alt Hacker News> 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.
They all differ pointwise by an infinitesmal! This flexibility is a feature, not a bug.
Of course, I agree with you that mathematical progress should proceed on rigorous grounds, and there is a lot to be proven here. But my point is mainly that this is so easy you can just go and see what happens in these cases yourself without much trouble. For applications you really don't have to care.
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?
(to be clear, I know what you're talking about, but our hypothetical high school student will have a lot more luck moving infinitesmals around, I guarantee it)
> 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?
For physics, just pick a representative, compute and do some sanity checks. Who is ZFC? =P
Just kidding. Anyway, as you probably know the deeper theory of nonstandard analysis has been worked out in detail already, so if you want to get stuck in the weeds there are answers out there.
Replies
> 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
This answer is my favorite. You got it =)