ABSTRACT
Are there any really believable philosophies of mathematics other than (possibly) Quine’s or perhaps as modifi ed in ways suggested by Colyvan (2001) and in some of his more recent work? Well there are some halfbelievable philosophies of mathematics but these strike me either as refutable or as profoundly unattractive as well as implausible. Logicism, which in a sense identifi ed mathematics with set theory but also identifi ed logic with set theory, got exploded by the realization that, as Quine has shown us, there is a clear and natural break between logic proper and set theory. Three considerations make the break in the same place. Taking logic to be fi rst order predicate logic (quantifi cation theory, as Quine called it), we have: (i) logic has no constant predicate, other than the identity predicate which is itself eliminable in application of the logic to a discourse with a fi nite vocabulary; but set theory has the set membership predicate. (ii) logic is complete, that is, all valid schemata are provably so from a fi nite set of axioms; whereas set theory is, as Gödel proved, incompletable; (iii) logic is not Platonistic in Quine’s sense, that is, it has variables and predicates but does not have expressions for, or quantifi cation over, abstract entities such as sets. I regard so-called higher order logic as in Quine’s witty phrase “set theory in sheep’s clothing.” (If we accept George Boolos’s plural quantifi cation, logic loses features (i) and (ii) but not (iii).) Thus logic and set theory have a clear break between them. Wittgenstein and Ramsey had persuaded Russell that logic was tautologous, and this profoundly depressed Russell. Early in life he had thought of mathematics as the study of a shining realm of Platonic entities. It was not nice for him to come to think that mathematics was a matter of saying in more and more complicated ways the same thing, namely nothing. Similarly it now amazes me how at Oxford in my younger days we were complaisant about thinking that mathematics was analytic. To be sure it seemed to solve the epistemological problem but at what a cost! Of course that a view is emotionally repellent is no argument against it, but Quine’s pointing out the clear break gives us a rational argument against the idea that mathematics is analytic. Indeed we can now argue that what was called “logicism” was really a form of Platonism. There is still the epistemological problem, and the indispensability argument plausibly solves that for all the mathematics needed for physics, which is most if not all of mathematics except for the remoter parts of set theory. What about nominalism? There seems to be considerable doubt whether Hartry Field’s nominalistic program can be achieved. He wants to show that all of physics could be stated without the mathematics, which he sees as a mere convenience. Even if it can, it still leaves the usefulness of mathematics for physics as a puzzling phenomenon. Moreover Field’s hoped-for achievement would seem to reduce to instrumentalism or fi ctionalism about mathematics: neither, even if true, a happy thought for the pure mathematician.
