‘Proof' Review: Finding Truth in Numbers

Wall Street Journal

a day ago

Thomas Jefferson's first draft of the Declaration of Independence read: 'We hold these truths to be sacred and undeniable . . . ' It was supposedly Benjamin Franklin who suggested instead announcing the truths to be 'self-evident,' as though they were fundamental mathematical axioms providing an incontestable foundation for the new republic.
The idea of self-evident truths goes all the way back to Euclid's 'Elements' (ca. 300 B.C.), which depends on a handful of axioms—things that must be granted true at the outset, such as that one can draw a straight line between any two points on a plane. From such assumptions Euclid went on to show, for example, that there are infinitely many prime numbers, and that the angles at the base of an isosceles triangle are equal. If the axioms are true, and the subsequent reasoning is sound, then the conclusion is irrefutable. What we now have is a proof: something we can know for sure.
Adam Kucharski, a professor of epidemiology at the London School of Hygiene & Tropical Medicine, takes the reader on a fascinating tour of the history of what has counted as proof. Today, for example, we have computerized proofs by exhaustion, in which machines chew through examples so numerous that they could never be checked by humans. The author sketches the development of ever-more-rarefied mathematics, from calculus to the mind-bending work on different kinds of infinity by the Russian-German sage Georg Cantor, who proved that natural integers (1,2,3 . . . ) are somehow not more numerous than even numbers (2,4,6 . . .), even though the former set includes all the elements of the latter set, in addition to the one that contains all odd numbers. My favorite example is the Banach-Tarski paradox, which proves that you can disassemble a single sphere and reconstitute it into two spheres of identical size. Climbing the ladder of proof, we can enter a wild realm where intuitions break down completely.
But proof, strictly understood, is only half the story here. Abraham Lincoln, Mr. Kucharski relates, taught himself to derive Euclid's proofs to give himself an argumentative edge in the courtroom and in Congress. Yet politics is messier than geometry; and so the dream of perfectly logical policymaking, immune to quibble, remains out of reach. What should we do, then, when a mathematical proof of truth is unavailable, but we must nonetheless act?