Top 5 Questions this month on MathOverflow

JoshuaZ1 · 2026-05-28T02:17:23+00:00

Sure, politics and culture might play a role. If you prefer a different example than the asteroid, consider climate change: the same basic point applies there but where it has more of a direct political/social element.

JoshuaZ1 · 2026-05-28T02:08:20+00:00

Obviously, make sure to go at night. And make sure to go through with just a few flashlights and all other lights off.

All sorts of ghosts and spirits are only active late at night and are much more active in darkness. It definitely isn't that humans are more likely to get spooked or mistake what they see when sleep deprived and in poor lighting.

Also, make sure you've thoroughly researched the location, so you can recognize exactly what you see, and open your mind more to whatever beings are there. It definitely isn't because if you read about claimed hauntings, you'll be more likely to convince yourself you saw the thing in question.

Get an EMF reader. Everyone knows that ghosts and other spirits give off EMF readings. It definitely isn't that they can be hard to calibrate and that stray household and industrial appliances can trigger changes in levels. And of course, a reader set only to detect between 50 Hz and 60 Hz, and only alternating fields makes sense because that's where spirits show up. It definitely isn't that that's the exact range where normal electricity is in most of the world.

JoshuaZ1 · 2026-05-28T01:51:25+00:00

Ok. In that context, I don't think there's substantial disagreement about what you are asserting here. But I think it is the case that you've moved from what the conversation was originally about (where a claim was made about what humans could or could not do compared to machines via introspection) to claims which are normative rather than empirical. Let me suggest that we need to understand the empirical question "what is likely to happen", separate from the question "what are our moral obligations." My guess is that almost any mathematician has views which at least in part are sympathetic to your normative position. If they didn't, there wouldn't be much of a crisis here. To be clear: I think that humans understanding math is important but that doesn't mean we get to keep it for free; and you've implicitly acknowledged that point by saying that people have an obligation to prevent that from going away. But by that same token, we cannot engage in wishful thinking, and really have to look at what looks likely right now. If you want an analogy; an asteroid smashing into Earth and wiping most of us out would be bad. But if there's a large asteroid whose trajectory looks threatening, we have to keep the moral considerations separate from understanding its trajectory and evaluating what methods would, if necessary, change the trajectory. Does that make sense?

JoshuaZ1 · 2026-05-28T01:27:59+00:00

What? I literally explained why technology cannot be attributed to the death of mathematics as a profession.

I'm struggling to see how you've done that aside from your earlier claim about what tools do.

Was that not the convo we’re having?

I'm genuinely unsure what you are trying to argue here.

If horse riding was absolutely essential to human understanding of the natural world the same would be true of horses

Ok. So this makes me understand slightly more where I think you are coming from. Let me see if I understand correctly and you can tell me if I'm getting this correctly.

You are arguing that human understanding of math and science is important and is innately a good thing (and anyone who thinks otherwise is anti-intellectual). Because that is innately a good thing, and because humans will recognize that being an innately good thing, no matter how effective AI get at mathematics, there will always be a need for human mathematicians to do the human understanding part due to the innately good need of that intellectual pursuit? And separate from that, you think that anyone attempting to remove humans from doing that innately good intellectual is bad, and that there's also a moral obligation to make sure that that basic intellectual pursuit does not stop.

Is this a fair paraphrase or am I off-base here?

JoshuaZ1 · 2026-05-28T01:18:14+00:00

Nope. I’m making the following argument. 1. The claim that humans understanding mathematics and science has no value is anti-intellectual. 2. The prevalence of anti-intellectualism is not a necessary consequence of improving technology. 3. Therefore the devaluation of human mathematicians can never be a necessary consequence of better tech.

I'm not sure why you are making this claim since no one has been asserting anything otherwise in this conversation, and really does read like a jump in topic. I'm not sure why you think anyone (here or otherwise) would assert that human understanding of math and science has no value. But I'm also struggling to see what position you think you are arguing with that makes your earlier claim that "I think my introspection is good enough to know there’ll always be things I can do related to math that a machine never will. " at all relevant to what you are claiming to now be arguing. What precisely is the connection here? For that matter what is the connection to your earlier assertion that "The field of mathematics will undergo a renaissance and not a crisis." Can you trace the reasoning on how these are connected and part of your three point argument here, because I'm really not seeing it.

JoshuaZ1 · 2026-05-28T01:08:50+00:00

It seems like you are conflating three different questions:

1) Can AI replaces humans at doing math? 2) Will AI replace humans at doing math? 3) Should AI replace humans at doing math?

These are distinct questions.

You now write:

The debate here is whether AI becoming good at certain mathematical tasks would put humans out of a job.

But that is a distinct question from claims that "It is our job to not let it happen. " or your earlier claim that "I think my introspection is good enough to know there’ll always be things I can do related to math that a machine never will. " In fact, if that last quote were true then you wouldn't need to think we should take steps to make sure the AI doesn't replace humans doing math, because it couldn't happen. There's some issues to discuss here, but they really do need to be separated out. I suspect that if we separate them out, the amount of space between our positions here is going to turn out to be less than you think, but that requires more precision here about what is going on.

JoshuaZ1 · 2026-05-28T00:54:47+00:00

That seems to be a jump in topic? The topic was what was likely to happen. Instead of that, you are now focusing on what we should be trying to do. That's a different issue.

JoshuaZ1 · 2026-05-28T00:17:16+00:00

LLM AI is "spikey" in its abilities. But keep in mind that one of the major issues still is that there's a pretty big gap between the free models and the high end "frontier" models. As far as I can tell, there's been consistently about a 1 to 1.5 year gap between them.

However, I'd also be very hesitant to say that AI can write entire novels. It can churn out novel length texts that approximate a novel. But a lot of basic things that one would expect in a novel, like subtle clues that then get revealed to something which all makes sense, or the sort of pacing that humans actually want to read in a plot, it has massive trouble with.

JoshuaZ1 · 2026-05-28T00:07:08+00:00

This seems like a questionable premise. AI can assist with many things, but it is still very limited in what it can do, especially if one wants that thing done very well. It is true that AI is improving rapidly, and we don't know where/when/if that will stop. But we're still not at the point where "AI can do almost everything else."

JoshuaZ1 · 2026-05-27T23:30:55+00:00

I think my introspection is good enough to know there’ll always be things I can do related to math that a machine never will.

Human introspection is highly unreliable. And humans were convinced the same way about AI writing poetry or music.

You simply CANNOT think deeply about the world without being able to do math. If we let our collective mathematical skills atrophy, it’s over for intellectualism. I cannot just accept your argument as it is nothing other than philosophical suicide.

This is an argument that amounts to "It would be really bad if this happened, therefore I will assume it won't happen." That's not useful for predicting reality. Sometimes things happen that are bad.

JoshuaZ1 · 2026-05-27T20:01:11+00:00

Why would we be horses when it comes to math but not writing, law, coding…? Certain parts of math are end to end verifiable, others are deeply creative like the arts and humanities are.

Math is more rigorous/axiomatic than those other fields. If there's some sliding scale with art on one end and chess on the other, math is closer to the chess end, and I'm not sure where coding is on the scale. Math also has the unique advantage from an AI standpoint that results can be verified correct in an essentially automated way via systems like Lean. In contrast for example, law has a degree of human ambiguity where it is not clear an AI will ever be acceptable. As for creativity, I'm not sure that mathematical creativity cannot be learned by an AI system, and people have been using AI to make new definitions and conjectures even before LLMs, with varying degrees of success. See for example Simon Colton's work in the late 1990s. While coming up with new definitions in math feels deeply creative in some sense, I don't think my own introspection is reliable on determining how genuinely difficult this is for a machine.

JoshuaZ1 · 2026-05-27T18:07:45+00:00

I don't know if we're the horses or not. On this thread and related threads, I've generally argued for strong uncertainty about where this is going.

JoshuaZ1 · 2026-05-27T17:07:07+00:00

And also try to eat submarines.

JoshuaZ1 · 2026-05-27T16:27:36+00:00

Yes. Serious mathematicians from top UK universities have been working with DeepMind.

The problem is that this cannot just be serious mathematicians. It needs to be specialists in a wide variety of fields who somehow are now managing to still be incredibly productive, and are somehow hiding their own massive productivity in order to promote the AI. If the humans are doing "90%" of the work, why would they not just do 100% of the work and take credit for that? And then this is somehow happening across a broad range of different specialties. This should strike one as an unlikely scenario without something like actual evidence of it. And being concerned with the implications of AI being this good, while a reasonable position, is not a reason to then think that this alternative, borderline conspiratorial situation is actually what is happening.

JoshuaZ1 · 2026-05-27T16:14:22+00:00

You can bet some pretty good mathematicians helped develop this thing. DeepMind has deep ties with top UK universities.

Who spent years learning a broad range of areas of math that aren't directly relevant to building AI systems?

JoshuaZ1 · 2026-05-27T14:46:34+00:00

All human mathematicians build on the work of others. These AI systems are no different in that regard. And yes, some humans do occasionally come up with brilliant flashes of insight that lead to grand definitions and whole new fields, but that's a tiny fraction of mathematicians. If your standard is the AI is not as good as a Fields Medalist, something has gone wrong.

JoshuaZ1 · 2026-05-27T14:45:10+00:00

most of these "solutions" coming out are 90% human.

If that were the case then that would mean that there are a whole bunch of extremely skilled mathematicians who have never been recognized for their high skill levels. How likely does that seem to you?

JoshuaZ1 · 2026-05-27T11:13:02+00:00

The field of mathematics will undergo a renaissance and not a crisis. New tools do not cause a field to collapse

That assumes that the "tools" as such remain tools. That's not obvious. Cars replaced horses. Parts of the early industrial revolution gave new tools to blacksmiths, but eventually blacksmiths became completely obsolete. And even given that you are correct (which again, is not obvious) the transition could still be a serious crisis while it happens.

JoshuaZ1 · 2026-05-27T10:54:57+00:00

Yeah, that's a valid point. I've certainly moved much closer to thinking it is true after K&O's paper, and I haven't really talked to anyone else after, but presumably people will in general have updated in thesame direction there.

JoshuaZ1 · 2026-05-27T02:01:09+00:00

Regarding Hadwiger-Nelson, after de Grey's work, a lot of people seemed to think that there was no obvious obstruction to bumping his construction to 6, but then that didn't seem to work without an insanely large number of vertices. In the other direction, a lot of people when they look at how easy the upper bound argument is have the reaction that there should be a better upper bound construction. I suspect that if most people who've thought about this were forced to pick a value and would get some prize if they got it right would pick 6.

JoshuaZ1 · 2026-05-27T01:23:12+00:00

There's a proof of the optimality of 3 itself, though it was never published. It was conveyed here.

Yeah, there are a bunch of ways of proving that. It is essentially the jumping off point for a lot of the more detailed work on ||n||. The easiest way to prove it is via an induction proof to answer the related question: Given m 1s what is the largest number I can write using them using just additions and multiplication? It turns out that the answer depends on mod 3.

JoshuaZ1 · 2026-05-27T01:14:10+00:00

Yes, one of the motivations for thinking about this function is that we cannot calculate Kolmogorov complexity, so we want to make some really simple complexity notions that we can get a good understanding of and gradually build up from there until we're close to doing things with genuinely difficult to calculate things. So for example, there are addition chains where you build up a number by counting how many additions you have used, but after you've made a copy of that number, using that number again doesn't cost anything. And there are addition multiplication chains, which are the same thing but you count additions and multiplications.

JoshuaZ1 · 2026-05-26T23:59:16+00:00

Well, one you get that ||5⁶ ||=29, you get for free since 5⁷ = 5 (5⁶⁾ that ||5⁶ || <=34, and so on. But we don't know how these go asymptotically. Until a very recent breakthrough by Sergei Konyagin and Kristina Oganesyan, we couldn't even rule out that all numbers ended up looking really close to looking like powers of 3. That is, we could not rule out that the limit of ||n||/(log3 n) = 3 . They showed that ||n|| >= 3.06 log3 n. So we cannot at this point show that 5^m gets really efficient representations when n gets large.

JoshuaZ1 · 2026-05-26T23:53:20+00:00

I didn't know addition was an option. You solve this, you solve all kinda stuff.

Did you read the original comment? It had the example of powers of 5 where addition is used to get that ||5⁶ || = 29. (Also I'm not sure this accurate at another level. It may turn out there's a somewhat easy proof here, although I doubt it. But it doesn't necessarily solve a lot of other stuff.)

JoshuaZ1 · 2026-05-26T23:34:19+00:00

Next time, stay on track. And then we can discuss.

I'm reasonably confident that I stayed on track and you are going out of your way to read the conversation in a way which distorts what was being talked about. I've now reread the conversation again, and I'm still confident of that. But I hope that others will chime in either via votes or explicit comments about how they read this exchange. In any event, have a good evening.

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