Philosophy of Mathematics

What it asks

Mathematics is the most secure knowledge there is. It is also knowledge of things that are nowhere, do nothing, and could not be photographed.

That combination is the problem, and Benacerraf stated it as a dilemma in 1973. The natural account of what mathematical statements MEAN takes them at face value, as being about numbers and sets. The natural account of how anyone could KNOW anything requires some causal contact with what is known. Numbers cannot enter into causal contact with anyone. So the two natural accounts cannot both be kept, and every position in the field is a choice about which to give up.

This branch has no parent among the sciences. Mathematics is not an empirical science, and its philosophy is not a department of philosophy of science.

The recurring problems

Access. If numbers exist independently of us, how did we find out about them? If they do not, why is mathematics so hard, and why does it resist our preferences so completely?

Application. Mathematics developed for its own sake turns out, repeatedly, to describe the physical world with unreasonable precision. That is a fact requiring explanation, and no position explains it comfortably.

Which objects. The number two can be modelled in set theory in several equally good ways, and nothing decides between them. Benacerraf argued that if numbers were objects, one of those would have to be correct - so perhaps numbers are positions in a structure rather than objects at all.

Incompleteness. Godel showed that no consistent system rich enough for arithmetic proves all arithmetical truths. There are therefore mathematical truths with no proof, which is a difficulty for anyone who wants truth in mathematics to mean provability.

How it developed

Plato's answer, that mathematical objects are real and grasped by the intellect rather than the senses, is still the position everything else is defined against.

The modern field begins with the attempt to put mathematics on secure foundations. Frege tried to derive arithmetic from logic alone; Russell's paradox in 1901 showed his system was inconsistent, in a letter Frege received while the second volume was in press.

Three programmes followed. Logicism continued the reduction to logic. Hilbert's formalism treated mathematics as the manipulation of symbols and sought a finitary consistency proof. Brouwer's intuitionism held that mathematical objects are mental constructions, which requires rejecting proof by contradiction for infinite domains.

Godel's results in 1931 ended Hilbert's programme in its original form and damaged the others. The field then turned from foundations to the questions Benacerraf sharpened, and to the Quine-Putnam indispensability argument: that since our best physical theories quantify over mathematical objects, anyone who believes the physics is committed to the mathematics.

The positions within it

Platonism, nominalism, fictionalism, structuralism, formalism, intuitionism and logicism are the standing positions, all in the Schools section.

Structuralism, on which mathematics studies patterns rather than objects, is the most discussed current answer, largely because it dissolves the which-object problem rather than solving it.

Where it stands now

Structuralism and fictionalism are the two most active programmes, and both are attempts to keep mathematical practice intact while giving up something the traditional picture assumed.

A newer strand takes mathematical PRACTICE as its object: what mathematicians count as a good proof, why some proofs are found explanatory and others merely convincing, and what role computers can have in proof. Those questions have answers that can be investigated, which the traditional ones largely do not.