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

Samuelson, however, hedged his personal bets - by putting some of his own money in Berkshire Hathaway.

The engine driving the Kelly system is the "law of large numbers." In a 1713 treatise on probability, Swiss mathematician Jakob Bernoulli propounded a law that has been misunderstood by gamblers (and investors) ever since.

Shannon's most radical insight was that meaning was irrelevant.

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 story of the Kelly system is a story of secrets - or if you prefer, a story of entropy.

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