Quotations by Stan Ulam

In many cases, mathematics is an escape from reality. The mathematician finds his own monastic niche and happiness in pursuits that are disconnected from external affairs. Some practice it as if using a drug. Chess sometimes plays a similar role. In their unhappiness over the events of this world, some immerse themselves in a kind of self-sufficiency in mathematics. (Some have engaged in it for this reason alone.)
Adventures of a Mathematician (New York 1976).

Any good idea can be stated in fifty words or less.

Knowing what is big and what is small is more important than being able to solve partial differential equations.

The infinite we shall do right away. The finite may take a little longer.
Quoted in D MacHale, Comic Sections (Dublin 1993)

Do not lose your faith. A mighty fortress is our mathematics. Mathematics will rise to the challenge, as it always has.
Quoted in G S Hall Educational Problems

What exactly is mathematics? Many have tried but nobody has really succeeded in defining mathematics; it is always something else. Roughly speaking, people know that it deals with numbers, figures, with relations, operations, and that its formal procedures involving axioms, proofs, lemmas, theorems have not changed since the time of Archimedes.

... there's nothing new under the sun - everything can be traced back to Archimedes or even earlier.

To exist (in mathematics), said Henri Poincaré, is to be free from contradiction. But mere existence does not guarantee survival. To survive in mathematics requires a kind of vitality that cannot be described in purely logical terms.
Mark Kac and Stanislaw Ulam, Mathematics and Logic (1968)

Rota's personality is compatible with mine
Adventures of a Mathematician (New York 1976). See Rota's comment about Ulam