It's a bit frustrating that smooth infinitesimal analysis isn't mentioned even once, even as the author lists other systems that add infinitesimals to real numbers. It's a much better system to develop calculus on. It features infinitesimals as normal everyday mathematical objects, rather than the hack that hyperreal numbers are.
Of course, I know that something like SIA would never be adopted. The main problem is that it is based on intuitionistic logic rather than classical logic. As such, it requires new intuitions that may not be appropriate to develop while studying calculus (it would work if it were a middle school topic). This is unfortunate, because those intuitions would make calculus much simpler and remove a large number of edge cases (a great deal of quirks with calculus are actually quirks in classical logic in disguise)
However, it was not even cited! And it was not cited most likely because the author never heard about it (even though he hedged with "and other systems"), even though the author spent a great deal to explain how teaching calculus with infinitesimals (that's what differentials are) is much simpler and easier to understand than epsilon-gama limits.
A book on SIA, that not only develop multivariate calculus but also builds classical mechanics using the same infinitesimal arguments of Newton and Leibniz, but within a rigorous mathematical setting (well that's just a free sample containing the table of contents, but the book itself is available elsewhere)
https://api.pageplace.de/preview/DT0400.9780511368400_A23677...
The smooth infinitesimal analysis assumption that all functions are infinitely differentiable indeed removes a lot of edge cases, but only in the way that the C assumption that all memory accesses are within bounds removes a lot of edge cases. Just as a C programmer may have difficulty telling a valid C program from a meaningless sequence of symbols, so does SIA make it difficult to tell a valid assumption from a paradoxical statement that would allow defining a non-differentiable function.
If you have a correct calculation in SIA involving infinitesimals squaring to zero, it can easily be translated into the classical setting using functions that have their magnitude bounded above by a quadratic in a neighborhood of zero, but with the advantage that if you try it on a non-differentiable function, you'll merely fail to prove the quadratic upper bound instead of getting nonsensical results.
Such a "quadratic bound" approach (in brief f'(x) exists if there exist a constant C and a neighborhood of zero where for all h in the neighborhood |f(x + h) - f(x) - f'(x)h| ≤ Ch²) could actually be adopted over the typical limit of (f(x + h) - f(x))/h without abandoning classical logic, but considering the difficulty of proving the bound for many functions of interest, it might not make much of a difference over just assuming differentiability either way.
The difference is that in SIA you can't "try it on a non-differentiable function" because in SIA, such a thing doesn't exist.
It's not like the C case where out of bounds access is assumed to not exist, but they can happen anyway and if they do, results are catastrophic. In SIA an analogous catastrophe doesn't happen - it's as if it had a type checker that will reject nonsensical results.
Such a "quadratic bound" approach (in brief f'(x) exists if there exist a constant C and a neighborhood of zero where for all h in the neighborhood |f(x + h) - f(x) - f'(x)h| ≤ Ch²) could actually be adopted over the typical limit of (f(x + h) - f(x))/h without abandoning classical logic, but considering the difficulty of proving the bound for many functions of interest, it might not make much of a difference over just assuming differentiability either way.
Now this I don't understand. I think that what changes in SIA isn't just the definition of a derivative, but being able to use infinitesimals to simplify the calculations
I completely agree. I have Bell's book on the infinitesimal approach and it is infinitely (hah) more intuitive (hah again) than epsilon-delta limit foundations. It trades a heady second order logical statement for simple algebra.
There's also really no excuse not to use it anymore since category theory has provided some of the missing rigor. I think there's a reason that Leibniz et al essentially started with this basis.
It trades a heady second order logical statement for simple algebra.
Also, other comment here says they wish calculus were more visual. Newton-style proofs were actually geometric, and the proofs in the book are like that too. You build triangles etc. with infinitesimal sides, and the math works
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.
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?
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.
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.
Comments
It's a bit frustrating that smooth infinitesimal analysis isn't mentioned even once, even as the author lists other systems that add infinitesimals to real numbers. It's a much better system to develop calculus on. It features infinitesimals as normal everyday mathematical objects, rather than the hack that hyperreal numbers are.
Of course, I know that something like SIA would never be adopted. The main problem is that it is based on intuitionistic logic rather than classical logic. As such, it requires new intuitions that may not be appropriate to develop while studying calculus (it would work if it were a middle school topic). This is unfortunate, because those intuitions would make calculus much simpler and remove a large number of edge cases (a great deal of quirks with calculus are actually quirks in classical logic in disguise)
However, it was not even cited! And it was not cited most likely because the author never heard about it (even though he hedged with "and other systems"), even though the author spent a great deal to explain how teaching calculus with infinitesimals (that's what differentials are) is much simpler and easier to understand than epsilon-gama limits.
Anyway let me drop some links
An one-page motivation (explains what it is all about) https://publish.uwo.ca/~jbell/invitation%20to%20SIA.pdf
A 14 page exposition https://arxiv.org/abs/0805.3307
Wikipedia article https://en.wikipedia.org/wiki/Smooth_infinitesimal_analysis
A book on SIA, that not only develop multivariate calculus but also builds classical mechanics using the same infinitesimal arguments of Newton and Leibniz, but within a rigorous mathematical setting (well that's just a free sample containing the table of contents, but the book itself is available elsewhere) https://api.pageplace.de/preview/DT0400.9780511368400_A23677...
The smooth infinitesimal analysis assumption that all functions are infinitely differentiable indeed removes a lot of edge cases, but only in the way that the C assumption that all memory accesses are within bounds removes a lot of edge cases. Just as a C programmer may have difficulty telling a valid C program from a meaningless sequence of symbols, so does SIA make it difficult to tell a valid assumption from a paradoxical statement that would allow defining a non-differentiable function.
If you have a correct calculation in SIA involving infinitesimals squaring to zero, it can easily be translated into the classical setting using functions that have their magnitude bounded above by a quadratic in a neighborhood of zero, but with the advantage that if you try it on a non-differentiable function, you'll merely fail to prove the quadratic upper bound instead of getting nonsensical results.
Such a "quadratic bound" approach (in brief f'(x) exists if there exist a constant C and a neighborhood of zero where for all h in the neighborhood |f(x + h) - f(x) - f'(x)h| ≤ Ch²) could actually be adopted over the typical limit of (f(x + h) - f(x))/h without abandoning classical logic, but considering the difficulty of proving the bound for many functions of interest, it might not make much of a difference over just assuming differentiability either way.
The difference is that in SIA you can't "try it on a non-differentiable function" because in SIA, such a thing doesn't exist.
It's not like the C case where out of bounds access is assumed to not exist, but they can happen anyway and if they do, results are catastrophic. In SIA an analogous catastrophe doesn't happen - it's as if it had a type checker that will reject nonsensical results.
Now this I don't understand. I think that what changes in SIA isn't just the definition of a derivative, but being able to use infinitesimals to simplify the calculations
I completely agree. I have Bell's book on the infinitesimal approach and it is infinitely (hah) more intuitive (hah again) than epsilon-delta limit foundations. It trades a heady second order logical statement for simple algebra.
There's also really no excuse not to use it anymore since category theory has provided some of the missing rigor. I think there's a reason that Leibniz et al essentially started with this basis.
Also, other comment here says they wish calculus were more visual. Newton-style proofs were actually geometric, and the proofs in the book are like that too. You build triangles etc. with infinitesimal sides, and the math works
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.
Could you expand on that? In which way nonstandard analysis is more rigorous or elementary?
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.
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.