From: Nicolas Alexander Schmidt <zero@fromzerotoinfinity.xyz> To: Homotopy Type Theory <HomotopyTypeTheory@googlegroups.com> Subject: [HoTT] Why did Voevodsky find existing proof assistants to be 'impractical'? Date: Sun, 27 Oct 2019 15:41:34 +0100 Message-ID: <e491d38b-b50a-6172-dca9-40d45fe1b6d2@fromzerotoinfinity.xyz> (raw) 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.

next reply indexThread overview:18+ messages / expand[flat|nested] mbox.gz Atom feed top2019-10-27 14:41 Nicolas Alexander Schmidt [this message]2019-10-27 17:22 ` Bas Spitters 2019-11-03 11:38 ` Bas Spitters 2019-11-03 11:52 ` David Roberts 2019-11-03 19:13 ` Michael Shulman 2019-11-03 19:45 ` Valery Isaev 2019-11-03 22:23 ` Martín Hötzel Escardó 2019-11-04 23:20 ` Nicolas Alexander Schmidt 2019-11-04 18:42 ` Kevin Buzzard 2019-11-04 21:10 ` Michael Shulman 2019-11-04 23:26 ` David Roberts 2019-11-05 15:43 ` Daniel R. Grayson 2019-11-05 20:29 ` Yuhao Huang 2019-11-06 23:59 ` Daniel R. Grayson 2019-11-05 23:14 ` Martín Hötzel Escardó 2019-11-06 0:06 ` Stefan Monnier 2019-11-11 18:26 ` Licata, Dan 2019-11-03 7:29 ` Michael Shulman

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