tag:blogger.com,1999:blog-11295132.post5859707800583893899..comments2019-07-19T01:35:14.385-07:00Comments on A Neighborhood of Infinity: A pictorial proof of the hairy ball theoremsigfpehttp://www.blogger.com/profile/08096190433222340957noreply@blogger.comBlogger5125tag:blogger.com,1999:blog-11295132.post-9405893920737826292016-12-23T15:11:57.795-08:002016-12-23T15:11:57.795-08:00Thanks a lot for this article! It helps me underst...Thanks a lot for this article! It helps me understood the intuition behind the otherwise impenetrable proofs. I found that this situation happens quite often in Math. The formal proofs just look like a bunch of symbols randomly put together. There are a few learning aids, they all talk about the same thing, the same thing that I don't understand. So after I have really determined to put everything together, I really got it. And then I find the concept, the core, to be quite simple. And then I realize that all the formal rigor was to say things exactly. However I think it's meaningful to put the beautiful core idea upfront, to get people interested. If we put rigor first, 95% people will give up before they actually understand the theorem. <br /><br />We need more articles like this for the modern math!<br /><br /><br /><br />Justin Z.https://www.blogger.com/profile/08878757829659893274noreply@blogger.comtag:blogger.com,1999:blog-11295132.post-86671348688420557272012-12-01T11:36:40.975-08:002012-12-01T11:36:40.975-08:00A typical constructive variant of this sort of the...A typical constructive variant of this sort of theorem would be say that for every epsilon there is a vector whose magnitude is less than epsilon, which is usually strong enough to work with.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-11295132.post-13494761668579086752012-11-18T20:39:00.577-08:002012-11-18T20:39:00.577-08:00I was thinking about constructive versions of this...I was thinking about constructive versions of this when I hinted that there's a procedure for searching for zeros based on computing winding numbers around various regions. Like the intermediate value theorem I'd expect it to fail in a constructively. But many of these kinds of results have discrete versions that I suspect do still work constructively and there may be a version of this winding number argument that still works in that case.sigfpehttps://www.blogger.com/profile/08096190433222340957noreply@blogger.comtag:blogger.com,1999:blog-11295132.post-42379679847805735182012-11-18T19:17:56.568-08:002012-11-18T19:17:56.568-08:00I bet this theorem fails in the effective topos, w...I bet this theorem fails in the effective topos, where there is a continuous map on the disc which has no fixed points (so it violates Brouwer's fixed point theorem).Andrej Bauerhttp://andrej.com/noreply@blogger.comtag:blogger.com,1999:blog-11295132.post-87694540666275813112012-11-18T19:16:12.540-08:002012-11-18T19:16:12.540-08:00This comment has been removed by the author.Profesor Umnikhttps://www.blogger.com/profile/03377063988492869441noreply@blogger.com