3 ms·

Look you're coming at this from a mathematics perspective and worrying about every single detail. There's value in this, obviously, but it's unnecessary in prac

Look you're coming at this from a mathematics perspective and worrying about every single detail. There's value in this, obviously, but it's unnecessary in practical use, in the same way that I don't have to explain Dedekind cuts to kids before they start to work with real numbers. Nor do I explain measure theory to beginners before they start integrating stuff. > What if you picked N^2? or 2^N? or some upper limit set whose existence is stronger than choice? I'm not sure what you mean about picking N to be an upper limit set. N is a hyperreal here, not an ordinal. There aren't really set theoretic difficulties, you can easily construct a model of the hyperreals in ZFC. It doesn't matter what representative you pick for your Heaviside function - so long as it differs pointwise from the standard Heaviside function by infinitesmals you will get a delta function by differentiating it in NSA. And continuing to differentiate it will give you the higher multiple moments. This is what I meant in my previous response. It's useful to have the choice because depending on what you want to model you can have non-standard functions that "go to infinity" twice as fast as other functions, for instance. Taking the equivalence class destroys that information, which is sometimes useful, and sometimes not. If all you care about is a small computation you can just pick a representative and move on, I don't think it's a big deal. Anyway, I'm going to leave this discussion now - we're not really bickering over anything important to my mind. Use the tool you like! Personally I'm having fun playing with NSA right now. Thanks for your time.

shwouchk 2y ago
My point is that as a practical tool to simplify calculations, i see no value in NSA. The specific infinite results you get depend on the representative you choose and if you want to calculate integrals which you have yet to define, you need to work harder. On the other hand, if you care about modeling specific infinities for applications you are very likely also someone who needs and wants the actual theory. As someone with mathematical background indeed i repeat that i think this theory is interesting and worth studying. Just not that it simplifies anything.
- deleted 2y ago
  [deleted]
- bubblyworld 2y ago
  Yeah, you make some good points. I think at the end of the day both can work - anything you can do in the standard analysis setting can in principle be done in NSA too by the transfer theorem, it's just less travelled ground and maybe the requisite techniques aren't as well known. Integration is very similar, by the way - you just do Riemann sums with a hyperfinite number of partitions of infinitesmal width. It sounds weird, and it is, but it makes it very easy to understand why integrating f(x) against a delta function gives you f(0), for instance, without having to justify it with limits or a bunch of deeper theory.

shwouchk 2y ago

My point is that as a practical tool to simplify calculations, i see no value in NSA. The specific infinite results you get depend on the representative you choose and if you want to calculate integrals which you have yet to define, you need to work harder. On the other hand, if you care about modeling specific infinities for applications you are very likely also someone who needs and wants the actual theory. As someone with mathematical background indeed i repeat that i think this theory is interesting and worth studying. Just not that it simplifies anything.

deleted 2y ago
[deleted]
bubblyworld 2y ago
Yeah, you make some good points. I think at the end of the day both can work - anything you can do in the standard analysis setting can in principle be done in NSA too by the transfer theorem, it's just less travelled ground and maybe the requisite techniques aren't as well known. Integration is very similar, by the way - you just do Riemann sums with a hyperfinite number of partitions of infinitesmal width. It sounds weird, and it is, but it makes it very easy to understand why integrating f(x) against a delta function gives you f(0), for instance, without having to justify it with limits or a bunch of deeper theory.