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- Hacker News
- I'm not a mathematician but AFAICT AI in mathematics only helped for paradigmatic works, either verification of well explained, and even researched, conjectures. Ideas that have been labelled as potentially interesting, are verifiable, and in a well known area. It is already quite useful... but it is NOT about paradigm change. It's not about revolutionary ideas. It's about being able to slightly more efficiently do more of of the same.
It is precisely NOT about big questions but rather potentially covering and thus cornering away all the "small" questions.
PS: already argued for something similar in recent Ergos related work, see comment history.
by utopiah - Isn’t that somewhat similar to everywhere else AI is being employed? More of the same — just faster? We aren’t getting novel software architecture out of AI, those things are still important, but it does help us rule out bad things (bugs, security vulnerabilities, etc) and help us focus on the important things. In that vein, AI could help mathematicians by ruling things out faster.by jadar
- In my field that sometimes involves some applied mathematics, it definitely help with busy work, giving raw ideas or writing some chain in latex so I don’t have to spell everything in details myself in the docby blks
- The big question is: why is government (or society) funding mathematics? What justifies this use of resources?
If it's utility, then why doesn't AI deliver utility? Isn't this an argument for AI?
One can expand this to various kinds of utility. Is knowing mathematical statements are true useful? In the ability to produce proofs useful? Does being able to prove carry over to other kinds of reasoning? Is producing a cadre of math-capable people useful in wartime? Is national prestige of value? There's a pro-AI reading of all these.
It it's beauty not utility, then why isn't math funding over with art funding, along with hills covered with acres of fabric, motorized embalmed animal carcasses, and religious symbols in bottles of urine?
by pfdietz - It's R&D. If you've listened to machine learning, it's heavily based around manifolds, statistics, linear algebra, learning theory, etc. All of which were once pure math.
The entire field of cryptography is built up on number theory, a field which famously a major contributor chose because it could never be useful [for war].
by lambdacity - It's amazing how much attention this issue has gotten. What is lost in the hype is no AI can tell you if a proof is correct. An AI can produce a convincing looking proof, but it can have a subtle but critical error or make an assumption that is unfounded. Thus, it ultimately comes down to humans. A mathematician has to craft the prompt, and mathematician to interpret/check the results. Also, these programs are very expensive and propitiatory. They are not like the commercial AI that regular people use. It takes considerable prompting and trial an error to solve even Olympiad/Putnam problems, and tons of work by humans pouring over the results to see if it's correct. For every Erdos problem that captures the headlines, there are many where it failed or untold hours of prompting and token burn to get that result, and manhours verify it.by paulpauper
- Please read the article. You've ignored proof checkers.by golly_ned
- But just imagine...
(edit: lol didn't realize the sibling comment below is essentially my comment)
by ares623 - AI can't yet come up with any new ideas to make the inductive leap to solve a math problem. New ideas are what get the accolades and using an old idea just means the original author missed something. We are still at the author missed something stage that AI is doing today.
It can definitely be a good research assistant though
- I don't think you understand the workflow. Terrence Tao has done a lot of work using them in conjunction with LEAN.
You aren't having the AI check the proof, you interactively work on the same LEAN proof, handing off between the AI assistant and having LEAN check it and provide feedback for both of you when there's a mistake.
by treyd - There's yet another major issue of the centralization of power and knowledge:
> Some worry about the accessibility of AI tools. Traditionally, mathematicians have required little more than intuition, training, and a pen and paper to advance their field. If this slow, deliberative process is no longer valued by society, and particularly by research funders, then mathematics could become an elitist activity, only practiced by select organizations that can afford to work with proprietary AI models.
This can be true of anything LLMs do better than existing options. Because the best LLMs require enormous resources to develop, access to them can be very limited. Right now they are priced for market share. What happens when your small law firm attorney, or self-representation, goes up against a large firm with LLM expertise? Can the kid from the poor high school compete with the kid from the rich school with premium LLM access, in mathematics for example?
by mmooss - always has been
the poor kid always had disadvantages, had to help the family, while the rich kid could focus on the math, and maybe get into a good math place with family help
by nok22kon - The article poses if AI will be a tool, a collaborator or an oracle. Why not all 3?
If mathematics is human understanding of logical consequences, understanding is the priority. But if AI proves something we can't understand but can utilize, that is a different sort of useful.
We are getting awfully close to "the answer of the universe is 42" and having it not be a joke...
by lubujackson - I don’t know about “close”, but there are certainly results in math that are considered deep because they require the use of a “Hard Theorem” at some point. That kind of building on top of something Very Difficult is still possible without understanding the “Very Difficult” part. I’d say a lot of not-amazing math is built by believing the platform works but not being able to built it yourself.
I couldn’t build an internal combustion engine or even a plastic box, so maybe there’s nothing wrong with this approach.
by fn-mote - Human mathematicians could become “priests to oracles.”
Priests were interpreting the oracles (at least at a place like Delphi) according to the context of the people asking the questions aka participating in politics of that ancient times.
Subjectivity was a feature and I’m not sure that fits to mathematics though.
I wonder if mathematics as a science field moves more into engineering or if a different branch will emerge that is closer to that because to the point of the article, science is about understanding not just results.
by cpard - Human mathematicians could become “priests to oracles.”
This is a decidedly anti-enlightenment statement.
by therobots927 - I think we’re going to find out the hard way that the proofs left to solve are very much not elegant.by rirze
- couldn't God have created a more orderly universe for us? this is ridiculousby stringfood
- Turns out you have to be Terence Tao to know when an LLM is right or wrongby glouwbug
- Yeah, so much for AI making mathematicians obsolete.by paulpauper
- is the similar statement true for coding as well?
i.e. You have to be a good engineer to understand the well generated LLM code and a program
by bijowo1676 - You don't have to be as good as the model you are overseeing, but it sure helps, otherwise you will only be able to evaluate partial claims, missing mistakes, and potentially the big picture.by esafak
- "I imagine my work could be completed with AI assistance in a matter of days—maybe hours."
Would some one with tokens to burn mind checking that statement out and post back. Be sure to use long dashes too.
by gerdesj - Beyond the technical aspects, this new technology also leads to fundamental social changes.
In particular, until now, mathematics were one of the rare sciences were great scientists could emerge from any country with a good education system.
With the raise of strong AI tools, only scientists in rich countries with access to those tools might be able to advance faster on the most difficult problems like the millennium problems.
Mathematics might become like experimental sciences were you need to build expensive machines to make further progress, such as nuclear fusion.
Actually, even now, the strongest models in mathematics are only available to a few engineers and a few mathematicians selected by Openai and Google.
by guyomes - > mathematics were one of the rare sciences were great scientists could emerge from any country with a good education system.
But would they be able to realize their potential without further (mostly monetary and institutional) support from their country? In that way, things now are not so different from before. However, the story will probably change in a few years' time.
Computational intelligence obviates the role of the human in making capital reproduce and multiply. Mathematics as its own field of study might become pointless other than as a hobby - an endeavour reserved for those entitled to the machine's output. What the machine develops in the place of mathematics may not be recognizable enough to even share the same label - that is, if it even uses mathematical principles rather than just finding correlations that are 'good enough' for its use.
by dag100 - The use of computers in mathematics has been somewhat controversial from the very start.
There are of course all the computer-assisted proofs (see 4 color theorem), as well as the partially-assisted ones (see Viazovska et al on packing problems in dimensions 8, 24). But even finding a solution numerically, then rigorously verifying its properties can leave a lingering sense of incompleteness, of a gap in understanding. I like this one quote by (allegedly) Wigner that illustrates it well:
"It is nice to know that the computer understands the problem, but I would like to understand the problem, too."
by fiforpg - To bluntly put it in a nutshell, and state the obvious:
If you don’t understand the problem you can’t be sure that the computer does.
by jackyinger - Reminds me of a quote from Tsoding
> “Programming is understanding. If you don't understand what you are doing, you are not programming. You are generating text.”
Perhaps a proof without understanding is just generating numbers.
by slopinthebag - Mathematics has always been an experimental science to some extent. While Newton, Euler and Gauss would spend a lot time calculating numerical approximations by hand, modern mathematicians have been doing the same using computers and software. And once an a clear picture emerge about what’s going, you can start to formalize that and attempt to prove and communicate your results in the standard definition, proposition, lemma, theorem scheme. (Btw there is even a journal called Experimental Mathematics devoted to this approach).
I don’t see that LLMs will fundamentally change this, but rather accelerate the speed of mathematical research.
Some computer generated proofs might of course be hard to understand, but at least their existence gives another data point work with.
Doing Mathematics is more than proving something, that’s just the end of a long road spent pondering at one’s desk about how things could work out.
- > but I would like to understand the problem, too
But why should it be the case that this is always possible?
It's entirely reasonable that the set of useful mathematical proofs is a proper superset of human intelligible useful proofs.
In fact, to argue the contrary would imply there is something incredibly remarkable about human cognition.