From mboxrd@z Thu Jan 1 00:00:00 1970 Return-Path: X-Spam-Checker-Version: SpamAssassin 3.4.2 (2018-09-13) on inbox.vuxu.org X-Spam-Level: X-Spam-Status: No, score=-0.9 required=5.0 tests=DKIM_SIGNED,DKIM_VALID, DKIM_VALID_AU,DKIM_VALID_EF,FREEMAIL_FORGED_FROMDOMAIN,FREEMAIL_FROM, HEADER_FROM_DIFFERENT_DOMAINS,HTML_MESSAGE,MAILING_LIST_MULTI, RCVD_IN_DNSWL_NONE autolearn=ham autolearn_force=no version=3.4.2 Received: from mail-wr1-x439.google.com (mail-wr1-x439.google.com [IPv6:2a00:1450:4864:20::439]) by inbox.vuxu.org (OpenSMTPD) with ESMTP id e5c23635 for ; Mon, 4 Nov 2019 18:43:08 +0000 (UTC) Received: by mail-wr1-x439.google.com with SMTP id a15sf10868206wrr.0 for ; Mon, 04 Nov 2019 10:43:08 -0800 (PST) ARC-Seal: i=2; a=rsa-sha256; t=1572892988; cv=pass; d=google.com; s=arc-20160816; b=QCUKhcW2lsdSoM8g1HlHp8TIPqenLTbt40+kBBxSoHbb7HCE4y/brj8FI/kvF7w19a CP/TKb9X9XuuU+5gGKrcmr170aCKuTQfCg6QbO4wf2X5w7LT1DgA36mss1n9LMUVjS02 JT+dmGhAV6DyEuOG/7xzO8rtrVYJ8NAO1cQ8sxOptuZQACCUnImOmRgrWVQ+4JC78D/I 4vDAOLsAv65qDRdLd6ex49kc6XT7uNQ6W5y8u1zDEq2fPWGNntLMAag2DvdCNCiQQMAR /B2857r8otsxapemEaqXWB5/d2p1FFiRyVGSncQ9o0NegZbIPgXKR7en0hhh+AJSt0eL 82RA== ARC-Message-Signature: i=2; a=rsa-sha256; c=relaxed/relaxed; d=google.com; s=arc-20160816; h=list-unsubscribe:list-archive:list-help:list-post:list-id :mailing-list:precedence:cc:to:subject:message-id:date:from :in-reply-to:references:mime-version:sender:dkim-signature :dkim-signature; bh=bQUYXWRMw0CWk+iK/0AKCsrBysMHfz1JrdGQkqeCfpw=; b=LLYWQe9cLq/gH5Vk2zWtg37KrXgap5ZP0d+ZKGiPoFeP6mSufaspdS2RYw5zSp9BKm slk36+/mjrIJU0L9mzn5JRtTEZmSQbVmNh0XAvQ5lJy3fxr+75DiOX7+QCsziZHoOmjK 3XouNi+/wVW2vOxJBjQXQe4r9DJF1ohstpO2n+Ze8lab4yo/M86KgaXtT4hf0ijTMe5P gl2VI5d6v4rusKTT6GObbr/mzxgcGEZiCafdL6QUGNo1tVC8OG5aWB9gF4u3NA29d5K2 eaysY8j2YLXgTaVvkjY8zwvEu4W5iGYVMMUJS6zQwJZiNwXArouRqlcs9X1QxvBU2zKv lApQ== ARC-Authentication-Results: i=2; gmr-mx.google.com; dkim=pass header.i=@gmail.com header.s=20161025 header.b=JleMM0o0; spf=pass (google.com: domain of kevin.m.buzzard@gmail.com designates 2a00:1450:4864:20::22e as permitted sender) smtp.mailfrom=kevin.m.buzzard@gmail.com; dmarc=pass (p=NONE sp=QUARANTINE dis=NONE) header.from=gmail.com DKIM-Signature: v=1; a=rsa-sha256; c=relaxed/relaxed; d=googlegroups.com; s=20161025; h=sender:mime-version:references:in-reply-to:from:date:message-id :subject:to:cc:x-original-sender:x-original-authentication-results :precedence:mailing-list:list-id:list-post:list-help:list-archive :list-unsubscribe; bh=bQUYXWRMw0CWk+iK/0AKCsrBysMHfz1JrdGQkqeCfpw=; b=X9JBYKtongqb7Md6OScTrsuSRL/8Akm8ifxUIzZICiL1HBgxikXcV0rLOQtFa7Kc5x mMVJUDoCLNJCMO64qE3Vt4f2QU03Yyf8/14i/UwZip+CJncbz4XuCWFVuetk5ttDAhMK 6Wm9zjiK94brHDcLgONhVf5JH/brWT0Bmx/fK/iWV5mbwmB5tcsTtbpIr7EYuojHuGOq VuafO9i6Cyf+fbpC4tH7LSRH3MnyIAYCZUL9QgY6gqKc7YuGiVC4q3nZN+phklTUk3Jj pPWagD92D+jw+n0rUSDOkKmlpnrpybdfWFG6pRvM9KrDbkffJuPt6PGpGImN8/yRdPOf Cpfg== DKIM-Signature: v=1; a=rsa-sha256; c=relaxed/relaxed; d=gmail.com; s=20161025; h=mime-version:references:in-reply-to:from:date:message-id:subject:to :cc:x-original-sender:x-original-authentication-results:precedence :mailing-list:list-id:list-post:list-help:list-archive :list-unsubscribe; bh=bQUYXWRMw0CWk+iK/0AKCsrBysMHfz1JrdGQkqeCfpw=; b=qwt6YFw9Tp1/m+WaKMchVWvNt4YYK+5K5J/O4WR53tpsqH9IghU83fn660hnzS3uxz iJOPmU4p+nrfqGUPEIPlFGCUHhcG4e4uwsFGzpD/g3UpQ2Zw3LE4/BPzAGSEGfST+WIT a9q9BhpNNc7Y8RHtgzfRnOFD8spNZZtagfiOiKNVsjMEcfrWa0LHC5ahfMhDMAgM3E1D V7ok3n1VIcRTxcBOTGBVbNCLMJH9QrHTOvtP8R1aNwuGrbWaoCsm4ABvGnlabkd/Os/G 8SeopRYOJFdZiQHOhcGgxlXn/V0I3PD3DR9KCuP+s2ECXGdRhuGhJyMIq9JG4/SiRk8q p9Cg== X-Google-DKIM-Signature: v=1; a=rsa-sha256; c=relaxed/relaxed; d=1e100.net; s=20161025; h=sender:x-gm-message-state:mime-version:references:in-reply-to:from :date:message-id:subject:to:cc:x-original-sender :x-original-authentication-results:precedence:mailing-list:list-id :x-spam-checked-in-group:list-post:list-help:list-archive :list-unsubscribe; bh=bQUYXWRMw0CWk+iK/0AKCsrBysMHfz1JrdGQkqeCfpw=; b=uONPbM+FnTj7ny/F5/+v1meLdM+VZ0oy01oA1KRVvDh9/LpxRv41b/6XPguyfRhBxh SBMXLrV4NeS9ylsaf60+L+bsXsV1aRhLFArdBwiGfGciV1G5p7uHn0QOGDgk3id7CYHA XO48nfUR+JsoujXBMJDWVU/OdV3jzHtFdfKJnfiq74Ue9hFEHZyOeo3CfjR9D6ISQXrP kZn4g5fhVt5d3tKdmEM5z7lFm5JelgbNqSpzRIcfMOe6lkbh6S656G6YX0xaof2XFDDS gseDbMF7Zmq709Fys31DXwID9z9ORSZOfTHzS4RfcT4EPKhWn5aJr078L5d5m/pRMhSA OQxA== Sender: homotopytypetheory@googlegroups.com X-Gm-Message-State: APjAAAWm/zFvG7Ua0xrCh7sp15Yoj5BDrJ49uKTxWJPLp1wdK0HG47dq Vbid5nvhj2dz9cHctSEOpWQ= X-Google-Smtp-Source: APXvYqwrUd8hErTdHF01De855qBxJLCGlvOLfMgmxarRrq5hx1BhANcD5eerJA/Hn5s+COE/hP+Nsg== X-Received: by 2002:a5d:4382:: with SMTP id i2mr25068741wrq.387.1572892988307; Mon, 04 Nov 2019 10:43:08 -0800 (PST) X-BeenThere: homotopytypetheory@googlegroups.com Received: by 2002:a1c:6085:: with SMTP id u127ls102972wmb.5.gmail; Mon, 04 Nov 2019 10:43:07 -0800 (PST) X-Received: by 2002:a05:600c:21c4:: with SMTP id x4mr449363wmj.172.1572892987504; Mon, 04 Nov 2019 10:43:07 -0800 (PST) ARC-Seal: i=1; a=rsa-sha256; t=1572892987; cv=none; d=google.com; s=arc-20160816; b=QRMO+XRFasnZLWRNHsNCkz7llyPdP0RB/lKVUyvGTynmWkqR5/19vzTV2xmdwwz10U GLF6EbHOT4GFbZBdm4j5PxcwBiqccPjDQNRy2cjhH82fg3DYMK/jiq68hYVJcjdLLJng GFA+yBmS9C/bVLQ+uVa1A6N9U1xOmAM1huaEO4vCs40Qan/HNUvXIiT/09c1Zhf6kR7Z x5uPOkwzK15BYpS85+TVK94ArpP7YknAQprPQ8OPHm4HUOZPxLFmum+oAfYuC7ZFvmIS MkMC+JTWKKIq4R3XFNNwyFpicNWsAQVx3oXtsdX6nhGPJG+/KFBMzv0Srz1NbZSVEarg zRNQ== ARC-Message-Signature: i=1; a=rsa-sha256; c=relaxed/relaxed; d=google.com; s=arc-20160816; h=cc:to:subject:message-id:date:from:in-reply-to:references :mime-version:dkim-signature; bh=ZJMGanv9hrW/W027cVBUhjFNf5Jmekhh04yxX8smKlo=; b=1DeDm638pmYxdCPlRmqpbQ7vAOgERuNrGAHMvyquvTzHM5yEhYhcL3M40Rcw7+z10I kHZEkaY7lwGs/RqNZTl+BE5NyJ+mnXdzYYombuX1hDg5MNcVwMPVOiSL4ZUOCWABQXhD dNujk8sr4rM9SHtRYiExTRA1xnDIPyBe8IWUqkORn2aX//BUxlP8gv/A3PAuLPYRXSog rBhuA9l8fgNxLvsXkGm9V8YpRm4KqMc/VYhj1xyE5BoJ/VcoDuqpxPY9lK1QkVymbOPh 61Orb4PCkOgoI53q63mjF9lqXCNzmvtKQOrw3SJnVWlK9+FsfwyisBVxIKDP4K6p6xZ1 aV6Q== ARC-Authentication-Results: i=1; gmr-mx.google.com; dkim=pass header.i=@gmail.com header.s=20161025 header.b=JleMM0o0; spf=pass (google.com: domain of kevin.m.buzzard@gmail.com designates 2a00:1450:4864:20::22e as permitted sender) smtp.mailfrom=kevin.m.buzzard@gmail.com; dmarc=pass (p=NONE sp=QUARANTINE dis=NONE) header.from=gmail.com Received: from mail-lj1-x22e.google.com (mail-lj1-x22e.google.com. [2a00:1450:4864:20::22e]) by gmr-mx.google.com with ESMTPS id w10si243797wru.4.2019.11.04.10.43.07 for (version=TLS1_3 cipher=TLS_AES_128_GCM_SHA256 bits=128/128); Mon, 04 Nov 2019 10:43:07 -0800 (PST) Received-SPF: pass (google.com: domain of kevin.m.buzzard@gmail.com designates 2a00:1450:4864:20::22e as permitted sender) client-ip=2a00:1450:4864:20::22e; Received: by mail-lj1-x22e.google.com with SMTP id m9so18785248ljh.8 for ; Mon, 04 Nov 2019 10:43:07 -0800 (PST) X-Received: by 2002:a2e:9112:: with SMTP id m18mr20524568ljg.75.1572892987033; Mon, 04 Nov 2019 10:43:07 -0800 (PST) MIME-Version: 1.0 References: In-Reply-To: From: Kevin Buzzard Date: Mon, 4 Nov 2019 18:42:50 +0000 Message-ID: Subject: Re: [HoTT] Why did Voevodsky find existing proof assistants to be 'impractical'? To: Michael Shulman Cc: David Roberts , Bas Spitters , homotopytypetheory , Nicolas Alexander Schmidt Content-Type: multipart/alternative; boundary="000000000000c30d2c059689af8c" X-Original-Sender: kevin.m.buzzard@gmail.com X-Original-Authentication-Results: gmr-mx.google.com; dkim=pass header.i=@gmail.com header.s=20161025 header.b=JleMM0o0; spf=pass (google.com: domain of kevin.m.buzzard@gmail.com designates 2a00:1450:4864:20::22e as permitted sender) smtp.mailfrom=kevin.m.buzzard@gmail.com; dmarc=pass (p=NONE sp=QUARANTINE dis=NONE) header.from=gmail.com Precedence: list Mailing-list: list HomotopyTypeTheory@googlegroups.com; contact HomotopyTypeTheory+owners@googlegroups.com List-ID: X-Google-Group-Id: 1041266174716 List-Post: , List-Help: , List-Archive: , --000000000000c30d2c059689af8c Content-Type: text/plain; charset="UTF-8" On Sun, 3 Nov 2019 at 19:13, Michael Shulman wrote: > But does univalence a la Book HoTT *actually* make it easier to reason > about such things? I think this is a really interesting and important question. I guess David was referring to my scheme fail of 2018. I wanted to formalise the notion of a scheme a la Grothendieck and prove that if R was a commutative ring then Spec(R) was a scheme [I know it's a definition, but many mathematicians do seem to call it a theorem, in our ignorance]. I showed an undergraduate a specific lemma in ring theory ( https://stacks.math.columbia.edu/tag/00EJ) and said "that's what I want" and they formalised it for me. And then it turned out that I wanted something else: I didn't have R_f, I had something "canonically isomorphic" to it, a phrase we mathematicians like to pull out when the going gets tough and we can't be bothered to check that any more diagrams commute. By this point it was too late to turn back, and so I had to prove that 20 diagrams commuted and it wasn't much fun. I then got an MSc student to redo everything using universal properties more carefully in Lean and it worked like a dream https://github.com/ramonfmir/lean-scheme. A lot of people said to me at the time "you wouldn't have had this problem if you'd been using HoTT instead of DTT" and my response to this is still the (intentionally) provocative "go ahead and define schemes and prove that Spec(R) is a scheme in some HoTT system, and show me how it's better; note that we did have a problem, but we solved it in DTT". I would be particularly interested to see schemes done in Arend, because it always felt funny to me using UniMath in Coq (and similarly it feels funny to me to do HoTT in Lean 3 -- in both cases it could be argued that it's using a system to do something it wasn't designed to do). I think it's easy to theorise about this sort of thing but until it happens in practice in one or more of the HoTT systems I don't think we will understand the issue properly (or, more precisely, I don't think I will understand the issue properly). I have had extensive discussions with Martin Escardo about HoTT and he has certainly given me hope, but on the Lean chat I think people assumed schemes would be easy in Lean (I certainly did) and then we ran into this unexpected problem (which univalence is probably designed to solve), so the question is whether a univalent type theory runs into a different unexpected problem -- you push the carpet down somewhere and it pops up somewhere else. I know this is a HoTT list but the challenge is also open to the HOL people like the Isabelle/HOL experts. In contrast to HoTT theories, which I think should handle schemes fine, I think that simple type theory will have tremendous problems defining, for example, tensor products of sheaves of modules on a scheme, because these are dependent types. On the other hand my recent ArXiv paper with Commelin and Massot https://arxiv.org/abs/1910.12320 goes much further and formalises perfectoid spaces in dependent type theory. I would like the people on this list to see this as a challenge. I think that this century will see the rise of the theorem prover in mathematics and I am not naive enough to think that the one I currently use now is the one which is guaranteed to be the success story. Voevodsky was convinced that univalence was the right way to do modern mathematics but I'm doing it just fine in dependent type theory and now he's gone I really want to find someone who will take up the challenge and do some scheme theory in HoTT, but convincing professional mathematicians to get interested in this area is very difficult, and I speak as someone who's been trying to do it for two years now [I recommend you try the undergraduates instead, anyone who is interested in training people up -- plenty of undergraduates are capable of reading the definition of a scheme, if they know what rings and topological spaces are] To get back to the original question, my understanding was that Voevodsky had done a bunch of scheme theory and it had got him a Fields medal and it was this mathematics which he was interested in at the time. He wanted to formalise his big theorem, just like Hales did. Unfortunately he was historically earlier, and his mathematics involved far more conceptual objects than spheres in 3-space, so it was a much taller order. All the evidence is there to suggest that over the next 15 or so years his interests changed. The clearest evidence, in my mind, is that there is no definition of a scheme in UniMath. Moreover his story in his Cambridge talk https://www.newton.ac.uk/seminar/20170710113012301 about asking Suslin to reprove one of his results without using the axiom of choice (46 minutes in) kind of shocked me -- Suslin does not care about mathematics without choice, and the vast majority of mathematicians employed in mathematics departments feel the same, although I'm well aware that constructivism is taken more seriously on this list. I think it is interesting that Voevodsky failed to prove a constructive version of his theorem, because I think that some mathematics is better off not being constructive. It is exactly the interaction between constructivism and univalence which I do not understand well, and I think that a very good way to investigate it would be to do some highly non-constructive modern mathematics in a univalent type theory. Kevin PS many thanks to the people who have emailed me in the past telling me about how in the past I have used "HoTT", "univalence", "UniMath", interchangeably and incorrectly. Hopefully I am getting better but I am still keen to hear anything which I'm saying which is imprecise or incorrect. > It allows us to write "=" rather than "\cong", but > to construct such an equality we have to construct an isomorphism > first, and to *use* such an equality we have to transport along it, > and then we get lots of univalence-redexes that we have to manually > reduce away. My experience formalizing math in HoTT/Coq is that it's > much easier if you *avoid* turning equivalences into equalities except > when absolutely necessary. (I don't have any experience formalizing > math in a cubical proof assistant, but in that case at least you > wouldn't have to manually reduce the univalence-redexes -- although it > seems to me you'd still have to construct the isomorphism before you > can apply univalence to make it an equality.) > > On Sun, Nov 3, 2019 at 3:57 AM David Roberts > wrote: > > > > Forget even higher category theory. Kevin Buzzard now goes around > telling the story of how even formally proving (using Lean) things in > rather elementary commutative algebra from EGA that are stated as > equalities was not obvious: the equality is really an isomorphism arising > from a universal property. Forget trying to do anything motivic, when > algebra is full of such equalities. This is not a problem with univalence, > of course. > > > > David > > > > On Sun, 3 Nov 2019, 10:08 PM Bas Spitters > wrote: > >> > >> There's also VV homotopy lambda calculus, which he later abandoned for > MLTT: > >> > https://www.math.ias.edu/~vladimir/Site3/Univalent_Foundations_files/Hlambda_short_current.pdf > >> > >> On Sun, Oct 27, 2019 at 6:22 PM Bas Spitters > wrote: > >>> > >>> I believe it refers to his 2-theories: > >>> https://www.ias.edu/ideas/2014/voevodsky-origins > >>> > >>> On Sun, Oct 27, 2019 at 3:41 PM Nicolas Alexander Schmidt < > zero@fromzerotoinfinity.xyz> wrote: > >>>> > >>>> In [this](https://www.youtube.com/watch?v=zw6NcwME7yI&t=1680) 2014 > talk > >>>> at IAS, Voevodsky talks about the history of his project of "univalent > >>>> mathematics" and his motivation for starting it. Namely, he mentions > >>>> that he found existing proof assistants at that time (in 2000) to be > >>>> impractical for the kinds of mathematics he was interested in. > >>>> > >>>> Unfortunately, he doesn't go into details of what mathematics he was > >>>> exactly interested in (I'm guessing something to do with homotopy > >>>> theory) or why exactly existing proof assistants weren't practical for > >>>> formalizing them. Judging by the things he mentions in his talk, it > >>>> seems that (roughly) his rejection of those proof assistants was based > >>>> on the view that predicate logic + ZFC is not expressive enough. In > >>>> other words, there is too much lossy encoding needed in order to > >>>> translate from the platonic world of mathematical ideas to this formal > >>>> language. > >>>> > >>>> Comparing the situation to computer programming languages, one might > say > >>>> that predicate logic is like Assembly in that even though everything > can > >>>> be encoded in that language, it is not expressive enough to directly > >>>> talk about higher level concepts, diminishing its practical value for > >>>> reasoning about mathematics. In particular, those systems are not > >>>> adequate for *interactive* development of *new* mathematics (as > opposed > >>>> to formalization of existing theories). > >>>> > >>>> Perhaps I am just misinterpreting what Voevodsky said. In this case, I > >>>> hope someone can correct me. However even if this wasn't *his* view, > to > >>>> me it seems to be a view held implicitly in the HoTT community. In any > >>>> case, it's a view that one might reasonably hold. > >>>> > >>>> However I wonder how reasonable that view actually is, i.e. whether > e.g. > >>>> Mizar really is that much more impractical than HoTT-Coq or Agda, > given > >>>> that people have been happily formalizing mathematics in it for 46 > years > >>>> now. And, even though by browsing the contents of "Formalized > >>>> Mathematics" one can get the impression that the work consists mostly > of > >>>> formalizing early 20th century mathematics, neither the UniMath nor > the > >>>> HoTT library for example contain a proof of Fubini's theorem. > >>>> > >>>> So, to put this into one concrete question, how (if at all) is > HoTT-Coq > >>>> more practical than Mizar for the purpose of formalizing mathematics, > >>>> outside the specific realm of synthetic homotopy theory? > >>>> > >>>> > >>>> -- > >>>> > >>>> Nicolas > >>>> > >>>> > >>>> -- > >>>> You received this message because you are subscribed to the Google > Groups "Homotopy Type Theory" group. > >>>> To unsubscribe from this group and stop receiving emails from it, > send an email to HomotopyTypeTheory+unsubscribe@googlegroups.com. > >>>> To view this discussion on the web visit > https://groups.google.com/d/msgid/HomotopyTypeTheory/e491d38b-b50a-6172-dca9-40d45fe1b6d2%40fromzerotoinfinity.xyz > . > >> > >> -- > >> You received this message because you are subscribed to the Google > Groups "Homotopy Type Theory" group. > >> To unsubscribe from this group and stop receiving emails from it, send > an email to HomotopyTypeTheory+unsubscribe@googlegroups.com. > >> To view this discussion on the web visit > https://groups.google.com/d/msgid/HomotopyTypeTheory/CAOoPQuRQPMkCFKYtAbB%2BpNK90XtFk%2BaVT_aY59U_-9t17sBzeA%40mail.gmail.com > . > > > > -- > > You received this message because you are subscribed to the Google > Groups "Homotopy Type Theory" group. > > To unsubscribe from this group and stop receiving emails from it, send > an email to HomotopyTypeTheory+unsubscribe@googlegroups.com. > > To view this discussion on the web visit > https://groups.google.com/d/msgid/HomotopyTypeTheory/CAFL%2BZM_%3D%3DiLS16Vy7sGiEqNkCxOMYL4j%2BZFqKv5uJ-ivkuemEg%40mail.gmail.com > . > > -- > You received this message because you are subscribed to the Google Groups > "Homotopy Type Theory" group. > To unsubscribe from this group and stop receiving emails from it, send an > email to HomotopyTypeTheory+unsubscribe@googlegroups.com. > To view this discussion on the web visit > https://groups.google.com/d/msgid/HomotopyTypeTheory/CAOvivQz47kSm9WbKDmUsndrpAJMkNwiWmVABqOFrVqyTOvSAbw%40mail.gmail.com > . > -- You received this message because you are subscribed to the Google Groups "Homotopy Type Theory" group. To unsubscribe from this group and stop receiving emails from it, send an email to HomotopyTypeTheory+unsubscribe@googlegroups.com. To view this discussion on the web visit https://groups.google.com/d/msgid/HomotopyTypeTheory/CAH52Xb3s0%2BvweUaSQBMBNLa5mRc9F1jrsg2sSoFmcE_4%3DdAt1w%40mail.gmail.com. --000000000000c30d2c059689af8c Content-Type: text/html; charset="UTF-8" Content-Transfer-Encoding: quoted-printable


=
On Sun, 3 Nov 2019 at 19:13, Michael = Shulman <shulm= an@sandiego.edu> wrote:
But does univalence a la Book HoTT *actually* make it easier= to reason
about such things?=C2=A0

I think this is a= really interesting and important question.

I gues= s David was referring to my scheme fail of 2018. I wanted to formalise the = notion of a scheme a la Grothendieck and prove that if R was a commutative = ring then Spec(R) was a scheme [I know it's a definition, but many math= ematicians do seem to call it a theorem, in our ignorance]. I showed an und= ergraduate a specific lemma in ring theory (https://stacks.math.columbia.edu/t= ag/00EJ) and said "that's what I want" and they formalise= d it for me. And then it turned out that I wanted something else: I didn= 9;t have R_f, I had something "canonically isomorphic" to it, a p= hrase we mathematicians like to pull out when the going gets tough and we c= an't be bothered to check that any more diagrams commute. By this point= it was too late to turn back, and so I had to prove that 20 diagrams commu= ted and it wasn't much fun. I then got an MSc student to redo everythin= g using universal properties more carefully in Lean and it worked like a dr= eam https://github.com= /ramonfmir/lean-scheme. A lot of people said to me at the time "yo= u wouldn't have had this problem if you'd been using HoTT instead o= f DTT" and my response to this is still the (intentionally) provocativ= e "go ahead and define schemes and prove that Spec(R) is a scheme in s= ome HoTT system, and show me how it's better; note that we did have a p= roblem, but we solved it in DTT". I would be particularly interested t= o see schemes done in Arend, because it always felt funny to me using UniMa= th in Coq (and similarly it feels funny to me to do HoTT in Lean 3 -- in bo= th cases it could be argued that it's using a system to do something it= wasn't designed to do). I think it's easy to theorise about this s= ort of thing but until it happens in practice in one or more of the HoTT sy= stems I don't think we will understand the issue properly (or, more pre= cisely, I don't think I will understand the issue properly). I have had= extensive discussions with Martin Escardo about HoTT and he has certainly = given me hope, but on the Lean chat I think people assumed schemes would be= easy in Lean (I certainly did) and then we ran into this unexpected proble= m (which univalence is probably designed to solve), so the question is whet= her a univalent type theory runs into a different unexpected problem -- you= push the carpet down somewhere and it pops up somewhere else.

I know this is a HoTT list but the challenge is also open = to the HOL people like the Isabelle/HOL experts. In contrast to HoTT theori= es, which I think should handle schemes fine, I think that simple type theo= ry will have tremendous problems defining, for example, tensor products of = sheaves of modules on a scheme, because these are dependent types. On the o= ther hand my recent ArXiv paper with Commelin and Massot https://arxiv.org/abs/1910.12320 goes much f= urther and formalises perfectoid spaces in dependent type theory. I would l= ike the people on this list to see this as a challenge. I think that this c= entury will see the rise of the theorem prover in mathematics and I am not = naive enough to think that the one I currently use now is the one which is = guaranteed to be the success story. Voevodsky was convinced that univalence= was the right way to do modern mathematics but I'm doing it just fine = in dependent type theory and now he's gone I really want to find someon= e who will take up the challenge and do some scheme theory in HoTT, but con= vincing professional mathematicians to get interested in this area is very = difficult, and I speak as someone who's been trying to do it for two ye= ars now [I recommend you try the undergraduates instead, anyone who is inte= rested in training people up -- plenty of undergraduates are capable of rea= ding the definition of a scheme, if they know what rings and topological sp= aces are]

To get back to the original question= , my understanding was that Voevodsky had done a bunch of scheme theory and= it had got him a Fields medal and it was this mathematics which he was int= erested in at the time. He wanted to formalise his big theorem, just like H= ales did. Unfortunately he was historically earlier, and his mathematics in= volved far more conceptual objects than spheres in 3-space, so it was a muc= h taller order. All the evidence is there to suggest that over the next 15 = or so years his interests changed. The clearest evidence, in my mind, is th= at there is no definition of a scheme in UniMath. Moreover his story in his= Cambridge talk https://www.newton.ac.uk/seminar/20170710113012301 about asking Sus= lin to reprove one of his results without using the axiom of choice (46 min= utes in) kind of shocked me -- Suslin does not care about mathematics witho= ut choice, and the vast majority of mathematicians employed in mathematics = departments feel the same, although I'm well aware that constructivism = is taken more seriously on this list. I think it is interesting that Voevod= sky failed to prove a constructive version of his theorem, because I think = that some mathematics is better off not being constructive. It is exactly t= he interaction between constructivism and univalence which I do not underst= and well, and I think that a very good way to investigate it would be to do= some highly non-constructive modern mathematics in a univalent type theory= .

Kevin

PS many thank= s to the people who have emailed me in the past telling me about how in the= past I have used "HoTT", "univalence", "UniMath&q= uot;, interchangeably and incorrectly. Hopefully I am getting better but I = am still keen to hear anything which I'm saying which is imprecise or i= ncorrect.

=C2=A0
It allows us to write "=3D" rather than= "\cong", but
to construct such an equality we have to construct an isomorphism
first, and to *use* such an equality we have to transport along it,
and then we get lots of univalence-redexes that we have to manually
reduce away.=C2=A0 My experience formalizing math in HoTT/Coq is that it= 9;s
much easier if you *avoid* turning equivalences into equalities except
when absolutely necessary.=C2=A0 (I don't have any experience formalizi= ng
math in a cubical proof assistant, but in that case at least you
wouldn't have to manually reduce the univalence-redexes -- although it<= br> seems to me you'd still have to construct the isomorphism before you can apply univalence to make it an equality.)

On Sun, Nov 3, 2019 at 3:57 AM David Roberts <droberts.65537@gmail.com> wrote:=
>
> Forget even higher category theory. Kevin Buzzard now goes around tell= ing the story of how even formally proving (using Lean) things in rather el= ementary commutative algebra from EGA that are stated as equalities was not= obvious: the equality is really an isomorphism arising from a universal pr= operty. Forget trying to do anything motivic, when algebra is full of such = equalities. This is not a problem with univalence, of course.
>
> David
>
> On Sun, 3 Nov 2019, 10:08 PM Bas Spitters <b.a.w.spitters@gmail.com> wrot= e:
>>
>> There's also VV homotopy lambda calculus, which he later aband= oned for MLTT:
>> https://www.math.ias.edu/~vladimir/Site3/Univalent_Foundations_files/Hla= mbda_short_current.pdf
>>
>> On Sun, Oct 27, 2019 at 6:22 PM Bas Spitters <b.a.w.spitters@gmail.com&g= t; wrote:
>>>
>>> I believe it refers to his 2-theories:
>>> https://www.ias.edu/ideas/2014/voevodsk= y-origins
>>>
>>> On Sun, Oct 27, 2019 at 3:41 PM Nicolas Alexander Schmidt <= zero@fromz= erotoinfinity.xyz> wrote:
>>>>
>>>> In [this](https://www.you= tube.com/watch?v=3Dzw6NcwME7yI&t=3D1680) 2014 talk
>>>> at IAS, Voevodsky talks about the history of his project o= f "univalent
>>>> mathematics" and his motivation for starting it. Name= ly, he mentions
>>>> that he found existing proof assistants at that time (in 2= 000) to be
>>>> impractical for the kinds of mathematics he was interested= in.
>>>>
>>>> Unfortunately, he doesn't go into details of what math= ematics he was
>>>> exactly interested in (I'm guessing something to do wi= th homotopy
>>>> theory) or why exactly existing proof assistants weren'= ;t practical for
>>>> formalizing them. Judging by the things he mentions in his= talk, it
>>>> seems that (roughly) his rejection of those proof assistan= ts was based
>>>> on the view that predicate logic + ZFC is not expressive e= nough. In
>>>> other words, there is too much lossy encoding needed in or= der to
>>>> translate from the platonic world of mathematical ideas to= this formal
>>>> language.
>>>>
>>>> Comparing the situation to computer programming languages,= one might say
>>>> that predicate logic is like Assembly in that even though = everything can
>>>> be encoded in that language, it is not expressive enough t= o directly
>>>> talk about higher level concepts, diminishing its practica= l value for
>>>> reasoning about mathematics. In particular, those systems = are not
>>>> adequate for *interactive* development of *new* mathematic= s (as opposed
>>>> to formalization of existing theories).
>>>>
>>>> Perhaps I am just misinterpreting what Voevodsky said. In = this case, I
>>>> hope someone can correct me. However even if this wasn'= ;t *his* view, to
>>>> me it seems to be a view held implicitly in the HoTT commu= nity. In any
>>>> case, it's a view that one might reasonably hold.
>>>>
>>>> However I wonder how reasonable that view actually is, i.e= . whether e.g.
>>>> Mizar really is that much more impractical than HoTT-Coq o= r Agda, given
>>>> that people have been happily formalizing mathematics in i= t for 46 years
>>>> now. And, even though by browsing the contents of "Fo= rmalized
>>>> Mathematics" one can get the impression that the work= consists mostly of
>>>> formalizing early 20th century mathematics, neither the Un= iMath nor the
>>>> HoTT library for example contain a proof of Fubini's t= heorem.
>>>>
>>>> So, to put this into one concrete question, how (if at all= ) is HoTT-Coq
>>>> more practical than Mizar for the purpose of formalizing m= athematics,
>>>> outside the specific realm of synthetic homotopy theory? >>>>
>>>>
>>>> --
>>>>
>>>> Nicolas
>>>>
>>>>
>>>> --
>>>> You received this message because you are subscribed to th= e Google Groups "Homotopy Type Theory" group.
>>>> To unsubscribe from this group and stop receiving emails f= rom it, send an email to HomotopyTypeTheory+unsubscribe@googleg= roups.com.
>>>> To view this discussion on the web visit htt= ps://groups.google.com/d/msgid/HomotopyTypeTheory/e491d38b-b50a-6172-dca9-4= 0d45fe1b6d2%40fromzerotoinfinity.xyz.
>>
>> --
>> You received this message because you are subscribed to the Google= Groups "Homotopy Type Theory" group.
>> To unsubscribe from this group and stop receiving emails from it, = send an email to HomotopyTypeTheory+unsubscribe@googlegroups.co= m.
>> To view this discussion on the web visit = https://groups.google.com/d/msgid/HomotopyTypeTheory/CAOoPQuRQPMkCFKYtAbB%2= BpNK90XtFk%2BaVT_aY59U_-9t17sBzeA%40mail.gmail.com.
>
> --
> You received this message because you are subscribed to the Google Gro= ups "Homotopy Type Theory" group.
> To unsubscribe from this group and stop receiving emails from it, send= an email to HomotopyTypeTheory+unsubscribe@googlegroups.com.
> To view this discussion on the web visit = https://groups.google.com/d/msgid/HomotopyTypeTheory/CAFL%2BZM_%3D%3DiLS16V= y7sGiEqNkCxOMYL4j%2BZFqKv5uJ-ivkuemEg%40mail.gmail.com.

--
You received this message because you are subscribed to the Google Groups &= quot;Homotopy Type Theory" group.
To unsubscribe from this group and stop receiving emails from it, send an e= mail to HomotopyTypeTheory+unsubscribe@googlegroups.com. To view this discussion on the web visit https://group= s.google.com/d/msgid/HomotopyTypeTheory/CAOvivQz47kSm9WbKDmUsndrpAJMkNwiWmV= ABqOFrVqyTOvSAbw%40mail.gmail.com.

--
You received this message because you are subscribed to the Google Groups &= quot;Homotopy Type Theory" group.
To unsubscribe from this group and stop receiving emails from it, send an e= mail to = HomotopyTypeTheory+unsubscribe@googlegroups.com.
To view this discussion on the web visit https:/= /groups.google.com/d/msgid/HomotopyTypeTheory/CAH52Xb3s0%2BvweUaSQBMBNLa5mR= c9F1jrsg2sSoFmcE_4%3DdAt1w%40mail.gmail.com.
--000000000000c30d2c059689af8c--