I recently posted a link to an explanation of the philosophy of mathematics. While I thought that Balaguer’s explanation was very good, I also remarked that I don’t find the philosophy of mathematics to be useful. In this post, I’ll say why I don’t find it useful.

Toward the end of his explanation, Balaguer presents the following argument for platonism:

- Semantic platonism is true–i.e., ordinary mathematical sentences like ‘2 + 2 = 4’ and ‘3 is prime’ are straightforward claims about abstract objects (or at any rate, they purport to be about abstract objects). Therefore,
- Mathematical sentences like ‘2 + 2 = 4’ and ‘3 is prime’ could be true only if platonism were true–i.e., only if abstract objects existed. But
- Mathematical sentences like ‘2 + 2 = 4’ and ‘3 is prime’ are true. Therefore,
- Platonism is true.

Balaguer, who says he is a fictionalist and not a platonist, questions step 3 in that argument. However, it seems to me that step 2 is already mistaken. People simply do not use “true” in the way that step 2 supposes.

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