The Act of Creation

by Arthur Koestler

  (d) Explanation by explicit analogy -- its validity depending on whether it is arrived at by selective or Procrustean methods.

  (e) Implicit understanding, when the phenomenon is recognized as an instance of a relation which has been abstracted but cannot be made verbally explicit.

  (f) The same as (e) plus a verbal label. The abstracted pattern can now be named but not otherwise verbally described, ('sweet', 'pungent', 'beautiful' -- visceral, kinesthetic, aesthetic experiences).

  (g) Explicit verbal explanations and definitions which sound precise and convincing, but where the codes to which they refer contain some hidden axiom, idée reçue, unwarranted assumption.

  (h) Over-explicit, rigid definitions which explain away problems as meaningless by taking the verbal components of the symbolic model to pieces -- forgetting that the 'exact' sciences have always operated with fuzzy concepts, that good cooks work in dirty kitchens, and that the sterilization of verbal concepts leads to sterility.

  Other headings could be interpolated into this list. Compared with the relatively few levels of understanding in the rat and even the chimpanzee, man's explanatory hierarchies represent a veritable tower of Babel; not merely because they reach higher, but because there are more finely graded levels between the unconscious processes at the base, and the abstract symbolism at the top.

  Thus instead of talking of insight and understanding as all-or-nothing processes, and making verbal explanation a test for passing school exams, we should proceed by more cautious statements, such as: Johnnie has now understood that a phenomenon P is a particular instance of a general relation R which he can name; he has also understood that R is a particular instance of S, which he can also name. He may further have grasped that S is a particular instance of T which he has abstracted but which he cannot verbalize; or it may dawn on him that experiences of the type S have something in common, and are perhaps particular instances of some general relation T, which, however, he has not yet abstracted.

  It follows that the degree of clarity and penetration of Johnnie's understanding must not be judged by the 'absolute height' he has reached in any 'vertical' abstractive hierarchy, but by the mastery he has attained on his own particular level. This depends on the factors already discussed, where the multi-dimensionality of experience (the intersection of several abstractive hierarchies in it) was taken for granted. Thus a garage mechanic may have a more complete understanding of the structure and function of motor cars than a theoretical physicist, in spite of the latter's more extended abstractive hierarchies; and an experienced Nanny may know more about children than an experimental psychologist. 'Vertical' progress in abstraction is of primary importance in the theoretical sciences only, but not in other domains of experience which are of greater significance to the majority. This may be the reason why the abstractive hierarchies were built up so very slowly in the learning process of the human species -- although the native equipment for it was given millennia ago -- and are acquired at an equally hesitant rate by the child.

  Theoretically the building of the tower of Babel, of hierarchies of abstractions, can go on indefinitely, or until the most general patterns of events are subsumed as particular instances under one all-embracing law -- a lapis philosophorum, or the unified field equations which Einstein hoped to find. But in fact individuals and cultures have their own ceilings of abstraction, where their quest for ultimates reaches saturation point -- in theism, pantheism, vitalism, mechanism, or Hegelian dialectics. In less exalted domains the ceiling can be surprisingly low. Some primitive languages have words for particular colours but no word for 'colour' as a class. The abstraction of Space and Time as categories independent from the objects which occupy them (i.e. from duration and extension) is only some three hundred years old; so are the concepts of mass, force, etc. The slow, fumbling emergence of abstract concepts which in retrospect appear so self-evident, is best illustrated by the beginning of mathematics -- a domain where pure abstraction seems to reign supreme.

  The Dawn of Mathematics

  To quote Russell's famous dictum once more, 'it must have required many ages to discover that a brace of pheasants and a couple of days were both instances of the number two'. In fact, evidence indicates that the discovery was not made in one fell swoop, but in several hesitant steps; and when it was achieved, some cultures were quite content to stop there and rest on their glories: Australian aborigines have only three number-words in their vocabulary: one, two, and many. [19] Most European languages show the traces of this stage of development: the Latin "ter" means both 'three times' and 'many' (cf. 'thrice blest').

  At the earliest stage the number concept is not yet abstracted from the objects which are numbered: 'two-ness' is a feature situated in particular twosome objects, not a general relation. Language bears witness to this 'embeddedness': a 'brace' of pheasants is not a 'pride' of lions; a 'pair', when married, is a 'couple', when engaged in singing, a 'duo'. In some primitive languages not only the number two but all numerals adhere to the type of object counted; in the Timshian tongue of New Guinea there are seven different classes of number-words referring, respectively, to flat objects, round objects, long objects, people, canoes, and measures; the seventh, used for counting in general, was the latest to develop. [20]

  Children go through a similar stage; Koffka mentions several three-year-olds who understood and used the words 'two apples', but did not understand 'two eyes', 'two ears'. One child, over four, when asked by his grandfather, 'How many fingers have I?', replied, 'I don't know; I can only count my own fingers.' There is an old joke about the new arithmetic teacher who, when he asked the class, 'How many oranges would Johnnie have if . . . etc.', received the indignant reply, 'Please, sir, we have only learned to count in apples'. The number-matrix, once adherent to the object-matrix, has gained such lofty independence, that their re-union is experienced as a bisociation of incompatibles.

  The next step is the abstraction of individual numbers, which are not yet regarded as parts of a continuous series. The first 'personalized' number-concept abstracted by primitive and child alike is almost invariably the number two. Next follow the concepts 'one' and 'many'. Some cultures, as mentioned, stop there; others retain traces of this stage in their languages; Hebrew and Greek have retained separate grammatical forms for the singular, the dual, and the plural. Koffka mentions a child who played with combinations of 'two and one', 'two and two', etc., until early into its fifth year; only then did the number-word 'three' become firmly established.

  These first individual number-concepts are only semi-abstract; they emerge as it were reluctantly from the womb, and retain for a long time the umbilical cord which attaches them to concrete objects or favourite symbols. In some primitive languages the word for five is 'hand', for ten 'two hands', Each number primarily refers to some such 'model collection' of practical or mystical significance: the four cardinal points, the Holy Trinity, the magic Pentagram, Each number has its preferential connotation, its personality and individual profile; it is as yet unrelated to other numbers and does not form a continuous series with them, The number sense of Otto Koehler's birds who can identify at a glance object-collections up to seven, and the same faculty of human subjects (to whom heterogeneous objects are shown on a screen for a time too short for counting) give us some idea of the character of our own earliest number-concepts, They could be described as qualities rather than as quantities in a graded series; the identification of numbers in experiments where counting is excluded, consists apparently in recognizing the quasi-Gestalt quality of 'fiveness' (I say 'quasi' because the objects are distributed at random and do not provide coherent figural unity). In other words, each of the first individual numbers up to perhaps seven or eight, is represented by a separate matrix -- its associative connotations -- and a perceptual analyser-code which enables us to recognize 'fiveness' directly, at a glance. The analyser probably works by scanning, as in the perception of triangles and squares; but this process is automatic and unc
onscious, as opposed to conscious counting.

  Thus the first 'personalized' number-concepts 'do not constitute a homogeneous series, and are quite unsuited to the simplest logical or mathematical operation'. [21] Those first operations are, apparently, carried out not by counting, but by matching the collection of objects to be counted against 'model collections' of pebbles, notches cut into a stick, knots made in a string, and above all the fingers and toes. The 'model collections' are usually those to which the individual number-concept originally referred. The earliest model collections seem to have been pebbles; to calculate' is derived from calculus, meaning pebble; to tally, from "talea", cutting. Relics of other model-collections abound in our weights and measures: feet, yards, furlongs, chains, bushels, rods. The Ayepones in Australia hunt wild horses; when they return from an excursion nobody asks them how many horses they have caught but 'How much space will they occupy?'. Even Xerxes counted his army by this method -- at least, if we are to believe Herodotus:

  All the fleet, being now arrived at Doriscus, was brought by its captains at Xerxes' command to the beach near Doriscus . . . and hauled up for rest. In the meanwhile, Xerxes numbered his army at Doriscus. What the number of each part of it was I cannot with exactness say, for there is no one who tells us that; but the count of the whole land army showed it to be a million and seven hundred thousand. The numbering was done as follows: a myriad [10,000] men were collected in one place, and when they were packed together as closely as might be, a line was drawn round them; this being drawn, the myriad was sent away, and a wall of stone built on the line reaching up to a man's navel; which done, others were brought into the walled space, till in this way all were counted. [22]

  It seems that as a general rule matching precedes counting in the most varied cultures.

  The next great advance was the integration of individual numbers into a homogeneous series -- the transition from cardinal to ordinal numbers, from 'fiveness' to 'the fifth'. The activity of counting seems to originate in the spontaneous, rhythmic motor activities of the small child: kicking, stamping, tapping, with his hands and feet; and the repetitive imitation of patterned series of nonsense syllables: 'Eeny meeny miny mo' -- a kind of pseudo-counting. Even more important is perhaps the spontaneous, rhythmical stretching of fingers and tapping with the fingers. Here was the ideal 'model collection' out of which, in the course of something like a hundred thousand years, the skill of finger-counting must have emerged. Danzig [23] calls attention to a subtle distinction:

  In his fingers man possesses a device which permits him to pass imperceptibly from cardinal to ordinal number. Should he want to indicate that a certain collection contains four objects he will raise or turn down four fingers simultaneously; should he want to count the same collection he will raise or turn down these fingers in succession. In the first case he is using his fingers as a cardinal model, in the second as an ordinal system. Unmistakable traces of this origin of counting are found in practically every primitive language.' A fascinating account of counting methods in primitive societies can be found in Lévy-Bruhl, How Natives Think (1926, Chapter V).

  I have tried to re-trace the first two steps at the base of the mathematical hierarchy. The first was the slow and hesitant abstraction of individual number concepts from the concrete objects to which they relate; the second was the abstraction of the sequential relation between numbers, which establishes the basic rule of the mathematical game: counting. A posteriori it would seem that the road now lay open to the logical deduction of the whole body of the theory of numbers; in fact each advance required the exercise of creative imagination, jumping over hurdles, following up crazy hunches, and overcoming mental blocks. Centuries of stagnation alternated with periods of explosive progress; discoveries were forgotten and re-discovered; within the same individual, brilliant insights could be followed by protracted snowblindness. It took several hundred years until the Hindu invention of zero was accepted in Christian Europe; Kepler detested and never accepted the 'coss' -- i.e. algebraic notation; his teacher Maestlin showed the same hostility towards Napier's logarithms. Progress in the apparently most rational of human pursuits was achieved in a highly irrational manner, epitomized by Gauss' 'I have had my solutions for a long time, but I do not yet know how I am to arrive at them'. The mind, owing to its hierarchic organization, functions on several levels at once, and often one level does not know what the other is doing; the essence of the creative act is bringing them together.

  The Dawn of Logic

  Let us turn to the genesis of logical codes -- and take as an example the so-called law of contradiction in its post-Kantian formulation: A is not not-A. To disregard this law used to be considered as a mortal sin against rationality; chief among the sinners were primitives and children, with their notorious imperviousness to contradiction -- plus all of us who dream at night being A and not-A in a single breath.

  Now in order to tell A from not-A, I must discriminate between them. Once I have discriminated between them 'A is not not-A' becomes tautologous, and you cannot sin against a tautology. But discrimination, as we saw, is a function of relevance. Functionally irrelevant differences between experiences may go either entirely unnoticed, or may be noticed but not retained, or they may be implicitly retained without arousing the need for explicit discriminatory responses, verbal or otherwise.

  Once upon a time I had a sheep farm in North Wales, and my Continental friends kept addressing their letters to: Bwylch Ocyn, Blaenau Ffestiniog, near Penrhyndeudraeth, England. The postman, a Gaelic patriot, was much aggrieved. Had he consulted Lord Russell (who was my neighbour and lived in Llan Ffestiniog), he would no doubt have learned that since Wales is not-England and Ffestiniog is Wales, it followed that foreigners had a pre-logical mentality and were unable to understand the law of contradiction. Thus, if the criteria of relevance of X, determined by X's patterns of motivation, values, and knowledge, are significantly different from Y's, then Y's behaviour must necessarily appear to X as irrational and 'indifferent to contradiction'. Hence the mass of misinterpretations which missionaries have put on the mentality of primitives, and grownups on the mental world of the child.

  To the primitive mind the most significant relations between persons, objects, and events are of magical character; in totemistic societies the existence of a magic link is assumed between members of the group and the totem. The Bororo tribe in northern Brazil, whose totem animal is the red arara, a kind of parakeet, affirm that they are red araras. Naturally, the Bororo can see the difference between a red bird and his fellow tribesman; but when referring to his conviction that both participate in a mystic unity, the difference between them is treated as irrelevant -- just as the child who calls all pointed things 'nose', chooses to ignore the difference between noses and shoes as irrelevant for its purpose. The difference between primitive and modern mentality is not that the former is indifferent to contradiction, but that statements which appear as contradictory to one, do not appear so to the other, because each mentality abstracts and discriminates along different dimensions of experience or 'gradients of relevance', determined by different motivations. This applies not only to so-called 'primitive' cultures (which, of course, are often far from primitive). European thought in the Middle Ages, and Aristotelian physics in particular, appear to us full of glaringly evident self-contradictions. The same applies to the philosophical systems of Buddhism and Hinduism, which do not discriminate between object and subject, perceiver and perceived, and in which the value of the discriminatory act itself is discredited by the dogma of the unity of opposites. [24] Vice versa, if we tried to see ounelves through the eyes of a Buddhist or medieval Christian, our notion that random events exert a decisive influence on an ordered and lawful universe would appear as self-contradictory. To them -- as to the pre-Socratians -- apparent coincidences were the vital gaps in the trivial web of physical causation through which the deus ex machina manifested its will; these gaps caused a kind of porousness in the texture of reali
ty through which destiny could infiltrate. In the modern European's universe, our critics would say, the figure-background relation in the porous texture is reversed.

  In the magic world of the child, physical causation and abstract categories play an equally subordinate and uncertain part, and cannot be regarded as a test for contradiction. When a child makes contradictory statements, for instance, 'the sun is alive because it gives light' and 'the sun is not alive because it has no blood', this is simply due to the fact that the word 'alive' was learned before the concept of aliveness was formed; it is a case, as so often found at that age, of a symbol in search of a referent. Piaget, from whose experiments with a child of nine I have been quoting, emphasized that children are apt to forget their previous judgements and then give a contradictory one. Yet obviously the word 'alive' is used on each of the two occasions with a different meaning, based on different criteria: in the first case on the discrimination between hot and cold bodies (A and not-A), in the second between bodies with and without blood (B and not-B). Thus there is no contradiction between the two statements, only confusion regarding the meaning of the word 'alive'; what the child intended to say was: 'the sun is alive in so far as it is hot, but not alive in so far as it has no blood'. Implicit discrimination between contrasting alternatives is often blurred in the explicit statements of the child owing to its linguistic inexperience.

  The child's attitude to its experiences is discriminatory within its framework of relevant relations, and its apparent contradictions are due partly to the fact that its scales of relevant values are different from the adult's, partly to the inadequacy of its symbolic equipment. But although the child experiences certain facts and relations as mutually exclusive and reacts accordingly, the relational concept of 'contradiction' itself is only abstracted at a much later stage of development; just as the child uses names before the name-thing relation as such is abstracted.