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I may not have used the term “Platonism” correctly. I’m not a great admirer of abstract philosophy, including as it applies to mathematics. My simplistic view is that if one believes mathematical truths are “discovered”, that’s Platonism, while if one believes mathematical truths are “created”, that’s the main alternative… and I’m not even clear what it should be called. I hope that’s not hopelessly naive, but I’m not all that interested in the argument to begin with. As you point out, mathematics is a pleasurable pursuit, and that is justification enough.

It does seem to me that the whole Langlands program, which is looking for very general correspondences between diverse mathematical structures, is amimed at “discovering” something, because the whole problem is trying to pin down rigorously exactly what it “is” that’s being searched for. Mathematicians keep trying to posit different structures in the hope that everything will then fall into place. That’s a “creative” phase, but in mathematics and other endeavors, most creative ideas don’t actually work, or don’t work well enough. Nothing much is accomplished unless the idea makes real progress on a problem, sort of how Edison kept trying things until he had an incandescent lamp that worked well enough. And that was the actual “discovery”. Given the technology of his time, Edison discovered a good solution to the problem; he didn’t create it. The solution was there before he thought of it and tried it. Better solutions have since been discovered, by a long process of trial and error and the development of quite different approaches.

To paraphrase Bill Clinton, it all depends on what one means by “is” (or “exists”). Debating that issue has kept philosophers in business for thousands of years. Mathematics has a way of settling its important questions sooner or later. Philosophy doesn’t seem to. Physics has a similar attitude towards philosophy. Do quarks exist? Who cares? The QFT works, very well, up to a point anyhow (and despite lack of a rigorous mathematical foundation). Someday a theoretical physicist will try something new that will actually be an improvement.

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