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]*.

- Logicism
- Most closely associated with Frege and Russell , logicism is the doctrine that Mathematics is (or is "reducible to") Logic.
- Intuitionism
- 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.
- Formalism
- This school is primarily associated with David Hilbert .
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.

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:

- Category theoretic foundations
- Revitalised Constructive approaches