Timeline for What was Weierstrass's counterexample to the Dirichlet Principle?
Current License: CC BY-SA 2.5
18 events
| when toggle format | what | by | license | comment | |
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| Jul 28, 2023 at 2:11 | answer | added | paul garrett | timeline score: 4 | |
| Nov 30, 2011 at 12:09 | answer | added | A. Loos | timeline score: 2 | |
| Oct 15, 2010 at 4:12 | history | edited | Andrey Rekalo |
Tag added
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| Oct 14, 2010 at 22:54 | comment | added | Will Jagy | I was unable to find your email address. If you look up mine at $$ $$ ams.org/cml $$ $$ and email me I can send you a pdf of a 22-page article by Marvin Greenberg, M.A.A. Monthly, March 2010, pages 198-219, called "Old and New Results in the Foundations of Elementary Plane Euclidean and Non-Euclidean Geometries. At least it would give you some sense of how the Archimedean field axiom fits into this, specifically what happens when it does not apply. Plus many nice references listed. | |
| Oct 14, 2010 at 20:12 | comment | added | Jeremy Shipley | Thanks, found a download of the Courant reference you gave below and am finding that particularly helpful. I'm more philosopher than mathematician. I'm onto this topic because of philosophical questions raised by Stewart Shapiro in "Categories, Structures, and the Frege-Hilbert Correspondence". My geometry and general topology are ok, but analysis (especially complex analysis) is a slog. I'm trying to say something about Hilbert's mathematical motivations for the philosophical and foundational positions he takes, but maybe you're right to suggest I'm a bit taken in by itsallconnectedism. | |
| Oct 14, 2010 at 19:56 | comment | added | Will Jagy | I don't see how to define a minimal surface in the case of $\mathbf F^3,$ where $\mathbf F$ is an arbitrary Euclidean field. It is not possible in general to define an area function for all interesting regions in $\mathbf F^2.$ You ought also to read the new 4th edition of Euclidean and Non-Euclidean Geometries by Marvin Jay Greenberg. Any similarities between the two problems seem to me, at best, expressions of Hilbert's general concern with firming up foundations after various disturbances caused by Cantor's set theory and disputes over the Axiom of Choice. | |
| Oct 14, 2010 at 19:42 | vote | accept | Jeremy Shipley | ||
| Oct 14, 2010 at 18:44 | answer | added | Andrey Rekalo | timeline score: 26 | |
| Oct 14, 2010 at 18:38 | comment | added | Jeremy Shipley | This sort of just thickens the plot for me. It is known that Dehn's notes include a unpublished proof of the Jordan curve theorem (see Guggenheimer, 1977) dated 1899 using only incidence and order axioms. Again, I'm straining at my horizons here but the minimal surface problems are posed using Jordan curves, right? I can't believe that Hilbert's concerns were purely geometric, especially because of his algebraic understanding of the axioms. | |
| Oct 14, 2010 at 17:46 | comment | added | Will Jagy | No, conterexamples to the Dirichlet Principle concern Plateau's problem mostly, initial repairs by Courant rather than Hilbert, see Richard Courant, "Dirichlet's Principle, Conformal Mapping, and Minimal Surfaces" Dover Publications, or Lectures on Minimal Surfaces, vol. 1 by J. C. C. Nitsche. The other stuff, non-Archimedean fields, is completely unrelated, see Robin Hartshorne, Geometry: Euclid and Beyond. This concerns models for non-Euclidean geometry without using continuity axioms at all, no real numbers in sight. | |
| Oct 14, 2010 at 17:36 | answer | added | Willie Wong | timeline score: 9 | |
| Oct 14, 2010 at 17:17 | history | edited | Jeremy Shipley | CC BY-SA 2.5 |
fixed Weirstrass to Weierstrass
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| Oct 14, 2010 at 17:13 | comment | added | Spencer | On why the Dirichlet principle approach to Riemann Mapping would be bad I remember Tom Korner mentioning both 'Weierstrass' and something like 'the domain obtained by taking a ball in $\mathbb{R}^3$ and pushing in the North Pole with a pencil so that it has a sharp point pointing in on itself' in close-by setences, but I don't know/can't remember if this is actually the Weierstrass example. | |
| Oct 14, 2010 at 17:12 | comment | added | Jeremy Shipley | Not in my library, so I'll have to interlibrary loan it. In the mean time I'd appreciate any help. | |
| Oct 14, 2010 at 17:07 | comment | added | Jeremy Shipley | en.wikipedia.org/wiki/Dirichlet%27s_principle | |
| Oct 14, 2010 at 17:07 | comment | added | Willie Wong | Have you looked at Monna's book Dirichlet's Principle: A Mathematical Comedy of Errors and Its Influence on the Development of Analysis? If I remember correctly he had a short description of Weierstrass's counterexample. | |
| Oct 14, 2010 at 17:05 | comment | added | Spencer | It might just be me being silly but could you be a little bit more specific as to what the exact claim that the counterexample is a counterexample to is? | |
| Oct 14, 2010 at 16:59 | history | asked | Jeremy Shipley | CC BY-SA 2.5 |