The Three Philosophies

“Platonists, such as Kurt Gödel, hold that numbers are abstract,necessarily existing objects, independent of the human mind” (Anglin(1994) p. 218)
“Formalists, such as David Hilbert (1862-1943), hold thatmathematics is no more or less than mathematical language. It is simplya series of games…” (Anglin (1994) p. 218)
“Intuitionists, such as L. E. J. Brouwer (1882-1966), hold thatmathematics is a creation of the human mind. Numbers, like fairy talecharacters, are merely mental entities, which would not exist if therewere never any human minds to think about them.” (Anglin (1994) p. 218)

W. S. Anglin [1994] (1996). Mathematics. A Concise History and Philosophy, Corrected 2nd printing edition, New York: Springer Verlag. ISBN 0387942807.  Chapter 39 Foundations contains concise descriptions, for the 20th century, of Platonism (with respect to Gödel), Formalism (with respect to Hilbert), and Intuitionism (with respect to Brouwer).