The Cost of Knowing

The sentence Pierre de Fermat scrawled in a margin around 1637 took a moment to write. The knowledge it gestured toward took three hundred and fifty-eight years of the hardest mathematical labor humanity has ever expended to produce. That ratio—between the cost of asserting and the cost of justifying—is one of the most eloquent figures in the history of thought, and it is the figure that the age of AI has made urgent at planetary scale. A large language model can produce in a second what would take a person a day; it can generate a thousand plausible answers while a human is still formulating the question. The great temptation of this abundance is to mistake the cheap, fast claim for the expensive, slow thing it resembles—to treat the conjecture, produced instantly, as though it carried the warrant that only proof, produced laboriously, can supply. Fermat’s margin is the standing rebuke to that