A reality completely independent of the spirit that conceives it, sees it, or feels it, is an impossibility. A world so external as that, even if it existed, would be forever inaccessible to us.

If nature were not beautiful, it would not be worth knowing, and if nature were not worth knowing, life would not be worth living

What is it indeed that gives us the feeling of elegance in a solution, in a demonstration?

Hypotheses are what we lack the least.

So is not mathematical analysis then not just a vain game of the mind? To the physicist it can only give a convenient language; but isn't that a mediocre service, which after all we could have done without; and, it is not even to be feared that this artificial language be a veil, interposed between reality and the physicist's eye? Far from that, without this language most of the initimate analogies of things would forever have remained unknown to us; and we would never have had knowledge of the internal harmony of the world, which is, as we shall see, the only true objective reality.

Mathematicians do not study objects, but the relations between objects.

Ideas rose in clouds; I felt them collide until pairs interlocked, so to speak, making a stable combination.