Poincarae's Philosophy: From Conventionalism to Phenomenology

By Elie Zahar | Go to book overview

6
Formalism, Intentionality,
or Platonism?

6.1 THE PROBLEM OF MATHEMATICAL ONTOLOGY

In the previous chapter, we saw that in his philosophy both of logic and of mathematics, Russell started out by being a Platonist. He subsequently regarded these two disciplines not as having any specific content, but as describing the most general forms of thought. He was then driven to despair by the realisation that some of his basic assumptions, namely the axioms of infinity, of choice, and of reducibility did not have a strictly logical character. So he finally looked upon his system as an inductive hypothesis to be treated on a par with astronomy. Thus he had been, in turn, a Platonist, a logicist and an empiricist.

We also noted that in his long debate with Russell, Poincaré adopted a Kantian or rather a quasi-Kantian attitude. As is well known. Kant held logic to be a completed discipline which had been brought to a close by its inventor, Aristotle. As for mathematics, it was supposed to construct all of its concepts in the pure forms of sensibility: space and time; the former was taken to found geometry, and the latter, arithmetic. Having rejected the idea that we possess an apriori spatial intuition, Poincaré took the geometrical postulates to be conventions or implicit definitions of a system of primitive concepts and relations. Also unlike Kant, he held not only arithmetic but also logic to be based on the intuition of time: in the same way that mathematical induction is founded on the notion of temporal succession, logic rests on operations of classification conceived as processes carried out over finite time-intervals. Only if they prove to be immutable, or at any rate ultimately stable, do classifications legitimately define their intended objects.

In Chapter 1, we examined the way in which phenomenology enables us to circumscribe the empirical basis of the sciences. In the present chapter. we shall analyse the sense in which Husserl's concept of intentionality might help towards reconstructing Poincaré's views about logic and about mathematical ontology; for as repeatedly remarked above. Poincaré did not

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