“Take, for instance, numbers: when you count, you count “things,” but “things” have been invented by human beings for their own convenience. This is not obvious on the earth's surface because, owing to the low temperature, there is a certain degree of apparent stability. But it would be obvious if one could live on the sun where there is nothing but perpetually changing whirlwinds of gas. If you lived on the sun, you would never have formed the idea of “things,” and you would never have thought of counting because there would be nothing to count. In such an environment, even Hegel's philosophy would seem to be common sense, and what we consider common sense would appear as fantastic metaphysical speculation.

Such reflections have led me to think of mathematical exactness as a human dream, and not as an attribute of an approximately knowable reality. I used to think that of course there is exact truth about anything, though it may be difficult and perhaps impossible to ascertain it. Suppose, for example, that you have a rod which you know to be about a yard long. In the happy days when I retained my mathematical faith, I should have said that your rod certainly is longer than a yard or shorter than a yard or exactly a yard long. Now I should admit that some rods can be known to be longer than a yard and some can be known to be shorter than a yard, but none can be known to be exactly a yard, and, indeed, the phrase “exactly a yard” has no definite meaning.

Exactness, in fact, was a Hellenic myth which Plato located in heaven. He was right in thinking that it can find no home on earth. To my mathematical soul, which is attuned by nature to the visions of Pythagoras and Plato, this is a sorrow. I try to console myself with the knowledge that mathematics is still the necessary implement for the manipulation of nature. If you want to make a battleship or a bomb, if you want to develop a kind of wheat which will ripen farther north than any previous variety, it is to mathematics that you must turn. Mathematics, which had seemed like a surgeon's knife, is really more like the battle-ax. But it is only in applications to the real world that mathematics has the crudity of the battle-ax. Within its own sphere, it retains the neat exactness of the surgeon's knife.

The world of mathematics and logic remains, in its own domain delightful; but it is the domain of imagination. Mathematics must live, with music and poetry, in the region of man-made beauty, not amid the dust and grime of the world below.”

— Bertrand Russell, Portraits From Memory And Other Essays (1950), Essay: V. Beliefs: Discarded and Retained, pp. 40-1