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One form of Platonism entails the actual existence of mathematical objects. Godel and others were known to feel intuitively that they were working with and actually perceiving entities different from, but every bit as "real" as, physical objects.

Other mathematicians have described their arrival at unexpected results as feeling like "experimental discovery." (Realism in Mathematics, Penelope Maddy) More to the point, I understand that there are now purely "experimental" branches of mathematics using computers, though I'm not sure what is involved here. This would seem to remove some of the subjectivity associated with talk of mathematical Platonism.

Have computers indeed enabled mathematicians to "experiment" in ways more analogous to the physical sciences? Does this constitute a stronger argument for the "reality" of mathematical objects? If we talk about "experimental mathematics" is this a form of Platonism or is it cognitive naturalism? Or either one? Odd, I had always imagined these two as complete opposites.

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Conerning the reality of mathematical objects:

Go[e]del and others were known to feel intuitively that they were working with and actually perceiving entities different from, but every bit as "real" as, physical objects.

That's quite clear to me: If you are thinking day and night about a certain mathematical problem and its components, you become fixed to these mathematical entities. They displace all other objects in your daily life.

Mathematicians employ computers a bit differently than physicists or astrophysists do. The latter can use computers to create a virtual reality. The virtual reality may obey different fundamental physical laws than out real world. Hence it is not only simulating our world but also creating different new worlds.

Mathematicians implement algorithms to compute examples conforming to the mathematical theory in question, e.g. algebraic geometry. The computer can calculate more complex examples than could be done manually. But qualitatively, it just excutes the same algorithms as the mathematician. Hence the computer does not change our stance to the ontology of mathematical objects.

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  • Thanks, good answer. That "the computer does not change our stance to the ontology of mathematical objects" is directly to the point. But even with zero knowledge in this area, I'm not entirely convinced. I thought there were brute calculations that really could not be done "manually" and could "falsify" some conjecture, hence something closer to "experimental method." In a sense the computer is detecting "something." But I find it pretty complicated, and need to give it more thought and study.
    – Nelson Alexander
    Commented Nov 7, 2015 at 20:40
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    Today all mathematical algorithms implemented in computers are still developed by mathematicians. Hence there is no qualitative difference between executing the algorithm by a computer or by the mathematicain himself. Of course, the computer can calculate examples which are out of reach for the mathematician due to the time the latter would need. And therefore these examples reveal some properties of the mathematical theory in question which could not have been detected by hand - or possible only at a later time.
    – Jo Wehler
    Commented Nov 7, 2015 at 20:58

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