A quotes list created by Lee Sonogan
Defender of transfinite numbers his work (1862-1943) helped in our current understanding of the axiomatization of geometry and formalism we use today. Also solving Gordan’s Problem with his finiteness theorem, notable students mentored by him were Edward Kasner and John von Neumann. While his main influence was Immanuel Kant leading an example before him.
- “We must know. We will know.”
- “The infinite! No other question has ever moved so profoundly the spirit of man.”
- “Sometimes it happens that a man’s circle of horizon becomes smaller and smaller, and as the radius approaches zero it concentrates on one point. And then that becomes his point of view.”
- “Mathematics knows no races or geographical boundaries; for mathematics, the cultural world is one country.”
- “If I were to awaken after having slept for a thousand years, my first question would be: has the Riemann Hypothesis been proven?”
- Keep computations to the lowest level of the multiplication table.
- If we do not succeed in solving a mathematical problem, the reason frequently consists in our failure to recognize the more general standpoint from which the problem before us appears only as a single link in a chain of related problems. After finding this standpoint, not only is this problem frequently more accessible to our investigation, but at the same time we come into possession of a method which is applicable also to related problems.
- To new concepts correspond, necessarily, new signs. These we choose in such a way that they remind us of the phenomena which were the occasion for the formation of the new concepts.
- “The art of doing mathematics consists in finding that special case which contains all the germs of generality.”
- “But above all I wish to designate the following as the most important among the numerous questions which can be asked with regard to the axioms: To prove that they are not contradictory, that is, that a definite number of logical steps based upon them can never lead to contradictory results.”
- Every kind of science, if it has only reached a certain degree of maturity, automatically becomes a part of mathematics.
- One can measure the importance of a scientific work by the number of earlier publications rendered superfluous by it.
- “We hear within us the perpetual call: There is the problem. Seek its solution.”
- In mathematics, as in any scientific research, we find two tendencies… [T]he tendency toward abstraction seeks to crystallize the logical relations inherent in the maze of material in a systematic and orderly manner. On the other hand, the tendency toward intuitive understanding fosters a more immediate grasp of the objects… a live rapport with them… which stresses the concrete meaning of their relations. …[I]ntuitive understanding plays a major role in geometry. …[S]uch concrete intuition is of great value not only for the research worker, but… for anyone who wishes to study and appreciate the results of research in geometry.
- [O]ur purpose is to give a presentation of geometry… in its visual, intuitive aspects. With the aid of visual imagination we can illuminate the manifold facts and problems… beyond this, it is possible… to depict the geometric outline of the methods of investigation and proof, without… entering into the details… In this manner, geometry being as many-faceted as it is and being related to the most diverse branches of mathematics, we may even obtain a summarizing survey of mathematics as a whole, and a valid idea of the variety of problems and the wealth of ideas it contains. Thus a presentation of geometry in large brushstrokes… and based on the approach through visual intuition, should contribute to a more just appreciation of mathematics by a wider range of people than just the specialists.
- [M]athematics is not a popular subject… The reason for this is to be found in the common superstition that [it] is but a continuation… of the fine art of arithmetic, of juggling with numbers. [We] 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[s] about a greater enjoyment of mathematics, by making it easier… to penetrate the essence of mathematics without… a laborious course of studies.
- The various branches of geometry are all interrelated closely and quite often unexpectedly. This shows up in many places in the book. Even so… it was necessary to make each chapter…self-contained… We hope that… we have rendered each chapter taken by itself… understandable and interesting. We want to take the reader on a leisurely walk… in the big garden that is geometry, so that each may pick for himself a bouquet to his liking.
More fun notes include his own version called Hilbert Space. Sharing a theorem with Albert Einstein himself defining more on general relativity. And finally a more technical origin behind Episilion calculus.