Euclid · Elements
What does it mean to prove something?
That time, that place
Alexandria had its Museum and its Library, built by the Ptolemies on the ambition of gathering every book in the world. Egyptian surveying, Babylonian calculation, and Greek geometry piled up in one place.
Why this question
The trouble was not that knowledge was scarce but that it was tangled. No one could say whether this fact followed from that one or simply stood beside it. Euclid's question was: what must we grant at the start, so that everything else can be pulled out of it?
What was found
He starts from very little: a handful of definitions saying what a point and a line are, and five rules that look obvious — such as that only one straight line passes through two points.
From these he draws out 465 propositions in order, each proved using only what came before. Grant the first five and you have no way left to refuse the rest. That is what proof is. Not persuading someone, but leaving them nowhere to stand.
Mathematics was a collection of useful procedures
The thirteen books of the Elements — a textbook for over two thousand years
Only the fifth rule never looked quite obvious. Two millennia later, those who tried discarding it opened an entirely different geometry