus documents. As soon as a collection of these became large enough, a filing and finding problem appeared. This is still with US. Our methods of solving it forrn a spectrum whose endpoints are the classification of items and the co-ordinate index. The first of these consists in breaking d o m the whole range of infomation with which the collection deals (and this may be coternunous with the whole extent of knodedge) into mutudly exclusive and collectively exhaustive sections. The sections are then broken down In a similar way, until a sufficiently large group of sufficiently fine divisáons has been achieved. The material to be f2ed is then placed in position according to its main subject matter, this being the subject represented by one (and only one) of the available divisions. This approach encourages the keeper of the records to put the major sciences, technologies and arts into a suitable order. The choice of order rerlects the persond views of the classifier, strongly affected by the informed opinion of his time. This method is of old and respectable lineage. The second technique is of more recent origin. It consists of making a list of the elements of knowledge and using a search mechanism to bring them together to describe the material sought. This material - books, papers, files, pictures and indeed anything else which is indexed - is stored in any convenient order, generally a numerical order whicb corresponds to the order in which it was acquired. The elernents of knowled- d i c h are used to find it, however, cannot be so treated. They must be given a suitable arrangement. %is time, however, the keeper of the records is not faced with a probliem of analysing the great kingdom of infomation into its constituent disciplines. Instead, he or she is concerned with finding positions for notions whieh are to be used in synthesis.

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77

e elements of knowledge may be stored alphabetically g to their names, so that a sort of dictionary of search words if f o m e d ; or they may be arranged ing to any convenient special grouping (for exarnple, heme of notions used in an index of personnel records cted by the administralive needs of the organization cerned); or, lastly, they may be both classified and emed with a range of knowledge which is wide enough views on the structure of the world to be brought into nto a suitable, entirely generd order. This last rny own investigation of the holotheme. -ordinate indexhg 4s a recent development, arrange the ePements of knowledge in a er is old, perhaps even older than the anrangeTbe First Ghapter of Genesis come heaven and earth, light and ness, sea and land; plants follow; then, in order, come birds and animals of the land; man appears ai the end of the sequence. This is not presented in the f o m 'first came trophysics and geology, then optics, oceanography and ineralogy, followed by botany, three branches of zoology end9, It is to do .evith individual rnents of notions reveal a strong drive an order of Importante, generally startlng rds, and mnning through the vegetable and to reach mankind, tkne realm of ideas, and at of the supernatural. This is in general an increasing power, increasing commaterials are inert; plants can do litde; animals can do something; mankind can do much, especiafly with the aid of reason; a god can do e v e ~ t h i n g .

The Greeks In the West, it is with the Greeks tliat we first find a new

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The Fabric of Rno wledge

type of general patternaproposed. This arises from speculation about how the world is made: it is a constmctive pattem, and as such it is the ancestor of al1 later studies of integrative levels. Thales of Miletus held that al1 things were forms of the element water, which could become solid to varying degees, forming earth and rock, or could pass throuci;h mist to air and so to the even more pure aether and thus to fire. Anaximenes, somewhat later, chose air as bis elemental substance. Anaximander, between the two, fixed on an indeteminate substance from which air, water, earth and fire separated out, and into which, by a sort of cancellation, they could return. Thus a considerable school of philosophers, two thousand five hundred years ago, more or less, bcgan t o look at the ultimate stmcture of things and selected various types of matter as their starting point. At the Same time, in tbe Same tradition of constmctive pattem, Pythagoras concluded that al1 things were numbers. Thus Ivithin little more than three generations a division of view between those who start with physics and those who start with mathematics becarne established. This division still continues to trouble us. For example, the work of the Glassiiication Research Group, already mentioned, starts with what is called 'level P: fu~damentalparticles', placing mathematical notions much later, amongst the mentefacts. My own analysis, by contrast, follows the views of the Pythagoreans: that is where my investigations led me. As it happened, the Pythagoreans ran into trouble whcn they found irrational numbers (the very name, irrational, is due to them). They could not accept these, and in consequente found it impossible to make a neat connection between numbers and geometry. In the lanpage of holothemics, they failed to move easily from the set-theoretic t o the spatial level. About three generations after Pythagoras, Zeno studied mathematical problems and reached a conclusion which, when I first read of it, led me to compare his work with the conclusions indicated by the relation code. Zeno was

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79

med with comparisons between geometv and arithHe decided, as Burnet puts Bt (Greek Philosophy, n, 1914) that 'geometry cannot be reduced to arithso long as the number one is regarded as the beginning

anguage of the relation code sequences, it amounts to saying t points, in the spatial level, conespond to instances of in the set-theoretic level. This is tme: the rank and grade uence of each is 0.000. It used to concern me, a little, that eared in a g a d e vvliich I knew as the grade of units; and fire (which was Heraclitus' choice for the ry rnaterial) were all thought of as substances; they fonned the numberless materials of

unsplittable atorns, which were arranged in differing patterns according to the substances which they cornposed. Democritus was stil1 alive, though oPd, when Plato founded the Acaderny in Athens. Plato developed a theory which brought together the idea of a smal] number of different elements (air, water, earth, fire and a rather rnysterious fifth), the concept of atoms, and various principles of geometry. These he connected by associating atoms of the vanous elernents with the five regular solids: the triangular pyramid was the shape of the atorn of fire; the octohedron was that of the atom of air; the icosahedron was the shape of the atom of water; the cube was that of the atom of earth; and the most refined substance, associated with the farthest boundaries of the universe, had atoms in the shape of dodecahedra. From the v i e v o i n t of holothemies, the interest in this is the connection between the spatial Pevel and the domain of physlcs. Plato went further: the regular solids can be built up from triangles, and these include the half square whose hypoteneuse is incornrnensurable with its other two sides. Here Is

an irrationd number of Ste sort which perturbed the Pythagorems. Here is a connection with the set-theoretic level. Thls Sleory even provides for sornething akln to chemica] reactions: an octohedral atom of air provides enough trimgles of the right type to form two atorns of fire. The propession of levds upward frorn the physical' omitted the cellular: in the absence of the microscope it is hard to see how it could have done othertvise. Aristotle, who studied under Blato and later set up his ovvn school at the Lyceum, was the son of a physician and put great effort into the study of living things, but he coiald hardly start at a stage lower than that of organs which could be seen on dissection. Lucsetlus, w210 lived in Italy sorne two hundred m d fifty years later, w o t e (in "he Nature of the Universe') that 'whatever is seen to be sentient is nevertheless composed of atorns tbat are insentient', thus emphasizing a view which tied living m t e r i d s to fundamental unsplittable particles. Such assertions, however, &d not go far to cxplain how inanimate matter was prevailed upon to produce the effects of life and the rnind. Aristotle had concluded that a fifth element, the pneuma, was involved, in addition to air and water, earth and fire. One is reminded again of the fifth regular solid. Among the substances formed out of pneuma, thcre are vegetative, sensitive and rational substances, corresponáing to the naturd, vitd and rational spirits of the Stoics, whose developing system of phillosophy lasted for some five hundred years from its beginning shortly after the life of Aristotle. Here again we meet the sequence plant beast - mankind. Galen, at the end of this period, held this view, and it was passed on to the thinkers of the mlddle ages.

Mediaeual do modem The tendency for mathematics to appear first in systematic treatments of knowledge continued and continues today: &e present analysis in the language or relation codes is an

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In CZassification and Indexing in Science (London, dition, 1959), B.C. Vickery lists more than twenty of classifications of the sciences and technologies. w how hese have been viewed, throughout the as more or less homogeneous blocks of knowledge om the whole of human experience and moved place to place as may best appeal to the classifier oncemed. They also show that, in this selection at least, hernes beginning with mathematics or with logic followed by mathematics outnumber the others by more than two to one. In 1140 Hugh of Saint Victor began with mathematics and geometry; Roger Bacon did the same in 1250. Both these schoolmen also adopted the pattern 'mineral - vegetable aninial' higher in their arrangements. So did nine-tenths of the remainder of the examples. This is a very persistent pattern, and indeed it is perfectly useful even when the classifier regards botany and zoology as equds in the same integrative level or domain. The principle of the Great Chain of Being, starting with minerals, excelling in durability, followed by plants, excelling in growth, and then by animals, excelling in the bodily senses, and then by mankind, whose excellence is reason, is clearPy stated in E.M.W. Tillyard? 7%e Elizabethan World Picture (London, 1943). The scherne continues through nine ranks of angels to the Deity. The reign of Elizabeth I just entered the seventeenth century; it was about two thousand years since Plato" day. During al1 this period there was no increase in knowledge sufficient to make any major change in either approach to the arrangement of ideas - the classification of main subjects or the finding of a sequence in the chain of individual objects. Object sequences continued to suggest subject sequences to writers of encyclopaedias and other accounts of the world. They still do. Uses for object sequences, however, were few. Gauring the next two hundred years, this situation was to change. Many phllosophers and scientists, trapped in their

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own languages after the decline in the use of Eatin, feBt the need for developing a new universal speech for use in international learned discourse. This need was felt even by those who still wrote in Eatin themselves. Arnong those who were attracted by the idea, Francis Bacon, Bewton, Descartes, Eeibniz and Wilkins may be mentioned. A universal lanpage demanded an arrangement of notions which irnitated reality with sufficient precision to be a reliable basis for the new t o n p e . In 1668, Wilkins tried to meet this need with his Essay towards a Real Character and a Philosophical Language. This contained a remarkable arrangement of ideas covering an immense range of knowledge. In passing, it incorporated the ubiquitous mineral - vegetable - animal sequence. Wilkins included qualities, actions and relations, together with much about the rnind. At the foot of each branch of hls classification of notions, a set of synonyms or closely related words is found. Nearly two hundred years later than the Essay, in 1852, Peter Mark Roget publlshed his Thesaurus of English Words and Phrases. He had been working on it since about 1805. Like the work of Wilkins, it is an arrangement of notions, round each of d i c h a collection of related words is clustered. In his introduction, Roget mentions various purposes his work might serve, including the standardization of the English 'tongue, the forming of a polyglot lexicon by using the Thesaurus to arrange words in other lanpages, the analysfs of ideas, and the development of a universal or phllosophic speech with a view to removing the barriers of comrnunication between &e different nations to mankind. These are al1 secondary purposes, however. The pnncipal aim is to provide a pattern of ideas in which similar notions are brought together, as an aid to literary composition. Roget starts his work with abstract relations and nurnber, followed by space, inorganic matter, plants and animals and mankind. Then he travels through sensations, the intellect, the comrnunication of ideas and individual and social

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i o n , where m a n y concepts o f t h e c o m m u n d and nationd are found. We t h e n proceeds through persond, symetic and moral affections t o a concluding section o n n: superhuman beings, doctrines, sentiments, acts and utions. ' I h e arrangement is so effective t h a t w h e n it was ~ s e dmore t h a n a c e n t u v after its first appearance its n e w t o r reported that some f i f t y thousand n e w entries had een made in it Mrithout destroying its framework, which was til1 b o t h workable and comprehensive. At t h e time o f its first lication, Boole9s Laws of Thought and J o h n Newland's o f octaves for t h e chemicd elements were still t w o years e future. Seven years were t o pass before Darwin hed his @i@n of Species, twelve before Mendeleev ced his periodic table, nineteen b e f o r e Rutherford, w h o split t h e atorn, was b o m . During t h e second half o f t h e n i n e t e e n h century (starting, b y a co-incidente, i n 1 8 5 2 ) Herbed Spencer began t o put forward an early theory o f i n t e g r a ~ v elevels based o n t h e idea o f progressive movement f r o m t h e simple t o t h e complex: evolution under its most general aspect. I n First Principles ( L o n d o n , slxth edition, 1900), n o w f o d i f i e d b y t h e w o r k o f Darwin and Wallace, h e gave as examples t h e development o f t h e earth f r o m a h o t b d l o f gases, t h e differentiation o f plants and animals f r o m simplex ancestors, and t h e developm e n t o f supranational bodies o u t o f nations. Me emphasized that society is an organism, comparing it - and contrasting it - w i t h t h e indlvidud animal. This aspect o f t h e matter is s h o w wel1 b y t h e relation code sequences: t h e very pattem o f units and zeros in t h e codes for t h e cytomechanic and biornorphic levels is t o b e lound embedded i n t h e code pattems o f t h e c o m m u n d and t h e nationd. Pt is easy t o treat a road system and a cardiovascular systern as serving a similar purpose at different levels; t h e same applies t o a telephone system and a nervous systern. This use o f t h e c o n c e p h f evolution as a tool t o produce an ordenng o f notions cornes close t o present-day views o n

integratlve levels. Above the g a d e of the photon, at least, it seems that more complex entities and attributes have appeared - indeed, could not help but appear - later than simpler ones. Though the redization of this has led to some dimensiond fdlacies (as dready mentioned) the fact of a generd movement seems to be certain. Sorne thirty years younger than Spencer, Melv2 Dewey c m e t o the problem of arranging knowledge from the side c l the practicd librarian. His approach was that of the classifier: knowledge was divided into ten main classes - generdia, philosopliy, religion, sociolo-y, Pingeiisties, pure science, applied science, secreation, literature md history. The first edition of the Declmal Classificatlon whieh he created appexed in 1876. It was too early for sufficient knowledge to have been amassed for a fuU statement of integrat3ve levels to be worked out; and even if this had been available there is rio assurance that It would have been adopted as a means of producing subject order, Whether, and (if ss) how, to use integrative levels for this purpose h a not yet been determined, and may have t o await a uilder understandkg of the pattem which is now appearing. At any rate, Athin the smdl section dlotted to the sciences, something Pike a series of levels appears in Dewey's wsrk: mathernatics is iollowed by astronomy and physics; chcmistry and earth sciences come next; Snen after pdaeontolo-y eome Sie biolo;ical sciences - botany first, zoology next. Applied sciences, which form the next main class, rnay be tliought of as rnainily eoncemed Mrlth industrial and commercid activitles at & e communal level; much the Same may be said of reereation except that here we are concerned G t h leisure-time actlvitiies. The class concemed uilth Interature would, in relatlon-code parlance, be found largely in the class of specid ideas. The social sciences, in the highest domain in the mundane class of notions, are placed M o r e pure sciences and languages by Dewey: the present holothemie malysis would place most of them (for exarnple, poPiticaE science, economics, law, public adminis-

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above the cornmunal (industnal, commercial, recthe advent of the twentieh century, the contrast an order of subjects and an order of elementay ns becarne a matter for study. Brian Vickery, in the d, comments on the lectures of Richardreferente is to E.G. IPichardson, Classfication, licleoretical and Practical, New York, 3rd edition, 1930). Rickiardson remarked that 'the order of the sciences is the order of things', and made it clear that by 'things' he meant objects, activities, qualities, ideas and, it seems, in general dl the sorts of term and relation here called ksemantic types9. He pursued the matter of arranging things, defined thus cornprehensively, and gave in essence a series of levels be@nning with that of sub-atomic particles and proceeding hrough molecules and cells to men and societies. He held the view that to classify subjects rather than objects was 'a profound theoretical and practica1 mistake, leading to endless confusion'. Gertainly anythlng can be a subject, and c m then reach forward and backward through the holotheme until its boundaries are lost and it is tangled with every other suibgect in the book. Let us make a science: first, wnte a word down at random; next, add the suffix bology'; behold, a subject is bom. Interest in the constructive order of noLions becomes more and more apparent as work on organizing knowledge proceeds in the twentieth century. The actual name, "ntegrative level9, does not seem to have been used before 1937, when Joseph Needham introduced it in his Herbert Spencer Lecture at Oxford, wkiich was entitled 'Integrative Levels: a Revduation of the I&a of Progress9. This is printed in hls book, Time, the Refreshing River (Eondon, 1943). Needham mentions just one earlies use of the word 'level9 in this context (S. Mexmder9s Space, Time and Deity, London, 1927). Much of his article deals with the rìghtness of Spencer's views on the organization of society. Ne reliates

86

The Fabric of ~ n o w l e d g e

these to the existence of a level concerned vvith nations, discussing improved internal nationd structures and the development of higher fonns of society which bning nations together - that is, discussing the contents of the higher rank of &e nationd level. Towards the end of his account, he restates the "ant vista of evolution', in particular noting the parallel development of plants and animals and referring to mind as 'a ahenomnon of high organizationd level'. Here he agrees, even in the wording chosen, Mrith the placing of mind as a phenomenon in a high position (though not, of course, the highest). So brief an account of the precursors of the relation-code zipproach to the holotheme must be supplemented. Stephen Toulrnin and June Goodfield7s ishe Architecture of Matter (London, 1962) contains a thorough survey of views and inquiries about how the world is made, from the prac.ticd wisdorn of an mcient craftsmen to the conclusions of the present day. Ets epilogue expsesses the authors9 view: "he most far-reaching outcome we can hope for frorn twentiethcentury matter-theory - which includes both quantum mechanics and molecular biology, as wel1 as hdf a dozen other specialities - is a comrnon system of fundamental concepts, embracing material systems at every level., 1x1 rny more hopeful moments I like to think that the relation code systern, based on arranging ideas according to their structure, may turn out to be such a unifying instrument.

The principles of a co-ordinale index have dready been descnibed, and we have seen that If such an index is desipned for handing documents with a wide range of subject-matter then its thesaurus (its collection of elements of knowledge) must be equdly wide. The relation code principle can be used as a basis for such an extensive thesaums, either as it stands or as a hidden classification of notions governing the relations

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between ideas, when these are recorded in the retrievd mechmism in dphabetlc order of the names they are $ven. In practice, most thesauri used for this purpose are dphabetic, so a mems of relating the ideas is most helpful. It is remarkable how far asunder the alphabet can place related notions. There are many reasons for recording the connectlons between ideas in such a list. For example, the troubles caused by the existence of synonyms and Inomonyms m s t be overcome; proper rdations between words of broader and narrower meaning must be established, and means of building up more complex notions from those of the simpIer sort must be made safe. In the absence of proper knowledge of how to assemble ideas, they may combine in undesirable ways and lead the index to find documents which do not satisfy tbe search questions. It was work on problems of this nature which first led me to study and identify the senantic types. It led me to Sie distinction between generics and collectives, and it provided me vvith an approach to the definition of integralive levels and fomative stages by noling the limits of semanlic factoring. Semantic factoring ( k n o m to linpists as componentid mdysis) can be wel1 s h o w by means of an example. Consider the notion "choolgirl" This can be analysed into the notions >uvenile7, 'human', Yeminine' and "pupillary'. The first three of these make up the notion "rl', which is not quite as specific as kchool$rlY. "he first two f o m 'child9, whicb is even less specific. To add >uplllary9 to this is to form 'schoolchlld9, a notion which has LPie s m e degree of specificity as 'girI9. If it is carned far enough, semantic f a c t o h g leads to the recovery of notions which cannot be fudher factored. At this point, dictionanes tend to rely on synonyms ("Jvenile: young9, says the Goncise Oxford). Encyclopaedias may try the sort of ostensive definltion which provides a picture of the thing (or other term) concerned. In addition, or

dtematlvely, they may try verbal description; but this leads to circdar definition if it is not made up entirely of notions of a lower gade than the one defined. h fact, to keep a definition sequence in being, directed unifomly from the simple to the complex, constrnctive descriptions are unavoidable once the most generd ideas in a stage have been factored out from the more complicated. Many co-ordinale indexes do not handle subject matter, but directly index objects or events - chernicals, machines, people, places, accidents, sales, illnesses, deliveries of goods. The work of framing cliassified Ests of features for use in these was another source of infomation about semantic types. It showed the distinction between passive entities such as chemicds, machines, people and places, tvhich were described by passive features - qudities - and active entities such as accidents, sales, illnesses and deliveries, whose descxiptors were active features - modes. W e n features of the sart used in these indexes were used in thesauri as elements of subject matter they did not change their behaviour. Experience of their use in both varieties of index llay behind their appearance in the relation code sequences tvhich formalize the full pattern of the holotheme. It is instmctive to see how a co-ordinale index which directly relates items to their features, may be absorbed into an index of the topics treated in a document. The items and the features, to-&er with the connections between them, form what is knovun as a data field. Such a field rnay be very complicated, contalning much information about intricate matters, and of course it is descnbed in words - u-ie words which are canadates for inclusion in the thesaums of notions whlch rnay be used as an aid to search, Indeed, the subject matter of any docurnent is a data field or a linked set of these, and when the document becomes an Item in a co-ordhate index then &e items in its included fields of textud matter become features of its contents, recorded agalnst k Thelr change of status is signified by the use of the

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'about'. An index o f objects m a y record, ag-ainst t h e h o m as t h e s k y , t h e feature that it is blue. A n index ocuments m a y record, against t h e i t e m k n o m as t h e m m e n t concemed, t h e features that it is abeut t h e s k y and o u t blueness. Much o f t h e trouble encountered b y those w h o apply co-ordinate indexing t o t e x t u d matter arises because an essentially two-dimensional pattem is thus comressed i n t o a single dinnension under t h e item, &e docue n t , t o which it b e l o n g .

Data fields T h e behaviour o f a data field can b e exarnined b y putting marks at t h e places where its items (represented b y rows or columns) cross its features (represented b y columns or rows). Signds o f this sort m a k e it clear which items have which features. Each i t e m is reveded as a member o f m a n y different subsets o f t h e full set o f features, while each feature is a m e d e r o f m a n y different subsets o f t h e full set o f items. All t h e numericd and set-theoretic relations o f t h e lowest integrative level m a y b e s h o w b y t h e use o f such a field (it is, o f course, a field i n t h e strict mathematicd sense), and this is yet another source o f t h e remarkable contribution made b y co-ordinate indexing t o &c theory o f intcgrativc levels. It exhibited &e pattern o f t h e ground level. Indeed, t h e reason w h y a data field, embodied i n a computer m e m o r y or a set o f punched feature cards or i n a flat tray visible index or any other device, m a y represent any collection o f notions whatever, is n o w clear. T h e relation pattem o f the lowest integrative level, t o whlch its data field behaviour corresponds, is embodied i n t h e p a t t e m o f al1 higher levels. T h e operations and conditions o f t h e lower rank o f this lowest level c m d l b e demonstrated b y t h e use o f a data field diapram Mrithout using any subsets which are transverse t o each other. Since items can b e treated as subsets o f t h e full set o f features, while features can b e treated as subsets o f t h e

FIGURE 6: A simple field of five items and six features. Another example is analysed more extensively in Appendix A To help relate the data field t o everyday life, the items may be thought of as plants and animals, according t o the foIlowing scheme: 0 : crow (features) A : animal kingdom (items) 1 : canary B : plant kingdom 2 : Brimstone C : bird butterfly D : insect 3 : buttercup E : yellow 4 : snowdrop F : corvine Under this scheme a blob at the intersection of row 3 with column E indicates that the buttercup is yellow.

In a data field a feature is recognisable as a subset of items: thus the subset 0 , 1, 2 (shown as three blobs in column A) represents feature A. (This of course leaves any knowledge obtained from sources outside the field out of account.) As an example of a set-theoretic condition found in the field, the condition of containing may be taken. Feature A contains feature G which contains feature F, as may be seen by inspection.

full set o f items, either dimension o f t h e field m a y b e used. Appendix A $ves an example. T h e usual procedure is t o consider t h e fuD set o f items and t o examine t h e subsets o f items (which dl,in t h e diagram, lie parallel t o each other). T h u s o n e subset m a y b e s h o w t o inélude another, t o overlap another, t o b e identical w i t h another. T h e condition o f membership, however, cannot b e s h o w in this way. It is a transversive condition, as d r e a d y mentioned, and it calls for t h e use o f t w o m u t u d l y transverse subsets i f it Is t o b e demonstrated. It is found at t h e start o f t h e higher rank o f t h e level, being a between-set (as opposed t o a ntithin-set) relation. On t h e attributive side, multiplication is a typical transversive or between-set relation. I n a data field it appears as an operation u p o n t w o sets or subsets o f whlch one is taken f r o m one dimension o f t h e field while t h e other is taken f r o m t h e other. T h e result o f t h e rnultiplication is t h e number o f connections, crossings, between t h e members o f one set or subset and t h e members o f t h e other. These crossings carry t h e marks, standing for data units, t o show whether or n o t t h e i t e m concerned has t h e feature which crosses i t at that point. T h u s t h e multiplication together o f t h e natura1 numbers o f t h e t w o sets which m a k e a data field produces t h e nurnber o f data units it contains. In passkg, i t m a y b e noted that t h e condition o f m e h e r s h i p ( t h e first transverse condition i n t h e series o f conditions o f set theory) is imeoexive. This has a bearing o n Russel19s famous paradox. T h e paradox runs as follows: 'Consider t h e set o f dl sets h i c h are n o t nnembers o f themselves: i f i t is a mernber o f itself, t h e n i t is n o t a m e m b e r o f itself; and i f i t is n o t a m e m b e r o f itself, t h e n it must b e a member o f itself.' However, since membership is irrenexive, n o set is a niember o f itself, and i n consequente t h e paradox does n o m o r e than ask U S t o consider t h e set o f al1 sets. W e cannot t h e n suppose rihis set t o b e a member o f itself, ouring t o t h e properties o f membership; and i f w e think o f it as finite, t h e n w e must suppose either that i t is n o t a set or that

In a co-ordinate index are independent) &e concepts Mrdll be unanchored m d wil1 drift into 21 sorts of w o n g arrangernents. Another problem arises frorn the way in which translation (say, from English to French) alters the order in which notions are placed in any alphabetie thesaurus. Fortunately, change of language does not affect relation code sequences. Julius Caesar is to be found sornewhere in positlon 0.101.1.000.011, or 05103, In any tongue. Yet another concern of the co-ordinate indexer is to seek sm assurance that no imporlant notion is ornitted from the thesaurus, and that effeetive definitions are available, t o aid understanding and to control the secular drift of rneaning which occurs to words as custorns change and personalities alter. 1Regcalar schems for definlng ideas are vduable Inere. The notions used in a co-ordinale index may be of any degree of complexity. There Is nothing in principle to stop the indexer using such a concept as %he effect of gift stamp trading on &e profit ratio of dl-night automobile service stations', nor is there anything to stop languages developing single words for such Pdeas. In supplying rules for constructing and positioning complicated notions of this kind, the hoPothemic pattern may provide a service for suibJect classiflcations of the analy$c type. Large numbers of subject mangements break down the field of knowledge into ideas of at least this degree of complexity, The mles for forrning such notions, perrnuling their elements, work on the relation codes, though subject classificatlons are very difl'erent from co-ordinale indexes. In the realm of mathematics the relation code sequences rnay have many applications. They appear t o form a rernarkLebly neat classlfication of that subject; and, if this is so, they may offer clues both to the solution of special problenis and to the teaching of the subject as a whole. b r example, a textbook on set m d naamber theory coulid be pattemed on the relation code sequences. Such a work rnight dso help to tidy up sorne of the present nomencBature of tbe

discipline: for exaniple, what is here cdled a condition is called a relation in most mathernaticd textbooks, Mjith the result that the mathematician is left with no word to cover both together, The reliation code sequences rnay have even vvider application than is found in mathematics if they are exarnined in the context of the whole school and university cuniculum. The h o l o t h e ~ cpattern is a map of knovvledge. As interdisciplinary studies grow in importante and as schools develop the techniques of project work in which their pupils are assisted by resource centres, the older dlvisions between subjects become blurred. Then a new set of p~idelinesis needed to replace the vanishing boundaries. The framework of levels and grades may wel1 serve such a purpose. If it proves valid, it wil1 remain firm no matter how the temportary partitions Mrithin it are arranged and reananged t o suit acadernic needs. I recdl a simple example of this application of pidelines in the realm of curriculum development. IIt is possible to divide the subjects taught at the average school Tor teenagers into those which involve physleal and nmentd skills - woodwork, lanpages, tPngonornet~,sIvlmming - and those which are concerned witli describing a pattem - the orgmization of govemment, the sequence of history, the stmcture of the atom, the constniction of the hurnan body. Obviously the two types overlap: languages involve gatterns of gammar, the study of biology involves handing a microscope. However, if the largely descriptive element of education is considered on its o m , it may be compared, subject by subject, ~ t the h levels of the holotheme. This is a reveding exercise. When I carried it out in respect of rny o m children I found that their schooling dealt tvith the structure of things at ever). level excegt the commund. Sets and shapes were dealt with by mathernatics, particles, atoms and molecules by physics or chemistry, cells and living beings by biology, and the structure of government both national and local by various comlainations of history, civics and current affairs. By

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contmst, t h e stmcture o f a n y large adrriinlstrative b o d y below that o f local government was conspicuously absent. It was n o teacher's duuty t o explain that an industrial company has a quality control section, a cash flow system, a maintenmce department, a time o f f i c e , besides +he more obvious production lines m d sales offices. Cornpanies, clubs, professional associations and other rnain units o f t h e communal level, with thelr departmental sub-units, were onritted, or dealt cvith only b y t h e way in connection Mnth other subjects. Incidentally, I suspect that a very useful analysis o f management science could b e based o n t h e layout o f t h e comrnunal level. So far as t h e school curriculum is concerned, I concluded that in m y children" case Sie organizations i n which t h e y are most likely t o eani their livings are t h e ones least likely t o have been explained t o thern, even b y t h e harassed ani3 probabliy press-pnged careers adviser. T h e pattern o f t h e holotheme m a y b e o f use i n linpuistlcs also. Pf it is t h e business o f language t o irnitate things as t h e y h then the are, and i f t h e pattem corresponds wel1 ~ 4 t reality, student o f language rnay l e a m m u c h frorn t h e code, and t h e student o f t h e code rnay leasn m c h f r o m t h e behaviour o f languages. It m a y turn o u t that t h e binary pattern I describe can act as a neutral referente against which t h e codes w e k n o w as lianpuages m a y b e measured. T h e question rnay come t o b e , h o w does this or that t y p e o f speech imitate t h e anangement o f notions in t h e ho9otheme? W h a t does it omit? What does it alter? H o w are its nouns, adjectlves and verbs related t o entities, attributes, operations? Most especidly, t h e relaaon code appeals t o m e as a unifying device, an arrangement of t h e elements o f knowledge which in fact possesses t h e desirable properties á listed at t h e start. It is infinitely hospltable: every place i n i t can b e subdivided as necessary. It is easily taught: although i t contains m u c h that is new, there are n o g e a t difficulties i n l e m i n g t h e binary pattern it ernbodies or t h e classes, domains, levels, ranks and &e like which i t contains.

Although the properlies of relations rnay not be common knowjedge to all today, our schools are introducing work on set t h e o y En the new mathematies, and laying a foundation upon vvhich the series of integralive levels may in time corne to seem a natura1 edifice. It is independent of persond opinion: given the mles, an Arabic-speaking Eskimo living in Peru Gil, from standard textbooks, place the concepts of photosynthesis, Tarzan of the Apes, the omega minus particle and the aceeleration of desved dernand in the Same positions as wil1 a Bantu-speaking Russian in Bengal: not because he or she has refenied to the schedules or the decisions of a central authoriky but because that is where the notions fit. It is based on a sirnple repetitive pattem: little is simpler than binary, and b i n q is certckinly vvidely-knom. Perhaps I should add that the binary pattern came of itself during my inquiry. I started out with no sort of idea as to what I might find. If l had begun sixty years ago, &en little was knovvn of set-theory or the contents of the subatomie level or the things to be found inside a living cel1 1 should have had a harder time of it, and rnight wel1 have ended in boredom or despair tuming to the attractions of chess or politics or fishing. I was fortunate in my decade, which gave me the mateials for the m r k , and in my family, &o allowed me to play with symbols when I should have been rnovving the l a m , and in my Xocd authority, whose many public librarles provided me urith books by the dozen.

M e n I look at this wholc stmcture of thought P realize how much of it has been found or checked by the process of comparison. Tf a gap is found in the operations on subsets, it may be filled by cornparison urith the operations on numbers (supposing these to be k n o w ) at the same grade and Ievel. Other, more distant coqarisons S e to be found dso. P sernember seeing a film on television, made by time-Papse

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(P 7

photography, which had the strange effect of shouring people in &e gulse of a substance, seerningly p u q e d by r e p l a r pulses across roads controlled by traffic lights, dispersing on leaving the entrances to subways, d r a m frorn dl quarters to the gates of stadia by an unseen force. It seerned that the city breathed them as a lung breathes molecules of the substances in alr. For amusement, L checked the integrative levels. The nurnber of grades between a city and its people is exactly the same as the number between a lung and its air! I hought: the city breathing people in and out: what a tremendous poetic image. I was often one of those people in the city of London. I kriiew how the unhappy and the frustrated might fee1 in the presence of this huge impartid simile. Yet it is no criticism of modern life. Time-lapse photography of the migration of peoples in the nomadic days when northern Europe was swamp and forest would have s h o m exactly the same effect. There is an exciting literary study to be carried out on the similanties of the levels. It is so like the doctriae of correspondentes, rnacrocosm and microcosm, of the first Elizabethan days: in principle, at least. As individuals, we belong to neither: we inhabit a mesocosm from which the view is vast in al1 directions. The connection between a city and its people, a Lung and its air, leads to another: can a sort of slide mle be made, graduated with operations (for exarnple, those of mathematics) in their holothemic order, so that the holotheme may be displaced in relation to itself Just as a slide rule displaces a scale of numbers? Such an arrangement could have put the city opposite to the lung, dlowing 'people' to be read off against 'air'. If the operations of mathematics were recorded on the slide in this way, the unit g a d e could be set anyvvhere dong the holothemic series, and higher grades would then have their appropriate mathematics in register against h e m , on the assumption that the notions opposite the origin were to be taken as units for the purpose in hand. This would certainly put mathematical group theory cor-

rectly opposite to the omega minus particle whose existence it predicted, the flow of traffic . opposite to liquid Row, money against energy (and indeed, money is well treated as social energy) - al1 for suitable displacements of the pattern. However attractive this course of speculation may be, it is probably time to stop. Once more a summar). may be useful. Since about the midde of the twentieth century there has been a considerable gowth of interest in the question of levels of integration: What levels exist, and how are they related to each other? At the same time, the development of the major sciences has reached a point at which answers to these questions appear to be possible. If it is compared with the relation code approach, other work in this field seems to introduce unnecessary complexity, arising mainly from two enticing but nnistaken views: an atternpt to found a level on the rnind, and perhaps to incorporate mathernatics and other abstractions into this in tbe ,guise of mentefacts; and an attempt to found a series of levels on spatial or temporal dimensions. All this has been done in the course of a conscious search for constructive pattern, but Ivithout making use of the science of pattern: set and number theory. The present effort is but the most recent manifestation of a very old study. The history of this displays two lines of development, each of two strands. The search for an arrangement of notions according to their structure sterns from the work of the Greeks two Siousand or more years ago; the search for an arrangement based on their importante, as measured by their r e d or irnaginary power, sterns from an even older source: it is to be found as far back as we can descry. Both rnethods have been applied in two ways: to break down tbe field of knowledge into subjects, and to arrange single ideas - the elements of knowledge - in order. When attempts are made to use a constructive order for the elements of knowledge it becomes possible t o seek laws relating integrative levels to each other; the development of such laws rnay be expected to lead to valuable results in

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rnany different fields. These include co-ordinate indexing, from which a great deal of the present research has sprung; they also include linguistics, mathematics, educational technology and possibly o h e r sciences and technologies where patterns are of especid significanee. The set of relation code sequences has the usefut properties of being infinitely hospitable, easily taught, simple in its essence and independent of persond opinion. The rnethod used for foming it c m be applied by all who care to try it: the same order of major types of idea wil1 always result. It is this replicability which distinpishes it from other schernes for organizing knowledge. It may perhaps appear too cut and dned, too facile, too pat an answer, in comparison ~ 4 t hthe xtraordinary richness and diversity of the universe we inhabit: its very regularities may make it seem hard to accept. Uet consider the deeps and the abstruse complexities of mathematies, the nested pattenis of nurnbers, naturd, rational, red and complex, each rnoving out to infinity four passages to infinity even before that great domain is conquered. Then reflect that this is but the groundwork. Beyond .lies the realm of physics; and beyond this again lies that of the life sciences; and beyond these lie the social sciences; and even this is no more than part of the land t o be explored. For above the class of al1 rnundane ideas there stAtches the class of special notions, c o q r e h e n d e d by the imagination, interpreting the mundane for reereation, guidmee, insight, understanding. This is the vista which opens from our simple binary pattern. For me, this simplicity is wealtli enough.

Appendix A A DATA FIELD AND ITS GONTENTS The following is a typical diagram of a data field, as used in the course of lectures on indexing to audiences d r a m frorn such widely differing disciplines as medicine, personnel administration, librarianship, örganization and methods work, teaching and engineering. In al1 these cases and in many others the problems of recording information for later retrieval and processing arise. When lecturing on such a topic, the items and features of the field are $ven particular narnes to correspond with the familiar activities of the audience. Here they are simply represented by letters (for the features) and numerds (for the item). The reader may find it helphl to think of the items as people and of the features as their characteristics: for example, artful, bashful, cheerful, dark, energetic and feminine. In the field, a blob is used to indicate that an item possesses a feature. Thus each colurn of such marks indicates the particular subset of the full set of items which possesses the feature represented by the column. Six subsets are s h o w , corresponding to the six features: clearly, others are possible if the field is extended. Similarly, each row shows the subset of the fuU. set of features which is possessed by the item represented by the row. Thus the relations which are found between subsets, within full sets, can be s h o w by using either dimension of the field. Relations between full sets are transversive, and can only be s h o w by using both dimensions,

Appendix A Here is the sample field:

FEATURES F

"he relation code sequences described in Ghapter 2 were developed as a result of studying the properties of the operations and conditions to be found in such a field as this. The following are examples of the relations concerned: disjunetion : subsets A and E are disjoint (note how the subsets of items are named by the use of their defining features). In the other dimension, subsets 1 and 2 are disjoint. Other examples of disjunction can be found in the field. containing : subset A contains subset C, which contains subset D; in the other dimension subset 1 contains subset 5 which contains subset 0. overlap : subset B overlaps subset C; subset O overlaps subset 3. Other examples of overlap are to be found; these two instances show every possibility (the members

102

The Fabric of Knowledge of the full set may appear in both, either, or neither of the subsets).

intersection : the intersection of subset C with subset F is subset D. addition : the natural nurnber of subset A is 4 (it has four members); that of subset E is 2; the addition of the two is 6, the number of members in the full set concerned. Note that this works because the two subsets are disjoint. subtraction : the natural number ofA is 4;and that of Cis 3; and the result of subtracting 3 frorn 4 is l, the number of mernbers in the larger subset (A) which are not in the smaller (C). Note that this works because A contains C. membership : unlike the preceding examples, this relation cannot be shown k t h o u t using both dimensions of the field: it is transversive. Item 3 is a rnember of subset B (we may say, item 3 is a member of feature B, or subset 3 is a member of subset B, or even 'person 3 is bashful' or 'bashful is person 3', which is poetic, like 'blue was the sea'). multiplication : i$ we multiply the number of items, 6, by the number of features, 6, we obtain the nurnber of cells, crossings or data units in the field: 36. Here again two dimensions are employed: multiplication is transversive. These exarnples could be extended in many ways, but this is sufficient to demonstrate the source of some of the methods used for exploring the set-theoretic level.

EXAMPLES OF PLACEMENT The follovving few notions have been defined very briefly and given code sequences according to the relation code pattern, as further examples of the method. There is a great deal t o be leamt about it still. Problems arise even when the indexer is concemed vvith apparently simple ideas - for e x a q l e , the concept of a trade name, or a googly, or a sundowner (either meaning), or stamp trading, or the Impressionist Movement. Generally the difficulties appear because some ideas are multiple collectives. However, there seems to be no reason why rules for arrangng the elements of such notions may not be agreed: a typical rule would be to place these in holothemic order - number before dimension (and, vyithin dimension, space before time), energy before matter, and so on up the scale. Meanwhile, here are some uncomplicated instances. A M E R G E R : The

nierging of two or more organizations to forrn a single new organization.

The notion is mundane, between such organizations as industrial campanies and consequently in the communal level, upper rank. It is not dissoclative, it is commutative, 1t is idempotent (a company is always merged with itself). It is an active entitive term, a phenomenon. Its

The Fabric of Kno wledge

104

relation code sequence is therefore 0.1 10.1.01 1.1 1l, or 06 137 in octal. A R A I N W A T E R S Y S T E M : A series of single-piece parts

such as gutters, swan-necks, domspouts and the like, channelling rainwater from the roof of a building to a drain or soakaway. This is not a system in the relation code sense. The notion is m n d m e , between single-piece parts V4thout movement, and consequently is in the cytomechanic level, lovver rank. It is a series and is consequendy transitive, not symmetrie and not reflexive. It is a passive entitive t e m , a thing. Its relation code sequence is uherefore 0.100.0.100.011, or 04043 in octal. G A U S A LTTY : Being-a-cause-of.

There are many variants: here we take being-an-immediate-physical-cause-of. The notion is mundme, occurring as soon as energy is available at the sub-atomic level, Power rank. Being immediate, transitivity is irrelevant; it is not syrnrnetric and it is not reflexive (no notion is a cause of itself). It is passive, as opposed to the operation of causing, which is active. It is entitive and it is a relation. Its relation code sequence is therefore 0.010.0.000.010, or 02002 in octal.

Another method of placing ideas displays the relation code sequence step by step as it is built up, thus: U T O P I A : An

irnaginary country.

1 : special 111 : national level l : upper rank, concerned ~ 4 t h entire nations 000 : a unit vvithin this rank

Appendix B

105

011 : a thing 1.111.1.000.011 or, in octal, 17103. P N O T O S Y N T H E S I S : Formation of complex substances

out of water and carbon dioxide, by the aid of chlorophyll acted on by light O : mundane

0 11 : molecular level 1 : upper rank (rehuiring molecules) 101 : a serial process 1l 1 : a phenomenon 0.011.1.101.111 or,inoctal, 03157 There is obviously a great deal of room for discussion, argument and disagreement in defining and placing ideas. For example, in the notion 'bble' to be regarded as a quality of materials (at molecular level) or as a phenomenon, the vibration of photons (at subatornic level)? It is clearly possible t o accept both meanings. M e n divergent views appear on matters of this sort it is generally possible to accept both, because in practice they simply point out different meanings for the same word. Deeper philosophic differences may be found, however, when rnental feelings, ernotions, sensations and the like are given homes, as ideas, in the brain at the l o m r biomorphic rank. People with an especidly strong belief in the existente of a mental level may find it hard to co-operate with the relation code pattern t o the extent of treating an idea as a pattern of states in a group of cells in the brain (or as sornething similar - here again we trespass on the very boundaries of knodedge). For my part, I am Glling to pretend that the pattem is right, and to try to fill it in, as far as is possible, so as t o see what it looks Iike when it is wel1 supplied with properly placed ideas. The pretence is easy because in many regions of the holotheme - in mathematics, for instance - I think it is

right in fact and not in pretence alone. Others may be less sure; but however we feel, it is only by going dong with it for a while that we can leam enough about it to judge it fairly, I have the impression &at, to an even greater extent than is noma1 in work with ideas, the examination of the relation code pattem wil1 cal1 for careful study of the arts and sciences concerned. M a t this wil1 reveal, in the end, who knows? At any rate, we shall all be rernarkably wel1 informed.

PREVIOUS NOTIES 8 N TNE LEVELS My inquiry into the pattern of integrative levels has been a long-dram-out affair with few progress reports. In December 1964 the Joumal of Documentation published a letter of mine in which I described the general pattern of four major domains and eight levels, together G t h the chain of types of notion mnning upward from units through assemblies and higher grades of notion til1 more advmced units appear. At that time I used the n m e %ystern9for what P now think of as a 'series9, and "combine' for what now seems best named a 'systern9. Also, I used names (for some of the semantic types) which differ from those used in the main text of this book. Otherwise, there is little fundamental differente between the pattern then put forward and that of the holotheme as I now see it. Here is the letter: Dear Sirs, 'Integrative Levels' I was pleased t o see a reference, o n pages 157 and 158 of your last issue (September 1964), t o work I have been c a q i n g out on the subject of integrative levels and semantic types. The reference appears as part of the Glassification Research Group's Bulletin No. 8. It gives provisional lists of levels and types, presented t o the Group as work in progress. I should not Iike t o think that any of your readers might now take these and try to build on them, because most of the ideas they contain have failed to prove themselves in practice. Only the set of integrative levels from the cellular t o the national remains resistant t o my attempts t o prove it ill-chosen. From my own point of view, however, the lists have done their Job well. They have made it possible t o formulate rules for the

detection of integrative levels and for the ways in which a higher level may be built up from a lower. Also, they have made it possible t o produce a theory of semantic types which is far more internally consistent than the hypothesis underlying the provisional work. Your readers can probably see for themselves many of the insufficiencies of the lists. Point-events, for instance, are nor fundamental, since to discuss them we need many specialized ideas - for example, those of set theory, which are therefore earlier. This is not to say that the ideas given in the lists are not ideas for which we must find places. The point is that they do not seem to be related t o each other in the way the lists indicate. Thek relations are at the Same time more complex (there are more semantic types than those which are given) and more simple ( h e k pattern is more systematic than the lists appear to show). It is perhaps unfair to pull down earlier work Mthout offering a replacement. My present view is that the list of integrative levels ought t o run more like this than Iike the provisional one:

1. set-theoretic, logical spatial, geometrie

logic and mathematics

3. subatomic

4. atomic and molecular physics 'hemistry 5. cellular biomorphic biology and artefacts

7. communal 8. national

social sciences

Also, it seems t o me that the type-structures (for example, atoms, cells, people) of one level come togethgr t o form those of the next by a more or less regular chain which can be summarized: unit (the type-structiare) assembly (several, acting together) system (several, ac&ng serially) combine (several, acting serially ~ 4 t feedback) h subunit of next level ('organ') ("interlevel') assernbly (of organs) system (of organs) combine (of organs) unit (the type-structure of the next level) Possible examples of this are: a mitochondrion is an organ of a cel!, definhg an interlevel; a flower is an organ of a glanc a personnel department is an organ of a company; the circulatory system is a system of an animal; the road system is a system of a natfon; a molecule is an assembly of atoms. This anrangement is, of course, as much at risk as the original lists

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The Fabric of Knowledge

to fee1 it should be $ven as wide a circulation as the earlier and cruder version. I hope you wil1 forgive so long a letter. Uours faithfully, J.L. JOLLEU

Almost dl this malysis held good, so fa as I could tell, when I continued with my study. During the next few months, the princlpal change in Sie patten? was a straightfomard addition to it: the four sernantic types became eiglit when the four categodes of relation were catered for. This led to the developnnent of the binary triads for the types, and was reinforced when set theory proved to hold a matheniatical basis for the whole arnagement. In 1967, the Classification Society Bulletin published an article ("he Pattem of Meaning') In which P put fonvard the basic scheme of relation codes. To reprint it al1 here would be wasteful repetition; but its. introduction may serve as an alternative presentation of rny approach to the subject, and its conclusion may round off the picture. THE PATTERN OF MEANING Nitherto, each attempt to make an orderly list of the concepts we use in daity Iife has been founded on the views of one learned man, or of a group of learned men, as t o where each idea may best be fitted into the pattern. However judiciously made, these arrangements are essentially subjective: they are collections of decisions with which we may not always agree. FOPexample, there seems no compelling reason for the social sciences to precede mathematics in the Universal Decimal Glassification, nor for the idea of Camembert cheese t o appear (as it does, in the revised Roget's Thesaurus) under the heading 'Space: motion with reference to direction'. By and large - give a little and take a little - we know what we mean by the social sciences or Camembert cheese; but that knowledge gives US no clue as t o where we may find these concepts in a classification or a thesaurus. For this, we must look in the p a d e or index t o the work. It would be pleasant if, instead, we could tell where an idea should be found merely by considering it in itself, armed with nothing more than a knowledge of its properties. Such a process would place ideas much as the periodic table places the chemica1 elements: they would hold their positions because, $ven the rules, they could hold no others.

Appendix G

11 1

The perioaic table provides a hint as t o how we might set about find4ng a pattern of this type. Pt arranges the elements according t o the structures of the a t o m which comrpose them - structures which determine the behaviour of the atoms, including the behaviour which fkst suggested the arrangement of the table t o Newtands and Mendeleev. A sequence of ideas which is based in sorne way upon structure may enable us t o understand them better, t o see why they act as they do, t o predict this behaviour, and - if there are gaps in the pattern - t o hazard a p e s s as t o which types of idea may one day be found. INTEGRATIVE LEVELS The simplest way t 0 apply this principle is t o consider the obvious, ordinary things of the world around us. Houses are made of bricks: trees have a trunk each, together with branches and leaves; a chair has legs, a seat and a back; a bicycle has wheels and handlebars. The method may be extended -houses, factories, roads, offices, help t o form a town; towns, rivers, hills, fields, valleys make a county, a regon, a nation entire. Towards the minuscule, flowers are made of cells, the cells contain plastids, the plastids in %heirturn have a complicated structure. These observations lead us to a rule which may be helpful: ideas are t o be placed after those which form them and before those which they form. A brief examination of the textbooks then produces a sequence of the following sort: particles - a t o m - molecules - organelles - cells - organs of the body people - communities - nations. Side-chains branch out from this arrangement and then rejoin it: some cells f o m animals other than human, some are embodied in plants, and yet the comrnunity known as a farm subsumes all these and so returns thern to the main series at a more complex level. Another side-chain, it seems, begins earlier: molecules may form non-living substances and so precede a series of artefacts, single-piece parts which may be brought together t o make hand tools, prime movers, power houses, aircraft. Again these may return t o the main series at the level of a community: a farm, for example, includes its tractors and its combine harvesters. This approach appears that of a specialist in naivety; but courage may be taken from those who have already followed the course. These pioneers have given the name 'integrative levels' t o the deprees of complexity in the build-up of the world. It seems worth enquiring, is there a special type of idea which acts as a major stage in this build-up? Gan we, for instance, choose a few things which appear t o mark important steps, places of provisional completion in the sequence? If so, c m we find in them any special properties which make them appropriate t o these places? IE we End such ideas, we may cal1 them the units of the levels they occupy. Indeed, we may name the levels by referente t o these types of unit.

Being self-centred, w e may take living beings (plants and animals, b u t especially people) as defining a level, and exarnine t h e m from a mechanic's point o f view. W e see that they consist o f many interacting systems which keep t h e m i n balance with their surroundings, enabling t h e m t o persist even i n conditions o f moderate adversity, t o heal themselves i f n o t t o o badly injured, and generally t o resist for a while t h e ravages o f time. This m a y also b e a doctor's or a biologist's viewpoint, though t o many it would seem t o miss out t h e desires, interests, emotions, habits and skills which make each person, and even each less advanced animal, different f r o m &herest. This property o f stability, thought o f as d process o f continuous Iceedback o f information allowing self-righting mechanisms t o come into play, has been christened homeostasis. Where else may it be found? It is trite i n the studies o f history and the social sciences t o remark o n t h e self-perpetuating property o f communities. 1Evex-y member o f a Trade Union may, i n due course, leave that b o d y and b e succeeded b y a newcomer; an industrial or commercial company may retain its identity - and boast o f i t - aAer three hundred years o f existence; churches, colleges, societies o f all types display this e f f e c t . When t h e y are highly organized, specialised systems can b e discerned within them: an industrial c o q a n y bas its production system, its personneI system, its internal flow o f information leading t o its conhol centres and away f r o m them, its purchasing and sales organs, its maintenmce departments. I f these all function well, it maintains and even improves its place i n t h e economy. Below t h e h u m n being, t h e other types o f animal and the plant, t h e living cel1 may b e thought of as having self-balancing propertjes. Offered a proper nuhient medium, it keeps going. Making adjustments in thought t o cater for t h e lower level o f complexity, and glossing over months o f ponderuig i n a brief sentence, the next object t o b e taken as meeting our requirements is t h e molecule. Curiously, atoms (except those o f inert gases) d o n o t seem t o qualify; the exceptions appear t o act as single-atom molecules, just as at higher levels there are single-cel1 animals and one-man companies. Below t h e molecule, the s u b - a t o ~ cparticle fits the specification: electrons and protons, the simplest o f these (even though t h e y , t o o , may prove t o have a substructure) perform miracles o f longevity. It is arguable whether w e may ask what pbotons are made o f , i n the Same sense that w e may ask what a house is made o f ; b u t it is possible t o descrïbe t h e m and their behaviour in highly sophisticated ways, and &e description uses t h e concepts o f rnatheraitics. S h e essential mass and solidity w e associate with most o f t h e things o f higher levels has vanished here; b u t ideas o f shape, distribution in

Appendix C

113

space and t h e like remain. W e m a y t h i n k o f t h e m as the ideas o f geometry. T h e rule, t h a t concepts fit after t h e concepts which f o r m t h e m , m a y n o w b e interpreted t o m e a n that t h e y fit after t h e concepts w e need i n order t o speak o f t h e m . A sequence o f ideas used i n m a t h e m t i c s is required, which m a y b e expected t o end at t h e advanced point where t h e ideas o f subatomic physics, gravity and t h e rest begin. T h i s t h r o w s t h e enquiry back t o a study o f t h e basis o f m t h e m a t i c s . I n d u e course t h e idea o f shape vanishes i n t h e same w a y that t h e idea o f mass l o o k its Right, and leaves us w i t h t h e abstractions o f set theory as our sole remaining companions. W e reach t h e set w i t h only o n e member. T h e n w e reach t h e e m p t y set. T h e n w e stop. Gingerly, w e m a y conclude that there are t w o more integrative levels here - o n e o f geometry t a k e n as concerned w i t h space, and one o f set theory. T h e levels n o w total eight: t h e set-theoretic, t h e geometric, t h e sub-atomic and t h e molecular followed b y that concerned w i t h cells, that concerned w i t h plants and animals, that concerned w i t h communities and that concerned w i t h nations. These group themselves neatly i n pairs, t w o o f which are t h e province of logic and niathematics, t w o are concerned w i t h physics and chemistry, t w o w i t h t h e l i f e sciences and t w o w i t h t h e social sciences. T h e amival o f living things appears exactly i n t h e middle o f t h e sequence. [ T h e article t h e n introduces and explains t h e principle o f relation codes, and concludes as follows] : T o present evidence o f a repetitive, teachable pattern i n t h e concepts o f daily l i f e is o n l y t h e start o f a programme o f research, n o t its conclusion. A good deal m r e is k n o w n about t h e holotheme than has b e e n described i n this paper - t o choose at random, systems o f notation have b e e n worked o u t , based o n eight digits, t o reduce t h e relation codes t o manageable proportions. Studies o f t h e placing o f ideas i n complicated fields o f w o r k , such as industrial management, have b e e n undertaken. Relation tables have b e e n assembled, and t h e concepts o f n u m b e r theory - rings, groups, fields and t h e like - have b e e n put alongside t h e formative stages o f t h e lowest integrative level. Considerably more specialised terminology than is used here has b e e n adopted. U e t all this, and more besides, remains a Reabite i n comparison w i t h t h e e f f o r t still t o b e made. What value rnay b e gained f r o m such an e f f o r t ? A good deal, certainly, and m u c h o f i t i n ways w h i c h have n o t yet b e c o m e clear. A n example o f t h e value it is already k n o w n t o possess m a y b e t a k e n f r o m t h e field o f co-ordinate indexing, f r o m which t h e impulse t o t h e present s t u d y arose. Co-ordinate indexing is o n e o f t h e responses t o the current

information explosion. B y i t , a librarian means a m e t h o d o f bringing together any number o f independent ideas which, separately or i n concord, describe t h e subject matter o f a document. Wsually, b u t n o t invariably, h e also implies that t h e ideas are represented b y punched feature cards, o n e card per concept o f subject matter, each card bearing u p t o t e n thousand numbered positions representing documents. T o stack - say - five cards is t o co-ordinate five concepts. T h e documents which have t h e five features i n c o m m o n appear i n t h e f o r m o f holes w h i c h pass through t h e stack i n t h e relevant numbered positions. T h i s m e t h o d o f search is extremely quick and Rexible (and o f interest t o t h e statistician also, for t h e density o f holes i n t h e cards is a measure o f t h e likelihood that a d o c u m e n t o r other i t e m wil1 possess t h e feature concerned). O n t h e other hand, it is a f f e c t e d b y t h e w a y t h e concepts can join themselves together: b y problerns concerned with h o m o n y m s , synonyrns, multiple items, separation o f monadic terms, and t h e like. Its efficiency is also dependent o n t h e order i n w h i c h t h e cards which represent t h e concepts are stored. Here, t h e s t u d y o f the pattern o f meaning has m u c h t o contribute. T h e proper classification o f ideas brings s y n o n y m together, separates h o m o n y m s , shows which t e r m are monadic, provides a storage order. In other directions, if the pattern here developed holds good, e f f e c t s m a y b e felt i n the field o f education, i n t h e shaping a f general t e x t b o o k s o n science, i n t h e presentation o f t h e relations between subjects i n t h e curricula developed i n secondary schools. In t h e wilder Bights o f fancy, it is possible t o imagine t h e pattern guiding research i n places where little is k n o w n . T h e realni betweeri t h e q u a n t u m o f energy and t h e subatotnic particle is n o t yet well illuminated. T h e pattern hints at assernblies, systems, combines (whatever these m a y b e at that integ-rative level) b e t w e e n t h e t w o . Wil1 something like tbis b e discovered? In this case, perhaps w e had better wait and see.

SUNLM[ARUOF THE RELATION CODE The holotheme is divided into two perccption classcs: O : mundane, containing notions commonly ageed to comespond to redity, such as 'chair', 'man', 2tal19, 'geen9, 'sunshine', 'rain'. 1 : special, containing notions which are matters of faitlh, hypothesis or fiction, such as Taradise9, "hlogiston9, "liver Twist'. Each class is divided into two kingdoms: O : Inanimate, divided into two domains: O : rnathematics, divided into two integrative levels: 0 : set-theoretic 1 : spatial 1 : physicd sciences, divided into two integrative levels: 0 : subatornic 1 : molecular I : animate, divided into two domains: O : life sciences, divided into two integrative levels: 0 : cytomechanic 1 : biomorphic I : social sciences, divided into two integrative levels: 0 : communal 1 : national Each integrative level is divided Into two ranks:

O : lower, concemed with notions found within the main units found in the level (for exarnple, within moleedes, vvithin animals) 1 : upper, concemed with notions found between the main units found in the level (for exmple, between molecules, between animals) Each rank is divided into two groups: O : intransitive, divided into two stages: O : non-symmetric, divided into two formative grades: O : irreflexive (for exmple, units) P : reflexive (for example, simples) 1 : symmetrie, divided into two fomative grades: O : irreflexive (for exmple, assemblies) 1 : reflexive (for example, niixtures) l : transitive, divided into two stages: O : non-symmatric, divided into two formative grades: O : irreflexive (for example, series) 1 : reflexive (for example, sheets) 1 : symmetric, divided into two formative grades: O : irreflexive (for example, system) 1 : reflexive (for example, stretches) Each formative g a d e is divided into two varieties: O : passive, divided into two sorts: O : attributive, divided into two semantic types: O : relalions (states) l : terms (qualities) 1 : entitive, divided into two semantic types: O : relations (structures) 1 : t e m s (things) P : active, divided into two sorts: O : attributive, divided into two semantic types: 0 : relations (actions) 1 : t e m s (modes) 1 : entitive, divided into two semantic types:

Appendix D

117

0 : relations (activities) 1 : tenns (phenomena) Tbe complete pattem is summarized as consisting of two classes, each of eight levels, each of two ranks, each of eight grades, each of eigi-rt types.

GENERAL INDEX This index contains very few references to the placing of specific ideas: for these, the reader should refer to the placement index which follows. actions : defined 34; their code sequence 35 active ideas : introduced 32; and system theory 75 activities : defined 34; noted 109 adjectives : related to qualities and modes 33; and integrative levels 92; and the holotheme 95 aggregative levels : described 68; mentioned 74; compared with the relation code treatment of substances 71, 72 algebra: and the representation of geometry 55 Anaximander: and constructive order 78 Anaximenes : and constructive order 78 Aristotle : and constructive order 80 artefacts : their integrative level 29,30 assemblies : as avariety of idea, 25; their code sequence 28,31 associative property : of relations, 47; referred to the dissociative property (which see), 47. astrophysics : in the holotheme 72 attributive ideas : introduced 32 Austen, D. : on the theory of integrative levels 74

Bacon, R. : and the position of mathematics in the holotheme 81 Bertalanffy, L. von : and genera1 system theory 75 binary notation : mentioned 14; for perception classes 18; for integrative levels 24, 30; for formative ranks, 31; 32; for formative grades 28,31,32,54; for semantic types 35; for full code sequences 3 7 ;for relations in set and number theory 42, 43,48,50,51 binary pattern : in the holotheme 22,23,24; and integrative levels 22, 24; and unit ideas 22; and formative grades 28, 54; its vaiidity 59; its nested structure 63,83 Burnet, J. : on Zeno's doctrine of zero 79 chains of ideas : initia1 chains, displayed 19, 20, 29; and integrative levels 21, 22, 23, 24, 30, 111; and formative grades and ranks 28, 31, 54; and semantic tyDes 35; and relations 43, 48, 50, 51, 54; and the Elizabethan scheme (Great Ghain of Being) 81; in Roget's

120

Cen era l Index

defined 47,48 documentation : and the arrangement of ideas 12 domains : of mathematical, physical, life and social sciences 23, 24; in relation code summary 115 education : and the arrangement of ideas 12; and curriculum development 94; and the holothemic pattern 94 energy : its arrival in the sequence of integrative levels 24 entitive ideas : introduced 32 events : related to objects 35; and co-ordinate indexes 88 evolution : and constructive order 83,84 extensives : introduced 35 faith : in the special perception class 15,38 feature cards : and co-ordinate indexing 1l 4 features : in the data field 88, 89, 90,100,101,102 feedback : and systems 25 Feibleman, J.K. : on the laws of the levels 69 fiction : in the special perception class 1 5 , 3 8 field (mathematical) : in the holotheme 52, 53, 54; and the set-theoretic level 53 formative grades : introduced 15; described 16, 2 7 ;in diagram 1 7, 28; their binary and octal notation 28,30 formative ranks : introduced 15; described 16, 26; in diagram 17, 28 formative stages : described 26,27 Galen : and sequences of notions 80 generics : described 37; and inte-

grative levels 38 Genesis: as an example of placement of ideas 7 7 geometry : and algebra 55; and Plato's constructive order 79; and Zeno 79 Goodfield, J. : and integrative levels 86 Great Ghain of Being : as an ordering device 8 1 group (mathematical) : in the holotheme 52,53,54 Weraclitus : and constructive order 79 holotheme : entire set of notions 19; summarised 38, 39; its repetitive (nested) pattern 63 ; and its laws 68, 69, 70; its constructive order 19, 39, 64, 71, 72, 73, 74, 77, 78, 79,80, 81, 84, 85, 92, 111, 112, 113; and genera1 systems theory 75; and the correspondences 97 holothemics : defined 14; and Plato's constructive order 79 homeostasis : in the maintenance of units in the holotheme 21, 26,112 hypothesjs: in the special perception class 1 5 , s8 ideas : universa1 ordering of, 12, 82; general pattern of 15, 16, 17, 38, 39, 62, 64, 115, 116, 117; chains of 19, 20; their placement exeinplified 31, 32, 3 7 , 45, 103, 104, 105; their placement rules 6 1 , 6 2 , 6 3 ; and the schemes of Roget 81 ; and Wilikins 82; idempotency : defined 46 infinity : its appearance in an integative level 28; and reflexiveness 45 integers : higher than natura1 nurnbers 49;& subsets 50; and g o u p s 52, 54; and polynomials 53

General Index integrative levels : introduced 15; described I 6, in diagram 17; and their contents 21, 22, 23, 24, 30, 158; and unit ideas 21, 22, 23; of machines and other artefacts 29, 30; named 30; and relation code sequences 44; Feibleman's work on 68,69,70; and their laws 69, 70 ;arrival of new 70, 71; early theory of 83; and evolution 83, 89; and co-ordinale indexing 89, 92 ; and the study of linguistics 95; and management science 95; and the study of mathernatics 93,94; and curriculum development 94; and chains of ideas 21, 22,23,111 inational numbers : in the holotheme 55; and the Pythagoreans 78 kingdoms : animate and inanimate 23,24 Leucippus : and conshuctive order 79 life : its appearance in the holotheme 23,24 life sciences : their domain in the holotheme 23,24,113 linguistics : and the arrangement of ideas 12; and semantic factoring (componential analysis) 87; and constructive order 95; and the relation code sequences 95 logic : views of its position in the holotheme 81 Lucretius :and constructive order 80 machines : their integative level 29,30 management science : and constructive order 95; and relation code sequences 95 mathematics : providing the sup-

121

port of a theory 14; its hoIothemic position 23, 24,65, 66, 78, 81, 113; as a basis for placing ideas 40-58; its internal structure as a basis for its classification 58, 59, 93; its application to the holotheme summarised 62, 64; and its nomenclature 47, 93, 94; amd comespondences in the holotheme 97 mathematical structures : in the holotheme 52,53,54,64 matrices : and complex numbers 55 mental level : in the holotheme, as afallacy 66,68,98,105 mentefacts : and their treatment in the holotheme 66,78,98 mind : and the holotheme 66,67, 105 mixtures : described 26; in table 28; and relation code sequences 44,45 ;and refîexiveness 45 modes : defined 33, 69; divided into datals and duratives, 35; and co-ordinale indexes 88 mundane perception class : its contents described 15, 18; in diagram 17, 24 and Dewey 84 names : Eor formative grades 28, 31 ; for integrative levels 35; for semantic types 32, 33, 34, 35; for mathematical relations 93, 94 natura1 numbers : and subsets 49, 50; andsemigroups 52,54 Needham J. and integrative levels 85 nested structure of holotheme : noted 63 83; its effect on data fields 89 nouns : and the holotheme 95 numbers : relations between 49, 55, 51; and mathematical structures 5 1-56

122

General Index

processes : introduced 35; and coobjects : defined 18; in chain of notions 19,20 ordinate indexes 88 octal notation : for integrative psychocentric fallacy : described levels 24, 30; for formative 66; occurrences of 68, 98, 105 grades 28, 30, 31, 32; for full psychological level : in holotheme, as a fallacy 66,68,98,105 code sequences 3 7 operations : defined 34; of set punched feature cards : and cotheory 40, 4 6 4 9 ; and their ordinate indexing 114 relation code sequences 48; on Py thagoras : and constructive nurnbers 50, 51 ; and the data order 78; and irrational field89,91,101,102 '\urnbers 78 order : and series 25, 28; its appearance in an integrative qualities : defined 33; divided int0 level 28; and the definition of positionals and extensives 35; integers and rational numbers and co-ordinate indexes 88; 52 mentioned 109 orthogonality : in the holotheme rational numbers : and sets 50; and 74,75 mathematical fields 52, 54; and passive ideas : introduced 32 Peano axioms : and the holotheme rational fractions 53 56 reality : and the mundane percepperception class : described 15,16, tion class 35,38 18;indiagram 17,24 reflexiveness : defined 40,41; and periodic table : and integrative infinity 45; and substances 45 levels 110, 111 ; as the 'law of Reiser, O.L. : and integrative octaves' 83 levels 74 phenomena : defined 33, 69; divi- r e l a t i o n code sequences : ded into events and processes described 42, 43; and code sequences for ranks and grades 35; andco-ordinates indexes 88 physical measurables : in the holo44; for conditions of set theory theme 57,58; code sequence for 43; for operations of set theory 58 48; and translation 93; and physical sciences : in the holoco-ordinate indexing 86, 87, 88, 93; and mathematics 93; theme 23,24,71,72,78,113 placement : of ideas exemplified and curricula 94; and linguis31, 32, 37, 45, 48, 103, 104, tics 95; and management science 95; and the data field 105; rules for, 60, 61, 62; and semantic types 48 101; and mental levels 105; Plato : and constructive order 79, surnmary 115, 116, 117 80; and geometry 79; and atoms relations : informing ideas 16; 79; and irrational numbers 80 giving internal construction to Polynornials : in the holotheme terms 33, 34; represented by 53; and vector spaces 53; and verbs 33; operations and conditions 34; their properties 40, integers 53 41; and set theory 40-49; and positionals: introduced 35 numbers 49-51;and thepattern prediction : by the use of the of the holotheme related to the pattern of integrative levels 60

General i n d e x data field 89,90,91, 100, 101, 102 religieus ideas : and the special perception class 15 Richardson E.C. : and constructive order 85 ring (mathematical) : in the holotheme 52,53,54 Roget, B.M. : and his Thesaurus as an arrangement of ideas 82,110 Rusell, B. : paradox 9 1 Saint Victor, Hugh of : and the position of mathematics in the holotheme 81 semantic factoring : described 87; and definition series 88 semantic types : introduced 15; described 16; in diagram 17; discussed 32,33; named 32,33, 34, 35; notation for 35; and the placement of ideas 48; and linguistics 95; and the views of E.G. Richardson 85; and the work of the Clasification Research Group 108 semigroup(mathematica1) : in the holotheme 52,53,54 series : as a variety of idea, 25; in table 28; and relation code sequences 45 set theory : and the theory of the holotheme 14, 66; names an integrative level 30; and relations 40-49; and the dimensional fallacy 68,70,71; and the arrivd of new integrative levels 70, 71; and data fields 89, 90, 91,100,101,102 sheets : as a variety of idea 27; in table 28; and reflexiveness 45 simples : as a variety of idea 2 7 ;in table 28; and refiexiveness 45 social sciences : in the holotheme 23,24,113 space : its arrival in the holotheme 24

123

special perception class : its contents described 15, 18; in diagram 17, 24; and supracultural levels 69 Spencer, N.: and the theory of integrative levels 83 states : as avariety of idea 34 Stoics : mentioned 80 stretches : as a variety of idea 27; in table 28; and refiexiveness 45 structures : as a variety of idea 34 subassemblies : as a variety of idea 26;in table 28 subrnixtures : as a variety of idea 28 subseries : as a variety of idea 26; in table 28 subsheets : as a variety of idea 28 subsimples : as avariety of idea28 substances : described 26, 27; and refiexiveness 45 substretches : as a variety of idea 28 subsystems : as a variety of idea 26;in table 28 subunits : as a variety of idea 26; in table 28 systems : as a variety of idea 25;in table 28 systems theory : mentioned 26; and the holotheme 75; and active ideas 75 symmetry : its appearance in an integrative level 28; defined 41 taxonyms : described 37; and integrative levels 38 terms : and relations 33 Thales: and constructive order 78 theology : and perception classes 18 thesaurus: by Roget 82, 83 ; in co-ordinate indexes 86, 87,88, 93 things : divided into objects and substances 32; and co-ordinate indexes 88; mentioned P09

Index Tillyard, E.M.W. : and the Elizabethan world picture 81 Toulmin, S. : and integrative levels 86 ~ransitivity: defined 43. translation : and relation code sequences 93 transversiveness : defined 41 ; in the data field 91,100 units : as a variety of idea 21, 22, 27;inchainsof ideas21,22;and integrative levels 21, 22, 23; in table 28; mentioned 108 Universal Decirnal Classification : and the order of ideas 110

vector space : in the holotheme 52, 53,54; andpolynonnials 53;and complex numbers 53 verbs : and relations 33 Vickery, B.C. : and classifications 81, 86 Wilkins, Bishop : and the arrangement of ideas in a universal language 82 Zeno : and constructive order 78; and the concept of zero 78,79 zero : and Zeno 78,79

PLACEMENT INDEX m i s index contains ideas whose position in the holotheme is discussed in the text. It does not contain other notions used in the discussion: for these the reader should refer to the genera1 index. In deciding whether to admit an idea to the placement index, the rule has generally been to do so even if the idea is only a passing example and is not treated in any depth. The question has been sinnply, 'Does something in the text relate this notion to others?' If this test has been passed, only excessive triteness or repetition have barred a notion's entry. Thus the notion of wheels does not appear, since the text, in discussing a constructive pattern, provides only the information that they are parts of bicycles. The notion of bicycles appears, however, though not referred to page 111 where the wheels are mentioned. The reference is to page 30, where an explanation is given for the concept appearing in the domain of the life sciences. As to repetition: the notion of an atom is not referred to page 11 1, although it appears there in a constructive chain. The reason is that the Same information is $ven on page 21, to which the reader is directed. absence 4 3 , 4 4 abstractions 15, 67 accidents 33 acid radical 16 acidity 69 70 action 57, 58 addition, 37, 50, 51, 54, 101, 102 adjunction 48 adjustable dwices 29,30 atmosphere 72 air 39, 71, 78, 79, 80, 109 aiimentary canal 26 alliance 20, 22 angels 8 1 angle 35 angular momentum 58 animals 20, 22, 25, 26, 29, 30, 30, 66, 67, 77, 80, 81, 82,83,

108 antiques 38 antiseptics 37, 109 argument of complex numbers 56 arithmetic 79 arithmetic mean 36 army 36 artefacts 29, 30, 108 asemblies 25,28, 31, 71, 108 atmosphere 72 atomic nucleus 19 atomic orbits 25 a t o m 16, 19, 20, 21, 24, 26, 30, 31, 32, 37, 39,68, 79, 80, 108 average 36 averaging 50, 54, 56 bakers 109

124

Genera1 Index

Tillyard, E.M.W. : and the Elizabethan world picture 81 Toulmin, S. : and integrative levels 86 transitivity : defined 41 translation : and relation code sequences 93 transversiveness : deiïned 41 ; in the data field 91,100 units : as a variety of idea 21, 22, 27; in chains of ideas 21,22; and integrative levels 2 1, 22, 23 ;in table 28; mentioned 108 Universal Decirnal Classification : and the order of ideas 110

vector space : in the holotheme 52, 53,54; andpolynornials 53;and complex nurnbers 53 verbs : and relations 33 Vickery, B.G. : and classifications 81, 86 WilkUns, Bishop : and the arrangement of ideas in a universal language 82 Zeno : and constructive order 78; and the concept of zero 78,79 zero : and Zeno 78,79

PLACEMENT INDEX This index contains ideas whose position in the holotheme is discussed in the text. It does not contain other notions used in the discussion: for these the reader should refer to the general index. In deciding whether to admit an idea to the placement index, the rule has generally been to do so even if the idea is only a passing example and is nol treated in any depth. The question has been simply, 'Does something in the text relate this notion to others?' If this test has been passed, onfy excessive triteness or repetition have barred a notion's entry. Thus the notion of wheels does not appear, since the text, in discussing a constructive pattern, provides only the information that they are parts of bicycles. The notion of bicycles appears, however, though not referred to gage 111 where the wheels are mentioned. The referente is to page 30, where an explanation is given for the concept appearing in the domain of the life sciences. As to repetition: the notion of an atom is not referred to page 1 l 1, although it appears there in a constructive chain. The reason is that the Same information is given on page 21, to which the reader is directed. absence 4 3 , 4 4 abstractions 15, 67 accidents 33 acid radical 16 acidity 69 70 action 57, 58 addition, 37, 50, 51, 54, 101, 102 adjunction 48 adjustable devices 29, 30 atmosphere 72 air 39, 71, 78, 79, 80, 109 alimentary canal26 alliance 20, 22 angels 8 1 angle 35 angular momentum 58 animals 20, 22, 25,26,29, 30, 30, 66, 67, 77, 80, 81, 82, 83,

l08 antiques 38 antiseptics 37, 109 argument of complex numbers 56 arithmetic 79 arithmetic mean 36 army 36 artefacts 29, 30, 108 asemblies 25, 28, 31, 71, 108 atmosphere 72 atomic nucleus 19 atomic orbits 25 atoms 16, 19, 20, 21, 24, 26, 30, 31, 32, 37, 39,68, 79,80, 108 average 36 averaging 50, 54, 56 bakers 109

126

Placement Index

Battle of Salamis 33 (being) bent round 34 (being) beside 34 bicycle 30 bids 70, 71, 74, 77 blue 33, 105 bol1 weevil 29 boundlessness 28 boy 109 brain, 25, 105 braking system 25 breakdowns 33 bricks 19 Bumble, Mr 18 butchers 109 Camembert cheese 110 Canada 31,32,37 candestick makers 109 carbon 17, 71 carbon atom 19,20, 71 cardiovascular system 20, 83 causality 104 cells 1 9 , 2 0 , 2 1 , 2 2 , 2 4 , 3 0 , 6 0 central nervous system 25 cheap 33 child 87 chlorophyll molecule 20 chloroplast 20 chloroplast Iayer 20 church triumphant 18 cinematograph film 27 circle 56 circulatory system 108 colour 69 coming of the cocquecigrues 1 8 comrnand structure 36 (having) common factors 60 communities 22, 24, 102, 108 companies 21, 108 completeness 2 1 completion 28 complex numbers 53,55 congruente 50, 54 conservation 48 (being) contained in 34 containing 42,43, 51, 71,90, 101

copper 26,109 copper atom 26,31, 32, 109 counter nand 48 counter nor 48 cufflinks 25 cultures (social) 68, 69 cyclic addition 50, 54 cyclic subtraction 50 dance 33 darkness 77 degreasing section 20 deity 8 1 department 20, 21,22,30,108 diamond 7 1 digestive system 25, 26, 32 disjunction 43,44,46, 101 (being) divisible 54 division 50, 51, 54,55, 57 division by a multiple 50, 51, 54 droplets 26 dog 37 door 29 door frame 29 dodecahedron 79 downspout l 0 4 earth (element) 39, 78, 79 earth (planet) 72, 77, 83 economics 22, 84 electricity 26, 73 electrons 21, 25, 26, 30, 37, 38, 72 emotions 67, 105 energy 23,24,57, 58,97, 109 engines 29,30 equality 50,54 equation of circle 56 exchange 35 exclusion 43,44 exploding 34 explosions 34 extracting the root 55 eyes 25 factories 20, 33 faith 15, 18

!nt Index falling 34 fankasies 15 farms 29 feature cards : and co-ordinate indexing 1 14 feathers 70, 71,74 feelings 67, 105 feminine 87 fibre (muscle) 19 fiction 15, 18 field (mathematical) 52 fiue 39,78, 79,80 fish 77 Bow 97 fiowers 2 1 force 57, 58 (being) fractional 50, 54,92 fractions 52 France 31,32 full sets 22, 30 general secretary 2 1 geometric figures 55 (forming the) geornetric mean 56 geometry 22, 79 girl 8 7 God 77,81 gravity 57, 58 @eing) greater than 50, 51,54 (being) greater than or equal to 50,54 group (mathematica!) 52,54,9 7 gutters 104 habits 67 halogens 37 hmesting machinery 29 heart 19 heat 35 heaven 77 helium 17 hereafter 18 hexagons 25 (forming the) highest common factor 50, 54 hinges 29 history 84

homeostasis 21,29 honesty 69 hosepiping 27 hours per mile 36 houses 19 human beings 18, 20, 29, 67, 77, 80,87 human groups 20 hydrogen ions 70 hydrogen molecules 25 hydrosphere 72 hypergeometric series 76 hypoteneuse 79 icosahedron 79 identicality 43, 44,9 1 ignition system 25 (being) imaginary (of numbers) 56 impulses (mental) 67 incidents 33 inclusion 4 3 , 4 4 , 9 1 infinity 2 8 (forming the) inner product 56 integers 49, 50, 52, 53, 55 intersection 47,48, 54, 101 inversion about unity 50, 51, 54 inversion about zero 50, 51,54 involution 55 iodine 37,38 iron l 7 konmongery 38 irrational numbers 55, 78, 80 Julius Caesar 93 juvenile 8 7, 109 Krebs cycle 36 land 7 7 latches 29 law 84 lawnmowers 30 Iayer (chloroplast) 20 leaf cel1 20 leaves 21, 111 leaderless groups 25 length 57, 58,69, 109

128

Placement Index

life 23, 24 life sciences 22, 23, 24 light 69, 77 linear spaces 22, 30 lines 22, 23, 30 linguistice 84 lipoprotein membrance 19 liquid Row 9 7 literature 84 lithosphere 72 living beings 22 local authorities 22,24, 30 locks 29 logic 81,108 machines 21,30 magnesium atoms 20 magnetisrn 73 mallorn trees 18 mankind 67, 77, 80, 85 masculine 109 mass 57, 58, 109 mathematics 22, 23, 24, 40-56, 66, 78, 81,84, 108, 110 matrices 36, 55 matrix conformation 55 matter 22 meetings 34 member of set 22, 24 membership 43,91, 102 mernbrane (lipoprotein) 19 memories 67 mentefacts 66, 78 methyl group 19, 20 merger 104, 105 microscope 30 midges 25 miles per hour 36 mind 66, 67, 82, 86, 105 mist 26, 78 mitochondrion 19, 20, 60, 108 molecular biology 86 molecules 16, 19, 20, 21, 22, 24, 25, 26,30,85, 108 moment of inertia 58 momentum 58 monaco 70

money 9 7 multiplication 50-54, 57,91, 102 muscle, 19 muscle cells 19, 20 muscle fibre 19 nand 48 nations 18, 20, 21, 22, 24, 30,32, 37, 83,86,108 natura1 numbers 49, 52, 5 6 , 9 1 natura1 spirits 80 (being) negative 50, 54 negative natura1 numbers 49, 50, 52 neighbourhoods 20 nervous systems 25, 83 non-membership 4 3 nor 48 north pole 35, 109 northern 33 nucleus (atomic) 19 nullity 56 numbers 15,49-56,69, 78, X2 octohedron 79 omega minus particle 97 orbits of atoms 2 5 , 3 7 order 25, 28 organelles 20-24, 30, 80, 108 organisations 30 organs 20,22,24,30,80,108 orientation 35 orthogonality 56 Osiris 18 Ottawa University 31 overlap 43,44,45,46,48, 71,91, 101 paint 38 painting department 20 particles (subatomic) 20, 21, 22, 24,30,78,85 personnel department 108 pets 3 7 philosophy 84 phlogiston 18 phospholipid molecules 19

ent Index photons 21, 22, 23, 24, 30, 69, 72, 73, 84, 105 photosynthesis 105 physical sciences 22, 23, 24, 57, 58, 78, 84, 108 plants 18, 20, 22, 26, 29, 30, 66, 67, 77, 80, 81, 82, 83, 108 plastics 109 plastids 1 11 pliers 30 plywood 27 pneuma 80 points 22,23,24, 26,30, 79 political behaviour 22, political science 84 political theories 36 polynornials 53, 55 (being) positive 50, 54 positive natura1 numbers 49, 50, 52 (being a) power of 55 powers 57, 58 precedence 25 presence 4 3 , 4 4 (being) prime 54 prime movers 1 11 production line 20, 25 productive 3 professors 3 8 (being) properly divisible 50, 54 protons 70 psychology 66 public administration 84 public relations officers 38 pupillary 8 7 pyramids 79 Quantum theory 73, 86 quarks 73 races 34 Ragnarok 18 rain 33 rainwater system 25, 104 rational fractions 53 rational numbers 50, 52, 53, 55 rational spkits 80

129

real numbers 53, 55 recigrocal membership 4 3 , 4 4 reciprocity 28 recreation 84 redemption 18 reincarnation 1 8 relative complementation 48, 51 (being) relatively prime 50, 54 religion 18, 77, 81, 83,84 restriction 48 (being at) right angles 56 ring (mathematical) 52, 54 road system 83, 108 rocks 38, 78 root extraction (mathematical) 55 rope 27 running 34 Ruritania 3 1 schoolgirl 87 scissors 29 sea 3 semigroup (mathernatics) 52, 54 sensations 67, 82, 105 sets 22, 24,40-49 shells 29 sine waves 23, 73 Singapore 70 social behaviour 22 social sciences 22, 23, 24, 68, 84, 108,110 single-piece parts 29, 30, 104 spinal cord 25 space 22,23, 24, 26, 82 squares 23 staining agents 37 stock in trade 38 stress (physics) 58 sub-atornic particles 20, 22 sub-atomic physics 22 subsets 40-49 subtraction 50, 51, 54,55, 102 subtraction of a sum 50, 51, 54 succession 50, 52, 54, 56 sun 29 supracultural levels 69 supranational bodies 83, 86 surface tension 57, 58

Placement Index surprise 24 swan neck 25 symmetry 26 telephone system 25, 83 temperature 35 theology 18 Thor 15 tiles 38 time 57,58,109 tin 17 tools 30 tortoises 29 tractors 29 trade 35 trade unions 2 1 traffic 3 5, 9 7 transmission system 25 trees 18 triangles 79 traingular 33 tumbles 16 union 48, 54 United Nations 20 unity 50, 56, 79

universe 7 2 Utopia, 18,31,104,105 vectors 55 vector spaces 52, 53,54 vibïation 69, 105 vital spirits 18, 80 voters 25 war 35 water 26,39, 68, 78, 79 wavelengtb 69 weapons 38 (being) whole 50, 54 whole numbers 52 wind 33, 109 yeast 27, 109 yeast cells 109 yeast plants 27 young 87,109 zero 56, 79 zinc 17