TODAY IN SCIENCE HISTORY ®  •  TODAYINSCI ®
Celebrating 24 Years on the Web
Find science on or your birthday

Today in Science History - Quickie Quiz
Who said: “Politics is more difficult than physics.”
more quiz questions >>
Home > Dictionary of Science Quotations > Scientist Names Index D > Tobias Danzig Quotes

Tobias Danzig
(19 Feb 1884 - 9 Aug 1956)

Russo-German-American mathematician.

Science Quotes by Tobias Danzig (7 quotes)

I recall my own emotions: I had just been initiated into the mysteries of the complex number. I remember my bewilderment: here were magnitudes patently impossible and yet susceptible of manipulations which lead to concrete results. It was a feeling of dissatisfaction, of restlessness, a desire to fill these illusory creatures, these empty symbols, with substance. Then I was taught to interpret these beings in a concrete geometrical way. There came then an immediate feeling of relief, as though I had solved an enigma, as though a ghost which had been causing me apprehension turned out to be no ghost at all, but a familiar part of my environment.
— Tobias Danzig
In Tobias Dantzig and Joseph Mazur (ed.), 'The Two Realities', Number: The Language of Science (1930, ed. by Joseph Mazur 2007), 254.
Science quotes on:  |  Apprehension (26)  |  Being (1276)  |  Bewilderment (8)  |  Cause (561)  |  Complex (202)  |  Complex Number (3)  |  Concrete (55)  |  Creature (242)  |  Desire (212)  |  Dissatisfaction (13)  |  Emotion (106)  |  Empty (82)  |  Enigma (16)  |  Environment (239)  |  Familiar (47)  |  Feeling (259)  |  Fill (67)  |  Geometry (271)  |  Ghost (36)  |  Illusory (2)  |  Immediate (98)  |  Impossible (263)  |  Initiate (13)  |  Interpret (25)  |  Lead (391)  |  Magnitude (88)  |  Manipulation (19)  |  Mystery (188)  |  Number (710)  |  Patently (4)  |  Recall (11)  |  Relief (30)  |  Remember (189)  |  Restless (13)  |  Restlessness (8)  |  Result (700)  |  Solve (145)  |  Substance (253)  |  Susceptible (8)  |  Symbol (100)  |  Teach (299)  |  Turn (454)  |  Turn Out (9)  |  Way (1214)

In the history of the discovery of zero will always stand out as one of the greatest single achievements of the human race.
— Tobias Danzig
Number: the Language of Science (1930), 35.
Science quotes on:  |  Achievement (187)  |  Discovery (837)  |  Great (1610)  |  Greatest (330)  |  History (716)  |  Human (1512)  |  Human Race (104)  |  Race (278)  |  Single (365)  |  Stand (284)  |  Stand Out (5)  |  Will (2350)  |  Zero (38)

Neither in the subjective nor in the objective world can we find a criterion for the reality of the number concept, because the first contains no such concept, and the second contains nothing that is free from the concept. How then can we arrive at a criterion? Not by evidence, for the dice of evidence are loaded. Not by logic, for logic has no existence independent of mathematics: it is only one phase of this multiplied necessity that we call mathematics.
How then shall mathematical concepts be judged? They shall not be judged. Mathematics is the supreme arbiter. From its decisions there is no appeal. We cannot change the rules of the game, we cannot ascertain whether the game is fair. We can only study the player at his game; not, however, with the detached attitude of a bystander, for we are watching our own minds at play.
— Tobias Danzig
In Number: The Language of Science; a Critical Survey Written for the Cultured Non-Mathematician (1937), 244-245.
Science quotes on:  |  Appeal (46)  |  Arbiter (5)  |  Arrive (40)  |  Ascertain (41)  |  Attitude (84)  |  Call (781)  |  Change (639)  |  Concept (242)  |  Contain (68)  |  Criterion (28)  |  Decision (98)  |  Detach (5)  |  Dice (21)  |  Evidence (267)  |  Existence (481)  |  Fair (16)  |  Find (1014)  |  First (1302)  |  Free (239)  |  Game (104)  |  Independent (74)  |  Judge (114)  |  Loaded (4)  |  Logic (311)  |  Mathematics (1395)  |  Mind (1377)  |  Necessity (197)  |  Nothing (1000)  |  Number (710)  |  Objective (96)  |  Phase (37)  |  Play (116)  |  Player (9)  |  Reality (274)  |  Rule (307)  |  Study (701)  |  Subjective (20)  |  Supreme (73)  |  Watch (118)  |  World (1850)

Our school curricula, by stripping mathematics of its cultural content and leaving a bare skeleton of technicalities, have repelled many a fine mind.
— Tobias Danzig
In Number, the Language of Science: A Critical Survey Written for the Cultured Non-mathematician (1930), vii.
Science quotes on:  |  Bare (33)  |  Content (75)  |  Cultural (26)  |  Curriculum (11)  |  Fine (37)  |  Leave (138)  |  Mathematics (1395)  |  Mind (1377)  |  Repel (2)  |  School (227)  |  Skeleton (25)  |  Strip (7)  |  Technicality (5)

The arithmetization of mathematics … which began with Weierstrass … had for its object the separation of purely mathematical concepts, such as number and correspondence and aggregate, from intuitional ideas, which mathematics had acquired from long association with geometry and mechanics. These latter, in the opinion of the formalists, are so firmly entrenched in mathematical thought that in spite of the most careful circumspection in the choice of words, the meaning concealed behind these words, may influence our reasoning. For the trouble with human words is that they possess content, whereas the purpose of mathematics is to construct pure thought. But how can we avoid the use of human language? The … symbol. Only by using a symbolic language not yet usurped by those vague ideas of space, time, continuity which have their origin in intuition and tend to obscure pure reason—only thus may we hope to build mathematics on the solid foundation of logic.
— Tobias Danzig
In Tobias Dantzig and Joseph Mazur (ed.), Number: The Language of Science (1930, ed. by Joseph Mazur 2007), 99.
Science quotes on:  |  Acquire (46)  |  Acquired (77)  |  Aggregate (24)  |  Association (49)  |  Avoid (123)  |  Begin (275)  |  Behind (139)  |  Build (211)  |  Careful (28)  |  Choice (114)  |  Circumspection (5)  |  Conceal (19)  |  Concealed (25)  |  Concept (242)  |  Construct (129)  |  Content (75)  |  Continuity (39)  |  Correspondence (24)  |  Entrench (2)  |  Firmly (6)  |  Foundation (177)  |  Geometry (271)  |  Hope (321)  |  Human (1512)  |  Idea (881)  |  Influence (231)  |  Intuition (82)  |  Language (308)  |  Latter (21)  |  Logic (311)  |  Long (778)  |  Mathematics (1395)  |  Mean (810)  |  Meaning (244)  |  Mechanic (120)  |  Mechanics (137)  |  Most (1728)  |  Number (710)  |  Object (438)  |  Obscure (66)  |  Opinion (291)  |  Origin (250)  |  Possess (157)  |  Pure (299)  |  Purely (111)  |  Purpose (336)  |  Reason (766)  |  Reasoning (212)  |  Separation (60)  |  Solid (119)  |  Space (523)  |  Spite (55)  |  Symbol (100)  |  Tend (124)  |  Thought (995)  |  Time (1911)  |  Trouble (117)  |  Use (771)  |  Vague (50)  |   Karl Weierstrass, (10)  |  Word (650)

The harmony of the universe knows only one musical form - the legato; while the symphony of number knows only its opposite - the staccato. All attempts to reconcile this discrepancy are based on the hope that an accelerated staccato may appear to our senses as a legato.
— Tobias Danzig
…...
Science quotes on:  |  Accelerate (11)  |  Appear (122)  |  Attempt (266)  |  Base (120)  |  Discrepancy (7)  |  Form (976)  |  Harmony (105)  |  Hope (321)  |  Know (1538)  |  Musical (10)  |  Number (710)  |  Opposite (110)  |  Reconcile (19)  |  Sense (785)  |  Staccato (2)  |  Symphony (10)  |  Universe (900)

The mathematician may be compared to a designer of garments, who is utterly oblivious of the creatures whom his garments may fit. To be sure, his art originated in the necessity for clothing such creatures, but this was long ago; to this day a shape will occasionally appear which will fit into the garment as if the garment had been made for it. Then there is no end of surprise and delight.
— Tobias Danzig
Number: the Language of Science (1930), 231.
Science quotes on:  |  Art (680)  |  Creature (242)  |  Delight (111)  |  Designer (7)  |  End (603)  |  Fit (139)  |  Garment (13)  |  Long (778)  |  Mathematician (407)  |  Necessity (197)  |  Oblivious (9)  |  Surprise (91)  |  Usefulness (92)  |  Will (2350)


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 |
Thank you for sharing.
- 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.