One of the most fruitful approaches in mathematics is to flip back and forth between geometric and algebraic views of a problem. I think this works so well because these are actually handled by two different parts of the brain on a physical level; spatial reasoning is separate from language processing. Cytoarchitecture shows these regions have different "textures;" the local details of the way neurons are wired together are simply different in these different regions of the brain, in the same way a CNN and a transformer have different topologies. Thus, by flipping problems from geometry to algebra and vice versa, we're able to bring an entirely different cognitive style to bear on a problem. For example, the proof of Monge's Theorem by moving to 3D and visualizing not three circles, but three spheres sitting on a table with a book on top of them and then pointing out that the intersection of two planes is a line. What is pages of unintuitive symbol pushing turns into something a child can understand. Going the other way, things like the angle addition formulas or the quadratic formula, which are quite hard to prove geometrically, become quite simple if you use a little algebra.

Current-gen LLMs are still relatively weak at visual reasoning; see the Vision Language Models are Blind paper, for example, or the ARC-AGI benchmark. So that's one way humans can stay ahead of the agents, at least for now.

Are we going to see less publicly shared science? With private actors or governments restricting access to AI resources to a few scientists and keeping new knowledge to themselves.

Advancing science in the open was the best strategy when there was real advantage to share the load with every brain on the planet willing to give a try at science, but if a computer can match or surpass the collective output of the entire human scientific community the equation will change.

It's a sad outlook.

Feels a lot like building software from bottom - once you get the building blocks defined right, the action, or the program, are trivial to express. When doing it from the top-down, you write the program using the building blocks you haven't defined yet, and you might end up with overly specific building blocks, needing other blocks for expressing different behaviors.

When you do the bottom-up building blocks right, new behavior is easy to express with them. Essentially, you are building up the language to reach the problem. Or making a DSL, whatever definition you like best.

HN history

  +-- 2mo before by sdfrew
  |   4 points / 1 comments
  |   The Fall of the Theorem Economy
  |   https://news.ycombinator.com/item?id=47862472
  |
  +-- 2mo before by fuglede_
  |   3 points / 1 comments
  |   The Fall of the Theorem Economy
  |   https://news.ycombinator.com/item?id=47891494
  |
  +-- 2mo before by mathgenius
  |   2 points / 0 comments
  |   The Fall of the Theorem Economy
  |   https://news.ycombinator.com/item?id=47909751
  |
  +-- 2mo before by delis-thumbs-7e
  |   15 points / 4 comments
  |   David Bessis on AI destroying mathematics
  |   https://news.ycombinator.com/item?id=47985962
  |
  +-- 1mo before by magoghm
  |   4 points / 0 comments
  |   The Fall of the Theorem Economy
  |   https://news.ycombinator.com/item?id=48084737
  |
  +-- 1mo before by cubefox
  |   2 points / 0 comments
  |   The Fall of the Theorem Economy
  |   https://news.ycombinator.com/item?id=48089716
  |
  +-- 1mo before by cubefox
  |   5 points / 0 comments
  |   The Fall of the Theorem Economy
  |   https://news.ycombinator.com/item?id=48152469
  |
  +-- 1mo before by tmp10423288442
  |   4 points / 1 comments
  |   The Fall of the Theorem Economy
  |   https://news.ycombinator.com/item?id=48214866
  |
  `-- this submission by varjag
      58 points / 7 comments
      The Fall of the Theorem Economy
      https://news.ycombinator.com/item?id=48758048

I can understand why this is a major concern for mathematicians. They got into their field because they love the beauty of mathematics, and the intellectual satisfaction of understanding non-obvious insights. But to put it crudely, this sounds like a you problem. As someone who isn't a mathematician, the main value I get out of math is its practical applications in science and technology. And their practical applications in human life. I have zero understanding of the math behind cryptography, but I still deeply appreciate the practical value they have provided humanity.

If AI systems start churning out accurate theorem-proofs, and we are able to use those theorems to build things that improve human quality of life, it doesn't bother me one bit that those theorems have not been understood by humans. If this offends your aesthetics, you are certainly entitled to your opinion and your preferences, but that does not make it a societal problem

What would happen if a non-human layer of mathematics emerged on top of human mathematics? In this article, the distinction between Mathlib and Mathslop might be a precursor to that.

If models advance enough in the future, and new definitions, compressions, and representational forms that are convenient for AI-to-AI communication emerge, what would happen then? Would mathematics split into Human-facing and Machine-facing branches?

"I was in Switzerland", "I was invited to a talk", "I started a machine learning company", look at me bro.

A wood-worker could do the same argument, there's the "official" wood-working word of perfect joinery and beautifully finished tables one can buy, but behind it there's the "secret" messy human element, the art, the craft, the mistakes and hard-ships, the elevation of human skills and imagination, the creation of whole new types of wood-working inventions and techniques, the perpetuation of millenia-old traditions, the teaching, the joy of selling to a happy customer, etc.

But now comes techo-capitalism, division of labor, you cut that piece a that piece over and over, you operate that machine, you won't even see the finished table, fuck your human element, we want that profit !