LOGICAL FORMALISM AND AXIOMATIC METHOD

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After the more or less evident bankruptcy of the  different systems developed to examine the relations of maths to reality or to the great categories of thought, it look, at the beginning of the century, as if the attempt had just about been abandoned to conceive of maths as a science characterized by a definitely specified purpose and method; instead, there was a tendency to look upon maths as "as  collection of disciplines based on particular, exactly specified concepts"  interrelative  by "a thousand roads of communication", allowing the methods of  any these disciplines to fertilize one or more of the others.

Today we believe, however, that the internal evolution of math science has, in spite of appearance, brought about a closer unity among its different parts, so as to create something like a central nucleus that is more coherent than it has ever been. The essential aspect of this evolution has been the systematic study of the relations existing between different math theories, and which has led to what is generally known as the "axiomatic method".

The words "formalism" and "formalistic method" are also often used; but it is important to be on one's guard from the start against the confusion which may  be caused by the use of these ill-defined words, and which is but too frequently made use of by the opponents of the axiomatic method. Everyone knows that superficially maths appears as this "long chain of reasons" of which Descartes  spoke; every math theory is a concatenation of propositions, each one derived  from the preceding ones in conformity with the rules of a logical system, which is essentially the one codified, since the time of Aristotle, under the name of I "formal logic", conveniently adapted to the particular aims of the mathematician.

It is, therefore, a meaningless truism to say that this "deductive reasoning" is a unifying principle for maths. So superficial a remark can certainly not account for the evident complexity of different math theories, not any more than one  could, for example, unite physics and biology into a single science on the ground ; that both use the experimental method.

The method of reasoning by means of chains of syllogisms is nothing but a transforming mechanism applicable just as well to one set of premises as to an other; it could not serve therefore to characterize these premises. In other words,  it is the external form which the mathematician gives to his thought, the vehicle I which makes it accessible to others, in short, the language suited to maths; this is all, no further significance should be attached to it.

Indeed, every mathematician knows that a proof has not really been "understood" if one has done nothing more than verifying step by step the correctness of the deductions of which it is composed, and has not tried to gain a clear insight into the ideas which have led to the construction of this particular chain of deductions in preference to every other one.

To lay down the rules of this language, to set up its vocabulary and to clarify its syntax, all that is, indeed, extremely useful; indeed, this constitutes one aspect of the axiomatic method, the one that can properly be called logical formalism (or "logistics" as it is sometimes called). But we emphasize that it is but one aspect of this method, indeed, the least interesting one.

What the axiomatic method sets as its essential aim, is exactly that which logical formalism by itself can not supply, namely, the profound intelligibility of  maths. Just as the experimental method starts from a priori belief in the permanence of natural laws, so the axiomatic method has its cornerstone in the conviction that, not only is maths not a randomly developing concatenation of syllogisms, but neither is it a collection of more or less "astute" tricks arrived at by  lucky combinations, in which purely technical cleverness wins the day.

Where the superficial observer sees only two, or several, quite distinct theories, lending one another "unexpected" support through the intervention of a  mathematician of genius, the axiomatic method teaches us to look for the deeplying reasons for such a discovery, to find the common ideas of these theories,  buried under the accumulation of details properly belonging to each of them, to  bring these ideas forward and to put them in their proper light.

The concept of the axiomatic method was not formed at once; rather, it is a stage in an evolution, which has been in progress for more than a half-century, and which has not escaped serious opposition, among philosophers as well as among mathematicians themselves. Many of the latter have been unwilling for a long time to see in axiomatics anything else than futile logical hairsplitting not capable of fructifying any theory whatever.

This critical attitude can probably be accounted for by a purely historical accident. The first axiomatic treatments and those which caused the greatest stir (those of arithmetic by Dedekind and Peano, those of Euclidean geometry by Hilbert) dealt with univalent theories, i. е., theories which are entirely determined by their complete system of axioms; for this reason they could not be applied to any theory except the one from which they had been extracted (quite contrary to what we have seen, for instance, for the theory of groups). If the same had been true for all other structures, the reproach of sterility brought against the axiomatic method, would have been fully justified.

There also occurred, especially at the beginning of axiomatics, a whole crop  of  monster-structures entirely without applications, their sole merit was that of ,1 lowing the exact bearing of each axiom, by observing what happened if one omitted or changed it. There was, of course, a temptation to conclude that these were the only results that could be expected from the axiomatic method.

But the further development of the method has revealed its power; and the repugnance which it still meets here and there, can only be explained by the natural difficulty of the mind to admit, in dealing with a concrete problem, that a form of intuition, which is not suggested directly by the given elements (and which often can be arrived at only by a higher and frequently difficult stage of abstraction), can turn out to be equally fruitful.

As concerns the objections of the philosophers, they are related to a domain, on which for reasons of inadequate competence we must guard ourselves from entering; the great problem of the relations between the empirical world and the math world. We do not consider here the objections which have arisen from the application of the rules of formal logic to the reasoning in axiomatic theories; these are connected with logical difficulties encountered in the theory of sets. Suffice it to point out that these difficulties can be overcome in a way which leaves neither the slightest qualms nor any doubt as to the correctness of the reasoning.

That there is an intimate connection between experimental phenomena and math structures, seems to be fully confirmed in the most unexpected manner by the recent discoveries of contemporary physics. But we are completely ignorant as the underlying reasons for this fact (supposing that one could indeed attribute a meaning to these words) and we shall, perhaps, always remain ignorant of them. There, certainly, is one observation which might lead the philosophers to the greater circumspection on this point in the future: before the revolutionary developments of modern physics, a great deal of effort was spent on trying to  drive maths from experimental truths, especially from immediate space intuitions. It turned out, however, that this intimate connection was nothing more than an a fortuitous contact of two disciplines whose real connections are much more deeply hidden than could have been supposed a priori.

From the axiomatic point of view, maths appears thus as a storehouse of abstract forms — the math structures; and it so happens — without our knowing why  -  that certain aspects of empirical reality fit themselves into these forms, as it though a kind of preadaptation, of course, it cannot be denied that most of these forms had originally a very definite intuitive content; but it is exactly by deliberately throwing out this content, that it has been possible to give these forms all the power which they were capable of displaying and to prepare them for new interpretations and for the development of their full power.

It is only in this sense of the word "form" that one can call the axiomatic method a "formalism". The unity which it gives to maths is not the armour of formal logic, the unity of a lifeless skeleton; it is the nutritive fluid of an organism to which all the great math thinkers since Gauss have contributed, all those who, in the words of Lejeune-Dirichlet, have always labored to "sub­stitute ideas for calculations".