Shape: The Hidden Geometry of Information, Biology, Strategy, Democracy, and EverythingElse

by Jordan Ellenberg

Euclid’s Elements

“No man can talk well unless he is able first of all to define to himself what he is talking about.

What Lincoln took from Euclid was the idea that, if you were careful, you could erect a tall, rock-solid building of belief and agreement by rigorous deductive steps, story by story, on a foundation of axioms no one could doubt: or, if you like, truths one holds to be self-evident. Whoever doesn’t hold those truths to be self-evident is excluded from discussion.

the Gettysburg Address, where he characterizes the United States as “dedicated to the proposition that all men are created equal.” A “proposition” is the term Euclid uses for a fact that follows logically from the self-evident axioms, one you simply cannot rationally deny.

have given up newspapers in exchange for Tacitus and Thucydides, for Newton and Euclid; and I find myself much the happier.

The paradox of education: what we most admire we put in a box and make dull.

The point of geometry in high school, they said, was to train up the student’s mind in the habits of strict deductive reasoning.

The ultimate reason for teaching kids to write a proof is not that the world is full of proofs. It’s that the world is full of non-proofs, and grown-ups need to know the difference.

“Surely, we can all agree.” Whenever you see this, you should at all costs not be sure that all agree on what follows. You are being asked to treat something as an axiom, and if there’s one thing we can learn from the history of geometry, it’s that you shouldn’t admit a new axiom into your book until it really proves its worth.

Always be skeptical when someone tells you they’re “just being logical.”

integrity, the principle that one does not say a thing unless one has justified, fair and square, that one has the right to say it. Geometry is a form of honesty.

The geometer Henri Poincaré, in a 1905 essay, identifies intuition and logic as the two indispensable pillars of mathematical thought.