20th Century Schools in the Philosophy of Mathematics

For a large part of this century the Philosophy of Mathematics has been commonly represented as a debate between three schools of thought, all originating around the turn of the Century. A good selection of papers by the original protagonists may be found in Benaceraff & Putnam's Philosophy of Mathematics. Each of these schools adopted a different approach to resolving the crisis in the foundations of mathematics which arose from emergence of contradictions in set theory and in Frege's Grundgesetze der Arithmetik.

During the second half of the 20th Century various writers have tried to broaden the scope of the Philosophy of Mathematics and/or to criticise the foundational schools en-masse. A selection of papers from these writers has been gathered together, giving the appearance of an anti-foundational movement, by Thomas Tymoczko in [Tymoczko98].

Most closely associated with Frege books and Russell books, logicism is the doctrine that Mathematics is (or is "reducible to") Logic.
Most closely associated with Brouwer, this school is characterised by a wholesale rejection of the methods characteristic of classical set theory, particularly of the treatment of infinity pioneered by Cantor.
This school is primarily associated with David Hilbert books. It is often characterised as the view that logic and mathematics are mere formal games and have a legitimacy independent of the semantic content of these formalisms, provided only that we can be reassured of the consistency of the formal systems.

Hilbert's "programme" for resolving the paradoxes was to seek a "finitary" consistency proof for the whole of classical mathematics. This is normally held to have been shown impossible by Gödel's second incompleteness theorem, however an element of uncertainty about what is meant by "finitary" makes this not absolutely conclusive.

Current Foundational Debates

It may have been that during the center part of this century there was established for a while a curious stand-off. At this time both Logicism and Formalism were held to have failed, Gödel's incompleteness results having contributed in both cases, but intuitionism remained intact. So philosophically intuitionism reigned supreme.

Mathematicians on the other hand, insofar as they considered these matters, probably remained fomalist or logicist in inclination.

In the latter half of the century pressure on these classical paradigms have grown from several sources. To the dissent of philosophers has been added dissent from mathematicians who have found fault with classical set theory as a foundation, or who have doubted the need for any foundation at all. Increasingly theoretical computer science has entered into this arena, and has tended to be a radical influence.

Opposed to classical set theory as a foundation for mathematics we must now consider:

These are by no means completely separate, and the interconnections make the pressure greater.

Category Theory and The Foundations of Mathematics

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