alt Hacker News> They all differ pointwise by an infinitesmal! This flexibility is a feature, not a bug.
No, they do not differ by an infinitesimal. You picked an arbitrary infinite N and found the derivative to be N/4. What if you picked N^2? or 2^N? or some upper limit set whose existence is stronger than choice? You get a different derivative every time and they all differ by an infinity between them. Good luck explaining that to high school students.
Moreover, working with equivalence relations is never a feature of any theory. Having to prove independence from representative at every step is not a feature, as you clearly demonstrate by making the mistake above.
> I've done a course in analysis that covered distributions, but your reply made me chuckle. You've told me that distributions are just as simple and then proceeded to dump paragraphs of jargon at me. L^infinity? Dual space? Support? Penrose defined? Inner product via integrals?
All concepts that are simple to define and understand. Majority of physicists likely understand well. Those that don’t, could.
Paragraphs of jargon? i’ve rigorously proven and justified my further assertions, at a similar level to OP and above what i’ve seen in some physics lectures.
I deliberately decided to avoid defining the above “jargon” terms after considering doing that to avoid extending the already long comment. I decided this because they are simple and a curious mind could quickly understand them by browsing wikipedia.
You’re welcome to ignore them and to go and compute derivatives and integrals just as mechanistically as in NSA (and I repeat, we haven’t even mentioned integrals in NSA. Good luck defining what are measurable functions on the hyperreals to your hypothetical AP high school students).
And to boot we never have to deal with any quantities that are not real measurable numbers. Anything we care about we can compute, more easily (integrals? integrals??) this way.
That is not to say that this isn’t an interesting theory that should be studied - just that it is quite the opposite of a “simplified” approach to general functions
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.