> 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?
> 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.
> 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.
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.
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.
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?