Re: [HoTT] doing "all of pure mathematics" in type theory

[Kevin Buzzard <kevin.m.buzzard@gmail.com>]

On Thu, 30 May 2019 at 18:14, Michael Shulman <shulman@sandiego.edu> 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
> <escardo.martin@gmail.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.
> >
> >
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