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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 exac

• saithound 19d ago
  > 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.
  • xyzzyz 19d 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 18d 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 17d 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.