Timeline for The definition of ${}^{\circ} \mathcal{HT}^{\mathcal{D}\text{-}\Theta^{\pm \mathscr{ell}}}$ in Inter-universal Teichmüller theory
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| Dec 3 at 22:53 | comment | added | categoricalequivalent | So now my speculation mathematicians already may reach, is how Mochizuki and their peers in Kyoto use these concepts which is defined correctly from even set theoretic perspective is under the cover. As a result, even if these definitions definitely make sense, arguments where they are used, are not requiring all information of isomorphisms and will be requiring just one isomorphism, then other isomorphisms will be get rid of. Hence, I want for them to show the point where it's suggested to use groupoids' or so isomorphisms' information. If there is no misconception of me, I'll edit my answer. | |
| Dec 3 at 22:52 | comment | added | categoricalequivalent | As an example, thinking of the moduli stack of vector bundles, $Aut(V)$, an automorphism group of a vector bundle $V$, is a general linear group and so we can't choose one isomorphism, to avoid losing this information. In this case, it goes the right way to see all isomorphisms. | |
| Dec 3 at 22:52 | comment | added | categoricalequivalent | I appreciate for you reading thoroughly my comments! First, I feel you make sense; if Mochizuki defines ${}^{\circ} \mathcal{HT}^{\mathcal{D}\text{-}\Theta^{\pm \mathscr{ell}}}$ as the groupoid comprised of each ${}^{m} \mathcal{HT}^{\mathcal{D}\text{-}\Theta^{\pm \mathscr{ell}}}$, there is no problem set theoretically.(Although as far as I know, I feel some odd to use 'up to isomorphism' and it's not used the word 'polyisomorhism', at this situation, like the paper you mentioned don't do them.) | |
| Dec 1 at 3:27 | comment | added | Syu Gau | @categoricalequivalent Also, let me mention that, one guiding idea of Mochizuki's IUTT is Bogomolov's Proof of the Geometric Version of the Szpiro Conjecture (see kurims.kyoto-u.ac.jp/~motizuki/…). I really recommend to read this before getting into any details of IUTT. So you can interpret his writing more faithfully. | |
| Dec 1 at 3:22 | comment | added | Syu Gau | @categoricalequivalent Let me mention that, it is not only in the IUTT series, indeed, Mochizuki (and many people) use words like "something determined up to isomorphism" to mean the moduli stack, or in-another-words the type, or the groupoid, of that "something". Namely, it means something AND the isomorphisms. | |
| Dec 1 at 3:12 | comment | added | Syu Gau | @categoricalequivalent I see the point. $\{H: H' \sim H\}$ is definitely not what Mochizuki works with. Instead, as a category-lover, I would rather interpret it as a groupoid of Hodge theaters whose set of isomorphisms are given by $\Xi$. The whole point of introducing the notion ${}^{\circ} \mathcal{HT}^{\mathcal{D}\text{-}\Theta^{\pm \mathscr{ell}}}$ is to treat the isomorphic Hodge theaters ${}^{m} \mathcal{HT}^{\mathcal{D}\text{-}\Theta^{\pm \mathscr{ell}}}$ simultaneously. For this purpose, it is important to use the isomorphisms in $\Xi$ rather than arbitrary isomorphisms. | |
| Nov 30 at 2:23 | comment | added | categoricalequivalent | Hence, $\textit{a class of Hodge theatres obtained by relating them with the given set of isomorphisms}$ you wrote, which means if I am right to understand it $\{H: H' \sim H$ for an isomorphism $\sim \quad \in$ a polyisomorphism, a set of isomorphisms $\}$, is completely, set-theoretically equal to the isomorphism class, which leads to the fact that Dr. Scholze and Dr. Stix proved contradiction using only one isomorphism between fundamental groups. Therefore from a personal perspective they can't resolve the ABC conjecture until they represent the proper meaning of 'up to polyisomorpism'. | |
| Nov 30 at 2:22 | comment | added | categoricalequivalent | If a model is isomorphic to another one as for an isomorphism, then even if we choose another isomorphism, both two remains isomorphic to each other. (As an aside, note that if we change metamodel, this is not the case; it could be occurred that vector spaces $V$ and $W$ are isomorphic relation at a set theoretic model but not isomorphic at another one.) | |
| Nov 30 at 2:21 | comment | added | categoricalequivalent | @SyuGau Thank you for your comments and apologise for that I have no time to think it. Let you explain if I have misconception. As a set-theoretic fact, an isomorphism class of an object is the set of all objects being isomorphic to the one. Therefore, if we got rid of the assumption about thinking a set of isomorphisms and replaced it to just one isomorphism, it would be just the isomorphism class. Then, if we think the case your assumption what happened? | |
| Nov 6 at 1:31 | comment | added | Syu Gau | In your vector space analogy, the class $[V]$ is NOT the isomorphism class of ALL vector spaces that are isomorphic to $V$. Rather, it is a class $V^\bullet$ whose members are related via given isomorphism. Or, more down to earth, what Mochizuki wants to say is just "treat all ${}^{m} \mathcal{HT}^{\mathcal{D}\text{-}\Theta^{\pm \mathscr{ell}}}$ as a single object ${}^{\circ} \mathcal{HT}^{\mathcal{D}\text{-}\Theta^{\pm \mathscr{ell}}}$ via all the possible isomorphisms in $\Xi$". | |
| Nov 6 at 1:26 | comment | added | Syu Gau | I see the point. By "Hodge theater determined up to isomorphism", Mochizuki doesn't mean an isomorphism class of Hodge theater. Rather than that, he means a class of Hodge theaters obtained by relating them with the given set of isomorphisms. | |
| Sep 30 at 0:46 | comment | added | categoricalequivalent | Although one choice we can opt for is we consider $\phi(V_1, V_2)$ as 'there is an isomorphism between $V_1$ and $V_2$', this means the same as selecting one isomorphism and so Mochizuki's paper leads to absurdity, since Mr. Scholze and Mr. Stix could prove that by choosing one isomorphism of fundamental groups, finally it caused contradiction. | |
| Sep 30 at 0:45 | comment | added | categoricalequivalent | I apologise my description possibly making you confusion. Here, $[V]$ is analogy to ${}^{\circ} \mathcal{HT}^{\mathcal{D}\text{-}\Theta^{\pm \mathscr{ell}}}$ and what I wrote is we can't define $[V]$ because 'we' don't know $\phi(V_1, V_2)$ to define $[V]$, using $\Xi$. Yeah, of course if one choose 'an' isomorphism, then using it we can define the isomorphic class since the isomorphism is a relation, which is a subset of $\mathit{Vect}$ $\times$ $\mathit{Vect}$ as well. However, set theoretically, using 'a set' of isomorphisms, how can we define $\phi(V_1,V_2)$ to define a relation? | |
| Sep 28 at 14:10 | comment | added | Syu Gau | Your formula $\phi(V_1,V_2)$ defines a different relation than the relation $V_1\sim V_2\iff V_1\cong V_2$. The point is: the data $\Xi$ is given and your isomorphism can only be taken from $\Xi$. | |
| May 22 at 6:46 | history | edited | YCor | CC BY-SA 4.0 |
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| May 18 at 6:19 | answer | added | categoricalequivalent | timeline score: 3 | |
| May 18 at 4:41 | comment | added | categoricalequivalent | @pastebee Thank you for your comment for my little ago question. Actually, I have thought about this question and DavidRoberts' comments when I have time, triaged my questions, of course learned about math. I prepare to post the answer myself to the question now, anyway, what I can say now is 'up to isomorphism' represents the justification of defining objects from set-theoretic foundationalistic view. For those who want to know the paper's situation from the view including me, I will write the meaning, so if you have time then check it. | |
| May 18 at 4:02 | comment | added | paste bee | @categoricalequivalent The thing is, the CPU shouldn't be relevant here, abstracting that away is the point of having a compiler. The phrase you quoted, "determined up to isomorphism", is not "inline assembly" - it's not trying to refer to the coding of the various objects as sets. Which means emitting an error about how the coding as sets doesn't work is a bug in your "compiler" (that targets ZFC); either there is some actual problem with the code that isn't about how to code it with sets and you should have said that, or there isn't and it should have compiled fine. | |
| May 18 at 1:15 | history | edited | categoricalequivalent | CC BY-SA 4.0 |
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| Jul 10, 2023 at 8:54 | history | edited | categoricalequivalent | CC BY-SA 4.0 |
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| Jul 1, 2023 at 3:58 | comment | added | David Roberts♦ | There are lots of different sorts of type theory, so you can forgiven if you're not familiar with them all! Especially as it's not your foundation of choice. :-) I hope it's helpful, and not discouraging. | |
| Jul 1, 2023 at 3:52 | comment | added | categoricalequivalent | In any case, I understood this is a philosophical question rather than a mathematical one. I appreciate your patience gets you to post many great comments. | |
| Jul 1, 2023 at 3:52 | comment | added | categoricalequivalent | Anyway, My philosophical position is, many programmers don't focus on what is CPU used, but there surely is there. Between you and me used CPU may be different though, CSS has compatibility and is implemented in some way. If not, there is no the CSS style sheet. Of course, although historically mathematics had been developed without any CPUs, I think the discovery that "mathematics is on a CPU" itself is now an immutable mathematical fact just like the Pythagorean theorem to unchanged. | |
| Jul 1, 2023 at 3:52 | comment | added | categoricalequivalent | I just had a quick look at Thorsten Altenkirch's "Should Type Theory Replace Set Theory as the Foundation of Mathematics?" before and he stated, "When saying Type Theory we mean HoTT." I assumed that "Type Theory" in the context of the comparison with ZFC meant "HoTT", but I'm sorry if I'm wrong. | |
| Jul 1, 2023 at 3:07 | comment | added | David Roberts♦ | And mathematics can be entirely founded in type theory (not HoTT, even simpler than that), and there's not a set in sight, and everything is just syntax. | |
| Jul 1, 2023 at 3:06 | comment | added | David Roberts♦ | No one in number theory or commutative algebra worries what set underlies the ring $R[X]$ any more than they worry what underlying set the natural numbers have. There's a famous paper "What numbers could not be" jstor.org/stable/2183530 that pushes back against the idea that numbers "are" sets. Before set theory, people did lots of amazing mathematics, and even today, most people don't even think much about set theory. It's like worrying about the underlying machine code CPU-level algorithm when trying to write a CSS style sheet. | |
| Jul 1, 2023 at 2:56 | comment | added | categoricalequivalent | I got it. I don't know if this is my lack of ability, my philosophical position, or my personality, but for example, I cannot say that a polynomial ring $R[X]$ exists if I don't know what kind of set the indeterminate element $X$ is(By the way, I find the definition of an indeterminate element as a set on page 97 of Lang's Algebra.) But at the same time, I understand that many people discuss about polynomials without being aware of what kind of set the indeterminate element $X$ is. Is that what you mean by "It's now how the matter proceeds."? | |
| Jul 1, 2023 at 2:39 | comment | added | David Roberts♦ | In particular, Mochizuki has acknowledged that a lot of the stuff in his IUT papers is needlessly general, and other people who have tried to make sense of it, for instance Taylor Dupuy, managed to write down bits of hypothetical mathematics that use standard mathematical definitions and objects, and very little trace of all the increasingly novel and baroque constructions of Mochizuki, while possibly and apparently capturing what was intended. If we knew for sure, we wouldn't “... have the ridiculous situation where ABC is a theorem in Kyoto but a conjecture everywhere else”—Frank Calegari. | |
| Jul 1, 2023 at 2:35 | comment | added | David Roberts♦ | OK. It's just that it seems like some of the confusions are coming from ideas that are much more preliminary that the complicated mess that Mochizuki built. You really shouldn't need to try to formally define as a set or class, in some chosen foundation, what you mean by the equivalence relation that defines equivalence classes. It would be like doing number theory research, and specifying how integers multiply by talking about the multiplication table as done in school maths. It's now how the matter proceeds. I'm not sure what to advise you here apart from that IUT might not be worth it? | |
| Jul 1, 2023 at 2:30 | comment | added | categoricalequivalent | Thank you very much for your patience with my question. As a matter of fact, I have already graduated from a mathematics department of a university in a not-so-famous country, and I have already studied equivalence relations, etc. The reason I am posting links to binary relations, etc. is to specify that I am thinking of binary relations as a set here, but I am sorry if this is unnecessary. | |
| Jul 1, 2023 at 2:17 | comment | added | David Roberts♦ | May I ask what your background is? The kind of detail you are providing makes me suspect that you are not far past learning things like equivalence classes and so on, but I apologise if this is mistaken. On a website for mathematics researchers, you really don't need to clarify what you mean by an binary relation by giving a link to a definition. | |
| Jul 1, 2023 at 0:45 | history | edited | categoricalequivalent | CC BY-SA 4.0 |
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| Jun 27, 2023 at 11:04 | comment | added | categoricalequivalent | In this situation, in a sense evidently, the definition that relations are a certain subset of a cartesian product, means what ${}^{\circ} \mathcal{HT}^{\mathcal{D}\text{-}\Theta^{\pm \mathscr{ell}}}$ doesn't go well, I think. My opinion is: as I wrote in my question, I think this object must be supposed to be the equivalence class by the equivalence relation $\Xi$. However, since $\Xi$ is a set of isomorphisms and so this is obviously even not a binary relation on $\mathcal{D}\text{-}\Theta^{\pm \mathscr{ell}}$ Hodge theaters. Therefore, the definition of this object doesn't make sense. | |
| Jun 27, 2023 at 9:49 | comment | added | David Roberts♦ | You don't need to go to such details. The actual encoding of functions in any foundation should have no impact on mathematics outside of actual foundational questions, like "is this function definable in an axiom system with such-and-such restricted quantifier complexity?" and the such like. So questions that involve thinking of isomorphisms as subsets of the cartesian product are not useful to try to unpack what is being claimed, here. | |
| Jun 27, 2023 at 9:05 | comment | added | categoricalequivalent | My understanding has been that if it could be defined on HoTT+Univalence axiom etc., it could always be translated (with additional axioms if necessary) as a definition in ZFC and vice versa, and hence, the definition that doesn't work on ZFC(even with additional axioms if necessary, again) doesn't work on other foundation as well. Is this wrong? | |
| Jun 27, 2023 at 9:05 | comment | added | categoricalequivalent | @DavidRoberts Thank you for your comment again (following my past question)! Actually, I am not much familiar with the foundations of mathematics, but my comprehension is the following: as you said, for example, I think one on the HoTT+Univalence axiom doesn't think of isomorphisms as subsets of the cartesian product, on the other hand, others on the ZFC(if necessary, plus the existence of large cardinals) think of such(and I am on ZFC and consider all mathematical objects as a set). However, I have thought this fact doesn't solve what can't be defined in ZFC. | |
| Jun 27, 2023 at 7:37 | comment | added | David Roberts♦ | Isomorphisms are only subsets of the cartesian product in one specific foundation. In others they are not. The mathematics here should really not depend on whether one uses ZFC or a structural foundation like ETCS+R. Mochizuki's uses of category theory, and in particular his ideas around "objects up to isomorphism" are rather non-standard, and in my eyes misleading. | |
| Jun 27, 2023 at 6:04 | history | asked | categoricalequivalent | CC BY-SA 4.0 |