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Home > Dictionary of Science Quotations > Scientist Names Index H > David Hilbert Quotes > Mathematics

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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.


David Hilbert Quotes on Mathematics (19 quotes)

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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.
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~~[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.
— 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.
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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.
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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
…...
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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.
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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).
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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).
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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”.
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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.
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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.
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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.
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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.”
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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.
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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.
— David Hilbert
In George Edward Martin, The Foundations of Geometry and the Non-Euclidean Plane (1982), 33.
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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.
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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.
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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.
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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.
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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.
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