 Central to Poincaré’s analysis of arithmetic is the principle of recurrence, or mathematical induction, which he considered to be a synthetic a priori principle that “would then seem so self-evident that we would not be able to conceive of the opposite proposition” (Poincaré 1902: 74; 2017: 42). The principle of mathematical induction is necessary to prove general statements about numbers and is thus a necessary tool in mathematics. Therefore the principle of mathematical induction is similar to a natural hypothesis or at least plays a similar role to that of a natural hypothesis, i. e. a mathematical structure, which is presupposed and “more fundamental than any particular systems of natural numbers” (Folina 2020: 293). Of course, there are important differences between mathematical induction and the principle of induction used in the natural sciences. The principle of mathematical induction gives certainty, while the inductive generalizations in the natural sciences give only probability (1902: 27; 2017: 3). The principle of mathematical induction is based on “the affirmation of the power of a mind that knows itself capable of conceiving the indefinite repetition of a particular action as soon as this action is possible” (1902: 41–42; 2017: 15), so that there can be no uncertainty. It can be argued that Poincaré is justified in including his analysis of arithmetic in Science and Hypothesis, that is, in his analysis of empirical science, given that there is a striking similarity between the role played by the principle of mathematical induction and the principle of induction in natural science, which “is as difficult to justify […] as it is to get along without it” (1905b: 176; 1913b: 345).