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

Samuelson spotted a mistake in Bacheliers work. Bachelier's model had failed to consider that stock prices cannot fall below zero.

People tend to be clueless about prices. Contrary to economic theory, we don't really decide between A and B by consulting our invisible price tags and purchasing the one that yields the higher utility, he says. We make do with guesstimates and a vague recollection of what things are “supposed to cost.”

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.

The ultimate compound return rate is acutely sensitive to fat tails.

The more improbable the message, the less "compressible" it is, and the more bandwidth it requires. This is Shannon's point: the essence is its improbability.

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