Celebrating 20 Years on the Web
TODAY IN SCIENCE HISTORY ®
Find science on or your birthday

Today in Science History - Quickie Quiz
Who said: “I believe that this Nation should commit itself to achieving the goal, before this decade is out, of landing a man on the moon and returning him safely to earth.”
more quiz questions >>
Home > Dictionary of Science Quotations > Scientist Names Index H > David Hilbert Quotes > Mathematics

Thumbnail of David Hilbert (source)
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 (21 quotes)

>> Click for 42 Science Quotes by David Hilbert

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.
Science quotes on:  |  Completely (135)  |  Difficult (247)  |  Effort (227)  |  Guide (98)  |  Hide (69)  |  Inaccessible (18)  |  Lest (3)  |  Mathematics (1333)  |  Mock (7)  |  Order (632)  |  Path (145)  |  Pleasure (179)  |  Post (6)  |  Problem (679)  |  Reminder (13)  |  Solution (269)  |  Study And Research In Mathematics (61)  |  Successful (123)  |  Truth (1062)  |  Ultimately (55)

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
…...
Science quotes on:  |  Clear (100)  |  Complete (204)  |  Consider (416)  |  Explain (322)  |  First (1284)  |  Man (2249)  |  Mathematics (1333)  |  Meet (32)  |  Street (23)  |  Theory (972)

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.
Science quotes on:  |  All (4107)  |  Axiom (63)  |  Become (815)  |  Conscious (45)  |  Deeper (4)  |  Dependent (24)  |  Formation (96)  |  Indirect (18)  |  Knowledge (1536)  |  Layer (40)  |  Leading (17)  |  Mathematics (1333)  |  Method (506)  |  More (2559)  |  Object (422)  |  Ripe (5)  |  Role (85)  |  Science (3880)  |  Scientific (940)  |  Scientific Thought (17)  |  Sign (58)  |  Soon (186)  |  Summon (10)  |  Theory (972)  |  Thought (954)  |  Unity (78)

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).
Science quotes on:  |  Critical (66)  |  Development (424)  |  Discipline (77)  |  Formal (34)  |  Mathematics (1333)  |  Naive (13)  |  Period (198)  |  Through (849)

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).
Science quotes on:  |  Abstraction (47)  |  Being (1278)  |  Correlate (6)  |  Find (999)  |  Hand (143)  |  Inherent (42)  |  Logical (55)  |  Material (353)  |  Mathematics (1333)  |  Maze (10)  |  Orderly (38)  |  Present (620)  |  Relation (157)  |  Seek (213)  |  Study (656)  |  Systematic (57)  |  Tendency (99)  |  Two (937)

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”.
Science quotes on:  |  Ignorance (240)  |  Mathematics (1333)

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.
Science quotes on:  |  Abstraction (47)  |  Actually (27)  |  Against (332)  |  All (4107)  |  Arithmetic (139)  |  Arithmetical (11)  |  Charge (60)  |  Completeness (19)  |  Comprehend (40)  |  Concept (221)  |  Condition (357)  |  Control (167)  |  Degree (275)  |  Equally (130)  |  Everywhere (95)  |  Examine (78)  |  Fault (55)  |  Field (365)  |  Foundation (173)  |  Fully (21)  |  High (363)  |  Knowledge (1536)  |  Least (75)  |  Mathematics (1333)  |  Method (506)  |  Methods (204)  |  Necessary (363)  |  Opinion (281)  |  Order (632)  |  Other (2236)  |  Proof (289)  |  Provide (69)  |  Require (219)  |  Rigour (21)  |  Study And Research In Mathematics (61)  |  Time (1877)

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.
Science quotes on:  |  All (4107)  |  Analogy (71)  |  Clearly (41)  |  Condition (357)  |  Connection (162)  |  Conscious (45)  |  Department (92)  |  Device (70)  |  Different (577)  |  Idea (845)  |  Indivisible (22)  |  Knowledge (1536)  |  Logical (55)  |  Mathematics (1333)  |  Nature Of Mathematics (80)  |  Numerous (68)  |  Opinion (281)  |  Organism (220)  |  Part (222)  |  Relationship (104)  |  Science (3880)  |  Similarity (31)  |  Still (613)  |  Variety (133)  |  Vitality (23)  |  Whole (738)

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.
Science quotes on:  |  According (237)  |  Certain (550)  |  Game (101)  |  Mark (43)  |  Mathematics (1333)  |  Meaningless (17)  |  Paper (183)  |  Play (111)  |  Rule (295)  |  Simple (406)

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.”
Science quotes on:  |  Importance (287)  |  Mathematics (1333)  |  Measurement (174)  |  Number (701)  |  Peculiar (113)  |  Publication (102)  |  Render (93)  |  Science (3880)  |  Superfluous (21)  |  Work (1351)

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.
Science quotes on:  |  Boundary (51)  |  Country (251)  |  Cultural (25)  |  Geographic (10)  |  Know (1519)  |  Knowledge (1536)  |  Mathematics (1333)  |  Race (268)  |  World (1778)

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.
Science quotes on:  |  Georg Cantor (6)  |  Creation (329)  |  Development (424)  |  Importance (287)  |  Mathematics (1333)  |  Paradise (13)  |  Set (394)  |  Set Theory (6)  |  Theory (972)

The art of doing mathematics consists in finding that special case which contains all the germs of generality.
— David Hilbert
Attributed, perhaps apocryphal. As given in Felix E. Browder, Norbert Wiener (1966), 65.
Science quotes on:  |  All (4107)  |  Art (657)  |  Case (99)  |  Consist (223)  |  Contain (68)  |  Containing (4)  |  Doing (280)  |  Find (999)  |  Finding (30)  |  Generality (45)  |  Germ (53)  |  Mathematics (1333)  |  Special (184)

The art of doing mathematics consists in finding that special case which contains all the germs of generality.
— David Hilbert
Stated without quotation marks, as “attributed apocryphally perhaps”, in M. Kac, 'Wiener and Integration in Function Spaces', Bulletin of the American Mathematical Society (Jan 1966), 72, No. 1, Part 2, 65.
Science quotes on:  |  All (4107)  |  Art (657)  |  Consist (223)  |  Doing (280)  |  Generality (45)  |  Germ (53)  |  Mathematics (1333)  |  Special (184)

The art of doing mathematics consists in finding that special case which contains all the germs of generality.
— David Hilbert
Attributed (apocryphally perhaps). As cited in Bulletin of the American Mathematical Society (1966), 72, 65.
Science quotes on:  |  All (4107)  |  Art (657)  |  Consist (223)  |  Doing (280)  |  Generality (45)  |  Germ (53)  |  Mathematics (1333)  |  Special (184)

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.
Science quotes on:  |  Branch (150)  |  Construction (112)  |  Develop (268)  |  Developed (11)  |  Disclosed (2)  |  Farther (51)  |  Harmonious (18)  |  Mathematics (1333)  |  More (2559)  |  Proceed (129)  |  Relation (157)  |  Science (3880)  |  Separate (143)  |  Theory (972)  |  Uniform (19)  |  Unsuspected (7)

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.
Science quotes on:  |  Already (222)  |  Based (10)  |  Bridge (47)  |  Build (204)  |  Contemporary (30)  |  Culture (143)  |  Entire (47)  |  Exploitation (14)  |  Figure (160)  |  Form (960)  |  Foundation (173)  |  Galileo Galilei (125)  |  Intellectual (255)  |  Intermediary (2)  |  Language (293)  |  Learn (632)  |  Learned (235)  |  Link (43)  |  Linking (8)  |  Mathematics (1333)  |  More (2559)  |  Nature (1928)  |  Observation (560)  |  Penetration (18)  |  Practice (204)  |  Reliable (12)  |  Serve (59)  |  Sign (58)  |  Speak (232)  |  Theory (972)  |  Thought (954)  |  Tool (117)  |  Understand (607)

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.
Science quotes on:  |  Call (769)  |  Conviction (98)  |  Find (999)  |  Hear (139)  |  Ignorance (240)  |  Incentive (9)  |  Mathematics (1333)  |  Perpetual (57)  |  Powerful (139)  |  Problem (679)  |  Pure (292)  |  Reason (744)  |  Seek (213)  |  Solution (269)  |  Solve (130)  |  Study And Research In Mathematics (61)  |  Worker (31)

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.
Science quotes on:  |  Advance (280)  |  All (4107)  |  Answer (366)  |  Assist (9)  |  Become (815)  |  Branch (150)  |  Cast (67)  |  Casting (10)  |  Complicated (115)  |  Department (92)  |  Development (424)  |  Early (186)  |  Easily (35)  |  Embrace (46)  |  Extension (59)  |  Finally (26)  |  Find (999)  |  Hand In Hand (4)  |  Impossible (253)  |  Individual (404)  |  Ingrained (5)  |  Invention (377)  |  Investigator (67)  |  Knowledge (1536)  |  Let (61)  |  Mathematics (1333)  |  Method (506)  |  Methods (204)  |  More (2559)  |  Point (580)  |  Point Out (8)  |  Possible (554)  |  Real (149)  |  Same (157)  |  Science (3880)  |  Sharp (14)  |  Simple (406)  |  Simpler (8)  |  Single (354)  |  Study And Research In Mathematics (61)  |  Theory (972)  |  Thoroughly (67)  |  Time (1877)  |  Tool (117)  |  Understand (607)  |  Understanding (514)  |  Various (200)  |  Will (2354)

[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.
Science quotes on:  |  Achievement (180)  |  Activity (210)  |  Fine (33)  |  Genius (285)  |  Human (1470)  |  Intellectual (255)  |  Mathematics (1333)  |  Product (160)  |  Purely (110)  |  Set (394)  |  Set Theory (6)  |  Supreme (71)  |  Theory (972)

~~[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.
Science quotes on:  |  Automatically (5)  |  Become (815)  |  Certain (550)  |  Degree (275)  |  Kind (557)  |  Known (454)  |  Mathematics (1333)  |  Maturity (14)  |  Part (222)  |  Reach (281)  |  Science (3880)


See also:

Carl Sagan Thumbnail 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) -- Carl Sagan
Quotations by:Albert EinsteinIsaac NewtonLord KelvinCharles DarwinSrinivasa RamanujanCarl SaganFlorence NightingaleThomas EdisonAristotleMarie CurieBenjamin FranklinWinston ChurchillGalileo GalileiSigmund FreudRobert BunsenLouis PasteurTheodore RooseveltAbraham LincolnRonald ReaganLeonardo DaVinciMichio KakuKarl PopperJohann GoetheRobert OppenheimerCharles Kettering  ... (more people)

Quotations about:Atomic  BombBiologyChemistryDeforestationEngineeringAnatomyAstronomyBacteriaBiochemistryBotanyConservationDinosaurEnvironmentFractalGeneticsGeologyHistory of ScienceInventionJupiterKnowledgeLoveMathematicsMeasurementMedicineNatural ResourceOrganic ChemistryPhysicsPhysicianQuantum TheoryResearchScience and ArtTeacherTechnologyUniverseVolcanoVirusWind PowerWomen ScientistsX-RaysYouthZoology  ... (more topics)
Sitewide search within all Today In Science History pages:
Visit our Science and Scientist Quotations index for more Science Quotes from archaeologists, biologists, chemists, geologists, inventors and inventions, mathematicians, physicists, pioneers in medicine, science events and technology.

Names index: | A | B | C | D | E | F | G | H | I | J | K | L | M | N | O | P | Q | R | S | T | U | V | W | X | Y | Z |

Categories index: | 1 | 2 | A | B | C | D | E | F | G | H | I | J | K | L | M | N | O | P | Q | R | S | T | U | V | W | X | Y | Z |
- 100 -
Sophie Germain
Gertrude Elion
Ernest Rutherford
James Chadwick
Marcel Proust
William Harvey
Johann Goethe
John Keynes
Carl Gauss
Paul Feyerabend
- 90 -
Antoine Lavoisier
Lise Meitner
Charles Babbage
Ibn Khaldun
Euclid
Ralph Emerson
Robert Bunsen
Frederick Banting
Andre Ampere
Winston Churchill
- 80 -
John Locke
Bronislaw Malinowski
Bible
Thomas Huxley
Alessandro Volta
Erwin Schrodinger
Wilhelm Roentgen
Louis Pasteur
Bertrand Russell
Jean Lamarck
- 70 -
Samuel Morse
John Wheeler
Nicolaus Copernicus
Robert Fulton
Pierre Laplace
Humphry Davy
Thomas Edison
Lord Kelvin
Theodore Roosevelt
Carolus Linnaeus
- 60 -
Francis Galton
Linus Pauling
Immanuel Kant
Martin Fischer
Robert Boyle
Karl Popper
Paul Dirac
Avicenna
James Watson
William Shakespeare
- 50 -
Stephen Hawking
Niels Bohr
Nikola Tesla
Rachel Carson
Max Planck
Henry Adams
Richard Dawkins
Werner Heisenberg
Alfred Wegener
John Dalton
- 40 -
Pierre Fermat
Edward Wilson
Johannes Kepler
Gustave Eiffel
Giordano Bruno
JJ Thomson
Thomas Kuhn
Leonardo DaVinci
Archimedes
David Hume
- 30 -
Andreas Vesalius
Rudolf Virchow
Richard Feynman
James Hutton
Alexander Fleming
Emile Durkheim
Benjamin Franklin
Robert Oppenheimer
Robert Hooke
Charles Kettering
- 20 -
Carl Sagan
James Maxwell
Marie Curie
Rene Descartes
Francis Crick
Hippocrates
Michael Faraday
Srinivasa Ramanujan
Francis Bacon
Galileo Galilei
- 10 -
Aristotle
John Watson
Rosalind Franklin
Michio Kaku
Isaac Asimov
Charles Darwin
Sigmund Freud
Albert Einstein
Florence Nightingale
Isaac Newton


by Ian Ellis
who invites your feedback
Thank you for sharing.
Today in Science History
Sign up for Newsletter
with quiz, quotes and more.