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> Unpopular treatments of Calculus that only deal with computable numbers exist and produce the same results for physics. I'd really appreciate if you named th

by creata 6y ago

> Unpopular treatments of Calculus that only deal with computable numbers exist and produce the same results for physics. I'd really appreciate if you named them. I've seen synthetic differential geometry, which isn't nearly well-developed enough for use in all of physics, and I've seen nonstandard analysis, which obviously doesn't do away with the uncomputable reals at all. > there is no way in which the use of mathematics by physics serves as justification for the reality I am not saying the standard reals are real. I am saying that they are more convenient than any existing alternative, and fulfil the current needs of all of physics. I am also saying that the parent's claim that "uncomputable numbers … in the guise of string theory obtained a death grip on fundamental physics" is absurd, since string theory and uncomputable reals are pretty much unrelated.

btilly 6y ago
I was thinking intuitionistic math, such as smooth infinitesmal analysis. And yes, they are less developed than classical analysis, and harder to work with. But should not produce different results.
- Jweb_Guru 6y ago
  As someone who does lots of constructive math, intuitionistic math can't reproduce a large number of results on the reals and this is a kind of well-known thing. Even basic notions of continuity can't be reproduced in the same way and as a result there are multiple competing formalisms. It is possible that there are alternative ways to reproduce results of interest to physics but that's far from obvious.
- creata 6y ago
  The world of SIA and SDG is a very, very different world to that of classical analysis. If you think that they "should not produce different results", then (a) you're either wrong or incredibly optimistic regarding our ability to carefully rephrase the statements of classical analysis in a way that works constructively. (b) what would be the point of a harder-to-use system that offers exactly the same results, anyway? The whole reason I find SDG interesting is that it will eventually be simpler to use, and because it provides an entirely fresh of the number line.

by creata 6y ago

btilly 6y ago

I was thinking intuitionistic math, such as smooth infinitesmal analysis. And yes, they are less developed than classical analysis, and harder to work with. But should not produce different results.

Jweb_Guru 6y ago
As someone who does lots of constructive math, intuitionistic math can't reproduce a large number of results on the reals and this is a kind of well-known thing. Even basic notions of continuity can't be reproduced in the same way and as a result there are multiple competing formalisms. It is possible that there are alternative ways to reproduce results of interest to physics but that's far from obvious.
creata 6y ago
The world of SIA and SDG is a very, very different world to that of classical analysis. If you think that they "should not produce different results", then (a) you're either wrong or incredibly optimistic regarding our ability to carefully rephrase the statements of classical analysis in a way that works constructively. (b) what would be the point of a harder-to-use system that offers exactly the same results, anyway? The whole reason I find SDG interesting is that it will eventually be simpler to use, and because it provides an entirely fresh of the number line.