True by Definition

In 1924, with great self-confidence, Hans Reichenbach wrote,

It has become customary to reduce a controversy about the logical status of mathematics to a controversy about the logical status of the axioms. Nowadays one can hardly speak of a controversy any longer. The problem of the axioms of mathematics was solved by the discovery that they are definitions, that is, arbitrary stipulations...

With just as much self-assurance, A.J. Ayer claimed that the truths of pure mathematics are “analytic” in the sense that each follows “simply from the definitions of the terms contained in it.” Despite the bravado of its early adherents, this position has subsequently fallen out of favor. Critics have alleged that the position was refuted by Gödel's Incompleteness Theorems, that it is obscure what “analytic” and “definition” are supposed to mean, that there are no “analytic” truths, or that the axioms cannot be definitions.

True By Definition revisits this debate. Thomas Donaldson introduces the distinction between analytic and synthetic truths, and explains the controversy around this distinction. He defends a radical new position, developing an abductive, empirical case for the consistency of the basic axioms of arithmetic.

The result is a thoroughgoing form of empiricism about arithmetic, on which the basic principles of arithmetic are analytic yet a posteriori. True By Definition contributes to discussions of fundamentality and metaphysical grounding, suggesting that there is a parallelism between relations of analytic entailment among sentences and relations of metaphysical ground among facts.

Thomas Donaldson is Associate Professor at Simon Fraser University. His research explores questions in metaphysics, epistemology, and the philosophy of mathematics. He lives in Burnaby, British Columbia with his spouse and two children.