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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

#61

Post by ben_w »

lstodd wrote: post #48
eru wrote: post #27

Earlier quoted context omitted.

> 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”. Math is an incredibly broad field. I mean, you don't expect a traffic engineer to understand anything about nuclear reactors, do you? Yet, they are all 'career engineers'.

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.

By that standard, I could've been an astronaut at that age.

Newton's laws of motions are not hard. Making a rocket that doesn't kill the occupant, is.

Re: Why Erdős Problems Are Falling to AI

#62

Post by glitchc »

tumdum_ wrote: post #9
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…

Latest presentation of Terence Tao on what current advancements in AI mean for math discusses (among other things) those issues: https://teorth.github.io/tao-web/slides/age-of-ai-icm-2026.p...

Thanks for sharing. Regrettably his conclusion on proof exposition increasing in importance reflects the broad trend in every field: When someone (or something) else is doing the actual work, all that's left for the original folks is to find a way to sell it. Pretty soon mathematics will be awash in marketing with slogans such as "I am a famous mathematician, I checked this proof and it looks legit. Trust me." Absolutely nothing wrong with that scenario right? Except for the logically fallacious appeal to authority and the inevitable corruption over time.

Re: Why Erdős Problems Are Falling to AI

#64

Post by glitchc »

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…

My perception is different. I start from the axiom that AI is a massive compressed corpus of knowledge. That it can find solutions suggests to me that the solutions were already known, but simply lacked publicity. This is less about discovery and more about pattern-matching.

Re: Why Erdős Problems Are Falling to AI

#65

Post by arjie »

ComputerPerson wrote: post #2

The mathematics are above my intellectual capacities, but I still find the man and his lifestyle fascinating. Though I admit his heart attack probably wasn't a coincidence, sadly. I wonder if national, institutional, or otherwise "eccentric" sponsorships (encouraging a similar migrant-madman approach to academic cultivation) of some of the folks on HN wouldn't lead to meaningful discoveries in CS. I often see comment…

When Gwern needed a fellowship to come to SF he got one. When he had his “great idea” it was funded. The truly unique people out there find a way. Any government bureaucrat will just find the Jason Ardays of the world.

Re: Why Erdős Problems Are Falling to AI

#66

Post by ben_w »

lstodd wrote: post #59
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 can explain what the void coefficient is, why it is a dangerous simplification, why neutron moderation is needed, how it is typically controlled, why BWR reactors are inherently less stable than PWR. etc. it's simple really. All the while I have never had anything to do with nuclear power professionally or studied specifically it. I might have overstated a bit, but by 9th grade (15 year old) this is what was taught…

> why neutron moderation is needed

What you learned, was it more like:

1. "You need to slow down neutrons so they can react"

or 2. "Here's the graphs of how the neutron absorption and scattering cross sections vary with neutron temperature for H-1, H-2, H-3, Be-9, C-12, O-16, Fe-54, Fe-56, Fe-57, U-233, U-235, U-238, Pu-239, …"

If it was the former, you didn't learn "nuclear engineering".

Re: Why Erdős Problems Are Falling to AI

#67

Post by mlinhares »

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?

The number of geniuses in every field that congregate in HN is incredible, too bad they're here responding to comments instead of revolutionizing nuclear energy, math or medicine.

Re: Why Erdős Problems Are Falling to AI

#68

Post by sumanthvepa »

lstodd wrote: post #59
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 can explain what the void coefficient is, why it is a dangerous simplification, why neutron moderation is needed, how it is typically controlled, why BWR reactors are inherently less stable than PWR. etc. it's simple really. All the while I have never had anything to do with nuclear power professionally or studied specifically it. I might have overstated a bit, but by 9th grade (15 year old) this is what was taught…

I'm pretty confident that no 15 year old would have learned about void coefficients if Chernobyl hadn't happened.

Re: Why Erdős Problems Are Falling to AI

#69

Post by somenameforme »

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…

I find some interesting parallels in chess, which often has lots of analogs with math to begin with. But chess went from a pure human endeavor to one where supercomputers aided by world class players finally managed to eek out a slightly suspicious win against a world champion (approximately where we are now in math) and to now a days - where your phone could easily crush the world's strongest player, who is also probably the strongest player of all time.

The way the chess world adapted this was initially to try to understand the machine. After all chess, like math, is complete information - so you can easily see the computers 'thoughts' in terms of the exact moves its saying are best in a variation and how it might respond to any other idea. But it quickly became clear that this wasn't working so well.

Players would regularly get positions that the computer says 'and black wins' and then proceed to lose it convincingly, simply because the positions were so extremely weird and difficult to play that even if it might be technically winning, it's the sort of position where you're walking a fine line with lots of complex moves to find. Humans aren't computers and even the best of us can't play like one in weird positions.

Now a days they're taken more in balance. The computer's evaluation of a position is probably about as good as you can get, but playability matters much more in practical terms. Knowing the eval of a position doesn't really matter if you don't understand the position. Knowing the answer can help with understanding (for instance computers have radically reshaped and improved human understanding of space in chess as we noticed computers obsessing over it) but I think the days of 'oh the computer says it's winning, so I should be able to take it from here' are near to gone.

Re: Why Erdős Problems Are Falling to AI

#70

Post by kaashif »

glitchc wrote: post #64
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…

My perception is different. I start from the axiom that AI is a massive compressed corpus of knowledge. That it can find solutions suggests to me that the solutions were already known, but simply lacked publicity. This is less about discovery and more about pattern-matching.

Why does being a compressed corpus of knowledge mean it can't discover things? What if it has knowledge of techniques to discover proofs?

It does and that's what's happening.

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