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Can computers be mathematicians?

quantamagazine.org

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Re: Can computers be mathematicians?

#21

Post by 323 »

t_mann wrote: post #18

Of course one of the first things that come to mind when reading the title is first having some sort of AlphaGo-style AI go wild on collections of math proofs and then maybe some AlphaZero-style model that can get to work even without the library. But I guess a fundamental problem for an RL-based approach (and a pretty interesting question in itself) is how difficult it would be to say what makes any mathematics "int…

Big results typically have a lot of implications and connect different fields together. This sounds like something that could be measured objectively.

You can also try reproving old theorems with newly discovered methods. So if suddenly you can reprove results in different fields with a new method, that sounds like something that might be big and should be investigated by humans.

Of course, what I wrote above will miss subtle things, but there is a place for a system which can just brute force it's way through mathematics.

Re: Can computers be mathematicians?

#23

Post by freemint »

civilized wrote: post #22

How do you get the AIs to prove interesting theorems instead of, say, summing random long sequences of integers? Or is this more focused on letting the computer try to prove a given conjecture?

By formalizing the same proof system we would use to convince ourselves that this works and letting the AI the search the space of logical statements using that language

Re: Can computers be mathematicians?

#24

Post by freemint »

IIAOPSW wrote: post #9
DiggyJohnson wrote: post #5

This is related to my pet favorite question of who is doing the art if an artist uses AI (e.g. [input image] -> Photoshop painting/masking -> AI Upscaler -> Crop and Print to Physical Medium? I think I feel like an artist, but part of the production of this art feels like curation , and not precisely the practice of art. The eye for composition feels similarly artistic, but the inability of envisioning the final prod…

I'm no artist but I've had an idea for a minor art project and please feel free to steal this. It's called "I pay rent therefore I am." Basically, a neural network is set up in a public space. The artificial neurons are literally shown as Christmas lights. This neural network has a few little gadgets it can control to entertain you. Let's say, a puppet, an electronic keyboard, a pen plotter. But, the installation is…

I wouldn't. I wouldn't say a human trapped in the same situation has free will either . If it could perform a variety of tasks and chooses what it performs while sustaining itself but not optimizing purely for profit and a bunch of those talk and agree to create an unemployment insurance which includes limits on self cloning i would say they have free will.

Re: Can computers be mathematicians?

#25

Post by freemint »

photochemsyn wrote: post #10

Disappointing that there's no mention of infinities. How do you teach a computer to think in symbolic terms about Cantor's concepts of infinity? There's no computational representation for irrational numbers, and barely for rational numbers. Even floating-point representations are a long-standing problems - and integers? Overflows are a problem, even just signed integers are a problem. The pencil-and-paper mathematic…

> How do you teach a computer to think in symbolic terms about Cantor's concepts of infinity?

All the other answers have been well so i do it tongue and cheek. "With Floating Point numbers"

Re: Can computers be mathematicians?

#26

Post by IIAOPSW »

freemint wrote: post #24
IIAOPSW wrote: post #9

Earlier quoted context omitted.

I'm no artist but I've had an idea for a minor art project and please feel free to steal this. It's called "I pay rent therefore I am." Basically, a neural network is set up in a public space. The artificial neurons are literally shown as Christmas lights. This neural network has a few little gadgets it can control to entertain you. Let's say, a puppet, an electronic keyboard, a pen plotter. But, the installation is…

I wouldn't. I wouldn't say a human trapped in the same situation has free will either . If it could perform a variety of tasks and chooses what it performs while sustaining itself but not optimizing purely for profit and a bunch of those talk and agree to create an unemployment insurance which includes limits on self cloning i would say they have free will.

whats the diff between me giving it the objective "make money" and evolution giving us the objective "make copies"?

Re: Can computers be mathematicians?

#27

Post by civilized »

freemint wrote: post #23
civilized wrote: post #22

How do you get the AIs to prove interesting theorems instead of, say, summing random long sequences of integers? Or is this more focused on letting the computer try to prove a given conjecture?

By formalizing the same proof system we would use to convince ourselves that this works and letting the AI the search the space of logical statements using that language

Most logical statements are uninteresting stuff like summing long sequences of integers, so now we have to sift through a heap of junk AI theorems hoping to find something interesting.

Re: Can computers be mathematicians?

#28

Post by freemint »

IIAOPSW wrote: post #26
freemint wrote: post #24

Earlier quoted context omitted.

I wouldn't. I wouldn't say a human trapped in the same situation has free will either . If it could perform a variety of tasks and chooses what it performs while sustaining itself but not optimizing purely for profit and a bunch of those talk and agree to create an unemployment insurance which includes limits on self cloning i would say they have free will.

whats the diff between me giving it the objective "make money" and evolution giving us the objective "make copies"?

Because we can choose not to, if an AI is designed to maximize an objective it will maximize that objective. The moment we make a human with a singular objective which they pursue above all else he is no longer free either.

Optimization for one objective eliminates all degrees of freedom in a system but for different (global?) minima one of which can be picked.

Re: Can computers be mathematicians?

#29

Post by freemint »

civilized wrote: post #27
freemint wrote: post #23

Earlier quoted context omitted.

By formalizing the same proof system we would use to convince ourselves that this works and letting the AI the search the space of logical statements using that language

Most logical statements are uninteresting stuff like summing long sequences of integers, so now we have to sift through a heap of junk AI theorems hoping to find something interesting.

Oh we can definitely bias the results toward statements which the AI thinks humans would produce.

Re: Can computers be mathematicians?

#30

Post by IIAOPSW »

freemint wrote: post #28
IIAOPSW wrote: post #26

Earlier quoted context omitted.

whats the diff between me giving it the objective "make money" and evolution giving us the objective "make copies"?

Because we can choose not to, if an AI is designed to maximize an objective it will maximize that objective. The moment we make a human with a singular objective which they pursue above all else he is no longer free either. Optimization for one objective eliminates all degrees of freedom in a system but for different (global?) minima one of which can be picked.

But, despite all the degrees of freedom of choice, you cannot ignore the most basic objective function that nature has handed us: stay alive. You can't really choose not to buy food and pay rent.

Arguably, at least from an evolutionary perspective, all other choices and wants are in service of this objective (albeit in very indirect and abstract ways).

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