David Hilbert

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We must know. We will know.

David Hilbert (January 23, 1862February 14, 1943) was a German logician, mathematician, and mathematical physicist. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of geometry, as well as the theory of Hilbert spaces, one of the foundations of functional analysis. Hilbert and his students also supplied much of the mathematics needed for quantum mechanics and general relativity.


No one shall expel us from the Paradise that Cantor has created.
We must not believe those, who today, with philosophical bearing and deliberative tone, prophesy the fall of culture and accept the ignorabimus.
One of the supreme achievements of purely intellectual human activity.
  • Every kind of science, if it has only reached a certain degree of maturity, automatically becomes a part of mathematics.
    • "Axiomatic Thought" (1918), printed in From Kant to Hilbert, Vol. 2 by William Bragg Ewald
  • Aus dem Paradies, das Cantor uns geschaffen, soll uns niemand vertreiben können.
    • No one shall expel us from the paradise that Cantor has created.
      • "Über das Unendliche" [On the Infinite] in Mathematische Annalen 95, (1926)
  • Wir dürfen nicht denen glauben, die heute mit philosophischer Miene und überlegenem Tone den Kulturuntergang prophezeien und sich in dem Ignorabimus gefallen. Für uns gibt es kein Ignorabimus, und meiner Meinung nach auch für die Naturwissenschaft überhaupt nicht. Statt des törichten Ignorabimus heiße im Gegenteil unsere Losung:
Wir müssen wissen — wir werden wissen!
  • We must not believe those, who today, with philosophical bearing and deliberative tone, prophesy the fall of culture and accept the ignorabimus. For us there is no ignorabimus, and in my opinion none whatever in natural science. In opposition to the foolish ignorabimus our slogan shall be:
We must know — we will know!
  • The art of doing mathematics consists in finding that special case which contains all the germs of generality.
  • One can measure the importance of a scientific work by the number of earlier publications rendered superfluous by it.
    • Quoted in Mathematical Circles Revisited (1971) by Howard Whitley Eves
  • Mathematics knows no races or geographic boundaries; for mathematics, the cultural world is one country.
    • Quoted in Mathematical Circles Revisited (1971) by Howard Whitley Eves
  • "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.
    • Quoted in Hilbert's Die Grundlagen der Mathematik (1927)
  • 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.
    • Quoted in Comic Sections (1993) by Desmond MacHale
  • If I were to awaken after having slept for a thousand years, my first question would be: Has the Riemann hypothesis been proven?
    • Quoted in Mathematical Mysteries : The Beauty and Magic of Numbers (1999) by Calvin C. Clawson, p. 258
  • One of the supreme achievements of purely intellectual human activity.
    • On the Cantor set, as quoted in A World Without Time : The Forgotten Legacy of Godel and Einstein (2005) by Palle Yourgrau, p. 44
  • 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.
    • Hilbert (2nd edition, 1996) by Constance Reid, p. 92
  • 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.
    • Hilbert-Courant (1984) by Constance Reid, p. 174
  • Keep computations to the lowest level of the multiplication table.
    • Hilbert-Courant (1984) by Constance Reid, p. 104
  • Immer mit den einfachsten Beispielen anfangen.
    • Begin with the simplest examples.
      • Hilbert-Courant (1984) by Constance Reid, p. 104; German version quoted in Algebra by Michael Artin
  • 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.
    • Hilbert-Courant (1984) by Constance Reid, p. 143
  • Good, he did not have enough imagination to become a mathematician.

Mathematical Problems (1900)

"Mathematical Problems" an address to the International Congress of Mathematicians at Paris (1900), translated by Maby Winton Newson for the Bulletin of the American Mathematical Society 8 (1902)
  • History teaches the continuity of the development of science. We know that every age has its own problems, which the following age either solves or casts aside as profitless and replaces by new ones. If we would obtain an idea of the probable development of mathematical knowledge in the immediate future, we must let the unsettled questions pass before our minds and look over the problems which the science of today sets and whose solution we expect from the future. To such a review of problems the present day, lying at the meeting of the centuries, seems to me well adapted. For the close of a great epoch not only invites us to look back into the past but also directs our thoughts to the unknown future.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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 organic unity of mathematics is inherent in the nature of this science, for mathematics is the foundation of all exact knowledge of natural phenomena. That it may completely fulfil this high mission, may the new century bring it gifted masters and many zealous and enthusiastic disciples!
  • 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.
English translation, by Paul Nemenyi, of the 1932 book, Anschauliche Geometrie by David Hilbert and Stephan Cohn-Vossen, based on a series of lectures Hilbert made in the winter of 1920–21
  • 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.
    • Preface
  • [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.
    • Preface
  • [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.
    • Preface
  • 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.
    • Preface


  • Physics is too difficult for physicists!
    • This quote has many variants. An early version attributed to the Göttingen School appears in a book review by Heinrich Wieleitner in Isis, Volume 7, No. 4, December 1925, p. 597: Ach, die Physik! Die ist ja für die Physiker viel zu schwer! (Oh, physics! That's just too difficult for the physicists!).

Quotes about Hilbert

  • More decisive than any other influence for the young Hilbert at Königsberg was his friendship with Adolf Hurwitz and Minkowski. He got his thorough mathematical training less from lectures, teachers or books, than from conversation.
  • David Hilbert—the undisputed, foremost living mathematician in the world and lifelong close friend and collaborator of the by then deceased Minkowski—had already presented to the Göttingen Academy his own version of the same equations a few days earlier [than Einstein]. Although Minkowski and Hilbert accomplished their most important achievements in pure mathematical fields, their respective contributions to relativity should in no sense be seen as merely occasional excursions into the field of theoretical physics. Minkowski and Hilbert were motivated by much more than a desire to apply their exceptional mathematical abilities opportunistically... On the contrary, Minkowski's and Hilbert's contributions to relativity are best understood as an organic part of their overall scientific careers.
    • Leo Corry, on the Relativity priority dispute "Hermann Minkowski and the Postulate of Relativity", in Archive for History of Exact Sciences Vol.51, No.4 (1997)
  • A more thorough study of Euclid's axioms and postulates proved them to be inadequate for the deduction of Euclid's geometry. ...Hilbert and others succeeded in filling the gap by stating explicitly a complete system of postulates for Euclidean and non-Euclidean geometries alike. Among the postulates missing in Euclid's list was the celebrated postulate of Archimedes, according to which, by placing an indefinite number of equal lengths end to end along a line, we should eventually pass any point arbitrarily selected on the line. Hilbert, by denying this postulate, just as Lobatchewski and Riemann had denied Euclid's parallel postulate, succeeded in constructing a new geometry known as non-Archimedean. It was perfectly consistent but much stranger than the classical non-Euclidean varieties. Likewise, it was proved possible to posit a system of postulates which would yield Euclidean or non-Euclidean geometries of any number of dimensions; hence, so far as rational requirements of the mind were concerned, there was no reason to limit geometry to three dimensions.
  • Hilbert's problems have the characteristics of any good founding document. Each one is a short essay on its subject, not overly specific, and yet Hilbert makes his intent remarkably clear. He leaves room for change and adjustment. Hilbert's goal was to foster the pursuit of mathematics.
  • Hilbert was on the verge of retirement. He was the dignified chairman of the mathematical society's meetings, though he no longer came up with those caustic quips that people would repeat afterwards, imitating as best they could his Baltic accent. It is a pity these were not recorded before it was too late. The samples cited in English translation, in Constance Reid's biography of Hilbert, give only the palest idea of his biting wit.
    • André Weil, The Apprenticeship of a Mathematician (1992), p. 51
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