It is usually supposed that Poincaré’s Science and Hypothesis contains a unified view of mathematics and physical science.  But its defense of a role for intuition in arithmetic does not fit well with the conventionalism Poincaré advocates elsewhere in the book.  After bringing out the conflict, I argue that the most usual way of resolving it does not succeed.  That is to suppose the sciences are arranged in a hierarchy such that arithmetic is presupposed by geometry, which is presupposed by mechanics, etc.  On the usual reading, Poincaré takes arithmetic to depend on an a priori intuition which underlies the notion of natural number (and with it the principle of mathematical induction), and is thereby seen to underlie all science.  In contrast, I maintain that Poincaré conceives mathematical reasoning as a general type, of which the justification of arithmetical notions is just one instance, distinct from its application to geometry.  The sense in which intuition is foundational for all science is that it helps us to decide on conventions, by showing them to be appropriate in light of our experience.  So Poincaré’s account of arithmetic has a place in his overall view of science, just a different place than is usually supposed.