Mathematics has a threefold purpose. It must provide an instrument for the study of nature. But this is not all: it has a philosophical purpose, and, I daresay, an aesthetic purpose.

It is with logic that one proves; it is with intuition that one invents.

Later generations will regard Mengenlehre (set theory) as a disease from which one has recovered.

It is through science that we prove, but through intuition that we discover.

. . . by natural selection our mind has adapted itself to the conditions of the external world. It has adopted the geometry most advantageous to the species or, in other words, the most convenient. Geometry is not true, it is advantageous.

All of mathematics is a tale about groups.

One does not ask whether a scientific theory is true, but only whether it is convenient.