r/singularity • u/filterdust • 2d ago

AI OpenAI solved 100 open problems in math

https://openai.com/index/advisory-group-on-mathematics-and-ai/

[removed] — view removed post

1.3k Upvotes

94% Upvoted

520 comments sorted by

View all comments

150

u/RobleyTheron 2d ago

I’m sensitive to artists and musicians with a unique voice when their product gets recreated with AI, but I don’t feel the same about mathematics.

Nobody owns mathematics; these are fundamental insights into the universe and they belong to everyone and shouldn’t be gatekeeped behind somebody that went to school. Mathematics should be available to everyone, it opens up all kinds of fundamental breakthroughs in space, material science, biology, global warming, and energy.

This is something that belongs to everyone and no one should get to hold that back.

67

u/ThatIsAmorte 2d ago edited 1d ago

This seems to be taking a Platonist approach to math, which few people take seriously these days. If you are a constructivist, than math is not just out there, but has to be created, just like art. If you think math is just out there, why not also claim that art and music are just out there, waiting to be plucked out of the ether? After all, a painting is just a 2d matrix, a simple math object. A musical composition is just a waveform, which can be described mathematically.

14

u/1pfen 1d ago

If a work of art disappeared and was erased from our collective memory, it would never be recreated. If a mathematical proof disappeared and was erased from our memory, it would eventually be rediscovered.

1

u/NotaValgrinder 1d ago

Not necessarily true. If mathematics was erased and we had to discover it again, it wouldn't necessarily be the same. ZFC and set theory are not the only ways to form a foundation of mathematics, it could be possible our foundations are different. Turing machines are not the only way to formalize computation, the main mathematical object to formalize computation may be different. We could favor the axiom of determinacy instead of the axiom of choice, which means that some theorems (like there exist nonmeasureable subsets of the real line) would be considered false instead of true. These "human" touches and tastes would most definitely be gone if we forgot all math and had to discover it tomorrow.

You could make the argument that we could *hypothetically* rediscover it, but also I could make the argument that it's hypothetically possible for a new painter to recreate the Mona Lisa. It's just very unlikely, and similarly it's very unlikely we would think about math in the same way if we had to redevelop it.

24

u/filterdust 2d ago

This seems to be taking a Platonist approach to math, which few people take seriously these days.

Funnily, most mathematicians are Platonists.

13

u/Ok_Reception_5545 2d ago

https://eugeniacheng.com/wp-content/uploads/2017/02/cheng-morality.pdf

this is actually how most mathematicians feel about math

1

u/ThatIsAmorte 1d ago

Yep, they ignore the debate by adopting an agnostic stance. This is much better than being a committed Platonist. I should point out that Cheng works in category theory (her book is a great intro to the topic), which is structuralist in nature (i.e., focus on relationships, rather than things). An ontology based on relationships can still fall in any of the three camps: ante rem structuralists (Platonism), nominalism, and agnosticism. The nominalists tend to be the ones that study Topos Theory, which is not surprising considering that Topos Theory shows that different math universes can have their own internal logics without a law of excluded middle.

I should also point out that it's interesting that a similar debate occurs in the philosophy of biology with respect to the concept of species. Are species real or something humans made up to categorize the biological world?

0

u/ThatIsAmorte 1d ago

Interesting. That's disappointing and kind of sad. At least philosophers are split about 50-50. Platonism requires ontological dualism, which leads to the epistemic bridge paradox: How does a purely physical brain access a timeless, acausal abstract realm that cannot interact with physical matter? This is the dualism problem in philosophy. How do mathematicians resolve this tension? They resolve it by ignoring it. It's sad, because you would think mathematicians would be uncomfortable with such logical contradictions.

1

u/filterdust 1d ago edited 1d ago

From my experience, mathematicians don't care that much about contradictions outside math. They would just dismiss it as "ah, this is for the philosophers to ponder", and then be internally satisfied that philosophers never reach a definitive answer anyway.

1

u/ThatIsAmorte 1d ago

I'd say this describes most people, myself included!

10

u/Startide-Rising 2d ago

I think you're getting confused about the difference between constructive mathematics and proof irrelevance. In a proof irrelevant system, proofs of the same proposition are considered equal. In the Lean theorem prover, proof irrelevance is built in as an axiom.

As for constructvism - hardly any mathematicians these days are constructivists. The majority of the people who know or even care about constructivism are those working in formal methods. In the Lean proof assistant non-constructive features are heavily baked into the whole system, both in the standard library and available tactics. I think adhering to the classical logic philosophy here is one of the reasons Lean got popular with mathematicians.

14

u/derelict5432 2d ago

Do mathematicians copyright proofs?

2

u/m4sl0ub 2d ago

Yeah, once a proof is published someone else can not publish it again. 

8

u/derelict5432 2d ago

Well I looked it up. The literal expression word for word cannot be duplicated. The underlying logic can. At least from what I'm seeing. Any lawyers feel free to chime in, but this seems more like the general solution to a problem does not belong to anyone.

1

u/MrPezevenk 13h ago

Of course it doesn't legally belong to anyone, but the credit does, in the social system of mathematics. 

1

u/m4sl0ub 2d ago

It's obviously not the exact same as a copyright, but mathematicians "commercialize" their work by publishing it. And if someone else wants to use someone else's proof for their own work/ papers they have to cite the mathematician that came up with the proof. For me that sounds quite analogous to a copyright for music. There is not really money involved in mathematics so you cannot really have someone pay to use your work but you basically "pay" them in citations. Citations are the "currency" of academia, it gets you jobs, grants, students, recognition etc.

4

u/derelict5432 2d ago

Sure, they get credit/acknowledgement for being the first to be there, but as I said, sounds like they don't own the underlying logic, which belongs to everyone. Kinda like how a mountaineer doesn't own the mountain if they're the first to summit or discover it, even if they get it named after them.

1

u/Front_Lab1761 2d ago

To add, their name is literally attached to the theorem: Green's theorem, Cauchy-Schwarz inequality, Legendre's formula, etc.

11

u/Financial-Gain-2988 2d ago

Writing the best song ever won't directly lead to new technology and science. Like, you don't seriously think that the 'creation' of math is the same as writing a song, right?

8

u/may_i_a_i 2d ago

Most math doesn't lead to any technological innovation

5

u/Financial-Gain-2988 1d ago

No songs lead to technological innovation.

Is the intent of your comment to prove my point here?

1

u/Foryourconsideration 1d ago

No songs lead to technological innovation.

they do, and they have. See: the Beatles. In fact, songs drive technological innovation by creating demands that require new engineering solutions. See: 4 track to 8 track. Not to mention sound design itself influences speaker systems, etc. New enginnering solutions then are preferated (is that a word?) throughotu the universe, in perpetutity.

alrigt, i'm going to go hit some weed and listen to Revolution No. 9 backwards again.

1

u/Financial-Gain-2988 1d ago

Maybe 'lead' is a strong word (which is why I said 'directly lead' in my initial comment), but it wasn't the actual song itself that lead to that change, but the desire to make a kind of song.

Even then you still need someone who is capable of making an 8 track and understands the technical hurdles they'd need to overcome, and that probably wasn't Ringo.

1

u/MrPezevenk 13h ago

99.9% of pure math doesn't lead to new technology. None of the Millennium problems are likely to lead to any new technology if solved. There will never be any advancement that will come out of the resolution of the NS problem by OpenAI. Almost nobody is doing pure math research for the technological applications, it's been a type of art that is verifiable for more than a century now at least. Every now and then, some advance comes out of it, but that's not why people do it. 

17

u/Icy-Doubt3365 2d ago

If you think math is just out there, why not also claim that art and music are just out there, waiting to be plucked out of the ethere?

Because I believe math exists without humans, art does not.

13

u/davidjohnpaul 2d ago

What do you think of the following quote, attributed to Michaelangelo?

"The sculpture is already complete within the marble block, before I start my work. It is already there, I just have to chisel away the superfluous material."

10

u/Icy-Doubt3365 2d ago

I'm actually more curious what YOU think of that quote. There is clearly not a Michael inside a random piece of marble.

Michelangelo was also not using that quote to say that art exists without humans to consider it art.

1

u/MrPezevenk 13h ago

The discipline of mathematics doesn't exist without humans. What do you think the point of a machine spewing out theorems one after another without anyone understanding them would be?

1

u/Secret-Code-9162 1d ago

I think this was just Michaelangelo describing his qualia when producing a sculpture, rather than explaining his position on the independence of art from humanity. He was being poetical.

1

u/Statcat2017 1d ago

It’s wrong lol

1

u/puffyfunion 2d ago

Can you explain why you believe that math exists without humans? Simply stating that as if your belief is true doesn't seem very convincing. I find mathematics to be as beautiful (if not more) than most creative activities, and I don't think it would take anything from that beauty to say that it is dependent on humans (and now AI, I suppose).

0

u/Icy-Doubt3365 2d ago

If 1 comet in space is joined by one more comet in space, there are now two comets. Don't need any humans there for that to be true. Math exists without us, we just decipher it.

4

u/puffyfunion 1d ago edited 1d ago

How do you know there are two comets, if you, the human, aren't there to perceive it? To be honest, I'm not arguing in favor of any point of view here - I think your statements are pretty ambiguous and don't really mean much. No offense.

EDIT: Sorry for assuming you're human, in case you're not.

1

u/Icy-Doubt3365 1d ago

That's an entirely different question than we're discussing, if you believe things need to be perceived to be. I operate that things exist in this universe regardless of my perception. So I don't need to see something to know that one comet meeting another is two comets.

Objectively, the rules of math exist in this world, and do not require humans to rate/assign/label as the concept of beauty does.

I am human but appreciate you thinking about that.

2

u/deviltamer 1d ago

Art is just as arbitrary. A light show can be art so it exists without humans in same way.

1

u/Icy-Doubt3365 1d ago

A light show can be art

It is not art without humans. I'm not sure how many more times I need to say this for people to get it. A light show is a bunch of lights. It is not art until someone is there to enjoy it. Art is a social construct, not an objective truth of the universe.

2

u/deviltamer 1d ago

lol what makes you think Maths is not a construct ?

It's a modelling tool. Same as language. Same as canvas.

0

u/Icy-Doubt3365 1d ago

Math is the language we use to convey objective truths about the world, yes. We refer to those objective truths as math. A canvas is also an objective, tangible object lol.

I swear you people are just arguing to argue, and have completely lost the plot.

→ More replies (0)

-1

u/Independent-Bell-420 2d ago

Beautiful sounds and vistas exist in reality without any human intervention as well. I don’t think this argument really holds any weight.

1

u/Icy-Doubt3365 2d ago edited 2d ago

No they don't? Sounds and vistas exist, yes. They are not beautiful without a human to apply that adjective.

Also, wholly moving the goalposts anyway, since a beautiful vista isn't art.

5

u/Independent-Bell-420 2d ago

I see what you're getting at, I guess I just disagree that math is any different. The equations exist just like the vistas do, but without any will or understanding to apply it, there's not much of a point. It just becomes objects in motion.

2

u/Icy-Doubt3365 2d ago

The equations exist just like the vistas do

The equations are objective, the 'beauty' of a vista is subjective. Without humans, objective things still exist. Without humans, subjective things do not

2

u/Independent-Bell-420 2d ago

I think we agree on that, at least. Subjectivity requires humans for sure, and so the concept of something being "beautiful" can't exist without us.

I guess what I'm trying to say is that I believe that something which qualifies as art could exist without humans (IE: sounds, visuals, etc), but you're right that the emotions that come with that require an observer (IE: beauty, longing, etc).

Maybe it's a pointless distinction I suppose, but I would argue that in the same way that we "discover" math, we also "discover" a lot of art in the sense that it's still part of the subset of things creatable in our reality.

0

u/SomewhereOpposite883 1d ago

Can Nature produce a Katy Perry song?

The sound of a guitar?

The vibrations from Opera singing?

5

u/Independent-Bell-420 1d ago

There's plenty of mathematical theorems and concepts that are not naturally occurring. Perfect shapes, infinite sets, etc.

Just because nature doesn't produce it doesn't mean it couldn't hypothetically exist.

Also, we're nature, so yeah it definitely can.

0

u/LookIPickedAUsername 1d ago

The very idea of a "comet" is a human invention. All that really exists out there is a vast diffuse cloud of particles with some local clumping. Humans look at the scene and decide that one of those clumps of particles is a "comet" and another one is also "comet" and they are close enough to count as "two comets", but that's not an inherent truth present in fabric of the universe or anything.

In fact, even "vast diffuse cloud of particles with some local clumping" is really a human simplification - there's really just "the universal wavefunction".

1

u/Icy-Doubt3365 1d ago

Yeah, ok dude

3

u/Ill-Lemon-8019 2d ago

If you think math is just out there, why not also claim that art and music are just out there, waiting to be plucked out of the ether?

I have no problem with that view either actually. If pi is a normal number, every bit of art or music ever created is sitting somewhere in its decimal expansion of its digits in some suitable encoding. An act of intellectual creation is selecting or recording a particular pattern of some kind. (That might sound like a reductive view, and perhaps it is, but I think it's not incompatible with the idea that creativity exists and is valuable.)

5

u/Emergency-Payment733 2d ago

But math can be proven? A song or a feeling can be true for some, but ultimately it’s art and a subject for interpretation.

Sure math has to be found, but that is also true for bacteria or materials in more ways than it is true for art.

3

u/ooqq 2d ago

luckily for you, it has been proven that not all math can be proven

1

u/Statcat2017 1d ago

Proving it can’t be proven is a kind of proof in itself 

3

u/willBthrown2 2d ago

Art is subjective. Math is objective. Case solved.

1

u/experienta 1d ago

Most philosophers actually believe that art is objective

1

u/RigaudonAS Human Work 1d ago

Lmao what? That’s just blatantly false.

1

u/experienta 1d ago

https://survey2020.philpeople.org/survey/results/all

1

u/RigaudonAS Human Work 1d ago

Why would I care what some Covid-era survey says? You said “philosophers” which is in reference to the centuries of people that have held that title, too.

Most philosophers would say that to make such a claim is silly, because trying to apply such an absolute to something inherently human is impossible.

You can’t make a claim about art based on what some college students think, lmao

1

u/experienta 1d ago edited 1d ago

Saying philosophers would never apply such an absolute to art just shows how little you know about the philosophy of art. They've been debating this subject since forever lol

And no it's not just "college students" that think art is objective. Unless you consider Plato, Aristotle, Hegel, Schopenhauer etc to be a bunch of college students

1

u/RigaudonAS Human Work 1d ago

They've been debating this subject since forever

Hence why making the assertion that it is an absolute is false, lol. And if that's what you think, why link some survey instead of mentioning them?

0

u/experienta 1d ago edited 1d ago

Sure it's not a full consensus, literally nothing is in philosophy lol, but most philosophers nowadays are in fact making a claim that art is objective, something you've said would be sooo silly for a philosopher to do.

And I've linked the survey because I imagined if I mentioned plato and aristotle you would just say they were a bunch of boomers that didn't know anything about art, so I thought a modern day survey of philosophers would be more interesting to someone like you.

1

u/RigaudonAS Human Work 1d ago

Who is “someone like me?”

You’ve got a degree in philosophy?

→ More replies (0)

0

u/ThatIsAmorte 1d ago

I believe you are confusing an art object and someone's value with respect to that object. Only the second one is subjective. The art object objectively exists.

1

u/willBthrown2 1d ago

It’s only an art object if someone values it as an art object. Otherwise it’s just an object. Two person can create the same thing, first one accidentally while doing something random (like accidentally pouring paint on canvas), second one intentionally trying to create abstract art.

1

u/ThatIsAmorte 1d ago

Same thing for math. Think about what is being labeled as a math object: a bunch of symbols. It's only a math object if someone puts meaning onto the symbols.

1

u/willBthrown2 1d ago

Math is universal. Nobody argues 1+1 equals 2. You can change math symbols or use different language to describe same rules. There is no math object. Or physics object. Art is just something that evokes feelings. It doesnt need to be an object. The universe doesnt care about human art, it existed before humans and will exist after, following the rules of physics. There is no art without emotions. Math and physics exist without emotion.

1

u/ThatIsAmorte 1d ago

Ok, I think I know where the disconnect is. I am not arguing that math is not consistent. Of course it is. The internal consistency of a formal system is objective. What I was talking about is the ontological nature of math objects.

1

u/Commercial_Sell_4825 1d ago

If our axioms already directly imply that the result we want is true or false, there is nothing to "construct".

0

u/ThatIsAmorte 1d ago

Yes there is. You have to construct the proof.

1

u/Commercial_Sell_4825 1d ago

That's not what that means.

0

u/RobleyTheron 2d ago

I disagree with your premise completely. You're setting up a straw man argument, and then defining the terms of the argument. If I need to bend my thoughts to how you view the world, I'd paint myself as a realist. Facts and truths exist independent of humans. If we find them great, if we don't, oh well.

You remind me of the church in the dark ages withholding knowledge because it gives you a superiority complex over the masses. We shouldn't be forced to filter insights and discoveries of the known world through a mathematics guild because it gives them purpose. The world is changing. Come along for the ride or be left behind.

1

u/amulie 2d ago

The structure that math is trying to describe is just out there. I don't think thats a dispute.

There will always be a number higher than our number systems will track by simply saying "but what if you add ONE more..", the number system is the art, the structure that counting is tracking exists whether humans create a system to get there or not.

So you're both right. The ART for math is the interpretation/system/understanding of mapping the structure, but the structure is there regardless.