THE EVOLUTION OF MATHEMATICAL CONCEPTS

Maybe the prototypical shepherd just kept a heap of pebbles handy by his cave, one for each sheep, to make sure by MATCHING that at the end of the day he had his whole flock together � the first occurrence of the mathematical arrow. The next guy paid attention to the pecking order among his charges and chose his pebbles accordingly. And then � much later � one with a poetic twist of mind gave individual names to his sheep and picked pebbles to match their personalities in looks, color, shape and mood so that, if one went missing, he could tell by looking at the leftover pebble which one of his flock to search for where, according to the culprit's specific idiosyncrasies. Finally, with all that time on their hands, some of the shepherds started creating poetry or inventing music, others projected and extrapolated their minds into higher realms of mathematics � and started wondering.

Here is the beginning of mathematics, not only arithmetic, the whole works, structures (you start grouping your flock, and those groups will interact), mappings and probably even the concept of infinity, "what if those ewes keep lambing and lambing till I run out of pebbles...". Pretty soon these concepts become PHENOMENA and begin evolving in interaction with their creators and with the uses they are put to.

When the ANTHROPOLOGIST has told his story and the PHENOMENO- LOGIST has had a look at how a mathematician's mind works it is for the NEURO PHYSIOLOGISTS to figure out what is going on in the brain of those shepherds and their descendants. The PSYCHOLOGY of mathematical activity � and obsession � also deserves attention and is bound to shed light on the mystery of the prodigy.

The view of mathematical "objects" as mental constructs forever caught up in a dynamic process of evolution was succinctly articulated by L.E.J. Brouwer, the Dutch topologist who, during the first quarter of this century, founded the school of INTUITIONISM as the most compelling alternative to PLATONISM. Occasionally Intuitionism is accused of leading into solipsism. But the understanding of mathematical intuition as a sense for charting one's way around an environment including fellow creatures implies that its tools, the concepts, must be evolving by joint and competing efforts of a community. Very much in keeping with what I understand is Stanislas Dehaene's view. With Brouwer I believe in preverbal mathematical perception, where by perception I mean an activity, a process of "seeing as", picking out of patterns and imposing frames of reference.

Friedrich Wilhelm Nietzsche (1844-1900) had a keen understanding of the anthropological evolution of mathematics and rational thinking. His Der Wille zur Macht (The Will to Power, 1887) contains poignantly expressed insights into the genesis of the laws of Logic, many of them anticipating Intuitionism!

George Lakoff's stress on image schemes and conceptual metaphors is compelling, especially his suggestion of "expansion to abstract mathematics by metaphorical projections from our sensory-motor experience". Yes we do have mathematical bodies! On a primordially homogenous environment we impose a grid commensurate in size and compatible in shape with our bodies as we know them from direct experience. One step further, we project our bodies beyond what is immediately perceivable, spurred on by a tenacious intention "to make sense of it all". Have you ever noticed how many mathematicians are rock climbers? The process of mulling over a mathematical problem displays a striking similarity to that of surveying a cliff before the ascent; of visualizing and comparing alternate routes, from the big lines of ridges, ledges and chimneys down to the details of toe and finger holds, and then weighing possibilities of what might be encountered beyond the visible; all in perfectly focused concentration, projecting ahead, extrapolating, performing so-called "Gedankenexperimente" (thought experiments) and sensing them throughout one's bones and muscles. And finally setting off to break trail through the folds of a brain!

Already in 1623 Blaise Pascal articulated in his Pens�es (thoughts) the observation that the abstract schemata we impose on the world in order to interact meaningfully with it are shaped by the experience of our bodies. [4]

During the last half century the evolution of so-called CATEGORY THEORY out of algebraic topology has developed a dynamic language of diagrams in which the abstract concepts of universal algebra find their natural habitat. [5] "Diagram chasing" � a systematic form of hand waving � is a way of making sense of the abstract structural and conceptual under-pinnings of mathematics, including Arithmetic and Geometry, Logic and set theory, as well as of the juxtaposition between discrete and continuous phenomena. It turns out that Topoi, a particularly prolific species of categories, have the structure of intuitionistic Logic � an amazing corroboration of INTUITIONISM. F. W. Lawvere at SUNY Buffalo, a pioneer in the field since the early sixties, and his associates are beginning to make significant contributions to cognitive science.

As to PLATONISM, whether deliberately or inadvertently, most mathematicians still act and talk as if they were dealing with objects that are part and parcel of the furniture of their Universe. I do it myself, and so does George Lakoff when he refers to the straight line and the reals. It is such a convenient make-believe stance, not to be confounded, however, with the deep allegorical truths revealed in the poetry of Plato's dialogues.

But there is more to be said when we stop to contemplate what we call REALITY. Think how often a writer will create characters only to find them taking on a life of their own, doing things or getting into trouble that their creator had not intended for them at all. So, the positive integers are mental constructs. They are tools shaped by the use they are intended for. And through that use they take on a patina of reality! Nor do they rattle about in isolation. They interrelate, they pick up individual personalities through interaction, by their position in the natural ordering, by splitting into primes, by what they are good for, in what contexts they play what roles.

And before we know it we have a problem on our hands like Fermat's Last Theorem! Its statement can be explained to every child, using a bit of hand waving and the ever handy dots. Through generations the belief in its truth had grown for ever more entrenched. No counter example was found, but no proof was in sight either until Andrew Wiles [6] succeeded in blazing the final trail to the goal through abstract territory, rugged and disconnected in places and prepared by the toil of his peers in others. To the experts the proof is illuminating, but not to the ordinary mathematician in the street. By now our tools are so highly developed that they bring us information about our own creations that we cannot fathom with the unaided mathematical senses, even though it may concern situations whose meaning we can understand perfectly well. In physics and astronomy we are used to similar situations: our instruments can reach physical phenomena way beyond the reach of our physical bodies. The interpretations of these messages from beyond are encoded in theories of our own construction.

The method of FORMALIZATION is by now widely accepted, used and discussed. But it has limitations and is trailing some baffling "non-standard" phenomena in its wake. In order to put these into proper perspective a technical digression is needed.

FORMAL THEORIES

Allowing the articulation of "axioms" and of rules of deduction governing their use, the systematic construction of formal languages leads to FORMALIZED THEORIES consisting of theorems, i.e., wellformed sentences (wfs' for short) obtained from axioms by chains of deductions according to those rules.

A formal proof is a finite sequence of wfs' starting with axioms, hanging together by the formal rules and ending with the theorem proved by it. The formalized theory itself becomes a topic for theoretical investigation since it is bound to have properties that go beyond what we put into it. Will it be formally consistent in the sense that the negation of a theorem will never show up as a theorem too? Is it formally complete ,i.e., does every sentence have a proof unless its negation has one? These are typical problems for the meta-theory.

The choice of axioms is not arbitrary. We are guided by common sense of mathematical perception, by criteria that deserve investigations to which the EDGE group seems to be making valuable contributions. As we acquire and develop intuitive concepts of sets, spaces, geometries, algebraic structures and all the rest, we try to grasp them by characteristic properties and are led to basic postulates.

Occasionally sustained experience reveals that the original construction was not fully determinate, that the axioms are not complete. They don't suffice to pin down the intended concept uniquely. Some sentence A � Euclid's fifth postulate for instance � is left undecided by what was considered an axiomatic characterization of the concept � of, say, a geometry. Both A and its negation not-A are formally consistent with the axioms. Well, for some purposes it is useful to assume Euclid's parallel axiom for geometry, or well-foundedness for sets, at other times it may be handy to deal with bottomless sets or crooked squares. The tools are evolving as we are using, refining and adjusting them. Such experiences that at first look like failures deepen conceptual understanding and expand mathematical horizons.

The situation of the arithmetic N over the natural numbers 0,1,2,3,... and that of the ordered field R of the reals are more subtle. In both cases we "know exactly" what structure we have in mind, there is no question of bifurcation of concepts. Yet in the case of N a complete axiomatization founders on the requirement of effectiveness while, even though completely formalizable, the elementary theory of R, has so-called non-standard models, as does every theory of an infinite structure.