Employees at the companies that do this stuff know exactly what these regulations are for and are violating them quite deliberately. They don't actually think it's "government bureaucracy" whatever they tell the public.
The article claims that "for the first time, string theory is testable," when in reality:
* the tests here concern particles that aren't actually known to exist yet
* lots of variants of string theory can be invalidated by the (non)existence of particles at particular masses or with particular properties
* moreover, this kind of invalidation has already happened on many occasions
* as usual it's only some models of string theory that are ruled out
So the headline and article are both kind of rubbish. In this case I wouldn't even say it's just a misleading headline because the article acts as though some (or even lots of) models of string theory getting hypothetically ruled out by a particle having a certain mass is a brand new thing.
Yeah people don't seem to get that the whole point of having a tightly checked kernel is so you don't have to care so much about the rest of it. Tactic heavy proofs have been "slop" long before LLMs got involved, and they lean heavily on the kernel rejecting nonsense.
People are very desperate to try to claim that something that's somewhat obviously morally wrong is actually highly nuanced, because it makes them feel uncomfortable.
I would disagree on the "obviously morally wrong". What's the alternative for the books if they were "saved" from the fate of being scanned and shredded? How many rare books have you personally brought in the last year? Not everything that's ever created needs to be preserved forever.
Yes, clearly it's completely absurd to think that shredding rare books to more cheaply train LLMs is anything but a moral good, which is why there is this entire thread is full of people jumping through hoops to explain how it's technically legal (and therefore fine) and "you wouldn't have bought those books anyway" (I suppose we don't have the choice anymore!) and, most amusingly, "they're actually becoming digitized and searchable this way" (are LLMs stochastic parrots or aren't they?).
Given that in the US we dispose of 640,000 tons of books each year, this attempt to inflate outrage is just ridiculous. Books are not things of intrinsic moral value; they are morally neutral physical objects.
To the extent the information in a book is something to preserve, this scanning is a good thing.
I think maybe a better way of explaining it would be that an uninformative proof by definition needs to be based on proving that the set under consideration must be inhabited without ever defining an object in that set. This generally means you must show the set is inhabited by exploring some abstract properties of the set itself. A single counterexample, by contrast, by itself is a direct proof that the set is inhabited, so you don't necessarily learn any other interesting properties about the set. So it's not really about constructive vs. non-constructive, I think it's closer to e.g. the idea that point-free stuff tends to be more beautiful and meaningful than pointed stuff (which I think most mathematicians would agree with and which really has nothing to do with intuitionism per se).
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.
Because you literally don't have to actually understand the proofs, just the definitions and proposition chain. Most of the code is going to be proving auxiliary lemmas or building up internal definitions that aren't needed to understand the final proposition. Then all you have to do is make sure the proof doesn't use any axioms and you're good. Being able to confidently do this is why proof assistants like Lean are really not just another programming language.
(That's not to say there's no value to making the proof themselves nicer--compilation time and reusability can actually be a really big deal in formalized mathematics!--but it's way less important than it is in software).
I think the constructive position is basically that people's entire issue with lack of excluded middle being absent is just that people like being able to say "P" instead of "~~P" because it sounds better, considering you can prove ~~P for all the classical propositions that use excluded middle.
> people like being able to say "P" instead of "~~P" because it sounds better
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.)
Ah, I didn't realize this was a generational thing. I am definitely a "new" intuitionist, so that probably greatly influences my perspective. I suppose that before results like this, the setoid model, etc. were known constructivism was indeed a much more hardline position to have to take!
As a constructivist: we don't disagree :) We just distinguish between "don't disagree" and "agree." Constructive mathematics says it's fine if you want to claim that there's not no counterexample -- you just can't use that in a situation that demands an actual counterexample (like an algorithm that produces a result). This tends to guide people towards looking for results that don't require this kind of indirection, since they apply more broadly and in more kinds of logics -- orthodox constructive results are kind of a lowest common denominator of consistent truth and remain broadly compatible with most axioms, while nonconstructive results often fail in particular models. Which seems like a pretty sane stance to me, but maybe I'm too thoroughly indoctrinated to see how unreasonable it is :P
(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).
Maybe I was unclear. The part that I meant a constructivist would disagree was the subjective part: "This kind of proof (existence of a counterexample) provides more insight into the problem than simply finding a counterexample."
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!?
A direct counterexample is more "informative" in a very literal sense (its truth value doesn't collapse). But the extra proof relevant content we can use here is not that large -- all it means in this case is that we can directly compute the object and its Jacobian, a well as two points evaluating to the same result. That's nice, but it's not that interesting by itself unless I can use the exact constructed form to prove other interesting stuff (and we can! Most of the followup results that immediately followed from the disrpoof come from being able to directly transform this object into counterexamples to other conjectures; if we didn't have constructive proofs of thoe counterexamples, we wouldn't have such procedures). But being more interesting than a completely uninformative counterexample still doesn't mean it's inherently interesting or enlightening. If anything I'd indeed argue constructive arguments are generally less mysterious and magical than nonconstructive proofs -- in some sense, the constructive proof pulls back the curtain and shows you where the trick is.
You can go through my commenter history and know I'm no fan of LLMs. I don't overstate LLM capabilities and am highly skeptical of them in general. 5.6 Pro is genuinely pretty good at certain kinds of math problems that just require trying out lots and lots of solutions, mostly because it's stubborn and can run a bunch of instance in parallel. It is NOT good at coming up with unique ideas or recognizing when its proof approach is doomed, and if the correct approach isn't in its "bag of tricks" for tackling a specific kind of problem, it is not going to get it without a lot of guidance. That said: I 100% believe that it's solved the problems people are claiming that it solved.
The way you should read this is (IMO) not that LLMs have somehow achieved AGI, but that a lot of mathematical research is more about knowing a huge amount of mathematical background, being stubborn, and getting lucky with an approach than it is about brilliant insight. Many people who don't think of themselves as particularly mathematically gifted could have made progress on these problems if they were given enough time and were interested enough. What's notably different about 5.6 (and born out in benchmark after benchmark) is that it does seem to genuinely "reason" through stuff at all -- without that, persistence is pretty worthless because the LLM just goes wildly off the rails if it's put to work for long enough (5.6 itself will still do this if it can't find an answer in a reasonable amount of time).
