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Why Erdős Problems Are Falling to AI

quantamagazine.org

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Re: Why Erdős Problems Are Falling to AI

#131

Post by stouset »

HarHarVeryFunny wrote: post #101
lacunary wrote: post #84

Earlier quoted context omitted.

> are using known math to solve them rather than inventing anything new. Isn't a new proof new math? If not, what qualifies as new math?

You could regard existing mathematical techniques as a set of Lego pieces. If you build a proof just using those pre-existing Lego pieces then it may be a new proof (a new Lego model), but it's not a new Lego piece - a new piece of mathematical machinery.

To be fair, very few proofs involve the creation of new machinery wholesale.

Re: Why Erdős Problems Are Falling to AI

#132

Post by jiggawatts »

neutrinobro wrote: post #55
lstodd wrote: post #48

Earlier quoted context omitted.

I would expect anyone calling themselves an engineer of any field to understand basics of fission power generation. Come on, it's 7th (school) grade material.

Basics yes, details no, and if you are deep in a mathematical proof the details matter. For example, I seriously doubt you will find an average 7th grader (or even a professional engineer outside of the nuclear power field) able to give you a good explanation or even definition of the void coefficient, and how it may interact with the fuel temperature coefficient of reactivity for a particular reactor design. Do you?

I had a similar conversation with a global warming sceptic friend.

It turns out that the "more CO2 = more warmer" model is hilariously simplified, and the real modelling gets into the weeds to put it mildly! There's a NASA database of the super-high-resolution absorption and emission spectra of every isotopic combination of every common molecule and ions, excited states, and more! It turns out that most of the forcing is determined by the behaviour of the upper atmosphere at high latitudes where the air is so thin that exotic excited states can persist for appreciable durations, and are made in large amounts by absorption of UV light. The "glancing angle" of the sunlight near the poles also means that even minor constituents participate in the exchange of IR radiation. Then, then, the simulation has to be run in many thin slices because air is so opaque to IR radiation that it bounces many times on the way up and down, and of course, the isotopic mixes (and excited fractions) are inconsistent between layers.

An insanely complex supercomputer model is required to come up with even a rough estimate of the actual warming.

Re: Why Erdős Problems Are Falling to AI

#133

Post by fragmede »

neutrinobro wrote: post #55
lstodd wrote: post #48

Earlier quoted context omitted.

I would expect anyone calling themselves an engineer of any field to understand basics of fission power generation. Come on, it's 7th (school) grade material.

Basics yes, details no, and if you are deep in a mathematical proof the details matter. For example, I seriously doubt you will find an average 7th grader (or even a professional engineer outside of the nuclear power field) able to give you a good explanation or even definition of the void coefficient, and how it may interact with the fuel temperature coefficient of reactivity for a particular reactor design. Do you?

I am sure there's at least one seventh grader who knows about void coefficients for a particular reactor design because of one very particular RBMK reactor. Average, no, but the existence of a void coefficient has been popularized by a television series.

Re: Why Erdős Problems Are Falling to AI

#134

Post by guenthert »

jiggawatts wrote: post #132
neutrinobro wrote: post #55

Earlier quoted context omitted.

Basics yes, details no, and if you are deep in a mathematical proof the details matter. For example, I seriously doubt you will find an average 7th grader (or even a professional engineer outside of the nuclear power field) able to give you a good explanation or even definition of the void coefficient, and how it may interact with the fuel temperature coefficient of reactivity for a particular reactor design. Do you?

I had a similar conversation with a global warming sceptic friend. It turns out that the "more CO2 = more warmer" model is hilariously simplified, and the real modelling gets into the weeds to put it mildly! There's a NASA database of the super-high-resolution absorption and emission spectra of every isotopic combination of every common molecule and ions, excited states, and more! It turns out that most of the forcin…

And it's a huge distraction. Afaik (and I'm no climate researcher), the increased albedo of greater cloud coverage due to greater vaporization because of higher temperatures is the only known negative feedback effect. Everything else, be it reduced albedo due to less ice cover, increased release of Methane due to melting permafrost, increased release of CO2 due to more forest fires, etc. only accelerates global warming. So "more CO2 = more warmer" might be hilariously simplified, but it ain't wrong.

Only those who try to maximize profits while skirting the risk of a revolt care whether global average temperature will be 1.2 or 1.8K above pre-industrial average in ten years. For the rest it's already too warm, the damage is already plainly visible.

We don't need better models to predict future warming; we dragged our feet long enough that we can now look at historic data to see where it's going.

Re: Why Erdős Problems Are Falling to AI

#135

Post by chrisjj »

svachalek wrote: post #40
serial_dev wrote: post #10

Earlier quoted context omitted.

I think it's the scientist version of "I vibecoded ten apps this weekend (at one point I'll have real users too)" . Because it's math, it's all mysterious and genuinely impressive, but in the end, if no human cares about it (apart from attention grabbing "it's so over" tweets and articles), does it really matter?

Nah, people are vibe coding lots of apps that no one cares about, but this is more like going through Stack Overflow and answering the most upvoted issues that don't have answers. These are published problems that have prior interest, for decades usually, and a few snarky internet comments don't undo that.

> this is more like going through Stack Overflow and answering the most upvoted issues that don't have answers.

Except OpenAI isn't showing any upvotes.

Re: Why Erdős Problems Are Falling to AI

#136

Post by scotty79 »

throwatdem12311 wrote: post #6

When OpenAI posted about their 10 breakthroughs, I saw lots of career research mathematicians say things mostly along the lines of “I don’t understand any of this it’s way over my head”. Are we missing the forest for the trees here? If a math problem falls in the forest but nobody is around to understand it does it make a sound? How can we possibly make use of these breakthroughs if we don’t understand them? How coul…

> How can we possibly make use of these breakthroughs if we don’t understand them? How could we ever make anything useful with them?

Isn't that what people from more practical sciences said about math anyways?

All math is eventually applied math.

Re: Why Erdős Problems Are Falling to AI

#137

Post by RugnirViking »

delhanty wrote: post #58

It seems that AIs are really good at finding counterexamples now. Even if progress by AIs in proving conjectures lags, it seems likely that AIs collectively will, in the next few years, find counterexamples to nearly all the Erdős (and other) conjectures that are actually false and also provably false. That means we will able to assume that nearly all the remaining conjectures are either true or undecidable. Surely,…

> That means we will able to assume that nearly all the remaining conjectures are either true or undecidable

I mean, we could. But it doesn't really make sense to think that AIs are perfect at finding counterexamples, just because they are good (or even better than us). My general opinion is that they are orthagonally intelligent, that is, they are intelligent in an entirely different way to the way that people are. They are undoubtably clever, but the distance between when they are better than us at their best skill (or even most skills) and when they are better than us in all aspects is going to be MASSIVE.

Re: Why Erdős Problems Are Falling to AI

#138

Post by anzuhoeren »

delhanty wrote: post #58

It seems that AIs are really good at finding counterexamples now. Even if progress by AIs in proving conjectures lags, it seems likely that AIs collectively will, in the next few years, find counterexamples to nearly all the Erdős (and other) conjectures that are actually false and also provably false. That means we will able to assume that nearly all the remaining conjectures are either true or undecidable. Surely,…

Counterexamples can be decidable and arbitrarily ugly or difficult to find. If you know anything about mathematics it's trivially simple things can yield extremely complex structures. And that can absolutely include terse conjectures whose solutions are in fact decidable but only with proofs that would consume more than a bit of memory for every particle in the universe. Not logically undecidable, but physically impossible to prove in our physical universe.

Further down the scale are ones that are decidable only by machines that we would never have the wherewithal to construct, even though they could physically be constructed with the material we have to work with.

Re: Why Erdős Problems Are Falling to AI

#139

Post by oliculipolicula »

eru wrote: post #116
oliculipolicula wrote: post #103

Earlier quoted context omitted.

No. Mathematicians who considered themselves experts on exactly those conjectures were bewildered. > And then -- the kicker -- something that I personally spent a couple years on in grad school, leading to some of my proudest work: quantum parallel repetition theorems ... > Is there some broader context or theory within which this would've been the obvious thing to do? What other results can be proven using these tec…

> we can however be optimistic, oddly. If you believe Feynman when he says that you can't explain to a child something you don't understand.. then we can see that ChatGPT has no idea of just what it has done! Why would that be optimistic? Seems like the pessimistic reading to me, if anything. Or are you having a bout of Schadenfreude?

It means that humans can, for now, still do something that chatgpt can't. Distill [proofs. Even if some may hesitate call that understanding.]

Compare the various summaries that humans have written on the Jacobian counterexample, to the chat logs that were archived.

Re: Why Erdős Problems Are Falling to AI

#140

Post by eru »

oliculipolicula wrote: post #139
eru wrote: post #116

Earlier quoted context omitted.

> we can however be optimistic, oddly. If you believe Feynman when he says that you can't explain to a child something you don't understand.. then we can see that ChatGPT has no idea of just what it has done! Why would that be optimistic? Seems like the pessimistic reading to me, if anything. Or are you having a bout of Schadenfreude?

It means that humans can, for now, still do something that chatgpt can't. Distill [proofs. Even if some may hesitate call that understanding.] Compare the various summaries that humans have written on the Jacobian counterexample, to the chat logs that were archived.

Have you tried asking ChatGPT / Claude etc for a good summary?
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