81–88 of 88 posts
Earlier quoted context omitted.
I hope you check out my book, "Calculus from the Ground Up": https://www.amazon.com/Calculus-Ground-Jonathan-Laine-Bartle...
Thanks for the pointer; will definitely take a look at it. Btw - Your name seemed familiar and then i remembered, you are also the author of "Programming from the Ground Up".
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 th…
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?
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 th…
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 ba…
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
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 th…
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 par…
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
Earlier quoted context omitted.
If we're going the hyperreal route, I quite like Goldblatt's GTM Lectures on the Hyperreals. You have to augment it with a paper or two if you want to work with other nonstandard objects, but when I was doing my graduate work it was the resource I kept going back to for clarity.
This looks like the most in-depth resource on the topic that I've seen so far; thanks for adding it! One of the reasons that I'm partial to the hyperreals is because it's such a natural thing, in the context of mathematical history, to extend the number system when that system isn't expressive enough to solve the problems we want to solve. The limit-based approach seems clumsy in comparison.
1. For every natural, you can find a path of that length.
2. Therefore (nonstandard chicanery), for every hypernatural you can find a hyperpath of that hyperlength. Pick one for some infinite hypernatural.
3. Restricting that hyperpath to the original graph yields the infinite normal path you were looking for.
Whole problems melt away entirely as soon as you don't have to worry about clumsy "limit-based" approaches.
Earlier quoted context omitted.
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?
Earlier quoted context omitted.
> 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…
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.
Earlier quoted context omitted.
My point is that the ideas of Sequences/Convergence/Infinity (Heine's definition) provides a better intuition than the epsilon/delta limits of function definition. The former is discrete so you could literally take any interval and demonstrate how an infinite sequence of real numbers within that interval can converge to a "limit". The student can now understand the idea of a "difference" i.e. a finite change that get…
I'm in the opposite camp. Most of the calculus foundation actually depends on continuity. And one thing that has changed recently: mobile phones and zooming. Every child now has an almost instinctive understanding of being able to find the correct zoom level on a map so that some feature can fit completely on the screen. And now the key insight: you can zoom-in on a continuous function indefinitely. So the epsilon-de…
Some thoughts about epsilon and delta (this is a pretty good AMS article on the subtleties involved) - https://blogs.ams.org/matheducation/2019/08/19/some-thoughts...
Teaching limits of sequences before limits of functions in Calculus? - https://matheducators.stackexchange.com/questions/3952/teach...
Advantages of the sequence definition of limits - https://mathoverflow.net/questions/105920/advantages-of-the-...