3 ms·

proofs by counterexample are effective but ultimately unsatisfying. they get you to an answer but they don't help help you understand and bend you r mind into s

proofs by counterexample are effective but ultimately unsatisfying. they get you to an answer but they don't help help you understand and bend you r mind into seeing how the math works and lead you on to the new set of questions. and for now as long humans are going to judge of what counts as an elegant or illuminating proof, there's going to be work for human mathematicians

taneq 2mo ago
Maybe I’m just not pure enough but I find the whole concept of proof by counterexample to be elegant, and I don’t see why proving that something must be true is superior to proving that it can’t be false.
- bananaflag 2mo ago
  You mean proof by contradiction, which is something different.
  - taneq 2mo ago
    Hmm, I think I might have conflated the two. Thanks!
- chowells 2mo ago
  It's elegant if all you're concerned with is whether a conjecture is true or false. Answered, move along! But mathematics is not a collection of facts. Mathematics is the study of abstraction. And what do you learn from a single data point? What can you abstract from that? That's why just being a counterexample isn't really interesting. There has to be more than "counterexample" for there to be something to abstract. Was it generated from an analysis of the problem? Can the counterexample be generalized to explore the problem further? Is the counterexample a surprise in a way that suggests something is missing from current understanding? Being a counterexample doesn't mean that something isn't interesting to a mathematician. But it's also not the interesting part.
  - ATMLOTTOBEER 2mo ago
    You’re smuggling in a frame here which isn’t obviously true: that mathematics is not just a collection of facts
    - chowells 2mo ago
      You have a weird definition of "smuggling". I say it outright. Because I follow it up with a description of the actual practice of math: it's the study of abstraction. That's not a frame. That's just what math is.
      - im3w1l 2mo ago
        I think the nature of mathematics is an interesting question without one clean answer. To present a radically different view of mathematics: It's a game of string transformations, where the goal is to produce specific strings given a set of rules. The (syntactically valid) strings would correspond to statements, a producible string a theorem, and the production the proof.
        chowells 2mo ago
        This is a description of one portion of math... one that's very easy to get tunnel vision towards when undertaking a very formal undergraduate mathematical education. And that's especially true if it was alongside a computer science education, which is precisely the branch of math concerned with formal systems being used in calculation. I've done both of those things. I know what you get taught. But I've kept my math education going for the 25 years since then. I've talked to practicing mathematicians about what they do. I've learned a lot about the scope of math. As an aside: most people really dislike it when I say that they should be much more precise about different numerical systems. The integers are not a subset of the rationals. They are entirely different constructions, but there is an isomorphism between integers and a subset of the rationals that preserves the integers' ring structure within that subset of the rationals and a few other aesthetic concerns. You can see why no one wants to communicate like this, even if they acknowledge it's technically correct. So I know all about pushing symbols around. But I also know that pushing symbols around isn't the whole story. Pushing symbols around is only useful as a final check. Do you want to validate that 1+2=3? Pushing symbols around can help. But how do you decide that the ideas behind 1, 2, 3, +, and = are worth having precise and compact representations? Math doesn't just use formal systems to generate proofs. It's not enough for symbols to be arranged neatly according to some rules. Math is also the process of creating the sets of symbols and their rules and communicating to other people why this set of rules and symbols is interesting. What ideas get preserved when you are working with this system? What is it an abstraction over?
mb7733 2mo ago
Are you thinking of proof by contradiction, which is rejected by constructionism? [Dis]proof by counterexample is the most straightforward way to show a statement to be false. What better way is there to disprove a general statement like 'all x are y' than finding an 'x' that isn't 'y'?
- SpicyLemonZest 2mo ago
  It’s very straightforward, but it often doesn’t (and here didn’t) fully satisfy the curiosity that was embedded in the original problem. Why did the Jacobian conjecture seem to be true? Is there some underlying symmetry that’s very slightly broken? Or perhaps there’s all kinds of counterexamples, and the intuitive pattern is only real for certain kinds of functions which happen to predominate in our intuition. Then how should we adjust our intuitions to better capture the space of possible polynomial functions?
  - mb7733 2mo ago
    Those are all good questions, but I don't really understand what alternative you or the OP are looking for actually resolving an untrue conjecture, besides a counter example.
    - SpicyLemonZest 2mo ago
      I would frame it differently. The existence of compact counterexamples to a true-seeming conjecture suggests that there’s some deeper understanding waiting to be discovered. Fuzz testing for theorems, if that makes sense. I hope mathematicians in 2036 will be able to explain in detail why the Jacobian conjecture was false and identify which similar, true conjectures the community’s intuition was pointing towards.
      - nopinsight 2mo ago
        I suspect an AI, possibly a successor to current LLMs, will achieve that by the early 2030s. It might help illuminate many mysteries in math and beyond for us all.
remus 2mo ago
I would only agree partially. There are counterexamples that are not illustrative, but it is fairly common that in thinking about how to construct a counterexample you gain a more thorough understanding of the original problem and at least one fundamental issue which prevents the conjecture from being true.
- fuglede_ 2mo ago
  Right; see Lakatos. In its roughest form, you study the structure of whatever counterexamples you find, add those as (negated) preconditions to your proposition, rinse and repeat until you have a true statement. If the proposition remains useful, you now have a new definition.
delecti 2mo ago
You could spend the rest of your life coming up with conjectures that look elegant but are ultimately false. Disproof by counterexample only works if it's false, and we shouldn't be satisfied with a false conjecture to begin with.
moralestapia 2mo ago
Not much worth in understanding a statement that is wrong and has been shown wrong. Unless you want to spend time "proving" that 2 * 2 = 1.
throwaway676712 2mo ago
Counterexamples are literally the only way to show a "for all" statement is false. (Non-constructive proofs by contradiction work by showing a counterexample must exist.) Also, 'brute-force' style attacks where one simply feeds the input into the computer and it yields a solution are nothing new and certainly predate LLMs: https://en.wikipedia.org/wiki/Euler%27s_sum_of_powers_conjecture https://en.wikipedia.org/wiki/Euler%27s_sum_of_powers_conjec... Hell, one could even go further into the past and refer to the thankless work of pre-computer era mathematicians who sweated over manual calculations in order to disprove various prime related conjectures: https://en.wikipedia.org/wiki/Mersenne_conjectures https://en.wikipedia.org/wiki/Mersenne_conjectures You seem to have an objection to non-intutionist mathematics in general, a position that was once held by many an illustrious mathematician but is relatively fringe in the contemporary academic community. Mathematical facts don't have to be intellectually satisfying or make sense to you, the human, rather it is up to you to wrap your mind around discovered mathematical facts.
linzhangrun 2mo ago
> and for now as long humans are going to judge of what counts as an elegant or illuminating proof, there's going to be work for human mathematicians Considering ChatGPT was released only three and half years ago, and LLMs could do high school math only less than two years ago, I think this "for now" will not last very long.
- bluGill 2mo ago
  > Considering ChatGPT was released only three and half years ago ... Or are they approaching an asymptotic limit?
veunes 2mo ago
Elegance may not remain exclusively human forever but usefulness probably requires more than correctness
reinitctxoffset 2mo ago
Seems like a breakdown on the incentives / imperatives in the field? I hope that's not an over bold guess from a non-mathematician. Couldn't people in principle continue to study a problem that's only been shown to break at one point? Prove something adjacent, or slightly weaker, or elaborate the counter example into a powerful explanatory framework?
SonOfLilit 2mo ago
The counterexample in the news cycle today helps better understand how the math works. I can't think of one that doesn't.

taneq 2mo ago

Maybe I’m just not pure enough but I find the whole concept of proof by counterexample to be elegant, and I don’t see why proving that something must be true is superior to proving that it can’t be false.

bananaflag 2mo ago
You mean proof by contradiction, which is something different.
- taneq 2mo ago
  Hmm, I think I might have conflated the two. Thanks!
chowells 2mo ago
It's elegant if all you're concerned with is whether a conjecture is true or false. Answered, move along! But mathematics is not a collection of facts. Mathematics is the study of abstraction. And what do you learn from a single data point? What can you abstract from that? That's why just being a counterexample isn't really interesting. There has to be more than "counterexample" for there to be something to abstract. Was it generated from an analysis of the problem? Can the counterexample be generalized to explore the problem further? Is the counterexample a surprise in a way that suggests something is missing from current understanding? Being a counterexample doesn't mean that something isn't interesting to a mathematician. But it's also not the interesting part.
- ATMLOTTOBEER 2mo ago
  You’re smuggling in a frame here which isn’t obviously true: that mathematics is not just a collection of facts
  - chowells 2mo ago
    You have a weird definition of "smuggling". I say it outright. Because I follow it up with a description of the actual practice of math: it's the study of abstraction. That's not a frame. That's just what math is.
    - im3w1l 2mo ago
      I think the nature of mathematics is an interesting question without one clean answer. To present a radically different view of mathematics: It's a game of string transformations, where the goal is to produce specific strings given a set of rules. The (syntactically valid) strings would correspond to statements, a producible string a theorem, and the production the proof.
      - chowells 2mo ago
        This is a description of one portion of math... one that's very easy to get tunnel vision towards when undertaking a very formal undergraduate mathematical education. And that's especially true if it was alongside a computer science education, which is precisely the branch of math concerned with formal systems being used in calculation. I've done both of those things. I know what you get taught. But I've kept my math education going for the 25 years since then. I've talked to practicing mathematicians about what they do. I've learned a lot about the scope of math. As an aside: most people really dislike it when I say that they should be much more precise about different numerical systems. The integers are not a subset of the rationals. They are entirely different constructions, but there is an isomorphism between integers and a subset of the rationals that preserves the integers' ring structure within that subset of the rationals and a few other aesthetic concerns. You can see why no one wants to communicate like this, even if they acknowledge it's technically correct. So I know all about pushing symbols around. But I also know that pushing symbols around isn't the whole story. Pushing symbols around is only useful as a final check. Do you want to validate that 1+2=3? Pushing symbols around can help. But how do you decide that the ideas behind 1, 2, 3, +, and = are worth having precise and compact representations? Math doesn't just use formal systems to generate proofs. It's not enough for symbols to be arranged neatly according to some rules. Math is also the process of creating the sets of symbols and their rules and communicating to other people why this set of rules and symbols is interesting. What ideas get preserved when you are working with this system? What is it an abstraction over?

mb7733 2mo ago

Are you thinking of proof by contradiction, which is rejected by constructionism? [Dis]proof by counterexample is the most straightforward way to show a statement to be false. What better way is there to disprove a general statement like 'all x are y' than finding an 'x' that isn't 'y'?

SpicyLemonZest 2mo ago
It’s very straightforward, but it often doesn’t (and here didn’t) fully satisfy the curiosity that was embedded in the original problem. Why did the Jacobian conjecture seem to be true? Is there some underlying symmetry that’s very slightly broken? Or perhaps there’s all kinds of counterexamples, and the intuitive pattern is only real for certain kinds of functions which happen to predominate in our intuition. Then how should we adjust our intuitions to better capture the space of possible polynomial functions?
- mb7733 2mo ago
  Those are all good questions, but I don't really understand what alternative you or the OP are looking for actually resolving an untrue conjecture, besides a counter example.
  - SpicyLemonZest 2mo ago
    I would frame it differently. The existence of compact counterexamples to a true-seeming conjecture suggests that there’s some deeper understanding waiting to be discovered. Fuzz testing for theorems, if that makes sense. I hope mathematicians in 2036 will be able to explain in detail why the Jacobian conjecture was false and identify which similar, true conjectures the community’s intuition was pointing towards.
    - nopinsight 2mo ago
      I suspect an AI, possibly a successor to current LLMs, will achieve that by the early 2030s. It might help illuminate many mysteries in math and beyond for us all.

remus 2mo ago

I would only agree partially. There are counterexamples that are not illustrative, but it is fairly common that in thinking about how to construct a counterexample you gain a more thorough understanding of the original problem and at least one fundamental issue which prevents the conjecture from being true.

fuglede_ 2mo ago
Right; see Lakatos. In its roughest form, you study the structure of whatever counterexamples you find, add those as (negated) preconditions to your proposition, rinse and repeat until you have a true statement. If the proposition remains useful, you now have a new definition.

delecti 2mo ago

You could spend the rest of your life coming up with conjectures that look elegant but are ultimately false. Disproof by counterexample only works if it's false, and we shouldn't be satisfied with a false conjecture to begin with.

moralestapia 2mo ago

Not much worth in understanding a statement that is wrong and has been shown wrong. Unless you want to spend time "proving" that 2 * 2 = 1.

throwaway676712 2mo ago

Counterexamples are literally the only way to show a "for all" statement is false. (Non-constructive proofs by contradiction work by showing a counterexample must exist.) Also, 'brute-force' style attacks where one simply feeds the input into the computer and it yields a solution are nothing new and certainly predate LLMs: https://en.wikipedia.org/wiki/Euler%27s_sum_of_powers_conjecture https://en.wikipedia.org/wiki/Euler%27s_sum_of_powers_conjec... Hell, one could even go further into the past and refer to the thankless work of pre-computer era mathematicians who sweated over manual calculations in order to disprove various prime related conjectures: https://en.wikipedia.org/wiki/Mersenne_conjectures https://en.wikipedia.org/wiki/Mersenne_conjectures You seem to have an objection to non-intutionist mathematics in general, a position that was once held by many an illustrious mathematician but is relatively fringe in the contemporary academic community. Mathematical facts don't have to be intellectually satisfying or make sense to you, the human, rather it is up to you to wrap your mind around discovered mathematical facts.

linzhangrun 2mo ago

> and for now as long humans are going to judge of what counts as an elegant or illuminating proof, there's going to be work for human mathematicians Considering ChatGPT was released only three and half years ago, and LLMs could do high school math only less than two years ago, I think this "for now" will not last very long.

bluGill 2mo ago
> Considering ChatGPT was released only three and half years ago ... Or are they approaching an asymptotic limit?

veunes 2mo ago

Elegance may not remain exclusively human forever but usefulness probably requires more than correctness

reinitctxoffset 2mo ago

Seems like a breakdown on the incentives / imperatives in the field? I hope that's not an over bold guess from a non-mathematician. Couldn't people in principle continue to study a problem that's only been shown to break at one point? Prove something adjacent, or slightly weaker, or elaborate the counter example into a powerful explanatory framework?

SonOfLilit 2mo ago

The counterexample in the news cycle today helps better understand how the math works. I can't think of one that doesn't.