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.

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.

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.

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.

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.

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.

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).

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.

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.

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?

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.

https://acornprover.org

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!

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?

Ask HN: What’s the single best resource for learning Lean (beyond the official docs)?

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?

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.

Lean is amazing for collaboration because anyone can contribute to a proof and their work be automatically verified.

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.

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.