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I think "Smooth Infinitesimal Analysis" is actually a bad idea because I think that the combination of Robinson's Nonstandard Analysis with little/big O notatio

by ogogmad 1mo ago

I think "Smooth Infinitesimal Analysis" is actually a bad idea because I think that the combination of Robinson's Nonstandard Analysis with little/big O notation gives you nilpotent infinitesimals in a more rigorous and elementary way. In fact, in light of this, I'm starting to wish logicians would stop promoting SIA/SDG.

nextaccountic 1mo ago
> I think "Smooth Infinitesimal Analysis" is actually a bad idea because I think that the combination of Robinson's Nonstandard Analysis with little/big O notation gives you nilpotent infinitesimals in a more rigorous and elementary way. Could you expand on that? In which way nonstandard analysis is more rigorous or elementary?
- ogogmad 1mo ago
  Nonstandard analysis reduces to just the Compactness Theorem of First-Order Logic. One happy consequence of this is that it's compatible with classical logic, unlike SIA/SDG. The seemingly very intuitive arguments in Bell's book "a primer of infinitesimal analysis" are only heuristics. It's also disturbing how the nilsquare "infinitesimals" in SIA/SDG satisfy δ^2 = ε^2 = 0 but NOT necessarily εδ = 0! By contrast, you can more easily obtain nilpotent infinitesimals by using NSA and combining it with little-o notation (on hyperreal numbers, and NOT on functions and their asymptotic growth rates). Then the geometric arguments in Bell's "a primer of infinitesimal analysis" can still be carried out, but are more easy to unravel back to core definitions.
  - nextaccountic 1mo ago
    > The seemingly very intuitive arguments in Bell's book "a primer of infinitesimal analysis" are only heuristics. This kinds of kill the whole idea. The book is written as if the intuitive arguments were actually rigorous; what's heuristic about them? I mean the whole point of SIA is that you really can have a triangle where one side is infinitesimal, because infinitesimals are part of the number system, it's not something extra.

by ogogmad 1mo ago

nextaccountic 1mo ago

> I think "Smooth Infinitesimal Analysis" is actually a bad idea because I think that the combination of Robinson's Nonstandard Analysis with little/big O notation gives you nilpotent infinitesimals in a more rigorous and elementary way. Could you expand on that? In which way nonstandard analysis is more rigorous or elementary?

ogogmad 1mo ago
Nonstandard analysis reduces to just the Compactness Theorem of First-Order Logic. One happy consequence of this is that it's compatible with classical logic, unlike SIA/SDG. The seemingly very intuitive arguments in Bell's book "a primer of infinitesimal analysis" are only heuristics. It's also disturbing how the nilsquare "infinitesimals" in SIA/SDG satisfy δ^2 = ε^2 = 0 but NOT necessarily εδ = 0! By contrast, you can more easily obtain nilpotent infinitesimals by using NSA and combining it with little-o notation (on hyperreal numbers, and NOT on functions and their asymptotic growth rates). Then the geometric arguments in Bell's "a primer of infinitesimal analysis" can still be carried out, but are more easy to unravel back to core definitions.
- nextaccountic 1mo ago
  > The seemingly very intuitive arguments in Bell's book "a primer of infinitesimal analysis" are only heuristics. This kinds of kill the whole idea. The book is written as if the intuitive arguments were actually rigorous; what's heuristic about them? I mean the whole point of SIA is that you really can have a triangle where one side is infinitesimal, because infinitesimals are part of the number system, it's not something extra.