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I think most mathematicians are more likely to be Platonists than formalists. Formalism is also a real position in philosophy of math, but a minority one for I
by samth 10y ago
I think most mathematicians are more likely to be Platonists than formalists. Formalism is also a real position in philosophy of math, but a minority one for I think two reasons. One is that the math we've developed seems significant enough and effective enough to deserve more explanation than just "it's one of many possible formal systems". Second, if you take a straightforward formalist approach, then there aren't really any interesting questions in philosophy of math, so people who think that are less likely to become philosophers of math.
- cokernel 10y agoIt's possible that a mathematician with philosophical commitments is more likely to be a Platonist than a formalist, but I suspect that most mathematicians are neither ("Philosophy is for the philosophers"). The so-called "naive" view that most mathematicians may be described as having ("I'm discovering something real when I prove a theorem", "Mathematics is objective", and so on) certainly sounds similar to the language used by Platonists, but I have yet to run into a mathematician who is comfortable with the logical conclusions of a strict Platonism. In particular, there are epistemological problems with Platonism (how can we know anything about non-physical entities?) that cause problems in the justification of mathematics and would be nice to be able to ignore.
- samth 10y agoI think we mostly agree on the empirical issues -- mathematicians usually don't care about the philosophical issues, and espouse naive Platonism which they haven't thought through. However, I don't think it's right to say that they're formalist just because they wouldn't agree, upon reflection, with Platonism's conclusions. I (like a lot of computer scientists) am a formalist, but it's not fair to say that other people secretly agree with me because I think Platonism has major flaws.