The Act of Creation (89 page)

 

 

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 unconscious, 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