"Outcounterexampled": there's a neologism worthy of German.
Outcounter example: I was playing baseball, and when the outcounter advanced to three, it was the other team's turn to play.
Human mathematicians are being outcounterexampled
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Re: Human mathematicians are being outcounterexampled
#172Earlier quoted context omitted.
vibe counterexamplemaxxing
Better as one word "vibecounterexamplemaxxing"
vibe-counter example: 1, 2, miss a few, 99, 100.
or that viral AI short example of a vibe-counter: Okay, you start with 1, and then I'll say 2, and then you'll say 3, and we'll continue like that. Great idea. I'll start with 1, and then you'll say 2, and then I'll say 3. That's right, I start with 1, and then you say 2...
Re: Human mathematicians are being outcounterexampled
#173I wonder if at some point mathematicians will be over-flooded with proofs to check and eventually some over confident false claim will make it into math. Maybe in the future the work of Mathematicians will be like the ones of SWEs with AI, check thousands of lines of AI generated proof and find the subtle errors
That's why the author refused to read a proof from someone he knew until it was formalized in Lean.
Re: Human mathematicians are being outcounterexampled
#174Re: Human mathematicians are being outcounterexampled
#175Earlier quoted context omitted.
You’re smuggling in a frame here which isn’t obviously true: that mathematics is not just a collection of facts
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.
Re: Human mathematicians are being outcounterexampled
#176When 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…
This is probably part of why machines are doing so well at counterexamples. They have no aesthetic commitment to the conjecture and no embarrassment about producing something ugly
That's not why.
It's because counterexamples are easy compared to proofs which require new mathematics.
GenAI is great at combining existing things in new ways (interpolation). It's terrible at creating new things from scratch (extrapolation).
Re: Human mathematicians are being outcounterexampled
#177How 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?
I am sure finding holes in existing proofs is what counts as "another little annoying detail". Some people are probably relieved when they have failed to prove a hypothesis and someone finds a counterexample.
Re: Human mathematicians are being outcounterexampled
#178Earlier quoted context omitted.
The other Naomi, Naomi Klein, wrote an entire book that uses Wolf's "transformation" as a structural scaffold to explore the rise of the new right/MAGA etc. etc. It's called "Doppelganger".
Which electronics youtuber Naomi got disappeared in China for showing bad parts of china? Was that a third Naomi?
Naomi Wu aka Sexy Cyborg - she did a lot of electronics work, and showcasing tech
Re: Human mathematicians are being outcounterexampled
#179Earlier quoted context omitted.
They're trained on human data. I would expect them to emulate human biases as closely as possible.
It's more complex than that, especially as post training is often goal based.
I wouldn't have expected that there was post training specifically on the issue of looking for proofs vs counter examples. But it might be that other post training has a side effect of making AIs better at looking for counter examples. I wonder if these agents are overall less biased and more rational than humans. Can you expand on what you mean by goal based training?
Re: Human mathematicians are being outcounterexampled
#180Earlier quoted context omitted.
They're trained on human data. I would expect them to emulate human biases as closely as possible.
Your comment stopped me in my tracks a little bit. Is a 'bias' in a piece of writing generally a property of word to word choice and sentence to sentence construction or is it something more nebulous? Especially in terms of the appreciation of mathematics and someone's hesitance about publishing a mathematical argument they think is ugly or brute forced in some way.
> Is a 'bias' in a piece of writing generally a property of word to word choice and sentence to sentence construction or is it something more nebulous?
You might be fascinated when you read the story of Golden Gate Claude: https://www.anthropic.com/news/golden-gate-claude