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

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.

The best strategy is one that offers the highest compound return consistent with no risk of going broke.

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 ultimate compound return rate is acutely sensitive to fat tails.

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.