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Lean was a gamechanger for me as someone who has a "hobby" level interest in abstract mathematics. I don't have the formal education that would have cultivated the practice and repetition needed to just know on a gut level the kinds of formal manipulations needed for precise and accurate proofs. but lean (combined with its incredibly well designed abbreviation expansion) gives probably the most intuitive way to manipulate formal mathematical expressions that you could hope to achieve with a keyboard.
It provides tools for discovering relevant proofs, theorems, etc. Toying around with lean has actively taught me math that I didn't know before. The entire time it catches me any time I happen to fall into informal thinking and start making assumptions that aren't actually valid.
I don't know of any way to extract the abbreviation engine that lean plugins use in the relevant editors for use in other contexts, but man, I'd honestly love it if I could type \all or \ne to get access to all of the mathematical unicode characters trivially. Or even extend it to support other unicode characters that I might find useful to type.
Bessis [1] argues that formalism - or loosely math writing - is foundational to clarifying intuition/meaning in a way that natural language cannot. Imagine it as a scalpel carving out precise shapes from the blur of images we carry thereby allowing us to "see" things we otherwise cannot.
I am curious to try out lean to understand how definitions in lean are able to operationally capture meaning in an unambiguous manner.
[1] https://www.amazon.com/Mathematica-Secret-World-Intuition-Cu...
For mathematics and certain fields, that is true. But the formalism matters, and as some have argued, the Fregean style that came to dominate in the 20th century is ill-suited for some fields, like linguistics. One argument is that linguists using this style inevitably recast natural language in the image of the formalism. (The traditional logical tradition is better suited, as its point of departure is the grammar of natural language itself.)
No formalism is ontologically neutral in the sense that there is always an implied ontology or range of possible ontologies. And it is always important to make a distinction between the abstractions proper to the formalism and the object of study. A common fallacy involves reifying those abstractions into objects of the theory, at least implicitly.
It is interesting that you argue for formalism using a metaphor in natural language, rather than use a mathematical/data oriented argument. I find the metaphor pleasing in a way that I suspect a more data driven argument would not be.
I mean, if you understand leans system then you understand the formal manipulation needed for precise and accurate proofs. Most mathematical papers are rather handwavy about things and expect people to fill in the formalism, which is not always true, as we have seen
I've been excited about Lean for years, not because of correctness guarantees, but because it opens the door to doing maths using software development methods.
Libraries of theorems and mathematical objects, with well defined abstractions that are ergonomic to apply in target use cases. Accompanied by good documentation, focused less on how the theorems are proven (how the functions are implemented), and more on what to use them for and how. With proper version control and package management.
I believe that all these practices could vastly improve collaboration and research velocity in maths, as much or more than AI, although they are highly complementary. If maths is coding, AI will be much better at it, and AI will be more applicable to it.
As a a hobbyist mathematician / type theorist, chatgpt et al are great at 'looking up' theorems that you want to exist but that you may not have read about yet. It's also good at connecting disparate areas of math. I don't think lean subsumes AI. Rather, lean allows you to check the AI proof. ChatGPT genuinely does have a knack for certain lines of thought.
LLMs and Lean are orthogonal, neither subsumes either.
They both can be useful or harmful, do to their respective strengths and trade offs.
PAC/statistical learning is good at needles in the haystack problems assuming that the tail losses, simplicity bias, and corpus representation issues are acceptable and you understand that it is fundamentally existential quantification and control for automation bias etc…
Lean is a wonderful collection of concepts and heuristics but due to Rice and Gödel etc… will not solve all problems with software development.
How Gödel’s second incompleteness theorem shows that you can prove anything, without that proof being meaningful is a lens into that.
It is horses for courses, and remember that even in sub-TC total functional programming, proving and arbitrary functions is very hard, while constructing one is far more tractable.
Even those proofs don’t demonstrate semantic correctness. History is riddled with examples of people using powerful tools that elegantly explain flawed beliefs.
The 2009 crash and gaussian copula as an example.
Get all the value you can out of these tools, but use caution, especially in math, where superficially similar similarities often have conflicting conventions, constraints, and assumptions.
Obviously if you problem is ergotic with the Markov property, both will help, but Automated theorem proving and PAC learning will never be a meta theory of the other IMHO.
Out of curiosity, does anyone know the mathematicians actively leaning into AI + Lean?
> I've been excited about Lean for years, not because of correctness guarantees, but because it opens the door to doing maths using software development methods.
> Libraries of theorems and mathematical objects, with well defined abstractions that are ergonomic to apply in target use cases. Accompanied by good documentation, focused less on how the theorems are proven (how the functions are implemented), and more on what to use them for and how.
How is any of that different from what we had in math before Lean?
I have proven quite a few theorems in Lean (and other provers) in my life, and the unfortunate reality is that for any non-trivial math, I still have to figure out the proof on paper first, and can only then write it in Lean. When I try to figure out the proof in Lean, I always get bogged down in details and loose sight of the bigger picture. Maybe better tactics will help. I'm not sure.
> Maybe better tactics will help. I'm not sure.
I don't see why they would.
If anyone is curious about the phenomenon, the second problem in session 7 at https://incredible.pm/ [ ∀x.(r(x)→⊥)→r(f(x)) ⟹ ∃x.r(x)∧r(f(f(x))) ] is one where the proof is straightforward, but you're unlikely to get to it by just fooling around in the prover.
In principle, LLMs can do this already. If you ask Claude to express this in simple words, you will get this translation of the theorem:
"If applying f to things makes them red whenever they're not already red, then there must exist something that is red AND stays red after applying f twice to it."
You will be able to interact like this, instead of using tactics.
It's good to remind yourself of Bill Thurston's points:
https://www.ams.org/journals/bull/1994-30-02/S0273-0979-1994...
I love the analogy in David Bessis's wonderful book Mathematica (nothing to do with Wolfram). We all know how to tie our shoes. Now, write in words and symbols to teach someone how you tie your shoes. This is what a proof is.
Often even people with STEM degrees confuse what mathematicians do with the visible product of it - symbols and words on a page. While the formalism of mathematics has immense value for precision, and provides a "serialization language" (to borrow a CS analogy), it would be akin to confusing a Toaster with the Toaster manual, or shoelaces with the instructions.
I started having a much easier time with mathematics when I realized and got comfortable with this idea. In hindsight it should've been obvious to me as a programmer - when I'm building something, I don't ideate in terms of individual lines of code, after all
As a former professional mathematician: the benefits mentioned in the article (click-through definitions and statements, analyzing meta trends, version control, ...) do not seem particularly valuable.
The reason to formalize mathematics is to automate mathematical proofs and the production of mathematical theory.
Rado Kirov shows that formalization transforms how mathematicians think about structure and collaboration. My work begins from the same premise, but in the world of programming and system software. I aim to bring formal structure to programming itself, treating algorithms, operating systems, and programming languages as subjects that can be expressed with the same rigor as mathematics.
I just released my treatise yesterday, at https://leanpub.com/elementsofprogramming
Elements of Programming presents programming as a mathematical discipline built on structure, logic, and proof. Written in the style of Euclid’s Elements, it defines computation through clear axioms, postulates, and propositions. Each book develops one aspect of programming as a coherent system of reasoning.
Book I establishes identity, transformation, and composition as the foundations of computation.
Book II introduces algebraic structures such as categories, functors, and monads.
Book III unites operational and denotational semantics to show that correctness means equivalence of meaning.
Book IV formalizes capability-based security and verification through invariants and confinement.
Book V connects type theory with formal assurance, explaining how types embody proofs.
Book VI extends these ideas into philosophy and ethics, arguing that software expresses human intention and responsibility.
I want to respond to each of his points one by one
> powering various math tools
I don't think going through a math proof like they were computer programs is a good way to approach mathematics. In mathematics I think the important thing is developing a good intuition and mental model of the material. It's not a huge problem if the proof isn't 100% complete or correct if the general approach is good. Unlike programming, where you need a program to work 99.9% of the time, you have to pay close attention to all the minute details.
> analyzing meta-math trends
I'm highly skeptical of the usefulness of this approach in identifying non-trivial trends. In mathematics the same kinds of principles can appear in many different forms, and you won't necessarily use the same language or cite the same theorems even though the parallels are clear to those who understand them. Perhaps LLMs with their impressive reasoning abilities can identify parallels but I doubt a simple program would yield useful insights.
> Basically, the process of doing math will become more efficient and hopefully more pleasant.
I don't see how his points make things more efficient. It seems like it's adding a bunch more work. It definitely doesn't sound more pleasant.
Noob question here.
Say I'm wanting to formalize a proof. How do I know that what I'm writing is actually a correct formulation?
If it gets more complicated, this problem gets worse. How do I know the thing it is checking is actually what I thought it was supposed to check?
I guess this is a bit like when you write a program and you want to know if it's correct, so you write some tests. But often you realize your tests don't check what you thought.
You don't know. Even with the best theorem provers, your definitions are still trusted. The best way I've found to help with this is to keep your definitions simple, and try to use them to do things (e.g. can you use your definition to solve other problems, does it work on some concrete examples, etc).
A point that is maybe not obvious to people who have not done mathematics at a high level or done “new” mathematics, is that often you end of changing your theorem or at least lemmas and definitions while figuring out the proof. That is, you have something you want to prove, but maybe it is easier to proving something more general or maybe your definitions need to change slightly. Anecdotally, during a project I spend perhaps a year figuring out exactly the right definition for a problem to be able to prove it. Of course, this was a very new thing. For well-know areas it is often straight forward, but at the frontier, both definitions and theorems often change as your proceed and understand the problem better.
Being sure that you are proving the right thing is something that can never be formally guaranteed.
In a lot of cases you can get far by locally proofreading the definitions.
Trying to formally prove something and then failing is a common way people find out they forgot to add an hypothesis.
Another pitfall is defining some object, but messing up the definitions, such that there's actually no object of that kind. This is addressed by using test objects. So suppose you define what a ring is, then you also prove that real numbers and polynomials are examples of the thing you defined.
Related: Terrence Tao discussing Lean (programming language for formalising mathematical proofs) on Lex podcast (starts 1h20m): https://www.youtube.com/watch?v=HUkBz-cdB-k&t=1h20m10s
I know nothing of mathematics but found it fascinating, especially the idea that if outside information changes that affects your proof, you can have the Lean compiler figure out which lines of your proof need updating (instead of having to go over every line, which can take days or more).
Another reason to formalize math is that formalized proofs become training material for automated mathematics.
Ultimately we want all of the math literature to become training material, but that would likely require automated techniques for converting it to formalized proofs. This would be a back-and-forth thing that would build on itself.
Much of the argument is the same as for the initial push to formalize mathematics in the late 19th century. Formalisms allow for precision and help reduce errors, but the most important change was in how mathematicians were able to communicate, by creating a shared understanding.
Computerized mathematics is just another step in that direction.
Imho it was always "computerized", they just didn't have a computer. To me the approaches used in the early 20th century look like people defining a simple VM then writing programs that "execute" on that VM.
Is there, somewhere, a list of theorems that were considered proved and true for a while, but after attempts at formalization the proof was invalidated and the theorem is now unknown or disproved?
There was a question [1] on mathoverflow about this with a couple of interesting answers and comments.
[1] https://mathoverflow.net/questions/291158/proofs-shown-to-be...
I’m not a mathematician, so could someone explain the difference in usage between Lean and Coq?
On a surface level my understanding is that both are computer augmented ways to formalize mathematics. Why use one over the other? Why was Lean developed when Coq already existed?
I think the difference is mostly cultural. The type theories of Lean and Rocq are fairly close, with the exception that Lean operates with definitional proof irrelevance as one of the default axioms. This causes Lean to lose subject reduction and decidability of definitinal equality as properties of the language.
Many people in the Rocq community see this as a no-go and some argue this will cause the system to be hard to use over the long run. In the Lean community, the interest in type theory is at a much lower level, and people see this as a practical tradeoff. They recognize the theoretical issues show up in practice, but so infrequently that having this axiom is worth it. I consider this matter to be an open question.
If you look at what's being done in the communities, in Lean the focus is very much on and around mathlib. This means there's a fairly monolithic culture of mathematicians interested in formalizing, supplemented with some people interested in formal verification of software.
The Rocq community seems much more diverse in the sense that formalization effort is split over many projects, with different axioms assumed and different philosophies. This also holds for tooling and language features. It seems like any problem has at least two solutions lying around. My personal take is that this diversity is nice for exploring options, it also causes the Rocq community to move slower due to technical debt of switching between solutions.
Lean has a good library of formalized mathematics, but lacks code extraction (you cannot generate a program from the proofs it constructs). So it is more suitable and highly used by mathematicians to prove theorems.
Coq has always focused on proving program correctness, so it sees lots of use by computer scientists. It also does code extraction, so after you prove a program correct in Coq you can generate a fast version of that program without the proof overhead.
I think that (most) mathematicians were not that interested in formal proof until quite recently (as opposed to computer scientists), and most of the interest in lean has been self-reinforcing, namely there is a (relatively speaking) huge library of formally verified mathematics. So now basically anyone who cares about formal verification as a tool for mathematics is working in lean. There are of course numerous techincal differences which you can read about if you google coq vs lean.
Rocq is ancient and has some longstanding UX problems. It is pleasant to try making a new code base.
This is kinda like asking, why write Clang when we already had GCC? Or, why making Python if we already have Perl?
It's good to have some competition for these things, Rocq I believe felt the heat and has been also doing some good things in recent years.
Lean has much better UX to be frank. Rocq is fine, but if I were to start formalising today, I'd pick Lean.
Ask HN: What’s the single best resource for learning Lean (beyond the official docs)?
Assuming you have some math background but no Lean background: https://adam.math.hhu.de/#/g/leanprover-community/nng4
All the good resources are listed here: https://lean-lang.org/learn/
I recommend the natural number game (also mentioned above) for a casual introduction to the mathematics side, just to get a feeling.
If you are serious about learning lean, I recommend Functional Programming in Lean for learning it as a programming language and Theorem Proving in Lean for learning it as a proof assistant
For anyone that's interested in formalizing mathematics but wished there was an easier way to do it, I've been working on a different sort of theorem prover recently.
The idea is that there's a small AI built into the VS Code extension that will fill in the details of proofs for you. Check it out if you're interested in this sort of thing!
Lean is amazing for collaboration because anyone can contribute to a proof and their work be automatically verified.
I think the analogy between JavaScript and TypeScript is not 100% because although JavaScript has some quirks in its design, it is fully consistent. My biggest issue with math is symbols that are reused to mean different things in different contexts. It makes maths more time-consuming to learn and makes it difficult to jump between different fields.
Personally, at times, I struggled with the dual nature of mathematics; its extreme precision in meaning combined with vague and inconsistent use of symbols is challenging... Especially frustrating when learning something new and some symbols that you think you understand turn out to mean something else; it creates distrust towards maths itself.
> While Paulson focuses on the obvious benefit of finding potential errors in proofs as they are checked by a computer, I will discuss some other less obvious benefits of shifting to formal math or “doing math with computers”
From https://news.ycombinator.com/item?id=44214804 sort of re: Tao's Real Analysis formalisms:
> So, Lean isn't proven with HoTT either.
The issue is when people conflate formalism with truth itself. I know a lot of people who reject anything that isn't under the umbrella of "stuff that is formalised", even if it can be formalised but was simply not presented as formalised in its first incarnation.
Not super convinced by this analogy. Tooling and convenience feel secondary in math. If formalization doesn't help us do better mathematics, not just more structured mathematics, I'm pretty skeptical these benefits will justify the cost.
I don't get the point about trivial proofs. Can't you just tell Lean to assume something is true and then get on with the rest of the interesting part?
You can but that ruins the fun and also misses the point. How do you know your "trivial" theorem is actually trivial? Proofs are mechanized to increase our trust into them, and it defeats the point if you have to still manually review a myriad of helper lemmas.