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Human mathematicians are being outcounterexampled

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Re: Human mathematicians are being outcounterexampled

#71

Post by vlovich123 »

> A few days earlier I had got an email from a professor in the maths department here at Imperial, expressing surprise that some of our graduate students were paying $200 per month to access models such as Sol and Fable. He said that he thought that these people were crazy. I did not immediately respond. But after meeting with Andrew I emailed the professor back and told him that in my opinion, any PhD student who was not paying $200 per month to access these tools was crazy. In fact during the workshop I learnt from Harvard PhD student Bryan Wang that Harvard were already giving free Fable access to all PhD students, post-docs and faculty at Harvard.

Yeah, given how much it accelerates grad students to produce meaningful output more quickly, why wouldn’t you make an investment of $2400/student/year. Seems like pennies overall.

Re: Human mathematicians are being outcounterexampled

#72

Post by ATMLOTTOBEER »

chowells wrote: post #41
taneq wrote: post #22

Earlier quoted context omitted.

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.

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.…

You’re smuggling in a frame here which isn’t obviously true: that mathematics is not just a collection of facts

Re: Human mathematicians are being outcounterexampled

#73

Post by fatcatsbestcats »

VladVladikoff wrote: post #17

A lot of this math is beyond my comprehension, but it often seems to talk of proofs of theorems. What I want to know is if we continue on this accelerated AI mathematics trajectory, will we eventually be discovering new forms of math that will in turn have some applications down the line in engineering or biomedicine etc? I guess what I’m asking is are we on the cusp of a huge breakthrough for humanity, or largely ju…

It’s possible. Compressed sensing is one example of what I suppose you could call a new form of math with applications in biomedicine. It can be used to significantly shorten the time required to obtain a MRI scan, which can improve the patient experience and enable more patients to access MRIs.

https://en.wikipedia.org/wiki/Compressed_sensing

Re: Human mathematicians are being outcounterexampled

#74

Post by bee_rider »

How do mathematicians view counter examples? Is it like an unexpected result in the physical sciences: annoying in the moment but potentially stupendously important as it reveals some inaccuracy in the current models? Or is it more like a bug report in coding… probably just, another little annoying detail?

Re: Human mathematicians are being outcounterexampled

#75

Post by AndrewKemendo »

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.

I couldn’t get the video to work. Here’s the relevant clip I found on youtube.

https://youtu.be/3uRCcEoGWxs

Re: Human mathematicians are being outcounterexampled

#76

Post by madhadron »

bee_rider wrote: post #74

How do mathematicians view counter examples? Is it like an unexpected result in the physical sciences: annoying in the moment but potentially stupendously important as it reveals some inaccuracy in the current models? Or is it more like a bug report in coding… probably just, another little annoying detail?

Counterexamples are clarifying. For everyone condition in a proof, it's really handy to have a maximally simple, memorable counterexample that makes it fail because it violates that condition. Mathematicians tend to walk around with a bestiary of counterexamples in their heads. It also makes it really easy to recover a theorem because you try to sketch out the statement, and the spiky, memorable counterexamples jump out of your memory and you add conditions to constrain the domain away from them.

Re: Human mathematicians are being outcounterexampled

#77

Post by nopinsight »

SpicyLemonZest wrote: post #49
mb7733 wrote: post #43

Earlier quoted context omitted.

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.

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.

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.

Re: Human mathematicians are being outcounterexampled

#79

Post by [deleted] »

mb7733 wrote: post #43
SpicyLemonZest wrote: post #32

Earlier quoted context omitted.

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 h…

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.

[deleted]

Re: Human mathematicians are being outcounterexampled

#80

Post by adw »

vlovich123 wrote: post #71

> A few days earlier I had got an email from a professor in the maths department here at Imperial, expressing surprise that some of our graduate students were paying $200 per month to access models such as Sol and Fable. He said that he thought that these people were crazy. I did not immediately respond. But after meeting with Andrew I emailed the professor back and told him that in my opinion, any PhD student who wa…

The living-costs stipend for an EPSRC PhD student is around £20k so it's about ten percent of that... big commitment for a student to make!
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