Talk:Brouwer fixed-point theorem: Difference between revisions

Revision as of 00:01, 12 March 2020

Latest revision as of 23:36, 8 March 2024

(11 intermediate revisions by 8 users not shown)

Line 1:

{{WikiProject banner shell|class=B|vital=yes|1=

{{maths rating

{{WikiProject Mathematics|importance=high}}

|field=topology

|importance=high

|class=B

|vital=

|historical=

}}

{{User:MiszaBot/config

Line 16 ⟶ 12:

}}

{{Archive box|auto=yes}}

== Accessible proof==

Line 21 ⟶ 18:

Courant and Robbins provide an accessible proof.

<small><span class="autosigned">—Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[User:198.144.199.xxx|198.144.199.xxx]] ([[User talk:198.144.199.xxx|talk]] • [[Special:Contributions/198.144.199.xxx|contribs]]) 30 August 2001</span></small><!-- Template:Unsigned -->

== 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?--[[User:192.35.35.36|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...''"

<br>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.

<br>See for example [https://mathoverflow.net/questions/202811/does-the-brouwer-fixed-point-theorem-admit-a-constructive-proof this discussion] in [[MathOverflow]]. [[User:נחי|Nachi]] ([[User talk:נחי|talk]]) 16:58, 4 February 2018 (UTC)

: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. &mdash;&nbsp;Carl <small>([[User:CBM|CBM]]&nbsp;·&nbsp;[[User talk:CBM|talk]])</small> 17:25, 4 February 2018 (UTC)

::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". [[User:נחי|Nachi]] ([[User talk:נחי|talk]]) 20:25, 4 February 2018 (UTC)

== Citation style ==

Line 50 ⟶ 32:

: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. &ndash;[[User:Deacon Vorbis|Deacon Vorbis]]&nbsp;([[User Talk:Deacon Vorbis|carbon]]&nbsp;&bull;&nbsp;[[Special:Contributions/Deacon Vorbis|videos]]) 00:32, 11 March 2020 (UTC)

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

<blockquote>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."</blockquote>

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. <!-- Template:Unsigned --><small class="autosigned">—&nbsp;Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[User:Natkuhn|Natkuhn]] ([[User talk:Natkuhn#top|talk]] • [[Special:Contributions/Natkuhn|contribs]]) 01:06, 8 May 2020 (UTC)</small> <!--Autosigned by SineBot-->

:Oops, yeah, that's a good catch. This probably deserves some looking into. &ndash;[[User:Deacon Vorbis|Deacon Vorbis]]&nbsp;([[User Talk:Deacon Vorbis|carbon]]&nbsp;&bull;&nbsp;[[Special:Contributions/Deacon Vorbis|videos]]) 01:19, 8 May 2020 (UTC)

== Did Brouwer offer the first proof for continuous functions? ==

In the book ''History of Topology'' by James on pages 273-274: "Bohl's theorem is also equivalent to the Brouwer theorem. Bohl's theorem was published in 1904, with a proof that required that f be differentiable. Brouwer published his fixed point theorem, for continuous functions on the 3-ball, in 1909. When the first proof for the n-ball, with f '''differentiable''', appeared in print a year later, in an appendix by '''J. Hadamard''' to a text by Tannery, the theorem was called the 'Brouwer Fixed Point Theorem', which suggests that the result was already famous by that time. It is not known in what year Brouwer made his discovery and, apparently, communicated it to other mathematicians in an informal manner. The first published proof of the general case, that is, for '''continuous''' functions on the n-ball, was by '''Brouwer''' himself in 1912."

In the book ''Brouwer Degree. The Core of Nonlinear Analysis'' by Mawhin on page 393: "In 1910, Jules Tannery published the second volume of the second edition of his book ‘Introduction à la théorie des fonctions de variables réelles’, for the time and still now a very modern presentation of analysis, introducing Weierstrass’ rigor in France. This volume two ended with a Note of Jacques Hadamard, connected in the following way to Tannery’s book material: 'The proof, following M. Ames, of Jordan’s theorem on closed curves without double point is based upon the concept of order of a point or, equivalently on the consideration of the variation of the argument. The generalization to the case where the dimension is larger than two is given by the Kronecker index. It is a now classical notion, mainly since the publication of the Traité d’Analyse of Mr. Picard (T. I, p. 123; T. II, p. 193). It has received new applications in various recent works. My aim is to present here some of them. All the following reasonings [...] only use the continuity of the considered functions.'"

The previous quote from Mawhin's book has a strange omission ([...]) that may indicate that Hadamard did not write down a proof for the continuous case. '''Can someone please check out if Hadamard proved the general case for differentiable functions and not for continuous functions?'''

:'''Comments.''' Please sign your posts with <nowiki>~~~~</nowiki> and, per [[WP:TPG]], add comments at the ''end'' of the talk page. Please stop edit-warring to insert your own [[WP:POV|point of view]]. There is already a discussion of the history of the FPT in the section [[Brouwer fixed-point theorem#First proofs]]. Bohl's proof applied to three dimensions. There are several historical accounts of the FPT, notably "A history of algebraic and differential topology, 1900–1960" by [[Jean Dieudonné]]. [[User:Mathsci|Mathsci]] ([[User talk:Mathsci|talk]]) 17:43, 21 February 2022 (UTC)