True by Definition

Abstract

Some of the logical empiricists defended the claim that the basic mathematical axioms are definitions and that the theorems of mathematics are consequently analytic. 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. It introduces the distinction between analytic and synthetic truths and explains the controversy around the distinction in the philosophy of mathematics and beyond. The book also defends a radical new position. It is argued that, in certain contexts, the axioms of arithmetic have the status of definitions; however, to establish the propriety of these definitions, one must establish that they are consistent. The book develops an abductive, empirical case for the consistency of the basic axioms of arithmetic. It argues that the best explanation of the empirical utility of mathematics requires the consistency of the axioms. The result is a thoroughgoing form of empiricism about arithmetic, on which the basic principles of arithmetic are analytic yet a posteriori. The book also 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.