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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?"

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"Intuition of pure number" is the intuition Poincare inherited from Kant's a priori form of perception in time. Kant recognized two forms of perception that produce synthetic a priori, and therefore "rigorous", knowledge, space an time. The former gives rise to the geometric intuition, and the latter to the arithmetical one. However, after the discovery of non-Euclidean geometries the fallibility of geometric intuition became a consensus in 19th century. Accordingly, Helmholtz and Poincare amended the Kant's view by suggesting that geometric intuition is not specific enough to single out the Euclidean geometry, and after Riemann not even the geometries of constant curvature, at most it supports any locally Euclidean one. So by the end of the century the only "rigorous" intuition left was the a priori synthesis of multitude in time, the arithmetical "intuition of pure number". See how Kant explains why 7+5=12 based on this intuition.

I am not sure what the OP means by "intuition related to the knowledge of the properties of numbers", but it sounds more like Hilbert's view than Poincare's. Hilbert thought that we have no Kantian intuition of numbers as such, but rather intuition about manipulating symbols that purportedly represent them, i.e. intuition of symbolic properties. Since not just arithmetic, but all of mathematics, can be seen as symbolic manipulation of formal theories, Hilbert came up with the idea of grounding all of mathematics in the a priori synthetic knowledge of symbols. That was the famous Hilbert programme, see Was there a Kantian influence on Hilbert's formalist programme?

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The terms - Intuition, Pure Intuition and Logical Form - at least in philosophy are associated with Kantian metaphysics; this reading is confirmed by:

From the SEP article on Poincare:

Balancing the empirical element, there is a strongly apriorist element in Poincares philosophical views; for he argued that intuition provides an a priori epistemological foundation for mathematics.

His views about intuition descend from Kant, whom he explicitly defends.

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