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.

[-- Attachment #1: Type: text/plain, Size: 3429 bytes --] 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/CAOoPQuTfkp%3DPNeYE8bpO20APnTBdpzqJNfUekE5ECrr0vD5cww%40mail.gmail.com. [-- Attachment #2: Type: text/html, Size: 4657 bytes --] <div dir="ltr"><div>I believe it refers to his 2-theories:<br></div><div><a href="https://www.ias.edu/ideas/2014/voevodsky-origins">https://www.ias.edu/ideas/2014/voevodsky-origins</a></div></div><br><div class="gmail_quote"><div dir="ltr" class="gmail_attr">On Sun, Oct 27, 2019 at 3:41 PM Nicolas Alexander Schmidt <<a href="mailto:zero@fromzerotoinfinity.xyz">zero@fromzerotoinfinity.xyz</a>> wrote:<br></div><blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left:1px solid rgb(204,204,204);padding-left:1ex">In [this](<a href="https://www.youtube.com/watch?v=zw6NcwME7yI&t=1680" rel="noreferrer" target="_blank">https://www.youtube.com/watch?v=zw6NcwME7yI&t=1680</a>) 2014 talk<br> at IAS, Voevodsky talks about the history of his project of "univalent<br> mathematics" and his motivation for starting it. Namely, he mentions<br> that he found existing proof assistants at that time (in 2000) to be<br> impractical for the kinds of mathematics he was interested in.<br> <br> Unfortunately, he doesn't go into details of what mathematics he was<br> exactly interested in (I'm guessing something to do with homotopy<br> theory) or why exactly existing proof assistants weren't practical for<br> formalizing them. Judging by the things he mentions in his talk, it<br> seems that (roughly) his rejection of those proof assistants was based<br> on the view that predicate logic + ZFC is not expressive enough. In<br> other words, there is too much lossy encoding needed in order to<br> translate from the platonic world of mathematical ideas to this formal<br> language.<br> <br> Comparing the situation to computer programming languages, one might say<br> that predicate logic is like Assembly in that even though everything can<br> be encoded in that language, it is not expressive enough to directly<br> talk about higher level concepts, diminishing its practical value for<br> reasoning about mathematics. In particular, those systems are not<br> adequate for *interactive* development of *new* mathematics (as opposed<br> to formalization of existing theories).<br> <br> Perhaps I am just misinterpreting what Voevodsky said. In this case, I<br> hope someone can correct me. However even if this wasn't *his* view, to<br> me it seems to be a view held implicitly in the HoTT community. In any<br> case, it's a view that one might reasonably hold.<br> <br> However I wonder how reasonable that view actually is, i.e. whether e.g.<br> Mizar really is that much more impractical than HoTT-Coq or Agda, given<br> that people have been happily formalizing mathematics in it for 46 years<br> now. And, even though by browsing the contents of "Formalized<br> Mathematics" one can get the impression that the work consists mostly of<br> formalizing early 20th century mathematics, neither the UniMath nor the<br> HoTT library for example contain a proof of Fubini's theorem.<br> <br> So, to put this into one concrete question, how (if at all) is HoTT-Coq<br> more practical than Mizar for the purpose of formalizing mathematics,<br> outside the specific realm of synthetic homotopy theory?<br> <br> <br> --<br> <br> Nicolas<br> <br> <br> -- <br> You received this message because you are subscribed to the Google Groups "Homotopy Type Theory" group.<br> To unsubscribe from this group and stop receiving emails from it, send an email to <a href="mailto:HomotopyTypeTheory%2Bunsubscribe@googlegroups.com" target="_blank">HomotopyTypeTheory+unsubscribe@googlegroups.com</a>.<br> To view this discussion on the web visit <a href="https://groups.google.com/d/msgid/HomotopyTypeTheory/e491d38b-b50a-6172-dca9-40d45fe1b6d2%40fromzerotoinfinity.xyz" rel="noreferrer" target="_blank">https://groups.google.com/d/msgid/HomotopyTypeTheory/e491d38b-b50a-6172-dca9-40d45fe1b6d2%40fromzerotoinfinity.xyz</a>.<br> </blockquote></div> <p></p> -- <br /> You received this message because you are subscribed to the Google Groups "Homotopy Type Theory" group.<br /> To unsubscribe from this group and stop receiving emails from it, send an email to <a href="mailto:HomotopyTypeTheory+unsubscribe@googlegroups.com">HomotopyTypeTheory+unsubscribe@googlegroups.com</a>.<br /> To view this discussion on the web visit <a href="https://groups.google.com/d/msgid/HomotopyTypeTheory/CAOoPQuTfkp%3DPNeYE8bpO20APnTBdpzqJNfUekE5ECrr0vD5cww%40mail.gmail.com?utm_medium=email&utm_source=footer">https://groups.google.com/d/msgid/HomotopyTypeTheory/CAOoPQuTfkp%3DPNeYE8bpO20APnTBdpzqJNfUekE5ECrr0vD5cww%40mail.gmail.com</a>.<br />

I know hardly anything about Mizar, so I can't comment on it. And I don't know for sure exactly what Voevodsky meant (and perhaps no one now does). But I am pretty sure that he was not thinking about synthetic homotopy theory at the time, because the possibility of synthetic homotopy theory wasn't really even imagined until later on (specifically, with the advent of higher inductive types). Instead, I expect he was thinking about category theory and higher category theory. The advantage of univalence in such contexts is that it formalizes the isomorphism-invariant behavior of category theory, incorporating it directly into the logical structure of the foundations so that you don't have to worry about whether your theorems are invariant under isomorphism, or prove explicitly that they are. Is this a substantial advantage over a ZFC-based system? Maybe, maybe not. (One might argue that it's not much of an advantage at all until univalence becomes "computational", as with cubical type theories, so that transporting along it can be done automatically by the proof assistant.) But there are other points to consider. Firstly, before even getting to univalence, there is a difference between membership-based set theories and type theories, which is more closely analogous to the assembly language / high level programming language divide. Type-theoretic foundations for mathematics, like strongly/statically typed programming languages, provide automatic "error-checking" for the user, catching various kinds of mistakes at "compile time", and allow more intelligent compiler optimization and inference due to the more informative nature of types. Dependent type theories are even better at this. And just as the advantages of static typing are not belied by the fact that a lot of people happily write code in dynamically typed languages, so the advantages of typed mathematics for formalization are not belied by the fact that some mathematicians are happy to do without them. It's worth noting that many of the recent high-profile formalizations of *recent* mathematical results, such as the four-color theorem, the odd-order theorem, and the Kepler conjecture, use type-theoretic proof assistants like Coq and HOL rather than set-theoretic ones like Mizar. From this perspective, the advantage of HoTT/UF is that it "fixes" certain infelicities in previously existing type theories. Now we understand quotients better and have more tools for doing without setoids; now we know what the equality type of the universe is; now we are better at computing with function extensionality and propositional extensionality; etc. So HoTT/UF gives us the benefits of a type-theoretic foundation while ameliorating some of the traditional disadvantages of that approach. But in my own opinion (which is, I believe, rather different from Voevodsky's, at least in emphasis), all of this is kind of beside the point. It's like arguing about whether or not my laptop can play movies better than an 80s-era VCR; it overlooks the main point that my laptop does *so much more* than just play movies. The real advantage of type-theoretic, and homotopy-type-theoretic, foundations, is that they have a plethora of categorical and higher-categorical models. Proving a theorem once, in constructive homotopy type theory, automatically entails the "internal" truth of the corresponding theorem in all higher toposes. I think this is a much more important selling point than whether or not we get a more practical system for formalizing plain old set-based mathematics. Mike On Sun, Oct 27, 2019 at 7:41 AM 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. 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[-- Attachment #1: Type: text/plain, Size: 3752 bytes --] 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 <b.a.w.spitters@gmail.com> 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. [-- Attachment #2: Type: text/html, Size: 5364 bytes --] <div dir="ltr"><div>There's also VV homotopy lambda calculus, which he later abandoned for MLTT:</div><div><a href="https://www.math.ias.edu/~vladimir/Site3/Univalent_Foundations_files/Hlambda_short_current.pdf">https://www.math.ias.edu/~vladimir/Site3/Univalent_Foundations_files/Hlambda_short_current.pdf</a></div></div><br><div class="gmail_quote"><div dir="ltr" class="gmail_attr">On Sun, Oct 27, 2019 at 6:22 PM Bas Spitters <<a href="mailto:b.a.w.spitters@gmail.com">b.a.w.spitters@gmail.com</a>> wrote:<br></div><blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left:1px solid rgb(204,204,204);padding-left:1ex"><div dir="ltr"><div>I believe it refers to his 2-theories:<br></div><div><a href="https://www.ias.edu/ideas/2014/voevodsky-origins" target="_blank">https://www.ias.edu/ideas/2014/voevodsky-origins</a></div></div><br><div class="gmail_quote"><div dir="ltr" class="gmail_attr">On Sun, Oct 27, 2019 at 3:41 PM Nicolas Alexander Schmidt <<a href="mailto:zero@fromzerotoinfinity.xyz" target="_blank">zero@fromzerotoinfinity.xyz</a>> wrote:<br></div><blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left:1px solid rgb(204,204,204);padding-left:1ex">In [this](<a href="https://www.youtube.com/watch?v=zw6NcwME7yI&t=1680" rel="noreferrer" target="_blank">https://www.youtube.com/watch?v=zw6NcwME7yI&t=1680</a>) 2014 talk<br> at IAS, Voevodsky talks about the history of his project of "univalent<br> mathematics" and his motivation for starting it. Namely, he mentions<br> that he found existing proof assistants at that time (in 2000) to be<br> impractical for the kinds of mathematics he was interested in.<br> <br> Unfortunately, he doesn't go into details of what mathematics he was<br> exactly interested in (I'm guessing something to do with homotopy<br> theory) or why exactly existing proof assistants weren't practical for<br> formalizing them. Judging by the things he mentions in his talk, it<br> seems that (roughly) his rejection of those proof assistants was based<br> on the view that predicate logic + ZFC is not expressive enough. In<br> other words, there is too much lossy encoding needed in order to<br> translate from the platonic world of mathematical ideas to this formal<br> language.<br> <br> Comparing the situation to computer programming languages, one might say<br> that predicate logic is like Assembly in that even though everything can<br> be encoded in that language, it is not expressive enough to directly<br> talk about higher level concepts, diminishing its practical value for<br> reasoning about mathematics. In particular, those systems are not<br> adequate for *interactive* development of *new* mathematics (as opposed<br> to formalization of existing theories).<br> <br> Perhaps I am just misinterpreting what Voevodsky said. In this case, I<br> hope someone can correct me. However even if this wasn't *his* view, to<br> me it seems to be a view held implicitly in the HoTT community. In any<br> case, it's a view that one might reasonably hold.<br> <br> However I wonder how reasonable that view actually is, i.e. whether e.g.<br> Mizar really is that much more impractical than HoTT-Coq or Agda, given<br> that people have been happily formalizing mathematics in it for 46 years<br> now. And, even though by browsing the contents of "Formalized<br> Mathematics" one can get the impression that the work consists mostly of<br> formalizing early 20th century mathematics, neither the UniMath nor the<br> HoTT library for example contain a proof of Fubini's theorem.<br> <br> So, to put this into one concrete question, how (if at all) is HoTT-Coq<br> more practical than Mizar for the purpose of formalizing mathematics,<br> outside the specific realm of synthetic homotopy theory?<br> <br> <br> --<br> <br> Nicolas<br> <br> <br> -- <br> You received this message because you are subscribed to the Google Groups "Homotopy Type Theory" group.<br> To unsubscribe from this group and stop receiving emails from it, send an email to <a href="mailto:HomotopyTypeTheory%2Bunsubscribe@googlegroups.com" target="_blank">HomotopyTypeTheory+unsubscribe@googlegroups.com</a>.<br> To view this discussion on the web visit <a href="https://groups.google.com/d/msgid/HomotopyTypeTheory/e491d38b-b50a-6172-dca9-40d45fe1b6d2%40fromzerotoinfinity.xyz" rel="noreferrer" target="_blank">https://groups.google.com/d/msgid/HomotopyTypeTheory/e491d38b-b50a-6172-dca9-40d45fe1b6d2%40fromzerotoinfinity.xyz</a>.<br> </blockquote></div> </blockquote></div> <p></p> -- <br /> You received this message because you are subscribed to the Google Groups "Homotopy Type Theory" group.<br /> To unsubscribe from this group and stop receiving emails from it, send an email to <a href="mailto:HomotopyTypeTheory+unsubscribe@googlegroups.com">HomotopyTypeTheory+unsubscribe@googlegroups.com</a>.<br /> To view this discussion on the web visit <a href="https://groups.google.com/d/msgid/HomotopyTypeTheory/CAOoPQuRQPMkCFKYtAbB%2BpNK90XtFk%2BaVT_aY59U_-9t17sBzeA%40mail.gmail.com?utm_medium=email&utm_source=footer">https://groups.google.com/d/msgid/HomotopyTypeTheory/CAOoPQuRQPMkCFKYtAbB%2BpNK90XtFk%2BaVT_aY59U_-9t17sBzeA%40mail.gmail.com</a>.<br />

[-- Attachment #1: Type: text/plain, Size: 4939 bytes --] 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 <b.a.w.spitters@gmail.com> 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 <b.a.w.spitters@gmail.com> > 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 > <https://groups.google.com/d/msgid/HomotopyTypeTheory/CAOoPQuRQPMkCFKYtAbB%2BpNK90XtFk%2BaVT_aY59U_-9t17sBzeA%40mail.gmail.com?utm_medium=email&utm_source=footer> > . > -- 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. [-- Attachment #2: Type: text/html, Size: 7151 bytes --] <div dir="auto">Forget even higher category theory. Kevin Buzzard now goes around telling the story of how even formally proving <span style="font-family:sans-serif">(using Lean)</span> 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.<div dir="auto"><br></div><div dir="auto">David</div></div><br><div class="gmail_quote"><div dir="ltr" class="gmail_attr">On Sun, 3 Nov 2019, 10:08 PM Bas Spitters <<a href="mailto:b.a.w.spitters@gmail.com">b.a.w.spitters@gmail.com</a>> wrote:<br></div><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex"><div dir="ltr"><div>There's also VV homotopy lambda calculus, which he later abandoned for MLTT:</div><div><a href="https://www.math.ias.edu/~vladimir/Site3/Univalent_Foundations_files/Hlambda_short_current.pdf" target="_blank" rel="noreferrer">https://www.math.ias.edu/~vladimir/Site3/Univalent_Foundations_files/Hlambda_short_current.pdf</a></div></div><br><div class="gmail_quote"><div dir="ltr" class="gmail_attr">On Sun, Oct 27, 2019 at 6:22 PM Bas Spitters <<a href="mailto:b.a.w.spitters@gmail.com" target="_blank" rel="noreferrer">b.a.w.spitters@gmail.com</a>> wrote:<br></div><blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left:1px solid rgb(204,204,204);padding-left:1ex"><div dir="ltr"><div>I believe it refers to his 2-theories:<br></div><div><a href="https://www.ias.edu/ideas/2014/voevodsky-origins" target="_blank" rel="noreferrer">https://www.ias.edu/ideas/2014/voevodsky-origins</a></div></div><br><div class="gmail_quote"><div dir="ltr" class="gmail_attr">On Sun, Oct 27, 2019 at 3:41 PM Nicolas Alexander Schmidt <<a href="mailto:zero@fromzerotoinfinity.xyz" target="_blank" rel="noreferrer">zero@fromzerotoinfinity.xyz</a>> wrote:<br></div><blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left:1px solid rgb(204,204,204);padding-left:1ex">In [this](<a href="https://www.youtube.com/watch?v=zw6NcwME7yI&t=1680" rel="noreferrer noreferrer" target="_blank">https://www.youtube.com/watch?v=zw6NcwME7yI&t=1680</a>) 2014 talk<br> at IAS, Voevodsky talks about the history of his project of "univalent<br> mathematics" and his motivation for starting it. Namely, he mentions<br> that he found existing proof assistants at that time (in 2000) to be<br> impractical for the kinds of mathematics he was interested in.<br> <br> Unfortunately, he doesn't go into details of what mathematics he was<br> exactly interested in (I'm guessing something to do with homotopy<br> theory) or why exactly existing proof assistants weren't practical for<br> formalizing them. Judging by the things he mentions in his talk, it<br> seems that (roughly) his rejection of those proof assistants was based<br> on the view that predicate logic + ZFC is not expressive enough. In<br> other words, there is too much lossy encoding needed in order to<br> translate from the platonic world of mathematical ideas to this formal<br> language.<br> <br> Comparing the situation to computer programming languages, one might say<br> that predicate logic is like Assembly in that even though everything can<br> be encoded in that language, it is not expressive enough to directly<br> talk about higher level concepts, diminishing its practical value for<br> reasoning about mathematics. In particular, those systems are not<br> adequate for *interactive* development of *new* mathematics (as opposed<br> to formalization of existing theories).<br> <br> Perhaps I am just misinterpreting what Voevodsky said. In this case, I<br> hope someone can correct me. However even if this wasn't *his* view, to<br> me it seems to be a view held implicitly in the HoTT community. In any<br> case, it's a view that one might reasonably hold.<br> <br> However I wonder how reasonable that view actually is, i.e. whether e.g.<br> Mizar really is that much more impractical than HoTT-Coq or Agda, given<br> that people have been happily formalizing mathematics in it for 46 years<br> now. And, even though by browsing the contents of "Formalized<br> Mathematics" one can get the impression that the work consists mostly of<br> formalizing early 20th century mathematics, neither the UniMath nor the<br> HoTT library for example contain a proof of Fubini's theorem.<br> <br> So, to put this into one concrete question, how (if at all) is HoTT-Coq<br> more practical than Mizar for the purpose of formalizing mathematics,<br> outside the specific realm of synthetic homotopy theory?<br> <br> <br> --<br> <br> Nicolas<br> <br> <br> -- <br> You received this message because you are subscribed to the Google Groups "Homotopy Type Theory" group.<br> To unsubscribe from this group and stop receiving emails from it, send an email to <a href="mailto:HomotopyTypeTheory%2Bunsubscribe@googlegroups.com" target="_blank" rel="noreferrer">HomotopyTypeTheory+unsubscribe@googlegroups.com</a>.<br> To view this discussion on the web visit <a href="https://groups.google.com/d/msgid/HomotopyTypeTheory/e491d38b-b50a-6172-dca9-40d45fe1b6d2%40fromzerotoinfinity.xyz" rel="noreferrer noreferrer" target="_blank">https://groups.google.com/d/msgid/HomotopyTypeTheory/e491d38b-b50a-6172-dca9-40d45fe1b6d2%40fromzerotoinfinity.xyz</a>.<br> </blockquote></div> </blockquote></div> <p></p> -- <br> You received this message because you are subscribed to the Google Groups "Homotopy Type Theory" group.<br> To unsubscribe from this group and stop receiving emails from it, send an email to <a href="mailto:HomotopyTypeTheory+unsubscribe@googlegroups.com" target="_blank" rel="noreferrer">HomotopyTypeTheory+unsubscribe@googlegroups.com</a>.<br> To view this discussion on the web visit <a href="https://groups.google.com/d/msgid/HomotopyTypeTheory/CAOoPQuRQPMkCFKYtAbB%2BpNK90XtFk%2BaVT_aY59U_-9t17sBzeA%40mail.gmail.com?utm_medium=email&utm_source=footer" target="_blank" rel="noreferrer">https://groups.google.com/d/msgid/HomotopyTypeTheory/CAOoPQuRQPMkCFKYtAbB%2BpNK90XtFk%2BaVT_aY59U_-9t17sBzeA%40mail.gmail.com</a>.<br> </blockquote></div> <p></p> -- <br /> You received this message because you are subscribed to the Google Groups "Homotopy Type Theory" group.<br /> To unsubscribe from this group and stop receiving emails from it, send an email to <a href="mailto:HomotopyTypeTheory+unsubscribe@googlegroups.com">HomotopyTypeTheory+unsubscribe@googlegroups.com</a>.<br /> To view this discussion on the web visit <a href="https://groups.google.com/d/msgid/HomotopyTypeTheory/CAFL%2BZM_%3D%3DiLS16Vy7sGiEqNkCxOMYL4j%2BZFqKv5uJ-ivkuemEg%40mail.gmail.com?utm_medium=email&utm_source=footer">https://groups.google.com/d/msgid/HomotopyTypeTheory/CAFL%2BZM_%3D%3DiLS16Vy7sGiEqNkCxOMYL4j%2BZFqKv5uJ-ivkuemEg%40mail.gmail.com</a>.<br />

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 <droberts.65537@gmail.com> 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 <b.a.w.spitters@gmail.com> 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 <b.a.w.spitters@gmail.com> 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.

[-- Attachment #1: Type: text/plain, Size: 7975 bytes --] 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 <shulman@sandiego.edu>: > 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 <droberts.65537@gmail.com> > 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 <b.a.w.spitters@gmail.com> > 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 <b.a.w.spitters@gmail.com> > 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. 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/CAA520ftr3b4cGi4vXdmJL-GbAVksV9ggrTAmqZ4E75P-ch1hVQ%40mail.gmail.com. [-- Attachment #2: Type: text/html, Size: 11338 bytes --] <div dir="ltr"><div>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 <i>properties</i> 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.</div><br clear="all"><div><div dir="ltr" class="gmail_signature" data-smartmail="gmail_signature"><div dir="ltr"><div><div>Regards,</div><div dir="ltr">Valery Isaev<br></div></div></div></div></div><br></div><br><div class="gmail_quote"><div dir="ltr" class="gmail_attr">вс, 3 нояб. 2019 г. в 22:13, Michael Shulman <<a href="mailto:shulman@sandiego.edu">shulman@sandiego.edu</a>>:<br></div><blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left:1px solid rgb(204,204,204);padding-left:1ex">But does univalence a la Book HoTT *actually* make it easier to reason<br> about such things? It allows us to write "=" rather than "\cong", but<br> to construct such an equality we have to construct an isomorphism<br> first, and to *use* such an equality we have to transport along it,<br> and then we get lots of univalence-redexes that we have to manually<br> reduce away. My experience formalizing math in HoTT/Coq is that it's<br> much easier if you *avoid* turning equivalences into equalities except<br> when absolutely necessary. (I don't have any experience formalizing<br> math in a cubical proof assistant, but in that case at least you<br> wouldn't have to manually reduce the univalence-redexes -- although it<br> seems to me you'd still have to construct the isomorphism before you<br> can apply univalence to make it an equality.)<br> <br> On Sun, Nov 3, 2019 at 3:57 AM David Roberts <<a href="mailto:droberts.65537@gmail.com" target="_blank">droberts.65537@gmail.com</a>> wrote:<br> ><br> > 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.<br> ><br> > David<br> ><br> > On Sun, 3 Nov 2019, 10:08 PM Bas Spitters <<a href="mailto:b.a.w.spitters@gmail.com" target="_blank">b.a.w.spitters@gmail.com</a>> wrote:<br> >><br> >> There's also VV homotopy lambda calculus, which he later abandoned for MLTT:<br> >> <a href="https://www.math.ias.edu/~vladimir/Site3/Univalent_Foundations_files/Hlambda_short_current.pdf" rel="noreferrer" target="_blank">https://www.math.ias.edu/~vladimir/Site3/Univalent_Foundations_files/Hlambda_short_current.pdf</a><br> >><br> >> On Sun, Oct 27, 2019 at 6:22 PM Bas Spitters <<a href="mailto:b.a.w.spitters@gmail.com" target="_blank">b.a.w.spitters@gmail.com</a>> wrote:<br> >>><br> >>> I believe it refers to his 2-theories:<br> >>> <a href="https://www.ias.edu/ideas/2014/voevodsky-origins" rel="noreferrer" target="_blank">https://www.ias.edu/ideas/2014/voevodsky-origins</a><br> >>><br> >>> On Sun, Oct 27, 2019 at 3:41 PM Nicolas Alexander Schmidt <<a href="mailto:zero@fromzerotoinfinity.xyz" target="_blank">zero@fromzerotoinfinity.xyz</a>> wrote:<br> >>>><br> >>>> In [this](<a href="https://www.youtube.com/watch?v=zw6NcwME7yI&t=1680" rel="noreferrer" target="_blank">https://www.youtube.com/watch?v=zw6NcwME7yI&t=1680</a>) 2014 talk<br> >>>> at IAS, Voevodsky talks about the history of his project of "univalent<br> >>>> mathematics" and his motivation for starting it. Namely, he mentions<br> >>>> that he found existing proof assistants at that time (in 2000) to be<br> >>>> impractical for the kinds of mathematics he was interested in.<br> >>>><br> >>>> Unfortunately, he doesn't go into details of what mathematics he was<br> >>>> exactly interested in (I'm guessing something to do with homotopy<br> >>>> theory) or why exactly existing proof assistants weren't practical for<br> >>>> formalizing them. Judging by the things he mentions in his talk, it<br> >>>> seems that (roughly) his rejection of those proof assistants was based<br> >>>> on the view that predicate logic + ZFC is not expressive enough. In<br> >>>> other words, there is too much lossy encoding needed in order to<br> >>>> translate from the platonic world of mathematical ideas to this formal<br> >>>> language.<br> >>>><br> >>>> Comparing the situation to computer programming languages, one might say<br> >>>> that predicate logic is like Assembly in that even though everything can<br> >>>> be encoded in that language, it is not expressive enough to directly<br> >>>> talk about higher level concepts, diminishing its practical value for<br> >>>> reasoning about mathematics. In particular, those systems are not<br> >>>> adequate for *interactive* development of *new* mathematics (as opposed<br> >>>> to formalization of existing theories).<br> >>>><br> >>>> Perhaps I am just misinterpreting what Voevodsky said. In this case, I<br> >>>> hope someone can correct me. However even if this wasn't *his* view, to<br> >>>> me it seems to be a view held implicitly in the HoTT community. In any<br> >>>> case, it's a view that one might reasonably hold.<br> >>>><br> >>>> However I wonder how reasonable that view actually is, i.e. whether e.g.<br> >>>> Mizar really is that much more impractical than HoTT-Coq or Agda, given<br> >>>> that people have been happily formalizing mathematics in it for 46 years<br> >>>> now. And, even though by browsing the contents of "Formalized<br> >>>> Mathematics" one can get the impression that the work consists mostly of<br> >>>> formalizing early 20th century mathematics, neither the UniMath nor the<br> >>>> HoTT library for example contain a proof of Fubini's theorem.<br> >>>><br> >>>> So, to put this into one concrete question, how (if at all) is HoTT-Coq<br> >>>> more practical than Mizar for the purpose of formalizing mathematics,<br> >>>> outside the specific realm of synthetic homotopy theory?<br> >>>><br> >>>><br> >>>> --<br> >>>><br> >>>> Nicolas<br> >>>><br> >>>><br> >>>> --<br> >>>> You received this message because you are subscribed to the Google Groups "Homotopy Type Theory" group.<br> >>>> To unsubscribe from this group and stop receiving emails from it, send an email to <a href="mailto:HomotopyTypeTheory%2Bunsubscribe@googlegroups.com" target="_blank">HomotopyTypeTheory+unsubscribe@googlegroups.com</a>.<br> >>>> To view this discussion on the web visit <a href="https://groups.google.com/d/msgid/HomotopyTypeTheory/e491d38b-b50a-6172-dca9-40d45fe1b6d2%40fromzerotoinfinity.xyz" rel="noreferrer" target="_blank">https://groups.google.com/d/msgid/HomotopyTypeTheory/e491d38b-b50a-6172-dca9-40d45fe1b6d2%40fromzerotoinfinity.xyz</a>.<br> >><br> >> --<br> >> You received this message because you are subscribed to the Google Groups "Homotopy Type Theory" group.<br> >> To unsubscribe from this group and stop receiving emails from it, send an email to <a href="mailto:HomotopyTypeTheory%2Bunsubscribe@googlegroups.com" target="_blank">HomotopyTypeTheory+unsubscribe@googlegroups.com</a>.<br> >> To view this discussion on the web visit <a href="https://groups.google.com/d/msgid/HomotopyTypeTheory/CAOoPQuRQPMkCFKYtAbB%2BpNK90XtFk%2BaVT_aY59U_-9t17sBzeA%40mail.gmail.com" rel="noreferrer" target="_blank">https://groups.google.com/d/msgid/HomotopyTypeTheory/CAOoPQuRQPMkCFKYtAbB%2BpNK90XtFk%2BaVT_aY59U_-9t17sBzeA%40mail.gmail.com</a>.<br> ><br> > --<br> > You received this message because you are subscribed to the Google Groups "Homotopy Type Theory" group.<br> > To unsubscribe from this group and stop receiving emails from it, send an email to <a href="mailto:HomotopyTypeTheory%2Bunsubscribe@googlegroups.com" target="_blank">HomotopyTypeTheory+unsubscribe@googlegroups.com</a>.<br> > To view this discussion on the web visit <a href="https://groups.google.com/d/msgid/HomotopyTypeTheory/CAFL%2BZM_%3D%3DiLS16Vy7sGiEqNkCxOMYL4j%2BZFqKv5uJ-ivkuemEg%40mail.gmail.com" rel="noreferrer" target="_blank">https://groups.google.com/d/msgid/HomotopyTypeTheory/CAFL%2BZM_%3D%3DiLS16Vy7sGiEqNkCxOMYL4j%2BZFqKv5uJ-ivkuemEg%40mail.gmail.com</a>.<br> <br> -- <br> You received this message because you are subscribed to the Google Groups "Homotopy Type Theory" group.<br> To unsubscribe from this group and stop receiving emails from it, send an email to <a href="mailto:HomotopyTypeTheory%2Bunsubscribe@googlegroups.com" target="_blank">HomotopyTypeTheory+unsubscribe@googlegroups.com</a>.<br> To view this discussion on the web visit <a href="https://groups.google.com/d/msgid/HomotopyTypeTheory/CAOvivQz47kSm9WbKDmUsndrpAJMkNwiWmVABqOFrVqyTOvSAbw%40mail.gmail.com" rel="noreferrer" target="_blank">https://groups.google.com/d/msgid/HomotopyTypeTheory/CAOvivQz47kSm9WbKDmUsndrpAJMkNwiWmVABqOFrVqyTOvSAbw%40mail.gmail.com</a>.<br> </blockquote></div> <p></p> -- <br /> You received this message because you are subscribed to the Google Groups "Homotopy Type Theory" group.<br /> To unsubscribe from this group and stop receiving emails from it, send an email to <a href="mailto:HomotopyTypeTheory+unsubscribe@googlegroups.com">HomotopyTypeTheory+unsubscribe@googlegroups.com</a>.<br /> To view this discussion on the web visit <a href="https://groups.google.com/d/msgid/HomotopyTypeTheory/CAA520ftr3b4cGi4vXdmJL-GbAVksV9ggrTAmqZ4E75P-ch1hVQ%40mail.gmail.com?utm_medium=email&utm_source=footer">https://groups.google.com/d/msgid/HomotopyTypeTheory/CAA520ftr3b4cGi4vXdmJL-GbAVksV9ggrTAmqZ4E75P-ch1hVQ%40mail.gmail.com</a>.<br />

[-- Attachment #1.1: Type: text/plain, Size: 8618 bytes --] This discussion reminds me of this question: https://mathoverflow.net/questions/336191/cauchy-reals-and-dedekind-reals-satisfy-the-same-mathematical-theorems/336233?noredirect=1#comment840649_336233 M. On Sunday, 3 November 2019 19:46:32 UTC, Valery Isaev wrote: > > 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 <shu...@sandiego.edu > <javascript:>>: > >> 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 <drober...@gmail.com >> <javascript:>> 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 <b.a.w...@gmail.com >> <javascript:>> 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 <b.a.w...@gmail.com >> <javascript:>> 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 < >> ze...@fromzerotoinfinity.xyz <javascript:>> 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 >> <javascript:>. >> >>>> 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 <javascript:> >> . >> >> 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 <javascript:> >> . >> > 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 <javascript:>. >> 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. 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/2ca6c47c-6196-4b45-b82c-db79b2b6568c%40googlegroups.com. [-- Attachment #1.2: Type: text/html, Size: 15499 bytes --] <div dir="ltr">This discussion reminds me of this question: https://mathoverflow.net/questions/336191/cauchy-reals-and-dedekind-reals-satisfy-the-same-mathematical-theorems/336233?noredirect=1#comment840649_336233<div><br></div><div>M.<br><br>On Sunday, 3 November 2019 19:46:32 UTC, Valery Isaev wrote:<blockquote class="gmail_quote" style="margin: 0;margin-left: 0.8ex;border-left: 1px #ccc solid;padding-left: 1ex;"><div dir="ltr"><div>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 <i>properties</i> 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.</div><br clear="all"><div><div dir="ltr"><div dir="ltr"><div><div>Regards,</div><div dir="ltr">Valery Isaev<br></div></div></div></div></div><br></div><br><div class="gmail_quote"><div dir="ltr">вс, 3 нояб. 2019 г. в 22:13, Michael Shulman <<a href="javascript:" target="_blank" gdf-obfuscated-mailto="OuRI0sBOAgAJ" rel="nofollow" onmousedown="this.href='javascript:';return true;" onclick="this.href='javascript:';return true;">shu...@sandiego.edu</a>>:<br></div><blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left:1px solid rgb(204,204,204);padding-left:1ex">But does univalence a la Book HoTT *actually* make it easier to reason<br> about such things? It allows us to write "=" rather than "\cong", but<br> to construct such an equality we have to construct an isomorphism<br> first, and to *use* such an equality we have to transport along it,<br> and then we get lots of univalence-redexes that we have to manually<br> reduce away. My experience formalizing math in HoTT/Coq is that it's<br> much easier if you *avoid* turning equivalences into equalities except<br> when absolutely necessary. (I don't have any experience formalizing<br> math in a cubical proof assistant, but in that case at least you<br> wouldn't have to manually reduce the univalence-redexes -- although it<br> seems to me you'd still have to construct the isomorphism before you<br> can apply univalence to make it an equality.)<br> <br> On Sun, Nov 3, 2019 at 3:57 AM David Roberts <<a href="javascript:" target="_blank" gdf-obfuscated-mailto="OuRI0sBOAgAJ" rel="nofollow" onmousedown="this.href='javascript:';return true;" onclick="this.href='javascript:';return true;">drober...@gmail.com</a>> wrote:<br> ><br> > 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.<br> ><br> > David<br> ><br> > On Sun, 3 Nov 2019, 10:08 PM Bas Spitters <<a href="javascript:" target="_blank" gdf-obfuscated-mailto="OuRI0sBOAgAJ" rel="nofollow" onmousedown="this.href='javascript:';return true;" onclick="this.href='javascript:';return true;">b.a.w...@gmail.com</a>> wrote:<br> >><br> >> There's also VV homotopy lambda calculus, which he later abandoned for MLTT:<br> >> <a href="https://www.math.ias.edu/~vladimir/Site3/Univalent_Foundations_files/Hlambda_short_current.pdf" rel="nofollow" target="_blank" onmousedown="this.href='https://www.google.com/url?q\x3dhttps%3A%2F%2Fwww.math.ias.edu%2F~vladimir%2FSite3%2FUnivalent_Foundations_files%2FHlambda_short_current.pdf\x26sa\x3dD\x26sntz\x3d1\x26usg\x3dAFQjCNGgR_wCvme0B7Sh5s7fCKpUG45TAg';return true;" onclick="this.href='https://www.google.com/url?q\x3dhttps%3A%2F%2Fwww.math.ias.edu%2F~vladimir%2FSite3%2FUnivalent_Foundations_files%2FHlambda_short_current.pdf\x26sa\x3dD\x26sntz\x3d1\x26usg\x3dAFQjCNGgR_wCvme0B7Sh5s7fCKpUG45TAg';return true;">https://www.math.ias.edu/~<wbr>vladimir/Site3/Univalent_<wbr>Foundations_files/Hlambda_<wbr>short_current.pdf</a><br> >><br> >> On Sun, Oct 27, 2019 at 6:22 PM Bas Spitters <<a href="javascript:" target="_blank" gdf-obfuscated-mailto="OuRI0sBOAgAJ" rel="nofollow" onmousedown="this.href='javascript:';return true;" onclick="this.href='javascript:';return true;">b.a.w...@gmail.com</a>> wrote:<br> >>><br> >>> I believe it refers to his 2-theories:<br> >>> <a href="https://www.ias.edu/ideas/2014/voevodsky-origins" rel="nofollow" target="_blank" onmousedown="this.href='https://www.google.com/url?q\x3dhttps%3A%2F%2Fwww.ias.edu%2Fideas%2F2014%2Fvoevodsky-origins\x26sa\x3dD\x26sntz\x3d1\x26usg\x3dAFQjCNEAo7GxDHRN5NR9FukMwKHtnnvj0g';return true;" onclick="this.href='https://www.google.com/url?q\x3dhttps%3A%2F%2Fwww.ias.edu%2Fideas%2F2014%2Fvoevodsky-origins\x26sa\x3dD\x26sntz\x3d1\x26usg\x3dAFQjCNEAo7GxDHRN5NR9FukMwKHtnnvj0g';return true;">https://www.ias.edu/ideas/<wbr>2014/voevodsky-origins</a><br> >>><br> >>> On Sun, Oct 27, 2019 at 3:41 PM Nicolas Alexander Schmidt <<a href="javascript:" target="_blank" gdf-obfuscated-mailto="OuRI0sBOAgAJ" rel="nofollow" onmousedown="this.href='javascript:';return true;" onclick="this.href='javascript:';return true;">ze...@fromzerotoinfinity.xyz</a>> wrote:<br> >>>><br> >>>> In [this](<a href="https://www.youtube.com/watch?v=zw6NcwME7yI&t=1680" rel="nofollow" target="_blank" onmousedown="this.href='https://www.youtube.com/watch?v\x3dzw6NcwME7yI\x26t\x3d1680';return true;" onclick="this.href='https://www.youtube.com/watch?v\x3dzw6NcwME7yI\x26t\x3d1680';return true;">https://www.youtube.<wbr>com/watch?v=zw6NcwME7yI&t=1680</a><wbr>) 2014 talk<br> >>>> at IAS, Voevodsky talks about the history of his project of "univalent<br> >>>> mathematics" and his motivation for starting it. Namely, he mentions<br> >>>> that he found existing proof assistants at that time (in 2000) to be<br> >>>> impractical for the kinds of mathematics he was interested in.<br> >>>><br> >>>> Unfortunately, he doesn't go into details of what mathematics he was<br> >>>> exactly interested in (I'm guessing something to do with homotopy<br> >>>> theory) or why exactly existing proof assistants weren't practical for<br> >>>> formalizing them. Judging by the things he mentions in his talk, it<br> >>>> seems that (roughly) his rejection of those proof assistants was based<br> >>>> on the view that predicate logic + ZFC is not expressive enough. In<br> >>>> other words, there is too much lossy encoding needed in order to<br> >>>> translate from the platonic world of mathematical ideas to this formal<br> >>>> language.<br> >>>><br> >>>> Comparing the situation to computer programming languages, one might say<br> >>>> that predicate logic is like Assembly in that even though everything can<br> >>>> be encoded in that language, it is not expressive enough to directly<br> >>>> talk about higher level concepts, diminishing its practical value for<br> >>>> reasoning about mathematics. In particular, those systems are not<br> >>>> adequate for *interactive* development of *new* mathematics (as opposed<br> >>>> to formalization of existing theories).<br> >>>><br> >>>> Perhaps I am just misinterpreting what Voevodsky said. In this case, I<br> >>>> hope someone can correct me. However even if this wasn't *his* view, to<br> >>>> me it seems to be a view held implicitly in the HoTT community. In any<br> >>>> case, it's a view that one might reasonably hold.<br> >>>><br> >>>> However I wonder how reasonable that view actually is, i.e. whether e.g.<br> >>>> Mizar really is that much more impractical than HoTT-Coq or Agda, given<br> >>>> that people have been happily formalizing mathematics in it for 46 years<br> >>>> now. And, even though by browsing the contents of "Formalized<br> >>>> Mathematics" one can get the impression that the work consists mostly of<br> >>>> formalizing early 20th century mathematics, neither the UniMath nor the<br> >>>> HoTT library for example contain a proof of Fubini's theorem.<br> >>>><br> >>>> So, to put this into one concrete question, how (if at all) is HoTT-Coq<br> >>>> more practical than Mizar for the purpose of formalizing mathematics,<br> >>>> outside the specific realm of synthetic homotopy theory?<br> >>>><br> >>>><br> >>>> --<br> >>>><br> >>>> Nicolas<br> >>>><br> >>>><br> >>>> --<br> >>>> You received this message because you are subscribed to the Google Groups "Homotopy Type Theory" group.<br> >>>> To unsubscribe from this group and stop receiving emails from it, send an email to <a href="javascript:" target="_blank" gdf-obfuscated-mailto="OuRI0sBOAgAJ" rel="nofollow" onmousedown="this.href='javascript:';return true;" onclick="this.href='javascript:';return true;">HomotopyTypeTheory+<wbr>unsubscribe@googlegroups.com</a>.<br> >>>> To view this discussion on the web visit <a href="https://groups.google.com/d/msgid/HomotopyTypeTheory/e491d38b-b50a-6172-dca9-40d45fe1b6d2%40fromzerotoinfinity.xyz" rel="nofollow" target="_blank" onmousedown="this.href='https://groups.google.com/d/msgid/HomotopyTypeTheory/e491d38b-b50a-6172-dca9-40d45fe1b6d2%40fromzerotoinfinity.xyz';return true;" onclick="this.href='https://groups.google.com/d/msgid/HomotopyTypeTheory/e491d38b-b50a-6172-dca9-40d45fe1b6d2%40fromzerotoinfinity.xyz';return true;">https://groups.google.com/d/<wbr>msgid/HomotopyTypeTheory/<wbr>e491d38b-b50a-6172-dca9-<wbr>40d45fe1b6d2%<wbr>40fromzerotoinfinity.xyz</a>.<br> >><br> >> --<br> >> You received this message because you are subscribed to the Google Groups "Homotopy Type Theory" group.<br> >> To unsubscribe from this group and stop receiving emails from it, send an email to <a href="javascript:" target="_blank" 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[-- Attachment #1: Type: text/plain, Size: 12458 bytes --] On Sun, 3 Nov 2019 at 19:13, Michael Shulman <shulman@sandiego.edu> wrote: > But does univalence a la Book HoTT *actually* make it easier to reason > about such things? I think this is a really interesting and important question. I guess David was referring to my scheme fail of 2018. I wanted to formalise the notion of a scheme a la Grothendieck and prove that if R was a commutative ring then Spec(R) was a scheme [I know it's a definition, but many mathematicians do seem to call it a theorem, in our ignorance]. I showed an undergraduate a specific lemma in ring theory ( https://stacks.math.columbia.edu/tag/00EJ) and said "that's what I want" and they formalised it for me. And then it turned out that I wanted something else: I didn't have R_f, I had something "canonically isomorphic" to it, a phrase we mathematicians like to pull out when the going gets tough and we can't be bothered to check that any more diagrams commute. By this point it was too late to turn back, and so I had to prove that 20 diagrams commuted and it wasn't much fun. I then got an MSc student to redo everything using universal properties more carefully in Lean and it worked like a dream https://github.com/ramonfmir/lean-scheme. A lot of people said to me at the time "you wouldn't have had this problem if you'd been using HoTT instead of DTT" and my response to this is still the (intentionally) provocative "go ahead and define schemes and prove that Spec(R) is a scheme in some HoTT system, and show me how it's better; note that we did have a problem, but we solved it in DTT". I would be particularly interested to see schemes done in Arend, because it always felt funny to me using UniMath in Coq (and similarly it feels funny to me to do HoTT in Lean 3 -- in both cases it could be argued that it's using a system to do something it wasn't designed to do). I think it's easy to theorise about this sort of thing but until it happens in practice in one or more of the HoTT systems I don't think we will understand the issue properly (or, more precisely, I don't think I will understand the issue properly). I have had extensive discussions with Martin Escardo about HoTT and he has certainly given me hope, but on the Lean chat I think people assumed schemes would be easy in Lean (I certainly did) and then we ran into this unexpected problem (which univalence is probably designed to solve), so the question is whether a univalent type theory runs into a different unexpected problem -- you push the carpet down somewhere and it pops up somewhere else. I know this is a HoTT list but the challenge is also open to the HOL people like the Isabelle/HOL experts. In contrast to HoTT theories, which I think should handle schemes fine, I think that simple type theory will have tremendous problems defining, for example, tensor products of sheaves of modules on a scheme, because these are dependent types. On the other hand my recent ArXiv paper with Commelin and Massot https://arxiv.org/abs/1910.12320 goes much further and formalises perfectoid spaces in dependent type theory. I would like the people on this list to see this as a challenge. I think that this century will see the rise of the theorem prover in mathematics and I am not naive enough to think that the one I currently use now is the one which is guaranteed to be the success story. Voevodsky was convinced that univalence was the right way to do modern mathematics but I'm doing it just fine in dependent type theory and now he's gone I really want to find someone who will take up the challenge and do some scheme theory in HoTT, but convincing professional mathematicians to get interested in this area is very difficult, and I speak as someone who's been trying to do it for two years now [I recommend you try the undergraduates instead, anyone who is interested in training people up -- plenty of undergraduates are capable of reading the definition of a scheme, if they know what rings and topological spaces are] To get back to the original question, my understanding was that Voevodsky had done a bunch of scheme theory and it had got him a Fields medal and it was this mathematics which he was interested in at the time. He wanted to formalise his big theorem, just like Hales did. Unfortunately he was historically earlier, and his mathematics involved far more conceptual objects than spheres in 3-space, so it was a much taller order. All the evidence is there to suggest that over the next 15 or so years his interests changed. The clearest evidence, in my mind, is that there is no definition of a scheme in UniMath. Moreover his story in his Cambridge talk https://www.newton.ac.uk/seminar/20170710113012301 about asking Suslin to reprove one of his results without using the axiom of choice (46 minutes in) kind of shocked me -- Suslin does not care about mathematics without choice, and the vast majority of mathematicians employed in mathematics departments feel the same, although I'm well aware that constructivism is taken more seriously on this list. I think it is interesting that Voevodsky failed to prove a constructive version of his theorem, because I think that some mathematics is better off not being constructive. It is exactly the interaction between constructivism and univalence which I do not understand well, and I think that a very good way to investigate it would be to do some highly non-constructive modern mathematics in a univalent type theory. Kevin PS many thanks to the people who have emailed me in the past telling me about how in the past I have used "HoTT", "univalence", "UniMath", interchangeably and incorrectly. Hopefully I am getting better but I am still keen to hear anything which I'm saying which is imprecise or incorrect. > 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 <droberts.65537@gmail.com> > 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 <b.a.w.spitters@gmail.com> > 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 <b.a.w.spitters@gmail.com> > 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. 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/CAH52Xb3s0%2BvweUaSQBMBNLa5mRc9F1jrsg2sSoFmcE_4%3DdAt1w%40mail.gmail.com. [-- Attachment #2: Type: text/html, Size: 16518 bytes --] <div dir="ltr"><div dir="ltr"><br></div><br><div class="gmail_quote"><div dir="ltr" class="gmail_attr">On Sun, 3 Nov 2019 at 19:13, Michael Shulman <<a href="mailto:shulman@sandiego.edu" target="_blank">shulman@sandiego.edu</a>> wrote:<br></div><blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left:1px solid rgb(204,204,204);padding-left:1ex">But does univalence a la Book HoTT *actually* make it easier to reason<br> about such things? </blockquote><div><br></div><div>I think this is a really interesting and important question.</div><div><br></div><div>I guess David was referring to my scheme fail of 2018. I wanted to formalise the notion of a scheme a la Grothendieck and prove that if R was a commutative ring then Spec(R) was a scheme [I know it's a definition, but many mathematicians do seem to call it a theorem, in our ignorance]. I showed an undergraduate a specific lemma in ring theory (<a href="https://stacks.math.columbia.edu/tag/00EJ" target="_blank">https://stacks.math.columbia.edu/tag/00EJ</a>) and said "that's what I want" and they formalised it for me. And then it turned out that I wanted something else: I didn't have R_f, I had something "canonically isomorphic" to it, a phrase we mathematicians like to pull out when the going gets tough and we can't be bothered to check that any more diagrams commute. By this point it was too late to turn back, and so I had to prove that 20 diagrams commuted and it wasn't much fun. I then got an MSc student to redo everything using universal properties more carefully in Lean and it worked like a dream <a href="https://github.com/ramonfmir/lean-scheme">https://github.com/ramonfmir/lean-scheme</a>. A lot of people said to me at the time "you wouldn't have had this problem if you'd been using HoTT instead of DTT" and my response to this is still the (intentionally) provocative "go ahead and define schemes and prove that Spec(R) is a scheme in some HoTT system, and show me how it's better; note that we did have a problem, but we solved it in DTT". I would be particularly interested to see schemes done in Arend, because it always felt funny to me using UniMath in Coq (and similarly it feels funny to me to do HoTT in Lean 3 -- in both cases it could be argued that it's using a system to do something it wasn't designed to do). I think it's easy to theorise about this sort of thing but until it happens in practice in one or more of the HoTT systems I don't think we will understand the issue properly (or, more precisely, I don't think I will understand the issue properly). I have had extensive discussions with Martin Escardo about HoTT and he has certainly given me hope, but on the Lean chat I think people assumed schemes would be easy in Lean (I certainly did) and then we ran into this unexpected problem (which univalence is probably designed to solve), so the question is whether a univalent type theory runs into a different unexpected problem -- you push the carpet down somewhere and it pops up somewhere else.<br></div><div><br></div><div>I know this is a HoTT list but the challenge is also open to the HOL people like the Isabelle/HOL experts. In contrast to HoTT theories, which I think should handle schemes fine, I think that simple type theory will have tremendous problems defining, for example, tensor products of sheaves of modules on a scheme, because these are dependent types. On the other hand my recent ArXiv paper with Commelin and Massot <a href="https://arxiv.org/abs/1910.12320">https://arxiv.org/abs/1910.12320</a> goes much further and formalises perfectoid spaces in dependent type theory. I would like the people on this list to see this as a challenge. I think that this century will see the rise of the theorem prover in mathematics and I am not naive enough to think that the one I currently use now is the one which is guaranteed to be the success story. Voevodsky was convinced that univalence was the right way to do modern mathematics but I'm doing it just fine in dependent type theory and now he's gone I really want to find someone who will take up the challenge and do some scheme theory in HoTT, but convincing professional mathematicians to get interested in this area is very difficult, and I speak as someone who's been trying to do it for two years now [I recommend you try the undergraduates instead, anyone who is interested in training people up -- plenty of undergraduates are capable of reading the definition of a scheme, if they know what rings and topological spaces are]<br></div><div><br></div><div>To get back to the original question, my understanding was that Voevodsky had done a bunch of scheme theory and it had got him a Fields medal and it was this mathematics which he was interested in at the time. He wanted to formalise his big theorem, just like Hales did. Unfortunately he was historically earlier, and his mathematics involved far more conceptual objects than spheres in 3-space, so it was a much taller order. All the evidence is there to suggest that over the next 15 or so years his interests changed. The clearest evidence, in my mind, is that there is no definition of a scheme in UniMath. Moreover his story in his Cambridge talk <a href="https://www.newton.ac.uk/seminar/20170710113012301">https://www.newton.ac.uk/seminar/20170710113012301</a> about asking Suslin to reprove one of his results without using the axiom of choice (46 minutes in) kind of shocked me -- Suslin does not care about mathematics without choice, and the vast majority of mathematicians employed in mathematics departments feel the same, although I'm well aware that constructivism is taken more seriously on this list. I think it is interesting that Voevodsky failed to prove a constructive version of his theorem, because I think that some mathematics is better off not being constructive. It is exactly the interaction between constructivism and univalence which I do not understand well, and I think that a very good way to investigate it would be to do some highly non-constructive modern mathematics in a univalent type theory.<br></div><div><br></div><div>Kevin</div><div><br></div><div>PS many thanks to the people who have emailed me in the past telling me about how in the past I have used "HoTT", "univalence", "UniMath", interchangeably and incorrectly. Hopefully I am getting better but I am still keen to hear anything which I'm saying which is imprecise or incorrect.<br></div><div><br></div><div> </div><blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left:1px solid rgb(204,204,204);padding-left:1ex">It allows us to write "=" rather than "\cong", but<br> to construct such an equality we have to construct an isomorphism<br> first, and to *use* such an equality we have to transport along it,<br> and then we get lots of univalence-redexes that we have to manually<br> reduce away. My experience formalizing math in HoTT/Coq is that it's<br> much easier if you *avoid* turning equivalences into equalities except<br> when absolutely necessary. (I don't have any experience formalizing<br> math in a cubical proof assistant, but in that case at least you<br> wouldn't have to manually reduce the univalence-redexes -- although it<br> seems to me you'd still have to construct the isomorphism before you<br> can apply univalence to make it an equality.)<br> <br> On Sun, Nov 3, 2019 at 3:57 AM David Roberts <<a href="mailto:droberts.65537@gmail.com" target="_blank">droberts.65537@gmail.com</a>> wrote:<br> ><br> > 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.<br> ><br> > David<br> ><br> > On Sun, 3 Nov 2019, 10:08 PM Bas Spitters <<a href="mailto:b.a.w.spitters@gmail.com" target="_blank">b.a.w.spitters@gmail.com</a>> wrote:<br> >><br> >> There's also VV homotopy lambda calculus, which he later abandoned for MLTT:<br> >> <a href="https://www.math.ias.edu/~vladimir/Site3/Univalent_Foundations_files/Hlambda_short_current.pdf" rel="noreferrer" target="_blank">https://www.math.ias.edu/~vladimir/Site3/Univalent_Foundations_files/Hlambda_short_current.pdf</a><br> >><br> >> On Sun, Oct 27, 2019 at 6:22 PM Bas Spitters <<a href="mailto:b.a.w.spitters@gmail.com" target="_blank">b.a.w.spitters@gmail.com</a>> wrote:<br> >>><br> >>> I believe it refers to his 2-theories:<br> >>> <a href="https://www.ias.edu/ideas/2014/voevodsky-origins" rel="noreferrer" target="_blank">https://www.ias.edu/ideas/2014/voevodsky-origins</a><br> >>><br> >>> On Sun, Oct 27, 2019 at 3:41 PM Nicolas Alexander Schmidt <<a href="mailto:zero@fromzerotoinfinity.xyz" target="_blank">zero@fromzerotoinfinity.xyz</a>> wrote:<br> >>>><br> >>>> In [this](<a href="https://www.youtube.com/watch?v=zw6NcwME7yI&t=1680" rel="noreferrer" target="_blank">https://www.youtube.com/watch?v=zw6NcwME7yI&t=1680</a>) 2014 talk<br> >>>> at IAS, Voevodsky talks about the history of his project of "univalent<br> >>>> mathematics" and his motivation for starting it. Namely, he mentions<br> >>>> that he found existing proof assistants at that time (in 2000) to be<br> >>>> impractical for the kinds of mathematics he was interested in.<br> >>>><br> >>>> Unfortunately, he doesn't go into details of what mathematics he was<br> >>>> exactly interested in (I'm guessing something to do with homotopy<br> >>>> theory) or why exactly existing proof assistants weren't practical for<br> >>>> formalizing them. Judging by the things he mentions in his talk, it<br> >>>> seems that (roughly) his rejection of those proof assistants was based<br> >>>> on the view that predicate logic + ZFC is not expressive enough. In<br> >>>> other words, there is too much lossy encoding needed in order to<br> >>>> translate from the platonic world of mathematical ideas to this formal<br> >>>> language.<br> >>>><br> >>>> Comparing the situation to computer programming languages, one might say<br> >>>> that predicate logic is like Assembly in that even though everything can<br> >>>> be encoded in that language, it is not expressive enough to directly<br> >>>> talk about higher level concepts, diminishing its practical value for<br> >>>> reasoning about mathematics. In particular, those systems are not<br> >>>> adequate for *interactive* development of *new* mathematics (as opposed<br> >>>> to formalization of existing theories).<br> >>>><br> >>>> Perhaps I am just misinterpreting what Voevodsky said. In this case, I<br> >>>> hope someone can correct me. However even if this wasn't *his* view, to<br> >>>> me it seems to be a view held implicitly in the HoTT community. In any<br> >>>> case, it's a view that one might reasonably hold.<br> >>>><br> >>>> However I wonder how reasonable that view actually is, i.e. whether e.g.<br> >>>> Mizar really is that much more impractical than HoTT-Coq or Agda, given<br> >>>> that people have been happily formalizing mathematics in it for 46 years<br> >>>> now. And, even though by browsing the contents of "Formalized<br> >>>> Mathematics" one can get the impression that the work consists mostly of<br> >>>> formalizing early 20th century mathematics, neither the UniMath nor the<br> >>>> HoTT library for example contain a proof of Fubini's theorem.<br> >>>><br> >>>> So, to put this into one concrete question, how (if at all) is HoTT-Coq<br> >>>> more practical than Mizar for the purpose of formalizing mathematics,<br> >>>> outside the specific realm of synthetic homotopy theory?<br> >>>><br> >>>><br> >>>> --<br> >>>><br> >>>> Nicolas<br> >>>><br> >>>><br> >>>> --<br> >>>> You received this message because you are subscribed to the Google Groups "Homotopy Type Theory" group.<br> >>>> To unsubscribe from this group and stop receiving emails from it, send an email to <a href="mailto:HomotopyTypeTheory%2Bunsubscribe@googlegroups.com" target="_blank">HomotopyTypeTheory+unsubscribe@googlegroups.com</a>.<br> >>>> To view this discussion on the web visit <a href="https://groups.google.com/d/msgid/HomotopyTypeTheory/e491d38b-b50a-6172-dca9-40d45fe1b6d2%40fromzerotoinfinity.xyz" rel="noreferrer" target="_blank">https://groups.google.com/d/msgid/HomotopyTypeTheory/e491d38b-b50a-6172-dca9-40d45fe1b6d2%40fromzerotoinfinity.xyz</a>.<br> >><br> >> --<br> >> You received this message because you are subscribed to the Google Groups "Homotopy Type Theory" group.<br> >> To unsubscribe from this group and stop receiving emails from it, send an email to <a href="mailto:HomotopyTypeTheory%2Bunsubscribe@googlegroups.com" target="_blank">HomotopyTypeTheory+unsubscribe@googlegroups.com</a>.<br> >> To view this discussion on the web visit <a href="https://groups.google.com/d/msgid/HomotopyTypeTheory/CAOoPQuRQPMkCFKYtAbB%2BpNK90XtFk%2BaVT_aY59U_-9t17sBzeA%40mail.gmail.com" rel="noreferrer" target="_blank">https://groups.google.com/d/msgid/HomotopyTypeTheory/CAOoPQuRQPMkCFKYtAbB%2BpNK90XtFk%2BaVT_aY59U_-9t17sBzeA%40mail.gmail.com</a>.<br> ><br> > --<br> > You received this message because you are subscribed to the Google Groups "Homotopy Type Theory" group.<br> > To unsubscribe from this group and stop receiving emails from it, send an email to <a href="mailto:HomotopyTypeTheory%2Bunsubscribe@googlegroups.com" target="_blank">HomotopyTypeTheory+unsubscribe@googlegroups.com</a>.<br> > To view this discussion on the web visit <a href="https://groups.google.com/d/msgid/HomotopyTypeTheory/CAFL%2BZM_%3D%3DiLS16Vy7sGiEqNkCxOMYL4j%2BZFqKv5uJ-ivkuemEg%40mail.gmail.com" rel="noreferrer" target="_blank">https://groups.google.com/d/msgid/HomotopyTypeTheory/CAFL%2BZM_%3D%3DiLS16Vy7sGiEqNkCxOMYL4j%2BZFqKv5uJ-ivkuemEg%40mail.gmail.com</a>.<br> <br> -- <br> You received this message because you are subscribed to the Google Groups "Homotopy Type Theory" group.<br> To unsubscribe from this group and stop receiving emails from it, send an email to <a href="mailto:HomotopyTypeTheory%2Bunsubscribe@googlegroups.com" target="_blank">HomotopyTypeTheory+unsubscribe@googlegroups.com</a>.<br> To view this discussion on the web visit <a href="https://groups.google.com/d/msgid/HomotopyTypeTheory/CAOvivQz47kSm9WbKDmUsndrpAJMkNwiWmVABqOFrVqyTOvSAbw%40mail.gmail.com" rel="noreferrer" target="_blank">https://groups.google.com/d/msgid/HomotopyTypeTheory/CAOvivQz47kSm9WbKDmUsndrpAJMkNwiWmVABqOFrVqyTOvSAbw%40mail.gmail.com</a>.<br> </blockquote></div></div> <p></p> -- <br /> You received this message because you are subscribed to the Google Groups "Homotopy Type Theory" group.<br /> To unsubscribe from this group and stop receiving emails from it, send an email to <a href="mailto:HomotopyTypeTheory+unsubscribe@googlegroups.com">HomotopyTypeTheory+unsubscribe@googlegroups.com</a>.<br /> To view this discussion on the web visit <a href="https://groups.google.com/d/msgid/HomotopyTypeTheory/CAH52Xb3s0%2BvweUaSQBMBNLa5mRc9F1jrsg2sSoFmcE_4%3DdAt1w%40mail.gmail.com?utm_medium=email&utm_source=footer">https://groups.google.com/d/msgid/HomotopyTypeTheory/CAH52Xb3s0%2BvweUaSQBMBNLa5mRc9F1jrsg2sSoFmcE_4%3DdAt1w%40mail.gmail.com</a>.<br />

Valery has a good point that transporting properties along an equivalence is definitely somewhere that Book HoTT could get you some mileage. But I suspect that a more significant advantage would come from using a cubical type theory in which transport computes (as long as the equivalence was defined constructively -- a good reason to care about being constructive even putting aside philosophy and internal languages). I'd be curious to know what those 20 diagrams were. FWIW, I think that nowadays Coq *is* designed for Book HoTT, certainly more than Lean 3 is. My understanding is that Lean 3 is actually technically incompatible with univalence, whereas over the past decade the Coq developers have incorporated various new features requested by the HoTT community to improve compatibility, and the HoTT Coq library is I believe one of the test suites that new Coq versions are tested against to ensure that breakage is dealt with on one side or another. I'm not sure how a proof assistant could be more designed for Book HoTT than modern Coq and Agda are. (Arend is not designed for Book HoTT, but for a flavor of HoTT that's partway to a cubical theory, with an interval type representing paths.) On Mon, Nov 4, 2019 at 10:43 AM Kevin Buzzard <kevin.m.buzzard@gmail.com> wrote: > > > > On Sun, 3 Nov 2019 at 19:13, Michael Shulman <shulman@sandiego.edu> wrote: >> >> But does univalence a la Book HoTT *actually* make it easier to reason >> about such things? > > > I think this is a really interesting and important question. > > I guess David was referring to my scheme fail of 2018. I wanted to formalise the notion of a scheme a la Grothendieck and prove that if R was a commutative ring then Spec(R) was a scheme [I know it's a definition, but many mathematicians do seem to call it a theorem, in our ignorance]. I showed an undergraduate a specific lemma in ring theory (https://stacks.math.columbia.edu/tag/00EJ) and said "that's what I want" and they formalised it for me. And then it turned out that I wanted something else: I didn't have R_f, I had something "canonically isomorphic" to it, a phrase we mathematicians like to pull out when the going gets tough and we can't be bothered to check that any more diagrams commute. By this point it was too late to turn back, and so I had to prove that 20 diagrams commuted and it wasn't much fun. I then got an MSc student to redo everything using universal properties more carefully in Lean and it worked like a dream https://github.com/ramonfmir/lean-scheme. A lot of people said to me at the time "you wouldn't have had this problem if you'd been using HoTT instead of DTT" and my response to this is still the (intentionally) provocative "go ahead and define schemes and prove that Spec(R) is a scheme in some HoTT system, and show me how it's better; note that we did have a problem, but we solved it in DTT". I would be particularly interested to see schemes done in Arend, because it always felt funny to me using UniMath in Coq (and similarly it feels funny to me to do HoTT in Lean 3 -- in both cases it could be argued that it's using a system to do something it wasn't designed to do). I think it's easy to theorise about this sort of thing but until it happens in practice in one or more of the HoTT systems I don't think we will understand the issue properly (or, more precisely, I don't think I will understand the issue properly). I have had extensive discussions with Martin Escardo about HoTT and he has certainly given me hope, but on the Lean chat I think people assumed schemes would be easy in Lean (I certainly did) and then we ran into this unexpected problem (which univalence is probably designed to solve), so the question is whether a univalent type theory runs into a different unexpected problem -- you push the carpet down somewhere and it pops up somewhere else. > > I know this is a HoTT list but the challenge is also open to the HOL people like the Isabelle/HOL experts. In contrast to HoTT theories, which I think should handle schemes fine, I think that simple type theory will have tremendous problems defining, for example, tensor products of sheaves of modules on a scheme, because these are dependent types. On the other hand my recent ArXiv paper with Commelin and Massot https://arxiv.org/abs/1910.12320 goes much further and formalises perfectoid spaces in dependent type theory. I would like the people on this list to see this as a challenge. I think that this century will see the rise of the theorem prover in mathematics and I am not naive enough to think that the one I currently use now is the one which is guaranteed to be the success story. Voevodsky was convinced that univalence was the right way to do modern mathematics but I'm doing it just fine in dependent type theory and now he's gone I really want to find someone who will take up the challenge and do some scheme theory in HoTT, but convincing professional mathematicians to get interested in this area is very difficult, and I speak as someone who's been trying to do it for two years now [I recommend you try the undergraduates instead, anyone who is interested in training people up -- plenty of undergraduates are capable of reading the definition of a scheme, if they know what rings and topological spaces are] > > To get back to the original question, my understanding was that Voevodsky had done a bunch of scheme theory and it had got him a Fields medal and it was this mathematics which he was interested in at the time. He wanted to formalise his big theorem, just like Hales did. Unfortunately he was historically earlier, and his mathematics involved far more conceptual objects than spheres in 3-space, so it was a much taller order. All the evidence is there to suggest that over the next 15 or so years his interests changed. The clearest evidence, in my mind, is that there is no definition of a scheme in UniMath. Moreover his story in his Cambridge talk https://www.newton.ac.uk/seminar/20170710113012301 about asking Suslin to reprove one of his results without using the axiom of choice (46 minutes in) kind of shocked me -- Suslin does not care about mathematics without choice, and the vast majority of mathematicians employed in mathematics departments feel the same, although I'm well aware that constructivism is taken more seriously on this list. I think it is interesting that Voevodsky failed to prove a constructive version of his theorem, because I think that some mathematics is better off not being constructive. It is exactly the interaction between constructivism and univalence which I do not understand well, and I think that a very good way to investigate it would be to do some highly non-constructive modern mathematics in a univalent type theory. > > Kevin > > PS many thanks to the people who have emailed me in the past telling me about how in the past I have used "HoTT", "univalence", "UniMath", interchangeably and incorrectly. Hopefully I am getting better but I am still keen to hear anything which I'm saying which is imprecise or incorrect. > > >> >> 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 <droberts.65537@gmail.com> 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 <b.a.w.spitters@gmail.com> 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 <b.a.w.spitters@gmail.com> 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. > 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/CAH52Xb3s0%2BvweUaSQBMBNLa5mRc9F1jrsg2sSoFmcE_4%3DdAt1w%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/CAOvivQy4zN1wc%3Dw5-%2Beu2hBwtbUC-gtzjuYabiZWdb4yKZ7NUw%40mail.gmail.com.

Dear Bas, Michael, David, Valery, Martín and Kevin, thank you all for your replies. When I posed my question, I didn't expect there to be so much room for debate. In particular, I am surprised that it seems not at all clear how much univalence actually helps the practicality of doing formal proofs (as opposed to any theoretical benefits). Kevin, I would like to second Michael's interest in seeing these 20 commutative diagrams. Moreover, I'd also be very interested in seeing your "spaghetti code" (quote from the slides of your Big Proof talk): it seems it should be informative to see where your initial approach went wrong, and how much these problems and their solution had anything to do at all with the formal system you were working in. Are your original files perhaps available somewhere? -- 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/bc1a186e-4d33-0296-4b1b-b09ee8188037%40fromzerotoinfinity.xyz.

Hi, Going back a step from DTT vs HoTT... > I think that simple type theory will have tremendous problems defining, for example, tensor products of sheaves of modules on a scheme, because these are dependent types. this is probably where VV found himself just under 20 years ago. I don't know what the state of play was around 2000, but certainly if he didn't know about Coq, then VV's options for formal proof systems were looking more like Mizar, MetaMath and similar, which are probably even worse than simple type-theory systems: imagine trying to formally define a scheme in the language of ZFC! Best regards, David David Roberts Webpage: https://ncatlab.org/nlab/show/David+Roberts Blog: https://thehighergeometer.wordpress.com On Tue, 5 Nov 2019 at 05:13, Kevin Buzzard <kevin.m.buzzard@gmail.com> wrote: > > > > On Sun, 3 Nov 2019 at 19:13, Michael Shulman <shulman@sandiego.edu> wrote: >> >> But does univalence a la Book HoTT *actually* make it easier to reason >> about such things? > > > I think this is a really interesting and important question. > > I guess David was referring to my scheme fail of 2018. I wanted to formalise the notion of a scheme a la Grothendieck and prove that if R was a commutative ring then Spec(R) was a scheme [I know it's a definition, but many mathematicians do seem to call it a theorem, in our ignorance]. I showed an undergraduate a specific lemma in ring theory (https://stacks.math.columbia.edu/tag/00EJ) and said "that's what I want" and they formalised it for me. And then it turned out that I wanted something else: I didn't have R_f, I had something "canonically isomorphic" to it, a phrase we mathematicians like to pull out when the going gets tough and we can't be bothered to check that any more diagrams commute. By this point it was too late to turn back, and so I had to prove that 20 diagrams commuted and it wasn't much fun. I then got an MSc student to redo everything using universal properties more carefully in Lean and it worked like a dream https://github.com/ramonfmir/lean-scheme. A lot of people said to me at the time "you wouldn't have had this problem if you'd been using HoTT instead of DTT" and my response to this is still the (intentionally) provocative "go ahead and define schemes and prove that Spec(R) is a scheme in some HoTT system, and show me how it's better; note that we did have a problem, but we solved it in DTT". I would be particularly interested to see schemes done in Arend, because it always felt funny to me using UniMath in Coq (and similarly it feels funny to me to do HoTT in Lean 3 -- in both cases it could be argued that it's using a system to do something it wasn't designed to do). I think it's easy to theorise about this sort of thing but until it happens in practice in one or more of the HoTT systems I don't think we will understand the issue properly (or, more precisely, I don't think I will understand the issue properly). I have had extensive discussions with Martin Escardo about HoTT and he has certainly given me hope, but on the Lean chat I think people assumed schemes would be easy in Lean (I certainly did) and then we ran into this unexpected problem (which univalence is probably designed to solve), so the question is whether a univalent type theory runs into a different unexpected problem -- you push the carpet down somewhere and it pops up somewhere else. > > I know this is a HoTT list but the challenge is also open to the HOL people like the Isabelle/HOL experts. In contrast to HoTT theories, which I think should handle schemes fine, I think that simple type theory will have tremendous problems defining, for example, tensor products of sheaves of modules on a scheme, because these are dependent types. On the other hand my recent ArXiv paper with Commelin and Massot https://arxiv.org/abs/1910.12320 goes much further and formalises perfectoid spaces in dependent type theory. I would like the people on this list to see this as a challenge. I think that this century will see the rise of the theorem prover in mathematics and I am not naive enough to think that the one I currently use now is the one which is guaranteed to be the success story. Voevodsky was convinced that univalence was the right way to do modern mathematics but I'm doing it just fine in dependent type theory and now he's gone I really want to find someone who will take up the challenge and do some scheme theory in HoTT, but convincing professional mathematicians to get interested in this area is very difficult, and I speak as someone who's been trying to do it for two years now [I recommend you try the undergraduates instead, anyone who is interested in training people up -- plenty of undergraduates are capable of reading the definition of a scheme, if they know what rings and topological spaces are] > > To get back to the original question, my understanding was that Voevodsky had done a bunch of scheme theory and it had got him a Fields medal and it was this mathematics which he was interested in at the time. He wanted to formalise his big theorem, just like Hales did. Unfortunately he was historically earlier, and his mathematics involved far more conceptual objects than spheres in 3-space, so it was a much taller order. All the evidence is there to suggest that over the next 15 or so years his interests changed. The clearest evidence, in my mind, is that there is no definition of a scheme in UniMath. Moreover his story in his Cambridge talk https://www.newton.ac.uk/seminar/20170710113012301 about asking Suslin to reprove one of his results without using the axiom of choice (46 minutes in) kind of shocked me -- Suslin does not care about mathematics without choice, and the vast majority of mathematicians employed in mathematics departments feel the same, although I'm well aware that constructivism is taken more seriously on this list. I think it is interesting that Voevodsky failed to prove a constructive version of his theorem, because I think that some mathematics is better off not being constructive. It is exactly the interaction between constructivism and univalence which I do not understand well, and I think that a very good way to investigate it would be to do some highly non-constructive modern mathematics in a univalent type theory. > > Kevin > > PS many thanks to the people who have emailed me in the past telling me about how in the past I have used "HoTT", "univalence", "UniMath", interchangeably and incorrectly. Hopefully I am getting better but I am still keen to hear anything which I'm saying which is imprecise or incorrect. > > >> >> 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 <droberts.65537@gmail.com> 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 <b.a.w.spitters@gmail.com> 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 <b.a.w.spitters@gmail.com> 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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[-- Attachment #1.1: Type: text/plain, Size: 1554 bytes --] Re: "To get back to the original question, my understanding was that Voevodsky had done a bunch of scheme theory and it had got him a Fields medal and it was this mathematics which he was interested in at the time. He wanted to formalise his big theorem, just like Hales did." I think he was more interested in formalizing things like his early work with Kapranov on higher categories, which turned out to have a mistake in it. He once told me that he wasn't interested in formalizing his proof of Bloch-Kato, because he was sure it was right. (I should have asked him at the time how he could be so sure!) Re: "The clearest evidence, in my mind, is that there is no definition of a scheme in UniMath." That's sort of accidental. In early 2014, expecting to speak at an algebraic geometry in the summer, he mentioned that one idea he had for his talk would be to formalize the definition of scheme in UniMath and speak about it. I think he was distracted from that by thinking about C-systems. The UniMath project aims at formalizing all of mathematics, so the definition of scheme is one of the next things that (still) needs to be done. -- 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/cda95637-0ab0-4897-8e38-b5ebb288a658%40googlegroups.com. [-- Attachment #1.2: Type: text/html, Size: 1913 bytes --] <div dir="ltr">Re: "To get back to the original question, my understanding was that Voevodsky had done a bunch of scheme theory and it had got him a Fields medal and it was this mathematics which he was interested in at the time. He wanted to formalise his big theorem, just like Hales did."<br><div><br></div><div>I think he was more interested in formalizing things like his early work with Kapranov on higher categories, which turned out to have a mistake in it. He once told me that he wasn't interested in formalizing his proof of Bloch-Kato, because he was sure it was right. (I should have asked him at the time how he could be so sure!)<br><br>Re: "The clearest evidence, in my mind, is that there is no definition of a scheme in UniMath."<br><br>That's sort of accidental. In early 2014, expecting to speak at an algebraic geometry in the summer, he mentioned that one idea he had for his talk would be to formalize the definition of scheme in UniMath and speak about it. I think he was distracted from that by thinking about C-systems. The UniMath project aims at formalizing all of mathematics, so the definition of scheme is one of the next things that (still) needs to be done.</div></div> <p></p> -- <br /> You received this message because you are subscribed to the Google Groups "Homotopy Type Theory" group.<br /> To unsubscribe from this group and stop receiving emails from it, send an email to <a href="mailto:HomotopyTypeTheory+unsubscribe@googlegroups.com">HomotopyTypeTheory+unsubscribe@googlegroups.com</a>.<br /> To view this discussion on the web visit <a href="https://groups.google.com/d/msgid/HomotopyTypeTheory/cda95637-0ab0-4897-8e38-b5ebb288a658%40googlegroups.com?utm_medium=email&utm_source=footer">https://groups.google.com/d/msgid/HomotopyTypeTheory/cda95637-0ab0-4897-8e38-b5ebb288a658%40googlegroups.com</a>.<br />

[-- Attachment #1.1: Type: text/plain, Size: 4927 bytes --] > > He once told me that he wasn't interested in formalizing his proof of > Bloch-Kato, because he was sure it was right. (I should have asked him at > the time how he could be so sure!) > Oh this is interesting... do you remember when this conversation was happening? Because in these slides ( https://www.math.ias.edu/vladimir/sites/math.ias.edu.vladimir/files/2014_09_Bernays_3%20presentation.pdf) he said "Next year I am starting a project of univalent formalization of my proof of Milnor’s Conjecture using this formalization of set theory as the starting point." (Page 11) 在 2019年11月5日星期二 UTC-8上午7:43:06，Daniel R. Grayson写道： > > Re: "To get back to the original question, my understanding was that > Voevodsky had done a bunch of scheme theory and it had got him a Fields > medal and it was this mathematics which he was interested in at the time. > He wanted to formalise his big theorem, just like Hales did." > > I think he was more interested in formalizing things like his early work > with Kapranov on higher categories, which turned out to have a mistake in > it. He once told me that he wasn't interested in formalizing his proof of > Bloch-Kato, because he was sure it was right. (I should have asked him at > the time how he could be so sure!) > > Re: "The clearest evidence, in my mind, is that there is no definition of > a scheme in UniMath." > > That's sort of accidental. In early 2014, expecting to speak at an > algebraic geometry in the summer, he mentioned that one idea he had for his > talk would be to formalize the definition of scheme in UniMath and speak > about it. I think he was distracted from that by thinking about > C-systems. The UniMath project aims at formalizing all of mathematics, so > the definition of scheme is one of the next things that (still) needs to be > done. > 在 2019年11月5日星期二 UTC-8上午7:43:06，Daniel R. Grayson写道： > > Re: "To get back to the original question, my understanding was that > Voevodsky had done a bunch of scheme theory and it had got him a Fields > medal and it was this mathematics which he was interested in at the time. > He wanted to formalise his big theorem, just like Hales did." > > I think he was more interested in formalizing things like his early work > with Kapranov on higher categories, which turned out to have a mistake in > it. He once told me that he wasn't interested in formalizing his proof of > Bloch-Kato, because he was sure it was right. (I should have asked him at > the time how he could be so sure!) > > Re: "The clearest evidence, in my mind, is that there is no definition of > a scheme in UniMath." > > That's sort of accidental. In early 2014, expecting to speak at an > algebraic geometry in the summer, he mentioned that one idea he had for his > talk would be to formalize the definition of scheme in UniMath and speak > about it. I think he was distracted from that by thinking about > C-systems. The UniMath project aims at formalizing all of mathematics, so > the definition of scheme is one of the next things that (still) needs to be > done. > 在 2019年11月5日星期二 UTC-8上午7:43:06，Daniel R. Grayson写道： > > Re: "To get back to the original question, my understanding was that > Voevodsky had done a bunch of scheme theory and it had got him a Fields > medal and it was this mathematics which he was interested in at the time. > He wanted to formalise his big theorem, just like Hales did." > > I think he was more interested in formalizing things like his early work > with Kapranov on higher categories, which turned out to have a mistake in > it. He once told me that he wasn't interested in formalizing his proof of > Bloch-Kato, because he was sure it was right. (I should have asked him at > the time how he could be so sure!) > > Re: "The clearest evidence, in my mind, is that there is no definition of > a scheme in UniMath." > > That's sort of accidental. In early 2014, expecting to speak at an > algebraic geometry in the summer, he mentioned that one idea he had for his > talk would be to formalize the definition of scheme in UniMath and speak > about it. I think he was distracted from that by thinking about > C-systems. The UniMath project aims at formalizing all of mathematics, so > the definition of scheme is one of the next things that (still) needs to be > done. > -- 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/0ef61665-eafd-40a0-8592-11bdd277d10b%40googlegroups.com. [-- Attachment #1.2: Type: text/html, Size: 5943 bytes --] <div dir="ltr"><blockquote class="gmail_quote" style="margin: 0px 0px 0px 0.8ex; border-left: 1px solid rgb(204, 204, 204); padding-left: 1ex;">He once told me that he wasn't interested in formalizing his proof of Bloch-Kato, because he was sure it was right. (I should have asked him at the time how he could be so sure!)<br></blockquote><div><br></div><div>Oh this is interesting... do you remember when this conversation was happening? Because in these slides (<a href="https://www.math.ias.edu/vladimir/sites/math.ias.edu.vladimir/files/2014_09_Bernays_3%20presentation.pdf">https://www.math.ias.edu/vladimir/sites/math.ias.edu.vladimir/files/2014_09_Bernays_3%20presentation.pdf</a>) he said "Next year I am starting a project of univalent formalization of my proof of Milnor’s Conjecture using this formalization of set theory as the starting point." (Page 11)<br></div><div> <br></div>在 2019年11月5日星期二 UTC-8上午7:43:06，Daniel R. Grayson写道：<blockquote class="gmail_quote" style="margin: 0;margin-left: 0.8ex;border-left: 1px #ccc solid;padding-left: 1ex;"><div dir="ltr">Re: "To get back to the original question, my understanding was that Voevodsky had done a bunch of scheme theory and it had got him a Fields medal and it was this mathematics which he was interested in at the time. He wanted to formalise his big theorem, just like Hales did."<br><div><br></div><div>I think he was more interested in formalizing things like his early work with Kapranov on higher categories, which turned out to have a mistake in it. He once told me that he wasn't interested in formalizing his proof of Bloch-Kato, because he was sure it was right. (I should have asked him at the time how he could be so sure!)<br><br>Re: "The clearest evidence, in my mind, is that there is no definition of a scheme in UniMath."<br><br>That's sort of accidental. In early 2014, expecting to speak at an algebraic geometry in the summer, he mentioned that one idea he had for his talk would be to formalize the definition of scheme in UniMath and speak about it. I think he was distracted from that by thinking about C-systems. The UniMath project aims at formalizing all of mathematics, so the definition of scheme is one of the next things that (still) needs to be done.</div></div></blockquote><br>在 2019年11月5日星期二 UTC-8上午7:43:06，Daniel R. Grayson写道：<blockquote class="gmail_quote" style="margin: 0;margin-left: 0.8ex;border-left: 1px #ccc solid;padding-left: 1ex;"><div dir="ltr">Re: "To get back to the original question, my understanding was that Voevodsky had done a bunch of scheme theory and it had got him a Fields medal and it was this mathematics which he was interested in at the time. He wanted to formalise his big theorem, just like Hales did."<br><div><br></div><div>I think he was more interested in formalizing things like his early work with Kapranov on higher categories, which turned out to have a mistake in it. He once told me that he wasn't interested in formalizing his proof of Bloch-Kato, because he was sure it was right. (I should have asked him at the time how he could be so sure!)<br><br>Re: "The clearest evidence, in my mind, is that there is no definition of a scheme in UniMath."<br><br>That's sort of accidental. In early 2014, expecting to speak at an algebraic geometry in the summer, he mentioned that one idea he had for his talk would be to formalize the definition of scheme in UniMath and speak about it. I think he was distracted from that by thinking about C-systems. The UniMath project aims at formalizing all of mathematics, so the definition of scheme is one of the next things that (still) needs to be done.</div></div></blockquote><br>在 2019年11月5日星期二 UTC-8上午7:43:06，Daniel R. Grayson写道：<blockquote class="gmail_quote" style="margin: 0;margin-left: 0.8ex;border-left: 1px #ccc solid;padding-left: 1ex;"><div dir="ltr">Re: "To get back to the original question, my understanding was that Voevodsky had done a bunch of scheme theory and it had got him a Fields medal and it was this mathematics which he was interested in at the time. He wanted to formalise his big theorem, just like Hales did."<br><div><br></div><div>I think he was more interested in formalizing things like his early work with Kapranov on higher categories, which turned out to have a mistake in it. He once told me that he wasn't interested in formalizing his proof of Bloch-Kato, because he was sure it was right. (I should have asked him at the time how he could be so sure!)<br><br>Re: "The clearest evidence, in my mind, is that there is no definition of a scheme in UniMath."<br><br>That's sort of accidental. In early 2014, expecting to speak at an algebraic geometry in the summer, he mentioned that one idea he had for his talk would be to formalize the definition of scheme in UniMath and speak about it. I think he was distracted from that by thinking about C-systems. The UniMath project aims at formalizing all of mathematics, so the definition of scheme is one of the next things that (still) needs to be done.</div></div></blockquote></div> <p></p> -- <br /> You received this message because you are subscribed to the Google Groups "Homotopy Type Theory" group.<br /> To unsubscribe from this group and stop receiving emails from it, send an email to <a href="mailto:HomotopyTypeTheory+unsubscribe@googlegroups.com">HomotopyTypeTheory+unsubscribe@googlegroups.com</a>.<br /> To view this discussion on the web visit <a href="https://groups.google.com/d/msgid/HomotopyTypeTheory/0ef61665-eafd-40a0-8592-11bdd277d10b%40googlegroups.com?utm_medium=email&utm_source=footer">https://groups.google.com/d/msgid/HomotopyTypeTheory/0ef61665-eafd-40a0-8592-11bdd277d10b%40googlegroups.com</a>.<br />

[-- Attachment #1.1: Type: text/plain, Size: 1901 bytes --] On Monday, 4 November 2019 18:43:08 UTC, Kevin Buzzard wrote: > > It is exactly the interaction between constructivism and univalence which > I do not understand well, and I think that a very good way to investigate > it would be to do some highly non-constructive modern mathematics in a > univalent type theory > Regarding *old* mathematics, you have the well-ordering principle proved in UniMath (from the axiom of choice, of course). Regarding your doubt about the interaction, we have that univalence is orthogonal to constructivism. In fact, univalence is not *inherently* constructive. It was hard work to find a constructive interpretation of univalence (which happens to rely on cubical sets as in homotopy theory). In particular (even if I lam fond of constructive mathematics, as you know), I work with univalence axiomatically, as a black box, rather than as a construction, in my (formal and informal) mathematical developments. And I do prefer to work with univalence-as-a-specification rather than univalence-as-a-construction. There is nothing inherently constructive about univalence. There is no a priori interaction between univalence and constructivism. There is only an a posteriori interaction, constructed by some of the constructively minded members of this list. The constructivity of univalence was an open problem for a number of year, and I would say that, even if it is solved via the cubical model, it is far from being fully understood. Best, Martin -- 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/cb153658-548a-4fe9-91ed-fc2e3db33723%40googlegroups.com. [-- Attachment #1.2: Type: text/html, Size: 2512 bytes --] <div dir="ltr"><br><br>On Monday, 4 November 2019 18:43:08 UTC, Kevin Buzzard wrote:<blockquote class="gmail_quote" style="margin: 0;margin-left: 0.8ex;border-left: 1px #ccc solid;padding-left: 1ex;"><div dir="ltr"><div dir="ltr"> It is exactly the interaction between constructivism and univalence which I do not understand well, and I think that a very good way to investigate it would be to do some highly non-constructive modern mathematics in a univalent type theory</div></div></blockquote><div><br></div><div>Regarding *old* mathematics, you have the well-ordering principle proved in UniMath (from the axiom of choice, of course). </div><div><br></div><div>Regarding your doubt about the interaction, we have that univalence is orthogonal to constructivism. </div><div><br></div><div>In fact, univalence is not *inherently* constructive. It was hard work to find a constructive interpretation of univalence (which happens to rely on cubical sets as in homotopy theory). In particular (even if I lam fond of constructive mathematics, as you know), I work with univalence axiomatically, as a black box, rather than as a construction, in my (formal and informal) mathematical developments. And I do prefer to work with univalence-as-a-specification rather than univalence-as-a-construction.</div><div><br></div><div>There is nothing inherently constructive about univalence. There is no a priori interaction between univalence and constructivism. There is only an a posteriori interaction, constructed by some of the constructively minded members of this list. The constructivity of univalence was an open problem for a number of year, and I would say that, even if it is solved via the cubical model, it is far from being fully understood. </div><div><br></div><div>Best,</div><div>Martin</div><div><br></div><div><br></div></div> <p></p> -- <br /> You received this message because you are subscribed to the Google Groups "Homotopy Type Theory" group.<br /> To unsubscribe from this group and stop receiving emails from it, send an email to <a href="mailto:HomotopyTypeTheory+unsubscribe@googlegroups.com">HomotopyTypeTheory+unsubscribe@googlegroups.com</a>.<br /> To view this discussion on the web visit <a href="https://groups.google.com/d/msgid/HomotopyTypeTheory/cb153658-548a-4fe9-91ed-fc2e3db33723%40googlegroups.com?utm_medium=email&utm_source=footer">https://groups.google.com/d/msgid/HomotopyTypeTheory/cb153658-548a-4fe9-91ed-fc2e3db33723%40googlegroups.com</a>.<br />

```
> members of this list. The constructivity of univalence was an open problem
> for a number of year, and I would say that, even if it is solved via the
> cubical model, it is far from being fully understood.
In my case, I still find it odd in a situation such as:
data Foo : Set -> Set where
bar : Foo UnaryNat
since transport supposedly allows us to take a proof of equivalence
between UnaryNat and BinaryNat and turn a `bar` into something of type
`Foo BinaryNat` although I can't see any way to directly construct an
object of this type.
Stefan
```

[-- Attachment #1.1: Type: text/plain, Size: 1139 bytes --] Sorry, at this point I don't remember precisely. On Tuesday, November 5, 2019 at 2:29:56 PM UTC-6, Yuhao Huang wrote: > > He once told me that he wasn't interested in formalizing his proof of >> Bloch-Kato, because he was sure it was right. (I should have asked him at >> the time how he could be so sure!) >> > > Oh this is interesting... do you remember when this conversation was > happening? Because in these slides ( > https://www.math.ias.edu/vladimir/sites/math.ias.edu.vladimir/files/2014_09_Bernays_3%20presentation.pdf) > he said "Next year I am starting a project of univalent formalization of my > proof of Milnor’s Conjecture using this formalization of set theory as the > starting point." (Page 11) > -- 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/3811fd43-0b84-4ac0-adcd-de638ae3ad57%40googlegroups.com. [-- Attachment #1.2: Type: text/html, Size: 2425 bytes --] <div dir="ltr">Sorry, at this point I don't remember precisely.<br><br>On Tuesday, November 5, 2019 at 2:29:56 PM UTC-6, Yuhao Huang wrote:<blockquote class="gmail_quote" style="margin: 0;margin-left: 0.8ex;border-left: 1px #ccc solid;padding-left: 1ex;"><div dir="ltr"><blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left:1px solid rgb(204,204,204);padding-left:1ex">He once told me that he wasn't interested in formalizing his proof of Bloch-Kato, because he was sure it was right. (I should have asked him at the time how he could be so sure!)<br></blockquote><div><br></div><div>Oh this is interesting... do you remember when this conversation was happening? Because in these slides (<a href="https://www.math.ias.edu/vladimir/sites/math.ias.edu.vladimir/files/2014_09_Bernays_3%20presentation.pdf" target="_blank" rel="nofollow" onmousedown="this.href='https://www.google.com/url?q\x3dhttps%3A%2F%2Fwww.math.ias.edu%2Fvladimir%2Fsites%2Fmath.ias.edu.vladimir%2Ffiles%2F2014_09_Bernays_3%2520presentation.pdf\x26sa\x3dD\x26sntz\x3d1\x26usg\x3dAFQjCNEc51jJXwQodsO-qY-qDVfkTXkyCw';return true;" onclick="this.href='https://www.google.com/url?q\x3dhttps%3A%2F%2Fwww.math.ias.edu%2Fvladimir%2Fsites%2Fmath.ias.edu.vladimir%2Ffiles%2F2014_09_Bernays_3%2520presentation.pdf\x26sa\x3dD\x26sntz\x3d1\x26usg\x3dAFQjCNEc51jJXwQodsO-qY-qDVfkTXkyCw';return true;">https://www.math.ias.edu/<wbr>vladimir/sites/math.ias.edu.<wbr>vladimir/files/2014_09_<wbr>Bernays_3%20presentation.pdf</a>) he said "Next year I am starting a project of univalent formalization of my proof of Milnor’s Conjecture using this formalization of set theory as the starting point." (Page 11)</div></div></blockquote></div> <p></p> -- <br /> You received this message because you are subscribed to the Google Groups "Homotopy Type Theory" group.<br /> To unsubscribe from this group and stop receiving emails from it, send an email to <a href="mailto:HomotopyTypeTheory+unsubscribe@googlegroups.com">HomotopyTypeTheory+unsubscribe@googlegroups.com</a>.<br /> To view this discussion on the web visit <a href="https://groups.google.com/d/msgid/HomotopyTypeTheory/3811fd43-0b84-4ac0-adcd-de638ae3ad57%40googlegroups.com?utm_medium=email&utm_source=footer">https://groups.google.com/d/msgid/HomotopyTypeTheory/3811fd43-0b84-4ac0-adcd-de638ae3ad57%40googlegroups.com</a>.<br />

For inductive families, one thing you can do is to think of them in terms of the translation to parametrized inductive types and identity types, so > data Foo (A : Set) : Set where > bar : Id U A UnaryNat -> Foo A in which case bar applied to the Id U BinaryNat UnaryNat that you get from univalence gives a Foo BinaryNat. A related perspective is to think of some transports as additional constructors for inductive families; see e.g. this approach to inductive families in cubical type theory https://www.cs.cmu.edu/~ecavallo/works/popl19.pdf -Dan > On Nov 5, 2019, at 7:06 PM, Stefan Monnier <monnier@iro.umontreal.ca> wrote: > >> members of this list. The constructivity of univalence was an open problem >> for a number of year, and I would say that, even if it is solved via the >> cubical model, it is far from being fully understood. > > In my case, I still find it odd in a situation such as: > > data Foo : Set -> Set where > bar : Foo UnaryNat > > since transport supposedly allows us to take a proof of equivalence > between UnaryNat and BinaryNat and turn a `bar` into something of type > `Foo BinaryNat` although I can't see any way to directly construct an > object of this type. > > > Stefan > -- 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/0CC0B8D7-3D66-42AD-A2D7-3B897A432B36%40wesleyan.edu.