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by threepipeproblm
3357 days ago
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Wonderful to know that Hilbert was a finitist, and that https://en.wikipedia.org/wiki/Finitism is an official camp. Without knowing too much about about the subject, I've vaguely wondered about this idea for a long time, now, but I figured it likely an un-respectable position. I think it's too bad that beginners are often shielded from controversies in foundations. Within the past few years I ran across an alternate approach to calculus, which if I recall correctly, achieves the same basic results, but without the same notion of infinitely small slices and so on... now I can't find it to link. |
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> I think it's too bad that beginners are often shielded from controversies in foundations.
Mathematicians generally shouldn't worry about foundations, as it is the intention of foundations to stay hidden -- except in cases where the foundation requires a rewrite of much of math, as in the case of Brouwer's intuitionism, but constructive math is pretty advanced anyway. Foundations are usually the concern of logicians, but from what little I've seen in online discussions, it seems like many logicians aren't interested in philosophy, either, and that's a real shame.
> I ran across an alternate approach to calculus, which if I recall correctly, achieves the same basic results, but without the same notion of infinitely small slices and so on
There's constructive analysis[1], which recreates analysis within the framework of constructive math (by finite means etc.), and there's also non-standard calculus[2] (which I know nothing about) that makes treat infinitesimals as actual numbers, which may be the opposite of what you meant, but maybe not -- I saw something about there being constructive versions of non-standard analysis, too.
[1]: https://en.wikipedia.org/wiki/Constructive_analysis
[2]: https://en.wikipedia.org/wiki/Non-standard_calculus