Thus far we have been introduced to two of the classical schools in the foundations of mathematics that developed at the turn of the century: Intuitionism and Hilbert's axiomatics (also known as Formalism). The third school to develop out of this era comes from Russell and Whitehead in which they tried to reduce mathematics to logic (refer to logicism). This school is probably the most formal of the three in that it tries to reduce all mathematical ideas to symbols and inference rules. Subsequently the Hilbert and Russell schools have converged and left a strong impression on modern mathematical logic, while Brouwer's school has been more influential in the fields of topology and category theory.

One important fact to note here is that these schools view mathematics from a Platonic point of view, considering mathematics as something outside of our personal experience, existing on its own. During the latter half of the 20th century however, this view has been challenged by philosopher/ mathematicians such as Imre Lakatos, Reuben Hirsch and Raymond Wilder. In their respective books "What is Mathematics, Really?" [1997] and "Mathematics as a Cultural System" [1981] Hirsch and Wilder analyse the idea of mathematics as being a cultural phenomena.

Some of the more interesting, and controversial, insights (i.e. if you agree with them) come from Lakatos's "Proofs and Refutations" [1963] wherein he expounds the idea of maths developing as a sequence of conjectures (theorems) followed by reduction of these conjectures to previous ones (proofs), and subsequently with criticisms (counterexamples) after which the process recurs. This idea is directly opposed to the more classical view of mathematics being a systematic accumulation of "eternal" truths.

*Aside:* Paul Bernays expresses the opinion that intuitionism is semi-platonic. Intuitionism assumes that the natural numbers exist as an idea outside of our direct perceptions, however it then tries to constructively build up all mathematics from here. Mathematics still seems to be viewed as an idea in Plato's sense, but the construction mechanism is less platonic.

(For the record, I personally think the main beauty of mathematics lies in it having an existence of its own, something akin to abstract poetry...)