I'm not a big fan of using nonstandard analysis for this. We're assuming the existence of arbitrary answers that we cannot ever produce.
For example, which function is eventually larger than the other?
(1 + sin(x)) * e^x + x
(1 + cos(x)) * e^x + x
In the ultrafilter, one almost certainly will be larger. In fact the ratio of the two will, asymptotically, approach a specific limit. Which one is larger? What is the ratio? That entirely depends on the ultrafilter.
Which means that we can accept the illusionary simplicity of his axiom about every predicate P(N), and it will remain simple right until we try to get a concrete and useful answer out of it.
I agree with this to some extent. Another perspective: think about the element [(1, 2, 3, 4, ...)] in the ultrafilter; let's call this omega. On some level, all of these questions are really just questions about what properties omega has: is it even or odd, prime or composite, etc. Simultaneously deciding all of these questions in a coherent way is equivalent to specifying an ultrafilter. Similarly, when we ask about some function f(x) being > g(x) asymptotically, we are basically asking if f(omega) > g(omega). This is just a different view of the same thing.
For instance, your question happens to be equivalent to asking whether sin(omega) > cos(omega), and thus if tan(omega) > 1. This is true iff the fractional part the hyperreal number omega/(2*pi) is between 1/8 and 5/8. Thus we have reduced the asymptotic statement to a question about an arithmetical property of one particular hyperreal number.
Choosing an ultrafilter basically involves simultaneously determining all properties of omega. There are different ultrafilters, each providing a different coherent "universe" which decides all possible predicates in a coherent way. That this is possible (with the axiom of choice) is highly interesting. However, it doesn't seem necessary for asymptotic analysis.
Of course, if there is some "canonical" or "most natural" ultrafilter to choose from, with some magical property universally deemed important, then it would settle your question and all such questions in a natural way.
Look for the comment in the article, after passing to a subsequence if necessary. The ultrafilter produces the necessary subsequence for any question that you ask, and will do so in such a way as to produce logically consistent answers for any combination of questions that you choose.
That is why the ultrafilter axiom is a weak version of choice. Take the set of possible yes/no questions that we can ask as predicates, such that each answer shows up infinitely often. The ultrafilter results in an arbitrary yet consistent set of choices of yes/no for each predicate.
The axioms demand that either one function is eventually dominated by the other, or both functions are of the same order. But which of these is the case will strongly depend on which subsequence you look at.
You may have missed the same subtlety that I did. Because pi is irrational, the functions are different at all integers. Therefore, in the total order, these two functions cannot have the same order.
That still doesn't resolve which one is larger though.
Well, as presented in Tao's post, the set Ω can be either the natural numbers or the real numbers. So I'm assuming the "subsequence" is a (perhaps uncountable?) set of real parameters, in the latter case.
Isn't this exactly what mathematics is about? You have a non-ordered set, you map it into a total ordered set, of course the new ordering won't be the same as the previous. Like taking a projection of the points of the 2d plane to an arbitrary line. You now get a total ordering, and you can do whatever. You do it if it helps you, and don't do it when it doesn't.
You can write down any set of hopefully consistent axioms, write down any set of definitions from them, and start proving theorems. The result will be mathematics. But not all mathematics is equally interesting.
People who look at asymptotic growth are interested in what happens for all, or occasionally almost all, large n. The possibility of this kind of total order is irrelevant, and therefore uninteresting to people who are interested in that. What Tao is doing is mathematics, but not mathematics of a kind that I, personally, like.
The total order on functions is not an end-goal in itself, but a step in a proof which provides useful results.
That's what mathematics is about. You work with something, then you work with something else. When you count kittens, you can't pet the numbers anymore, but the corresponding integers are still useful.
It is a theorem that any argument that can be made with nonstandard analysis (NSA), can also be made without it. The question is therefore whether NSA helps people's intuition enough to make it worthwhile.
In elementary Calculus, it really does help people's intuition. In fact it allowed us to formalize a lot of the intuitive arguments through which Calculus was originally built. In analysis, it has helped at least some people's intuition. See, for example, Robinson and Bernstein's proof of the invariant problem. However most people in analysis have found that it isn't that hard to translate the NSA version of such proofs into more familiar terminology, and they don't find the NSA version to help their intuition.
When we go as far afield as the asymptotic growth of functions, I don't see our intuition being helped much by NSA. I could be wrong - I would have been on the wrong side of the importance of oracles in cryptography on somewhat similar intuitions - but it remains my impression.
Since we know that these hyper real numbers are well defined we can teach them axiomatically to high school students the way Leibniz used them (and keep the explicit construction via filters to university students just like with a dedekind cut for reals)
I'm not a big fan of using nonstandard analysis for this. We're assuming the existence of arbitrary answers that we cannot ever produce.
For example, which function is eventually larger than the other?
In the ultrafilter, one almost certainly will be larger. In fact the ratio of the two will, asymptotically, approach a specific limit. Which one is larger? What is the ratio? That entirely depends on the ultrafilter.Which means that we can accept the illusionary simplicity of his axiom about every predicate P(N), and it will remain simple right until we try to get a concrete and useful answer out of it.
I agree with this to some extent. Another perspective: think about the element [(1, 2, 3, 4, ...)] in the ultrafilter; let's call this omega. On some level, all of these questions are really just questions about what properties omega has: is it even or odd, prime or composite, etc. Simultaneously deciding all of these questions in a coherent way is equivalent to specifying an ultrafilter. Similarly, when we ask about some function f(x) being > g(x) asymptotically, we are basically asking if f(omega) > g(omega). This is just a different view of the same thing.
For instance, your question happens to be equivalent to asking whether sin(omega) > cos(omega), and thus if tan(omega) > 1. This is true iff the fractional part the hyperreal number omega/(2*pi) is between 1/8 and 5/8. Thus we have reduced the asymptotic statement to a question about an arithmetical property of one particular hyperreal number.
Choosing an ultrafilter basically involves simultaneously determining all properties of omega. There are different ultrafilters, each providing a different coherent "universe" which decides all possible predicates in a coherent way. That this is possible (with the axiom of choice) is highly interesting. However, it doesn't seem necessary for asymptotic analysis.
Of course, if there is some "canonical" or "most natural" ultrafilter to choose from, with some magical property universally deemed important, then it would settle your question and all such questions in a natural way.
I don't think that's the case. They can both not have the property that it is eventually larger than the other.
No, it is the case.
Look for the comment in the article, after passing to a subsequence if necessary. The ultrafilter produces the necessary subsequence for any question that you ask, and will do so in such a way as to produce logically consistent answers for any combination of questions that you choose.
That is why the ultrafilter axiom is a weak version of choice. Take the set of possible yes/no questions that we can ask as predicates, such that each answer shows up infinitely often. The ultrafilter results in an arbitrary yet consistent set of choices of yes/no for each predicate.
Okay, yes, I see. But then it seems that O doesn't obey some very natural standard schools, and then what is it good for?
O is a total order, but functions aren't in any way a total order, so what's the point?
And now you see what I don't like about it!
The axioms demand that either one function is eventually dominated by the other, or both functions are of the same order. But which of these is the case will strongly depend on which subsequence you look at.
You may have missed the same subtlety that I did. Because pi is irrational, the functions are different at all integers. Therefore, in the total order, these two functions cannot have the same order.
That still doesn't resolve which one is larger though.
Well, as presented in Tao's post, the set Ω can be either the natural numbers or the real numbers. So I'm assuming the "subsequence" is a (perhaps uncountable?) set of real parameters, in the latter case.
Ah. Good point.
Isn't this exactly what mathematics is about? You have a non-ordered set, you map it into a total ordered set, of course the new ordering won't be the same as the previous. Like taking a projection of the points of the 2d plane to an arbitrary line. You now get a total ordering, and you can do whatever. You do it if it helps you, and don't do it when it doesn't.
You can write down any set of hopefully consistent axioms, write down any set of definitions from them, and start proving theorems. The result will be mathematics. But not all mathematics is equally interesting.
People who look at asymptotic growth are interested in what happens for all, or occasionally almost all, large n. The possibility of this kind of total order is irrelevant, and therefore uninteresting to people who are interested in that. What Tao is doing is mathematics, but not mathematics of a kind that I, personally, like.
That's not it.
The total order on functions is not an end-goal in itself, but a step in a proof which provides useful results.
That's what mathematics is about. You work with something, then you work with something else. When you count kittens, you can't pet the numbers anymore, but the corresponding integers are still useful.
It is a theorem that any argument that can be made with nonstandard analysis (NSA), can also be made without it. The question is therefore whether NSA helps people's intuition enough to make it worthwhile.
In elementary Calculus, it really does help people's intuition. In fact it allowed us to formalize a lot of the intuitive arguments through which Calculus was originally built. In analysis, it has helped at least some people's intuition. See, for example, Robinson and Bernstein's proof of the invariant problem. However most people in analysis have found that it isn't that hard to translate the NSA version of such proofs into more familiar terminology, and they don't find the NSA version to help their intuition.
When we go as far afield as the asymptotic growth of functions, I don't see our intuition being helped much by NSA. I could be wrong - I would have been on the wrong side of the importance of oracles in cryptography on somewhat similar intuitions - but it remains my impression.
Since we know that these hyper real numbers are well defined we can teach them axiomatically to high school students the way Leibniz used them (and keep the explicit construction via filters to university students just like with a dedekind cut for reals)
Here is the axiomatic approach in Julia and Lean https://github.com/pannous/hyper-lean