There are many intelligent and curious people reading HN, who might be tempted to spend time understanding it because its somewhat frequent appearance here gives the (false) impression that it plays some important role in modern mathematics.
I regret spending the (large) amount of time I did learning what it is about, only to realize that's not much actual content there, and I would hope my post saves other people that trouble. There is a lot of far more interesting, elegant, and useful mathematics out there, imo.
My background - I’m a programmer and like building programming languages. I’m interested in HoTT because I’m interested in type theory generally and also formalization of mathematics.
Most mathematicians don’t care much about these things, which is fine. Like most weird stuff, it’s good that some folks are doing it.
Lastly you might enjoy this thread about why formalization is interesting to another mathematician:
https://mobile.twitter.com/andrejbauer?prefetchtimestamp=160...
I want to clarify that I think formalizing mathematics is interesting, too! It's just entirely unclear to me that HoTT has any special advantages here (see link from my original post).
The supposed selling point of HoTT wrt to formalizing mathematics is higher level proofs that are more general and are less work to express, due to having this richer notion of equivalence in the type theory.
I think the post you linked is pretty good... it makes me wonder - at this point, what is holding back more formalization? Is it a cultural thing? (None of the people who are doing “interesting” math care about formalizing It) Is it a technical thing? (Proof assistants and their type theories aren’t good enough?) Are we just missing a big enough “standard library” of mathematics so mathematicians can get to the interesting parts sooner? It’s probably all of the above.
What I think might happen is it reaches a tipping point where enough of the pieces are in place to where it rapidly becomes normal or expected to formalize mathematics and we have awesome tools and “standard libraries” to make it good.
This is a bizarre thing to say. Learning any mathematics is a complete waste of time.
> gives the (false) impression that it plays some important role in modern mathematics.
What are you talking about?
HoTT is broadly regarded as an exciting and important new area of mathematics; your view is absolutely the minority view. I mean I don't think I'm going to be able to convince you otherwise if you genuinely think that there's "not much content" in HoTT, but there are loads of well-regarded and prominent mathematicians who hold the opposite view to yours.
Your claim that HoTT is "broadly regarded as an exciting and important new area of mathematics" is simply wrong. I suggest talking to a wider variety of mathematicians.
In particular, concerning "well-regarded and prominent mathematicians," Steve Simpson and Harvey Friedman are on record on the Foundations of Mathematics mailing list saying, essentially, that HoTT is a waste of time [0]. The other set theory experts I know agree with them but haven't said so publicly. My impression is that this is a fairly common view in the set theory and foundations community.
Further, Jacob Lurie, who is perhaps the most talented mathematician currently working in higher category theory, has also written he doesn't think highly of it [1].
[0] I can provide links if it you want, but if you search the archive these arguments are easy to find.
[1] View the comments on this blog post: https://mathematicswithoutapologies.wordpress.com/2015/05/13...
The set theory and foundations community, i.e. the small minority of mathematicians who manage to do effective work on mathematical foundations in ZFC, are precisely the people who we'd expect to be most anti HoTT. It's not necessarily wrong, but it's like concluding that OOP is pointless because you've talked to leading C programming experts.
You have missed my point. We generally don't study only the "useful" things; if we did most of the most important mathematics we have today would have gone unstudied.
> Your claim that HoTT is "broadly regarded as an exciting and important new area of mathematics" is simply wrong.
According to whom? You? You're making an argument about consensus regarding the importance of a given area, but I have seen nothing to indicate that the consensus among mathematicians is in line with your opinion.
> Steve Simpson and Harvey Friedman
They are in the minority. Again, if you're arguing about consensus you can't just point to one or two individuals and say "look! They agree with me!" For goodness' sake, the thread your referring to is well-known partially because it was controversial.
> Jacob Lurie [...] doesn't think highly of it
The content of Lurie's comment and yours are quite different.
> According to whom?
I've talked to a lot of very good mathematicians, including people who've studied and collaborated with people in the HoTT orbit, and people from many other subfields. I'm sorry I don't have exact quantitative data for you.
> They are in the minority. Again, if you're arguing about consensus you can't just point to one or two individuals and say "look! They agree with me!" For goodness' sake, the thread your referring to is well-known partially because it was controversial.
Again, I don't think they're in the minority. Also, there's way more than one relevant thread. That particular fight was litigated over years.
> The content of Lurie's comment and yours are quite different.
He makes many comments on that post. I invite you to read them all. Certainly, I'm not just regurgitating what he says there, but it does provide an example of a top mathematician expressing skepticism about the entire enterprise.
Yes, I was being facetious. My point is that by whatever measure pure maths is worth studying (i.e. those measure beyond "practical" applications) also obviously applies to HoTT.
> I've talked to a lot of very good mathematicians
You're an anonymous person on the internet, this is such a strange thing to fall back on. There are pretty standard ways of talking about how well-regarded/important certain subfields are in mathematics, "I've asked around" isn't one of them.
> I'm not just regurgitating what he says there
His comments express a substantially different point to yours. He does not say that HoTT is a waste of time, or that it "doesn't have much content".
Not all branches of pure mathematics are equally worthy of study. A few are essentially dead for various (good) reasons, for example.
> There are pretty standard ways of talking about how well-regarded/important certain subfields are in mathematics, "I've asked around" isn't one of them.
How should one figure out how well-regarded a subfield of mathematics is if not by asking members of the mathematical community?
> His comments express a substantially different point to yours. He does not say that HoTT is a waste of time, or that it "doesn't have much content".
His comments support the point I intended them to support, which is that Lurie is not a huge fan of HoTT. You made a remark about well-regarded mathematicians, and I gave you comments from a well-regarded mathematician.
Ok? What does this have to do with the point I was making?
My point was, pretty simply, that "usefulness" is not the only thing we consider when deciding whether something is worthy of study. (If it was we wouldn't have touched most of modern pure mathematics.) As a result, your notion that studying HoTT would be a waste of time because it's not "useful" is nonsense.
> How should one figure out how well-regarded a subfield of mathematics is if not by asking members of the mathematical community?
Oh come on, you really don't understand the point I'm making?
When talking about consensus it's not good enough to say "I've talked to a few people and they agree with me". Work on HoTT, measured by any kind of actual metric rather than "people I've spoken to (who I can't name by the way) say it's bad", is broadly well-regarded and seen as important. In terms of research funding, citation/publication metrics, or the opinion of most mathematicians.
> His comments support the point I intended them to support
No they do not. My objection is to you saying:
* That HoTT is a waste of time.
* That HoTT doesn't have much content.
* That HoTT is not broadly regarded by mathematicians as important or exciting.
Please do not move the goalposts.
> You made a remark about well-regarded mathematicians, and I gave you comments from a well-regarded mathematician.
What? I "made a remark" about well-regarded mathematicians? I said, specifically, "there are loads of well-regarded and prominent mathematicians who hold the opposite view to yours".
I also tried to stress that:
> if you're arguing about consensus you can't just point to one or two individuals
Please bear that in mind.
I've talked to more than "a few people," about this, and I think it's clear I've thought more about the sociology of this issue than just chatting with Tom Scanlon or whoever over a beer one night. I stand by what I said about that earlier, along with my comment that you ought to talk to a wider variety of mathematicians if you think there's some broad consensus that HoTT is important. (Start with the PDE and numerical analysis people, for example.)
Anyway, sticking purely to objective metrics, how many young mathematicians (postdocs) who work primarily on HoTT have ever been hired by R1 universities to tenure-track positions? Surely if this were seen as an important area, departments would want to snatch people in that field up, right? (I'd even accept examples of postdocs at these schools working on HoTT, though this represents a far less serious commitment by the department.)
This is (in principle – you'd have to collect it) publicly available information, which anyone reading this exchange can check on. (Look for assistant profs at schools listed as R1 on Wikipedia.) Maybe I'm forgetting someone, or not knowledgeable enough, but I can't really think of any.
(ER at Johns Hopkins was the first name that popped into my head, but she doesn't count because she has only a single HoTT paper among dozens, and it was written after she was hired.)
Another metric: how many HoTT papers are there in top mathematics journals? Annals, Inventiones, etc.?
- Mathematics is a very large field.
- For beginners, it is necessary to have wiser people than yourself put up some signposts.
I found the comment you're referring to very illuminating. I'm not naively assuming the author's view is the only one, or "correct", but s/he certainly like they have some familiarity with graduate/research level mathematics in relevant areas. So it was very helpful to see, at least from the point of view of some mathematicians, that this is not considered an important area.
I was left wondering whether the author of that comment understood the importance of this stuff in computer science (i.e. in academic approaches to programming languages, might be the right term, I don't know, I'm pretty ignorant of these areas myself), or whether they were only able to criticize it from the point of view of pure mathematics.
It is useful for my own work, and thus I study it, why isn't this as simple as that?