Posted by zvr 10 hours ago
"The product of mathematics is clarity and understanding. Not theorems, by themselves. Is there, for example any real reason that even such famous results as Fermat's Last Theorem, or the Poincaré conjecture, really matter? Their real importance is not in their specific statements, but their role in challenging our understanding, presenting challenges that led to mathematical developments that increased our understanding."
I suggest if one looks at the history of funding for mathematics and science, the product of these efforts is not understanding, but rather power. Funding went way up after WW2 when the war demonstrated that power flows from them. Math not only contributed to the scientific weapons of the way, but was directly used in operation planning (the birth of the field of Operations Research) as well as in cryptography.
The reason this matters is that AI is also a quintessential power-oriented technology. From the point of those providing the monetary lifeblood on which modern mathematical practice depends, the current math-AI discussion presents no issue worthy of concern.
For many who pursue mathematics, it is a refuge from ordinary life (or normal "power"). Within mathematics, there has historically been a tension between the "Babylonian" model of pure calculation and the "Greek" model of advancement through understanding. If current AI models subsume mathematics entirely, the Babylonian model will have "won" and the possibility of an informed citizenry will be in doubt.
The foundations of the WW2 technologies you cite were dependent on previous theoretical efforts (ex:relativity) to develop a good understanding.
Without understanding, you get brittle demos which fail as the environment or problem description changes.
But for the most part, math discovery relied more on human curiosity than on resources to "do math". Conversely, if people allocate lots of money to developing AI, that doesn't mean mathematicians have an obligation to take the money provide ROI to investors.
Getting funding can be quite difficult at times, so you'll see some portion of researchers (or mathematicians in this case) take the dollars they can get.
Yes, and your examples are exactly examples of what the GP quote is talking about.
Of course people paying money want applications, which includes "power" in your kind of reductive framing (applications to war being only one of many types of applications, or we could redefine any gradient provided by expanded understanding as "power", in which case the choice of word just seems melodramatic).
What we've also learned over the centuries, a lot more clearly in the last few, is that seemingly pointless or applicationless understanding can very quickly become useful. This is why it's clearly worth still funding pure math.
Mathematics seems to be entering an era where human + machine maximizes performance, much like chess in the 1990s. However, imagine a future where even talented mathematicians are nothing but noise in the machine (as is the case in chess now). A future where AI generates and verifies proofs without humans in the loop. Where the mathematics may be beyond human comprehension.
In that future, does it matter that early career mathematicians are inhibited by these developments? Perhaps not. Programming faces the same issue. As AI crawls up the competence ladder, does it matter that fewer people have opportunities to develop the skillset of a senior engineer? Perhaps not.
There's also the separate, less glamorous issue that people don't want to talk about, which is proof reliability. [0] If you have systems to help you formalise the problems and leave an algorithm or AI or whatever solve it in a verifiable way, that's a win for both the mathematicians and the rest of the world.
The deeper question is whether AI can replace the human role in deciding what mathematics should be done and what concepts matter. If that's automated, then yeah, we're screwed.
[0] https://lamport.azurewebsites.net/tla/proof-statistics.html
Isn't chess more popular than ever? Ai dominating the game didnt seen to matter
I don't think there's any computer system which autonomously come up with new directions in openings. As far as I know; a GM looks at stockfish's evaluation of the top x moves and analyzes one that hasn't been played a lot etc.
That would be AGI. My conjecture is that LLMs alone are not enough for that future. They are incredible, but AGI needs other breakthroughs.
In that sense, I think math is very different from chess or Go. Chess and Go are complete-information games with fixed rules and a fixed board. Math is open-ended.
“Current automated techniques can produce plausible but unreliable (or even incorrect) arguments which are difficult to distinguish from correct mathematical proofs.”
That seems like a problem for mathematics with or without AI.
Isn’t this a problem with human proofs as well?
“Many current models are also built on data obtained by systematically exploiting licenses and access arrangements that were not made with artificial intelligence in mind, or indeed by simply violating copyright protections”
Copyright? The copyright arguments have been hard to make in domains where copyright is much stronger, mathematical knowledge isn’t even subject to copyright.
“Technologies which affect the way in which mathematics is practiced may disturb the current system of incentives”
Resistance to change again.
“Proper evaluation is endangered if results are communicated through informal channels”
Gatekeeping again.
> or even writing a thesis, is like climbing Mt. Everest. A lot of the value is actually in the effort you put into it.
As an analogy, in the music industry, if you need a jingle written, you wouldn't care if someone spent five minutes or five years writing it. AI is now filling that formulaic space very well. It won't replace the top end of humans output but it completely outdoes all the boilerplate stuff humans take an age creating
Human proofs are themselves a kind of a proof of work. They certainly write flawed proofs, but you can expect a human author of a paper to have put in more effort--substantially more--than the human reader needs to verify it. Arguably, this asymmetry disappears for generated proofs.
Automated theorem provers help a bit here, but they don't eliminate the human verification cost.
> Can't all proofs be eventually broken down into their fundamental pieces and then it's clear as day if it's right or wrong?
The proof that there are 26 of these sporadic simple groups is part of the theorem known as the classification of finite simple groups, sometimes known as the “Enormous Theorem”.[1] It took over 100 mathematicians nearly 50 years and resulted in hundreds of papers. Even with that many mathematicians involved, there were still errors and revisions needed to the original proof. Some of the original authors are gradually publishing a somewhat simplified version of the proof but it’s still a massive effort.
[1] https://en.wikipedia.org/wiki/Classification_of_finite_simpl...
In combining the parts you have the correct answer to a question, but is it that question you want to know?
Consider a proof that in the future all people will be happy.
You can methodically show this to be true but at the same time inadvertently include a proof that the number of people in the future will be zero.
It doesn't make the claim wrong, it stays undoubtedly true. It's just not what you assume it means.
Even when the proof is produced by the llm in a formal system like Lean4 it may not be “honest”[2] and it can be hard to tell if the proof is very long and complex and especially if it includes highly specialized results from lots of different areas of maths. Llms can (and do) do this just fine, but for a human proof that would require a team each of which was specialized in a particular area. Those people are more likely to be able to cross-check each other.
[1] https://pubs.ams.org/ebooks/conm/098/ and https://en.wikipedia.org/wiki/Four_color_theorem
[2] An “honest” proof may contain bugs or errors but it does not constitute a deliberate attack on the proof system or the math libraries it uses. Systems like Lean aim to not incorrectly validate an honest proof with mistakes but don’t guarantee anything in the case of a proof being dishonest. This is the sense used here https://lean-lang.org/doc/reference/latest/ValidatingProofs/ .
Notably you don't seem to be looking at either the list of identified values or their recommendations to researchers in their use of LLMs, which would seem much more important to engage with in any non-shallow dismissal of the document as "feel[ing] like gate keeping and resistance to change".
It's also kind of a bad look (and actively harmful for discourse) for people working on AI to be so dismissive of fields actively engaging with how their field is changing due to AI. I haven't seen any other field engaging this actively with its possible futures, have you? Usually we seem to only get some extreme of over-hyped utopia, doomerism, or dismissal of everything as slop.
The Bourbaki group was (is) about documenting existing mathematics rigorously, not doing original research. This document is specifically about the future of mathematical research.
At least for me, in many cases I have achieved a much better understanding of various things after I studied the historical development of the ideas related to them.
Therefore I agree with the point "2." at "Potential Threats". For me a novel mathematical demonstration that is not presented in a way which disentangles its really new elements from the previously known elements, by proper quotation of all relevant older sources, has a value that is many times lower than that of a demonstration with proper attributions.
> This has been the result of months of community input about the fundamental values and goals of the mathematical community. In retrospect, these were questions we should have been systematically discussing years ago, but in any event the exercise was extremely valuable, and the end result is excellent. I wholeheartedly endorse the statements and recommendations in this declaration.
>I support this declaration. I have one small comment: the document notes that "Technologies which affect the way in which mathematics is practiced may disturb the current system of incentives." The current system of incentives seriously is flawed in many ways, and I don't think maintaining the status quo should be our goal. However, we should work to improve it, not let it be corrupted by outside forces, as has already been done for decades by university administrators, journal oligopolies, etc.
> 2. Don’t believe the hype
> 3. Regulate the artificial intelligence industry"Now, here, you see, it takes all the running you can do, to keep in the same place."
There is no moral or ethical obligation to disclose tool use. The disclosure in of itself presents an asymmetric disadvantage to the disclosee. Especially in this charged environment where large swathes of people are champing at the bit to discredit or diminish any effort that leverages these tools.
This system incentivizes people to hide tool use to gain a competitive advantage.
This moralistic grandstanding will be seen as a reactionary movement of people trying to cope with transformative technology.
Lie about tool use, don't admit it. Use it as you see fit and rely on your taste, expertise and best judgement.