Henri Poincaré's The Nature Of Mathematical Reasoning

In “The Nature of Mathematical Reasoning,” Henri Poincaré argues that mathematical facts are synthetic and a priori, because we can see the principle of induction is true by non-analytic methods. Exponents of logicism, such as Carl Hempel, would disagree with Poincaré because they would see induction as following analytically from the definition of natural numbers. However, this response fails to fully address the argument, because Hempel’s logical approach cannot capture human intuition about the infinity of the natural numbers. To show the synthetic a priori nature of mathematical statements, Poincaré distinguishes between general and particular statements about the natural numbers, arguing that the former are synthetic, while the latter …show more content…
Intuitively, the natural numbers are an infinite collection of objects, which we know exist because we can imagine how the process of starting at zero and taking successors continues forever; it is then evident that the principle of mathematical induction applies to this collection. In order to prove that the process of taking successors continues without end, Hempel is forced to take the existence of infinity as axiomatic. However, the axiom of infinity cannot be a mere logical law. We can imagine a self-consistent universe containing only finitely many objects (we can even imagine one with numbers that add and multiply, as in modular arithmetic); we cannot imagine a self-consistent universe that violates a genuine logical law, such as (p and q) implies p. We know that an infinity of objects exists, not through mere logic, but because we can imagine the infinite process of taking successors, using precisely the same intuitions that Poincaré uses to justify mathematical induction. In general, assuming the existence of infinities is fraught with danger, as Russell’s paradox shows, so doing so cannot be meaningless and analytic. We can legitimately accept the axiom of infinity, not because it is an analytic tautology, but because we have a special ability to imagine the …show more content…
Hempel does admit this: in a footnote, he states that his definitions do not reflect “what everybody has in mind” when they refer to the natural numbers, but they do give a means of “interpreting coherently and systematically” our “statements in science and everyday discourse.” However, the disconnect between his definition and what we mean by natural numbers is in actuality a serious blow to his project. Statements about our natural numbers are synthetic, not analytic; even if statements about Hempel’s natural numbers were analytic, the correspondence between statements about his numbers and statements about our numbers would be a synthetic fact, not an analytic one. The process of using induction to can be understood using Poincaré’s infinite chain of syllogisms, but not Hempel’s logic; as Poincaré says, “the difference must be