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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 expressi

by xyzzyz 11d ago

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.

saithound 10d ago
> 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 https://en.wikipedia.org/wiki/Reverse_mathematics [3] Potgieter: "Computable counter-examples to the Brouwer fixed point theorem", https://arxiv.org/abs/0804.3199 https://arxiv.org/abs/0804.3199 (2008).
- xyzzyz 9d ago
  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.
  - saithound 9d ago
    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.

by xyzzyz 11d ago

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.

saithound 10d ago

> 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 https://en.wikipedia.org/wiki/Reverse_mathematics [3] Potgieter: "Computable counter-examples to the Brouwer fixed point theorem", https://arxiv.org/abs/0804.3199 https://arxiv.org/abs/0804.3199 (2008).

xyzzyz 9d ago
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.
- saithound 9d ago
  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.