Poincarae's Philosophy: From Conventionalism to Phenomenology

By Elie Zahar | Go to book overview

5
The Foundations of Mathematics
According to Poincaré and Russell

5.1 AN OVERVIEW OF POINCARÉ'S PHILOSOPHY
OF MATHEMATICS

In his preface to Dugac's book on Dedekind's work, Dieudonne refers to the profound intuition which guided the creative work of great mathematicians like Poincaré, Lebesgue, and Dedekind. Dieudonne claims that their sound mathematical instinct “led them to be wary of the excessive freedom demanded by this [Cantorian] school, a freedom which appeared to be in danger of degenerating into licence” (Dugac 1976, Preface; my translation). In the present chapter, I propose to examine another of Poincaré's powerful instincts, namely his philosophical intuition with regard to the foundations of mathematics. We should note that his philosophy of mathematics has been largely—and unjustly—ignored mainly because it conflicts with his actual mathematical practice. In previous chapters, we noted that Poincaré not only failed to unify his philosophical views into a coherent system; but also that he frequently shifted his position, thus exhibiting the kind of 'opportunism' which was regarded by Einstein as indispensable to all great scientists. Putting it pedantically, the set of all opinions expressed by Poincaré at different points of his career is inconsistent. Let us however remember that this remark applies to practically all great philosophers, which explains why system-building on a grand scale has gone out of fashion; for such systems are likely at any moment to succumb to their inner contradictions. A good philosopher is nowadays taken to be someone who, by means of analyses guided by a central idea, illuminates or solves some important foundational problem. In this sense, Poincaré was not only a supreme mathematician but also a major philosopher; so that Popper was largely justified in calling him the greatest philosopher of science ever (see Bouveresse 1998, p. 191).

In his controversy with Couturat, Poincaré adopted the haughty and even sarcastic style of a great mathematician irritated by a philosopher's

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