According to Poincaré, induction is the perfect example of the kind of reasoning that characterizes mathematics; it bridges the gap between the finite and the infinite. Suppose that the number 1 has been defined, along with the operation that maps an integer x to x + 1. It is then easy to define the sum of two integers x and y. We next prove the properties of addition, first associativity and then commutativity: for example, we show that x + 1 = 1+x, and then that if, for some integer y, we have x + y = y + x, then we also have x + ( y + 1) = ( y + 1) + x. We therefore conclude that, for every y, x + y = y + x.
But what exactly is the nature of mathematical induction? Is it the only form of reasoning used to prove the properties of addition? Is it an axiom of logic, or merely a definition? Poincaré went on to think deeply about these questions, which remain relevant today, especially in mathematics education, where arithmetic, alongside geometry, plays a central role.
