tag:blogger.com,1999:blog-11295132.post3213030729419630796..comments2018-05-24T12:10:52.808-07:00Comments on A Neighborhood of Infinity: The Infinitude of the PrimesDan Piponihttps://plus.google.com/107913314994758123748noreply@blogger.comBlogger7125tag:blogger.com,1999:blog-11295132.post-43176968373902051102011-08-01T05:55:17.758-07:002011-08-01T05:55:17.758-07:00@sigfpe Andrej has definitely been one of my "...@sigfpe Andrej has definitely been one of my "people to watch" for a while!Marc Hamannhttps://www.blogger.com/profile/11526878435261617011noreply@blogger.comtag:blogger.com,1999:blog-11295132.post-42528490214435088972011-07-31T15:02:42.369-07:002011-07-31T15:02:42.369-07:00@Marc You can get an idea of how surprising this s...@Marc You can get an idea of how surprising this stuff is for mathematicians by the score I got here: http://mathoverflow.net/questions/19152/why-is-a-topology-made-up-of-open-sets<br /><br />I'm convinced there's a pile of interesting stuff to be mined from this when applied to physics, connected to some of the things Andrej Bauer says here: http://math.andrej.com/2008/08/13/intuitionistic-mathematics-for-physics/sigfpehttps://www.blogger.com/profile/08096190433222340957noreply@blogger.comtag:blogger.com,1999:blog-11295132.post-9967183186092587072011-07-31T14:51:36.610-07:002011-07-31T14:51:36.610-07:00@sigfpe
Thanks for the thoughtful response!
It s...@sigfpe<br /><br />Thanks for the thoughtful response!<br /><br />It sounds like "profound" is a partial synonym with "surprising", and surprising is, of course, always a bit subjective.<br /><br />I'm probably warped by the fact that I got into mathematics backwards: I went from computation to category theory, and only then started to learn set theory, topology, etc. I probably have an inverted sense of surprise in this regard. (I was more shocked when I discovered how much of mathematics goes on with no conscious consideration of computation. ;-) )Marc Hamannhttps://www.blogger.com/profile/11526878435261617011noreply@blogger.comtag:blogger.com,1999:blog-11295132.post-4709883491716510002011-07-31T13:37:30.243-07:002011-07-31T13:37:30.243-07:00@Marc Interesting.
I went through my mathematical...@Marc Interesting.<br /><br />I went through my mathematical education studying courses in point-set topology, algebraic topology and homotopy theory and at no point did anyone point out to me that there was any connection to computation. When I finally came across it a few years back I was blown away. I find it amazing to see the same language that I'd used to describe the deformation of rubber sheets being used to talk about computation. A connection between two different areas of mathematics like this is an example of what I'd call "profound". That's what a large part of mathematics is about, finding ever deeper structures that are found in more and more places.<br /><br />Along similar lines I'd say that the Curry-Howard isomorphism is also profound. It's another example of how the same language appear in two different places. In the case of Curry-Howard it's less profound, after all, a priori we expect many connections between logic and computing. But connecting rubber sheets and the discrete actions of machines is much more surprising.sigfpehttps://www.blogger.com/profile/08096190433222340957noreply@blogger.comtag:blogger.com,1999:blog-11295132.post-91999906890126945482011-07-31T08:19:19.295-07:002011-07-31T08:19:19.295-07:00@sigfpe I'm curious about your perception that...@sigfpe I'm curious about your perception that the connection between topology and computation is "profound".<br /><br />My own feeling is that, depending on your mathematical ontology, the connection is either just a "pun", or an identity, i.e. that topology is computation seen through a glass darkly.<br /><br />The thing that links them is fuzzy membership "along the edges". In computation, this is a direct consequence of the halting problem; in topology because of the limitation to a finite number of intersections.<br /><br />If you believe that mathematical objects "exist" independent of computations, then I would say this is a "pun". If you believe, as I do, that only mathematical objects that can be computed (perhaps for a slightly extended notion of "computed") exist, then the two notions are identical, and we just have two formalisms for the same phenomenon (useful perhaps, but not profound).<br /><br />Or perhaps, we just find different things "profound". ;-)Marc Hamannhttps://www.blogger.com/profile/11526878435261617011noreply@blogger.comtag:blogger.com,1999:blog-11295132.post-23691482363965981262011-07-25T11:45:25.474-07:002011-07-25T11:45:25.474-07:00@polus Yes, I know that paper, though I haven'...@polus Yes, I know that paper, though I haven't read all of it. I love the connection between computation and topology. I think it's really profound and it should be better known. These days, any time I see some point set topology, my first question is "how does this relate to computation?".sigfpehttps://www.blogger.com/profile/08096190433222340957noreply@blogger.comtag:blogger.com,1999:blog-11295132.post-62384133984935275232011-07-25T11:13:08.409-07:002011-07-25T11:13:08.409-07:00Connection between logic and topology is discusse...Connection between logic and topology is discussed in that nice paper: http://lambda-the-ultimate.org/node/4037<br /><br />(Maybe you already know about it)poluxhttps://www.blogger.com/profile/17812487450262251135noreply@blogger.com