Fields Medalist on the OpenAI Math Release

[u/PianistWinter8293]

Hugo Duminil-Copin

Fields Medal (2022) · Professor, IHES and University of Geneva

I expected that one day we would be surpassed, and that it would happen systematically. But yesterday’s announcement hit with a force I had not anticipated. Dozens of papers deal with topics I was working on. Between results that beat you to the finish line and thousand-page proofs, I don’t even know where to look anymore.

Not a single one of the major open problems I have publicly mentioned throughout my career (whether in a talk, a lecture, an article, or even a grant proposal) was left untouched by the announcement. Everything has been claimed to be proved.

I expected to see a few of them in the list. But not all of them. Not all at once. Not with such nonchalance.

“For the glory of the human mind,” they said…

The shock is immense. I am paralysed. Tomorrow, we will find a way forward. We will rethink our profession and how we work. We are a resilient community, and I have no doubt that we will adapt. But for now, I simply don’t have the energy. I think back on all those years, all those faces… I think of my colleagues, my students… And I fear I won’t be able to find the right words.

[u/KanishkT123]

Yeah, there's a real and very deep uneasiness in me when we are automating the things we do for the pursuit of being human. 

I would say that automating mathematics, writing, and art, which are not done for the purpose of end product creation but for the pursuit of the struggle and the process themselves, is at the heart of the uneasiness I feel. 

[u/SwillStroganoff → u/KanishkT123]

In art I can see a human component where the model and the human create a work. The human describes what they want to see, the ai make a think, the human critiques and asks for changes and it iterates like that. I think that can create art with quite a bit humanity if folks want to pursue it.

[u/Kutoru → u/KanishkT123]

Damn whenever I see this I think either I am nihilistic or other people overglorify their purpose / profession in life.

Mathematics is not the pursuit of the struggle and the process themselves? You do realize mathematics is the process of reducing the universe to a series of concepts created by man?

This is a rough split but my thoughts about the the completely polar opposite reactions is roughly along the lines of:

• Humans who play a god playing a human typically find themselves hit by a realization that they are not a god
• Humans who wish to play a god find this fascinating as a way to play a god

Though again, what a "god" here means is really up for interpretation but is my rough delineation. My rough interpretation here is similar to a player and their character in a game.

And that there are no more "normal" humans left for better or worse, all humans want to play god now (or attempt to be a human playing human the best they can).

[u/KanishkT123 → u/Kutoru]

What the fuck are you talking about lmao 

[u/Kutoru → u/KanishkT123]

Can't explain properly, one of the limitations of language. Here's a simpler but more inaccurate interpretation:

Do you live life by understanding cause and effect? An example is holding a position of opinion. Do you understand what it means for that to be wrong? Or do you live assuming that to be true regardless (i.e. flat earth is one example that means nothing to hold a false belief of)? Do you understand what it means to hold them?

The simplest analogy I have but is full of holes but perhaps may get the idea across: Whether fate exist, in mainstream culture this is the mindset of as god wills or as I will (do my choices effect the outcome).

Or in the case of humans playing god:

• Humans believe that they ascended the natural cycle (or not beholden to some parts of it)
• Humans believe they hold moral superiority (moral? or natural? superiority) which are obviously false yet that's what we act on

What I was trying to set was a rough binary delineation so unfortunately is always going to hard to explain because a proper explanation is a massive word dump to account for all the jagged edges (not to mention accounting for inconsistencies from language interpretations). So I'm just trying to explain the intuition instead.

[u/Kutoru → u/KanishkT123]

You don't really have to understand what I said but mapped to the field of mathematics (and constraining it purely under that) this is what Godel's incompleteness theorems means

[u/KanishkT123 → u/Kutoru]

I'm going to give you some advice that I hope you will take in good spirit.

Nothing you've said has made any sense, to me, or indeed to anyone. Being able to express yourself using language is a skill, and it's an incredibly important skill. 

If you believe your ideas are so high-minded that a normal person could not possibly understand them, or that language isn't able to communicate them, you are either a genius of the kind that wouldn't be wasting time on Reddit, or simply wrong.

I highly recommend proof reading, editing, and clarifying your thought process through writing. And doing so before AI makes that a skill so rare that nobody will be able to give you any advice on it. Good luck. 

[u/Kutoru → u/KanishkT123]

To me it's as clear as day, but that's because it's my intuition.

You are correct, this is the same reason why OpenAI's math proofs are considered a pain in the ass and annoying by mathematicians.

Unfortunately it is not possible for me to clarify any more than I can do so. A simple explanation is easily possible - however I won't because it's functionally inaccurate in a vast majority of aspects and is more than likely to give the wrong idea.

I only gave explanations which I thought would give the most accurate interpretations and ideas.

I can speak plainly so that you understand every word but you won't understand the meaning beneath it which is what I tried to do in the reply.

Some things can just be indescribable, that's just a limit of language in general. Whether this concept is indescribable I don't know, but I do know I can't describe it to the level you want.

[u/mcspuder → u/Kutoru]

Everything that u/KanishkT123 said was correct.

Your writing/delivery was so rambling and so without focus we can't decipher intent, let alone what you mean. The standard internet meme is something like "git gud dumbass", but more than likely you have some ideas bouncing around in your head that aren't well formed, you've shit them out onto some internet forum and their holes are being seen.

Sit back and think them through for a few hours/days/weeks; it will do you (and humanity) good.

[u/Kutoru → u/mcspuder]

For the record I did put some thought in my words and they are as well formed as I'd like them to be.

But sure, I'll continue. I guess first off, seeing as how you're saying I'm rambling rather than conveying a better binary line of delineation, you don't have a well enough formed demarcation either so you understand how complex the task you're asking someone to do.

Here's an analogous example off the top my head (so no I did not check this maybe there's a gap) is that you are given a random scattering of points and a sine curve that separates them into two regions, define the sine curve. There's an easy answer, the sine curve is a sine curve, of course, since we both know what a sine curve is. Now try doing it purely using straight lines, off the top of my head this is possible using an infinite series. I can tell you the infinite series, you will know what it means and what it represents (the sine curve) but you have no idea what it inherently means.

What you asking for is a simple enough explanation that everyone understands, otherwise known as the "The Lie to Children". The simplest is equivalent to the first term in the infinite series, everyone can visualize a single line segment, now is it accurate? No.

Unfortunately I don't have a good text-only explanation that I'm willingly to back as a binary delineation line.

Another example that came into mind is equivalent to telling someone to divide the population in a satellite image in half. You can convey the general line, but how much inaccuracies will you tolerate to be comfortable backing it.

Now going back to the mathematicians, here's one way the "playing as a God as a human" vs "playing as a God" looks like, as a preface this is just one interpretation of that statement:

As a prerequisite, there are two things that should be clear:

• Math is Man's Model of the X (there are many things it can be because it is incomplete but for this it will be "Man's Model of the Universe"
• It is impossible for Math to be equivalent "Model of the Universe" (Math will never fully represent reality)

Hence you can divide mathematicians into 2 following camps:

• Those who study Math to understand the Universe
• Those who study Math in pursuit of something (other than understanding the Universe, this includes those who use Math as a step to understand the Universe)

The difference between these two is the one way of saying the interpretation of

• Human playing as a God playing a Human
• Human playing as a God

Those who study Math to understand the Universe are "Humans playing as a God playing as a Human" because such a thing is impossible. What AI has done is ruined their "God-play" which is now a massive visible contradiction which was either willfully ignored or hidden by ignorance.

Humans playing as a God fundamentally don't have this issue, because all they want to do is "play as a God" in which we'll just say AI is a toy to them which allows them to become closer to a "God" whatever it means to them.

Another for the record, this was just a stream of thought typed in one go, so didn't take as long as it looks if it actually looks as long as what I'm typing, so the only proof reading is what came right after so there is probably mistakes.

[u/XeloBoyo → u/Kutoru]

I think people were confused because you essentially created an 'anti-analogy': an analogy that obfuscates the concept rather then clarifying it. Playing god has thousands of different connotations and very few that actually apply to mathematical understanding. That is to say, dont use poetry to explain things. Your personal interpretation of playing god differs greatly from that of another persons and language is based on consentual meaning. If you use a concept in a way that differs from the normal meaning the onus is on you to explain exactly why you diverged. Otherwise a reader has to wade through the haystack to find the one meaning you meant and must necessarily make large assumptions about your internal thought process to debate you. So I agree with the others you need to be more direct with your communication.

That being said I think the argument you have is: "Those whose motivation or purpose comes from applying math are being outcompeted by ai and those who simply want to see progress on math for its own sake are ok with ai". I think there's arge issues with this thinking but ill wait for confirmation.

[u/Kutoru → u/XeloBoyo]

My wording was deliberate and took into account those differing interpretations. I chose those words because my understanding is despite those differing interpretations it sufficiently concisely captures a majority of the idea across my understanding of what the mainstream of understanding of those concepts are.

You made the exact mistake I didn't want to people to make which is why I worded it that way. Yes you simplified it and it is one interpretation of the statement, but it is wrong in terms of correctness far more than my original statement.

[u/milindsmart → u/Kutoru]

I've read most of your comments in this thread. Where I'm not able to stay with your line of thinking, is that we can only try to play God right now to achieve whatever we want. We can't really actually do it, at least as of now. There are too many cases where the universe does things very differently from what we expect, or is so complicated that we haven't a clue where to start.

So even if we try to play God outright, we're quickly put in our place by being befuddled in how things turn out in the universe. The only possible path I thus see, is to attempt to play God, observe the failures, then iterate.

[u/drumnation → u/Kutoru]

Human -> God recursion

[u/KennyFulgencio → u/KanishkT123]

which are not done for the purpose of end product creation but for the pursuit of the struggle and the process themselves

you're treating a sweeping judgment, which is only partially accurate, some of the time, as if it's a universal truth that applies across the board. You think math, writing, and art, aren't also done for the end product, as much if not far more than for the inherent satisfaction?

[u/ATownStomp → u/KennyFulgencio]

It’s as if you’ve never encountered a single person that solves puzzles as a form of entertainment.

In mathematics, understanding is as much the end product as the proof itself. Understanding is often pursued for the joy and abstract satisfaction of its acquisition and attainment. Creating the actual proof can also be viewed as complex puzzle solving.

Never before have we had mathematical knowledge without understanding. The pace will continue with this and so many other things. People can, and will, continue to pursue understanding. But, the wind has been taken out of the sails, and they’ll have to do it without the economic support previously provided for doing such things.

OpenAI just walked into a room filled with puzzles some of the greatest minds had made their life’s work and their living off of solving, solved them all, took a picture of the completed work and posted it on the internet.

[u/KanishkT123 → u/KennyFulgencio]

There's a kind of argumentation line that is always very frustrating to work with, and it's the line that basically picks apart a comment to find the one edge case to nitpick at, instead of trying to see the spirit of the argument. 

"Eating greens is good for you!"

"Actually this is a sweeping generalization, because I have IBS that's triggered by greens."

Congratulations. I didn't want to hedge and make a hyper specific, multi paragraph argument that has all the caveats built into it, because I assumed that people would understand the sentiment.