Putnam, Hilary (1926–2016)

5. Mathematics and necessary truth

Putnam did significant work in mathematics, collaborating with Martin Davis and Julia Robinson in the late 1950s on proving the unsolvability of Hilbert’s tenth problem, which sought an algorithm deciding the solvability of diophantine equations. The proof was completed by Yuri Matiyasevich in 1970.

The nature of logical and mathematical truth has been one of Putnam’s ongoing concerns, yielding several different positions. Throughout, he rejects the standard alternatives, platonism and conventionalism. The former, is, he maintains, given twentieth-century physics, obsolete; the latter, empty: as Carroll, Wittgenstein and Quine pointed out, conventions cannot ground logic because logic is required for their application (see Mathematics, foundations of). In ‘It Ain’t Necessarily So’, Putnam proposed replacing necessary truth with the more flexible, context-dependent notion of relative necessity, in line with his suggestion, raised regarding QM, that logic is empirical. Later, in ‘Analyticity and Apriority’, he argued that at least some logical truths are constitutive of rationality and, as such, cannot be rationally criticized or revised. This view is further elaborated in Rethinking Mathematical Necessity, where Putnam represents logical truths as ‘formal presuppositions of thought’ rather than as truths in the ordinary sense (see Analytic and synthetic).