Interesting article, HERE. Some highlights below:
To the question: ‘Is mathematics about something?’ there are two answers: ‘Yes’ and ‘No’. Both are profoundly unsatisfying.
The ‘No’ answer, whose champions are known as nominalists, says that mathematics is just a language. On this view, it is just a way of talking about other things, or a collection of logical trivialities (as Singer claims), or a formal manipulation of symbols according to rules.
Nominalism might have a certain down-to-earth appeal, but further reflection suggests that it can’t be right. Although manipulation of symbols is useful as a technique, we also have a strong sense that mathematics makes objective discoveries about a terrain that is in some sense ‘out there’.
It seems as if pure mathematics reveals the topography of a region whose truths pre-existed investigation, even language.