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

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 assumption that anything true is knowable is the grandfather of paradoxes.

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

There is a deep connection between Bernoulli's dictum and John Kelly's 1956 publication. It turns out that Kelly's prescription can be restated as this simple rule: When faced with a choice of wagers or investments, choose the one with the highest geometric means of outcomes.

Your second ducat, like your second million, is never quite as sweet.

Bernoulli's real contribution was to coin a word. The word has been translated into English as "utility". It describes this subjective value people place on money.