Philosophy of Logic

The argument holds that if our best scientific theories quantify over mathematical objects and cannot be formulated without doing so, then we have the same reason to believe in those objects as in the theoretical entities of physics. Putnam presents this as a way of justifying mathematical realism without appeal to a special faculty of mathematical intuition. Mathematics earns its ontological standing through its role in explanation and prediction. The argument is usually associated with both Quine and Putnam, and it has become a standard starting point in the philosophy of mathematics, though critics dispute its premises.

Quine's Philosophy of Logic, published in 1970, treats logic as classical first-order logic and argues that deviant logics change the subject. Putnam's book shares Quine's view of ontological commitment through quantification, but it gives more direct attention to the realism of mathematics and the problem of how mathematical truth is known. The two books are often paired because they developed the same naturalistic outlook from slightly different directions. Readers should keep them distinct, since the shared title encourages confusion in citations and library searches.

Hartry Field, in Science Without Numbers, argued that physical theory can be reformulated without quantifying over mathematical objects, treating mathematics as a useful but dispensable fiction. If that project succeeds, the indispensability premise fails. Defenders of the argument have replied that Field's reconstruction covers only parts of physics and sacrifices simplicity and explanatory power. Others, such as Mark Colyvan, have defended the argument in more recent work. This exchange is the central line of dispute stemming from the argument Putnam helped popularize.

Not entirely. In later work he moved away from the strong realism about mathematical objects and explored alternatives, including an interpretation of mathematics in terms of modal facts rather than abstract entities, and he became critical of metaphysical realism more broadly. His later essays on the philosophy of mathematics and his internal realism reflect those changes. The book is therefore best read as a statement of an earlier position, and readers tracing his development should compare it with his subsequent papers and with the 1983 collection Realism and Reason.

Putnam discusses whether logical truths are true by convention, by meaning, or by virtue of how things are, and he treats the conventionalist answers with skepticism. He considers how logic relates to inquiry and whether even basic logical principles might be revised in light of evidence, a theme he explored further in his later writing on quantum logic. The discussion is compact, and some of its claims were developed in more technical detail elsewhere, so readers wanting a full treatment of the revisability of logic should consult those papers.

A common misreading takes it to prove that mathematical objects exist independently of any reasoning about them, as if it were a deductive proof. It is better described as an inference to the best explanation: given a naturalistic outlook and our best theories, commitment to mathematical entities is the most reasonable stance. Its force therefore depends on accepting those background assumptions, including confirmational holism, which critics have challenged. Treating it as a knock-down argument misses the premises on which its conclusion rests.