Serre wasn't there, but did a talk over Zoom from Switzerland where he now lives. I wasn't there either, but followed (and enjoyed) his talk over the YouTube stream. This was both a reminiscence of the mathematics of the 50s, and reflections on the role of counterexamples in mathematics.
Fun facts: Serre is still the youngest mathematician to be awarded the Fields medal, at 27. And Serre still publishes mathematics papers, the last one in 2025 (he was 99). A volume V of his Collected Papers (1998-2025) was published this summer by Springer. What a career!
I like this site of Mathematicians' biographies with some occasional quoted segments for extra flavor.
I also like how Serre wrote a book on linear representations of symmetry groups, because his wife needed a good exposition of the subject for her work on quantum chemistry, and that Serre described that as "fullfiling his duty as a husband" :-P
> I did not like, and did not understand, epsilons and deltas.
It's nice to have this perspective validated by someone like Serre! I felt like I was missing something when I first encountered that formalism. In fact, all of my introductory calculus classes sucked and turned me off of math for a few years.
Euler was a master manipulator of formal expressions; don't think it bothered him very much whether, e.g., an infinite series converged or not. (Unfortunately don't remember a source for this offhand -- maybe a leftover impression from having read ET Bell's book long ago? I also don't know how reliable Bell is.)
Cauchy is the one who introduced rigor to calculus with the formalization of limits, including epsilons and deltas, a full century after Leibniz and half a century after Euler.
Nonstandard analysis [0] [1] uses infinitesimals but is still completely rigorous. I haven't ever really used nonstandard analysis myself, but there's a fairly well-regarded textbook available online [2].
Also conceptually it feels just right to use nilpotents to probe the smooth structure. In a way nilpotents are violently smaller than even non standard analysis infinitesimals, as the laters’ powers are incredibly small but never vanishing.
Another way to see this is that it makes Taylor expansion exact by killing terms above a bound so it works naturally with the ecosystem surrounding it
Finally duals are very similar to complex in a way. i can be defined as root of X^2 + 1 = 0 even if it felt impossible initially, the dual number as a non nul solution of X^2 = 0 even if it is as counterintuitive.
There's no alternative that's significantly easier to understand and to use. The so-called "nonstandard analysis" hasn't caught on, because it's mostly the exact same arguments wrapped in slightly different language, not making them any simpler or shorter.
The language used by mathematicians is subject to constant evolution. 18th and 19th century results in analysis are not expressed and taught in the same way their original authors did it. Newton, Leibniz, Euler, Lagrange, Fourier, Riemann -- none of them expressed their results in terms of epsilons and deltas. These only caught on in the second half of 19th century, and they did so, because they were a better tool to rigorously prove the ideas.
New terminology inventions that make the subjects easier to understand take the field by storm. Some of the relatively recent examples are category theory, homological algebra, or, for that matter, the notion of sheafs, popularized by J.P. Serre himself. Mathematicians are very open to innovation, and intransigence is not the reason why we're stuck with epsilon-delta.
The reason is that nobody has yet come up with a better way of talking about these concepts. I repeatedly observe many people who seem to believe that their difficulty in understanding math stems from mathematicians gatekeeping their results. I think that this belief is just a coping mechanism. Mathematics is genuinely hard, and when people have trouble understanding something, it's easier to think that it's someone else's fault, rather than accepting one's own deficiencies.
> The so-called "nonstandard analysis" hasn't caught on, because it's mostly the exact same arguments wrapped in slightly different language
No. Let's take a nonstandard proof of the intermediate value theorem on [0,1] by Nelson.
By the transfer principle it is enough to prove this for a standard continuous function f on [0,1] with f(0)<0<f(1).
Take a finite subset of [0,1] containing every standard point. Colour its points blue, green, or red according to whether f is negative, zero, or positive at tha point.
The first point of the interval is blue and the last red. Hence either awe can find some green point, or we can find two neighbouring points that have different colours, the first blue and the second red.
In the first case there is a zero, so we are done. In the second, let the neighbouring points be p and q. By the completeness of the real numbers, every nonstandard real in [0,1] is infinitesimally close to exactly one standard real. So p and q are infinitesimally close to some standard real number, let's call it z.
Standard continuous functions send infinitesimally close points to infinitesimally close points. So f(p) and f(q) are both infinitesimally close to f(z). But f(p) is negative and f(q) is positive. The only standard number infinitesimally close to both positive and negative numbers is zero. Thus f(z) is zero. This proves the theorem.
You tell me, which standard proof is this? It's certainly not the nested interval proof. Not the supremum proof. Not the bisection proof in disguise. Which argument does it wrap in slightly different language? Can you point to a single textbook, course note or lecture that gives such an argument?
No. One could of course argue that this is not simpler/shorter than the usual arguments. But it is very different from them. Saying that it's the same arguments repackaged in a different language is just wrong, and detracts from an otherwise valid point.
This is the standard nested interval proof, you’re just replacing the limiting step of taking smaller and smaller intervals with the nonstandard way of expressing the same thing.
I'll be honest: your one sentence response tells me you did not read the proof above in any detail.
I chose Nelson's proof precisely because its construction is well-studied and well-understood. The same construction of a mesh containing all standard points, with the coloring forcing a tiny multicolored cell, extends from the interval to the triangle. In one dimension you get two adjacent differently colored points; in two dimensions you get an infinitesimal triangle whose three vertices have the three relevant colors. Taking their common standard part and applying continuity gives a short proof of Brouwer's fxied-point theorem on the triangle.
But it is well-understood (there's a whole field studying such questions [2]) that the nested interval proof of the Intermediate Value Theorem does not generalize to proving Brouwer's fixed point theorem on the triangle [1]. This fact can be derived from a computability argument as well [3].
Nelson's argument does generalize to prove Brouwer, so it's not the nested intervals argument. But really, nobody cares about these technical reasons. It's obvious to most math undergraduates that Nelson's proof is not the nested interval proof, the clear absence of any nested construction kinda gives it away. The only reason it was necessary to get technical is that you did not really inspect the proof before claiming it was nested intervals. The technical results cited above are just a formal way to show that any correspondence you might imagine between the two proofs is just not there.
[1] Shioji/Tanaka: "Fixed Point Theory in Weak Second-Order Arithmetic", Annals of Pure and Applied Logic v47, pp 167188 (1990).
What you just described is a classic proof of Brouwer's fixed point theorem using Sperner's lemma. The proof you cited earlier does not generalize to it on its own, the Sperner's lemma is a crucial combinatorial ingredient. It's crucial, because it only works on spaces with the topology of the triangle; you cannot perform the same argument on, say, an annulus.
In the standard formulation, you apply the Sperner's lemma to find smaller and smaller triangles, and apply compactness, precisely as in the standard proof of intermediate value theorem.
The rest of your post, where you quote reverse mathematics stuff, is completely irrelevant to the point I was making. Nothing I said is about what theorems follows from what axioms, but rather whether nonstandard analysis is meaningfully different, clearer, or more useful language than standard one. It is not.
You made a sweeping claim that nonstandard analysis arguments are the exact same arguments, wrapped in nonstandard langauge. I explained that (while your other claims about simplicity may be valid) this is not so and detracts from the rest of your points. I challenged you to defend your "same arguments" claim by finding any standard analysis textbook which teaches a standard language version of Nelson's argument as a proof of the IVT. Let me recap what happened since then:
1. Two comments ago you confidently claimed that Nelson's IVT proof is "the standard nested interval proof" with the limiting step written in nonstandard language. That is a straightforward claim about the structure of the proof, one that you didn't bother to substantiate, and that is straightforwardly false.
2. After I explained why it's false (Nelson's construction proves BFPT, which no nested interval type proof can do), you changed your response: now the Sperner lemma was a "crucial additional ingredient". But Nelson's combinatorial step, that opposite endpoint colors force a blue-red interval, _is_ the one-dimensional instance of the Sperner lemma (and indeed the base case when you prove Sperner's lemma for arbitrary dimensional simplices by induction; the analytic part is independent of dimension, once you find an infinitesimal multicolored simplex, you take its common standard part and apply continuity exactly as before).
3. Then you wrote this:
> In the standard formulation, you apply the Sperner's lemma to find smaller and smaller triangles, and apply compactness, precisely as in the standard proof of intermediate value theorem.
There is a standard proof of Brouwer via the Sperner lemma, and it is _also_ not of the same form as the standard nested interval proof of the IVT. In the nested interval proof, you find a sign-change interval, then find a smaller sign-change interval inside it, and so on. The intersection of all of these contains a point, and that's your zero. Nelson's proof does not do this, and neither does the standard proof of Brouwer via the Sperner lemma: you do not, and cannot, take a 3-color interior triangle, then find a smaller 3-color interior triangle inside it and so on. Even the first step would not work, since the inherited labelling does not satisfy the right boundary condition relative to the small triangle!
And this is also why the computability paper I cited ("where I quote reverse mathematics stuff" ;) was very much relevant. There can be no effective "nested triangle" proofs of the Brouwer fixed point theorem at all, because such a proof would let you compute a Brouwer fixed point, and there are examples of computable maps on the triangle without computable fixed points. If Nelson's IVT proof was the nested interval proof, then swapping in the higher-dimensional Sperner step would give a nested-type proof of BFPT. No such proof can exist. Since Nelson's argument proves the BFPT without any change to the analytic part, it is not a nested interval type argument.
You first misidentified Nelson's proof as nested intervals, and then treated Sperner as an additional ingredient even though the coloring step in Nelson's proof is already the corresponding Sperner argument. Those are both fairly serious misunderstandings about these proof. Given this, I don't think our exchange leaves readers with much confidence in your assessment of NSA's drawbacks and benefits. That's a disappointing outcome, as far as I'm concerned. There are good arguments to make that NSA adds little value to undergraduate education, such as simplicity or the difficulty of the prerequisites, and good conversations to be had about them. But "NSA proofs are the same proofs wrapped in a different language" is not one, and I wish you had just narrowed it instead of doubling down.
I never said the triangles in the standard proof are going to be nested, so your whole segue into reverse mathematics is, just like I said, irrelevant. The point of the argument is that you can find a sequence of triangles with differently colored vertices, the vertices of which all converge to the same point (thanks to compactness), which contradicts continuity of the retraction on the boundary. The nonstandard version of this is exactly the same argument, it just replaces the explicit limiting step that contradicts continuity with the an argument that uses the nonstandard formulation of continuity in terms of infinitesimals.
If that makes it easier for you to understand it, in the standard proof, you also color every point of the rectangle, with the color of the edge it retracts to (picking the colors of the vertices of the big triangle arbitrarily, just making sure that the color of each vertex is a color of one of the edges it belongs to, not one of the opposite edges). Then, an easy argument from continuity shows that no interior point will have points of three different colors arbitrarily close to it. Finally, applying Sperner's lemma as above proves that such point must nevertheless exist, obtaining contradiction with the existence of the retraction.
and then treated Sperner as an additional ingredient even though the coloring step in Nelson's proof is already the corresponding Sperner argument.
I don't understand what are you saying here. What I'm saying is that for the coloring proof of BFPT to work, whether clothed in standard or nonstandard language, you must perform a combinatorial argument that uses a topology of a triangle as a necessary ingredient, similar in shape to the proof of Sperner's lemma.
You opened with the claim that nonstandard analysis hasn't caught on because it's "mostly the exact same arguments wrapped in slightly different language". I pointed out that the arguments are in fact very distinct: e.g. Nelson's proof of the intermediate value theorem is something that any NSA student would see, but no standard textbook teaches IVT by a standard language counterpart of it.
One post later, you answered that Nelson's IVT argument is in fact the "standard nested interval proof" with the limiting step rewritten in nonstandard language.
That claim is simply wrong. Why? Because Nelson's construction straightforwardly generalises to Brouwer, while the nested intervals proofs cannot. The discussion of reverse mathematics / computability is not a tangent, it explains precisely why Nelson's proof can generalise to give the BFPT in two dimensions, whereas the nested intervals proofs (which you claim is the same) cannot.
You then brought up that the BFPT generalisation of Nelson's argument needs the Sperner lemma as "crucial additional ingredient". Now you make the same point again:
> What I'm saying is that for the coloring proof of BFPT to work, whether clothed in standard or nonstandard language, you must perform a combinatorial argument that uses a topology of a triangle as a necessary ingredient, similar in shape to the proof of Sperner's lemma.
Presumably you keep pointing this out because you think it justifies some claim like '1D Nelson is actually nested intervals with the limiting step recast in nonstandard language, even if the 2D generalization of Nelson is not'.
But it does not. The combinatorial content is the same, the 1-dimensional interval case uses the topology of the domain just as much as the 2-dimensional triangle case does. The 2D argument wouldn't work on the annulus, and the 1D version would not work on the union of two disjoint intervals. The Sperner lemma is present in 1D, and present in 2D. If instead your point is only that proving the BFPT requires a harder case of the Sperner lemma than IVT, then of course it does. But what relevance does that have to the original claim that Nelson's IVT proof is the nested-interval proof? The proof of the Sperner lemma is pure combinatorics, it does not involve any (standard or nonstandard) analysis.
Or have you changed your mind on your earlier claim that Nelson's proof is "is the standard nested interval proof"?
If so, I think that's great, and closes the thread on whether NSA is largely the same arguments, since even the first proofs of the basic results are different.
If you still think that it's the nested interval proof, well, I am not sure what else to say, apart from linking the literature which studies this exact question, that I've already done, and that you dismissed as a tangent.
Either way, this discussion went on for too long at this point, so I won't monitor it further.
A professor of mine had an anecdote of meeting Serre and complaining to him about Bourbaki style and how hard it is for students.
Serre's reply was "But we never wrote those books for students! We wrote them for researchers to have a handy reference for all proofs of basic results."
I'm reading and liking his book just called "Trees" (some bites), it is about group actions on trees, written in the '80. It talks about the modular group among others, the tree action also being used in Shai Haran work on the "real" prime (if the AIs let us dream the old way about classic problems). Serre et al graph of groups idea is very categorical. He could well have been present in the very seminars when Grothendiek was breweing what is now called the Grothendiek construction (absurdly, since G was a serial constructor). That wraps that graph of groups thing.
on slightly related note of longevity - Fred Haise, one of three Apollo 13 astronatuts celebrated 92nd birthday day before yesterday, Jim Lovell died at 97 and only one who lived short life was Jack Swigert who died at 51 (respiratory failure)
There was a conference yesterday in his honour in Paris: https://serre100.sciencesconf.org/
Serre wasn't there, but did a talk over Zoom from Switzerland where he now lives. I wasn't there either, but followed (and enjoyed) his talk over the YouTube stream. This was both a reminiscence of the mathematics of the 50s, and reflections on the role of counterexamples in mathematics.
Fun facts: Serre is still the youngest mathematician to be awarded the Fields medal, at 27. And Serre still publishes mathematics papers, the last one in 2025 (he was 99). A volume V of his Collected Papers (1998-2025) was published this summer by Springer. What a career!
Proving Hardy’s “A Mathematician’s Apology” wrong?
[dead]
I like this site of Mathematicians' biographies with some occasional quoted segments for extra flavor.
I also like how Serre wrote a book on linear representations of symmetry groups, because his wife needed a good exposition of the subject for her work on quantum chemistry, and that Serre described that as "fullfiling his duty as a husband" :-P
The European Mathematical Society just published an interview with Serre on the occasion of his birthday: https://euromathsoc.org/news/ems-magazine-no.-141:-celebrati...
My favorite part curled up in a footnote:
> AI told me that, among my books, this is the most difficult to read for students. I write for mathematicians, not for students.
Gépété raté.
EDIT:
Actually, I didn't know he bouldered!
> I did not like, and did not understand, epsilons and deltas.
It's nice to have this perspective validated by someone like Serre! I felt like I was missing something when I first encountered that formalism. In fact, all of my introductory calculus classes sucked and turned me off of math for a few years.
He describes the style he did like as "Euler's style, so to speak". What was Euler's style in this context? i.e. as opposed to epsilons and deltas?
Euler was a master manipulator of formal expressions; don't think it bothered him very much whether, e.g., an infinite series converged or not. (Unfortunately don't remember a source for this offhand -- maybe a leftover impression from having read ET Bell's book long ago? I also don't know how reliable Bell is.)
Edit: added semi-source
Cauchy is the one who introduced rigor to calculus with the formalization of limits, including epsilons and deltas, a full century after Leibniz and half a century after Euler.
What's the alternative for explaining those concepts that's still reasonably rigorous?
Nonstandard analysis [0] [1] uses infinitesimals but is still completely rigorous. I haven't ever really used nonstandard analysis myself, but there's a fairly well-regarded textbook available online [2].
[0]: https://en.wikipedia.org/wiki/Nonstandard_analysis
[1]: https://math.stackexchange.com/questions/51453/is-non-standa...
[2]: https://people.math.wisc.edu/%7Ehkeisler/keislercalc-06-03-2...
When I first learned non-standard analysis, my reaction was that we don't need the axiom of choice to find the derivative of x^2.
The formalism is very simple symbolically. But the mathematical machine behind it is very complex.
Various algebras of dual numbers are used in most automatic derivative routines.
This is treated more rigorously and generically in the subject of synthetic differential geometry.
wanted to say this.
Also conceptually it feels just right to use nilpotents to probe the smooth structure. In a way nilpotents are violently smaller than even non standard analysis infinitesimals, as the laters’ powers are incredibly small but never vanishing.
Another way to see this is that it makes Taylor expansion exact by killing terms above a bound so it works naturally with the ecosystem surrounding it
Finally duals are very similar to complex in a way. i can be defined as root of X^2 + 1 = 0 even if it felt impossible initially, the dual number as a non nul solution of X^2 = 0 even if it is as counterintuitive.
There's no alternative that's significantly easier to understand and to use. The so-called "nonstandard analysis" hasn't caught on, because it's mostly the exact same arguments wrapped in slightly different language, not making them any simpler or shorter.
The language used by mathematicians is subject to constant evolution. 18th and 19th century results in analysis are not expressed and taught in the same way their original authors did it. Newton, Leibniz, Euler, Lagrange, Fourier, Riemann -- none of them expressed their results in terms of epsilons and deltas. These only caught on in the second half of 19th century, and they did so, because they were a better tool to rigorously prove the ideas.
New terminology inventions that make the subjects easier to understand take the field by storm. Some of the relatively recent examples are category theory, homological algebra, or, for that matter, the notion of sheafs, popularized by J.P. Serre himself. Mathematicians are very open to innovation, and intransigence is not the reason why we're stuck with epsilon-delta.
The reason is that nobody has yet come up with a better way of talking about these concepts. I repeatedly observe many people who seem to believe that their difficulty in understanding math stems from mathematicians gatekeeping their results. I think that this belief is just a coping mechanism. Mathematics is genuinely hard, and when people have trouble understanding something, it's easier to think that it's someone else's fault, rather than accepting one's own deficiencies.
> The so-called "nonstandard analysis" hasn't caught on, because it's mostly the exact same arguments wrapped in slightly different language
No. Let's take a nonstandard proof of the intermediate value theorem on [0,1] by Nelson.
By the transfer principle it is enough to prove this for a standard continuous function f on [0,1] with f(0)<0<f(1).
Take a finite subset of [0,1] containing every standard point. Colour its points blue, green, or red according to whether f is negative, zero, or positive at tha point.
The first point of the interval is blue and the last red. Hence either awe can find some green point, or we can find two neighbouring points that have different colours, the first blue and the second red.
In the first case there is a zero, so we are done. In the second, let the neighbouring points be p and q. By the completeness of the real numbers, every nonstandard real in [0,1] is infinitesimally close to exactly one standard real. So p and q are infinitesimally close to some standard real number, let's call it z.
Standard continuous functions send infinitesimally close points to infinitesimally close points. So f(p) and f(q) are both infinitesimally close to f(z). But f(p) is negative and f(q) is positive. The only standard number infinitesimally close to both positive and negative numbers is zero. Thus f(z) is zero. This proves the theorem.
You tell me, which standard proof is this? It's certainly not the nested interval proof. Not the supremum proof. Not the bisection proof in disguise. Which argument does it wrap in slightly different language? Can you point to a single textbook, course note or lecture that gives such an argument?
No. One could of course argue that this is not simpler/shorter than the usual arguments. But it is very different from them. Saying that it's the same arguments repackaged in a different language is just wrong, and detracts from an otherwise valid point.
This is the standard nested interval proof, you’re just replacing the limiting step of taking smaller and smaller intervals with the nonstandard way of expressing the same thing.
> This is the standard nested interval proof
It is not.
I'll be honest: your one sentence response tells me you did not read the proof above in any detail.
I chose Nelson's proof precisely because its construction is well-studied and well-understood. The same construction of a mesh containing all standard points, with the coloring forcing a tiny multicolored cell, extends from the interval to the triangle. In one dimension you get two adjacent differently colored points; in two dimensions you get an infinitesimal triangle whose three vertices have the three relevant colors. Taking their common standard part and applying continuity gives a short proof of Brouwer's fxied-point theorem on the triangle.
But it is well-understood (there's a whole field studying such questions [2]) that the nested interval proof of the Intermediate Value Theorem does not generalize to proving Brouwer's fixed point theorem on the triangle [1]. This fact can be derived from a computability argument as well [3].
Nelson's argument does generalize to prove Brouwer, so it's not the nested intervals argument. But really, nobody cares about these technical reasons. It's obvious to most math undergraduates that Nelson's proof is not the nested interval proof, the clear absence of any nested construction kinda gives it away. The only reason it was necessary to get technical is that you did not really inspect the proof before claiming it was nested intervals. The technical results cited above are just a formal way to show that any correspondence you might imagine between the two proofs is just not there.
[1] Shioji/Tanaka: "Fixed Point Theory in Weak Second-Order Arithmetic", Annals of Pure and Applied Logic v47, pp 167188 (1990).
[2] https://en.wikipedia.org/wiki/Reverse_mathematics
[3] Potgieter: "Computable counter-examples to the Brouwer fixed point theorem", https://arxiv.org/abs/0804.3199 (2008).
What you just described is a classic proof of Brouwer's fixed point theorem using Sperner's lemma. The proof you cited earlier does not generalize to it on its own, the Sperner's lemma is a crucial combinatorial ingredient. It's crucial, because it only works on spaces with the topology of the triangle; you cannot perform the same argument on, say, an annulus.
In the standard formulation, you apply the Sperner's lemma to find smaller and smaller triangles, and apply compactness, precisely as in the standard proof of intermediate value theorem.
The rest of your post, where you quote reverse mathematics stuff, is completely irrelevant to the point I was making. Nothing I said is about what theorems follows from what axioms, but rather whether nonstandard analysis is meaningfully different, clearer, or more useful language than standard one. It is not.
You made a sweeping claim that nonstandard analysis arguments are the exact same arguments, wrapped in nonstandard langauge. I explained that (while your other claims about simplicity may be valid) this is not so and detracts from the rest of your points. I challenged you to defend your "same arguments" claim by finding any standard analysis textbook which teaches a standard language version of Nelson's argument as a proof of the IVT. Let me recap what happened since then:
1. Two comments ago you confidently claimed that Nelson's IVT proof is "the standard nested interval proof" with the limiting step written in nonstandard language. That is a straightforward claim about the structure of the proof, one that you didn't bother to substantiate, and that is straightforwardly false.
2. After I explained why it's false (Nelson's construction proves BFPT, which no nested interval type proof can do), you changed your response: now the Sperner lemma was a "crucial additional ingredient". But Nelson's combinatorial step, that opposite endpoint colors force a blue-red interval, _is_ the one-dimensional instance of the Sperner lemma (and indeed the base case when you prove Sperner's lemma for arbitrary dimensional simplices by induction; the analytic part is independent of dimension, once you find an infinitesimal multicolored simplex, you take its common standard part and apply continuity exactly as before).
3. Then you wrote this:
> In the standard formulation, you apply the Sperner's lemma to find smaller and smaller triangles, and apply compactness, precisely as in the standard proof of intermediate value theorem.
There is a standard proof of Brouwer via the Sperner lemma, and it is _also_ not of the same form as the standard nested interval proof of the IVT. In the nested interval proof, you find a sign-change interval, then find a smaller sign-change interval inside it, and so on. The intersection of all of these contains a point, and that's your zero. Nelson's proof does not do this, and neither does the standard proof of Brouwer via the Sperner lemma: you do not, and cannot, take a 3-color interior triangle, then find a smaller 3-color interior triangle inside it and so on. Even the first step would not work, since the inherited labelling does not satisfy the right boundary condition relative to the small triangle!
And this is also why the computability paper I cited ("where I quote reverse mathematics stuff" ;) was very much relevant. There can be no effective "nested triangle" proofs of the Brouwer fixed point theorem at all, because such a proof would let you compute a Brouwer fixed point, and there are examples of computable maps on the triangle without computable fixed points. If Nelson's IVT proof was the nested interval proof, then swapping in the higher-dimensional Sperner step would give a nested-type proof of BFPT. No such proof can exist. Since Nelson's argument proves the BFPT without any change to the analytic part, it is not a nested interval type argument.
You first misidentified Nelson's proof as nested intervals, and then treated Sperner as an additional ingredient even though the coloring step in Nelson's proof is already the corresponding Sperner argument. Those are both fairly serious misunderstandings about these proof. Given this, I don't think our exchange leaves readers with much confidence in your assessment of NSA's drawbacks and benefits. That's a disappointing outcome, as far as I'm concerned. There are good arguments to make that NSA adds little value to undergraduate education, such as simplicity or the difficulty of the prerequisites, and good conversations to be had about them. But "NSA proofs are the same proofs wrapped in a different language" is not one, and I wish you had just narrowed it instead of doubling down.
I never said the triangles in the standard proof are going to be nested, so your whole segue into reverse mathematics is, just like I said, irrelevant. The point of the argument is that you can find a sequence of triangles with differently colored vertices, the vertices of which all converge to the same point (thanks to compactness), which contradicts continuity of the retraction on the boundary. The nonstandard version of this is exactly the same argument, it just replaces the explicit limiting step that contradicts continuity with the an argument that uses the nonstandard formulation of continuity in terms of infinitesimals.
If that makes it easier for you to understand it, in the standard proof, you also color every point of the rectangle, with the color of the edge it retracts to (picking the colors of the vertices of the big triangle arbitrarily, just making sure that the color of each vertex is a color of one of the edges it belongs to, not one of the opposite edges). Then, an easy argument from continuity shows that no interior point will have points of three different colors arbitrarily close to it. Finally, applying Sperner's lemma as above proves that such point must nevertheless exist, obtaining contradiction with the existence of the retraction.
and then treated Sperner as an additional ingredient even though the coloring step in Nelson's proof is already the corresponding Sperner argument.
I don't understand what are you saying here. What I'm saying is that for the coloring proof of BFPT to work, whether clothed in standard or nonstandard language, you must perform a combinatorial argument that uses a topology of a triangle as a necessary ingredient, similar in shape to the proof of Sperner's lemma.
You opened with the claim that nonstandard analysis hasn't caught on because it's "mostly the exact same arguments wrapped in slightly different language". I pointed out that the arguments are in fact very distinct: e.g. Nelson's proof of the intermediate value theorem is something that any NSA student would see, but no standard textbook teaches IVT by a standard language counterpart of it.
One post later, you answered that Nelson's IVT argument is in fact the "standard nested interval proof" with the limiting step rewritten in nonstandard language.
That claim is simply wrong. Why? Because Nelson's construction straightforwardly generalises to Brouwer, while the nested intervals proofs cannot. The discussion of reverse mathematics / computability is not a tangent, it explains precisely why Nelson's proof can generalise to give the BFPT in two dimensions, whereas the nested intervals proofs (which you claim is the same) cannot.
You then brought up that the BFPT generalisation of Nelson's argument needs the Sperner lemma as "crucial additional ingredient". Now you make the same point again:
> What I'm saying is that for the coloring proof of BFPT to work, whether clothed in standard or nonstandard language, you must perform a combinatorial argument that uses a topology of a triangle as a necessary ingredient, similar in shape to the proof of Sperner's lemma.
Presumably you keep pointing this out because you think it justifies some claim like '1D Nelson is actually nested intervals with the limiting step recast in nonstandard language, even if the 2D generalization of Nelson is not'.
But it does not. The combinatorial content is the same, the 1-dimensional interval case uses the topology of the domain just as much as the 2-dimensional triangle case does. The 2D argument wouldn't work on the annulus, and the 1D version would not work on the union of two disjoint intervals. The Sperner lemma is present in 1D, and present in 2D. If instead your point is only that proving the BFPT requires a harder case of the Sperner lemma than IVT, then of course it does. But what relevance does that have to the original claim that Nelson's IVT proof is the nested-interval proof? The proof of the Sperner lemma is pure combinatorics, it does not involve any (standard or nonstandard) analysis.
Or have you changed your mind on your earlier claim that Nelson's proof is "is the standard nested interval proof"?
If so, I think that's great, and closes the thread on whether NSA is largely the same arguments, since even the first proofs of the basic results are different.
If you still think that it's the nested interval proof, well, I am not sure what else to say, apart from linking the literature which studies this exact question, that I've already done, and that you dismissed as a tangent.
Either way, this discussion went on for too long at this point, so I won't monitor it further.
He later participated in Bourbaki, who were known by their overly formal style, tough.
A professor of mine had an anecdote of meeting Serre and complaining to him about Bourbaki style and how hard it is for students.
Serre's reply was "But we never wrote those books for students! We wrote them for researchers to have a handy reference for all proofs of basic results."
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I'm reading and liking his book just called "Trees" (some bites), it is about group actions on trees, written in the '80. It talks about the modular group among others, the tree action also being used in Shai Haran work on the "real" prime (if the AIs let us dream the old way about classic problems). Serre et al graph of groups idea is very categorical. He could well have been present in the very seminars when Grothendiek was breweing what is now called the Grothendiek construction (absurdly, since G was a serial constructor). That wraps that graph of groups thing.
I especially liked the proof in that book that looks at the action of a group on its Cayley graph to show a subgroup of a free group is free.
"When I was 14 or 15, I used to look at these books, and study them"
Smartphones prevent most teenagers of this era to read a book. This may apply to lots of adults.
Smartphones and now AI...
I'm looking forward to Dario's essay begging for regulating the use of AI on education.
Oh wait, that goes against his economic interest.
Happy birthday “Tonton Serre”
on slightly related note of longevity - Fred Haise, one of three Apollo 13 astronatuts celebrated 92nd birthday day before yesterday, Jim Lovell died at 97 and only one who lived short life was Jack Swigert who died at 51 (respiratory failure)