The isomorphism invariance might be useful when you don't care about the actual stuff you get after transporting. If you have two different definitions of the same set, you can transport its *properties* along the isomorphism (that you still need to construct). For example, we can define rational numbers in two different ways: as arbitrary quotients and as reduced quotients. Then you can prove that they are isomorphic and that one of them gives you a field. Then you know that the other one is also a field. You may be interested in the actual definitions of the operations, but you dob't usually care about the actual proof of the properties. So, you construct the addition and the multiplication explicitly, but you get proofs that they determine a field "for free" from the other definition. If the proofs for one of the definitions is easier than for the other one, this might be a significant simplification. Regards, Valery Isaev вс, 3 нояб. 2019 г. в 22:13, Michael Shulman : > But does univalence a la Book HoTT *actually* make it easier to reason > about such things? 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. 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