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- Hacker News
- Wolfram gives a very important and sober take on the present state and future of pure math. I came away with the following key takeaways:
1. An essential goal of mathematics is human understanding. The computation of proof terms doesn't necessarily enrich human understanding. The proof of the four color theorem result is a good example, and formal verification/SAT solving gives many more: these are results that can be trusted up to our trust in the system used to produce them, and they can be used in practice, but they don't necessarily enrich our understanding. Imagine a computer with near infinite proof search powers set loose with the current human definitions, theorems, and understanding of mathematics. Suppose it constructs a proof for a new theorem at our mathematical frontier. The shortest such proof in terms of currently understood definitions and concepts could be so long and mechanical that the entire lineage of humans until the end of the universe could not finish reading it. So though it overlaps with the activity of mathematicians, this type of computational proof search is not mathematics as such. This is an important distinction that many people do not seem to grasp and some dismiss as cope.
2. The human activity of theory building, rendering otherwise monstrous proofs like the one I discussed above into light conceptual arguments a person can understand, appears at this time out of reach of models. Maybe they will do this in the future, but it is not yet the case. Human theory building drastically compresses the spaces of theorems and their proofs: this is why great theory builders like Groethendieck are so important to the field; grinding has its value too, but runs up against computational limits in both humans and computers. These limits are collapsed by the conceptual shortcuts created by theory builders.
Gowers has a nice and arguably better-grounded article on the mathematical capabilities of recent LLMs that I think is enlightening to read alongside Wolfram's bird's eye view of the implications of those capabilities: https://gowers.wordpress.com/2026/08/12/what-sort-of-maths-a...
by jaykru - What is a good exit strategy?by amelius
- Read by the author https://www.youtube.com/live/gPrWX8i1htM (with multiple mentions of the Wolfram language and such)by suopspaces
- Some arguments are based on "past experience..."
However, such experiences are not absolute truths and cannot be equated with the current situation.
by veexx103 - The reason we (as in university-funding society) need mathematicians is so that there's a steward over math that can help me, a complete layman, understand and apply mathematics. "Being a steward" will sometimes not look like much. Sometimes it's just going to be dicking around with LLM's.
But math is too important for it to be abandoned by humans and left to models. I, the tax-paying bonehead engineer, need mathematicians to exist. They can have their ego beaten up by Claude, sure, whatever, but they're not allowed to just close up shop in despair.
- I see two assumptions in the core argument, both of which are open to challenge.
1. AI can do proofs, but deciding which problems to solve, which math is useful, is by humans.
2. The math that's picked needs to be understandable by humans.
For #1, AI may be able to play a significant, if not a takeover, role for even figuring out what math is useful.
For #2, understandability by humans may be good for now, but could also turn out to be a significant constraint. Correctness is a goal, trust is an important requirement, human understandability may be an intermediary for that, but not necessarily the end goal.
In other words, the article may stand the current state of the art, but may not stand merely a couple years down the road.
by alok-g - Much more rejections due to more submissions, and AI will probably not help you (unless you solve a major open problem).
Even frontier models still struggle at proving small conjecturers despite all the hype about major breakthroughs. It really depends a lot on the prompt, the type of problem, among other factors. But AI does not suddenly make publishing in a journal easier, although it does make it easier to produce papers.
by paulpauper - > I’ve had the experience quite a few times now: I try to autoformalize something, and an AI will tell me “I did it; look, the proof checks out!” But, actually, in some sense it cheated: instead of formalizing what I intended, it found a (sometimes very squirrely) way to interpret what I asked so that it could successfully prove it. It’s often very hard to tell, though, that this is what happened—not least because the formalized versions of things (say as expressed in popular proof assistant systems) tend to be very low level, very verbose and very hard for us humans to understand.
This has largely been my experience in programming, too. Often when given a broad objective within an existing code base, even the frontier models seem to stand up a half dozen tests proving compliance and then add 3 new branches solving for those specific tests alone. My best guess is that our code base is quite far out of the distribution (and not for just good reasons). The agents are thus reduced to these tactics rather than extending and refactoring well known abstractions.
by piker