# What did Poincaré mean by intuition of pure number?

To what does Poincaré refer in his article Intuition and Logic in mathematics when he speaks about the intuition of pure number? He refers also to two other forms of intuition, besides the "intuition of pure number", namely, "analogical intution" and intuition which presupposes "mathematical induction". My guess is that he may refer to a sort of intuition related to the knowledge of the properties of numbers.

Here are some passages of the text in which he refers to the "intuition of the pure number".

"We have then many kinds of intuition; first, the appeal to the senses and the imagination; next, generalization by induction, copied, so to speak, from the procedures of the experimental sciences; finally, we have the intuition of pure number, whence arose the second of the axioms just enunciated, which is able to create the real mathematical reasoning [...]

I have shown above by examples that the first two can not give us certainty; but who will seriously doubt the third, who will doubt arithmetic? Now in the analysis of to-day, when one cares to take the trouble to be rigorous, there can be nothing but syllogisms or appeals to this intuition of pure number, the only intuition which can not deceive us. [...]

I have said above that there are many kinds of intuition. I have said how much the intuition of pure number, whence comes rigorous mathematical induction, differs from sensible intuition to which the imagination, properly so called, is the principal contributor. [...] Could we recognize with a little attention that this pure intuition itself could not do without the aid of the senses? [...]

It is the intuition of pure number, that of pure logical forms, which illumines and directs those we have called analysts. This it is which enables them not alone to demonstrate, but also to invent. By it they perceive at a glance the general plan of a logical edifice, and that too without the senses appearing to intervene. [...] Is there room for a new distinction, for distinguishing among the analysts those who above all use this pure intuition and those who are first of all preoccupied with formal logic?"

• You can see this post. Dec 7, 2015 at 18:22