At some time in the history of the universe, there were no human minds, and at some time later, there were. Within the blink of a cosmic eye, a universe in which all was chaos and void came to include hunches, beliefs, sentiments, raw sensations, pains, emotions, wishes, ideas, images, inferences, the feel of rubber, Schadenfreude, and the taste of banana ice cream.

Arithmetic is where the content lies, and not logic; but logic prompts certainty, and not arithmetic.

However good an argument in philosophy may happen to be, it is generally not good enough.

I do not know whether any of this is true. I am certain that the scientific community does not know that it is false.

The world of shapes, lines, curves, and solids is as varied as the world of numbers, and it is only our long-satisfied possession of Euclidean geometry that offers us the impression, or the illusion, that it has, that world, already been encompassed in a manageable intellectual structure. The lineaments of that structure are well known: as in the rest of life, something is given and something is gotten; but the logic behind those lineaments is apt to pass unnoticed, and it is the logic that controls the system.

No man is obliged to be what he might have been.

Aristotelian logic is massive and marmoreal, but every monument accumulates graffiti.