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Theorem: There are infinitely many prime numbers. Proof: [GPT-3's "proof"] Let p be a prime number. Then by definition, p is a positive integer that has no p

Theorem: There are infinitely many prime numbers. Proof: [GPT-3's "proof"] Let p be a prime number. Then by definition, p is a positive integer that has no positive divisors other than 1 and p. Now consider the integer q = p + 1. We will show that q is not a prime number, which will prove that there are infinitely many prime numbers. Since p is a positive divisor of q, q cannot be prime. [...another attempt with the same prompt] We know that there are infinitely many natural numbers. Let's suppose that there are only finitely many prime numbers. Then we can list them all: p_1, p_2, p_3, ..., p_n Now, consider the number N = p_1 p_2 p_3 ... p_n + 1 This number is greater than any of the prime numbers in our list, so it cannot be divisible by any of them. But it is also a natural number, so it must be prime. This contradicts our assumption that there are only finitely many prime numbers, so we must conclude that there are infinitely many prime numbers. ~~Personal remark: I had to give it the same prompt like 5 times to get a correct-ish proof. The first attempts came out like the first example.~~ The proof is still wrong.

ogogmad 5y ago
I keep asking GPT-3 to prove that the LR algorithm (for finding eigenvalues and eigenvectors) converges for PSD matrices. It keeps insisting that it's a form of gradient descent. Is that true?
deleted 5y ago
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gnulinux 5y ago
Both proofs are wrong, second one is closest. Second one should not claim that N is a prime (it likely isn't). It should say N is not divisible by any of p_i, and since due to Fun. Theo. of Arith. it is such that N = Sum {c_i q_i} where q_i are prime, and none of q_i in {p_i} which shows a finite list of primes is not possible construct.
brian_cloutier 5y ago
Interestingly, these attempts are about the same as what pops up when I try to remember the proof: - It's a proof by contradiction - The key step is in taking the finite list of primes, multiplying them together, and adding 1 I then try to flesh out the details, it might take a second to realize that this new number is also prime, and then a few moments more to remember the exact rationale why. Along the way the proof lives in a kind of superposition where I'm not clear on the exact details. The "proofs" you gave here seem to be serializations of a similar superposition! GPT-3 seems to remember the proof about as well as I do, but it's missing the final sanity check which tweaks the proof until all the pieces correctly fit together. In this case, you seem to be performing a version of this sanity check by running the prompt multiple times until a correct answer comes out. I wonder if it's possible to prove something more obscure using a similar process: GPT-3 comes up with ideas and the human sanity checks.
- ctoth 5y ago
  I believe this recent paper demonstrates a method for allowing these large language models to perform this "sanity check" automatically[0]. [0]: Self-Consistency Improves Chain of Thought Reasoning in Language Models https://arxiv.org/abs/2203.11171 https://arxiv.org/abs/2203.11171
- Banana699 5y ago
  >this new number is also prime Not necessarily, it might be composite, but in this case one of it's prime factors will necessarily not lie in the supposed list of primes, therefore also a contradiction. The first counter example to "If L := {P0,P1,..,Pn} is a list of primes, then prod(L)+1 is prime" is {2,3,5,7,11,13}, their product is 30030, and 30031 is a composite of 2 primes, none of which are in the list.
  - falcor84 5y ago
    It's somewhat silly semantics, but I believe it is a valid deductive step on the way to the contradiction - if the number is not divisible by any other prime, then it must be a new prime, ⊥.
    - ivegotnoaccount 5y ago
      The issue is that it is not divisible by any other prime *from the list*. The two cases (prime or composite) must be handled separately since they do not use the same logic to infer there is one more prime. For instance, 2 * 3 * 5 * 7 * 11 * 13 + 1 = 30031 = 59 * 509.
      - ravi-delia 5y ago
        But to get the contradiction, you assume a finite number of primes. As each of them does not divide the new one, the new one is not divisible by a prime. It seems like your method is some kind of induction? Which probably gets a little closer to the "reason" for it, but isn't the standard proof I've seen.
nonameiguess 5y ago
This isn't really the "human level mathematician" equivalent task anyway. A human mathematician's main purpose isn't to memorize and reproduce proofs generated by other people. It's to prove original results no one else has proven before. To remember and reproduce existing proofs, I just typed "proof infinitely many primes" into DuckDuckGo and it gave me plenty of correct results.
- karpierz 5y ago
  That's like saying "standing still" isn't a human-level sprinter's task. In principle, yes, nothing in the 100m sprint requires that you need to be able to stand still. In practice, I would be very skeptical of someone who can't stand claiming they can sprint.
- zardo 5y ago
  It's a human level mathematics student problem. If it can't determine that's it's proof is nonsense here there's little hope it could produce any worthwhile original work.
jameshart 5y ago
What does GPT-3 come up with if you ask it for a proof that there are a finite number of primes? Or that pi is rational? I guess it would stitch together some more seemingly sensible statements that also don’t quite add up to a rigorous proof?
deleted 5y ago
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supermdguy 5y ago
I'm actually taking a proofs class right now, and edit my Latex in VS Code with Copilot enabled. Its syntax is always perfect, but most of the time it produces stuff that doesn't make a ton of sense. There have been a few times when it gets the next couple of lines correct for repetitive proofs with a lot of "boilerplate", but it doesn't really make big logical/creative jumps.

ogogmad 5y ago

I keep asking GPT-3 to prove that the LR algorithm (for finding eigenvalues and eigenvectors) converges for PSD matrices. It keeps insisting that it's a form of gradient descent. Is that true?

deleted 5y ago

[deleted]

gnulinux 5y ago

Both proofs are wrong, second one is closest. Second one should not claim that N is a prime (it likely isn't). It should say N is not divisible by any of p_i, and since due to Fun. Theo. of Arith. it is such that N = Sum {c_i q_i} where q_i are prime, and none of q_i in {p_i} which shows a finite list of primes is not possible construct.

brian_cloutier 5y ago

Interestingly, these attempts are about the same as what pops up when I try to remember the proof: - It's a proof by contradiction - The key step is in taking the finite list of primes, multiplying them together, and adding 1 I then try to flesh out the details, it might take a second to realize that this new number is also prime, and then a few moments more to remember the exact rationale why. Along the way the proof lives in a kind of superposition where I'm not clear on the exact details. The "proofs" you gave here seem to be serializations of a similar superposition! GPT-3 seems to remember the proof about as well as I do, but it's missing the final sanity check which tweaks the proof until all the pieces correctly fit together. In this case, you seem to be performing a version of this sanity check by running the prompt multiple times until a correct answer comes out. I wonder if it's possible to prove something more obscure using a similar process: GPT-3 comes up with ideas and the human sanity checks.

ctoth 5y ago
I believe this recent paper demonstrates a method for allowing these large language models to perform this "sanity check" automatically[0]. [0]: Self-Consistency Improves Chain of Thought Reasoning in Language Models https://arxiv.org/abs/2203.11171 https://arxiv.org/abs/2203.11171
Banana699 5y ago
>this new number is also prime Not necessarily, it might be composite, but in this case one of it's prime factors will necessarily not lie in the supposed list of primes, therefore also a contradiction. The first counter example to "If L := {P0,P1,..,Pn} is a list of primes, then prod(L)+1 is prime" is {2,3,5,7,11,13}, their product is 30030, and 30031 is a composite of 2 primes, none of which are in the list.
- falcor84 5y ago
  It's somewhat silly semantics, but I believe it is a valid deductive step on the way to the contradiction - if the number is not divisible by any other prime, then it must be a new prime, ⊥.
  - ivegotnoaccount 5y ago
    The issue is that it is not divisible by any other prime *from the list*. The two cases (prime or composite) must be handled separately since they do not use the same logic to infer there is one more prime. For instance, 2 * 3 * 5 * 7 * 11 * 13 + 1 = 30031 = 59 * 509.
    - ravi-delia 5y ago
      But to get the contradiction, you assume a finite number of primes. As each of them does not divide the new one, the new one is not divisible by a prime. It seems like your method is some kind of induction? Which probably gets a little closer to the "reason" for it, but isn't the standard proof I've seen.

nonameiguess 5y ago

This isn't really the "human level mathematician" equivalent task anyway. A human mathematician's main purpose isn't to memorize and reproduce proofs generated by other people. It's to prove original results no one else has proven before. To remember and reproduce existing proofs, I just typed "proof infinitely many primes" into DuckDuckGo and it gave me plenty of correct results.

karpierz 5y ago
That's like saying "standing still" isn't a human-level sprinter's task. In principle, yes, nothing in the 100m sprint requires that you need to be able to stand still. In practice, I would be very skeptical of someone who can't stand claiming they can sprint.
zardo 5y ago
It's a human level mathematics student problem. If it can't determine that's it's proof is nonsense here there's little hope it could produce any worthwhile original work.

jameshart 5y ago

What does GPT-3 come up with if you ask it for a proof that there are a finite number of primes? Or that pi is rational? I guess it would stitch together some more seemingly sensible statements that also don’t quite add up to a rigorous proof?

deleted 5y ago

[deleted]

supermdguy 5y ago

I'm actually taking a proofs class right now, and edit my Latex in VS Code with Copilot enabled. Its syntax is always perfect, but most of the time it produces stuff that doesn't make a ton of sense. There have been a few times when it gets the next couple of lines correct for repetitive proofs with a lot of "boilerplate", but it doesn't really make big logical/creative jumps.