There's also VV homotopy lambda calculus, which he later abandoned for MLTT:--On Sun, Oct 27, 2019 at 6:22 PM Bas Spitters <b.a.w.spitters@gmail.com> wrote:I believe it refers to his 2-theories: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
--
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