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.

For example, it seems to me to be entirely true that Sherlock Holmes resided at 221B Baker street, while at the same time it is quite obvious that Sherlock Holmes never actually existed. Similarly, it seems entirely true that having one horn is what characterizes the unicorn, while it is also clear that there never were any unicorns.

In ordinary speech, existence is not a requirement for assertions of truth. And, it seems to me, mathematical talk is similar.

Of course mathematicians do say that the number 3 exists. However, it has always seemed to me that “exist” is used in a special sense within mathematics. To say that *x* exists, in the mathematical sense, is just to say that assuming the existence of *x* does not introduce any contradiction.

For myself, I’m a fictionalist. Many mathematicians are platonists. As to how we do mathematics, I do it much the same way that a platonist would. The issues that platonism are supposed to address do not seem particularly relevant to mathematics and how it is done.

Yes, there are mathematical aspects to platonism. A mathematical platonist is likely to say that there is a fact of the matter as to whether the continuum hypothesis is true, though we have not yet determined what is that fact. As a fictionalist, I am skeptical that there is such a “fact of the matter.”

Although platonism has mathematical implications, the type of argument often used within philosophy of mathematics, such as the one discussed above, does not seem at all compelling or even useful.