That's a good thing. It saves people wasting time trying to prove something they now know to be false, so that they can move on to other things to prove, it's a more fruitful use of humanity's time overall at least in the field of mathematics.
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
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.
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.
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?
This actually strikes me as a very formal perspective.
Considering it from an informal perspective, it's a bit more fuzzy isn't it. As you say there are many isomorphic things, and when we say The Integers it's not actually clear which one of them we mean. Maybe we mean one of them today and another tomorrow. Often times it doesn't matter, and so we don't clarify the question.
Like you could imagine defining the BootstrapNaturals then the BootstrapIntegers then the BootstrapRationals then use them to define the Reals. And then say that the Naturals, Integers and Rationals are defined as subsets of the Reals. This would be one way to put the common view of the Naturals as being a subset of the Reals on a solid formal foundation. It's rarely done ig because it's seen as obviously unproblematic to be a bit handwavy.
Another criticism of the common construction of numbers we could pose, inspired by object oriented programming, is that they fail at "information hiding". In programming an object should ideally not expose its internals. But in mathematics we may define 0 as say the empty set, making set operations on numbers syntactically valid which is kinda strange.
But yeah I think everyone has a sort of implicit understanding that 0 isn't actually the empty set. That it's merely a sort of hmm... thought experiment? That considering it 0 is a limited time offer, for the duration of the definition phase?
Maybe we come back to the isomorphism after all. "The natural numbers are something isomorphic to this set stuff I will now do"
[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'?
*statements only apply to maps whose domain is finite-dimensional
(Note that this is about excluded middle. There ARE constructive logics with interpretations of excluded middle, e.g. some forms of classical linear logic, but they do not play as nicely with other logics. Constructivists often reject even weak forms of choice for largely the same reasons--there are some forms of choice that are constructively valid in some logics, but these results often fail to hold true in more conventional logics. And the same is true for a whole host of related notions that proof assistants like Rocq reject by default, propositional extensionality (which says that two proofs of the same proposition are equal) and function extensionality (which says functions are equal whenever their results are equal on all the arguments in their domain -- which might seem obviously acceptable until you realize that it's false in most programming languages!) being prominent but much less discussed examples. It's all about remaining broadly compatible with lots of different types of reasoning, not because people think the reasoning is invalid per se).
I find it hard to believe that a constructivist would agree that across the board, proof of existence of X is more interesting than a construction of X. Isn't that the point!?
The constructivist will still object that you can’t, from that, conclude that “…therefore a counterexample must exist,” without actually providing a counterexample. But the general principle I was describing still applies - a proof often gives you insight that an example by itself doesn’t.
It depends on how old an intuitionist/constructionist you are. Back in the day, they were interested in logic as a description of correct reasoning. Brouwer saw LEM as a mistake in the foundations.
These days, the influence of formalization, including proof theory and model theory, has removed a lot of the teeth from that debate and made it possible to summarize as you have.
I studied this in the early 1980s, and my professor was definitely in the "this is a black and white issue" camp, although he came down on the classical side.
(Side note, I was once a back seat passenger in a car with my prof and Quine in the front seat, who was visiting at the time. Quine was famously committed to the idea that first order logic is the only kind worthy of the name.)
The part that I was referring to was the last statement from the OP: that "a proof of existence of a counterexample necessarily provides more insight than a counterexample". I can't imagine a constructivist would agree with that in general.
Whereas a counterexample, on its own, proves the proposition false but doesn't necessarily tell you anything else.
The real difference in the constructive case is that there are fewer classes of proposition for which a proof without witnesses is possible.
(Edit: side note, I didn't explicitly say "necessarily" in my original comment. I suppose there could be exceptions, although I'm struggling to think of an example. Constructively speaking, the ball is in your court!)
In this case, I think part of the problem is that there was kind of no good reason to think the Jacobian conjecture was true in > 2 dimensions other than it being kind of hard to find counterexamples. So a really interesting disproof would be one that, e.g., was able to exhaustively classify the counterexamples, or showed why it seemed in practice to be hard to come up with functions violating the conjecture. AFAIK, this doesn't really accomplish either of those things, not even after you learn the procedure that constructed the function -- it kind of tells you why we should have expected to find a counterexample but not how rare such counterexamples are.
Unless you want to spend time "proving" that 2 * 2 = 1.
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_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
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.
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.
Or are they approaching an asymptotic limit?
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?
P.S. It helps that in CS lots of theorems are about either inductive or coinductive definitions.