Talk:Brouwer fixed-point theorem

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Accessible proof

Courant and Robbins provide an accessible proof. —Preceding unsigned comment added by 198.144.199.xxx (talk • contribs) 30 August 2001

First proved by Bol?

According to Lyusternik Convex Figures and Polyhedra, the theorem was first proved by a Lettish mathematician named Bol. No references are provided. Anyone know what this is about?--192.35.35.36 00:08, 18 Feb 2005 (UTC)

"Constructive proof"?

The article sais "The first algorithm to construct a fixed point was proposed by H. Scarf." and also "Kellogg, Li, and Yorke turned Hirsch's proof into a constructive proof by observing that..."
I'm wondering if it's indeed a constructive proof, since Brower's theorem for one dimension is equivalent to intermediate value theorem, which does not admit a constructive proof.
See for example this discussion in MathOverflow. Nachi (talk) 16:58, 4 February 2018 (UTC)[reply]

This is a symptom of different people using "constructive" to mean different things. The paper by Kellog, Li, and York really is titled "A Constructive Proof of the Brouwer Fixed-Point Theorem and Computational Results". But they are working in numerical analysis, not in constructive mathematics. So perhaps all that they mean by 'constructive proof' is that their proof can be used to obtain a numerical algorithm to approximate a fixed point. I am not completely sure what they mean by constructive, though, as I look at their paper. They also assume that the map is not only continuous, but twice differentiable. In the sense of many branches of constructive mathematics, it is known that the fixed point theorem implies nonconstructive principles such as LLPO, and so the fixed point theorem is not constructive in the sense of those branches. — Carl (CBM · talk) 17:25, 4 February 2018 (UTC)[reply]

Thank you for the answer. It makes it clear.

Unless I completely not aware of the usual use of "constructive" in mathematics, I guess the best way to describe KLY version of Hirsch's proof is simply write "numerical algorithm" or "computable method", instead of "constructive". Also in the description of Scarf's proof the word "construct" should be replaced by "calculate". Nachi (talk) 20:25, 4 February 2018 (UTC)[reply]

Citation style

This article mixes parenthetical referencing with footnoted references. The parenthetical ones were there first, so according to WP:CITEVAR we'd have to use that until explicit consensus. However, it would be significantly easier to turn the couple of parenthetical ones into footnotes than about 50 footnotes into parentheticals. Can we form consensus to continued using footnoted references? – Finnusertop (talk ⋅ contribs) 19:59, 24 February 2019 (UTC)[reply]

I'm sure that'd be okay here. –Deacon Vorbis (carbon • videos) 00:04, 25 February 2019 (UTC)[reply]

Great. I've turned the remaining parentheticals into footnotes. – Finnusertop (talk ⋅ contribs) 00:10, 25 February 2019 (UTC)[reply]

Function mapping in closedness section

It is stated that the function f(x) = (x+1)/2 is a continous function from the open interval (-1,1) to itself. Is it not the case that the function maps from (-1,1) to (0,1)? Salomonaber (talk) 00:13, 11 March 2020 (UTC)[reply]

It doesn't claim (nor is it required) that the function is surjective, so what's there is correct and appropriate. The example could have even arranged for a bijection, but I don't think it matters much either way. –Deacon Vorbis (carbon • videos) 00:32, 11 March 2020 (UTC)[reply]

Highly skeptical that the remarks "said to have [been] added" by Brouwer are actually due to him

Brouwer is said to have added: "I can formulate this splendid result different, I take a horizontal sheet, and another identical one which I crumple, flatten and place on the other. Then a point of the crumpled sheet is in the same place as on the other sheet."

The citation is apparently from a French-language educational TV show (https://archive.is/20130113210953/http://archives.arte.tv/hebdo/archimed/19990921/ftext/sujet5.html). The remarks appear to be spoken by a fictional Brouwer trying to explain his result. The web page that this refers to gives no citation.

I would like to know who originally came up with the "crumpled paper theorem" explanation of the BFPT. It could have been Brouwer himself, but my guess is it was not. — Preceding unsigned comment added by Natkuhn (talk • contribs) 01:06, 8 May 2020 (UTC)[reply]

Oops, yeah, that's a good catch. This probably deserves some looking into. –Deacon Vorbis (carbon • videos) 01:19, 8 May 2020 (UTC)[reply]