> You wrote that "Learning any mathematics is a complete waste of time."
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".
> My point is that by whatever measure pure maths is worth studying (i.e. those measure beyond "practical" applications) also obviously applies to HoTT.
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.
> Not all branches of pure mathematics are equally worthy of study.
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
I think to develop this conversation further would require me explaining why mathematicians are interested in certain subfields of mathematics and not others (e.g. the dead ones), but that's far too big a project for a comment. Buzzard says a little bit about this in the blog post I linked, which is a highly compressed version of what I'd write; if you're curious, maybe read that.
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.?
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".