We must admit with humility that, while number is purely a product of our minds, space has a reality outside our minds, so that we cannot completely prescribe its properties a priori.

That this subject [of imaginary magnitudes] has hitherto been considered from the wrong point of view and surrounded by a mysterious obscurity, is to be attributed largely to an ill-adapted notation. If, for example, +1, -1, and the square root of -1 had been called direct, inverse and lateral units, instead of positive, negative and imaginary (or even impossible), such an obscurity would have been out of the question.

It is always noteworthy that all those who seriously study this science [the theory of numbers] conceive a sort of passion for it.

Mathematics is the queen of science, and arithmetic the queen of mathematics.

When I have clarified and exhausted a subject, then I turn away from it, in order to go into darkness again.

I protest against the use of infinite magnitude ..., which is never permissible in mathematics.

No contradictions will arise as long as Finite Man does not mistake the infinite for something fixed, as long as he is not led by an acquired habit of mind to regard the infinite as something bounded.