David Hilbert Quotes

"Mathematics is a presuppositionless science. To found it I do not need God, as does Kronecker, or the assumption of a special faculty of our understanding attuned to the principle of mathematical induction, as does Poincaré, or the primal intuition of Brouwer, or, finally, as do Russell and Whitehead, axioms of infinity, reducibility, or completeness, which in fact are actual, contentual assumptions that cannot be compensated for by consistency proofs.

If one were to bring ten of the wisest men in the world together and ask them what was the most stupid thing in existence, they would not be able to discover anything so stupid as astrology.

Immer mit den einfachsten Beispielen anfangen.

Begin with the simplest examples.

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.

I do not see that the sex of the candidate is an argument against her admission as a Privatdozent. After all, the Senate is not a bath-house.

An old French mathematician said: A mathematical theory is not to be considered complete until you have made it so clear that you can explain it to the first man whom you meet on the street. This clearness and ease of comprehension, here insisted on for a mathematical theory, I should still more demand for a mathematical problem if it is to be perfect; for what is clear and easily comprehended attracts, the complicated repels us.

Eine mathematische Theorie ist nicht eher als vollkommen anzusehen, als bis du sie so klar gemacht hast, daß du sie dem ersten Manne erklären könntest, den du auf der Straße triffst.

Physics is too difficult for physicists!

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.

[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.

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.

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.

But he (Galileo) was not an idiot,... Only an idiot could believe that scientific truth needs martyrdom — that may be necessary in religion, but scientific results prove themselves in time.

This conviction of the solvability of every mathematical problem is a powerful incentive to the worker. We hear within us the perpetual call: There is the problem. Seek its solution. You can find it by pure reason, for in mathematics there is no ignorabimus.