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It is an interesting piece but to claim that no heavy machinery is used is a bit disingenuous at best. You have defined some purely algebraic operation “differe

It is an interesting piece but to claim that no heavy machinery is used is a bit disingenuous at best. You have defined some purely algebraic operation “differentiation”. This operation involves a choice of infinitesimal. Is it trivial to show that the definition is independent of infinitesimal? especially if we are deriving at a hyperreal point? I doubt it and likely you would need to do more complicated set theoretic limits rather analytic limits. How do you calculate the integral of this function? Or even define it? Or rather functions, since it’s an infinite family of logistic functions? To even properly define this space you need to go quite heavily into set theory and i doubt many would find it simpler, even than working with distributions

Tainnor 2y ago
Even just defining the hyperreals and showing why statements about them are also valid for the reals needs to go through either ultrafilters (which are some rather abstract objects) or model theory. Of course you can just handwave all of that away but then I guess you can also do that with standard analysis.
- yuppiemephisto 2y ago
  There are theories like SPOT and Internal Set Theory that don’t require filters. Plus the ancient mathematicians did very well with just their intuition. And more to the point, I cared much more about building (hyper)number sense than some New Math “let’s learn ultrafilters before we’ve even done arithmetic”.
  - Tainnor 2y ago
    > Plus the ancient mathematicians did very well with just their intuition. They did. But they also got things wrong, such as thinking that pointwise limits are enough to carry over continuity (see here for this and other examples: https://mathoverflow.net/a/35558 https://mathoverflow.net/a/35558). Anyway, mathematics has changed as a discipline, we now have strong axiomatic foundations and they mean that we can, in principle, always verify whether a proof is correct.
    - yuppiemephisto 2y ago
      Certainly I'm glad we have better tools now, and know the rules. Sure beats the old days. As for verification, well I'm big into Lean 4 and it plays well with NSA since transferring theory and proofs between finite and infinite saves a lot of labor on the computer.
      - Tainnor 2y ago
        Oh I wasn't trying to say that NSA is in any way non-rigorous, only that if you make it rigorous it does require some machinery that is itself not terribly straightforward. As for whether that's worth it or not, I have no strong opinion.
bubblyworld 2y ago
The machinery of mathematics goes arbitrarily deep. I think the interesting thing here is that with relatively little training you can start to compute with these numbers, which is definitely not the case with analysis on distributions. Or put differently - here you can kinda ignore the deeper formalities and still be productive, whereas with distributions you actually need to sit down and pore over them before you can do anything. That said, I'm curious why infinitesmals never took off in physics. This kind of quick, shut-up-and-calculate approach seems right up their alley.
- shwouchk 2y ago
  > I think the interesting thing here is that with relatively little training you can start to compute with these numbers, which is definitely not the case with analysis on distributions. I don’t know, this feels like a math “hold my beer” moment. Math is infinitely deep and interconnected, but you have to start somewhere, on solid ground. I was not being facetious above - the issues that i mentioned above are actual problems when you make calculations. But let’s ignore those issues for a second. So you found the “derivative” of a single, arbitrary chosen representative of an infinite family of functions. What if you chose (tanh(Nx)+1)/2? What if you chose Logistic(N^2 x) instead of Logistic(N x)? You’d get different derivatives. In fact any function (up to additive constant) whose integral of the neighborhood of 0 is 1 would work there. What use are the values you are calculating if they reflect your choice and not anything inherent to the problem? As for distributions, i picked up and read a small 100 page penguin “leaflet” from the library during my undergrad that went through the subject rigorously (and with plenty of examples). It’s not that different from working rigorously with probability or real analysis. And at the end, in applications we indeed are usually interested in integrals, not derivatives which we have not even defined. At the end of the day, you have a [X=weak L^infinity(R)] function (heavyside). You look at the dual space and since we established don't really need the deep theory, believe me when i tell you that the correct space is the space of test function on R (X’=infinitely smooth, compact support, bounded integral). Each of those conditions is simple for our simple example of R. The inner product is via integral. Formally speaking elements of X are equivalence classes of sequences of functions and are not really defined pointwise, but neither was the NSA example. There we had to choose an arbitrary representative hyperreal function and here we may identify pointwise defined functions with the classes of the constant sequences of those functions. using integration by parts it is simple to show that <F,G’> = <F’,G> if F is continuously differentiable on G’s support. Let us formally define in this way the weak derivative for functions that are not traditionally differentiable, if such an element exists an is unique that satisfies all the integral relations. However note that differentiation is an linear isomorphism on the space of test functions and so weak derivative indeed exists and is unique. Furthermore We can also define elements of X poinwise by identifying F(x) with the limit <Txn,F> as n grows if it exists and is independent of the sequence Txn where Txn is a sequence of functions with support tending to {x} and constant integral 1. It is a simple exercise to show that for “normal” functions this holds, and by above we can poinwise define derivatives this way as well. What about our H(x)? it is an exercise to check that by pointwise we get what we should outside of 0. What about the derivative at 0? Well, do the exercise above with <T0n’,H> and we see that it is penrose undefined. Decidedly not even necessarily infinite, just undefined. However, integration by parts shows that <T,DH>=T(0) ie dirac delta at 0. Aside from all the theory that i kinda gave handwavingly much like OP in the post, the mechanics are simple integration by parts to get the only stuff that’s “real” here, which are the integrals. in NSA we haven’t even defined those. How will knowing what infinity i will get at 0 given an arbitrarily chosen representative for H help me? Do your results depend on ZFC? stronger axioms? At what level of infinity do we stop? You can brush aside the formalities but then what better is this approach than physicists?

Tainnor 2y ago

Even just defining the hyperreals and showing why statements about them are also valid for the reals needs to go through either ultrafilters (which are some rather abstract objects) or model theory. Of course you can just handwave all of that away but then I guess you can also do that with standard analysis.

yuppiemephisto 2y ago
There are theories like SPOT and Internal Set Theory that don’t require filters. Plus the ancient mathematicians did very well with just their intuition. And more to the point, I cared much more about building (hyper)number sense than some New Math “let’s learn ultrafilters before we’ve even done arithmetic”.
- Tainnor 2y ago
  > Plus the ancient mathematicians did very well with just their intuition. They did. But they also got things wrong, such as thinking that pointwise limits are enough to carry over continuity (see here for this and other examples: https://mathoverflow.net/a/35558 https://mathoverflow.net/a/35558). Anyway, mathematics has changed as a discipline, we now have strong axiomatic foundations and they mean that we can, in principle, always verify whether a proof is correct.
  - yuppiemephisto 2y ago
    Certainly I'm glad we have better tools now, and know the rules. Sure beats the old days. As for verification, well I'm big into Lean 4 and it plays well with NSA since transferring theory and proofs between finite and infinite saves a lot of labor on the computer.
    - Tainnor 2y ago
      Oh I wasn't trying to say that NSA is in any way non-rigorous, only that if you make it rigorous it does require some machinery that is itself not terribly straightforward. As for whether that's worth it or not, I have no strong opinion.

bubblyworld 2y ago

The machinery of mathematics goes arbitrarily deep. I think the interesting thing here is that with relatively little training you can start to compute with these numbers, which is definitely not the case with analysis on distributions. Or put differently - here you can kinda ignore the deeper formalities and still be productive, whereas with distributions you actually need to sit down and pore over them before you can do anything. That said, I'm curious why infinitesmals never took off in physics. This kind of quick, shut-up-and-calculate approach seems right up their alley.

shwouchk 2y ago
> I think the interesting thing here is that with relatively little training you can start to compute with these numbers, which is definitely not the case with analysis on distributions. I don’t know, this feels like a math “hold my beer” moment. Math is infinitely deep and interconnected, but you have to start somewhere, on solid ground. I was not being facetious above - the issues that i mentioned above are actual problems when you make calculations. But let’s ignore those issues for a second. So you found the “derivative” of a single, arbitrary chosen representative of an infinite family of functions. What if you chose (tanh(Nx)+1)/2? What if you chose Logistic(N^2 x) instead of Logistic(N x)? You’d get different derivatives. In fact any function (up to additive constant) whose integral of the neighborhood of 0 is 1 would work there. What use are the values you are calculating if they reflect your choice and not anything inherent to the problem? As for distributions, i picked up and read a small 100 page penguin “leaflet” from the library during my undergrad that went through the subject rigorously (and with plenty of examples). It’s not that different from working rigorously with probability or real analysis. And at the end, in applications we indeed are usually interested in integrals, not derivatives which we have not even defined. At the end of the day, you have a [X=weak L^infinity(R)] function (heavyside). You look at the dual space and since we established don't really need the deep theory, believe me when i tell you that the correct space is the space of test function on R (X’=infinitely smooth, compact support, bounded integral). Each of those conditions is simple for our simple example of R. The inner product is via integral. Formally speaking elements of X are equivalence classes of sequences of functions and are not really defined pointwise, but neither was the NSA example. There we had to choose an arbitrary representative hyperreal function and here we may identify pointwise defined functions with the classes of the constant sequences of those functions. using integration by parts it is simple to show that <F,G’> = <F’,G> if F is continuously differentiable on G’s support. Let us formally define in this way the weak derivative for functions that are not traditionally differentiable, if such an element exists an is unique that satisfies all the integral relations. However note that differentiation is an linear isomorphism on the space of test functions and so weak derivative indeed exists and is unique. Furthermore We can also define elements of X poinwise by identifying F(x) with the limit <Txn,F> as n grows if it exists and is independent of the sequence Txn where Txn is a sequence of functions with support tending to {x} and constant integral 1. It is a simple exercise to show that for “normal” functions this holds, and by above we can poinwise define derivatives this way as well. What about our H(x)? it is an exercise to check that by pointwise we get what we should outside of 0. What about the derivative at 0? Well, do the exercise above with <T0n’,H> and we see that it is penrose undefined. Decidedly not even necessarily infinite, just undefined. However, integration by parts shows that <T,DH>=T(0) ie dirac delta at 0. Aside from all the theory that i kinda gave handwavingly much like OP in the post, the mechanics are simple integration by parts to get the only stuff that’s “real” here, which are the integrals. in NSA we haven’t even defined those. How will knowing what infinity i will get at 0 given an arbitrarily chosen representative for H help me? Do your results depend on ZFC? stronger axioms? At what level of infinity do we stop? You can brush aside the formalities but then what better is this approach than physicists?