For the ceiling, he showed the count can never exceed n to the power 1 plus 1 divided by 3, about n^1.333. For the floor, he exhibited explicit point sets where the count reaches at least n to the power 1 plus c divided by log log n, a formula that grows faster than n but only just barely. The chasm between these two bounds — stretching across eight decades and defeating every attempt to close it — is exactly what an AI system cracked open in 2025.
But the burden of proof, he cautions, remains with the infinity doubters. If you could somehow prove experimentally that the physical universe is indeed finite, even the most ardent backers of the higher infinite would likely take a moment to pause and reflect.
Those early adopters found, to their surprise, not only that the models were good at puzzles, but that they could help break genuinely new ground. Soon, mathematicians were using AI to discover and prove new results, accomplishing in a day what would have once taken them weeks or months.
An irrational number — a number that cannot be expressed as a fraction, the ratio between two whole numbers — π unmoors our basic intuitions about reality with its disquieting whisper of an infinity beyond the grasp of reason.
4mo ago
Underscored — save the words that stop you in your tracks.
