At a bare minimum, understanding entails being able to detect an internal contradiction: a paradox.

Expectation is a statistical fiction, like having 2.5 children.

A bit is worth 10,000 basis points.

"Average" isn't so hot at the race track given those steep track takes. "Average" is pretty decent for stocks, something like 6 percent above the inflation rate. For a buy-and -hold investor, commissions and taxes are small.

Carl Friedrich Gauss, often rated the greatest mathematician of all time, played the market. On a salary of 1,000 thalers a year, Euler left an estate of 170,587 thalers in cash and securities. Nothing is known of Gauss's investment methods.

Paradox is thus a much deeper and universal concept than the ancients would have dreamed. Rather than an oddity, it is a mainstay of the philosophy of science.

The best paradoxes raise questions about what kinds of contradictions can occur-what species of impossibilities are possible.