@johncarlosbaez
> formalization tends to impose consensus

I'm not sure people are getting your point. The way I see it, formalization tends to make people subconsciously think in terms of The One And Only Truth -- and thus consensus.

But we know that in mathematics, there are always other paths, and truths follow from axioms / axiom schemas, which themselves can vary.

Or perhaps I'm missing your point, too.

@dougmerritt - You got my point. Working in Lean or any computer system for formalization, you need to submit to the already laid down approaches, or spend a lot of time rewriting things.

I added a quote from Kevin Buzzard to emphasize the problem:

Kevin Buzzard says "It [formalization? Lean?] forces you to think about mathematics in the right way."

But there's no such thing as "the" right way!

@johncarlosbaez @dougmerritt

This is just the beginning.

Current systems are the FORTRAN and Pascal of proof systems; they are for building pyramids--imposing, breathtaking, static structures built by armies pushing heavy blocks into place.

What we need is for someone to invent the Lisp of proof systems. Something that helps individuals to think new thoughts.

@maxpool @johncarlosbaez
Yes, well, moving past John's point:

Easier said than done. Current things like Lean are lots better than the systems of years ago, but -- do you have any specific ideas?

I used to follow that area of technology, but I somewhat burned out on it. For now, Terry Tao et al is getting good mileage out of Lean.

I suppose there's some analogy with the period of shift from Peano axioms to ZFC and beyond.

@johncarlosbaez @dougmerritt @MartinEscardo @JacquesC2 @pigworker Somewhat unexpectedly, I find myself on the same side as @xenaproject on this one, I suppose because I read "the right way" differently from @johncarlosbaez

Formalized mathematics makes us think "the right way" in the sense that it requires mental hygiene, it encourages better organization, it invites abstraction, and it demands honesty.

Formalized mathematics does not at all impose "One and Only Truth", nor does it "nail things down with rigidity" or "impose concensus". Those are impressions that an outsider might get by observing how, for the first time, some mathematicians have banded together to produce the largest library of formalized mathematics in history. But let's be honest, it's miniscule.

Even within a single proof assistant, there is a great deal of freedom of exploration of foundations, and there are many different ways to formalize any given topic. Not to mention that having several proof assistants, each peddling its own foundation, has only contributed to plurality of mathematical thought.

Current tools are relatively immature and do indeed steal time from creative thought to some degree, although people who are proficient in their use regularly explore mathematics with proof assistants (for example @MartinEscardo and myself), testifying to their creative potential.

Finally, any fear that Mathlib and Lean will dominate mathematical thought, or even just formalized mathematics, is a hollow one. Mathlib will soon be left in the dust of history, but it will always be remembered as the project that brought formalized mathematics from the fringes of computer science to the mainstream of mathematics.

@andrejbauer @johncarlosbaez @dougmerritt @JacquesC2 @pigworker @xenaproject

It is not inconceivable that one day in the not-too-distant future, proof assistants will be able to "understand" proofs written in sufficiently careful, informal, mathematical vernacular and translate it to a suitable formal language.

And this formal language doesn't need to be fixed. The mathematician just chooses a foundation, or, in case they don't care, they let the proof assistant choose a suitable one for the informal (but hopefully rigorous) mathematics at hand.

I don't mean AI, but people are certainly trying this with so-called AI nowadays (personally, I think this is the wrong approach, but **I don't want** this to become the subject of discussion here).

In any case, a person will need to check that the definitions and the statements of the theorems and constructions are correctly translated (*). Then the formal proofs obtained from informal proofs don't need to be checked by people.

(*) At least at the beginning. For example, we now trust that C compilers produce correct machine code and don't check it ourselves.

In any case, all of the above can happen only step by step, and currently we are at an important step, I think, where the first were in the 1960's by de Bruijn.

As I said before, I use proof assistants as smart blackboards. If I could get interactive help while I write in mathematical vernacular, I would immediately adopt this incredible new proof assistant.

And, I repeat, I don't mean the kind of non-help I get from ChatGPT, Gemini, Claude, DeepSeek, or what-you-have - I feel I help them rather than the other way round.

I mean the kind of help I already get in non-AI-based proof assistants

@MartinEscardo @andrejbauer @johncarlosbaez @dougmerritt @JacquesC2 @pigworker @xenaproject

It seems to me that anything that translates careful but informal mathematical vernacular into a formal language, must ipso facto be AI. But that doesn't mean that it would be the kind of AI that you describe as unhelpful at the end of your post. Rather, it means that AI encompasses much more than those things.

@TobyBartels It is interesting you say this.

Some things that were considered AI in the past are now deterministic algorithms.

For example, checking formal proofs with incomplete information (to be reconstructed by the checker) for correctness.

@andrejbauer @johncarlosbaez @dougmerritt @JacquesC2 @pigworker @xenaproject