What is the perfect video game that makes the user infinitely happy?

What is the perfect economy optimizing program?

What algorithm can solve political strife?

I think this might depend on the department, but I was at a pure math department last year, and struggling with my Linear Algebra textbook (written by the professor, incidentally, who was not a great communicator).

I consulted the machines, and learned, to my great delight, that linear algebra is used in like 20 different fields in the real world. It's "perhaps the most applied branch of mathematics in existence".

I complained in the group chat, that our didactic materials, specifically tasked with providing motivation and concrete examples, did not contain a single application, of this most richly applied field.

I was promptly pilloried, and shunned.

(Apparently that particular department was the wrong one, to ask a question like that!)

I suspect the value is in showing the potential that LLMs have in developing new breakthroughs.

Its the same with proofs. First time someone proves something gets a lot of credit. The second proof for the same theorem gets a lot less buzz.

But even then, math proofs mostly get buzz when its something famous or at least important. Proving a random lemma usually doesn't get much buzz.

No, the value is that Erdos's name is attached to it.

Lots of mathematicians prove things they don't publish, or their manuscripts get rejected - not because of a flaw in the proof but because no one cares about the theorem they proved.

And I'm sure it'll be the case with LLM models performing proofs. It'll be notable only when the theorem is a known one that people have had difficulty proving.

To go with your analogy, mathematicians care more about the source code of the program than about the result of the program. But I'm afraid that we will see things change with the increase of vibecoded proof slop. A black box proof is not as useful, even if it is correct.

What does it mean "new"? And, was it a difficult or trivial accomplishment?

A solution to a well known open math problem is both new and non-trivial- you know that many, very smart, very well trained human experts have dedicated time to the problem and haven't been able to solve it, despite good incentives.

Math is something humans invented and is a model, nothing else. There is no logic per se, but a model that works quite well for us.

I studied Math and CS as a very highly gifted and quickly found out, there is no beauty of Mathematical Logic, only humans approval of what they deem most accurate.

A good example is set theory. Cantor was not openly welcomed after he introduced his "theory" to others. In fact, he was received quite some pushback and hostility - this doesn't sound like someone received love the mathematical logic's way.

In fact, the story of Cantor is really a tragic one. He left math for quite some time, due to the pushback.

Only later humans accepted his theory and found it useful. Well, well, what is Mathematical Logic and what not is after all just broad consensus by humans.

And if you go deeper, you will hear more of these stories. Math is anything else but logic. Proofs are religious things, often so complicated, they are simply accepted as "approved by a committee". Many profs cannot really explain simple proofs, they refer to the textbook.

This doesn't sound like romance nor easily reproducible logic.

After all, we deal with human beings.

Or in other words I’d argue novelty is contextual and that these kinds of discoveries’ novelty will eventually wear off too but for right now it’s pretty cool that the “math discovery compiler” works well enough to do this (again imperfect analogy).

This isn't true using the level of originality you're implying with your software examples.

Technically speaking, many novel mathematics proofs are written all the time (quite a few textbook exercises are actually technically novel problems that have never been posed before they were written in a textbook!) that get absolutely no fanfare. Overwhelmingly though they are not very original or difficult and really just required a fairly routine combination of different pre-existing techniques, even if technically speaking that combination didn't exist before. Those textbook problems are hence easy and therefore not given much public attention even if they are technically novel problems.

Indeed over the course of developing a new mathematical result, many many novel results are glossed over to the extent that even their proofs are left out ("as an exercise for the reader") because they are fairly trivial.

This is true for the overwhelming majority of new software as well. A new CRUD program may, technically speaking, be novel, but it's almost certainly just a routine combination of different pre-existing things.

Mathematics open problems that are actually named are generally problems that have resisted the low hanging fruit of the most obvious combinations of pre-existing problems. When those are solved they are a big deal precisely because they usually require some novelty!

Similarly in software, if someone were to create a new kind of database that solves a variety of new classes of problems that current databases fail to solve that would be a big deal! Truly novel software is also perceived as a big deal.

Exactly, "clever". Isn't that the whole point?

As far as whether something like Lean could evaluate this proof: sure, if it were mechanized rigorously. But the amount of work that takes to do varies with both subject and complexity of result. In this case, from what other people are saying, the infrastructure for doing graph theory proofs like this isn't as built up as it is for some other areas of mathematics, so it might take a while.

Unfortunately in my experience that's not really the case. For me, very often GPT 5.5 (which was a good deal better than Opus at this kind of task) would just get stuck for long periods when working in a logic like Iris. It wouldn't necessarily outright prove nonsense, but it would vastly overclaim what it had proved and failed to get anywhere without a lot of hinting. 5.6 is hopefully a lot better about this.