"It remains to discuss briefly what general requirements may be justly laid down for the solution of a mathematical problem. I should say first of all, this: that it shall be possible to establish the correctness of the solution by means of a finite number of steps based upon a finite number of hypotheses which are implied in the statement of the problem and which must always be exactly formulated. This requirement of logical deduction by means of a finite number of processes is simply the requirement of rigor in reasoning."

“Every kind of science, if it has only reached a certain degree of maturity, automatically becomes a part of mathematics.”

“Good, he did not have enough imagination to become a mathematician.”

“athematics is not a popular subject... The reason for this is to be found in the common superstition that is but a continuation... of the fine art of arithmetic, of juggling with numbers. combat that superstition, by offering, instead of formulas, figures that may be looked at and that may easily be supplemented by models which the reader may construct. This book... bring about a greater enjoyment of mathematics, by making it easier... to penetrate the essence of mathematics without... a laborious course of studies.”

“A mathematical problem should be difficult in order to entice us, yet not completely inaccessible, lest it mock at our efforts. It should be to us a guide post on the mazy paths to hidden truths, and ultimately a reminder of our pleasure in the successful solution.”

“Mathematical science is in my opinion an indivisible whole, an organism whose vitality is conditioned upon the connection of its parts. For with all the variety of mathematical knowledge, we are still clearly conscious of the similarity of the logical devices, the relationship of the ideas in mathematics as a whole and the numerous analogies in its different departments. We also notice that, the farther a mathematical theory is developed, the more harmoniously and uniformly does its construction proceed, and unsuspected relations are disclosed between hitherto separate branches of the science. So it happens that, with the extension of mathematics, its organic character is not lost but only manifests itself the more clearly.”

“One of the supreme achievements of purely intellectual human activity.”