underscored

@underscored

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In science, as well as in other fields of human endeavor, there are two kinds of geniuses: the "ordinary" and the "magicians." An ordinary genius is a [person] that you and I would be just as good as, if we were only many times better. There is no mystery as to how his mind works. Once we understand what he has done, we feel certain that we, too, could have done it. It is different with the magicians. They are, to use mathematical jargon, in the orthogonal complement of where we are and the working of their minds is for all intents and purposes incomprehensible. Even after we understand what they have done, the process by which they have done it is completely dark.

— Mark Kac
3w ago

Pulsating beneath the story is a reminder that whether the poem is made of words or of mathematics, it is only ever made in the mind — that strange and solitary place where ideas are born like new stars out of the fertile cosmic darkness, molecules of thought pulled together by the gravity of feeling to become light.

1mo ago

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.

4mo ago

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.

6mo ago

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.

6mo ago

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.

7mo ago

Underscored — save the words that stop you in your tracks.

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