Putnam, Hilary (1926–2016)

5. Mathematics and necessary truth

Putnam did significant work in mathematics and computation theory. Among other achievements, he, collaborated 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 (polynomial equations, named after Diophantus of Alexandria, involving only integer constants and allowing only integer solutions). The proof was completed by Yuri Matiyasevich in 1970.

The nature of logical and mathematical truth has been one of Putnam’s ongoing concerns, and the subject of many of his papers. Like in other areas of philosophy, he has mostly been a realist about mathematics, seeing mathematical truths as objective, though rejecting platonism and its commitment to a distinct realm of mathematical objects. He strongly opposed conventionalism, the position favored by logical positivists. As Lewis Carroll, Wittgenstein and Quine pointed out, conventions cannot ground logic because logic is required for their application (see Mathematics, foundations of). Furthermore, mathematical truth, according to Putnam's early papers, is neither a priori nor irrevisable. In fact, it is comparable to empirical knowledge in both epistemic status and method. Realism about mathematics is thus supported by an argument similar to the 'no miracle' argument for scientific realism: It is the best explanation for the success of mathematically-based science. In other words, to the extent that empirical science (physics in particular) is based on mathematics, this mathematical basis is likewise vindicated. This realist argument from the indispensability of mathematics in empirical science—the indispensability argument, as it is often called-- has also been advocated by Quine. In It "Ain’t Necessarily So" (1962; reprinted in 1975a, Putnam proposed replacing necessary truth with the more flexible, context-dependent notion of relative necessity. Necessity is thus relativised to a framework of received assumptions; it is not absolutely irrevisable. This account of necessity is in line with Putnam's suggestion, raised in the context of discussing QM, that logic is empirical. Later, in "Analyticity and Apriority" (1979; reprinted in 1983), he argued that at least some logical truths are constitutive of rationality and, as such, cannot be rationally criticized or revised. The same view is further elaborated in "Rethinking Mathematical Necessity" (in Putnam 1994), where Putnam represents logical truths as ‘formal presuppositions of thought’ rather than as truths in the ordinary sense.