Hilary Putnam - Philosophy of Mathematics

Philosophy of Mathematics

Putnam made a significant contribution to philosophy of mathematics in the Quine–Putnam "indispensability argument" for mathematical realism. This argument is considered by Stephen Yablo to be one of the most challenging arguments in favor of the acceptance of the existence of abstract mathematical entities, such as numbers and sets. The form of the argument is as follows.

1. One must have ontological commitments to all entities that are indispensable to the best scientific theories, and to those entities only (commonly referred to as "all and only").
2. Mathematical entities are indispensable to the best scientific theories. Therefore,
3. One must have ontological commitments to mathematical entities.

The justification for the first premise is the most controversial. Both Putnam and Quine invoke naturalism to justify the exclusion of all non-scientific entities, and hence to defend the "only" part of "all and only". The assertion that "all" entities postulated in scientific theories, including numbers, should be accepted as real is justified by confirmation holism. Since theories are not confirmed in a piecemeal fashion, but as a whole, there is no justification for excluding any of the entities referred to in well-confirmed theories. This puts the nominalist who wishes to exclude the existence of sets and non-Euclidean geometry, but to include the existence of quarks and other undetectable entities of physics, for example, in a difficult position.

Putnam holds the view that mathematics, like physics and other empirical sciences, uses both strict logical proofs and "quasi-empirical" methods. For example, Fermat's last theorem states that for no integer are there positive integer values of x, y, and z such that . Before this was proven for all in 1995 by Andrew Wiles, it had been proven for many values of n. These proofs inspired further research in the area, and formed a quasi-empirical consensus for the theorem. Even though such knowledge is more conjectural than a strictly proven theorem, it was still used in developing other mathematical ideas.