The philosophy of mathematics antithetical to Platonism is formalism, allied to positivism. According to formalism, mathematics is really just an elaborate set of rules for manipulating symbols. By applying the rules to certain “axiomatic” strings of symbols, mathematicians go about “proving” certain other strings of symbols to be “theorems.”

The game of mathematics is, for some obscure reason, a useful game. Some strings of symbols seem to reflect certain patterns of the physical world. Not only is “2 + 2 = 4” a theorem, but two apples taken with two more apples make four apples.