cevi

Finitism doesn't escape anything, it just gives you the illusion of safety. Any intellectually honest thinker should accept the possibility that 10 is a nonstandardly large number.

Mochizuki's claimed proof of the abc conjecture was extremely unusual for the reason that nobody was able to extract a single useful idea from the argument. I was starting grad school when it came out, and my immediate visceral response was "if this is what number theory is going to look like in the future, then I will leave mathematics."

The current wave of AI slop mathematics might end up driving the next generation of mathematicians away from the subject for the same reason that Mochizuki would have convinced me to quit if his proof had been accepted by the community. Luckily, my professors had the taste to immediately recognize that it was garbage.

I grew up in a cult. Based on my experience, I believe that the most dangerous thing a human can do is to allow someone else to do their thinking for them.

Nothing makes me respect Terry Tao more than the line "I hate Jean Bourgain" handwritten into the margin of one of Bourgain's papers. IYKYK

When writing math papers, many (but unfortunately not all) mathematicians go through a post-processing step, where they take their ideas and proofs, and try to reduce them to simple and reusable core ideas that can be understood by the reader. Good writers will often also provide some representative examples that guided the proofs, explaining why various intermediate results can't be strengthened and why the proof can't be made much shorter without inventing new techniques. If AI-generated proofs were required to go through such a post-processing step before being published, that would go a long way towards improving the situation.

I saw it as a sort of science-fiction - imagine living in a world where the smartest intellectuals all struggled to solve basic exercises about graph theory. Really imagine living in such a world - would you not feel frustrated when you tried to explain this basic concept to these supposed experts and they just didn't get it? The main character must have felt like he was going crazy!

Are you also uncomfortable with the idea of flipping 256 unbiased coins independently?

There is no general procedure for computing upper bounds on busy beaver numbers (this can be proven). We haven't even come close to enumerating all of the interesting six-state Turing machines, so right now we don't even have a wild guess for an upper bound on BB(6).

I've only skimmed the paper, but this looks very nice: the construction is very simple (aside from the precise choices of the parameters), just the analysis to show that it works is difficult.

(I bet the construction can be refined - it feels like there is a semidefinite programming problem lurking in the background, so there is probably a way to mindlessly optimize things with an SDP solver once the proof technique is rephrased a bit.)

The consistency of ZFC is (presumably) a theorem of second order PA, and ZFC is unable to prove it (unless ZFC is inconsistent).