Comments on: HoTT Coq
http://www.metafilter.com/150345/HoTT-Coq/
Comments on MetaFilter post HoTT CoqTue, 09 Jun 2015 23:43:07 -0800Tue, 09 Jun 2015 23:43:07 -0800en-ushttp://blogs.law.harvard.edu/tech/rss60HoTT Coq
http://www.metafilter.com/150345/HoTT-Coq
<a href="https://www.quantamagazine.org/20150519-will-computers-redefine-the-roots-of-math/">Univalent Foundations Redefines Mathematics</a> - "When a legendary mathematician found a mistake in his own work, he embarked on a computer-aided quest to eliminate human error. To succeed, he has to <a href="http://homotopytypetheory.org/book/">rewrite the century-old rules</a> underlying all of mathematics." (<a href="http://www.metafilter.com/126041/Computerized-Math-Formal-Proofs-andamp-Alternative-Logic">previously</a>) <br /><br /><a href="http://www.ams.org/notices/201309/rnoti-p1164.pdf">Voevodsky's Univalence Axiom in Homotopy Type Theory</a>
<blockquote>One of Voevodsky's goals (<a href="https://intelligence.org/2014/02/21/john-baez-on-research-tactics/">as we understand it</a>) is that, in a not too distant future, mathematicians will be able to verify the correctness of their own papers by working <a href="http://math.andrej.com/2014/01/13/univalent-foundations-subsume-classical-mathematics/">within the system of univalent foundations</a> formalized in a proof assistant and that doing so will become natural even for pure mathematicians (the same way that most mathematicians now typeset their own papers in TeX). We believe that this aspect of the <a href="http://ncatlab.org/nlab/show/univalence+axiom">univalent foundations program</a> distinguishes it from other approaches to foundations by providing a practical utility for the working mathematician.</blockquote>
-<a href="http://www.science4all.org/le-nguyen-hoang/type-theory/">Type Theory: A Modern Computable Paradigm for Math</a>
-<a href="http://www.science4all.org/le-nguyen-hoang/homotopy-type-theory/">Homotopy Type Theory and Higher Inductive Types</a>
-<a href="http://www.science4all.org/le-nguyen-hoang/univalence/">Univalent Foundations of Mathematics</a>
also btw...
-<a href="https://www.youtube.com/watch?v=QNznD9hMEh0">James Simons interview</a>
-<a href="https://mathematicswithoutapologies.wordpress.com/2015/05/13/univalent-foundations-no-comment/">Univalent Foundations: "No Comment."</a> (<a href="http://math-frolic.blogspot.com/2015/05/set-theory-type-theory-hott-univalent.html">via</a>)
-<a href="https://terrytao.wordpress.com/career-advice/there%E2%80%99s-more-to-mathematics-than-rigour-and-proofs/">There's more to mathematics than rigour and proofs</a>post:www.metafilter.com,2015:site.150345Tue, 09 Jun 2015 22:40:35 -0800kliulessVladimirVoevodskymathmathsmathematicsprogramprogrammingprooflogicinductionunivalanceaxiomunivalentfoundationsgroupsetcategoryhomotopytypetheorycoqcodecomputationresearchrealnumbersnumbersystemdefinitionisomorphicidentityfunctionequivalenceequalsequalityequationBy: brambleboy
http://www.metafilter.com/150345/HoTT-Coq#6080795
Thanks. This goes on my "must read" list... though I doubt it will be a fast read!comment:www.metafilter.com,2015:site.150345-6080795Tue, 09 Jun 2015 23:43:07 -0800brambleboyBy: a lungful of dragon
http://www.metafilter.com/150345/HoTT-Coq#6080796
<em> And this reminds me of a quite different concern I have — this is MH speaking now, not the panelists — about computer-verifiable proofs: the prospect of technological lock-in. </em>
If there is an isomorphism that translates an existing human-written proof to a computer-written proof, isn't the same possible in the other direction, which eliminates "lock-in"?comment:www.metafilter.com,2015:site.150345-6080796Tue, 09 Jun 2015 23:46:45 -0800a lungful of dragonBy: NMcCoy
http://www.metafilter.com/150345/HoTT-Coq#6080803
Look at the proof of the four-color theorem for an example of the kind of thing they're concerned about, I think.comment:www.metafilter.com,2015:site.150345-6080803Wed, 10 Jun 2015 00:10:29 -0800NMcCoyBy: ethansr
http://www.metafilter.com/150345/HoTT-Coq#6080808
As I understand it, these proofs aren't exactly computer written, they are written in cooperation with a proof assistant. Coq is probably the one most associated with HoTT, but Nuprl is quite interesting as well. You provide input and then the proof assistant uses various "tactics" to try to prove what you've written based on existing proofs. When it gets stuck it asks for help by printing the statement it is stuck on, you give a hint and it tries again.
So they really are human readable, just requires a different set of skills. With a bit of UX work it may even end up being more human friendly than the classical approach.
Bonus: <a href="http://www.ams.org/notices/200811/tx081101382p.pdf">Four-Color Theorem via Coq</a>comment:www.metafilter.com,2015:site.150345-6080808Wed, 10 Jun 2015 00:16:35 -0800ethansrBy: DoctorFedora
http://www.metafilter.com/150345/HoTT-Coq#6080819
I hate this FPP's title, but also I love this FPP's titlecomment:www.metafilter.com,2015:site.150345-6080819Wed, 10 Jun 2015 01:12:17 -0800DoctorFedoraBy: No-sword
http://www.metafilter.com/150345/HoTT-Coq#6080831
I'm sorry, Doctor Fedora, Coq won't allow that statement as it stands. Univalence permits only love and the successor to love!comment:www.metafilter.com,2015:site.150345-6080831Wed, 10 Jun 2015 01:42:50 -0800No-swordBy: Nanukthedog
http://www.metafilter.com/150345/HoTT-Coq#6080911
Math Nerd!comment:www.metafilter.com,2015:site.150345-6080911Wed, 10 Jun 2015 05:28:34 -0800NanukthedogBy: 7segment
http://www.metafilter.com/150345/HoTT-Coq#6080983
<em>Set theory begins with the set containing nothing — the null set — which is used to define the number zero.</em>
Wait, are we talking about set theory, ZFC, or talking so far down to the reader as to imply they're the same thing? Sloppy article, f----- would not read again.comment:www.metafilter.com,2015:site.150345-6080983Wed, 10 Jun 2015 06:45:21 -08007segmentBy: Sangermaine
http://www.metafilter.com/150345/HoTT-Coq#6080989
Can someone explain this for idiots like me? What is this, why is it important, and is it as big of a deal as the articles seem to make it?comment:www.metafilter.com,2015:site.150345-6080989Wed, 10 Jun 2015 06:51:22 -0800SangermaineBy: destrius
http://www.metafilter.com/150345/HoTT-Coq#6080999
It's a bit like trying to find the Theory of Everything for mathematics, a foundation upon which everything else is built. Bertrand Russell tried to do it using set theory but failed. Type theory might be the answer.
An interesting way my lecturer put it, during his class on type theory, is that it turns out that mathematics (which can be said to be mostly founded upon set theory) is actually a sub-discipline of computer science (which via programming language theory is founded upon type theory / category theory).
<small>[disclaimer: IANA mathematician, but I'm somewhat familiar with type theory in relation to programming language theory and logic]</small>comment:www.metafilter.com,2015:site.150345-6080999Wed, 10 Jun 2015 06:58:57 -0800destriusBy: perihare
http://www.metafilter.com/150345/HoTT-Coq#6081083
The argument above is the <a href="http://en.wikipedia.org/wiki/Curry–Howard_correspondence">Curry-Howard isomorphism</a>, which draws a parallel between writing a program that produces a value with a specific type and creating a proof of a mathematical theorem represented by that type.comment:www.metafilter.com,2015:site.150345-6081083Wed, 10 Jun 2015 07:48:29 -0800perihareBy: symbioid
http://www.metafilter.com/150345/HoTT-Coq#6081329
I haven't read the article yet, but I've heard of Computer Aided Proofs and Coq in particular. At my new job, one of the interns is going to school for math and I asked him if he'd heard of Coq, which he hadn't yet, so I got to turn him on to the idea. Now, ask me what any of it means or how to use it and I'm lost, but hey - it made me feel a little smart for a few seconds at least ;)
Speaking of Type theory and Set theory... What's the difference between them? I thought that in many ways they were the same thing? I suppose I'll look into the Curry-Howard isomorphism. I'm sure I've seen it before in my meanderings, but didn't grok the details enough.comment:www.metafilter.com,2015:site.150345-6081329Wed, 10 Jun 2015 09:53:28 -0800symbioidBy: logopetria
http://www.metafilter.com/150345/HoTT-Coq#6081783
Hey, this is what I'm working on right now! Sorry to self-link, but if you're interested in learning about this stuff we have a <a href="http://www.bristol.ac.uk/media-library/sites/arts/research/homotopy-type/documents/hott-primer-part-one.pdf">Primer on Homotopy Type Theory</a> available as a free download at the webpage for our <a href="http://www.bristol.ac.uk/homotopy-type-theory/">research project</a> at the University of Bristol. Whereas the <a href="http://homotopytypetheory.org/book/">book</a> linked in the post (produced by the group at IAS Princeton mentioned in the Quanta article) is very technically advanced, our Primer is intended to be a very slow and gentle introduction to the elementary ideas of the subject and doesn't assume any prior knowledge. We spell out the core of the language of HoTT and explain how to think about it, and show how to use it to do some basic logic.comment:www.metafilter.com,2015:site.150345-6081783Wed, 10 Jun 2015 14:11:23 -0800logopetria