When Computers Write Proofs, What's the Point of Mathematicians?

Andrew Granville knows that artificial intelligence will profoundly change math. The programming language Lean already plays a role in theorem proving. That's why the University of Montreal number theorist has started talking to philosophers about the nature of mathematical proof — and how the discipline of mathematics might evolve in the age of AI.

Read the full article at Quanta Magazine:

How Close Are Computers to Automating Mathematical Reasoning?

#math #proof #computerscience

[AustinSmithProfile]

The shade he casually throws at physicists 😂

[lucaaaa]

I am learning math by myself at the moment and I have to say... as someone who hated math in school, now I see the beauty of it because I started learning it in a proof based way. I also learned programming and I feel like the logic of programming and programming languages has helped me gain a new look on math and I'm here for it

[lostmylaundrylist]

I find it funny that the worst case scenario for a mathematician seems to be to become more like a physicist. The worst case scenario for a physicist usually is becoming an engineer. As a physicist getting more seasoned, I find it sometimes actually refreshing to do something that might be useful during my lifetime. Indeed I am getting old it seems.

[mfourn]

"So who are we going to become ? We're going to become more like physicists, probably and say any old nonsense, and just hope the computer verifies it."

Savage hahaha

[diegobriaaresrac]

The proofs behind mathematical theorems can often carry more weight than the theorems themselves, revealing additional insights that remain a challenge to fully capture. Consequently, the future may see an intricate interdependence between humans and computers, each relying on the other's strengths to advance our collective knowledge.

[ceyhunay]

Imagine that in 10 years from now, AI is so advanced that some AI company comes along and says their new product was able to generate tens of thousands of new mathematical theorems and it keeps generating new ones exponentially based on ones it already proved. What do the mathematicians do then? Just understand and parse through these theorems, and write explanatory textbooks about hundreds of potentially new mathematical fields that just emerged? Life looks really dull when it's just catching up to AI.

[boudivv]

I think. That, we humanity, should always be able (trained) to rebuild everything from scratch. Universities should become the Guardians of this skill.

[yolanankaine]

I was struck by Andrew’s choice of words and manner of speaking. It shows how well his thoughts converge together and are translated into something remarkable. I have no doubt that years of mathematical experience forces one to transcend into a flow state of ideas.

[isaacwolford]

Well for us to even understand these proofs we would still need a great deal of familiarity with advanced mathematics and the underlying axioms. Computers could really just help us push the boundaries of mathematical discovery much further by laying down new foundations for understanding deeper theorems yet unsolved or even discovered. I think they could become a real good companion for the mathematician. Its an exciting time to be alive!

[FatLingon]

Eventually, when AI surpass us, some problems it will solve might even be too hard for us to understand.

[krustykrewe]

i believe this is a predicament which will encompass many professions as AI technology progresses

[theultimatereductionist]

I work in differential algebra. I am trying to prove that all polynomial (algebraic) ODEs can be solved, at least parametrically, via a finite sequence of linear ODEs by introducing new intermediate differential variables. I would love if computer software and hardware technology were at a point where it could help me prove what I want, but currently it's not.

[prettytrue-zjtj]

It's always "i'm scared for us, what about me..." we rarely think bigger than ourselves

[undeadz]

Looking back in history, for example at a big innovation like the computer, I feel confident in saying that although AI will play a bigger and bigger part, it will never be more than a tool. A future mathmatician will be able to expertly navigate this tool to find what he's looking for. Mathmaticians won't disappear, they'll evolve.

[abrahamanand]

Oh please. We are barely scratching the surface of mathematics. There are a trillion things we still don't know mathematically

[antoniusnies-komponistpian]

We definitely need a metric that measures the length of a proof in a way that minimizing it means to minimize the difficulty of reading it. This would help us to construct a graph of all the important mathematical statements connected with weighted arrows for the easiest proofs. Then we could systematically give computers the arrows we want to be shorter to find better proofs there.
This could help a lot with education in mathematics because it would help us to explain better and also from the student's perspective. We could just insert the beliefs that are already there, and the computer would find a proof from that point.

[axeldaguerre]

This though comes to my mind frequently and is not exclusive to mathematics. If we rely heavily on AI, years after years humans will loose the deep understanding of their "craft". I am just wondering if it's ok or not. It seems different than what came with computers before, but completely aware that most people from this age were certainly thinking the same.

[sunsetclub]

5:40 That physicist dig was very funny

... And also very true.

[aroundandround]

The most fundamental unsolved problem in computer science is equivalent to asking if deriving a proof is qualitatively harder than simply checking correctness of a proof.

We don’t know.

[hannesthiersen]

What will be the point of doing mathematics? For me the point is that mathematics is fun and interesting. AI can never take that away even if it might beat us to the punch by doing it by itself.