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

+∞

I certainly am. Of mathematics and everything.

@JacquesC2 @mc @cbaberle @johncarlosbaez @andrejbauer @dougmerritt @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.

@mc @cbaberle @johncarlosbaez @andrejbauer @dougmerritt @MartinEscardo @pigworker @xenaproject

You don't need to formalize all prior art to be able to do what you want in blackboard mode.

For example, @de_Jong_Tom and I used a result by Mike Shulman, which was published and also formalized in Rocq/HoTT, in TypeTopology.

We didn't want to formalize it (although I have been entertaining the possibility of formalizing it, to understand it better, in my own terms - formalizing for the sake of thinking about it).

So we just used it as a hypothesis to our desired result. That is, we proved "Shulman's result implies our result".

This doesn't establish the ultimate truth of our result in TypeTopology.

But it does allow us to keep using TypeTopology as a blackboard, without worrying about formalizing everything that every mathematician has done in this planet in order to carry on.

And without any lack of honesty. All assumptions are explicitly given. The adopted `--safe` flag guarantees that. There are no postulates. Just explicit assumptions.

@ohad @JacquesC2 @cbaberle @johncarlosbaez @andrejbauer @dougmerritt @pigworker @xenaproject

I have used it as a whiteboard to explain concepts to other people, but that's probably because we insist on clipping our students' wings by not teaching them the relevant mathematics.

I say 'students' here with a very broad interpretation, we are all students indefinitely.

@mc @cbaberle @johncarlosbaez @andrejbauer @dougmerritt @MartinEscardo @pigworker @xenaproject

My 5 might be too low.

@ohad @cbaberle @johncarlosbaez @andrejbauer @dougmerritt @MartinEscardo @pigworker @xenaproject

But then please stop saying things such as "I don't think formalizing mathematics at all makes us think the right way" or other claims about formalization.

If you want to speak with authority about formalization, then learn it first. And, yes, it is not easy or fast to learn it.

I just noticed I didn't answer your question. An hour is fine. More is more.

@MartinEscardo

Lean Game Server

(adam.math.hhu.de)

It's good to work a few examples to see how they spell out in a proof assistant. It's not difficult! It's fun!

I see proof assistants as a sort of calculator for assumptions. They really shine when you throw lots of stuff at them, like verifying computer programs. Which involves lots of boring stuff, and a proof assistant can help with that! If it's a burden then don't use a proof assistant.

@johncarlosbaez @MartinEscardo

I have no fear that *you* will be pressured into doing anything you don't want to, just because it's the hot new trend.

There are dozens of less famous but still interesting examples. I would never have written a paper about the Cobordism Hypothesis (and the still less finished Tangle Hypothesis and Generalized Tangle Hypothesis) if I had been thinking about formal definitions or proofs.

(That's an example that comes from outside analysis! But my other examples suggest analysis is a lot more future-leaning than algebra, in the sense of racking up debts against future formalization.)

- Interactive theorem proving and text-based representations are already uneasy bedfellows. Let's embrace the "interactive" part and move to true structure editing where proof data is stored as abstract syntax trees and the language provides an open protocol for interfacing with such ASTs, on top of which one can implement whatever UI one pleases. Text-based editing becomes just one such "view" of the underlying proof data.
- A lot of "ordinary" math comes down to: draw a structured graph (in *both* senses of the word "graph") of some sort, and make some conclusion based on its structure. By now there's plenty of existing work on compiling things like commutative diagrams/string diagrams/etc. to syntax trees, etc., and we can and *should* make use of this work to provide more convenient interfaces to ITPs, via protocols as above.
- Interactive theorem proving deserves interactive documentation. We should have an analogue of Jupyter/Mathematica Notebooks for ITPs where different editor UIs can be mixed and matched, data can be displayed in a variety of formats, etc.

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

This issue of fit (which is indeed related to linguistics) is why CASes have a huge leg up on ITPs regarding UX. But they also have their fit problems, never mind their correctness problems.

Mathematica's UI coupled with a nice ITP would be nice. That would need a complete rewrite of both halves to be viable. But that would be a vast improvement.

Then we'd be in a good position, as a community, to actually figure out the "next generation" system.

@cbaberle @johncarlosbaez @andrejbauer @dougmerritt @MartinEscardo @pigworker @xenaproject