C Chevalley: On Herbrand's thought
Jacques Herbrand was born in 1908. In 1925, at the end of his secondary studies, he took first place at the École Normale Supérieure. I entered the École myself in 1926, and immediately noticed the entirely special place he occupied among his comrades, due to the vigour and universality of his spirit. In 1928, he took first place in the Agrégation, and was able to stay a fourth year at the École, during which he completed his thesis, which contains the work in mathematical logic that he had been pursuing for two years. In 1929-1930 he did his military service; during that year he published two notes on the units of algebraic number fields. These notes form the basis of the new methods in class-field theory. He also finished the article 1931, which was a continuation of his thesis. He spent the academic year 1930-1931 in Germany: first in Berlin with von Neumann, where he continued his work in logic, notably comparing his results to those of Gödel; then he went to Hamburg and finally to Göttingen. He returned to France at the end of July and left immediately to indulge in his favourite sport, mountaineering. There he died in a fall, on July 27. Thus passed one whom a mathematician described as "one of the greatest of his generation". Here is an extract from a letter of Professor Courant to Herbrand's father:-
During his short stay in Göttingen and even before this from his works, we have come to respect your son as one of the most promising and, because of the results he had already obtained, prominent young mathematicians of the world.
Jacques Herbrand expressed himself rather little on the philosophical ideas relating to the problems of mathematical logic (see, however, the Introduction to his thesis). This is why I have eagerly taken the opportunity afforded by the University of Geneva to deliver several remarks on his ideas. I draw them from memories of the many conversations I had with him. One should remember that I have no other source and, having myself thought about these questions, my own opinions could have unconsciously infected my recollections. Given the agreement which there was in general between us, I hope that the distortions I might thus have involuntarily introduced will be minimal.
In our attempt to penetrate Herbrand's system of thought, we shall rely on the following quotation (1930):-
But it should not be hidden that perhaps the role of mathematics is merely to furnish us with arguments and forms, and not to find out which of these apply to which objects. Just as the mathematician who studies the equation of wave propagation no longer has to ask himself whether waves satisfy this equation in nature, so no longer in studying set theory or arithmetic should he ask whether the sets or numbers of which he intuitively thinks satisfy the hypotheses of the theory he is considering. He ought to concern himself with developing the consequences of these hypotheses and with presenting them in the most suggestive manner; the rest is the role of the physicist or the philosopher.
When we refer to the distinctions that Bernays and Fraenkel formulated very clearly in their lectures [Bernays 1934 and Fraenkel 1934] between Platonist and intuitionist (or Aristotelian) mathematicians, I believe that this quotation allows us to conclude immediately that there are no Platonist inclinations to be found in Herbrand's thought. Indeed, Platonism, no matter what form it appears in, always admits the existence of a given world ruled by purely rational laws. Consequently mathematics is quite naturally considered to be knowledge of this world by man. Its role is precisely to find the human arguments which fit this world and which let us penetrate its structure. It was this that Herbrand called into question.
If one abandons the Platonist view, one must admit that mathematical objectivity, no longer a sign of the existence of a rational world, is created by man. The processes of axiomatization and the formalist method are the most extreme points of this movement towards the objective. This is to say, and I believe that this is what Herbrand thought, that objectivity is attained only in a pure symbolism, in emptying symbols completely of all meaning. Objectivity and concrete reality, far from being synonyms, exclude each other.
We can now understand why Herbrand's thought, although not Platonist, was not intuitionist either. In fact, the Intuitionists do not disallow treating in mathematics objects that are simultaneously rational and real. Undoubtedly they do not believe that such objects are given a priori. But they construct these objects starting from an intuition, namely, temporal intuition, so that mathematical assertions represent for them the assertions one can make regarding intuitions about time. Only the assertions that can be translated in this manner are valuable. For Herbrand, such restrictions were without foundation, for he believed that no reasoning whatsoever concerning something given and concrete would be valuable from a purely mathematical point of view, nor all the more that it was necessary to limit oneself to such reasoning.
The same considerations apply to logic. Logic comprises a schema which is objective only insofar as it is purely formal. Were one to give a sense to the symbols appearing in the formulas of logic, were one to consider them as representing operations of thought, one could cause logic to lose its objectivity. This is why it is not surprising that different thinkers have different opinions on the value of the axioms of logic. If the system of forms of classical logic is repugnant to Brouwer's thought, for example, this does not mean that this logic is denuded of value; it is an assertion about Brouwer's thought. If Heyting's formalism agrees with Brouwer's thought, this means that this formalism is suitable for describing the datum that his thought comprises. But in any case a human thought remains incongruous in any formalism; there is the same relationship between a formal logic and a mode of thought as between a mathematical equation and a physical phenomenon.
In regard to this, let us recall another quotation from Herbrand (1930l):-
It can be said that many of the obscurities and discussions that have arisen in regard to the foundations of mathematics have their origin in a confusion between the 'mathematical' and the 'metamathematical' senses of terms.
These difficulties spring from one's wanting to treat purely symbolic formulas of the domain of mathematics as assertions relating to something given. They are not of a different nature from those which engendered the birth of the infinitesimal calculus, which one wished to exclude for philosophical reasons because one wanted to see a 'real' object in the differential. Similarly, today, certain people wish to exclude the principle of the excluded middle because they wish to see 'real' assertions in the propositions.
From the preceding considerations we must not conclude that a mathematical act was for Herbrand a 'gratuitous' sort of act. Undoubtedly it is possible to carry out mathematics with any axioms and any rules of reasoning whatsoever; but in reality, and Herbrand liked to insist on this point, rigor has in a sense two complementary faces: if it is first the requirement of formalism, with respect to the 'rules of the game', it is also, in the sense given it by Leonardo da Vinci, an attempt at an ever more perfect description of something given. This description comes about by interpreting the axioms by means of experimental concepts. Mathematical physics already shows that one approaches positivist schemata (the direct interpretation of sensation) only at the price of an increasing abstraction, which one can compare to a sort of magic by which man dominates the domain of sensation only in first completely leaving it and in passing to the pure world of mathematics.
Just as mathematical physics permits us to penetrate further and further into the structure of matter, logic allows us to describe something nearer yet to man than his sensations: his intellectual thought. Herbrand said to me one day, "I would like to construct a system that contains all present-day thoughts". This is the greatest demand one could make on a formal logic: it leads us to the very centre of the drama of Herbrand's thought, balancing between an investigation always more concrete and a formalism always more abstract. This drama was enacted in Herbrand's thought with a poignant intensity. Could it perhaps be a necessity of fate that where the spirit attains such a degree of violent purity, there death would be closest?
JOC/EFR August 2006
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