Caltrops - The Knuckle Shuffle





	[quote name="mark"][quote name="Zseni"] Guys like this have a precisely opposite attitude towards math - opposite from the correct path, which is to say mine. Here is a mathematician who wants math to be only about math, and wants to extend the outside world's methodology into the math bubble. I think everything lies inside the math bubble, and we are best server by extending the methodology of math <i>out</i>. But this sentence in particular - is he the janitor of the AMS?[/quote] Well, first note that this article was written on April 1st, 2002 so it might be a little facetious, but anyway: The line in the sand has been drawn: Do you accept that Appel and Haken's "Every planar map is four colorable" contains a legitimate proof? The fact it isn't traditional has thrown the philosophy of math into the harshest glare since Hilbert was shown to be the leader of a movement that had no purpose. Hibert's downfall/Godel's rise, that math is incomplete, shows that our attempts to justify all of mathematics in it's own terms are failed (or very limited) to begin with. While we have largely pretended otherwise, math in-itself cannot describe all of mathematics. In a great <a href="http://www.ams.org/notices/199701/comm-rota.pdf">talk</a> by Gian-Carlo Rota, he notes that Hilbert (I'm picking on him because he's both an easy target and he one of the greatest mathematicians ever) got almost all of his famous proofs wrong. They were so persuasive that they convinced other mathematicians on grounds that, if we believe that formalism is the only answer, must be wrong. But the proofs have been corrected and Hilbert's theorems stand so clearly something was right about them ther first time round. So while you're method isn't wrong, but I'm pretty sure it isn't right either. I'm sympathetic to it in this regard: accepting conjectures because we haven't found a counter-example seems scary and I like to be reassured by proofs, but then again, why can't we just accept Goldbach's conjecture? It's valid up to like 400 trillion or something and seems awfully difficult to prove. Experimental math has been hugely successful this last century and computer proofs are likely to become more common as many problems seem just too hard for humans to keep straight. In short: I believe 4 colouring is valid. It is not a traditional proof. I am forced to believe that the traditions are wrong or overly limited. mark[/quote]