Discussion summary
The discussion revolves around an essay on the economy of mathematics, with opinions on its political naivety and its comparison to other crafts. Participants mention the normalcy of academic conferences and the potential future of mathematics resembling a niche sport.
What the discussion says
- Some see the essay as naive politically.
- Many defend the normalcy of academic conferences.
- The analogy to woodworking and craftsmanship is debated.
- The essay is considered highly interesting by some.
- Concerns about the future of mathematics are expressed.
“It's like he's re-discovering alienation under capitalism.”
“Being invited to conference talks is normal for researchers.”
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- Hacker News
- Does the the (,1) conjecture paper in annnals of Math say 7 years between submission and acceptance? Insaneby j7ake
- The core thesis seems to be that the "real value" is not in producing/proving theorems, but in understanding them. AI might be good at producing and proving theorems, but it fails utterly at getting humans to understand them. Even worse, humans have no interest in working on theorems that have already been proven, so we end up with theorems that will never be understood by humans.
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
by whack - Very good! Thank you!!
Meta:
- Sad to see how well thought out + well written stuff like this only makes it to the HN front page by a fluke. (Below: https://news.ycombinator.com/item?id=48758048#48759391)
- @davidbessis: maybe look for an open alternative to Substack? (avoid bullshit, proprietary, gated access for quality content by humans).
by manuelz - Some previous submissions
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=48758048by octopus143 - Tangential but this article got me thinking.
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.
- Incredibly thoughtful. This essay gives that very rare sense of being well reasoned, gods at forest and trees, and sitting atop a shit ton of domain expertise.by mojosmojo
- People think mathematics is about proving theorems.
I think that's just an accident of history.
When we write software, we very seldom write proofs that our algorithms are correct. We just write tests, and we also run the algorithms and when they fail we know we have a bug and then we proceed to debug, fix, and add new tests (if we are disciplined, but most of us are). In time, by usage and testing, we gain confidence that our battle tested software is correct, mostly. And we tell people that we will never be 100% confident that any software is bug free. But that's a slight lie: if we wanted such confidence, we would start using provers, and create bug-free software. That possibility exists, but it's just extraordinarily expensive.
Well, in math that's the only possibility, and we use it. And it is indeed extraordinarily expensive, but it's also the cheapest among the alternatives. The alternatives are 2: be rigorous and do these proofs, or be sloppy and allow bugs to creep in, and allow an entire school of math to collapse like the Italian school of algebraic geometry [1].
There is one more alternative. If a particular math theorem has some applicability, then you write a program and use it in real life. In time you eliminate the bugs as much as you can, and you get to the steady state of "virtually bug free". At that point you don't have a solid proof that the theorem is correct, but in general you don't really care. Because you feel that a formal proof is just a thing one would pursue for getting academic satisfaction only.
[1] https://en.wikipedia.org/wiki/Italian_school_of_algebraic_ge...
by credit_guy - Greg Egan's description of how mathematics evolves into "truth mining" in his novel Diaspora is seeming more and more prescient. It essentially describes what mathematics would look like after formalization records all theorems discovered so far in a huge, collective database and proof assistants can instantly work out the details of a given proof. What remains of mathematics? According to Egan, visualization, intuition, and insight.
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.
by olooney