From: Kevin Buzzard <email@example.com> To: Michael Shulman <firstname.lastname@example.org> Cc: "Martín Hötzel Escardó" <email@example.com>, "Homotopy Type Theory" <HomotopyTypeTheory@googlegroups.com> Subject: Re: [HoTT] doing "all of pure mathematics" in type theory Date: Sun, 2 Jun 2019 18:49:12 +0100 Message-ID: <CAH52Xb0SKwTaKtkBdZgav3KPkrqGs-eYae11v103ErYU=2dCRQ@mail.gmail.com> (raw) In-Reply-To: <CAOvivQxAZEXiCSS5+ipOszo=JeymDXL3jyfuNFdY7AKxAZSS4g@mail.gmail.com> On Thu, 30 May 2019 at 18:14, Michael Shulman <firstname.lastname@example.org> wrote: > > Thanks Martin! Of course you are right that there are people who > don't need to be convinced and who derive benefit and pleasure from > their formalization efforts, and in our community we probably > encounter a lot of them. However, I still maintain that the *average* > working mathematician is not yet going to find such work useful or > rewarding for its own sake. If you go to, say, the U.S. Joint > Mathematics Meetings and stop a hundred mathematicians at random in > the lobby and ask whether they have ever formalized their work in a > proof assistant, how many do you think will say yes? I think this is absolutely right, and an important point. We can even go further: ask 100 mathematicians whether they have ever formalised *anything at all* in a proof assistant, and only epsilon will say yes. This is why I am focussing on the undergraduates. What we as a community can offer undergraduates is a guarantee that a proof is correct; one could regard this as "instant feedback". I think it is currently very difficult to answer the question I've been asked by several staff members -- "what is in it for me"? Some people might find this a negative approach, but if computer proof systems are introduced in undergraduate curricula, and the response to the question "what is the point of teaching these people type theory?" can be something like this: "what is the point of teaching them Banach spaces? Some will use it later on, many won't. Many students graduate, leave mathematics, and never really use these high-powered objects again. For those people, a course on type theory will teach them just as much as a course on Banach spaces -- it will just give them another domain where they can reason logically, learn some easy theorems, learn some deeper results, etc. Oh, and there's evidence that one day type theory will change the world in a way that Banach spaces probably will not". This is the line I am going to take in a couple of years' time when we are discussing changes to my university's curriculum. And then in a generation's time, one can hope that more than epsilon will say yes. Mathematical culture sometimes changes very slowly. We might be using different systems but I still believe that we are looking at the future, and if the current staff can't see it then we have to work on the future staff. The problem I find with this evil plan for world domination is that most senior people I talk to (across all of these systems) tend not to be employed by maths departments. This is a different issue, which needs different ideas, but those are really orthogonal to this conversation. The thing I know for sure is that there are modern maths undergraduates who grew up with computer games and who find the idea of turning their example sheet questions into levels of a computer game quite appealing. Kevin > > Perhaps I am too pessimistic. But let me in turn offer myself as an > example: in addition to being a homotopy type theorist, I have another > hat as a classical category theorist and homotopy theorist. When I > prove things in HoTT, I often formalize them in a proof assistant. > But when I prove things in classical mathematics, I very rarely > consider formalizing them. Only once have I formalized a proof in > classical category theory; the experience was more time-consuming and > less enjoyable than I expected, and I have no plans to do it again. > And I expect that as classical mathematicians go, I am (even when I > have my classical mathematician hat on) pretty far on the > sympathetic-to-formalization side of the spectrum. > > > > On Wed, May 29, 2019 at 12:05 PM Martín Hötzel Escardó > <email@example.com> wrote: > > > > Thanks for this discussion. I like it. > > > > Maybe I would like to argue with this point: > > > > On 28/05/2019 10:50, Michael Shulman wrote: > > > I think it's fairly hopeless to convince a classical mathematician > > > that they should put in a lot of work to convince a computer of the > > > truth of *something they already knew*. > > > > I am not sure why the person who started this thread, a mathematician, > > Kevin Buzzard, decided to put in such a lot of work, but did and he > > has (in Lean), with his students. > > > > But having interacted with a lot of students (from my institution, and > > from everywhere in the world, from maths, logic, computer science and > > philosophy departments (and once even a high school student in the > > UniMath School)), what I can say is that they are not trying to > > convince the computer. > > > > They are trying to convince themselves, using the computer to both > > check their understanding and record their understanding when > > the proof is complete. > > > > If I am allowed to speak for myself, I created a univalent library in > > Agda for the purpose of *doing something else*. However, it is nice to > > stare at the library and see everything developed from first > > principles. When presented with the mathematical literature, both as > > students and experienced mathematicians, we are never sure how far > > back one has to read until everything begins in a precise way. How > > much have we created, and how do all the different fields of > > mathematics interact with each other? When one records mathematics in > > the computer, this begins to become clear, or at least the asnwers to such > > questions become possible. > > > > We don't need to *convince* anybody. This will *happen*. And it is > > already happening. The students like it. This is my experience. > > > > M. > > > > > > -- > > 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 HomotopyTypeTheoryfirstname.lastname@example.org. > > To view this discussion on the web visit https://groups.google.com/d/msgid/HomotopyTypeTheory/c7197bc2-7fc5-4027-bed6-24b2d350950f%40googlegroups.com. > > For more options, visit https://groups.google.com/d/optout. > > -- > 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 HomotopyTypeTheoryemail@example.com. > To view this discussion on the web visit https://groups.google.com/d/msgid/HomotopyTypeTheory/CAOvivQxAZEXiCSS5%2BipOszo%3DJeymDXL3jyfuNFdY7AKxAZSS4g%40mail.gmail.com. > For more options, visit https://groups.google.com/d/optout. -- 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 HomotopyTypeTheoryfirstname.lastname@example.org. To view this discussion on the web visit https://groups.google.com/d/msgid/HomotopyTypeTheory/CAH52Xb0SKwTaKtkBdZgav3KPkrqGs-eYae11v103ErYU%3D2dCRQ%40mail.gmail.com. For more options, visit https://groups.google.com/d/optout.
next prev parent reply index Thread overview: 31+ messages / expand[flat|nested] mbox.gz Atom feed top 2019-05-25 10:12 Kevin Buzzard 2019-05-25 10:22 ` Steve Awodey 2019-05-25 12:23 ` Kevin Buzzard [not found] ` <B7D67BBA-5E0B-4438-908D-4EF316C8C1F1@chalmers.se> [not found] ` <CAH52Xb1Y=Xq=012v_-KSDUuwgnKpEp5qjrxgtUJf+qc_0RWJUg@mail.gmail.com> 2019-05-25 13:13 ` Fwd: " Kevin Buzzard 2019-05-25 13:34 ` Juan Ospina 2019-05-25 14:50 ` Noah Snyder 2019-05-25 15:36 ` Kevin Buzzard 2019-05-25 16:41 ` Noah Snyder 2019-05-26 5:50 ` Bas Spitters 2019-05-26 11:41 ` Kevin Buzzard 2019-05-26 12:09 ` Bas Spitters 2019-05-26 17:00 ` Kevin Buzzard 2019-05-27 2:33 ` Daniel R. Grayson 2019-06-02 16:30 ` Bas Spitters 2019-06-02 17:55 ` Kevin Buzzard 2019-06-02 20:46 ` Nicola Gambino 2019-06-02 20:59 ` Valery Isaev 2019-06-04 20:32 ` Michael Shulman 2019-06-04 20:58 ` Kevin Buzzard 2019-06-06 16:30 ` Matt Oliveri 2019-05-27 13:09 ` Assia Mahboubi 2019-05-28 9:50 ` Michael Shulman 2019-05-28 10:13 ` Nils Anders Danielsson 2019-05-28 10:22 ` Michael Shulman 2019-05-29 19:04 ` Martín Hötzel Escardó 2019-05-30 17:14 ` Michael Shulman 2019-06-02 17:49 ` Kevin Buzzard [this message] 2019-06-04 20:50 ` Martín Hötzel Escardó 2019-06-05 17:11 ` Thorsten Altenkirch 2019-05-28 15:20 ` Joyal, André 2019-05-27 8:41 ` Nils Anders Danielsson
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