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Human mathematicians are being outcounterexampled

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Re: Human mathematicians are being outcounterexampled

#241

Post by wizzwizz4 »

gowld wrote: post #193
wizzwizz4 wrote: post #2

Human mathematicians have been being out-counterexampled for at least two decades. The main difference, as I understand, is that (A) we now have a lot more compute to throw at such things, and (B) it is currently trendy to do so. But the sizes of counterexample we're seeing are around about what I'd expect pre-generative-AI counterexample search systems to be able to find. It's not easy to find a counterexample to th…

The word "just" is the mark of a coward. AI is only as good as a human mathemetician, which we don't have enough of? "just"? https://en.wikipedia.org/wiki/Jacobian_conjecture > The conjecture was first stated for two variables by Ludwig Kraus in 1884 [...] an example of a difficult question in algebraic geometry that can be understood using little beyond a knowledge of calculus. > The Jacobian conjecture is number 16…

This is a counterexample for the three-variable case, which hasn't been studied as much. The long-standing two-variable case remains open.

Re: Human mathematicians are being outcounterexampled

#242

Post by Jweb_Guru »

antonvs wrote: post #225
mb7733 wrote: post #217

Earlier quoted context omitted.

To roughly repeat myself from a sibling comment: I may have been unclear. I didn't claim that the only way to disprove something in a constructive system is to produce a counterexample. 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 ge…

I'm the OP. In the cases where a proposition can be refuted by a proof that doesn't involve counterexamples, by its nature that proof will tell you something about the reason that the proposition is false. 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 whi…

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.

Re: Human mathematicians are being outcounterexampled

#243

Post by vonneumannstan »

tantalor wrote: post #236
vonneumannstan wrote: post #207

Earlier quoted context omitted.

>GenAI is great at combining existing things in new ways (interpolation). It's terrible at creating new things from scratch (extrapolation). I think you believe a fallacy about how human cognition works if you think we actually do something different than interpolation

Then where do new ideas come from? Do the muses bestow divine creativity upon artists, poets, and musicians?

>Then where do new ideas come from? Do the muses bestow divine creativity upon artists, poets, and musicians?

Apparently thats what you believe if you think "creative" ideas just float into brains somehow.

Re: Human mathematicians are being outcounterexampled

#244

Post by 12_throw_away »

somenameforme wrote: post #199
bluGill wrote: post #182

Earlier quoted context omitted.

The more important lesson for those of us are are "old" is to not get stuck in things that were learned in the past when they are found false. If you are not yet old, remember this - it is very likely you will old in a few years.

It's more nuanced because many things are never really found false, or can't realistically be falsified. Planck was implicitly referencing the whole controversy over quantum mechanics, and how exactly to interpret it. None other than Einstein refused, all the way to his death, to accept a probabilistic interpretation in spite of all evidence pointing in that direction. He simply believed on a fundamental level that t…

> None other than Einstein refused, all the way to his death, to accept a probabilistic interpretation in spite of all evidence pointing in that direction. [...] Of course there even remains the possibility that at some point in the future he may even be proven correct

To the extent that decoherence is the currently favored interpretation of quantum mechanics, that's basically now. In that picture, god does not play dice with the universe, rather the observers are themselves the dice.

Re: Human mathematicians are being outcounterexampled

#246

Post by vlovich123 »

evenhash wrote: post #237
vlovich123 wrote: post #89

Earlier quoted context omitted.

Which is why the school should be covering it.

It would be better to just give the students the money and let them spend it as they need. If I were a student living on a measly $1600/month I would be livid if the school gave me a $200 raise in AI credits, instead of money to buy groceries or pay rent.

Cash might count as taxable income whereas this might just be part of your student account that there’s a deal on. In other words this isn’t fungible for a cash benefit - it’s either this or nothing not this or $200 in your pocket.

And while agency in finances is important, if this helps you finish your schooling faster, I suspect you might not care how you get access to a frontier AI model and having 10% of a theoretical $1800/mo carved out isn’t a huge deal.

Re: Human mathematicians are being outcounterexampled

#247

Post by ITVerticals »

Dove wrote: post #15

When I was in grad school, I had the opportunity to take a course from my adviser in which he discussed his current research and some open questions. It was a relatively accessible subject area and the questions were sometimes easy enough that we could meaningfully contribute. On one particular Friday afternoon, he stated a conjecture that he hoped was true, and invited us to try to help him prove or disprove it. It…

[flagged]

Re: Human mathematicians are being outcounterexampled

#248

Post by chowells »

im3w1l wrote: post #175
chowells wrote: post #103

Earlier quoted context omitted.

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.

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.

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?

Re: Human mathematicians are being outcounterexampled

#249

Post by peterfirefly »

zahlman wrote: post #63
angry_octet wrote: post #55

Earlier quoted context omitted.

A recording of a car crash: discovering on live radio/podcast that the central tenet of your book is wrong, and amateurishly so. Naomi Wolf 'death recorded' on BBC[1], skip to 5:51. After this the book was pulped and she had some sort of psychotic break during COVID and allied with ultra-right and COVID denialist loonies. [1] https://www.bbc.com/news/av/world-us-canada-48639663

Is the video available not through a proprietary player? > After this the book was pulped and she had some sort of psychotic break during COVID and allied with ultra-right and COVID denialist loonies. That's quite an extreme shift considering she had previously been a leading figure in third-wave feminism and an OWS activist.

Why? She went from crazy (that you sympathize with) to crazy (that you don't sympathize with). She was always crazy.

Re: Human mathematicians are being outcounterexampled

#250

Post by peterfirefly »

NooneAtAll3 wrote: post #141
somenameforme wrote: post #97

Earlier quoted context omitted.

The term she mistook for meaning a death/execution had been recorded was "death recorded". So it's a mistake that could be made similarly to how one might confidently assume that a "public school" is a school which is a school that is either government funded or open to anybody in the local area. More generally I think people are inappropriately confident in general. When you go through history just about every centu…

...what is public school then?

When rich people started having their kids educated with children of other rich people instead of at home by tutors.
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