I will give a talk under that title at Magic Postgraduate Student Conference 2009.
Are mathematical objects invented or discovered? Questions of that type are inevitably asked and answered by mathematicians in the course of their work. In most cases, the answers are not revealed publicly but retained for personal use. But even as questions like “what is a mathematical object?” are suppressed, their derivatives, like “What do we mean when we say that two objects are identical or when we say that two objects are equivalent?” are unavoidable in any formal mathematical discourse.
At an informal level, the situation is even more puzzling. I quote a Fields Medal winner, Timothy Gowers:
“The following informal concepts of mathematical practice cry out to be explicated:
beautiful, natural, deep, trivial, “right”, difficult, genuinely, explanatory …”
Without doubt, you use these words when you talk to your colleagues about mathematics: can you explain their meaning?
Unfortunately, philosophy of mathematics as an academic discipline fails to fulfil its basic function: it does not help mathematicians to develop a conceptual framework for their normal, day-to-day, professional discourse. In my talk, I will argue that mathematicians should take care of themselves and try to clarify “informal” aspects of their work.