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Comment by stepchowfun | Hacker News Reader

To the folks in this thread who think they know better than researchers at the forefront of foundational mathematics: I encourage you to learn type theory, and especially homotopy type theory. In doing so, you'll be forced to confront the various notions of judgmental and propositional equality. You'll see why it's useful to have a formal justification for treating isomorphic objects as equal (this is called "univalence"), which is something you may have even been doing informally/subconsciously all along.

In particular, homotopy type theory is a type theory, which means it looks like a programming language. So people here might find it a bit more familiar than, say, higher category theory. It'll be much easier if you know functional programming, so you may need to learn that first if you haven't already.

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This is not "foundational mathematics." This is about the foundations of algebraic topology. The experts in the foundations of mathematics as a whole (set theorists, logicians, model theorists, etc.) don't care about homotopy type theory. It's sort of pointless for the study of the foundations of mathematics. That is, they really just care about ZF or ZFC, because those are already formulated so that they are maximally useful for foundational investigations. (And the e.g. reverse mathematics people who want to work outside of ZF/ZFC already have more direct ways of doing so.)

Some idea of what people in foundations think about day-to-day can be gleaned by searching the Foundations of Math mailing list archive: https://cs.nyu.edu/pipermail/fom/

In particular: don't confuse higher category theory (of interest to the algebraic topology community at large) with homotopy type theory (which is not). Jacob Lurie has a rather pointed conversation with Urs Schreiber about this, linked to here: https://mathematicswithoutapologies.wordpress.com/2015/05/18....

edit: I'd love to hear from the downvoters why they disagree with this take. It's pretty standard, afaik.

> I'd love to hear from the downvoters why they disagree with this take. It's pretty standard, afaik.

Because it's obtuse. There are well-known ways of phrasing all of this stuff in set theory (or rather, "legitimate" varieties of set theory) should anyone really care, replicating type-theoretical constructions by adding increasingly large "inaccessible cardinals" to one's pre-existing notion of set. But it's a pointless exercise when the type-theoretical description of these foundations is so much easier to work with. It's only useful if one believes that "set theory" is the only legitimate language for talking about anything foundational, which is again quite silly.

In what sense is type theory "easier to work with"? For what purpose? ZFC exists largely because it's a useful framework for proving meta-mathematical theorems. In this sense, it is extremely convenient to work with.

Also, I'm not sure what "all of this stuff" refers to here re: your comment about inaccessible cardinals. You don't need large cardinals to do 99.99% of modern mathematics (essentially, everything but set/proof theory). ZFC is good enough (and you can usually get away with something weaker).

Frankly we are sick of set theorist traditionalists complaining about type theory, category theory, and other new things without actually bothering to understand them at all.

After multiple decades, it's clear you all are never going to bother to learn our stuff nearly as well as we bothered to learn your stuff. This "debate" is thus completely unproductive without any good agreed-upon foundation to argue form.

(How ironic!)

First, I'm not a set theorist. Second, one of the major motivations for introducing type theory in mathematics was to assist with computer-aided formalization efforts, which has had some success and has a promising future (e.g. the stuff the Lean community has been doing lately). This is great! I'm not complaining at all.

But I do think trumpeting HoTT as a new, improved foundation of mathematics evinces a misunderstanding of what people who work on foundations actually want out of a foundation of mathematics; or even more broadly, the mistaken belief that finding a new foundation of mathematics is an interesting project in the 21st century.

Lurie's comment here explains a bit more: https://mathematicswithoutapologies.wordpress.com/2015/05/13...

(And clearly, Lurie is not some ignorant traditionalist. There's a substantive disagreement that can't be reduced to people not wanting to learn "your stuff.")

> evinces a misunderstanding of what people who work on foundations actually want out of a foundation of mathematics

This is still gatekeeping who gets to use the word "foundations".

I will go read https://mathematicswithoutapologies.wordpress.com/2015/05/13 but my hopes are not high for a sudden breakthrough.

I'm sorry, but this is not gatekeeping; that term has an established meaning and mathematical history, and corresponds to a productive vein of research going back to Gödel (at least). If you'd like to argue that the study of foundations should be expanded to include the things that HoTT people are interested in, that's one thing. But can we at least agree it does not have a ton of relevance to the questions people have traditionally considered in foundational work? (This is almost by definition if we set it up so that it is mutually interpretable with ZFC.)

> going back to Gödel (at least)

See, that is the basic misunderstanding here. Foundations now is a wider topic area than it was in Gödel's day. It's like saying that automobiles run on internal combustion engines in a productive vein of manufacturing going back to Henry Ford (at least). Since those days though, there is this guy Elon who you might like to meet...

I absolutely agree that it's a wider, more sophisticated topic today. This doesn't mean that its main concerns have fundamentally changed. The Maddy article linked in a comment above is extremely clear explanation of this.

I don't see the Maddy link, but anyway try for something written in the current century.

Here, from three years ago: http://www.socsci.uci.edu/~pjmaddy/bio/What%20do%20we%20want...

Thanks, that looks interesting and I'll try to read it. But it appears to mostly be about philosophy, and it doesn't say much about what (mathematical) logicians are doing today.

I did start to read it last night, and finished today.

My understanding is that regular MLTT already can do all the bold things Set theory does, and HFTT (and other developments) aims to make "Proof Checker" better realized (no more "setoid hell").

Ultimately what us CS people care about is a unity of theory and practice. What drew us to computers int the first place was trying to make explicit and tangible ideas that previously were only laboriously communicated from one generation of big brains to the next in the ivory tower.

In that regards, I would say don't even look HoTT or fancy theory developments. Look at Lean, written by a non-type-theorist, and mathlib, written by a non-constructivist.

Lean and Mathilb are plainly plowing a head in ways that no set theory based tool has ever achieved, and covering all the set theory bold points in the process.

Perhaps a new more elegant type theory will replace Lean soon, but honestly I am not even worried about that. The important thing is making math much more accessible without loss of rigor with these tools. If better theories can follow rather than lead that methological revolution, that's fine with me,

As to Category Theory and essential guidance, I have a simple test :). As mathlib (or its sucesssors) matures, let us see how many proofs are refactored to use category theory! I don't want to poo-poo the deeper connections between type theory and cateogory theory, but merely using lots of category theory "library code" in a type theoretic proof assistant not itself designed with explicit reference to category theory is good enough a success story for me.

In conclusion:

- Risk Assessment: depends how many "large cardinal" type things we want (unlike the 1970s category theorists I personally don't really care)

  - Regular type theory: *check*
  - HoTT: *check*
  - Set theory: *check*, bonus points for more exotic large cardinals

- Generous Arena, Shared Standard, Meta- mathematical Corral

  - Regular type theory: *check*
  - Set theory: *check*
  - HoTT: WIP because culture shock of weird equality

- Proof Checker:

  - Set theory: *FAIL*
  - Regular type theory: *partial support*, full if we discard all constructive side goals (Lean)
  - HoTT: *check*! (Very recent cubical results

- Essential guidance:

  - Category theory does best, but can use as "library" rather than "language". good enough for now.

I have no objections to claiming success in formalizing mathematics! As I've said, Lean is a big win for type theory.

But formalizing proofs with computers is essentially separate from the rigorous study of the foundations of math (consistency, relative strength of theories, etc.).

You've complained above about set theorists not learning "your stuff," and now it's my turn to complain. You write: 'Risk Assessment: depends how many "large cardinal" type things we want (unlike the 1970s category theorists I personally don't really care).' Large cardinal axioms are essential to the study of the relative consistency strength of mathematical theories. You personally might not care about them, but accommodating them is not a negotiable detail for anything that purports to be useful in the study of foundations.

And yeah, I'm aware the HoTT people have ways around this, fine. My comment is more about the perspective conveyed by your remarks: if you want to have a seat at the table in a discussion about foundational issues, you need to understand basic facts like this. My suspicion is that you might have had bad interactions with set theorists because you don't share certain knowledge they take for granted, and then they get frustrated and end the conversation early. I truly believe that if you went and read a book on set theory and logic, and learned, e.g., wny large cardinals are important, you might understand better where I and other people are coming from.

Large cardinals are closely related to what type theorists call "universes". Again, there's a huge amount of equiconsistency and relative consistency results linking type- and set theories. To keep suggesting and implying that type theory cannot possibly be a language for talking about these things is so incredibly careless and obtuse that some people might call it actively misleading.

I'm not sure how that comment is responsive to my post. Sure, type theorists talk about universes, and sure, there are relative consistency results linking type and set theories. My claim is just that large cardinals (or some equivalent concept) are essentially necessary to order theories in consistency strength, so is hard to take a response of the form "it doesn't matter whether HoTT can talk about large cardinals" seriously.

> Large cardinal axioms are essential to the study of the relative consistency strength of mathematical theories. You personally might not care about them, but accommodating them is not a negotiable detail for anything that purports to be useful in the study of foundations.

It's my understanding that some larger cardinals are needed for foundations, but since the foundations problem was "solved" future work on large cardinals has continued, and that is more "more set theory for set theory's sake", than something needed to accommodate math other "regular" mathematicians are doing.

(With the possible exception of the Grothendiek and Voevodsky line of problems perhaps as the article relates --- separate and ancillary to Voevodsky's desire for a proof checker, and also separate to whether or not categorical methods alone can suffice.)

I actually don't know of type theory having "ways around this". My understanding is that while type theory doesn't appear to face fundamental obstacles here, set theory "has the lead" in making large cardinals (or their equivalents) due to being older, and the type theorists relative disinterest in this area has meant the gap is not closing.

Foundations goes back to Cantor and Frege, at least.

Set theorists have made a Whig History of sorts where everything leads to ZFC, but that is just their post-hoc reading of the intellectual history.

We can write Whig Histories too. See https://homepages.inf.ed.ac.uk/wadler/papers/frege/frege.pdf for example.

The fact is both narratives are wrong --- intellectual history is always more pluralistic and meandering than partisans want to admit in hindsight. We, as the underdogs, are ready to admit that. Can you all do the same?

That link talks almost entirely about implications for programming and not mathematics. I recommend the Penelope Maddy article linked in a comment above for a more in-depth discussion of the relevant history and aims of the field. She says things much more clearly than I can.

Yes, my point is "functional programming" and "ZFC" are both not predestined as the inevitable conclusion the ideas preceding them would lead up to.

I think Mike Shulman's side of the discussion in that thread is also very worth reading, and it's somewhat unfair to (perhaps unintentionally) characterize Lurie as the representative of mathematicians trying to talk sense into the homotopy type theorists. They have a very balanced discussion where each respectfully tries to understand where the other is coming from.

I'm desperately trying to find a lovely philosophical writeup on the different purposes to which a "foundation" is put, and why different foundationalists want those foundations for distinct purposes. I can't seem to hunt it down (Google has become steadily less helpful over time). But the thrust is, I don't think anybody is claiming that HoTT ought to replace ZFC for reverse mathematics in particular; it's "just" a capturing of some pervasive intuitions for concepts at the ground floor of the formalism itself. It takes a perspective that makes grappling with some concepts easier to approach. If you're already a deft hand with a different (very well-established) framework, as Lurie exhibits throughout, maybe it's worth it to keep using the entirely viable tools you have!

You write: "I don't think anybody is claiming that HoTT ought to replace ZFC for reverse mathematics in particular; it's "just" a capturing of some pervasive intuitions for concepts at the ground floor of the formalism itself. It takes a perspective that makes grappling with some concepts easier to approach. If you're already a deft hand with a different (very well-established) framework, maybe it's worth it to keep using the entirely viable tools you have!"

My claim is that "capturing [...] some pervasive intuitions" is not what ZFC is trying to do, or what people in foundations are interested in. Lurie puts the point well when he says a foundation "mean[s] a small number of concepts and assumptions from which the rest of mathematics can be derived." The purposes here are first, to figure out what exactly one must assume to get the rest of mainstream mathematics, and second, to provide a convenient base for doing metamathematics (since the fewer elementary operations/objects you have to reason with, the easier it is to write proofs). Capturing intuitions is completely orthogonal to this endeavor; no one is claiming that set-theoretic translations of everyday mathematical arguments capture the intuitive content of those arguments, because no one one intended ZFC to ever do that in the first place.

Similarly, no one is claiming that handing someone a Turing machine that performs quicksort is a great way to capture the intuition behind quicksort, or explain the algorithm in an understandable way. Turing machines exist as a special-purpose construct to probe the limits of computation, not to reproduce or communicate high-level algoritmhic reasoning. ZFC is like a Turing machine for (meta)math.

Yes, hence my desperate desire to relocate the article on what purposes a foundation may be put to. I say this as neutrally as possible, but I don't think "foundation" has such a singular meaning, and misattributing one "foundation's" goals to be those of another's is probably near the root of your frustration. (It would help my case to find the article; at present I can only relay the ghost of what I took from it.)

HoTT/UF share some purposes of a foundation with ZFC, and not others. Each has purposes the other does not. That's about the safest thing I can say with any faith that I'm doing it justice.

I understand this desire but I think people who believe this should develop a new word or description for what you're attempting to do. See my comment here: https://news.ycombinator.com/item?id=30817964

I want to emphasize I'm not policing semantics just for the sake of being ornery. My reaction is approximately the same as it would be to someone who sends me a graph theory paper and calls it number theory. These distinctions are useful.

Yes, I don't especially disagree; I also think words are most useful when they mean something consistent and precise. Nonetheless, I think it's worth separating the merely terminological disagreement from the deeper semantic disagreement. Please do see the article I found(!) in a cousin comment, by Penelope Maddy. I think it provides a very enlightening perspective on the ways in which ZFC and HoTT/UF both purport to be "foundations" and yet seek to perform different duties.

Thanks. I enjoyed the article you linked. I think the description on page 16 is a good summary of what most people working in foundations take the purpose of foundations to be.

Wouldn't it be amusing if a common english word like "foundation", usually referring to the thing on which buildings are constructed, is uniquely reserved for the all the kinds of things on which you can theoretically build all of mathematics, but where none of the people involved are actually trying to do that, and a different word has to be invented for a similar thing, because you got banned out of the club for trying to actually build mathematics on top of it.

I've finally found the article! It is "What do we want a foundation to do?", by Penelope Maddy.

http://www.socsci.uci.edu/~pjmaddy/bio/What%20do%20we%20want...

I need to reread it myself, but this is the article I was referencing.

There's a well-known line of research into the equiconsistency of set-theoretical and type-theoretical foundations, with many compelling results. So you actually can do "reverse mathematics" of type-theoretical foundations. You aren't losing anything.

I've read rather too much of the fascinating comments thread you've indirectly linked to, but this nugget by Mike Shulman in that thread seems especially relevant to the present one:

> If we distill it down from the categorical heights, this brings us back to John Baez’s pithy quote about equality: “Every interesting equation is a lie.” The only obviously true equation is x=x, but it carries no (or little) information. Any other equation, like x=y, is “a lie” because x and y are not the same thing; yet what is interesting about the equality is that they are nevertheless the same in some way. But knowing the equation doesn’t mean that we should forget all about y and use only x; the point of having the equation is that we can pass back and forth between x and y, according to which is most appropriate in any given context.

Not downvoting,in particular there are useful links in your post.

But note that by foundations one has atleast two different meanings. One meaning being examining the consistency of axioms or finding weakest set of axioms for a theorem.

The other meaning being - finding a foundation in which a given mathematical theory is naturally and easily described, analogus to designing a programming language in which the programs are easy to describe.

Most of the time in mathematics, this is done by defining high level concepts and working from there - no need to change the foundational axioms or rules of inference.

But sometimes we need more - for instance, if you read the book "Synthetic Differential Geometry", infinitesimals are naturally embedded in the theory by working in a cartesian closed category of sets. One can either embed this in ZFC using cumbersome models, or change the logic itself(remove the law of excluded middle).

The logic in HOTT deals with equality in a more careful way than the usual foundations. This is not restricted to algebraic topology alone - it certainly includes algebraic geometry and tackles a general issue which will plausibly have many more applications once the technology is available. The field of noncommutative geometry, in the style of Connes, is also about this as the problem of taking quotients is another way of the stating the issue with equality.

It is a work in progress and might take some time before bearing substantive fruit, but note that the current alternative is that any student learning higher level mathematics at the level of Lurie's books has to go through a huge amount of foundational material while the ideas themselves may not be intrinsically complicated. Instead, this complexity is an artifact of embedding the theory in current foundations like the SDG example mentioned above.

With a new foundations, there is the possibility that we can teach the material at this level to students except that there are some tweaks in foundations.

I don't have much to say here that I didn't already say above about the meta-mathematical issues. However, about object-level mathematical issues, you write: "It is a work in progress and might take some time before bearing substantive fruit, but note that the current alternative is that any student learning higher level mathematics at the level of Lurie's books has to go through a huge amount of foundational material while the ideas themselves may not be intrinsically complicated. Instead, this complexity is an artifact of embedding the theory in current foundations like the SDG example mentioned above."

Higher category theory is not the same thing as homotopy type theory. One does not need to learn anything about the foundations of mathematics to read and understand Lurie's work. As far as I know, issues of ZFC vs type theory simply do not arise. You can just pick up Higher Topos Theory and start reading, assuming you have a strong background in algebraic topology.

The linked article is only about higher category theory and making sure all of its claims are written down in a clear and rigorous fashion to be maximally useful to workers in the field. It has nothing to do with the foundations of mathematics as a whole.

> One does not need to learn anything about the foundations of mathematics to read and understand Lurie's work.

By foundations here, I didnt mean foundations of mathematics, rather the amount of material a student needs to read to become familiar with the basic definitions of what a space means. We can teach topology to undergrads, can we teach higher categorical notions of space? Even schemes themselves had a daunting reputation some time back, now we have derived stacks which involve infinite categories.

> You can just pick up Higher Topos Theory and start reading

Yes, but my point, as illustrated in SDG example, was that definitions can be easy or hard depending on which foundations you use. You havent mentioned whether you disagree with this and if you see Kock's book, I think you will agree.

When dealing with higher categories, a foundations which uses '=' symbol in a different way, with an equivalence relation(modelled on homotopy equivalence) playing a much more prominent role could make the theory much more natural.

Note, that I am not even emphasizing sets vs type theory here, but excluded middle and a good notion of equivalence(and equivalence between equivalences... like in homotopy theory) in the foundations.

Maybe a lot of this debate will get dissolved if we just use two different words for 'examining consistency of axioms' and 'naturality of theory as encoded in a given system' instead of using the same word - foundations - for both of them.

At the very least if discussions begin with an awareness of this distinct usage, there would be lot less talking past each other.

Urs Schreiber makes the same point about naturality here - https://golem.ph.utexas.edu/category/2016/03/foundations_of_...

1) Yeah, Lurie's books are advanced tracts aimed at specialists in algebraic topology. Of course there are going to be substantial prerequisites. This is sort of inevitable if you want to understand the motivating applications.

2) Why would we ever want to teach higher categorical notions of space to undergraduates? We can barely get them out in four years knowing real analysis and abstract algebra. The next stop would be the basic theory of manifolds, not higher topos theory.

3) I looked at Kock's book (the draft of the CUP book on his website). The definitions are more abstract than the ones I would give to undergrads, and consequently harder for them to understand. The appeal is not clear to me.

4) You don't have to change your foundations (meaning ZFC vs. type theory, etc.) to do higher category theory in a reasonable way. Again, go pick up Lurie's books. The exposition would not really be affected by whether you choose type theory or ZFC for a foundation. Let me know if you think there's a counterexample to this claim. (The exposition probably would be affected if you denied the law of excluded middle, for the worse; luckily Lurie doesn't do that.)

5) Foundations are sort of irrelevant here anyway. The point of ZFC is that you can forget about it when actually doing math. If you're thinking about foundations when reading Lurie's books, you're doing something wrong. (Consider: most mathematicians have zero opinion about foundational questions or concepts because they have no bearing on their day to day work.)

ZFC is the x86 assembly language of foundations. Saying that ZFC is good enough because you can do all of mathematics in it is like saying that we don't need Java/Rust/Python because x86 assembly is good enough because x86 assembly is Turing complete. True, but that's missing the point of what people are trying to do with higher level languages.

You can theoretically do everything you want in ZFC, but it's low level and a bit outdated and you don't actually want to write formal proofs in it. We know from experience that it is possible to translate the kinds of proofs that people do on paper into formal ZFC proofs, at least in theory. The mathematicians that study ZFC-like foundations are OK with that "in theory"-qualifier and are instead investigating the theoretical properties of the foundations.

The goals of (homotopy) type theory are very different: they are trying to be suitable for actually doing modern mathematics in, and computer proof assistants are being developed that make it possible to actually write down formal proofs in them, in the form of proof code. For this to work it needs to be possible to translate all the kinds of arguments that mathematicians write down on paper into formal proof steps. It is not sufficient to remark that it is theoretically possible to find some kind of translation: it must actually be done for the computer to accept the validity, and the burden must not lie on the mathematician writing down the proof. Instead, the system must make it easy to write down the steps that the mathematician wants to do. One of the kinds of arguments that mathematicians use on paper is that if A is isomorphic to B, then a theorem about A automatically gives you a corresponding theorem about B. Homotopy type theory is trying to formally enable this kind of reasoning.

In summary, the foundations people interested in ZFC have a very different goal than than the foundations people interested in (homotopy) type theory. The fact that the ZFC people are perfectly happy with ZFC and only care about ZFC doesn't mean much. The goals of the type theory people are not achieved by ZFC, not by a long shot.

Note that it is not the case that the general body of mathematicians cares about ZFC and does not care about HoTT. They care neither about HoTT nor about ZFC. The nod to ZFC that mathematicians give when pressed is mere lip service; most mathematicians would not even be able to list the axioms of ZFC. Only foundations people care about ZFC, and the kind of foundations that the ZFC people care about is irrelevant to the practice of general mathematics.

> ZFC is the x86 assembly language of foundations.

> The goals of (homotopy) type theory are very different: they are trying to be suitable for actually doing modern mathematics

> Only foundations people care about ZFC, and the kind of foundations that the ZFC people care about is irrelevant to the practice of general mathematics.

Exactly this, thanks!

I agree with you that this is really about the foundations of algebraic topology, but what makes you say that these are not the foundations of mathematics? In the community of computer-aided formalization, a lot of interest in modern type theory arose, because some things in algebraic topology just couldn't easily be done in the old First-order-logic + ZFC framework: For example, Kevin Buzzard once asked if Isabelle/HOL (which even uses higher-order-logic) can even be used to formalize schemes! (It turns out that it is possible if you do it in a special way with locales... but the Lean formalization still seems more elegant). Of course, there is a consistency proof of Leans type theory using inaccessible cardinals, so the construction of schemes carries over to ZFC+large cardinals, but then it is not very understandable anymore.

I agree with your point about HoTT not being useful to the algebraic topology community, but this seems orthogonal to its usefulness in formalization: I also would not expect Lean's type theory to be directly useful to working mathematicians. Instead it can make formalization easy or even possible: Even Jacob Lurie seems to acknowledge that it could be useful for that and only disputes that it does not bring new insights on an intuitive, informal level. My reading of the discussion is that some HoTT proponents see HoTT as a new language of mathematics (similar to category theory) and that Lurie does not believe that this perspective adds much, but that he does believe that one can use that perspective to understand the theorems.

In the linked discussion Voevodsky points out that "Calculus of Inductive Constructions, the type theory that we currently using to formalize mathematics in the univalent style [and basically what Lean uses] is surprisingly convenient for doing mathematics at the level of sets and categories (and maybe 2-categories). As for mathematics of higher h-levels it is, in practice, of very little use. Note that this is a great advance of what was before as before we did not even have a system for doing abstract mathematics both formally and conveniently at the level of sets." [0]

This highlights how crucial type theory is to do modern algebraic topology, even as he seems to claim that HoTT actually can not formalize Lurie's work? Still, he seems to agree that dependent type theories were a step in the right direction (unsurprisingly, since he founded HoTT) and that more work is needed for finding new foundations for mathematics. Indeed, the, as you write below, "mistaken belief that finding a new foundation of mathematics is an interesting project in the 21st century" is something that some people in foundations (and close to the Lean community) like Jeremy Avigad are actively pursuing: see e.g. https://arxiv.org/pdf/2009.09541.pdf

[0]: https://mathematicswithoutapologies.wordpress.com/2015/05/13...

> It turns out that it is possible if you do it in a special way with locales

What was special about it? From memory, the formalisation [1] proceeded exactly how you would expect. Locales are simply an Isabelle mechanism (in addition to type classes) through which hierarchies of structures are built up.

[1]: https://www.isa-afp.org/entries/Grothendieck_Schemes.html

Curious how approachable you think homotopy type theory is to people who think they know better than researchers? Simple type systems would be understandable (Haskell has type systems similar to System F and rust has an affine your system) but if anything being a programmer may add obstacles to understand HoTT since programmers (generally) have never had the opportunity to learn topology or anything else that HoTT builds on. Also not everything in HoTT has a clear computational interpretation, notably the univalence axiom.

That said I encourage everyone who's interested to investigate this but I don't think it's realistic without having a solid foundation in mathematics.

(And I also agree with the sibling comment that HoTT isn't really used as a foundation of mathematics.)

I don't think it's an easy journey—not because the material is intrinsically difficult, but because almost all resources are written for mathematicians and computer scientists. I think a "homotopy type theory for programmers" book/blog/playlist could be a big hit. I've been considering putting something like that together.

I view the relationship between HoTT and topology as similar to that of functional programming and category theory: you don't need to know the latter to learn the former, but having that extra background can certainly help.

I was also considering writing some introductory stuff about HoTT but I don't think it's possible to avoid talking about topology, homotopy theory, and category theory in depth like you can with, say Haskell and category theory. This is because the stuff that makes HoTT so cool cannot be separated from it's mathematical foundations. How would you explain truncated types without going fairly deep into the math? Or the circle type? Sure both of these could be explained as instances of higher inductive types but that explanation is missing a lot.

As already has been done with category theory, you need to target a specific audience, whether it is programmers, scientists, engineers, or mathematicians, and illustrate the concepts by giving examples from their area of expertise, if possible.

A bit more context can be found here. It was very enjoyable as a video https://youtu.be/WLkMBMUk48E

> You'll see why it's useful to have a formal justification for treating isomorphic objects as equal (this is called "univalence"), which is something you may have even been doing informally/subconsciously all along.

Would you be able to discuss how this compares to quotienting? It sounds similar, but I assume has very important differences. I have, very formally, been doing quotienting. I identify an equivalence relation, quotient it, prove that the properties and operations of interest are well-defined when lifted to the equivalence classes, and then work exclusively at the level of the equivalence classes and the lifted operations. Say, when I work exclusively with reals, which are equivalence classes of Cauchy sequences, or with cosets in group theory.

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I see where you're coming from, but it's hard to appreciate the subtle issues surrounding the notion(s) of equality without actually investing some time into doing formal mathematics. My claim is that type theory is a good avenue to explore this subject matter, since you can check your work and get instant feedback with a type checker.

Anyway, I sympathize with your sentiment that it can be hard to know whether something is worth learning without knowing if or how that knowledge will be useful to you. I feel that way about many topics myself.

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