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David Hilbert
(23 Jan 1862 - 14 Feb 1943)
German mathematician who presented the first complete set of axioms since Euclid, in his book, Foundations of Geometry. In his work, he made notable contributions to the formalistic foundations of mathematics.
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[Cantor’s set theory:] The finest product of mathematical genius and one of the supreme achievements of purely intellectual human activity.
— David Hilbert
As quoted in Constance Reid, Hilbert (1970), 176.
Meine Herren, der Senat ist doch keine Badeanstalt.
The faculty is not a pool changing room.
Indignant reply to the blatent sex discrimination expressed in a colleague’s opposition when Hilbert proposed appointing Emmy Noether as the first woman professor at their university.
The faculty is not a pool changing room.
Indignant reply to the blatent sex discrimination expressed in a colleague’s opposition when Hilbert proposed appointing Emmy Noether as the first woman professor at their university.
— David Hilbert
Quoted in A L Mackay, Dictionary of Scientific Quotations (1994).
Wir mussen wissen. Wir werden wissen.
We must know. We will know.
We must know. We will know.
— David Hilbert
Engraved on his tombstone in Göttingen. Lecture at Konigsberg, 1930. Gesammelte Abhandlungen, Vol. 3, 387, trans. Ivor Grattan-Guinness.
~~[No known source]~~ Every kind of science, if it has only reached a certain degree of maturity, automatically becomes a part of mathematics.
Eine jede Wissenschaft fällt, hat sie erst eine gewisse Reife erreicht, automatisch der Mathematik anheim.
Eine jede Wissenschaft fällt, hat sie erst eine gewisse Reife erreicht, automatisch der Mathematik anheim.
— David Hilbert
Webmaster has so far found no source for these verbatim words. (Can you help?) Expressed in totally different words, Hilbert expresses a similar idea in Address (11 Sep 1917), 'Axiomatisches Denken' delivered before the Swiss Mathematical Society in Zürich. See the quote that begins, “Anything at all that can be the object of scientific thought …”, on the David Hilbert Quotes page on this website.
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.
— David Hilbert
In Mathematical Problems', Bulletin American Mathematical Society, 8, 438.
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.
— David Hilbert
…...
After seeking in vain for the construction of a perpetual motion machine, the relations were investigated which must subsist between the forces of nature if such a machine is to be impossible; and this inverted question led to the discovery of the law of the conservation of energy, which, again, explained the impossibility of perpetual motion in the sense originally intended.
— David Hilbert
Opening of Lecture (1900), 'Mathematische Probleme' (Mathematical Problems), to the International Congress of Mathematicians, Paris. From the original German reprinted in David Hilbert: Gesammelte Abhandlungen (Collected Treatises, 1970), Vol. 3. For full citation, see the quote that begins, “This conviction of the solvability…”, on the David Hilbert Quotes page on this website.
Anything at all that can be the object of scientific thought becomes dependent on the axiomatic method, and thereby indirectly on mathematics, as soon as it is ripe for the formation of a theory. By pushing ahead to ever deeper layers of axioms … we become ever more conscious of the unity of our knowledge. In the sign of the axiomatic method, mathematics is summoned to a leading role in science.
— David Hilbert
Address (11 Sep 1917), 'Axiomatisches Denken' delivered before the Swiss Mathematical Society in
Zürich. Translated by Ewald as 'Axiomatic Thought', (1918), in William Bragg Ewald, From Kant to Hilbert (1996), Vol. 2, 1115.
As long as a branch of science offers an abundance of problems, so long it is alive; a lack of problems foreshadows extinction or the cessation of independent development.
— David Hilbert
In 'Mathematical Problems', Bulletin American Mathematical Society, 8, 438.
Before beginning I should put in three years of intensive study, and I haven’t that much time to squander on a probable failure.
— David Hilbert
Answering (1920) why he did not attempt a proof of Fermat's last theorem. As quoted, without citation, by Eric Temple Bell, Mathematics, Queen and Servant of Science (1951, 1961), 238. Collected in 'The Queen of Mathematics', The World of Mathematics (1956), Vol. 1, 510.
Besides it is an error to believe that rigour is the enemy of simplicity. On the contrary we find it confirmed by numerous examples that the rigorous method is at the same time the simpler and the more easily comprehended. The very effort for rigor forces us to find out simpler methods of proof.
— David Hilbert
'Mathematical Problems', Bulletin of the American Mathematical Society (Jul 1902), 8, 441.
Every mathematical discipline goes through three periods of development: the naive, the formal, and the critical.
— David Hilbert
Quoted in R Remmert, Theory of complex functions (New York, 1989).
From time immemorial, the infinite has stirred men's emotions more than any other question. Hardly any other idea has stimulated the mind so fruitfully. Yet, no other concept needs clarification more than it does.
— David Hilbert
In address (4 Jun 1925), at a congress of the Westphalian Mathematical Society in Munster, in honor of Karl Weierstrass. First published in Mathematische Annalen (1926), 95, 161-190. Translated by Erna Putnam and Gerald J. Massey as 'On the Infinite', collected in Paul Benacerraf (ed.) Philosophy of Mathematics: Selected Readings (1983), 185. Compare another translation elsewhere on this page, beginning, “The Infinite!…”.
Galileo was no idiot. Only an idiot could believe that science requires martyrdom—that may be necessary in religion, but in time a scientific result will establish itself.
— David Hilbert
As quoted, without citation, in Harold Eves, Mathematical Circles Squared (1971). Collected in Bill Swainson, The Encarta Book of Quotations (2000), 361.
Geometry is the most complete science.
— David Hilbert
Quotation noted by Ivor Grattan-Guinness from unpublished Hilbert lecture course on Geometry.
He who seeks for methods without having a definite problem in mind seeks for the most part in vain.
— David Hilbert
'Mathematical Problems', Bulletin of the American Mathematical Society (Jul 1902), 8, 444.
I have tried to avoid long numerical computations, thereby following Riemann’s postulate that proofs should be given through ideas and not voluminous computations.
— David Hilbert
In Report on Number Theory (1897). As given in epigraph, without citation, in Eberhard Zeidler and Juergen Quandt (trans.), Nonlinear Functional Analysis and its Applications: IV: Applications to Mathematical Physics (2013), 448.
If I were to awaken after having slept for a thousand years, my first question would be: Has the Riemann hypothesis been proven?
— David Hilbert
…...
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.
— David Hilbert
Quoted in D MacHale, Comic Sections (1993)
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
— David Hilbert
Geometry and the imagination (New York, 1952).
In mathematics there is no ignorabimus!
— David Hilbert
This is part of a longer quote that begins, “This conviction of the solvability…”, which has the full citation. See the David Hilbert Quotes page on this website. Note that ignorabimus (first-person plural future active indicative of the Latin verb ignoro) refers to the future: “we will not know” or “we will not be ignorant of”. Compare ignoramus, (first-person plural present active indicative of ignoro) meaning in the present, “we do not know” or “we are ignorant of”.
In order to comprehend and fully control arithmetical concepts and methods of proof, a high degree of abstraction is necessary, and this condition has at times been charged against arithmetic as a fault. I am of the opinion that all other fields of knowledge require at least an equally high degree of abstraction as mathematics,—provided, that in these fields the foundations are also everywhere examined with the rigour and completeness which is actually necessary.
— David Hilbert
In 'Die Theorie der algebraischen Zahlkorper', Vorwort, Jahresbericht der Deutschen Mathematiker Vereinigung, Bd. 4.
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.
— David Hilbert
In 'Mathematical Problems', Bulletin American Mathematical Society, 8, 478.
Mathematics is a game played according to certain simple rules with meaningless marks on paper.
— David Hilbert
Given as narrative, without quotation marks, in Eric Temple Bell, Mathematics, Queen and Servant of Science (1951, 1961), 21.
Mathematics is that peculiar science in which the importance of a work can be measured by the number of earlier publications rendered superfluous by it.
— David Hilbert
As stated in narrative, without quotation marks, in Joong Fang, Bourbaki (1970), 18, citing “as Hilbert declared at the end of his famous paper on the twenty-three unsolved problems.” Webmaster has not identified this in that paper, however. Also quoted, without citation, in Harold Eves, Mathematical Circles Revisited (1971), as “One can measure the importance of a scientific work by the number of earlier publications rendered superfluous by it.”
Mathematics knows no races or geographic boundaries; for mathematics, the cultural world is one country.
— David Hilbert
In H. Eves, Mathematical Circles Squared (1972). As cited in Anton Zettl, Sturm-Liouville Theory (2005), 171.
Meine Herren, 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 bathhouse.
Objecting to sex discrimination being the reason for rejection of Emmy Noether's application to join the faculty at the University of Gottingen.
Objecting to sex discrimination being the reason for rejection of Emmy Noether's application to join the faculty at the University of Gottingen.
— David Hilbert
Quoted in C. Reid Hilbert: With an appreciation of Hilbert's Mathematical Work by Hermann Weyl (1970), 143.
No one shall expel us from the paradise which Cantor has created for us.
Expressing the importance of Cantor's set theory in the development of mathematics.
Expressing the importance of Cantor's set theory in the development of mathematics.
— David Hilbert
In George Edward Martin, The Foundations of Geometry and the Non-Euclidean Plane (1982), 33.
One hears a lot of talk about the hostility between scientists and engineers. I don't believe in any such thing. In fact I am quite certain it is untrue... There cannot possibly be anything in it because neither side has anything to do with the other.
— David Hilbert
Quoted in A. Rosenfeld, Langmuir: The Man and the Scientist (1962), 57.
Physics is becoming too difficult for the physicists.
— David Hilbert
As quoted in Constance Reid, Hilbert (1970), 127.
Physics is much too hard for physicists.
— David Hilbert
As quoted in Constance Reid, Hilbert (2012), 127. Reid explains Hilbert’s goal of “the axiomatization of physics. … But this project required a mathematician.” Hilbert envisioned “A few fundamental physical phenomena should be set up as the axioms from which all observable data could then be derived by rigorous mathematical deduction as smoothly and as satisfyingly as the theorems of Euclid had been derived from his axioms.”
Since the examination of consistency is a task that cannot be avoided, it appears necessary to axiomatize logic itself and to prove that number theory and set theory are only parts of logic. This method was prepared long ago (not least by Frege’s profound investigations); it has been most successfully explained by the acute mathematician and logician Russell. One could regard the completion of this magnificent Russellian enterprise of the axiomatization of logic as the crowning achievement of the work of axiomatization as a whole.
— David Hilbert
Address (11 Sep 1917), 'Axiomatisches Denken' delivered before the Swiss Mathematical Society in
Zürich. Translated by Ewald as 'Axiomatic Thought', (1918), in William Bragg Ewald, From Kant to Hilbert (1996), Vol. 2, 1113.
The art of doing mathematics consists in finding that special case which contains all the germs of generality.
— David Hilbert
Given as “attributed (apocryphally perhaps)” and no further citation; stated without quotation marks, in M. Kac, 'Wiener and Integration in Function Spaces', Bulletin of the American Mathematical Society (Jan 1966), 72, No. 1, Part 2, 65. This issue of the Bulletin, subtitled 'Norbert Wiener
1894-1964', Felix E. Browder (ed.), was dedicated to the memory of Norbert Wiener.
The farther 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.
— David Hilbert
In 'Mathematical Problems', Lecture at the International Congress of Mathematics, Paris, (8 Aug 1900). Translated by Dr. Maby Winton Newson in Bulletin of the American Mathematical Society (1902), 8, 479.
The infinite! No other question has ever moved so profoundly the spirit of man; no other idea has so fruitfully stimulated his intellect; yet no other concept stands in greater need of clarification than that of the infinite.
— David Hilbert
Address in memory of Karl Weierstrass. As quoted in Journal of the University of Bombay (1933), 2, 201. Also in Tobias Dantzig, Number: The Language of Science (1937), 237. Also partially quoted as epigraph in in James Roy Newman, The World of Mathematics (1956), Vol. 3, 1593. which dates the address as 1921. Another translation for perhaps the same address ('On the Infinite'), in honor of Weierstrass, dates it as 4 Jun 1925, in Paul Benacerraf (ed.) Philosophy of Mathematics (1983), 183. See this alternate version elsewhere on this page, beginning, “From time immemorial…”.
The most suggestive and notable achievement of the last century is the discovery of Non-Euclidean geometry.
— David Hilbert
In George Edward Martin, The Foundations of Geometry and the Non-Euclidean Plane (1982), 72.
The tool which serves as intermediary between theory and practice, between thought and observation, is mathematics; it is mathematics which builds the linking bridges and gives the ever more reliable forms. From this it has come about that our entire contemporary culture, inasmuch as it is based on the intellectual penetration and the exploitation of nature, has its foundations in mathematics. Already Galileo said: one can understand nature only when one has learned the language and the signs in which it speaks to us; but this language is mathematics and these signs are mathematical figures.
— David Hilbert
Radio broadcast (8 Sep 1930). As quoted in Michael Fitzgerald and Ioan James, The Mind of the Mathematician (2007), 6-7.
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!
— David Hilbert
Ignorabimus as used here, means “we will not know” (which is slightly different from ignoramus meaning present ignorance, “we do not know”). In Lecture (1900), 'Mathematische Probleme' (Mathematical Problems), to the International Congress of Mathematicians, Paris. From the original German reprinted in David Hilbert: Gesammelte Abhandlungen (Collected Treatises, 1970), Vol. 3, 298, “Diese Überzeugung von der Lösbarkeit eines jeden mathematischer Problems ist uns ein kräftiger Ansporn während der Arbeit ; wir hören in uns den steten Zuruf: Da ist das Problem, suche die Lösung. Du kannst sie durch reines Denken finden; denn in der Mathematik gibt es kein Ignorabimus. English version as translated by Dr. Maby Winton Newson for Bulletin of the American Mathematical Society (1902), 8, 437-479. The address was first published in Göttinger Nachrichten is Nachrichten von der Königl. Gesellschaft der Wiss. zu Göttingen (1900), 253-297; and Archiv der Mathematik und Physik (1901), 3, No. 1, 44-63.
Who of us would not be glad to lift the veil behind which the future lies hidden; to cast a glance at the next advances of our science and at the secrets of its development during future centuries? What particular goals will there be toward which the leading mathematical spirits of coming generations will strive? What new methods and new facts in the wide and rich field of mathematical thought will the new centuries disclose?
— David Hilbert
Opening of Lecture (1900), 'Mathematische Probleme' (Mathematical Problems), to the International Congress of Mathematicians, Paris. From the original German reprinted in David Hilbert: Gesammelte Abhandlungen (Collected Treatises, 1970), Vol. 3. For full citation, see the quote that begins, “This conviction of the solvability…”, on the David Hilbert Quotes page on this website.
With the extension of mathematical knowledge will it not finally become impossible for the single investigator to embrace all departments of this knowledge? In answer let me point out how thoroughly it is ingrained in mathematical science that every real advance goes hand in hand with the invention of sharper tools and simpler methods which, at the same time, assist in understanding earlier theories and in casting aside some more complicated developments.
— David Hilbert
In 'Mathematical Problems', Lecture at the International Congress of Mathematics, Paris, (8 Aug 1900). Translated by Dr. Maby Winton Newson in Bulletin of the American Mathematical Society (1902), 8, 479. As quoted and cited in Robert Édouard Moritz, Memorabilia Mathematica; Or, The Philomath's Quotation-book (1914), 94-95. It is reprinted in Jeremy Gray, The Hilbert Challenge (2000), 282.
Quotes by others about David Hilbert (5)
But in the present century, thanks in good part to the influence of Hilbert, we have come to see that the unproved postulates with which we start are purely arbitrary. They must be consistent, they had better lead to something interesting.
In A History of Geometrical Methods (1940, reprint 2003), 423.
To the average mathematician who merely wants to know his work is securely based, the most appealing choice is to avoid difficulties by means of Hilbert's program. Here one regards mathematics as a formal game and one is only concerned with the question of consistency ... . The Realist position is probably the one which most mathematicians would prefer to take. It is not until he becomes aware of some of the difficulties in set theory that he would even begin to question it. If these difficulties particularly upset him, he will rush to the shelter of Formalism, while his normal position will be somewhere between the two, trying to enjoy the best of two worlds.
In Axiomatic Set Theory (1971), 9-15. In Thomas Tymoczko, New Directions in the Philosophy of Mathematics: an Anthology (), 11-12.
Plenty of mathematicians, Hardy knew, could follow a step-by-step discursus unflaggingly—yet counted for nothing beside Ramanujan. Years later, he would contrive an informal scale of natural mathematical ability on which he assigned himself a 25 and Littlewood a 30. To David Hilbert, the most eminent mathematician of the day, he assigned an 80. To Ramanujan he gave 100.
In The Man who Knew Infinity: A Life of the Genius Ramanujan (1975), 226.
Das ist nicht Mathematik, das ist Theologie!
This is not mathematics; this is theology.
[Remark about David Hilbert's first proof of his finite basis theorem.]
This is not mathematics; this is theology.
[Remark about David Hilbert's first proof of his finite basis theorem.]
Attributed. It does not seem to appear in Gordan’s written work. According to Colin McClarty, in 'Theology and its Discontents: the Origin of the Myth of Modern Mathematics' (2008), “The quote first appeared a quarter of a century after the event, as an unexplained side comment in a eulogy to Gordan by his long-time colleague Max Noether. Noether was a reliable witness, but he says little about what Gordan meant.” See Noether's obituary of Gordan in Mathematische Annalen (1914), 75, 18. It is still debated if the quote is pejorative, complimentary or merely a joke.
I found out at an early age that science is a haven for the timid, the freaks, the misfits. That is more true perhaps for the past than now. If you were a student in Göttingen in the 1920s and went to the seminar “Structure of Matter” which was under the joint auspices of David Hilbert and Max Born, you could well imagine that you were in a madhouse as you walked in. Every one of the persons there was obviously some kind of a severe case. The least you could do was put on some kind of a stutter. Robert Oppenheimer as a graduate student found it expedient to develop a very elegant kind of stutter, the "njum-njum-njum" technique. Thus, if you were an oddball you felt at home.
Answering the question, “Why did you choose science as your life’s work?” In 'Homo Scientificus According to Beckett," collected in William Beranek, Jr. (ed.)Science, Scientists, and Society, (1972), 135. Excerpted in Ann E. Kammer, Science, Sex, and Society (1979), 278.
See also:
- 23 Jan - short biography, births, deaths and events on date of Hilbert's birth.
- The Theory of Algebraic Number Fields, by David Hilbert, I.T. Adamson. - book suggestion.
- Booklist for David Hilbert.
In science it often happens that scientists say, 'You know that's a really good argument; my position is mistaken,' and then they would actually change their minds and you never hear that old view from them again. They really do it. It doesn't happen as often as it should, because scientists are human and change is sometimes painful. But it happens every day. I cannot recall the last time something like that happened in politics or religion.
(1987) -- 
