Voevodsky’s Mathematical Revolution
blogs.scientificamerican.com
blogs.scientificamerican.com
I doubt though that HoTT is the way this revolutionary change will happen. If you don't believe me, try reading the first few pages of chapter 2 of the HoTT book. It is basically total gibberish to me, and yes, this is because I am neither well versed in constructive type theory, nor in homotopy theory. I DO have though 10+ years of experience in interactive theorem proving, as well have a diploma in math where I learnt the basic notions of topology. Now, if this is gibberish to me, it will be gibberish to about 99% of all mathematicians out there .
A revolution does not happen by excluding 99% of its targeted population. HoTT will have to evolve into something much easier to understand with a clear value proposition to non-homotopy / non-type-theorists before I would even remotely consider it for providing the underpinnings of a system for interactive proof.
By the way, I don't have a PhD in mathematics, but I do have a PhD in interactive theorem proving.
read the article. HOmotopy Type Theory
If you'd said "programmer" instead of "mathematician," I'd agree with you. As it stands, though, those pages are reviewing just basic algebraic topology, which any practicing professional mathematician would know to some degree. These days, it's taught to top undergraduates and almost universally to first-year graduate students; it is also typically prominent material on PhD qualifying exams in mathematics.
1. Despite your claim that you are not versed in type theory, chapter 1 provides what I find, as a 21 year old graduate student in programming languages with a relatively standard undergraduate background in mathematics and then some, to be a clear and helpful explanation of Martin-Löf type theory, and a good exposition of background needed for chapter 2. Did you read it?
2. This book makes extremely clear that the "homotopy theory" that is developed towards the exposition of homotopy type theory is merely synthetic, which is to say that it considers homotopies as first class objects, rather than deriving them from their traditional topological underpinnings in a more analytic way. It might be useful to have a little understanding of point-set topology with maybe a little inkling of what's going on in algebraic topology to figure this out, but certainly it doesn't seem absolutely necessary, since the book's notion of homotopy is built from first principles. Are there specific points in chapter 2 that you find confusing?
3. It appears you have a doctorate in CS, specifically to do with interactive theorem proving. Almost every proof assistant I have come across either uses dependent types or higher-order logic, and it seems like in order to have earned a PhD in interactive theorem proving you might have had to have become familiar with at least one of these formalisms. Given that you should be comfortable in one of these domains, it doesn't seem like the material in the book is a huge leap. Could you speak a little bit more to what you worked on grad school?
I don't mean to come across as harsh, but your claim that this book excludes 99% of its target population seems false; at the very least I do not consider myself in the top 1% of people who might hope to consume this book.
Chapter 2 lost me at the point where infinity-groupoids entered the game. Until then everything was easy to understand. From then on I would have to seriously sit down and work with the book.
With regard to constructive criticism: I told you what I want, make it easier to understand, and tell me what the value is for me (a non-constructive classical applied mathematician / programmer / engineer). Yep, I did not tell you HOW to get there. I see how that can be annoying for a constructive mathematician.
So, maybe you can tell me: Why should I care about the univalence axiom as a non-constructivist ?
The emphasis that you are placing on the fact that type theory happens to be a constructive logic is perhaps causing you to miss the point. Homotopy type theory is not about advocating constructivism. The "big idea" is that HoTT is a foundation that computers can already reason about easily, based on the work that has already been invested into developing sufficiently powerful dependently typed languages (Coq, Agda). Because it is possible to formulate set theory, category theory, and even real numbers (all discussed in part 2 of the book) within the framework of homotopy type theory, it should be possible to extend these formulations to encompass more and more results from the rest of the mathematics. Because HoTT has already been shown to be implementable (in the form of a library for Coq), this means that any math that is done informally under homotopy type theory can be carried directly into a formal, machine-checkable series of theorems and proofs, in the form of a Coq development.
To expect this material to be readable and useful to every average Joe right away is asking far too much of any new idea in mathematics. This is cutting edge research, and there is still too much even the people closest to this material don't understand yet. As another comment on yours alluded to, at one time your equivalent in the 1700s would have written off calculus as indecipherable and judged it not likely to succeed as a result.
That mathematics can be done in a machine-checkable way was known long before HoTT. Nothing new there. For example, I've had no (particular) difficulty mechanising surreal numbers without using HoTT for this. Also real numbers can be done easily without HoTT. So again:
Why should I care about the univalence axiom?
If you cannot answer that, I am not sure you know what you are talking about. And also, maybe you are a little bit underestimating the links between HoTT and constructivism.
This has little to do with the reason for constructive logic in HoTT or any type theory. Models of type theory collapse if you add excluded middle and you cannot interpret the resulting structure in any particularly useful way. Without excluded middle you have more freedom in designing new axioms.
This is the reason why HoTT is constructive: You have a choice between classical logic and univalence. The latter turns out to be more useful.
Briefly, in Martin-Löf type theory equalities are very strong, but you do not have many tools for proving new equalities. Univalence is one answer to this problem. So the choice is roughly between "having more equalities" (univalence) or "making everything decidable" (excluded middle).
Additionally, in HoTT you can embed classical set theory at "h-level 1". Basically, you get a slightly weaker form of excluded middle + choice, which is nevertheless sufficient for building a model of classical ZFC. The reverse is not possible, as far as I know.
This means that classical mathematics still works as expected, so long as you have the assumption that you are working with things that are "set-like". Additionally you can make any type "set-like", since this amounts to manually collapsing its equality type.
However... the truth is that most of the time excluded middle is unnecessary (in a type theory). Having nice equality types is so much more useful that, honestly, you are not making an informed decision.
So let's say I am writing a theorem proving system based on HoTT (with univalence). Can I then do the classic proof of the irrationality of the square root 2 in that system? Then this would be classical mathematics as I expect it.
This is exactly the answer I expected. It was a little bit of a trap, I have to admit that.
I didn't ask for any proof of the irrationality of square root 2. I asked specifically for the classic proof. Which I cannot do. So it is not classic mathematics as usual.
An honest answer to this whole thread would have been: "No, you cannot do classical mathematics as you were used too. You have to give stuff up, but I think that you will gain other stuff in exchange, and in my opinion this other stuff is more valuable than the stuff you gave up."
What is meant by "even if it wasn't, it would still work" goes back to something he said earlier: type theory embeds an infinite hierarchy of axioms of choice and laws of excluded middles. If you want to do propositional-like reasoning in homotopy type theory, you can assume AC or LEM for homotopy (-1)-types, corresponding to propositional logic.
In type theory you are encouraged to drop the law of the excluded middle and the axiom of choice, because of the fact that doing so gives you potentially more expressive ways of doing things as we have said, but you have gotten the impression that you have to, which you don't.
Also, the claim in this thread was that the results from classical mathematics are provable using homotopy type theory, not that they are provable in the same way (though that holds as well, as I've said above; it's just that the mathematics might not look as clean as if you did it in a more idiomatic way). This kind of a value proposition is not exactly new: category theory loses certain axioms over set theory and mathematicians adapted to the point that category theory is now the language of modern algebra.
I want to point out that I suggested that you read the introduction to the book because it provides these same answers to the questions you are wondering about. I still suggest you do so, as it goes into more detail on what we have said here in a way that I am not able to quite as well.
From what I understand: If you are a type theorist / constructivist, then certain constructions will not be provably equivalent, although their classical counterparts are. Some of those classically trivial equivalences might be recovered though via the univalence axiom. Is that correct or totally wrong? Furthermore it seems to me from the discussions with fmap, that if I am a classical mathematician, I cannot use HoTT (unless I am willing to compromise and give up some of my power).
At this point we are mostly debating what "can use" means -- it's probably enough to say that unless you reframe your thinking, perhaps radically, probably it will be hard for you to be able to use HoTT to get work done. It is possible to do classical mathematics within HoTT, in the normal way, but it would not be very fun.
This is actually a great example, because it doesn't need excluded middle. Equality of rational numbers is decidable, which means that classical reasoning is provable. And yes, even if it wasn't, it would still work.
Edit: Ok, this was apparently confusing, so let me repeat it here. Yes, the same proof works. This form of excluded middle is entirely unproblematic.
In fact, the standard proof-by-contradiction that sqrt(2) is irrational is also intuitionistically valid. The important point for this case is that we're proving a negative property (irrationality), so this sort of proof by contradiction is fine.
The standard example of a classical proof I use is the one showing that there exist irrationals x and y such that x^y is rational. The dead easy proof asks whether x^y is rational for x=y=sqrt(2). If it is, we're done. Otherwise, we just take x=sqrt(2)^sqrt(2) and y=sqrt(2).
But note that we don't actually know for sure which x and y work, so this is definitely classical. An intuitionistic proof here would have to tell us directly what x and what y work.
"The thing that’s so remarkable about this new foundation is that the fundamental concepts are much closer to where ordinary mathematicians do their work. In the usual foundation, Zermelo-Frankel set theory, it takes an enormous amount of work just to build up the basic concepts and theorems that mathematicians rely on every day. The result is that if you want a computer to check your proofs, you have to teach it all that theory first. Essentially, you have to give it the same education you got — except that you have to do it in a far more exacting way. As a result, the only people so far who have used computer proof systems are computer scientists who specialize in it, and it takes them many years of effort to check a single new theorem. Georges Gonthier, for example, spent a decade checking the four-color theorem.
But this approach circumvents all that labor. Not only that, but the language the computer understands is far closer to natural mathematical language."
I should point out that the Four-Colour Theorem was formalised in Coq, not a ZF-based proof assistant. But that aside, these are pretty big claims. I'd like to hear what sort of basic theory is being talked about, and how the proofs are closer to natural language. On the stackexchange thread, ek comments:
"Anyone who has used Coq will tell you that the proofs that you produce often don't end up looking much like equivalent plain-English proofs at all."
But this doesn't strike me as fair. I'm a HOL Light user, and the de facto proof style is procedural. In that style, which I understand is preferred also in Coq, the user tells the machine how to compose an assortment of automated tools to break down and solve the problem, often performing computations to massage the goal into a suitable form. The scripts are usually completely unreadable, but that's by design. Those who want a more readable style of formalised proof are encouraged to use "declarative styles".
How do the claims made about HoTT's readability factor in here?
It's possible to create readable coq scripts. However, like with any kind of programming it takes effort. If you take a look at the proof of four colors theorem of fiet thompson they are quite readable and don't look like complete mess. They, however, use modified coq instead which is called ssreflect.
Some years ago I spent a serious amount of time reading topos theory, particularly local set theory, as given by the Mitchell-Benabou internal language of any topos, looking for a better approach to description logics.
I do believe topos theory provides a better foundations than set theory because the logic is inherently intuitionistic. I found it also provided a simpler explanation of independence results. However one of the things I believe is true of the proof theory is that there is no cut-elimination theorem, which is important to establish a sub-formula property.
Sorry to ramble here, let me ask specifically, are there connections between HoTT and topos theory? Or geometric logic? Your earlier comment on the synthetic nature made me think there might be.
Thank you
Edit: I more or less answered my own question by reading the introduction, the section on open problems, the possible connections are between HoTT and the higher toposes. Judging from the intro, this seems very readable.
Perhaps you are saying that it simply won't deliver, but you cannot possibly reach such a conclusion by reading 1.5 chapters out of 11.
I do not understand how the fact that currently most people do not know the relevant material is an argument against HoTT: of course nobody knows it: it's new! Did you expect "revolutions" to happen by telling people something they already know? Do you believe Grothendieck's work was a mistake because 99.9999% percent of mathematicians weren't Algebraic Geometers by the time EGA was published? Your argument makes no sense, maybe because you are comparing HoTT to Facebook, when in fact it's a mathematical achievement.
Also: if this is gibberish to me, it will be gibberish to about 99% of all mathematicians out there .
I seriously doubt that.
Now that is utter and complete nonsense. And that is why I am criticising HoTT. HoTT might be a beautiful mathematical theory. I am not competent in judging that, to be so, I would have to invest more time. But there is not a single shred of evidence that HoTT will improve automated, or for that matter interactive theorem proving. And I am pretty competent to judge that.
Read for example: http://homotopytypetheory.org/2013/03/08/homotopy-theory-in-...
So HoTT might solve a problem for constructivists working with Coq. Problems you can entirely avoid by not working constructively (or with Coq, for that matter). Therefore, at this point, there is no evidence at all that HoTT might improve the state of the art in interactive theorem proving.
That does not mean that this evidence might not come up at some point.
How can I be sure in my judgement? Well, I don't have to understand string theory to be sure at this point that it is not relevant to interactive theorem proving. I would change my mind if somebody points out evidence that string theory is actually of major relevance to interactive theorem proving.
Edit: I would even start learning string theory then.
Instead of mathematician, I mean: people who could potentially benefit from interactive theorem proving technology.
Basic notions of topology only touch a bit of the corners of HoTT.
Have you seen the Agda code on the HoTT github site? That might be closer to your language. Or poke around at Andrej Bauer's work, as he leans toward the computerization side a bit.
http://hottheory.files.wordpress.com/2013/03/hott-online-323...
I really wish more people would open source their books.
I studied a lot of mathematics, and I find this fascinating, but I've found it hard to read the literature on type theory, and mathematical logic also.
You can definitely just read the material and then come back and try the exercises in Coq later, or more tightly couple the two. I think either approach would work well, and it's up to personal preference.
The first chapter of HoTT is an introduction to Martin-Löf that I personally find quite intuitive, to the point that I'd say it's clear than most other expositions of the same material that I've tried to read.
It will be easier to understand dependent type theory if you're programmed in a dependently typed language, but I wouldn't say it's strictly necessary to understand dependent types theoretically. I have just a little bit of Coq experience and found myself comfortable with the HoTT presentation of dependent types.
http://cs.stackexchange.com/questions/19001/what-were-the-re...
Is automated theorem proving really this close to being this powerful?
So soon we'd be able to feed the proof of Poincare Conjecture to a computer and it'd be able to verify the proof? I was under the impression we were nowhere close to being able to do that.
We are definitely not even close. But getting mathematicians acquainted with HoTT is a good first step, I think. The book itself presents formulations of homotopy theory, set theory, category theory, and a constructive view of the real numbers, all within homotopy type theory, and on top of all this we could likely start building up a corpus of more results from topology, algebra, and analysis.
If you read your link carefully, you'll notice it's about provability via axioms, not verification.
Peace
Until I read your post, it never occurred to me you were confused by the ambiguous implications of "verification".
You remind me how easy it is, once you get the jargon and the concepts, to not actually write out everything into a form someone from the outside can understand.
In retrospect, it seems like something I should have been able to deduce, since it's kind of the only thing makes sense.
Missing such a theorem isn't exactly a deal-breaker though.
http://www.cse.unsw.edu.au/~liamoc/talk.pdf
They were presented at fpsyd.
Vladimir mentions that he had a course at Princeton where he learned Coq. Most probably this is the course: http://www.cs.princeton.edu/courses/archive/fall10/cos441/in...
And its based on on-line book Software Foundations http://www.cis.upenn.edu/~bcpierce/sf/toc.html which is really accessible for about anyone with small programming experience.
Why not archive and compress them in a tar.gz ?