There was a time, early in the twentieth century, in which mathematicians were passionately interested in the philosophy of mathematics. People were deeply concerned about what mathematics is, what sort of existence mathematical objects have, and their opinions on these questions actually influenced the mathematics they did.
There were three major points of view in the debate about the nature of mathematics. Theformalists argued (roughly: the short summaries that follow are really caricatures) that mathematics was really simply the formal manipulation of symbols based on arbitrarily-chosen axioms. The Platonists saw mathematics as almost an experimental science, studying objects that really exist (in some sense), though they clearly don't exist in a physical or material sense.
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