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

by Jordan Ellenberg

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, 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.” If they are talking about an economic policy or a culture figure whose behavior they deplore or a relationship concession they want you to make, and not a congruence of triangles, they are not “just being logical,” because they’re operating in a context where logical deduction—if it applies at all—can’t be untangled from everything else. They want you to mistake an assertively expressed chain of opinions as the proof of a theorem. But once you’ve experienced the sharp click of an honest-to-goodness proof, you’ll never fall for this again. Tell your “logical” opponent to go square a circle.