Formalizing the proof of PFR in Lean4 using Blueprint: a short tour

https://terrytao.wordpress.com/2023/11/18/formalizing-the-proof-of-pfr-in-lean4-using-blueprint-a-short-tour/

by georgehill • 3 years ago

99 37 3 years ago

terrytao.wordpress.com

lanstin2• 3 years ago

Ok, I'll download lean4 and read the tutorials. Everything else I've tried that Prof. Tao has suggested has worked well. I didn't really realize that proof software had advanced so far. I tried to do a project in undergrad to put a lot of ZFC into Prolog but it was just too early to work.

swagmoney1606• 3 years ago

Lean4 has been a pleasure to learn so far! Starting with the natural numbers game was great for me: https://adam.math.hhu.de/

Epa095• 3 years ago

In the last comment below he writes

The project has now completed its primary goals; the entire dependency graph is now green.

haskellandchill• 3 years ago

I don't follow how the proof text is generated from lean? Or is it not, they are doing the verification in lean and the proof text is completely separate.

Looking at the project more I see it's the latter. Turning lean tactics into readable proof text would be hard, as the lean phrase book he is keeping shows: https://docs.google.com/spreadsheets/d/1Gsn5al4hlpNc_xKoXdU6...

empath-nirvana• 3 years ago

There will probably be a time where most proofs are just done in Lean or some similar system and nobody bothers writing a "human readable" paper. Math will just become a kind of programming.

082349872349872• 3 years ago

I had been under the impression that mechanised proof assistants were much too low level for actual mathematicians (like asking programmers to explicitly allocate functional units and issue slots?) so it's interesting news that Prof. Tao finds mucking about with them worthwhile.

lanstin2• 3 years ago

I'm taking a masters in mathematics, and for me the steps are at least three-fold: understanding what you are trying to prove, which is mostly thinking about it and maybe reading or talking to people; picking up pen and paper and proceeding with a sketch of the essential points; typing the proof into LaTeX for a human readable homework :)

I suspect lean will change the thoughts a bit but not eliminate step one. Maybe step three becomes interactive with lean4 etc. and ends in a PR with green build, but the work does not seem likely to end there.

Put it this way: Terrence Tao is unbelievably better at maths than I am; Terrence Tao with lean4 is also unbelievably better at maths than I am with lean4. As far as the value embedded in all already proven mathematics, having it in mathlib is going to make it easy to use, but extending it in the way Prof. Tao can is still pretty hard.

As a side note, I've extended my experimenting with GPT4 from writing Go and emacs lisp to asking it questions about my classes and it does sound smart but will happily give fractional results for an application of Burnside's Lemma, and say things like:

"An example of a simple group without an element of order 2 is any non-abelian simple group of odd order. By the Feit-Thompson theorem, every finite group of odd order is solvable, which means that non-abelian simple groups of odd order do not exist. However, this theorem does not rule out the existence of infinite simple groups of odd order."

Now this answer is correct in describing how (Feit-Thompson theorem) to show that there are no simple groups of odd order, but "infinite simple groups of odd order" is pretty non-sensical. If the order is infinite, it is not odd.

One hopes future iterations will develop understanding in a way that this text lacks.

da39a3ee• 3 years ago

No, I don't think that is so. For one, you still have to decide what is worthwhile to prove. The subject matter of mathematics is very different from that of programming.

jhanschoo3• 3 years ago

Informal proof will always be a thing like how project specifications are always thing. It helps to have a traditional proof to outline the major milestones of your formal proof before actually delving into the formal proof.

zozbot234• 3 years ago

There are declarative systems like Mizar and Isar that do take something a bit closer to natural language proof text as input. Outputing actual text in natural language is just an exercise in pretty-printing (and of course the system must know and work with the natural-language name for every object and statement you're working with, otherwise the output gets clunky).

t_mann11• 3 years ago

I don't follow how the proof text is generated from lean? Or is it not, they are doing the verification in lean and the proof text is completely separate.

My understanding was that the Blueprint tool that he linked is what's responsible for the human-readable math sections shown in the screenshots. I'm not saying it's easy, but it should definitely be doable, basically a spreadsheet like that one on steroids.

alexpeattie• 3 years ago

The proof text is separate. Currently it's up to humans to ensure the proof text (written in LaTeX) properly matches the Lean proof.

Indeed, you can see in the article that the text of Theorem 7.2 omits for brevity details which are present in the non-pretty printed Lean proof (for example H is non-empty, K is a real number, etc.)

InfoSecErik• 3 years ago

Lean has been a really exciting thing for me to watch (as an amateur observer), it's gonna go right up to the limits that Godel found.

t_mann11• 3 years ago

I find it almost scary how casual Terry is about all this. Comparing this to my own humble attempts at graduate math, I feel like a caveman trying to rock a basic wheelbarrow who just gets a visit from a guy in a helicopter. And then that guy is stretching out his hand 'Here, try it out, it's fun, useful and not scary at all'.

It was always clear to me that formalization was the future of math ever since I heard about the four-color theorem. What I did not anticipate, and what Terry seems to take in stride here, was that we'd get an AI that can almost do the job of writing the math as well around the same time as formalization starts to gain traction. The fact that I get to learn about it from the most friendly and humble generational genius is just the cherry on top.

rq1• 3 years ago

Formalism and its “derivatives” died a long time ago already. Even Hilbert couldn’t do much about it.

That being said, constructive math is still a great idea.

t_mann11• 3 years ago

Not sure what you mean by that - what I meant by formalization was to really deduce every statement in a proof from basic axioms and rules. So basically what Prof Tao did in this blog post (except that the hard rote labor has all been abstracted away and handed off to a machine). This kind of formalization is hardly dead I'd say, in fact it's just starting to gain traction in mainstream mathematics, as Prof Tao's recent work shows (if it's not too much work for a Fields medal winner to formalize their proofs, then what excuse do mere mortal mathematicians have?). So what did you mean?

Edit: it's clear that the original four-color proof had nothing to do with this kind of formalization, it was just the debate that got me thinking about it. And tools like Lean4 aren't limited to constructive math afaik.

jhanschoo3• 3 years ago

My experience with copilot in lean is that it's just as useful as in programming. It guesses alright and works well for boilerplate, and structure and syntax, but you need to tune by hand whenever you need to do something significant or clever.

laichzeit0• 3 years ago

I find his comment on Copilot to be enlightening. He says it’s not always right, but he still finds it useful. Kinda like that old saying in stats “all models are wrong, but some models are useful”. Language models are really helpful when you have enough experience to be able to judge when it fails, I do feel sorry for novices who don’t have enough experience yet to do that.

t_mann11• 3 years ago

It arguably matters a lot less in this context - you don't need experience to see when Copilot has failed at Lean, the compiler will tell you anyway. But I'm not saying that understanding it yourself won't matter. In fact, I wouldn't be surprised if the gap between him and more mortal mathematicians actually widens with the availability of those tools.

Formalizing the proof of PFR in Lean4 using Blueprint: a short tour

Related Stories