tag:blogger.com,1999:blog-3722233.post4438251809304202082..comments2023-03-25T10:00:22.914-05:00Comments on Computational Complexity: The very first Ramseyian TheoremLance Fortnowhttp://www.blogger.com/profile/06752030912874378610noreply@blogger.comBlogger4125tag:blogger.com,1999:blog-3722233.post-39652836911908957822016-12-26T15:29:12.630-06:002016-12-26T15:29:12.630-06:00I can mention that my initial role was helping Bil...I can mention that my initial role was helping Bill deal with that German. I translated the paper but without full command of terms whose literal translation becomes oxymoronic in current English math usage, such as "whole rational number." Then Bill and I couldn't get past one point in the proof of the Irreducibility Theorem until Mark both did a fresh translation and pushed through the obstacle.KWReganhttps://www.blogger.com/profile/09792573098380066005noreply@blogger.comtag:blogger.com,1999:blog-3722233.post-27421900419362432732016-12-21T16:39:41.418-06:002016-12-21T16:39:41.418-06:00Hilbert's Cube lemma was in 1894
Schur was BOR...Hilbert's Cube lemma was in 1894<br />Schur was BORN in 1891. Most 3 year olds know very little combinatorics.<br />Ramsey was BORN in 1903.<br />So surely when Hilbert proved his Cube Lemma he could not have know of the connections.<br /><br />But I think you mean later. But again, no. There is no evidence<br />that Hilbert knew of Ramsey's work (was it that well known?)<br />or of Schur's work. <br />The connection to Ramsey's work would have been hard to <br />figure out- in fact, while we all think of Ramsey's theroem<br />and VDW's theorem and HCL as being parts of Ramsey Theory,<br />Mathematically Ramsey's theorem and HCL do not share anything except the notion of `when you color big enough stuff there is a nice monochromatic substructure'<br /><br />Schur's theorem also- it would have been hard to see the<br />connection.<br /><br />BUT- VDW's theorem YES- VDW implies HCL. And VDW was alive<br />at the right time. But I doubt Hilbert knew of it since VDW's<br />theorem was not really out there until it was publicized by<br />Khinchin in <br /><br />Three pearls of Number THeory<br />which was published in 1952- and<br />Hilbert died in 1943.<br /><br /><br />Three pearls:<br /><br />http://plouffe.fr/simon/math/Three%20Pearls%20of%20Number%20Theory%20-%20Khinchin.pdf<br /><br /><br />All this bring us back to my motivation for the paper--- this theorem seemed to be in danger of being lost. Hilbert's lack of interest is part of that reason. <br /><br /><br /><br />GASARCHhttps://www.blogger.com/profile/06134382469361359081noreply@blogger.comtag:blogger.com,1999:blog-3722233.post-77033210130647162532016-12-21T15:23:09.781-06:002016-12-21T15:23:09.781-06:00Bill, thanks for the blog post and to you and your...Bill, thanks for the blog post and to you and your coauthors for the article. I've read through the article once and have a question based on the end of the article.<br /><br />At the end you have some comments about the context that Hilbert was working in. In addition to Hilbert's cube lemma and Ramsey's original paper (connected with math. logic) there was Schur's theorem on sum free sets. Do we know if Hilbert knew of Ramsey's work? Schur's? Did he see the connection?<br /><br />Bill, Lance, thanks for the work you've put into the CC weblog and may you and yours have happy holidays and new year!MSShttps://www.blogger.com/profile/10782385200928123717noreply@blogger.comtag:blogger.com,1999:blog-3722233.post-70451052899025199222016-12-20T10:20:48.935-06:002016-12-20T10:20:48.935-06:00This fine Villarino/Gasarch/Regan preprint ("...This fine Villarino/Gasarch/Regan preprint ("Hilbert's Proof of His Irreducibility Theorem", arXiv:1611.06303) builds largely upon the theorems and methods of Victor Puiseux. <i>Computational Complexity</i> readers can find much further information regarding Victor Puiseux' persona and works in Étienne Ghys' just-released notes <i>A Singular Mathematical Promenade</i> (<a href="http://www.ams.org/open-math-notes" rel="nofollow">AMS Open Math Notes</a>, 2016).<br /><br />Beyond their intrinsic mathematical interest, Ghys' notes are notable too for their impeccable typographic quality and overall beauty of visual design … these notes make a wonderful holiday gift for the mathematicians in your life … especially if you don't mind that <i>A Singular Mathematical Promenade</i> is released under a Creative Commons License (and hence is entirely free-as-in-freedom, both economically and informatically).<br /><br />Gift-givers who prefer the physicality of paper can take inspiration from the works of the artist Caspar David Friedrich, whose paintings Ghys' notes features prominently, including in particular Friedrich's “Wanderer above the sea of fog” (p. ii) "Tree of crows" (p. 284), and "Man and woman contemplating the moon" (p. 286). <br /><br />The Wikipedia page "Humboldtian Science" — which features Friedrich's “Wanderer above the sea of fog" — then leads us to Andrea Wulf's just released biography <i>The Invention of Nature: Alexander von Humboldt's New World</i> (2016). Wulf's history-of-natural-science very beautifully complements (as it seems to me) Puiseux' beautiful mathematics and Friedrich's beautiful paintings. That is Wulf's (purchased) book, together with Ghys' (free-as-in-freedom) AMS Open Math Note, jointly make a wonderful gift.<br /><br />Can any <i>practical</i> benefits be associated to this rich banquet of mathematical beauty-and-truth? Yah, sure, you betcha! My own interest in the Puiseux-grounded mathematics that Ghys/Villarino/Gasarch/Regan discuss is associated to the still-mysterious role(s) of algebraic singularities in the tensor-network state-spaces that nowadays propagate throughout the global STEAM enterprise (see <i>e.g.</i> comments #84 and #88 in the recent <i>Shtetl Optimized</i> essay "My quantum computing cartoon with Zach Weinersmith"). Needless to say, it will be a long time before the Friedrich-style mathematical fog entirely lifts from this vast landscape of algebraic, geometric, informatic, combinatorical, computational, practical, and dynamical structures and capacities (both classical and quantum). <br /><br />We are all of us fortunate, in this holiday season, to be so diversely gifted with shared appreciations of beauty and truth; and we are fortunate too, to be blessed with plenty of Friedrich-style fog, that throughout decades to come will no doubt lift, to slowly unveil still more beauty and truth. <br /><br />Best wishes for a happy holiday season are hereby extended to all! :)John Sidleshttps://www.blogger.com/profile/16286860374431298556noreply@blogger.com