Jordan Ellenberg

Edna St. Vincent Millay’s sonnet “Euclid alone has looked on Beauty bare.”

At last I said, “Lincoln, you can never make a lawyer if you do not understand what demonstrate means;” and I left my situation in Springfield, went home to my father’s house, and staid there till I could give any propositions in the six books of Euclid at sight. I then found out what “demonstrate” means, and went back to my law studies.

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. I hear the echoes of Euclid in Lincoln’s most famous speech, 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.

Much of the material was known to mathematicians prior to Euclid’s time, but what’s radically new, and was instantly revolutionary, is the organization of that huge body of knowledge. From a small set of axioms, which were almost impossible to doubt,* one derives step by step the whole apparatus of theorems about triangles, lines, angles, and circles. Before Euclid—if there actually was a Euclid, and not a shadowy collective of geometry-minded Alexandrians writing under that name—such a structure would have been unimaginable. Afterward, it was a model for everything admirable about knowledge and thought.

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.

We encounter non-proofs in proofy clothing all the time, and unless we’ve made ourselves especially attentive, they often get by our defenses. There are tells you can look for. In math, when an author starts a sentence with “Clearly,” what they are really saying is “This seems clear to me and I probably should have checked it, but I got a little confused, so I settled for just asserting that it was clear.” The newspaper pundit’s analogue is the sentence starting “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

what made Lincoln special was that “it was morally impossible for Lincoln to argue dishonestly; he could no more do it than he could steal; it was the same thing to him in essence, to despoil a man of his property by larceny, or by illogical or flagitious reasoning.” What Lincoln had taken from Euclid (or what, already existing in Lincoln, harmonized with what he found in Euclid) was 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. They might have called him Geometrical Abe.

The geometer Henri Poincaré, in a 1905 essay, identifies intuition and logic as the two indispensable pillars of mathematical thought. Every mathematician leans in one direction or the other, and it is the intuition-leaners, Poincaré says, that we tend to call “geometers.”

This notion that the isosceles triangle is a palindrome, that it stays the same when written backward, records something I’ll bet your intuition also tells you—that the triangle is unchanged when you pick it up, flip it over, and lay it back down again in the same spot. Like a palindromic word, it has a symmetry.

It would be more modern in flavor to define an isosceles triangle as a palindromic one: a triangle you can pick up, flip over, and place back down, only to find it unchanged. That such a triangle has two equal sides and two equal angles is just about automatic. In this geometric world, Pappus’s proof would be the means of showing that a triangle with two equal sides was isosceles