The further a mathematical theory is developed, the more harmoniously and uniformly does its construction proceed, and unsuspected relations are disclosed between hitherto separated branches of the science.

Mathematical science is in my opinion an indivisible whole, an organism whose vitality is conditioned upon the connection of its parts.

If I were to awaken after having slept for a thousand years, my first question would be: Has the Riemann hypothesis been proven?

I do not want to presuppose anything as known. I see in my explanation in section 1 the definition of the concepts point, straight line and plane, if one adds to these all the axioms of groups i-v as characteristics. If one is looking for other definitions of point, perhaps by means of paraphrase in terms of extensionless, etc., then, of course, I would most decidedly have to oppose such an enterprise. One is then looking for something that can never be found, for there is nothing there, and everything gets lost, becomes confused and vague, and degenerates into a game of hide and seek.

In mathematics ... we find two tendencies present. On the one hand, the tendency towards abstraction seeks to crystallise the logical relations inherent in the maze of materials ... being studied, and to correlate the material in a systematic and orderly manner. On the other hand, the tendency towards intuitive understanding fosters a more immediate grasp of the objects one studies, a live rapport with them, so to speak, which stresses the concrete meaning of their relations.

The arithmetical symbols are written diagrams and the geometrical figures are graphic formulas.

I have tried to avoid long numerical computations, thereby following Riemann's postulate that proofs should be given through ideas and not voluminous computations.