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: “Nature does nothing in vain when less will serve; for Nature is pleased with simplicity and affects not the pomp of superfluous causes.”
more quiz questions >>
Home > Category Index for Science Quotations > Category Index M > Category: Mathematics

Mathematics Quotes (1395 quotes)
Math Quotes, Maths Quotes, Mathematical Quotes, Mathematick Quotes

Godfrey Harold Hardy quote “Languages die and mathematical ideas do not.”
background by Tom_Brown 6117, CC by 2.0 (source)

… how the real proof should run. The main thing is the content, not the mathematics. With mathematics one can prove anything.
Commenting on a mistake in some work from Planck, for which “the result is correct but the proof is faulty.” As quoted in Denis Brian, Einstein—A Life (1996), 76.
Science quotes on:  |  Content (75)  |  Main Thing (4)  |  Proof (304)  |  Prove (261)  |  Prove Anything (7)  |  Real (159)  |  Run (158)  |  Thing (1914)

… just as the astronomer, the physicist, the geologist, or other student of objective science looks about in the world of sense, so, not metaphorically speaking but literally, the mind of the mathematician goes forth in the universe of logic in quest of the things that are there; exploring the heights and depths for facts—ideas, classes, relationships, implications, and the rest; observing the minute and elusive with the powerful microscope of his Infinitesimal Analysis; observing the elusive and vast with the limitless telescope of his Calculus of the Infinite; making guesses regarding the order and internal harmony of the data observed and collocated; testing the hypotheses, not merely by the complete induction peculiar to mathematics, but, like his colleagues of the outer world, resorting also to experimental tests and incomplete induction; frequently finding it necessary, in view of unforeseen disclosures, to abandon one hopeful hypothesis or to transform it by retrenchment or by enlargement:—thus, in his own domain, matching, point for point, the processes, methods and experience familiar to the devotee of natural science.
In Lectures on Science, Philosophy and Art (1908), 26
Science quotes on:  |  Abandon (73)  |  Analysis (244)  |  Astronomer (97)  |  Calculus (65)  |  Class (168)  |  Colleague (51)  |  Complete (209)  |  Data (162)  |  Depth (97)  |  Devotee (7)  |  Disclosure (7)  |  Domain (72)  |  Elusive (8)  |  Enlargement (8)  |  Experience (494)  |  Experimental (193)  |  Exploration (161)  |  Fact (1257)  |  Facts (553)  |  Familiar (47)  |  Find (1014)  |  Forth (14)  |  Frequently (21)  |  Geologist (82)  |  Guess (67)  |  Harmony (105)  |  Height (33)  |  Hopeful (6)  |  Hypothesis (314)  |  Idea (881)  |  Implication (25)  |  Incomplete (31)  |  Induction (81)  |  Infinite (243)  |  Infinitesimal (30)  |  Internal (69)  |  Limitless (14)  |  Literally (30)  |  Located (2)  |  Logic (311)  |  Look (584)  |  Making (300)  |  Match (30)  |  Mathematician (407)  |  Merely (315)  |  Metaphor (37)  |  Method (531)  |  Microscope (85)  |  Mind (1377)  |  Minute (129)  |  Natural (810)  |  Natural Science (133)  |  Nature Of Mathematics (80)  |  Necessary (370)  |  Objective (96)  |  Observe (179)  |  Observed (149)  |  Order (638)  |  Other (2233)  |  Outer (13)  |  Peculiar (115)  |  Physicist (270)  |  Point (584)  |  Powerful (145)  |  Process (439)  |  Quest (39)  |  Regard (312)  |  Relationship (114)  |  Resort (8)  |  Rest (287)  |  Sense (785)  |  Speak (240)  |  Speaking (118)  |  Student (317)  |  Telescope (106)  |  Test (221)  |  Thing (1914)  |  Transform (74)  |  Unforeseen (11)  |  Universe (900)  |  Vast (188)  |  View (496)  |  World (1850)

… the reasoning process [employed in mathematics] is not different from that of any other branch of knowledge, … but there is required, and in a great degree, that attention of mind which is in some part necessary for the acquisition of all knowledge, and in this branch is indispensably necessary. This must be given in its fullest intensity; … the other elements especially characteristic of a mathematical mind are quickness in perceiving logical sequence, love of order, methodical arrangement and harmony, distinctness of conception.
In Treatise on Infinitesimal Calculus (1868), Vol. 8, 6.
Science quotes on:  |  Acquisition (46)  |  Arrangement (93)  |  Attention (196)  |  Branch (155)  |  Characteristic (154)  |  Conception (160)  |  Degree (277)  |  Different (595)  |  Element (322)  |  Employ (115)  |  Great (1610)  |  Harmony (105)  |  Indispensable (31)  |  Intensity (34)  |  Knowledge (1647)  |  Logical (57)  |  Love (328)  |  Methodical (8)  |  Mind (1377)  |  Must (1525)  |  Nature Of Mathematics (80)  |  Necessary (370)  |  Order (638)  |  Other (2233)  |  Perceive (46)  |  Process (439)  |  Quickness (5)  |  Reasoning (212)  |  Required (108)  |  Sequence (68)

… the three positive characteristics that distinguish mathematical knowledge from other knowledge … may be briefly expressed as follows: first, mathematical knowledge bears more distinctly the imprint of truth on all its results than any other kind of knowledge; secondly, it is always a sure preliminary step to the attainment of other correct knowledge; thirdly, it has no need of other knowledge.
In Mathematical Essays and Recreations (1898), 35.
Science quotes on:  |  Attain (126)  |  Attainment (48)  |  Bear (162)  |  Characteristic (154)  |  Correct (95)  |  Distinct (98)  |  Distinguish (168)  |  Express (192)  |  First (1302)  |  Follow (389)  |  Imprint (6)  |  Kind (564)  |  Knowledge (1647)  |  More (2558)  |  Nature Of Mathematics (80)  |  Need (320)  |  Other (2233)  |  Positive (98)  |  Preliminary (6)  |  Result (700)  |  Step (234)  |  Truth (1109)

… what is physical is subject to the laws of mathematics, and what is spiritual to the laws of God, and the laws of mathematics are but the expression of the thoughts of God.
In 'The Uses of Mathesis', Bibliotheca Sacra, 32, 523.
Science quotes on:  |  Expression (181)  |  God (776)  |  Law (913)  |  Nature Of Mathematics (80)  |  Physical (518)  |  Spiritual (94)  |  Subject (543)  |  Thought (995)

…indeed what reason may not go to Schoole to the wisdome of Bees, Aunts, and Spiders? what wise hand teacheth them to doe what reason cannot teach us? Ruder heads stand amazed at those prodigious pieces of nature, Whales, Elephants, Dromidaries and Camels; these I confesse, are the Colossus and Majestick pieces of her hand; but in these narrow Engines there is more curious Mathematicks, and the civilitie of these little Citizens more neatly sets forth the wisedome of their Maker.
In Religio Medici and Other Writings (1909), 17.
Science quotes on:  |  Amaze (5)  |  Ant (34)  |  Bee (44)  |  Camel (12)  |  Citizen (52)  |  Civility (2)  |  Curious (95)  |  Elephant (35)  |  Engine (99)  |  Indeed (323)  |  Little (717)  |  Majestic (17)  |  Maker (34)  |  More (2558)  |  Narrow (85)  |  Nature (2017)  |  Prodigious (20)  |  Reason (766)  |  School (227)  |  Set (400)  |  Spider (14)  |  Stand (284)  |  Teach (299)  |  Whale (45)  |  Wisdom (235)  |  Wise (143)

…nature seems very conversant with the rules of pure mathematics, as our own mathematicians have formulated them in their studies, out of their own inner consciousness and without drawing to any appreciable extent on their experience of the outer world.
In The Mysterious Universe (1930), 113.
Science quotes on:  |  Consciousness (132)  |  Conversant (6)  |  Drawing (56)  |  Experience (494)  |  Extent (142)  |  Inner (72)  |  Mathematician (407)  |  Nature (2017)  |  Outer (13)  |  Pure (299)  |  Pure Mathematics (72)  |  Rule (307)  |  Study (701)  |  World (1850)

...the source of all great mathematics is the special case, the concrete example. It is frequent in mathematics that every instance of a concept of seemingly generality is, in essence, the same as a small and concrete special case.
I Want to be a Mathematician: an Automathography in Three Parts (1985), 324.
Science quotes on:  |  Case (102)  |  Concept (242)  |  Concrete (55)  |  Essence (85)  |  Example (98)  |  Frequent (26)  |  Generality (45)  |  Great (1610)  |  Instance (33)  |  Seeming (10)  |  Seemingly (28)  |  Small (489)  |  Source (101)  |  Special (188)  |  Special Case (9)

‘I was reading an article about “Mathematics”. Perfectly pure mathematics. My own knowledge of mathematics stops at “twelve times twelve,” but I enjoyed that article immensely. I didn’t understand a word of it; but facts, or what a man believes to be facts, are always delightful. That mathematical fellow believed in his facts. So do I. Get your facts first, and’—the voice dies away to an almost inaudible drone—’then you can distort ‘em as much as you please.’
In 'An Interview with Mark Twain', in Rudyard Kipling, From Sea to Sea (1899), Vol. 2, 180.
Science quotes on:  |  Article (22)  |  Belief (615)  |  Delight (111)  |  Delightful (18)  |  Distort (22)  |  Distortion (13)  |  Do (1905)  |  Drone (4)  |  Enjoyment (37)  |  Fact (1257)  |  Facts (553)  |  Fellow (88)  |  First (1302)  |  Knowledge (1647)  |  Man (2252)  |  Please (68)  |  Pleasure (191)  |  Pure (299)  |  Pure Mathematics (72)  |  Reading (136)  |  Stop (89)  |  Time (1911)  |  Twelve (4)  |  Understand (648)  |  Understanding (527)  |  Word (650)

“Can you do Addition?” the White Queen said. “What's one and one and one and one and one and one and one and one and one and one?”
“I don't know,” said Alice. “I lost count.”
“She can’t do Addition,” the Red Queen interrupted.
Through the Looking Glass and What Alice Found There (1871, 1897), 189.
Science quotes on:  |  Addition (70)  |  Count (107)  |  Do (1905)  |  Know (1538)  |  Red Queen (3)  |  White (132)

“Every moment dies a man,/ Every moment one is born”:
I need hardly point out to you that this calculation would tend to keep the sum total of the world's population in a state of perpetual equipoise whereas it is a well-known fact that the said sum total is constantly on the increase. I would therefore take the liberty of suggesting that in the next edition of your excellent poem the erroneous calculation to which I refer should be corrected as follows:
'Every moment dies a man / And one and a sixteenth is born.” I may add that the exact figures are 1.167, but something must, of course, be conceded to the laws of metre.
Unpublished letter to Tennyson in response to his Vision of Sin (1842). Quoted in Philip and Emily Morrison, Charles Babbage and his Calculating Engines: Selected Writings by Charles Babbage and Others (1961), xxiii.
Science quotes on:  |  Calculation (134)  |  Course (413)  |  Erroneous (31)  |  Fact (1257)  |  Figure (162)  |  Follow (389)  |  Increase (225)  |  Known (453)  |  Law (913)  |  Man (2252)  |  Moment (260)  |  Must (1525)  |  Next (238)  |  Perpetual (59)  |  Poem (104)  |  Point (584)  |  Population (115)  |  Something (718)  |  State (505)  |  Statistics (170)  |  Sum (103)  |  Tend (124)  |  Total (95)  |  World (1850)

“I think you’re begging the question,” said Haydock, “and I can see looming ahead one of those terrible exercises in probability where six men have white hats and six men have black hats and you have to work it out by mathematics how likely it is that the hats will get mixed up and in what proportion. If you start thinking about things like that, you would go round the bend. Let me assure you of that!”
In The Mirror Crack’d (1962), 190.
Science quotes on:  |  Black (46)  |  Exercise (113)  |  Go Crazy (2)  |  Hat (9)  |  Likely (36)  |  Mix (24)  |  Probability (135)  |  Proportion (140)  |  Question (649)  |  See (1094)  |  Start (237)  |  Terrible (41)  |  Thing (1914)  |  Think (1122)  |  Thinking (425)  |  White (132)  |  Will (2350)  |  Work (1402)

“Obvious” is the most dangerous word in mathematics.
In The Queen of the Sciences (1938), 14.
Science quotes on:  |  Dangerous (108)  |  Most (1728)  |  Obvious (128)  |  Word (650)

[1155] Mechanics are the Paradise of mathematical science, because here we come to the fruits of mathematics.
Notebook E (1513), folio 8 back. In the original Italian: “La meccanica è il paradiso delle sciētie matematiche, perchè cō quella si viene al frutto matematico.” English and Italian in Jean Paul Richter (trans.), 'Philosophical Maxims: Of Mechanics', The Literary Works of Leonardo da Vinci (1883), Vol. 1, Part 2, 289, Aphorism 1155. [Note: da Vinci wrote ē=en, ō=on] Also translated as “Mechanics is the paradise of the mathematical sciences, because by means of it one comes to the fruits of mathematics,” in Edward McCurdy, The Notebooks of Leonardo Da Vinci (1939, 1958), Vol. 1, 613.
Science quotes on:  |  Fruit (108)  |  Means (587)  |  Mechanic (120)  |  Mechanics (137)  |  Paradise (15)

[1157] The man who blames the supreme certainty of mathematics feeds on confusion, and can never silence the contradictions of sophistical sciences which lead to an eternal quackery.
W. An. III. 241 a. From the original Italian: “Chi biasima la soma certezza della matematica, si pasce di confusione mai porrà silentio alle contraditioni delle soffistiche sciētie, colle quali s’inpara vno eterno gridore.” English and Italian in Jean Paul Richter (trans.), 'Philosophical Maxims: Of Mechanics', The Literary Works of Leonardo da Vinci (1883), Vol. 1, Part 2, 289, Aphorism 1157. [Note: Da Vinci writes ē=en.] Also translated beginning, “Those who condemn…”. Also seen translated as “Whoever despises the high wisdom of mathematics nourishes himself on delusion and will never still the sophistic sciences whose only product is an eternal uproar,” in Nicholas J. Rose Mathematical Maxims and Minims (1988).
Science quotes on:  |  Blame (31)  |  Certainty (180)  |  Condemn (44)  |  Confusion (61)  |  Contradiction (69)  |  Eternal (113)  |  Feed (31)  |  Lead (391)  |  Man (2252)  |  Never (1089)  |  Quackery (4)  |  Silence (62)  |  Sophism (2)  |  Supreme (73)

[1158] There is no certainty in sciences where one of the mathematical sciences cannot be applied, or which are not in relation with these mathematics.
Notebook G (c.1515), sheet 95 back. In the original Italian: “Nessuna certezza delle sciētie è, dove no si può applicare vna delle sciētie matematiche e che non son vnight con esse matematiche.” English and Italian in Jean Paul Richter (trans.), 'Philosophical Maxims: Of Mechanics', The Literary Works of Leonardo da Vinci (1883), Vol. 1, Part 2, 289, Aphorism 1158. [Note: da Vinci wrote ē=en; v=u] The following, found on the web, without citation, seems to be a paraphrase: “No human investigation can be called real science if it cannot be demonstrated mathematically.”
Science quotes on:  |  Applied (176)  |  Call (781)  |  Certainty (180)  |  Demonstrate (79)  |  Human (1512)  |  Investigation (250)  |  Real (159)

[Adams] supposed that, except musicians, everyone thought Beethoven a bore, as every one except mathematicians thought mathematics a bore.
The Education of Henry Brooks Adams: An Autobiography (1918), 80.
Science quotes on:  |  Beethoven (14)  |  Musician (23)  |  Thought (995)

[All phenomena] are equally susceptible of being calculated, and all that is necessary, to reduce the whole of nature to laws similar to those which Newton discovered with the aid of the calculus, is to have a sufficient number of observations and a mathematics that is complex enough.
Unpublished Manuscript. Quoted In Frank Edward Manuel and Fritzie Prigohzy Manuel, Utopian Thought in the Western World (1979, 2009), 493.
Science quotes on:  |  Aid (101)  |  Being (1276)  |  Calculus (65)  |  Complex (202)  |  Discover (571)  |  Enough (341)  |  Equally (129)  |  Law (913)  |  Nature (2017)  |  Necessary (370)  |  Sir Isaac Newton (363)  |  Number (710)  |  Observation (593)  |  Reduce (100)  |  Sufficient (133)  |  Whole (756)

[An appealing problem is] a combination of being fairly concrete—so one can understand concretely examples—and also connecting with a lot of other ideas. For example, you see the analysis in a minimal surface equation, but then you also realize it has connections with other geometric questions that are not just analysis. I am definitely very attracted to the idea that there are a lot of different facets in mathematics and seeing the connections.
From Allyn Jackson, 'Interview with Karen Uhlenbeck', part of Celebratio Mathematica on the celebratio.org website.
Science quotes on:  |  Analysis (244)  |  Attraction (61)  |  Biography (254)  |  Combination (150)  |  Concrete (55)  |  Connection (171)  |  Equation (138)  |  Facet (9)  |  Geometry (271)  |  Idea (881)  |  Problem (731)  |  Question (649)  |  Understanding (527)

[At high school in Cape Town] my interests outside my academic work were debating, tennis, and to a lesser extent, acting. I became intensely interested in astronomy and devoured the popular works of astronomers such as Sir Arthur Eddington and Sir James Jeans, from which I learnt that a knowledge of mathematics and physics was essential to the pursuit of astronomy. This increased my fondness for those subjects.
'Autobiography of Allan M. Cormack,' Les Prix Nobel/Nobel Lectures 1979, editted by Wilhelm Odelberg.
Science quotes on:  |  Acting (6)  |  Astronomer (97)  |  Astronomy (251)  |  Biography (254)  |  Debate (40)  |  Devour (29)  |  Sir Arthur Stanley Eddington (135)  |  Essential (210)  |  Extent (142)  |  Fondness (7)  |  High (370)  |  Interest (416)  |  Sir James Jeans (34)  |  Knowledge (1647)  |  Learning (291)  |  Outside (141)  |  Physic (515)  |  Physics (564)  |  Pursuit (128)  |  School (227)  |  Subject (543)  |  Tennis (8)  |  Work (1402)

[Before the time of Benjamin Peirce it never occurred to anyone that mathematical research] was one of the things for which a mathematical department existed. Today it is a commonplace in all the leading universities. Peirce stood alone—a mountain peak whose absolute height might be hard to measure, but which towered above all the surrounding country.
In 'The Story of Mathematics at Harvard', Harvard Alumni Bulletin (3 Jan 1924), 26, 376. Cited by R. C. Archibald in 'Benjamin Peirce: V. Biographical Sketch', The American Mathematical Monthly (Jan 1925), 32, No. 1, 10.
Science quotes on:  |  Absolute (153)  |  Alone (324)  |  Commonplace (24)  |  Country (269)  |  Department (93)  |  Exist (458)  |  Hard (246)  |  Height (33)  |  Leading (17)  |  Measure (241)  |  Mountain (202)  |  Never (1089)  |  Occurred (2)  |  Peak (20)  |  Benjamin Peirce (11)  |  Research (753)  |  Surrounding (13)  |  Thing (1914)  |  Time (1911)  |  Today (321)  |  Tower (45)  |  University (130)

[Cantor’s set theory:] The finest product of mathematical genius and one of the supreme achievements of purely intellectual human activity.
As quoted in Constance Reid, Hilbert (1970), 176.
Science quotes on:  |  Achievement (187)  |  Activity (218)  |  Fine (37)  |  Genius (301)  |  Human (1512)  |  Intellectual (258)  |  Product (166)  |  Purely (111)  |  Set (400)  |  Set Theory (6)  |  Supreme (73)  |  Theory (1015)

[Comte] may truly be said to have created the philosophy of higher mathematics.
In System of Logic (1846), 369.
Science quotes on:  |  Auguste Comte (24)  |  Create (245)  |  Higher Mathematics (7)  |  Mathematicians and Anecdotes (141)  |  Philosophy (409)  |  Say (989)  |  Truly (118)

[Euclid's Elements] has been for nearly twenty-two centuries the encouragement and guide of that scientific thought which is one thing with the progress of man from a worse to a better state. The encouragement; for it contained a body of knowledge that was really known and could be relied on, and that moreover was growing in extent and application. For even at the time this book was written—shortly after the foundation of the Alexandrian Museum—Mathematics was no longer the merely ideal science of the Platonic school, but had started on her career of conquest over the whole world of Phenomena. The guide; for the aim of every scientific student of every subject was to bring his knowledge of that subject into a form as perfect as that which geometry had attained. Far up on the great mountain of Truth, which all the sciences hope to scale, the foremost of that sacred sisterhood was seen, beckoning for the rest to follow her. And hence she was called, in the dialect of the Pythagoreans, ‘the purifier of the reasonable soul.’
From a lecture delivered at the Royal Institution (Mar 1873), collected postumously in W.K. Clifford, edited by Leslie Stephen and Frederick Pollock, Lectures and Essays, (1879), Vol. 1, 296.
Science quotes on:  |  Aim (175)  |  Alexandria (2)  |  Application (257)  |  Attain (126)  |  Beckoning (4)  |  Better (493)  |  Body (557)  |  Book (413)  |  Call (781)  |  Career (86)  |  Conquest (31)  |  Element (322)  |  Encouragement (27)  |  Euclid (60)  |  Extent (142)  |  Follow (389)  |  Following (16)  |  Form (976)  |  Foundation (177)  |  Geometry (271)  |  Great (1610)  |  Growing (99)  |  Guide (107)  |  Hope (321)  |  Ideal (110)  |  Knowledge (1647)  |  Known (453)  |  Man (2252)  |  Merely (315)  |  Mountain (202)  |  Museum (40)  |  Nearly (137)  |  Perfect (223)  |  Perfection (131)  |  Progress (492)  |  Reason (766)  |  Rest (287)  |  Sacred (48)  |  Scale (122)  |  School (227)  |  Scientific (955)  |  Scientific Thought (17)  |  Soul (235)  |  Start (237)  |  State (505)  |  Student (317)  |  Subject (543)  |  Thing (1914)  |  Thought (995)  |  Time (1911)  |  Truth (1109)  |  Two (936)  |  Whole (756)  |  Whole World (29)  |  World (1850)

[Experimental Physicist] Phys. I know that it is often a help to represent pressure and volume as height and width on paper; and so geometry may have applications to the theory of gases. But is it not going rather far to say that geometry can deal directly with these things and is not necessarily concerned with lengths in space?
[Mathematician] Math. No. Geometry is nowadays largely analytical, so that in form as well as in effect, it deals with variables of an unknown nature. …It is literally true that I do not want to know the significance of the variables x, y, z, t that I am discussing. …
Phys. Yours is a strange subject. You told us at the beginning that you are not concerned as to whether your propositions are true, and now you tell us you do not even care to know what you are talking about.
Math. That is an excellent description of Pure Mathematics, which has already been given by an eminent mathematician [Bertrand Russell].
In Space, Time and Gravitation: An Outline of the General Relativity Theory (1920, 1921), 14.
Science quotes on:  |  Already (226)  |  Application (257)  |  Beginning (312)  |  Care (203)  |  Concern (239)  |  Deal (192)  |  Do (1905)  |  Effect (414)  |  Experimental (193)  |  Experimental Physicist (11)  |  Form (976)  |  Geometry (271)  |  Know (1538)  |  Literally (30)  |  Nature (2017)  |  Necessarily (137)  |  Paper (192)  |  Physicist (270)  |  Pressure (69)  |  Proposition (126)  |  Pure (299)  |  Pure Mathematics (72)  |  Represent (157)  |  Say (989)  |  Significance (114)  |  Space (523)  |  Strange (160)  |  Subject (543)  |  Talking (76)  |  Tell (344)  |  Theory (1015)  |  Thing (1914)  |  Unknown (195)  |  Variable (37)  |  Want (504)

[I can] scarcely write upon mathematics or mathematicians. Oh for words to express my abomination of the science.
Lamenting mathematics whilst an undergraduate at Cambridge, 1818.
Quoted in John Gascoigne, Cambridge in the Age of Enlightenment (1989), 272.
Science quotes on:  |  Abomination (3)  |  Express (192)  |  Scarcely (75)  |  Undergraduate (17)  |  Word (650)  |  Write (250)

[I was advised] to read Jordan's 'Cours d'analyse'; and I shall never forget the astonishment with which I read that remarkable work, the first inspiration for so many mathematicians of my generation, and learnt for the first time as I read it what mathematics really meant.
In A Mathematician’s Apology (1940, reprint with Foreward by C.P. Snow 1992), 23.
Science quotes on:  |  Astonishment (30)  |  First (1302)  |  Forget (125)  |  Generation (256)  |  Inspiration (80)  |  Mathematician (407)  |  Never (1089)  |  Read (308)  |  Time (1911)  |  Work (1402)

[I]f in other sciences we should arrive at certainty without doubt and truth without error, it behooves us to place the foundations of knowledge in mathematics, in so far as disposed through it we are able to reach certainty in other sciences and truth by the exclusion of error. (c.1267)
Translation by Robert Burke, Opus Majus of Roger Bacon (1928), vol 1, 124. In Fred R. Shapiro, The Yale Book of Quotations (2006), 39.
Science quotes on:  |  Behoove (6)  |  Certainty (180)  |  Doubt (314)  |  Error (339)  |  Exclusion (16)  |  Foundation (177)  |  Knowledge (1647)  |  Other (2233)  |  Reach (286)  |  Through (846)  |  Truth (1109)

[In junior high school] I liked math—that was my favorite subject—and I was very interested in astronomy and in physical science.
Interview conducted on Scholastic website (20 Nov 1998).
Science quotes on:  |  Astronomy (251)  |  Favorite (37)  |  High (370)  |  Interest (416)  |  Junior (6)  |  Junior High (3)  |  Physical (518)  |  Physical Science (104)  |  School (227)  |  Subject (543)

[In mathematics] There are two kinds of mistakes. There are fatal mistakes that destroy a theory, but there are also contingent ones, which are useful in testing the stability of a theory.
In 'Ten Lessons I Wish I Had Been Taught', Indiscrete Thoughts (2008), 202.
Science quotes on:  |  Contingent (12)  |  Destroy (189)  |  Fatal (14)  |  Kind (564)  |  Mistake (180)  |  Stability (28)  |  Testing (5)  |  Theory (1015)  |  Two (936)  |  Useful (260)

[In mathematics] we behold the conscious logical activity of the human mind in its purest and most perfect form. Here we learn to realize the laborious nature of the process, the great care with which it must proceed, the accuracy which is necessary to determine the exact extent of the general propositions arrived at, the difficulty of forming and comprehending abstract concepts; but here we learn also to place confidence in the certainty, scope and fruitfulness of such intellectual activity.
In Ueber das Verhältnis der Naturwissenschaften zur Gesammtheit der Wissenschaft, Vorträge und Reden (1896), Bd. 1, 176. Also seen translated as “In mathematics we see the conscious logical activity of our mind in its purest and most perfect form; here is made manifest to us all the labor and the great care with which it progresses, the precision which is necessary to determine exactly the source of the established general theorems, and the difficulty with which we form and comprehend abstract conceptions; but we also learn here to have confidence in the certainty, breadth, and fruitfulness of such intellectual labor”, in Robert Édouard Moritz, Memorabilia Mathematica; Or, The Philomath’s Quotation-book (1914), 20. From the original German, “Hier sehen wir die bewusste logische Thätigkeit unseres Geistes in ihrer reinsten und vollendetsten Form; wir können hier die ganze Mühe derselben kennen lernen, die grosse Vorsicht, mit der sie vorschreiten muss, die Genauigkeit, welche nöthig ist, um den Umfang der gewonnenen allgemeinen Sätze genau zu bestimmen, die Schwierigkeit, abstracte Begriffe zu bilden und zu verstehen; aber ebenso auch Vertrauen fassen lernen in die Sicherheit, Tragweite und Fruchtbarkeit solcher Gedankenarbeit.”
Science quotes on:  |  Abstract (141)  |  Accuracy (81)  |  Activity (218)  |  Arrive (40)  |  Behold (19)  |  Care (203)  |  Certainty (180)  |  Comprehend (44)  |  Concept (242)  |  Confidence (75)  |  Conscious (46)  |  Determine (152)  |  Difficulty (201)  |  Exact (75)  |  Extent (142)  |  Form (976)  |  Forming (42)  |  Fruitfulness (2)  |  General (521)  |  Great (1610)  |  Human (1512)  |  Human Mind (133)  |  Intellectual (258)  |  Laborious (17)  |  Learn (672)  |  Logical (57)  |  Mathematics And Logic (27)  |  Mind (1377)  |  Most (1728)  |  Must (1525)  |  Nature (2017)  |  Nature Of Mathematics (80)  |  Necessary (370)  |  Perfect (223)  |  Place (192)  |  Proceed (134)  |  Process (439)  |  Proposition (126)  |  Pure (299)  |  Realize (157)  |  Scope (44)

[In the Royal Society, there] has been, a constant Resolution, to reject all the amplifications, digressions, and swellings of style: to return back to the primitive purity, and shortness, when men deliver'd so many things, almost in an equal number of words. They have exacted from all their members, a close, naked, natural way of speaking; positive expressions; clear senses; a native easiness: bringing all things as near the Mathematical plainness, as they can: and preferring the language of Artizans, Countrymen, and Merchants, before that, of Wits, or Scholars.
The History of the Royal Society (1667), 113.
Science quotes on:  |  Amplification (3)  |  Back (395)  |  Clarity (49)  |  Constant (148)  |  Countryman (4)  |  Deliver (30)  |  Digression (3)  |  Easiness (4)  |  Expression (181)  |  Language (308)  |  Member (42)  |  Merchant (7)  |  Native (41)  |  Natural (810)  |  Number (710)  |  Plainness (2)  |  Positive (98)  |  Primitive (79)  |  Purity (15)  |  Reject (67)  |  Rejection (36)  |  Resolution (24)  |  Return (133)  |  Royal (56)  |  Royal Society (17)  |  Scholar (52)  |  Sense (785)  |  Society (350)  |  Speaking (118)  |  Style (24)  |  Swelling (5)  |  Thing (1914)  |  Way (1214)  |  Wit (61)  |  Word (650)

[Karen] Uhlenbeck’s research has led to revolutionary advances at the intersection of mathematics and physics. Her pioneering insights have applications across a range of fascinating subjects, from string theory, which may help explain the nature of reality, to the geometry of space-time.
In news release, 'Mathematics’ Highest Prize Awarded to UT Austin’s Karen Uhlenbeck', UT News (19 Mar 2019) on website of University of Texas at Austin.
Science quotes on:  |  Geometry (271)  |  Physics (564)  |  Reality (274)  |  Research (753)  |  Space-Time (20)  |  String Theory (14)  |  Karen Uhlenbeck (7)

[Kepler] had to realize clearly that logical-mathematical theoretizing, no matter how lucid, could not guarantee truth by itself; that the most beautiful logical theory means nothing in natural science without comparison with the exactest experience. Without this philosophic attitude, his work would not have been possible.
From Introduction that Einstein wrote for Carola Baumgardt and Jamie Callan, Johannes Kepler Life and Letters (1953), 13.
Science quotes on:  |  Attitude (84)  |  Beautiful (271)  |  Clearly (45)  |  Comparison (108)  |  Experience (494)  |  Guarantee (30)  |  Johannes Kepler (95)  |  Logic (311)  |  Lucid (9)  |  Matter (821)  |  Mean (810)  |  Means (587)  |  Most (1728)  |  Natural (810)  |  Natural Science (133)  |  Nothing (1000)  |  Philosophy (409)  |  Possible (560)  |  Realize (157)  |  Theory (1015)  |  Truth (1109)  |  Work (1402)

[Mathematics is] the study of ideal constructions (often applicable to real problems), and the discovery thereby of relations between the parts of these constructions, before unknown.
In 'Mathematics', Century Dictionary.
Science quotes on:  |  Applicable (31)  |  Construction (114)  |  Definitions and Objects of Mathematics (33)  |  Discovery (837)  |  Ideal (110)  |  Often (109)  |  Part (235)  |  Problem (731)  |  Real (159)  |  Relation (166)  |  Study (701)  |  Unknown (195)

[Mathematics is] the study of the measurement, properties, and relationships of quantities and sets, using numbers and symbols.
Definition of Mathematics in William morris (ed.), American Heritage Dictionary (2000).
Science quotes on:  |  Measurement (178)  |  Number (710)  |  Property (177)  |  Quantity (136)  |  Relationship (114)  |  Set (400)  |  Study (701)  |  Symbol (100)

[Mathematics] has for its object the indirect measurement of magnitudes, and it proposes to determine magnitudes by each other, according to the precise relations which exist between them.
In The Positive Philosophy of Auguste Comte, translated by Harriet Martineau, (1896), Vol. 1, 40.
Science quotes on:  |  Accord (36)  |  According (236)  |  Definitions and Objects of Mathematics (33)  |  Determine (152)  |  Exist (458)  |  Indirect (18)  |  Magnitude (88)  |  Measurement (178)  |  Object (438)  |  Other (2233)  |  Precise (71)  |  Propose (24)  |  Relation (166)

[Mathematics] is an independent world
Created out of pure intelligence.
In The Prelude, Book 6, lines 186-187. [Preceding lines refer to “laws of Nature” and “a treatise of geometry.” Wordsworth did not use the word “mathematics”, which is added parenthetically to give context to the quote.]
Science quotes on:  |  Create (245)  |  Independent (74)  |  Intelligence (218)  |  Pure (299)  |  World (1850)

[Mathematics] is security. Certainty. Truth. Beauty. Insight. Structure. Architecture. I see mathematics, the part of human knowledge that I call mathematics, as one thing—one great, glorious thing. Whether it is differential topology, or functional analysis, or homological algebra, it is all one thing. … They are intimately interconnected, they are all facets of the same thing. That interconnection, that architecture, is secure truth and is beauty. That’s what mathematics is to me.
From interview with Donald J. Albers. In John H. Ewing and Frederick W. Gehring, Paul Halmos Celebrating 50 Years of Mathematics (1991), 13.
Science quotes on:  |  Algebra (117)  |  Analysis (244)  |  Architecture (50)  |  Beauty (313)  |  Call (781)  |  Certainty (180)  |  Glorious (49)  |  Great (1610)  |  Human (1512)  |  Insight (107)  |  Interconnection (12)  |  Knowledge (1647)  |  Mathematical Beauty (19)  |  Security (51)  |  See (1094)  |  Structure (365)  |  Thing (1914)  |  Truth (1109)

[Mathematics] is that [subject] which knows nothing of observation, nothing of experiment, nothing of induction, nothing of causation.
In 'The Scientific Aspects of Positivism', Fortnightly Review (1898) in Lay Sermons, Addresses and Reviews, (1872), 169.
Science quotes on:  |  Causation (14)  |  Experiment (736)  |  Induction (81)  |  Know (1538)  |  Nature Of Mathematics (80)  |  Nothing (1000)  |  Observation (593)  |  Subject (543)

[P]olitical and social and scientific values … should be correlated in some relation of movement that could be expressed in mathematics, nor did one care in the least that all the world said it could not be done, or that one knew not enough mathematics even to figure a formula beyond the schoolboy s=(1/2)gt2. If Kepler and Newton could take liberties with the sun and moon, an obscure person ... could take liberties with Congress, and venture to multiply its attraction into the square of its time. He had only to find a value, even infinitesimal, for its attraction.
The Education of Henry Adams: An Autobiography? (1918), 376.
Science quotes on:  |  Attraction (61)  |  Beyond (316)  |  Care (203)  |  Congress (20)  |  Enough (341)  |  Express (192)  |  Figure (162)  |  Find (1014)  |  Formula (102)  |  Infinitesimal (30)  |  Johannes Kepler (95)  |  Moon (252)  |  Motion (320)  |  Movement (162)  |  Multiply (40)  |  Sir Isaac Newton (363)  |  Obscure (66)  |  Person (366)  |  Politics (122)  |  Scientific (955)  |  Social (261)  |  Society (350)  |  Square (73)  |  Sun (407)  |  Time (1911)  |  Value (393)  |  World (1850)

[P]ure mathematics is on the whole distinctly more useful than applied. For what is useful above all is technique, and mathematical technique is taught mainly through pure mathematics.
In A Mathematician’s Apology (1940, reprint with Foreward by C.P. Snow 1992), 134.
Science quotes on:  |  Applied (176)  |  More (2558)  |  Pure (299)  |  Pure Mathematics (72)  |  Technique (84)  |  Through (846)  |  Useful (260)  |  Whole (756)

[Pure mathematics is] good to give chills in the spine to a certain number of people, me included. I don’t know what else it is good for, and I don’t care. But … like von Neumann said, one never knows whether someone is going to find another use for it.
In The Beauty of Doing Mathematics: Three Public Dialogues (1985), 49.
Science quotes on:  |  Care (203)  |  Certain (557)  |  Chill (10)  |  Find (1014)  |  Good (906)  |  Know (1538)  |  Never (1089)  |  Number (710)  |  People (1031)  |  Pure (299)  |  Pure Mathematics (72)  |  Spine (9)  |  Use (771)  |  John von Neumann (29)

[Referring to Fourier’s mathematical theory of the conduction of heat] … Fourier's great mathematical poem…
In W. Thomson and P. G. Tait, Treatise on Natural Philosophy. Reprinted as Principles of Mechanics and Dynamics (2000), 470.
Science quotes on:  |  Biography (254)  |  Conduction (8)  |  Baron Jean-Baptiste-Joseph Fourier (17)  |  Great (1610)  |  Heat (180)  |  Poem (104)  |  Theory (1015)

[Regarding mathematics,] there are now few studies more generally recognized, for good reasons or bad, as profitable and praiseworthy. This may be true; indeed it is probable, since the sensational triumphs of Einstein, that stellar astronomy and atomic physics are the only sciences which stand higher in popular estimation.
In A Mathematician's Apology (1940, reprint with Foreward by C.P. Snow 1992), 63-64.
Science quotes on:  |  Astronomy (251)  |  Atomic Physics (7)  |  Bad (185)  |  Einstein (101)  |  Albert Einstein (624)  |  Good (906)  |  Indeed (323)  |  More (2558)  |  Physic (515)  |  Physics (564)  |  Profitable (29)  |  Reason (766)  |  Stand (284)  |  Triumph (76)

[Student:} I only use my math book on special equations.
From movie Rock 'n' Roll High School (1979). Writers, Richard Whitley, Russ Dvonch and Joseph McBride. In Larry Langman and Paul Gold, Comedy Quotes from the Movies (2001), 359.
Science quotes on:  |  Book (413)  |  Equation (138)  |  Quip (81)  |  Special (188)  |  Student (317)  |  Use (771)

[The error in the teaching of mathematics is that] mathematics is expected either to be immediately attractive to students on its own merits or to be accepted by students solely on the basis of the teacher’s assurance that it will be helpful in later life. [And yet,] mathematlcs is the key to understanding and mastering our physical, social and biological worlds.
In editorial in Focus, a Journal of the Mathematical Association of America (1986), quoted in obituary by Eric Pace, New York Times (11 Jun 1992).
Science quotes on:  |  Accept (198)  |  Assurance (17)  |  Attractive (25)  |  Basis (180)  |  Biological (137)  |  Error (339)  |  Expect (203)  |  Helpful (16)  |  Immediately (115)  |  Life (1870)  |  Mastering (11)  |  Merit (51)  |  Physical (518)  |  Relevance (18)  |  Social (261)  |  Student (317)  |  Teacher (154)  |  Teaching (190)  |  Understanding (527)  |  Will (2350)  |  World (1850)

[The famous attack of Sir William Hamilton on the tendency of mathematical studies] affords the most express evidence of those fatal lacunae in the circle of his knowledge, which unfitted him for taking a comprehensive or even an accurate view of the processes of the human mind in the establishment of truth. If there is any pre-requisite which all must see to be indispensable in one who attempts to give laws to the human intellect, it is a thorough acquaintance with the modes by which human intellect has proceeded, in the case where, by universal acknowledgment, grounded on subsequent direct verification, it has succeeded in ascertaining the greatest number of important and recondite truths. This requisite Sir W. Hamilton had not, in any tolerable degree, fulfilled. Even of pure mathematics he apparently knew little but the rudiments. Of mathematics as applied to investigating the laws of physical nature; of the mode in which the properties of number, extension, and figure, are made instrumental to the ascertainment of truths other than arithmetical or geometrical—it is too much to say that he had even a superficial knowledge: there is not a line in his works which shows him to have had any knowledge at all.
In Examination of Sir William Hamilton's Philosophy (1878), 607.
Science quotes on:  |  Accurate (88)  |  Acknowledgment (13)  |  Acquaintance (38)  |  Afford (19)  |  Apparently (22)  |  Applied (176)  |  Apply (170)  |  Arithmetical (11)  |  Ascertain (41)  |  Ascertainment (2)  |  Attack (86)  |  Attempt (266)  |  Case (102)  |  Circle (117)  |  Comprehensive (29)  |  Degree (277)  |  Direct (228)  |  Establishment (47)  |  Evidence (267)  |  Express (192)  |  Extension (60)  |  Famous (12)  |  Figure (162)  |  Fulfill (19)  |  Geometrical (11)  |  Give (208)  |  Great (1610)  |  Greatest (330)  |  Ground (222)  |  Hamilton (2)  |  Hamilton_William (2)  |  Human (1512)  |  Human Intellect (32)  |  Human Mind (133)  |  Important (229)  |  Indispensable (31)  |  Instrumental (5)  |  Intellect (251)  |  Investigate (106)  |  Know (1538)  |  Knowledge (1647)  |  Law (913)  |  Line (100)  |  Little (717)  |  Mathematicians and Anecdotes (141)  |  Mind (1377)  |  Mode (43)  |  Most (1728)  |  Must (1525)  |  Nature (2017)  |  Number (710)  |  Other (2233)  |  Physical (518)  |  Prerequisite (9)  |  Proceed (134)  |  Process (439)  |  Property (177)  |  Pure (299)  |  Pure Mathematics (72)  |  Recondite (8)  |  Requisite (12)  |  Rudiment (6)  |  Say (989)  |  See (1094)  |  Show (353)  |  Study (701)  |  Subsequent (34)  |  Succeed (114)  |  Superficial (12)  |  Tendency (110)  |  Thorough (40)  |  Tolerable (2)  |  Truth (1109)  |  Unfitted (3)  |  Universal (198)  |  Verification (32)  |  View (496)  |  Work (1402)

[The infinitely small] neither have nor can have theory; it is a dangerous instrument in the hands of beginners [ ... ] anticipating, for my part, the judgement of posterity, I would dare predict that this method will be accused one day, and rightly, of having retarded the progress of the mathematical sciences.
Annales des Mathematiques Pures et Appliquées (1814-5), 5, 148.
Science quotes on:  |  Accusation (6)  |  Anticipation (18)  |  Beginner (11)  |  Danger (127)  |  Dangerous (108)  |  Dare (55)  |  Differentiation (28)  |  Infinity (96)  |  Instrument (158)  |  Judgment (140)  |  Method (531)  |  Posterity (29)  |  Predict (86)  |  Prediction (89)  |  Progress (492)  |  Retardation (5)  |  Small (489)  |  Theory (1015)  |  Will (2350)

[The] humanization of mathematical teaching, the bringing of the matter and the spirit of mathematics to bear not merely upon certain fragmentary faculties of the mind, but upon the whole mind, that this is the greatest desideratum is. I assume, beyond dispute.
Address (28 Mar 1912), Michigan School Masters' Club, Ann Arbor, 'The Humanization of the Teaching of Mathematics. Printed in Science (26 Apr 1912). Collected in The Human Worth of Rigorous Thinking: Essays and Addresses (1916), 62-63.
Science quotes on:  |  Assume (43)  |  Bear (162)  |  Beyond (316)  |  Certain (557)  |  Desideratum (5)  |  Dispute (36)  |  Faculty (76)  |  Fragmentary (8)  |  Greatest (330)  |  Matter (821)  |  Merely (315)  |  Mind (1377)  |  Spirit (278)  |  Teaching (190)  |  Whole (756)

[There is] one distinctly human thing - the story. There can be as good science about a turnip as about a man. ... [Or philosophy, or theology] ...There can be, without any question at all, as good higher mathematics about a turnip as about a man. But I do not think, though I speak in a manner somewhat tentative, that there could be as good a novel written about a turnip as a man.
In 'A Much Repeated Repetition', Daily News (26 Mar 1904). Collected in G. K. Chesterton and Dale Ahlquist (ed.), In Defense of Sanity: The Best Essays of G.K. Chesterton (2011), 84.
Science quotes on:  |  Distinctly (5)  |  Do (1905)  |  Good (906)  |  Human (1512)  |  Man (2252)  |  Manner (62)  |  Novel (35)  |  Philosophy (409)  |  Question (649)  |  Speak (240)  |  Story (122)  |  Tentative (18)  |  Theology (54)  |  Thing (1914)  |  Think (1122)  |  Turnip (3)  |  Writing (192)

[There is] some mathematical quality in Nature, a quality which the casual observer of Nature would not suspect, but which nevertheless plays an important role in Nature’s scheme.
From Lecture delivered on presentation of the James Scott prize, (6 Feb 1939), 'The Relation Between Mathematics And Physics', printed in Proceedings of the Royal Society of Edinburgh (1938-1939), 59, Part 2, 122.
Science quotes on:  |  Casual (9)  |  Important (229)  |  Nature (2017)  |  Nevertheless (90)  |  Observer (48)  |  Quality (139)  |  Role (86)  |  Scheme (62)  |  Science And Mathematics (10)  |  Suspect (18)

[There was] in some of the intellectual leaders a great aspiration to demonstrate that the universe ran like a piece of clock-work, but this was was itself initially a religious aspiration. It was felt that there would be something defective in Creation itself—something not quite worthy of God—unless the whole system of the universe could be shown to be interlocking, so that it carried the pattern of reasonableness and orderliness. Kepler, inaugurating the scientist’s quest for a mechanistic universe in the seventeenth century, is significant here—his mysticism, his music of the spheres, his rational deity demand a system which has the beauty of a piece of mathematics.
In The Origins of Modern Science (1950), 105.
Science quotes on:  |  17th Century (20)  |  Aspiration (35)  |  Beauty (313)  |  Century (319)  |  Clock (51)  |  Clockwork (7)  |  Creation (350)  |  Defective (4)  |  Deity (22)  |  Demand (131)  |  Demonstrate (79)  |  God (776)  |  Great (1610)  |  Intellectual (258)  |  Johannes Kepler (95)  |  Leader (51)  |  Mathematical Beauty (19)  |  Music (133)  |  Music Of The Spheres (3)  |  Mysticism (14)  |  Orderliness (9)  |  Orderly (38)  |  Pattern (116)  |  Quest (39)  |  Rational (95)  |  Reasonable (29)  |  Reasonableness (6)  |  Religious (134)  |  Scientist (881)  |  Significant (78)  |  Something (718)  |  Sphere (118)  |  System (545)  |  Universe (900)  |  Whole (756)  |  Work (1402)

[There] are still to be found text-books of the old sort, teaching Mathematics under the guise of Physics, presenting nothing but the dry husks of the latter.
A paper read at the Association for the Improvement of Geometrical Teaching (19 Jan 1889), 'The Vices of our Scientific Education', in Nature (6 Jun 1889), 40, 128.
Science quotes on:  |  Book (413)  |  Dry (65)  |  Husk (4)  |  Nothing (1000)  |  Old (499)  |  Physic (515)  |  Physics (564)  |  Still (614)  |  Teaching (190)  |  Textbook (39)

[Urbain Jean Joseph] Le Verrier—without leaving his study, without even looking at the sky—had found the unknown planet [Neptune] solely by mathematical calculation, and, as it were, touched it with the tip of his pen!
In Camille Flammarion, Astronomy (1914), 171.
Science quotes on:  |  Calculation (134)  |  Discovery (837)  |  Looking (191)  |  Neptune (13)  |  Pen (21)  |  Planet (402)  |  Sky (174)  |  Study (701)  |  Tip (2)  |  Touch (146)  |  Tribute (10)  |  Unknown (195)  |  Urbain-Jean-Joseph Le Verrier (4)

[W]hen Galileo discovered he could use the tools of mathematics and mechanics to understand the motion of celestial bodies, he felt, in the words of one imminent researcher, that he had learned the language in which God recreated the universe. Today we are learning the language in which God created life. We are gaining ever more awe for the complexity, the beauty, the wonder of God's most devine and sacred gift.
From White House Announcement of the Completion of the First Survey of the Entire Human Genome Project, broadcast on the day of the publication of the first draft of the human genome. Quoted in transcript on the National Archives, Clinton White House web site, 'Text of Remarks on the Completion of the First Survey of the Entire Human Genome Project' (26 Jun 2000).
Science quotes on:  |  Awe (43)  |  Beauty (313)  |  Celestial (53)  |  Complexity (121)  |  Discover (571)  |  Galileo Galilei (134)  |  Gift (105)  |  God (776)  |  Language (308)  |  Learn (672)  |  Learned (235)  |  Learning (291)  |  Life (1870)  |  Mathematical Beauty (19)  |  Mechanic (120)  |  Mechanics (137)  |  More (2558)  |  Most (1728)  |  Motion (320)  |  Planets (2)  |  Researcher (36)  |  Sacred (48)  |  Today (321)  |  Tool (129)  |  Understand (648)  |  Understanding (527)  |  Universe (900)  |  Use (771)  |  Wonder (251)  |  Word (650)

[We] can easily distinguish what relates to Mathematics in any question from that which belongs to the other sciences. But as I considered the matter carefully it gradually came to light that all those matters only were referred to Mathematics in which order and measurements are investigated, and that it makes no difference whether it be in numbers, figures, stars, sounds or any other object that the question of measurement arises. I saw consequently that there must be some general science to explain that element as a whole which gives rise to problems about order and measurement, restricted as these are to no special subject matter. This, I perceived was called “Universal Mathematics,” not a far-fetched asignation, but one of long standing which has passed into current use, because in this science is contained everything on account of which the others are called parts of Mathematics.
Rules for the Direction of the Mind (written 1628). As translated by Elizabeth Sanderson Haldane and George Robert Thomson Ross in The Philosophical Works of Descartes (1911, 1931), 13.
Science quotes on:  |  Account (195)  |  Arise (162)  |  Belong (168)  |  Call (781)  |  Carefully (65)  |  Consider (428)  |  Current (122)  |  Difference (355)  |  Distinguish (168)  |  Element (322)  |  Everything (489)  |  Explain (334)  |  Figure (162)  |  General (521)  |  Gradually (102)  |  Investigate (106)  |  Light (635)  |  Long (778)  |  Matter (821)  |  Measurement (178)  |  Must (1525)  |  Number (710)  |  Object (438)  |  Order (638)  |  Other (2233)  |  Pass (241)  |  Problem (731)  |  Question (649)  |  Rise (169)  |  Saw (160)  |  Sound (187)  |  Special (188)  |  Star (460)  |  Stars (304)  |  Subject (543)  |  Universal (198)  |  Use (771)  |  Whole (756)

Apud me omnia fiunt Mathematicè in Natura.
In my opinion, everything happens in nature in a mathematical way.
In letter (11 Mar 1640) to Père Marin Mersenne. English version inspired by a translation of the original Latin in German, “Nach meiner Ansicht geschieht alles in der Natur auf mathematische Art,” in René Descartes and Artur Buchenau (trans., ed.), René Descartes' Philosophische Werke (1905), 246. The Latin is often seen misquoted as “omnia apud me mathematica fiunt.” See context in longer quote that begins, “I have no doubt….” on the René Descartes Quotes page of this website.
Science quotes on:  |  Everything (489)  |  Happen (282)  |  Nature (2017)  |  Opinion (291)  |  Way (1214)

Ces détails scientifiques qui effarouchent les fabricans d’un certain âge, ne seront qu’un jeu pour leurs enfans, quand ils auront apprit dans leurs collèges un peu plus de mathématiques et un peu moins de Latin; un peu plus de Chimie, et un peu moins de Grec!
The scientific details which now terrify the adult manufacturer will be mere trifles to his children when they shall be taught at school, a little more Mathematics and a little less Latin, a little more Chemistry, and a little less Greek.
As quoted in 'Sketches From Life of Some Eminent Foreign Scientific Lecturers: Dumas', Magazine of Popular Science, and Journal of the Useful Arts (1836). Vol. 1, 177.
Science quotes on:  |  Certain (557)  |  Chemistry (376)  |  Child (333)  |  Children (201)  |  Detail (150)  |  Education (423)  |  Greek (109)  |  Latin (44)  |  Little (717)  |  Manufacturer (10)  |  More (2558)  |  Plus (43)  |  School (227)  |  Scientific (955)  |  Taught (4)  |  Terrify (12)  |  Trifle (18)  |  Will (2350)

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.]
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.
Science quotes on:  |  Basis (180)  |  Finite (60)  |  First (1302)  |  David Hilbert (45)  |  Joke (90)  |  Proof (304)  |  Theology (54)  |  Theorem (116)

Das Leben der Gotter ist Mathematik.
Mathematics is the Life of the Gods.
Attributed.
Science quotes on:  |  God (776)  |  Life (1870)

Every teacher certainly should know something of non-euclidean geometry. Thus, it forms one of the few parts of mathematics which, at least in scattered catch-words, is talked about in wide circles, so that any teacher may be asked about it at any moment. … Imagine a teacher of physics who is unable to say anything about Röntgen rays, or about radium. A teacher of mathematics who could give no answer to questions about non-euclidean geometry would not make a better impression.
On the other hand, I should like to advise emphatically against bringing non-euclidean into regular school instruction (i.e., beyond occasional suggestions, upon inquiry by interested pupils), as enthusiasts are always recommending. Let us be satisfied if the preceding advice is followed and if the pupils learn to really understand euclidean geometry. After all, it is in order for the teacher to know a little more than the average pupil.
In George Edward Martin, The Foundations of Geometry and the Non-Euclidean Plane (1982), 72.
Science quotes on:  |  Advice (57)  |  Advise (7)  |  Against (332)  |  Answer (389)  |  Ask (420)  |  Average (89)  |  Better (493)  |  Beyond (316)  |  Bring (95)  |  Certainly (185)  |  Circle (117)  |  Emphatically (8)  |  Enthusiast (9)  |  Euclidean (3)  |  Follow (389)  |  Form (976)  |  Geometry (271)  |  Give (208)  |  Imagine (176)  |  Impression (118)  |  Inquiry (88)  |  Instruction (101)  |  Interest (416)  |  Know (1538)  |  Learn (672)  |  Least (75)  |  Let (64)  |  Little (717)  |  Moment (260)  |  More (2558)  |  Non-Euclidean (7)  |  Occasional (23)  |  On The Other Hand (40)  |  Order (638)  |  Other (2233)  |  Part (235)  |  Physic (515)  |  Physics (564)  |  Precede (23)  |  Pupil (62)  |  Question (649)  |  Radium (29)  |  Ray (115)  |  Really (77)  |  Recommend (27)  |  Regular (48)  |  Wilhelm Röntgen (8)  |  Satisfied (23)  |  Say (989)  |  Scatter (7)  |  School (227)  |  Something (718)  |  Suggestion (49)  |  Talk (108)  |  Teacher (154)  |  Unable (25)  |  Understand (648)  |  Wide (97)  |  Word (650)  |  X-ray (43)

Je me rends parfaitement compte du desagreable effet que produit sur la majorite de l'humanité, tout ce qui se rapporte, même au plus faible dègré, á des calculs ou raisonnements mathematiques.
I am well aware of the disagreeable effect produced on the majority of humanity, by whatever relates, even at the slightest degree to calculations or mathematical reasonings.
From 'French Reply to Baron Czyllak' concerning the game at Monte Carlo, Monte Carlo Facts and Fallacies (1904), 290, originally published in L'Écho de la Mediterranée as a response to an earlier open letter by the Baron in the same magazine. Maxim defended his prior mathematical calculations about gambling games. At the end of his paper giving a cautionary mathematical analysis of 'The Gambler's Ruin', < a href="http://todayinsci.com/C/Coolidge_Julian/CoolidgeJulian-Quotations.htm">Julian Coolidge referenced this quotation, saying “it gives the best explanation which I have seen for the fact that the people continue to gamble.”
Science quotes on:  |  Aware (36)  |  Calculation (134)  |  Degree (277)  |  Disagreeable (5)  |  Effect (414)  |  Humanity (186)  |  Majority (68)  |  Plus (43)  |  Produced (187)  |  Reasoning (212)  |  Reasonings (2)  |  Slightest (2)  |  Whatever (234)

La chaleur pénètre, comme la gravité, toutes les substances de l’univers, ses rayons occupent toutes les parties de l’espace. Le but de notre ouvrage est d’exposer les lois mathématiques que suit cet élément. Cette théorie formera désormais une des branches les plus importantes de la physique générale.
Heat, like gravity, penetrates every substance of the universe, its rays occupy all parts of space. The object of our work is to set forth the mathematical laws which this element obeys. The theory of heat will hereafter form one of the most important branches of general physics.
From 'Discours Préliminaire' to Théorie Analytique de la Chaleur (1822), i, translated by Alexander Freeman in The Analytical Theory of Heat (1878), 1.
Science quotes on:  |  Branch (155)  |  Element (322)  |  Form (976)  |  General (521)  |  Gravity (140)  |  Heat (180)  |  Important (229)  |  Law (913)  |  Most (1728)  |  Obey (46)  |  Object (438)  |  Occupy (27)  |  Part (235)  |  Penetrate (68)  |  Physic (515)  |  Physics (564)  |  Plus (43)  |  Ray (115)  |  Set (400)  |  Space (523)  |  Substance (253)  |  Theory (1015)  |  Universe (900)  |  Will (2350)  |  Work (1402)

Les mathématique sont un triple. Elles doivent fournir un instrument pour l'étude de la nature. Mais ce n'est pas tout: elles ont un but philosophique et, j'ose le dire, un but esthétique.
Mathematics has a threefold purpose. It must provide an instrument for the study of nature. But this is not all: it has a philosophical purpose, and, I daresay, an aesthetic purpose.
La valeur de la science. In Anton Bovier, Statistical Mechanics of Disordered Systems (2006), 161.
Science quotes on:  |  Aesthetic (48)  |  Dire (6)  |  Instrument (158)  |  Must (1525)  |  Nature (2017)  |  Purpose (336)  |  Study (701)

Longtemps les objets dont s'occupent les mathématiciens étaient our la pluspart mal définis; on croyait les connaître, parce qu'on se les représentatit avec le sens ou l'imagination; mais on n'en avait qu'une image grossière et non une idée précise sure laquelle le raisonment pût avoir prise.
For a long time the objects that mathematicians dealt with were mostly ill-defined; one believed one knew them, but one represented them with the senses and imagination; but one had but a rough picture and not a precise idea on which reasoning could take hold.
La valeur de la science. In Anton Bovier, Statistical Mechanics of Disordered Systems (2006), 97.
Science quotes on:  |  Idea (881)  |  Image (97)  |  Imagination (349)  |  Long (778)  |  Object (438)  |  Picture (148)  |  Precise (71)  |  Reasoning (212)  |  Represent (157)  |  Sense (785)  |  Time (1911)

Mathematical Knowledge adds a manly Vigour to the Mind, frees it from Prejudice, Credulity, and Superstition.
In An Essay On the Usefulness of Mathematical Learning, (1701), 7.
Science quotes on:  |  Credulity (16)  |  Free (239)  |  Knowledge (1647)  |  Mind (1377)  |  Prejudice (96)  |  Superstition (70)  |  Vigour (18)

Mathematical truth has validity independent of place, personality, or human authority. Mathematical relations are not established, nor can they be abrogated, by edict. The multiplication table is international and permanent, not a matter of convention nor of relying upon authority of state or church. The value of π is not amenable to human caprice. The finding of a mathematical theorem may have been a highly romantic episode in the personal life of the discoverer, but it cannot be expected of itself to reveal the race, sex, or temperament of this discoverer. With modern means of widespread communication even mathematical notation tends to be international despite all nationalistic tendencies in the use of words or of type.
Anonymous
In 'Light Thrown on the Nature of Mathematics by Certain Aspects of Its Development', Mathematics in General Education (1940), 256. This is the Report of the Committee on the Function of Mathematics in General Education of the Commission on Secondary School Curriculum, which was established by the Executive Board of the Progressive Education Association in 1932.
Science quotes on:  |  Amenable (4)  |  Authority (99)  |  Caprice (10)  |  Church (64)  |  Communication (101)  |  Convention (16)  |  Despite (7)  |  Discoverer (43)  |  Episode (5)  |  Establish (63)  |  Expect (203)  |  Human (1512)  |  Independent (74)  |  International (40)  |  Life (1870)  |  Matter (821)  |  Mean (810)  |  Means (587)  |  Modern (402)  |  Multiplication (46)  |  Multiplication Table (16)  |  Nation (208)  |  Notation (28)  |  Permanent (67)  |  Personal (75)  |  Personality (66)  |  Place (192)  |  Race (278)  |  Relation (166)  |  Reveal (152)  |  Romantic (13)  |  Sex (68)  |  State (505)  |  Table (105)  |  Temperament (18)  |  Tend (124)  |  Theorem (116)  |  Truth (1109)  |  Type (171)  |  Use (771)  |  Validity (50)  |  Value (393)  |  Widespread (23)  |  Word (650)

Natura non facit saltum or, Nature does not make leaps… If you assume continuity, you can open the well-stocked mathematical toolkit of continuous functions and differential equations, the saws and hammers of engineering and physics for the past two centuries (and the foreseeable future).
From Benoit B. Mandelbrot and Richard Hudson, The (Mis)Behaviour of Markets: A Fractal View of Risk, Ruin and Reward (2004,2010), 85-86.
Science quotes on:  |  Assume (43)  |  Century (319)  |  Continuity (39)  |  Continuous (83)  |  Differential (7)  |  Differential Equation (18)  |  Engineering (188)  |  Equation (138)  |  Foreseeable (3)  |  Function (235)  |  Future (467)  |  Hammer (26)  |  Leap (57)  |  Natura Non Facit Saltum (3)  |  Nature (2017)  |  Open (277)  |  Past (355)  |  Physic (515)  |  Physics (564)  |  Saw (160)  |  Two (936)

Neumann, to a physicist seeking help with a difficult problem: Simple. This can be solved by using the method of characteristics.
Physicist: I'm afraid I don’t understand the method of characteristics.
Neumann: In mathematics you don't understand things. You just get used to them.
Attributed, as related by Dr. Felix Smith (Head of Molecular Physics, Stanford Research Institute) to author Gary Zukav, who quoted it in The Dancing Wu Li Masters: An Overview of the New Physics (1979, 2001), 208, footnote. The physicist (a friend of Dr. Smith) worked at Los Alamos after WW II. It should be noted that although the author uses quotation marks around the spoken remarks, that they represent the author's memory of Dr. Smith's recollection, who heard it from the physicist. Therefore the fourth-hand wording is very likely not verbatim. Webmaster finds Zukav's book seems to be the only source for this quote.
Science quotes on:  |  Characteristic (154)  |  Difficult (263)  |  Method (531)  |  Physicist (270)  |  Problem (731)  |  Simple (426)  |  Solution (282)  |  Thing (1914)  |  Understand (648)  |  Understanding (527)

Simplicibus itaque verbis gaudet Mathematica Veritas, cum etiam per se simplex sit Veritatis oratio. (So Mathematical Truth prefers simple words since the language of Truth is itself simple.)
Epistolarum astronomicarum liber primus (1596)
Science quotes on:  |  Language (308)  |  Simple (426)  |  Truth (1109)  |  Word (650)

The Annotated Alice, of course, does tie in with math, because Lewis Carroll was, as you know, a professional mathematician. So it wasn’t really too far afield from recreational math, because the two books are filled with all kinds of mathematical jokes. I was lucky there in that I really didn’t have anything new to say in The Annotated Alice because I just looked over the literature and pulled together everything in the form of footnotes. But it was a lucky idea because that’s been the best seller of all my books.
In Anthony Barcellos, 'A Conversation with Martin Gardner', The Two-Year College Mathematics Journal (Sep 1979), 10, No. 4, 241.
Science quotes on:  |  Best (467)  |  Book (413)  |  Lewis Carroll (48)  |  Course (413)  |  Everything (489)  |  Footnote (5)  |  Form (976)  |  Idea (881)  |  Joke (90)  |  Kind (564)  |  Know (1538)  |  Literature (116)  |  Look (584)  |  Lucky (13)  |  Mathematician (407)  |  New (1273)  |  Professional (77)  |  Pull (43)  |  Pull Together (2)  |  Recreation (23)  |  Say (989)  |  Tie (42)  |  Together (392)  |  Two (936)

Ultima se tangunt. How expressive, how nicely characterizing withal is mathematics! As the musician recognizes Mozart, Beethoven, Schubert in the first chords, so the mathematician would distinguish his Cauchy, Gauss, Jacobi, Helmholtz in a few pages.
In Ceremonial Speech (15 Nov 1887) celebrating the 301st anniversary of the Karl-Franzens-University Graz. Published as Gustav Robert Kirchhoff: Festrede zur Feier des 301. Gründungstages der Karl-Franzens-Universität zu Graz (1888), 29, as translated in Robert Édouard Moritz, Memorabilia Mathematica; Or, The Philomath’s Quotation-book (1914), 186-187. From the original German, “Ultima se tangunt. Und wie ausdrucksfähig, wie fein charakterisirend ist dabei die Mathematik. Wie der Musiker bei den ersten Tacten Mozart, Beethoven, Schubert erkennt, so würde der Mathematiker nach wenig Seiten, seinen Cauchy, Gauss, Jacobi, Helmholtz unterscheiden.” [The Latin words translate as “the final touch”. —Webmaster]
Science quotes on:  |  Beethoven (14)  |  Baron Augustin-Louis Cauchy (11)  |  Characterize (22)  |  Chord (4)  |  Distinguish (168)  |  Expressive (6)  |  First (1302)  |  Carl Friedrich Gauss (79)  |  Hermann von Helmholtz (32)  |  Karl Jacobi (11)  |  Mathematician (407)  |  Mathematics And Art (8)  |  Mathematics As A Fine Art (23)  |  Mozart (3)  |  Musician (23)  |  Page (35)  |  Recognize (136)  |  Schubert (2)

~~[Attributed, authorship undocumented]~~ Mathematical demonstrations are a logic of as much or more use, than that commonly learned at schools, serving to a just formation of the mind, enlarging its capacity, and strengthening it so as to render the same capable of exact reasoning, and discerning truth from falsehood in all occurrences, even in subjects not mathematical. For which reason it is said, the Egyptians, Persians, and Lacedaemonians seldom elected any new kings, but such as had some knowledge in the mathematics, imagining those, who had not, men of imperfect judgments, and unfit to rule and govern.
From an article which appeared as 'The Usefulness of Mathematics', Pennsylvania Gazette (30 Oct 1735), No. 360. Collected, despite being without clear evidence of Franklin’s authorship, in The Works of Benjamin Franklin (1809), Vol. 4, 377. Evidence of actual authorship by Ben Franklin for the newspaper article has not been ascertained, and scholars doubt it. See Franklin documents at the website founders.archives.gov. The quote is included here to attach this caution.
Science quotes on:  |  Capable (174)  |  Capacity (105)  |  Demonstration (120)  |  Discern (35)  |  Discerning (16)  |  Egyptian (5)  |  Elect (5)  |  Enlarge (37)  |  Exact (75)  |  Falsehood (30)  |  Formation (100)  |  Govern (66)  |  Imagine (176)  |  Imperfect (46)  |  It Is Said (2)  |  Judgment (140)  |  King (39)  |  Knowledge (1647)  |  Learn (672)  |  Learned (235)  |  Logic (311)  |  Mathematics And Logic (27)  |  Mind (1377)  |  More (2558)  |  New (1273)  |  Occurrence (53)  |  Persian (4)  |  Reason (766)  |  Reasoning (212)  |  Render (96)  |  Rule (307)  |  School (227)  |  Seldom (68)  |  Serving (15)  |  Strengthen (25)  |  Subject (543)  |  Truth (1109)  |  Unfit (13)  |  Use (771)

~~[Misattributed ?]~~ Mathematical discoveries, like springtime violets in the woods, have their season which no human can hasten or retard.
Webmaster believes this quote is likely a misattributed paraphrase. The subject quote is as given in Israel Kleiner, 'Thinking the Unthinkable: The Story of Complex Numbers (with a Moral)', Mathematics Teacher (Oct 1988), 81, No. 7, 590. In Kleiner’s paper, alongside the quote is a citation, thus: “(Kline 1972)?” Notice the appended question mark. The reference at the end of the paper gives: Morris Kline, Mathematical Thought from Ancient to Modern Times (1972), but without page number. Webmaster checked a later edition, Vol. 3 (1990), 861, in which Kline has an epigraph, with different wording about violets, attributed - not to János - but to his father, “Wolfgang Bolyai” (who is also known as Farkas Bolyai). Translator Abe Shenitzer wrote an ambiguous passage in Herbert Meschkowski, NonEuclidean Geometry (1964), 33. In a discussion posted in the NCTM online Math Forum in 1998, Shenitzer clarified that the proper reading is that the “violet talk” is a simile used in advice given by the father to his son. Note that in the passage, János (Johann/John) reports about that advice in narrative form. Thus, one should also note that even in the original language, perhaps the father’s words are not verbatim. See Farkas Bolyai Quotes on another page of this website.
Science quotes on:  |  Discovery (837)  |  Hasten (13)  |  Human (1512)  |  Misattributed (19)  |  Retard (4)  |  Season (47)  |  Spring (140)  |  Springtime (5)  |  Violet (11)  |  Wood (97)

~~[Misquote]~~ Fourier is a mathematical poem.
Seen in various books and on the web. This seems to be a misquote based on Kelvin’s reference to Fourier’s mathematical theory of the conduction of heat as “Fourier's great mathematical poem.” More information on the latter quote on the Lord Kelvin Quotes page on this website.
Science quotes on:  |  Baron Jean-Baptiste-Joseph Fourier (17)  |  Poem (104)

~~[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.
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 (821)  |  Certain (557)  |  Degree (277)  |  Kind (564)  |  Known (453)  |  Maturity (14)  |  Part (235)  |  Reach (286)

~~[No known source]~~ Medicine makes people ill, mathematics make them sad and theology makes them sinful.
Webmaster strongly doubts that this is an authentic quote. As yet, Webmaster has found no primary source, no citation, and not even an early publication with this quote. Best guess is that it is a recent viral fake or mistake. Can you help? Meanwhile, Webmaster includes this quote here only to add this caution. At best, likely, it is barely a joke, by anonymous.
Science quotes on:  |  Known (453)  |  Medicine (392)  |  People (1031)  |  Sadness (36)  |  Theology (54)

A general course in mathematics should be required of all officers for its practical value, but no less for its educational value in training the mind to logical forms of thought, in developing the sense of absolute truthfulness, together with a confidence in the accomplishment of definite results by definite means.
In 'Mathematics at West Point and Annapolis', United States Bureau of Education, Bulletin 1912, No. 2, 11.
Science quotes on:  |  Absolute (153)  |  Accomplishment (102)  |  Confidence (75)  |  Course (413)  |  Definite (114)  |  Develop (278)  |  Educational (7)  |  Form (976)  |  General (521)  |  Logical (57)  |  Mean (810)  |  Means (587)  |  Mind (1377)  |  Officer (12)  |  Practical (225)  |  Require (229)  |  Required (108)  |  Result (700)  |  Sense (785)  |  Thought (995)  |  Together (392)  |  Training (92)  |  Truthfulness (3)  |  Value (393)  |  Value Of Mathematics (60)

A chemist who does not know mathematics is seriously handicapped.
Quoted in Albert Rosenfeld, Langmuir: The Man and the Scientist (1962), 293.
Science quotes on:  |  Chemist (169)  |  Handicap (7)  |  Handicapped (7)  |  Know (1538)

A chess problem is genuine mathematics, but it is in some way “trivial” mathematics. However, ingenious and intricate, however original and surprising the moves, there is something essential lacking. Chess problems are unimportant. The best mathematics is serious as well as beautiful—“important” if you like, but the word is very ambiguous, and “serious” expresses what I mean much better.
'A Mathematician's Apology', in James Roy Newman, The World of Mathematics (2000), 2029.
Science quotes on:  |  Ambiguous (14)  |  Beautiful (271)  |  Best (467)  |  Better (493)  |  Chess (27)  |  Essential (210)  |  Genuine (54)  |  Important (229)  |  Ingenious (55)  |  Intricate (29)  |  Mean (810)  |  Move (223)  |  Original (61)  |  Problem (731)  |  Serious (98)  |  Something (718)  |  Surprise (91)  |  Trivial (59)  |  Unimportant (6)  |  Way (1214)  |  Word (650)

A formal manipulator in mathematics often experiences the discomforting feeling that his pencil surpasses him in intelligence.
In An Introduction to the History of Mathematics (1953, 1976), 354. This same idea was said much earlier by Ernst Mach (1893). See the quote that begins, “The mathematician who pursues his studies,” on the Ernst Mach Quotes page on this website.
Science quotes on:  |  Discomfort (4)  |  Experience (494)  |  Feeling (259)  |  Formal (37)  |  Intelligence (218)  |  Manipulator (5)  |  Pencil (20)  |  Surpass (33)

A fractal is a mathematical set or concrete object that is irregular or fragmented at all scales.
Cited as from Fractals: Form, Chance, and Dimension (1977), by J.W. Cannon, in review of The Fractal Geometry of Nature (1982) in The American Mathematical Monthly (Nov 1984), 91, No. 9, 594.
Science quotes on:  |  Concrete (55)  |  Definition (238)  |  Fractal (11)  |  Fragment (58)  |  Irregular (7)  |  Object (438)  |  Scale (122)  |  Set (400)

A good deal of my research in physics has consisted in not setting out to solve some particular problem, but simply examining mathematical quantities of a kind that physicists use and trying to fit them together in an interesting way, regardless of any application that the work may have. It is simply a search for pretty mathematics. It may turn out later to have an application. Then one has good luck. At age 78.
International Journal of Theoretical Physics (1982), 21, 603. In A. Pais, 'Playing With Equations, the Dirac Way'. Behram N. Kursunoglu (Ed.) and Eugene Paul Wigner (Ed.), Paul Adrien Maurice Dirac: Reminiscences about a Great Physicist (1990), 110.
Science quotes on:  |  Age (509)  |  Application (257)  |  Consist (223)  |  Deal (192)  |  Equation (138)  |  Fit (139)  |  Good (906)  |  Interesting (153)  |  Kind (564)  |  Luck (44)  |  Physic (515)  |  Physicist (270)  |  Physics (564)  |  Problem (731)  |  Research (753)  |  Search (175)  |  Setting (44)  |  Solve (145)  |  Together (392)  |  Trying (144)  |  Turn (454)  |  Use (771)  |  Way (1214)  |  Work (1402)

A good mathematical joke is better, and better mathematics, than a dozen mediocre papers.
In A Mathematician’s Miscellany (1953), reissued as Béla Bollobás, Littlewood’s Miscellany (1986), 24.
Science quotes on:  |  Better (493)  |  Good (906)  |  Joke (90)  |  Mediocre (14)  |  Paper (192)  |  Publication (102)

A good theoretical physicist today might find it useful to have a wide range of physical viewpoints and mathematical expressions of the same theory (for example, of quantum electrodynamics) available to him. This may be asking too much of one man. Then new students should as a class have this. If every individual student follows the same current fashion in expressing and thinking about electrodynamics or field theory, then the variety of hypotheses being generated to understand strong interactions, say, is limited. Perhaps rightly so, for possibly the chance is high that the truth lies in the fashionable direction. But, on the off-chance that it is in another direction—a direction obvious from an unfashionable view of field theory—who will find it?
In his Nobel Prize Lecture (11 Dec 1965), 'The Development of the Space-Time View of Quantum Electrodynamics'. Collected in Stig Lundqvist, Nobel Lectures: Physics, 1963-1970 (1998), 177.
Science quotes on:  |  Asking (74)  |  Available (80)  |  Being (1276)  |  Chance (244)  |  Class (168)  |  Current (122)  |  Direction (185)  |  Electrodynamics (10)  |  Expression (181)  |  Fashionable (15)  |  Field (378)  |  Find (1014)  |  Follow (389)  |  Generate (16)  |  Good (906)  |  High (370)  |  Hypothesis (314)  |  Individual (420)  |  Interaction (47)  |  Lie (370)  |  Limit (294)  |  Limited (102)  |  Man (2252)  |  New (1273)  |  Obvious (128)  |  Physical (518)  |  Physicist (270)  |  Possibly (111)  |  Quantum (118)  |  Quantum Electrodynamics (3)  |  Range (104)  |  Say (989)  |  Strong (182)  |  Student (317)  |  Theoretical Physicist (21)  |  Theory (1015)  |  Think (1122)  |  Thinking (425)  |  Today (321)  |  Truth (1109)  |  Understand (648)  |  Unfashionable (2)  |  Useful (260)  |  Variety (138)  |  View (496)  |  Viewpoint (13)  |  Wide (97)  |  Will (2350)

A great deal of my work is just playing with equations and seeing what they give.
Quoted in Frank Wilczek, ',The Dirac Equation'. Proceedings of the Dirac Centennial Symposium (2003), 45.
Science quotes on:  |  Biography (254)  |  Deal (192)  |  Equation (138)  |  Great (1610)  |  Playing (42)  |  Seeing (143)  |  Work (1402)

A great department of thought must have its own inner life, however transcendent may be the importance of its relations to the outside. No department of science, least of all one requiring so high a degree of mental concentration as Mathematics, can be developed entirely, or even mainly, with a view to applications outside its own range. The increased complexity and specialisation of all branches of knowledge makes it true in the present, however it may have been in former times, that important advances in such a department as Mathematics can be expected only from men who are interested in the subject for its own sake, and who, whilst keeping an open mind for suggestions from outside, allow their thought to range freely in those lines of advance which are indicated by the present state of their subject, untrammelled by any preoccupation as to applications to other departments of science. Even with a view to applications, if Mathematics is to be adequately equipped for the purpose of coping with the intricate problems which will be presented to it in the future by Physics, Chemistry and other branches of physical science, many of these problems probably of a character which we cannot at present forecast, it is essential that Mathematics should be allowed to develop freely on its own lines.
In Presidential Address British Association for the Advancement of Science, Sheffield, Section A, Nature (1 Sep 1910), 84, 286.
Science quotes on:  |  Adequate (50)  |  Advance (298)  |  Allow (51)  |  Application (257)  |  Branch (155)  |  Character (259)  |  Chemistry (376)  |  Complexity (121)  |  Concentration (29)  |  Cope (9)  |  Degree (277)  |  Department (93)  |  Develop (278)  |  Entirely (36)  |  Equip (6)  |  Equipped (17)  |  Essential (210)  |  Expect (203)  |  Forecast (15)  |  Former (138)  |  Freely (13)  |  Future (467)  |  Great (1610)  |  High (370)  |  Importance (299)  |  Important (229)  |  Increase (225)  |  Indicate (62)  |  Inner (72)  |  Interest (416)  |  Intricate (29)  |  Knowledge (1647)  |  Least (75)  |  Life (1870)  |  Mainly (10)  |  Mental (179)  |  Mind (1377)  |  Must (1525)  |  Open (277)  |  Other (2233)  |  Outside (141)  |  Physic (515)  |  Physical (518)  |  Physical Science (104)  |  Physics (564)  |  Preoccupation (7)  |  Present (630)  |  Probably (50)  |  Problem (731)  |  Purpose (336)  |  Range (104)  |  Relation (166)  |  Require (229)  |  Sake (61)  |  Specialize (4)  |  State (505)  |  Study And Research In Mathematics (61)  |  Subject (543)  |  Suggestion (49)  |  Thought (995)  |  Time (1911)  |  Transcendent (3)  |  True (239)  |  View (496)  |  Will (2350)

A great man, [who] was convinced that the truths of political and moral science are capable of the same certainty as those that form the system of physical science, even in those branches like astronomy that seem to approximate mathematical certainty.
He cherished this belief, for it led to the consoling hope that humanity would inevitably make progress toward a state of happiness and improved character even as it has already done in its knowledge of the truth.
Describing administrator and economist Anne-Robert-Jacques Turgot in Essai sur l’application de l’analyse à la probabilité des décisions rendues à la pluralité des voix (1785), i. Cited epigraph in Charles Coulston Gillispie, Science and Polity in France: The End of the Old Regime (2004), 3
Science quotes on:  |  Already (226)  |  Approximate (25)  |  Approximation (32)  |  Astronomy (251)  |  Belief (615)  |  Capable (174)  |  Certainty (180)  |  Character (259)  |  Cherish (25)  |  Cherishing (2)  |  Consoling (4)  |  Conviction (100)  |  Form (976)  |  Great (1610)  |  Happiness (126)  |  Hope (321)  |  Humanity (186)  |  Knowledge (1647)  |  Man (2252)  |  Moral (203)  |  Physical (518)  |  Physical Science (104)  |  Political (124)  |  Politics (122)  |  Progress (492)  |  State (505)  |  System (545)  |  Truth (1109)

A great part of its [higher arithmetic] theories derives an additional charm from the peculiarity that important propositions, with the impress of simplicity on them, are often easily discovered by induction, and yet are of so profound a character that we cannot find the demonstrations till after many vain attempts; and even then, when we do succeed, it is often by some tedious and artificial process, while the simple methods may long remain concealed.
Quoted in H. Eves, Mathematical Circles (1977) .
Science quotes on:  |  Arithmetic (144)  |  Attempt (266)  |  Character (259)  |  Charm (54)  |  Concealed (25)  |  Demonstration (120)  |  Derive (70)  |  Discover (571)  |  Do (1905)  |  Find (1014)  |  Great (1610)  |  Impress (66)  |  Induction (81)  |  Long (778)  |  Method (531)  |  Peculiarity (26)  |  Process (439)  |  Profound (105)  |  Proposition (126)  |  Remain (355)  |  Simple (426)  |  Simplicity (175)  |  Succeed (114)  |  Success (327)  |  Tedious (15)  |  Vain (86)

A large part of mathematics which becomes useful developed with absolutely no desire to be useful, and in a situation where nobody could possibly know in what area it would become useful; and there were no general indications that it ever would be so.
From Address (1954) to Princeton Alumni, 'The Role of Mathematics in the Sciences and in Society', published in A.H. Taub (ed.), John von Neumann: Collected Works (1963), Vol. 6, 489. As quoted and cited in Rosemary Schmalz,Out of the Mouths of Mathematicians: A Quotation Book for Philomaths (1993), 123.
Science quotes on:  |  Absolutely (41)  |  Become (821)  |  Desire (212)  |  Develop (278)  |  General (521)  |  Indication (33)  |  Know (1538)  |  Large (398)  |  Nobody (103)  |  Possibly (111)  |  Situation (117)  |  Useful (260)

A large part of the training of the engineer, civil and military, as far as preparatory studies are concerned; of the builder of every fabric of wood or stone or metal designed to stand upon the earth, or bridge the stream, or resist or float upon the wave; of the surveyor who lays out a building lot in a city, or runs a boundary line between powerful governments across a continent; of the geographer, navigator, hydrographer, and astronomer,—must be derived from the mathematics.
In 'Academical Education', Orations and Speeches on Various Occasions (1870), Vol. 3, 513.
Science quotes on:  |  Across (32)  |  Astronomer (97)  |  Boundary (55)  |  Bridge (49)  |  Build (211)  |  Builder (16)  |  Building (158)  |  City (87)  |  Civil (26)  |  Civil Engineer (4)  |  Concern (239)  |  Continent (79)  |  Derive (70)  |  Design (203)  |  Earth (1076)  |  Education (423)  |  Engineer (136)  |  Fabric (27)  |  Float (31)  |  Geographer (7)  |  Government (116)  |  Hydrographer (3)  |  Large (398)  |  Line (100)  |  Lot (151)  |  Metal (88)  |  Military (45)  |  Military Engineer (2)  |  Must (1525)  |  Navigator (8)  |  Powerful (145)  |  Preparatory (3)  |  Resist (15)  |  Run (158)  |  Stand (284)  |  Stone (168)  |  Stream (83)  |  Study (701)  |  Surveyor (5)  |  Training (92)  |  Wave (112)  |  Wood (97)

A marveilous newtrality have these things mathematicall, and also a strange participation between things supernaturall, immortall, intellectuall, simple and indivisible, and things naturall, mortall, sensible, componded and divisible.
John Dee
In Mathematicall Praeface to the Elements of Geometrie of Euclid of Megara (1570).
Science quotes on:  |  Compound (117)  |  Divisible (5)  |  Immortal (35)  |  Indivisible (22)  |  Intellectual (258)  |  Marvellous (25)  |  Mortal (55)  |  Natural (810)  |  Nature Of Mathematics (80)  |  Neutrality (5)  |  Participation (15)  |  Sensible (28)  |  Simple (426)  |  Strange (160)  |  Supernatural (26)  |  Thing (1914)

A mathematical argument is, after all, only organized common sense, and it is well that men of science should not always expound their work to the few behind a veil of technical language, but should from time to time explain to a larger public the reasoning which lies behind their mathematical notation.
In The Tides and Kindred Phenomena in the Solar System: The Substance of Lectures Delivered in 1897 at the Lowell Institute, Boston, Massachusetts (1898), Preface, v. Preface
Science quotes on:  |  Argument (145)  |  Behind (139)  |  Common (447)  |  Common Sense (136)  |  Explain (334)  |  Language (308)  |  Lie (370)  |  Men Of Science (147)  |  Notation (28)  |  Organized (9)  |  Reasoning (212)  |  Sense (785)  |  Technical (53)  |  Time (1911)  |  Veil (27)  |  Work (1402)

A mathematical point is the most indivisble and unique thing which art can present.
Letters, 21. 1817. In Robert Édouard Moritz, Memorabilia Mathematica (1914), 295.
Science quotes on:  |  Art (680)  |  Indivisible (22)  |  Most (1728)  |  Point (584)  |  Present (630)  |  Thing (1914)  |  Unique (72)

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.
In Mathematical Problems', Bulletin American Mathematical Society, 8, 438.
Science quotes on:  |  Completely (137)  |  Difficult (263)  |  Effort (243)  |  Guide (107)  |  Hide (70)  |  Inaccessible (18)  |  Lest (3)  |  Mock (7)  |  Order (638)  |  Path (159)  |  Pleasure (191)  |  Post (8)  |  Problem (731)  |  Reminder (13)  |  Solution (282)  |  Study And Research In Mathematics (61)  |  Successful (134)  |  Truth (1109)  |  Ultimately (56)

A mathematical proof should resemble a simple and clear-cut constellation, not a scattered cluster in the Milky Way.
In A Mathematician’s Apology (1940, 2012), 113.
Science quotes on:  |  Clear-Cut (10)  |  Cluster (16)  |  Constellation (18)  |  Cut (116)  |  Milky Way (29)  |  Proof (304)  |  Resemble (65)  |  Scattered (5)  |  Simple (426)  |  Way (1214)

A mathematical science is any body of propositions which is capable of an abstract formulation and arrangement in such a way that every proposition of the set after a certain one is a formal logical consequence of some or all the preceding propositions. Mathematics consists of all such mathematical sciences.
In Lectures on Fundamental Concepts of Algebra and Geometry (1911), 222.
Science quotes on:  |  Abstract (141)  |  Arrangement (93)  |  Body (557)  |  Capable (174)  |  Certain (557)  |  Consequence (220)  |  Consist (223)  |  Definitions and Objects of Mathematics (33)  |  Formal (37)  |  Formulation (37)  |  Logic (311)  |  Precede (23)  |  Proposition (126)  |  Set (400)  |  Way (1214)

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.
…...
Science quotes on:  |  Clear (111)  |  Complete (209)  |  Consider (428)  |  Explain (334)  |  First (1302)  |  Man (2252)  |  Meet (36)  |  Street (25)  |  Theory (1015)

A mathematical truth is timeless, it does not come into being when we discover it. Yet its discovery is a very real event, it may be an emotion like a great gift from a fairy.
…...
Science quotes on:  |  Being (1276)  |  Discover (571)  |  Discovery (837)  |  Emotion (106)  |  Event (222)  |  Fairy (10)  |  Gift (105)  |  Great (1610)  |  Real (159)  |  Timeless (8)  |  Truth (1109)

A mathematician who can only generalise is like a monkey who can only climb UP a tree. ... And a mathematician who can only specialise is like a monkey who can only climb DOWN a tree. In fact neither the up monkey nor the down monkey is a viable creature. A real monkey must find food and escape his enemies and so must be able to incessantly climb up and down. A real mathematician must be able to generalise and specialise. ... There is, I think, a moral for the teacher. A teacher of traditional mathematics is in danger of becoming a down monkey, and a teacher of modern mathematics an up monkey. The down teacher dishing out one routine problem after another may never get off the ground, never attain any general idea. and the up teacher dishing out one definition after the other may never climb down from his verbiage, may never get down to solid ground, to something of tangible interest for his pupils.
From 'A Story With A Moral', Mathematical Gazette (Jun 1973), 57, No. 400, 86-87
Science quotes on:  |  Attain (126)  |  Becoming (96)  |  Climb (39)  |  Creature (242)  |  Danger (127)  |  Definition (238)  |  Down (455)  |  Enemy (86)  |  Escape (85)  |  Fact (1257)  |  Find (1014)  |  Food (213)  |  General (521)  |  Generalization (61)  |  Ground (222)  |  Idea (881)  |  Incessant (9)  |  Interest (416)  |  Mathematician (407)  |  Modern (402)  |  Modern Mathematics (50)  |  Monkey (57)  |  Moral (203)  |  Must (1525)  |  Never (1089)  |  Other (2233)  |  Problem (731)  |  Pupil (62)  |  Real (159)  |  Routine (26)  |  Solid (119)  |  Something (718)  |  Specialization (24)  |  Tangible (15)  |  Teacher (154)  |  Think (1122)  |  Tree (269)  |  Up (5)  |  Verbiage (3)

A mind is accustomed to mathematical deduction, when confronted with the faulty foundations of astrology, resists a long, long time, like an obstinate mule, until compelled by beating and curses to put its foot into that dirty puddle.
As quoted in Arthur Koestler, The Sleep Walkers: A History of Man’s Changing Vision of the Universe (1959), 243, citing De Stella Nova in Pede Serpentarii (1606).
Science quotes on:  |  Accustom (52)  |  Accustomed (46)  |  Astrology (46)  |  Beat (42)  |  Compel (31)  |  Confront (18)  |  Curse (20)  |  Deduction (90)  |  Dirt (17)  |  Dirty (17)  |  Faulty (3)  |  Foot (65)  |  Foundation (177)  |  Long (778)  |  Mind (1377)  |  Mule (2)  |  Obstinate (5)  |  Resist (15)  |  Time (1911)

A modern branch of mathematics, having achieved the art of dealing with the infinitely small, can now yield solutions in other more complex problems of motion, which used to appear insoluble. This modern branch of mathematics, unknown to the ancients, when dealing with problems of motion, admits the conception of the infinitely small, and so conforms to the chief condition of motion (absolute continuity) and thereby corrects the inevitable error which the human mind cannot avoid when dealing with separate elements of motion instead of examining continuous motion. In seeking the laws of historical movement just the same thing happens. The movement of humanity, arising as it does from innumerable human wills, is continuous. To understand the laws of this continuous movement is the aim of history. … Only by taking an infinitesimally small unit for observation (the differential of history, that is, the individual tendencies of man) and attaining to the art of integrating them (that is, finding the sum of these infinitesimals) can we hope to arrive at the laws of history.
War and Peace (1869), Book 11, Chap. 1.
Science quotes on:  |  Absolute (153)  |  Aim (175)  |  Ancient (198)  |  Appear (122)  |  Arise (162)  |  Arising (22)  |  Arrive (40)  |  Art (680)  |  Attain (126)  |  Avoid (123)  |  Branch (155)  |  Chief (99)  |  Complex (202)  |  Concept (242)  |  Conception (160)  |  Condition (362)  |  Conform (15)  |  Continuity (39)  |  Continuous (83)  |  Correct (95)  |  Deal (192)  |  Differential (7)  |  Element (322)  |  Error (339)  |  Examine (84)  |  Find (1014)  |  Happen (282)  |  Historical (70)  |  History (716)  |  Hope (321)  |  Human (1512)  |  Human Mind (133)  |  Humanity (186)  |  Individual (420)  |  Inevitable (53)  |  Infinite (243)  |  Infinitesimal (30)  |  Innumerable (56)  |  Insoluble (15)  |  Integrate (8)  |  Law (913)  |  Man (2252)  |  Mind (1377)  |  Modern (402)  |  More (2558)  |  Motion (320)  |  Movement (162)  |  Observation (593)  |  Other (2233)  |  Problem (731)  |  Seek (218)  |  Separate (151)  |  Small (489)  |  Solution (282)  |  Solution. (53)  |  Sum (103)  |  Tendency (110)  |  Thing (1914)  |  Understand (648)  |  Unit (36)  |  Unknown (195)  |  Will (2350)  |  Yield (86)

A modern mathematical proof is not very different from a modern machine, or a modern test setup: the simple fundamental principles are hidden and almost invisible under a mass of technical details.
Unterrichtsblätter für Mathematik und Naturwissenschaften (1932), 38, 177-188. As translated by Abe Shenitzer, in 'Part I. Topology and Abstract Algebra as Two Roads of Mathematical Comprehension', The American Mathematical Monthly (May 1995), 102, No. 7, 453.
Science quotes on:  |  Detail (150)  |  Different (595)  |  Fundamental (264)  |  Hidden (43)  |  Invisible (66)  |  Machine (271)  |  Mass (160)  |  Modern (402)  |  Principle (530)  |  Proof (304)  |  Simple (426)  |  Technical (53)  |  Test (221)

A peculiar beauty reigns in the realm of mathematics, a beauty which resembles not so much the beauty of art as the beauty of nature and which affects the reflective mind, which has acquired an appreciation of it, very much like the latter.
From Berliner Monatsberichte (1867), 395. As translated in Robert Édouard Moritz, Memorabilia Mathematica; Or, The Philomath’s Quotation-book (1914), 185.
Science quotes on:  |  Acquire (46)  |  Acquired (77)  |  Affect (19)  |  Appreciation (37)  |  Art (680)  |  Beauty (313)  |  Mathematical Beauty (19)  |  Mathematics As A Fine Art (23)  |  Mind (1377)  |  Nature (2017)  |  Peculiar (115)  |  Realm (87)  |  Reflective (3)  |  Reign (24)  |  Resemble (65)

A professor … may be to produce a perfect mathematical work of art, having every axiom stated, every conclusion drawn with flawless logic, the whole syllabus covered. This sounds excellent, but in practice the result is often that the class does not have the faintest idea of what is going on. … The framework is lacking; students do not know where the subject fits in, and this has a paralyzing effect on the mind.
In A Concrete Approach to Abstract Algebra (1959), 1-2.
Science quotes on:  |  Art (680)  |  Axiom (65)  |  Class (168)  |  Conclusion (266)  |  Do (1905)  |  Effect (414)  |  Excellent (29)  |  Faint (10)  |  Fit (139)  |  Framework (33)  |  Idea (881)  |  Know (1538)  |  Lack (127)  |  Logic (311)  |  Mind (1377)  |  Paralyze (3)  |  Perfect (223)  |  Practice (212)  |  Professor (133)  |  Result (700)  |  Sound (187)  |  Student (317)  |  Subject (543)  |  Whole (756)  |  Work (1402)  |  Work Of Art (3)

A short, broad man of tremendous vitality, the physical type of Hereward, the last of the English, and his brother-in-arms, Winter, Sylvester’s capacious head was ever lost in the highest cloud-lands of pure mathematics. Often in the dead of night he would get his favorite pupil, that he might communicate the very last product of his creative thought. Everything he saw suggested to him something new in the higher algebra. This transmutation of everything into new mathematics was a revelation to those who knew him intimately. They began to do it themselves. His ease and fertility of invention proved a constant encouragement, while his contempt for provincial stupidities, such as the American hieroglyphics for π and e, which have even found their way into Webster’s Dictionary, made each young worker apply to himself the strictest tests.
In Florian Cajori, Teaching and History of Mathematics in the United States (1890), 265.
Science quotes on:  |  Algebra (117)  |  American (56)  |  Apply (170)  |  Arm (82)  |  Arms (37)  |  Broad (28)  |  Brother (47)  |  Capacious (2)  |  Cloud (111)  |  Communicate (39)  |  Constant (148)  |  Contempt (20)  |  Creative (144)  |  Dead (65)  |  Do (1905)  |  Ease (40)  |  Encouragement (27)  |  English (35)  |  Everything (489)  |  Favorite (37)  |  Fertility (23)  |  Head (87)  |  Hieroglyphic (6)  |  High (370)  |  Himself (461)  |  Invention (400)  |  Last (425)  |  Lost (34)  |  Man (2252)  |  Mathematicians and Anecdotes (141)  |  New (1273)  |  Night (133)  |  Often (109)  |  Physical (518)  |  Pi (14)  |  Product (166)  |  Provincial (2)  |  Pupil (62)  |  Pure (299)  |  Pure Mathematics (72)  |  Revelation (51)  |  Saw (160)  |  Short (200)  |  Something (718)  |  Strict (20)  |  Stupidity (40)  |  James Joseph Sylvester (58)  |  Test (221)  |  Themselves (433)  |  Thought (995)  |  Transmutation (24)  |  Tremendous (29)  |  Type (171)  |  Vitality (24)  |  Way (1214)  |  Winter (46)  |  Worker (34)  |  Young (253)

A superficial knowledge of mathematics may lead to the belief that this subject can be taught incidentally, and that exercises akin to counting the petals of flowers or the legs of a grasshopper are mathematical. Such work ignores the fundamental idea out of which quantitative reasoning grows—the equality of magnitudes. It leaves the pupil unaware of that relativity which is the essence of mathematical science. Numerical statements are frequently required in the study of natural history, but to repeat these as a drill upon numbers will scarcely lend charm to these studies, and certainly will not result in mathematical knowledge.
In Primary Arithmetic: First Year, for the Use of Teachers (1897), 26-27.
Science quotes on:  |  Belief (615)  |  Certain (557)  |  Certainly (185)  |  Charm (54)  |  Count (107)  |  Counting (26)  |  Drill (12)  |  Equality (34)  |  Essence (85)  |  Exercise (113)  |  Flower (112)  |  Frequent (26)  |  Fundamental (264)  |  Grasshopper (8)  |  Grow (247)  |  History (716)  |  Idea (881)  |  Ignore (52)  |  Incidental (15)  |  Knowledge (1647)  |  Lead (391)  |  Leave (138)  |  Leg (35)  |  Lend (4)  |  Magnitude (88)  |  Natural (810)  |  Natural History (77)  |  Number (710)  |  Numerical (39)  |  Petal (4)  |  Pupil (62)  |  Quantitative (31)  |  Reason (766)  |  Reasoning (212)  |  Relativity (91)  |  Repeat (44)  |  Require (229)  |  Required (108)  |  Result (700)  |  Scarce (11)  |  Scarcely (75)  |  Statement (148)  |  Study (701)  |  Subject (543)  |  Superficial (12)  |  Teach (299)  |  Teaching of Mathematics (39)  |  Unaware (6)  |  Will (2350)  |  Work (1402)

A superficial knowledge of mathematics may lead to the belief that this subject can be taught incidentally, and that exercises akin to counting the petals of flowers or the legs of a grasshopper are mathematical. Such work ignores the fundamental idea out of which quantitative reasoning grows—the equality of magnitudes. It leaves the pupil unaware of that relativity which is the essence of mathematical science. Numerical statements are frequently required in the study of natural history, but to repeat these as a drill upon numbers will scarcely lend charm to these studies, and certainly will not result in mathematical knowledge.
In Primary Arithmetic: First Year, for the Use of Teachers (1897), 26-27.
Science quotes on:  |  Belief (615)  |  Certainly (185)  |  Charm (54)  |  Counting (26)  |  Equality (34)  |  Essence (85)  |  Exercise (113)  |  Flower (112)  |  Fundamental (264)  |  Grasshopper (8)  |  Grow (247)  |  History (716)  |  Idea (881)  |  Ignore (52)  |  Knowledge (1647)  |  Lead (391)  |  Leg (35)  |  Magnitude (88)  |  Natural (810)  |  Natural History (77)  |  Number (710)  |  Numerical (39)  |  Pupil (62)  |  Quantitative (31)  |  Reasoning (212)  |  Relativity (91)  |  Required (108)  |  Result (700)  |  Scarcely (75)  |  Statement (148)  |  Study (701)  |  Subject (543)  |  Will (2350)  |  Work (1402)

A surprising proportion of mathematicians are accomplished musicians. Is it because music and mathematics share patterns that are beautiful?
In 'Introduction' contributed to Donald J. Albers and Gerald L. Alexanderson, More Mathematical People: Contemporary Conversations (1990), xi.
Science quotes on:  |  Accomplishment (102)  |  Beautiful (271)  |  Mathematician (407)  |  Music (133)  |  Musician (23)  |  Pattern (116)  |  Proportion (140)  |  Share (82)  |  Surprise (91)

A teacher of mathematics has a great opportunity. If he fills his allotted time with drilling his students in routine operations he kills their interest, hampers their intellectual development, and misuses his opportunity. But if he challenges the curiosity of his students by setting them problems proportionate to their knowledge, and helps them to solve their problems with stimulating questions, he may give them a taste for, and some means of, independent thinking.
In How to Solve It (1948), Preface.
Science quotes on:  |  Challenge (91)  |  Curiosity (138)  |  Development (441)  |  Drill (12)  |  Fill (67)  |  Give (208)  |  Great (1610)  |  Hamper (7)  |  Help (116)  |  Independent (74)  |  Intellectual (258)  |  Interest (416)  |  Kill (100)  |  Knowledge (1647)  |  Mean (810)  |  Means (587)  |  Misuse (12)  |  Operation (221)  |  Operations (107)  |  Opportunity (95)  |  Problem (731)  |  Proportionate (4)  |  Question (649)  |  Routine (26)  |  Setting (44)  |  Solve (145)  |  Stimulate (21)  |  Student (317)  |  Taste (93)  |  Teacher (154)  |  Thinking (425)  |  Time (1911)

A theory of physics is not an explanation; it is a system of mathematical oppositions deduced from a small number of principles the aim of which is to represent as simply, as completely, and as exactly as possible, a group of experimental laws.
As quoted in Philipp Frank, Modern Science and its Philosophy (1949), 15, which cites Théorie Physique; Son Objet—Son Structure (1906), 24.
Science quotes on:  |  Aim (175)  |  Complete (209)  |  Completely (137)  |  Deduce (27)  |  Exact (75)  |  Experimental (193)  |  Explanation (246)  |  Group (83)  |  Law (913)  |  Number (710)  |  Opposition (49)  |  Physic (515)  |  Physics (564)  |  Possible (560)  |  Principle (530)  |  Represent (157)  |  Simply (53)  |  Small (489)  |  System (545)  |  Theory (1015)

A theory with mathematical beauty is more likely to be correct than an ugly one that fits some experimental data. God is a mathematician of a very high order, and He used very advanced mathematics in constructing the universe.
In Scientific American (May 1963). As quoted and cited in The Hutchinson Encyclopedia of Science (1998), 468.
Science quotes on:  |  Advance (298)  |  Beauty (313)  |  Construct (129)  |  Correct (95)  |  Data (162)  |  Experimental (193)  |  Fit (139)  |  God (776)  |  High (370)  |  Likely (36)  |  Mathematical Beauty (19)  |  Mathematician (407)  |  More (2558)  |  Order (638)  |  Theory (1015)  |  Ugly (14)  |  Universe (900)

A thing is obvious mathematically after you see it.
Used as a filler, referring to Dean R.D. Carmichael, in Franz E. Hohn (ed.), Pi Mu Epsilon Journal (Fall 1956), 2, No. 5, 224. Carmichael was Dean of the Graduate School at the University of Illinois, from 1933 to his retirement in 1947. The journal was published at the University of Illinois. Webmaster has not yet found an earlier or primary source (can you help?) but would not rule out the quote being passed down by oral tradition at the university.
Science quotes on:  |  Obvious (128)  |  See (1094)  |  Thing (1914)

A troubling question for those of us committed to the widest application of intelligence in the study and solution of the problems of men is whether a general understanding of the social sciences will be possible much longer. Many significant areas of these disciplines have already been removed by the advances of the past two decades beyond the reach of anyone who does not know mathematics; and the man of letters is increasingly finding, to his dismay, that the study of mankind proper is passing from his hands to those of technicians and specialists. The aesthetic effect is admittedly bad: we have given up the belletristic “essay on man” for the barbarisms of a technical vocabulary, or at best the forbidding elegance of mathematical syntax.
Opening paragraph of 'The Study of Man: Sociology Learns the Language of Mathematics' in Commentary (1 Sep 1952). Reprinted in James Roy Newman, The World of Mathematics (1956), Vol. 2, 1294.
Science quotes on:  |  Advance (298)  |  Aesthetic (48)  |  Already (226)  |  Application (257)  |  Bad (185)  |  Barbarism (8)  |  Best (467)  |  Beyond (316)  |  Decade (66)  |  Discipline (85)  |  Dismay (5)  |  Effect (414)  |  Elegance (40)  |  Essay (27)  |  General (521)  |  Intelligence (218)  |  Know (1538)  |  Letter (117)  |  Man (2252)  |  Man Of Letters (6)  |  Mankind (356)  |  Passing (76)  |  Past (355)  |  Possible (560)  |  Problem (731)  |  Proper (150)  |  Question (649)  |  Reach (286)  |  Remove (50)  |  Significant (78)  |  Social (261)  |  Social Science (37)  |  Solution (282)  |  Specialist (33)  |  Study (701)  |  Syntax (2)  |  Technical (53)  |  Technician (9)  |  Two (936)  |  Understand (648)  |  Understanding (527)  |  Vocabulary (10)  |  Will (2350)

About the year 1821, I undertook to superintend, for the Government, the construction of an engine for calculating and printing mathematical and astronomical tables. Early in the year 1833, a small portion of the machine was put together, and was found to perform its work with all the precision which had been anticipated. At that period circumstances, which I could not control, caused what I then considered a temporary suspension of its progress; and the Government, on whose decision the continuance or discontinuance of the work depended, have not yet communicated to me their wishes on the question.
In The Ninth Bridgewater Treatise: A Fragment (1838), 186.
Science quotes on:  |  Astronomy (251)  |  Calculate (58)  |  Circumstance (139)  |  Circumstances (108)  |  Consider (428)  |  Construction (114)  |  Control (182)  |  Decision (98)  |  Depend (238)  |  Early (196)  |  Engine (99)  |  Government (116)  |  Machine (271)  |  Perform (123)  |  Period (200)  |  Portion (86)  |  Precision (72)  |  Print (20)  |  Printing (25)  |  Progress (492)  |  Question (649)  |  Science And Politics (16)  |  Small (489)  |  Suspension (7)  |  Table (105)  |  Temporary (24)  |  Together (392)  |  Work (1402)  |  Year (963)

Abstractness, sometimes hurled as a reproach at mathematics, is its chief glory and its surest title to practical usefulness. It is also the source of such beauty as may spring from mathematics.
In 'General Prospectus', The Development of Mathematics (1940, 2017), Chap. 1, 9.
Science quotes on:  |  Abstract (141)  |  Beauty (313)  |  Chief (99)  |  Glory (66)  |  Mathematical Beauty (19)  |  Practical (225)  |  Reproach (4)  |  Spring (140)  |  Useful (260)  |  Usefulness (92)

Abstruse mathematical researches … are … often abused for having no obvious physical application. The fact is that the most useful parts of science have been investigated for the sake of truth, and not for their usefulness. A new branch of mathematics, which has sprung up in the last twenty years, was denounced by the Astronomer Royal before the University of Cambridge as doomed to be forgotten, on account of its uselessness. Now it turns out that the reason why we cannot go further in our investigations of molecular action is that we do not know enough of this branch of mathematics.
In 'Conditions of Mental Development', Lectures and Essays (1901), Vol. 1, 115.
Science quotes on:  |  Abstruse (12)  |  Abuse (25)  |  Account (195)  |  Action (342)  |  Application (257)  |  Astronomer (97)  |  Branch (155)  |  Cambridge (17)  |  Denounce (6)  |  Do (1905)  |  Doom (34)  |  Enough (341)  |  Fact (1257)  |  Far (158)  |  Forget (125)  |  Forgotten (53)  |  Investigate (106)  |  Investigation (250)  |  Know (1538)  |  Last (425)  |  Molecular (7)  |  Most (1728)  |  New (1273)  |  Obvious (128)  |  Often (109)  |  Part (235)  |  Physical (518)  |  Reason (766)  |  Research (753)  |  Royal (56)  |  Sake (61)  |  Spring (140)  |  Study And Research In Mathematics (61)  |  Truth (1109)  |  Turn (454)  |  Turn Out (9)  |  University (130)  |  Useful (260)  |  Usefulness (92)  |  Uselessness (22)  |  Why (491)  |  Year (963)

After an honest day’s work a mathematician goes off duty. Mathematics is very hard work, and dons tend to be above average in health and vigor. Below a certain threshold a man cracks up; but above it, hard mental work makes for health and vigor (also—on much historical evidence throughout the ages—for longevity). I have noticed lately that when I am working really hard I wake around 5.30 a.m. ready and eager to start; if I am slack, I sleep till I am called.
In 'The Mathematician’s Art of Work' (1967), collected in Béla Bollobás (ed.), Littlewood’s Miscellany (1986), 195.
Science quotes on:  |  Age (509)  |  Average (89)  |  Call (781)  |  Certain (557)  |  Duty (71)  |  Eager (17)  |  Evidence (267)  |  Hard (246)  |  Hard Work (25)  |  Health (210)  |  Historical (70)  |  History (716)  |  Honest (53)  |  Longevity (6)  |  Man (2252)  |  Mathematician (407)  |  Mental (179)  |  Ready (43)  |  Sleep (81)  |  Start (237)  |  Tend (124)  |  Threshold (11)  |  Throughout (98)  |  Vigor (12)  |  Wake (17)  |  Work (1402)

Again and again in reading even his [William Thomson] most abstract writings one is struck by the tenacity with which physical ideas control in him the mathematical form in which he expressed them. An instance of this is afforded by … an example of a mathematical result that is, in his own words, “not instantly obvious from the analytical form of my solution, but which we immediately see must be the case by thinking of the physical meaning of the result.”
As given in Life of Lord Kelvin (1910), Vol. 2, 1136. The ellipsis gives the reference to the quoted footnote, to a passage in his Mathematical and Physical Papers, Vol. 1, 457. [Note: William Thomson, later became Lord Kelvin. —Webmaster]
Science quotes on:  |  Abstract (141)  |  Afford (19)  |  Analysis (244)  |  Control (182)  |  Express (192)  |  Form (976)  |  Idea (881)  |  Immediately (115)  |  Instantly (20)  |  Baron William Thomson Kelvin (74)  |  Mathematicians and Anecdotes (141)  |  Meaning (244)  |  Most (1728)  |  Must (1525)  |  Obvious (128)  |  Physical (518)  |  Reading (136)  |  Result (700)  |  See (1094)  |  Solution (282)  |  Tenacity (10)  |  Think (1122)  |  Thinking (425)  |  Understand (648)  |  Word (650)  |  Writing (192)

All science as it grows toward perfection becomes mathematical in its ideas.
In An Introduction to Mathematics (1911), 14. This is part of a longer quote that begins, “In modern times the belief that the ultimate explanation…”, on the Alfred North Whitehead Quotes page of this website.
Science quotes on:  |  Become (821)  |  Becoming (96)  |  Grow (247)  |  Growth (200)  |  Idea (881)  |  Perfection (131)

All science requires mathematics.
[Editors' summary of Bacon's idea, not Bacon's wording.]
These are not the exact words of Roger Bacon, but are from an editor's sub-heading, giving a summary for the topic of Chapter 2, for example, in Roger Bacon and Robert Belle Burke (ed.), Opus Maius (reproduction 2002), Vol. 1, Part 4, 117. Part 4 is devoted to a discourse on Mathematics. In its Chapter 1, as translated, Bacon states that 'There are four great sciences, without which the other sciences cannot be known nor a knowledge of things secured. ... Of these sciences the gate and key is mathematics.'
Science quotes on:  |  Idea (881)  |  Require (229)  |  Requirement (66)  |  Science Requires (6)  |  Summary (11)

All the effects of Nature are only the mathematical consequences of a small number of immutable laws.
From the original French, “Tous les effets de la nature ne sont que résultats mathématiques d'un petit noinbre de lois immuables.”, in Oeuvres de Laplace, Vol. VII: Théorie des probabilités (1847), Introduction, cliv.
Science quotes on:  |  Consequence (220)  |  Effect (414)  |  Immutable (26)  |  Law (913)  |  Nature (2017)  |  Number (710)  |  Small (489)

All the events which occur upon the earth result from Law: even those actions which are entirely dependent on the caprices of the memory, or the impulse of the passions, are shown by statistics to be, when taken in the gross, entirely independent of the human will. As a single atom, man is an enigma; as a whole, he is a mathematical problem. As an individual, he is a free agent; as a species, the offspring of necessity.
In The Martyrdom of Man (1876), 185-186.
Science quotes on:  |  Action (342)  |  Agent (73)  |  Atom (381)  |  Caprice (10)  |  Dependent (26)  |  Earth (1076)  |  Enigma (16)  |  Entirely (36)  |  Event (222)  |  Free (239)  |  Gross (7)  |  Human (1512)  |  Impulse (52)  |  Independent (74)  |  Individual (420)  |  Law (913)  |  Man (2252)  |  Memory (144)  |  Necessity (197)  |  Occur (151)  |  Offspring (27)  |  Passion (121)  |  Problem (731)  |  Result (700)  |  Single (365)  |  Species (435)  |  Statistics (170)  |  Whole (756)  |  Will (2350)

All the mathematical sciences are founded on relations between physical laws and laws of numbers, so that the aim of exact science is to reduce the problems of nature to the determination of quantities by operations with numbers.
from Faraday's Lines of Force (1856)
Science quotes on:  |  Aim (175)  |  Determination (80)  |  Law (913)  |  Nature (2017)  |  Number (710)  |  Operation (221)  |  Operations (107)  |  Physical (518)  |  Physical Law (15)  |  Problem (731)  |  Reduce (100)

All the modern higher mathematics is based on a calculus of operations, on laws of thought. All mathematics, from the first, was so in reality; but the evolvers of the modern higher calculus have known that it is so. Therefore elementary teachers who, at the present day, persist in thinking about algebra and arithmetic as dealing with laws of number, and about geometry as dealing with laws of surface and solid content, are doing the best that in them lies to put their pupils on the wrong track for reaching in the future any true understanding of the higher algebras. Algebras deal not with laws of number, but with such laws of the human thinking machinery as have been discovered in the course of investigations on numbers. Plane geometry deals with such laws of thought as were discovered by men intent on finding out how to measure surface; and solid geometry with such additional laws of thought as were discovered when men began to extend geometry into three dimensions.
In Lectures on the Logic of Arithmetic (1903), Preface, 18-19.
Science quotes on:  |  Algebra (117)  |  Arithmetic (144)  |  Best (467)  |  Calculus (65)  |  Course (413)  |  Deal (192)  |  Dimension (64)  |  Discover (571)  |  Discovery (837)  |  Doing (277)  |  Elementary (98)  |  Extend (129)  |  First (1302)  |  Future (467)  |  Geometry (271)  |  Human (1512)  |  Investigation (250)  |  Known (453)  |  Law (913)  |  Lie (370)  |  Machinery (59)  |  Measure (241)  |  Measurement (178)  |  Modern (402)  |  Modern Mathematics (50)  |  Number (710)  |  Operation (221)  |  Operations (107)  |  Present (630)  |  Pupil (62)  |  Reality (274)  |  Solid (119)  |  Surface (223)  |  Teacher (154)  |  Thinking (425)  |  Thought (995)  |  Track (42)  |  Understanding (527)  |  Wrong (246)

All the sciences have a relation, greater or less, to human nature; and...however wide any of them may seem to run from it, they still return back by one passage or another. Even Mathematics, Natural Philosophy, and Natural Religion, are in some measure dependent on the science of MAN; since they lie under the cognizance of men, and are judged of by their powers and faculties.
A Treatise on Human Nature (1739-40), ed. L. A. Selby-Bigge (1888), introduction, xix.
Science quotes on:  |  Back (395)  |  Greater (288)  |  Human (1512)  |  Human Nature (71)  |  Lie (370)  |  Man (2252)  |  Measure (241)  |  Natural (810)  |  Natural Philosophy (52)  |  Nature (2017)  |  Passage (52)  |  Philosophy (409)  |  Power (771)  |  Religion (369)  |  Return (133)  |  Run (158)  |  Still (614)  |  Wide (97)

All the truths of mathematics are linked to each other, and all means of discovering them are equally admissible.
In article by Jean Itard, 'Legendre, Adrien-Marie', in Charles Coulston Gillespie (ed.), Dictionary of Scientific Biography (1973), Vol. 8, 142.
Science quotes on:  |  Admissible (6)  |  Discovery (837)  |  Equal (88)  |  Equally (129)  |  Link (48)  |  Mean (810)  |  Means (587)  |  Other (2233)  |  Truth (1109)

All things began in Order, so shall they end, and so shall they begin again, according to the Ordainer of Order, and the mystical mathematicks of the City of Heaven.
In 'Garden of Cyrus', Religio Medici and Other Writings (1909), 229.
Science quotes on:  |  According (236)  |  Begin (275)  |  City (87)  |  End (603)  |  Heaven (266)  |  Mystical (9)  |  Order (638)  |  Thing (1914)

Almost everything, which the mathematics of our century has brought forth in the way of original scientific ideas, attaches to the name of Gauss.
In Zahlentheorie, Teil 1 (1901), 43.
Science quotes on:  |  Attach (57)  |  Century (319)  |  Everything (489)  |  Carl Friedrich Gauss (79)  |  Idea (881)  |  Mathematicians and Anecdotes (141)  |  Name (359)  |  Original (61)  |  Scientific (955)  |  Way (1214)

Although I was first drawn to math and science by the certainty they promised, today I find the unanswered questions and the unexpected connections at least as attractive.
In Warped Passages (2005), 65.
Science quotes on:  |  Attractive (25)  |  Certainty (180)  |  Connection (171)  |  Draw (140)  |  Find (1014)  |  First (1302)  |  Promise (72)  |  Question (649)  |  Today (321)  |  Unanswered (8)  |  Unexpected (55)

Although I was four years at the University [of Wisconsin], I did not take the regular course of studies, but instead picked out what I thought would be most useful to me, particularly chemistry, which opened a new world, mathematics and physics, a little Greek and Latin, botany and and geology. I was far from satisfied with what I had learned, and should have stayed longer.
[Enrolled in Feb 1861, left in 1863 without completing a degree, and began his first botanical foot journey.]
John Muir
The Story of My Boyhood and Youth (1913), 286.
Science quotes on:  |  Botany (63)  |  Chemistry (376)  |  Course (413)  |  Degree (277)  |  First (1302)  |  Geology (240)  |  Greek (109)  |  Journey (48)  |  Latin (44)  |  Learn (672)  |  Learned (235)  |  Little (717)  |  Most (1728)  |  New (1273)  |  Open (277)  |  Physic (515)  |  Physics (564)  |  Regular (48)  |  Thought (995)  |  University (130)  |  Useful (260)  |  World (1850)  |  Year (963)

Among all highly civilized peoples the golden age of art has always been closely coincident with the golden age of the pure sciences, particularly with mathematics, the most ancient among them.
This coincidence must not be looked upon as accidental, but as natural, due to an inner necessity. Just as art can thrive only when the artist, relieved of the anxieties of existence, can listen to the inspirations of his spirit and follow in their lead, so mathematics, the most ideal of the sciences, will yield its choicest blossoms only when life’s dismal phantom dissolves and fades away, when the striving after naked truth alone predominates, conditions which prevail only in nations while in the prime of their development.
From Die Entwickelung der Mathematik im Zusammenhange mit der Ausbreitung der Kultur (1893), 4. As translated in Robert Édouard Moritz, Memorabilia Mathematica; Or, The Philomath’s Quotation-Book (1914), 191-192. From the original German, “Bei allen Kulturvölkern ist die Blüthezeit der Kunst auch immer zeitlich eng verbunden mit einer Blüthezeit der reinen Wissenschaften, insbesondere der ältesten unter ihnen, der Mathematik.
Dieses Zusammentreffen dürfte auch nicht ein zufälliges, sondern ein natürliches, ein Ergebniss innerer Notwendigkeit sein. Wie die Kunst nur gedeihen kann, wenn der Künstler, unbekümmert um die Bedrängnisse des Daseins, den Eingebungen seines Geistes lauschen und ihnen folgen kann, so kann die idealste Wissenschaft, die Mathematik, erst dann ihre schönsten Blüthen treiben, wenn des Erdenlebens schweres Traumbild sinkt und sinkt und sinkt, wenn das Streben nach der nackten Wahrheit allein bestimmend ist, was nur bei Nationen in der Vollkraft ihrer Entwickelung vorkommt.”
Science quotes on:  |  Accidental (31)  |  Age (509)  |  Alone (324)  |  Ancient (198)  |  Anxiety (30)  |  Art (680)  |  Artist (97)  |  Blossom (22)  |  Civilized (20)  |  Coincidence (20)  |  Coincident (2)  |  Condition (362)  |  Development (441)  |  Dissolve (22)  |  Due (143)  |  Existence (481)  |  Fade (12)  |  Follow (389)  |  Golden (47)  |  Golden Age (11)  |  Ideal (110)  |  Inner (72)  |  Inspiration (80)  |  Lead (391)  |  Life (1870)  |  Listen (81)  |  Look (584)  |  Mathematics As A Fine Art (23)  |  Most (1728)  |  Must (1525)  |  Nation (208)  |  Natural (810)  |  Necessity (197)  |  People (1031)  |  Phantom (9)  |  Predominate (7)  |  Prevail (47)  |  Prime (11)  |  Pure (299)  |  Pure Science (30)  |  Relieve (6)  |  Spirit (278)  |  Strive (53)  |  Thrive (22)  |  Truth (1109)  |  Will (2350)  |  Yield (86)

Among the minor, yet striking characteristics of mathematics, may be mentioned the fleshless and skeletal build of its propositions; the peculiar difficulty, complication, and stress of its reasonings; the perfect exactitude of its results; their broad universality; their practical infallibility.
In Charles S. Peirce, ‎Charles Hartshorne (ed.), ‎Paul Weiss (ed.), Collected Papers of Charles Sanders Peirce (1931), Vol. 4, 197.
Science quotes on:  |  Broad (28)  |  Build (211)  |  Characteristic (154)  |  Complication (30)  |  Difficulty (201)  |  Exactitude (10)  |  Flesh (28)  |  Infallibility (7)  |  Mention (84)  |  Minor (12)  |  Peculiar (115)  |  Perfect (223)  |  Practical (225)  |  Proposition (126)  |  Reasoning (212)  |  Result (700)  |  Skeleton (25)  |  Stress (22)  |  Striking (48)  |  Universal (198)  |  Universality (22)

An announcement of [Christopher] Zeeman’s lecture at Northwestern University in the spring of 1977 contains a quote describing catastrophe theory as the most important development in mathematics since the invention of calculus 300 years ago.
In book review of Catastrophe Theory: Collected Papers, 1972-1977, in Bulletin of the American Mathematical Society (Nov 1978), 84, No. 6, 1360. Reprinted in Stephen Smale, Roderick Wong(ed.), The Collected Papers of Stephen Smale (2000), Vol. 2, 814.
Science quotes on:  |  Announcement (15)  |  Calculus (65)  |  Catastrophe (35)  |  Catastrophe Theory (3)  |  Development (441)  |  Important (229)  |  Invention (400)  |  Lecture (111)  |  Most (1728)  |  Quote (46)  |  Spring (140)  |  Theory (1015)  |  University (130)  |  Year (963)  |  Sir Erik Christopher Zeeman (6)

An astronomer must be the wisest of men; his mind must be duly disciplined in youth; especially is mathematical study necessary; both an acquaintance with the doctrine of number, and also with that other branch of mathematics, which, closely connected as it is with the science of the heavens, we very absurdly call geometry, the measurement of the earth.
Plato
From the 'Epilogue to the Laws' (Epinomis), 988-990. As quoted in William Whewell, History of the Inductive Sciences from the Earliest to the Present Time (1837), Vol. 1, 161. (Although referenced to Plato’s Laws, the Epinomis is regarded as a later addition, not by Plato himself.)
Science quotes on:  |  Absurd (60)  |  Acquaintance (38)  |  Astronomer (97)  |  Both (496)  |  Branch (155)  |  Call (781)  |  Connect (126)  |  Discipline (85)  |  Doctrine (81)  |  Earth (1076)  |  Geometry (271)  |  Heaven (266)  |  Heavens (125)  |  Measurement (178)  |  Mind (1377)  |  Must (1525)  |  Necessary (370)  |  Nomenclature (159)  |  Number (710)  |  Other (2233)  |  Study (701)  |  Wise (143)  |  Youth (109)

An essential [of an inventor] is a logical mind that sees analogies. No! No! not mathematical. No man of a mathematical habit of mind ever invented anything that amounted to much. He hasn’t the imagination to do it. He sticks too close to the rules, and to the things he is mathematically sure he knows, to create anything new.
As quoted in French Strother, 'The Modern Profession of Inventing', World's Work and Play (Jul 1905), 6, No. 32, 187.
Science quotes on:  |  Amount (153)  |  Analogy (76)  |  Create (245)  |  Do (1905)  |  Essential (210)  |  Habit (174)  |  Imagination (349)  |  Invent (57)  |  Inventor (79)  |  Know (1538)  |  Logical (57)  |  Man (2252)  |  Mathematician (407)  |  Mind (1377)  |  New (1273)  |  Rule (307)  |  See (1094)  |  Thing (1914)

An incidental remark from a German colleague illustrates the difference between Prussian ways and our own. He had apparently been studying the progress of our various crews on the river, and had been struck with the fact that though the masters in charge of the boats seemed to say and do very little, yet the boats went continually faster and faster, and when I mentioned Dr. Young’s book to him, he made the unexpected but suggestive reply: “Mathematics in Prussia! Ah, sir, they teach mathematics in Prussia as you teach your boys rowing in England: they are trained by men who have been trained by men who have themselves been trained for generations back.”
In John Perry (ed.), Discussion on the Teaching of Mathematics (1901), 43. The discussion took place on 14 Sep 1901 at the British Association at Glasgow, during a joint meeting of the mathematics and physics sections with the education section. The proceedings began with an address by John Perry. Langley related this anecdote during the Discussion which followed.
Science quotes on:  |  Back (395)  |  Book (413)  |  Boy (100)  |  Charge (63)  |  Colleague (51)  |  Difference (355)  |  Do (1905)  |  England (43)  |  Fact (1257)  |  Faster (50)  |  Generation (256)  |  German (37)  |  Incidental (15)  |  Little (717)  |  Master (182)  |  Mention (84)  |  Progress (492)  |  Reply (58)  |  River (140)  |  Row (9)  |  Say (989)  |  Studying (70)  |  Teach (299)  |  Teaching of Mathematics (39)  |  Themselves (433)  |  Train (118)  |  Unexpected (55)  |  Various (205)  |  Way (1214)  |  Young (253)

An old French geometer used to say that a mathematical theory was never to be considered complete till you had made it so clear that you could explain it to the first man you met in the street.
In Nature (1873), 8, 458.
Science quotes on:  |  Clear (111)  |  Complete (209)  |  Consider (428)  |  Explain (334)  |  First (1302)  |  French (21)  |  Geometer (24)  |  Man (2252)  |  Meet (36)  |  Never (1089)  |  Old (499)  |  Say (989)  |  Street (25)  |  Study And Research In Mathematics (61)  |  Theory (1015)

And as for Mixed Mathematics, I may only make this prediction, that there cannot fail to be more kinds of them, as nature grows further disclosed.
In Advancement of Learning (1605), Book 2. Collected in The Works of Francis Bacon (1765), Vol. 1, 61.
Science quotes on:  |  Disclosed (2)  |  Fail (191)  |  Grow (247)  |  Kind (564)  |  Mixed (6)  |  More (2558)  |  Nature (2017)  |  Prediction (89)

And having thus passed the principles of arithmetic, geometry, astronomy, and geography, with a general compact of physics, they may descend in mathematics to the instrumental science of trigonometry, and from thence to fortification, architecture, engineering, or navigation. And in natural philosophy they may proceed leisurely from the history of meteors, minerals, plants, and living creatures, as far as anatomy. Then also in course might be read to them out of some not tedious writer the institution of physic. … To set forward all these proceedings in nature and mathematics, what hinders but that they may procure, as oft as shall be needful, the helpful experiences of hunters, fowlers, fishermen, shepherds, gardeners, apothecaries; and in other sciences, architects, engineers, mariners, anatomists.
In John Milton and Robert Fletcher (ed.), 'On Education', The Prose Works of John Milton: With an Introductory Review (1834), 100.
Science quotes on:  |  Anatomist (24)  |  Anatomy (75)  |  Apothecary (10)  |  Architect (32)  |  Architecture (50)  |  Arithmetic (144)  |  Astronomy (251)  |  Compact (13)  |  Course (413)  |  Creature (242)  |  Descend (49)  |  Engineer (136)  |  Engineering (188)  |  Experience (494)  |  Fisherman (9)  |  Fortification (6)  |  Forward (104)  |  Gardener (6)  |  General (521)  |  Geography (39)  |  Geometry (271)  |  Helpful (16)  |  Hinder (12)  |  History (716)  |  Hunter (28)  |  Institution (73)  |  Leisure (25)  |  Life (1870)  |  Living (492)  |  Mariner (12)  |  Medicine (392)  |  Meteor (19)  |  Mineral (66)  |  Natural (810)  |  Natural Philosophy (52)  |  Nature (2017)  |  Navigation (26)  |  Other (2233)  |  Pass (241)  |  Philosophy (409)  |  Physic (515)  |  Physics (564)  |  Plant (320)  |  Principle (530)  |  Proceed (134)  |  Proceeding (38)  |  Read (308)  |  Science And Education (17)  |  Set (400)  |  Shepherd (6)  |  Tedious (15)  |  Trigonometry (7)  |  Writer (90)

Angling may be said to be so like the Mathematics that it can never be fully learnt; at least not so fully but that there will still be more new experiments left for the trial of other men that succeed us.
In The Complete Angler (1653, 1915), 7.
Science quotes on:  |  Angling (3)  |  Experiment (736)  |  Learning (291)  |  More (2558)  |  Never (1089)  |  New (1273)  |  Other (2233)  |  Still (614)  |  Succeed (114)  |  Succession (80)  |  Trial (59)  |  Will (2350)

Angling may be said to be so like the mathematics, that it can never be fully learnt.
In Izaak Walton and Charles Cotton, 'Walton to the Reader', The Complete Angler (1653, 1824), Vol. 1, lxv.
Science quotes on:  |  Angling (3)  |  Learn (672)  |  Never (1089)

Another advantage of a mathematical statement is that it is so definite that it might be definitely wrong; and if it is found to be wrong, there is a plenteous choice of amendments ready in the mathematicians’ stock of formulae. Some verbal statements have not this merit; they are so vague that they could hardly be wrong, and are correspondingly useless.
From 'Mathematics of War and Foreign Politics', in James R. Newman, The World of Mathematics (1956), Vol. 2, 1248.
Science quotes on:  |  Advantage (144)  |  Amendment (2)  |  Choice (114)  |  Corresponding (3)  |  Definite (114)  |  Find (1014)  |  Formula (102)  |  Hardly (19)  |  Mathematician (407)  |  Merit (51)  |  Ready (43)  |  Statement (148)  |  Stock (7)  |  Useless (38)  |  Vague (50)  |  Verbal (10)  |  Wrong (246)

Another characteristic of mathematical thought is that it can have no success where it cannot generalize.
In Eberhard Zeidler, Applied Functional Analysis: main principles and their applications (1995), 282.
Science quotes on:  |  Characteristic (154)  |  Generalize (19)  |  Success (327)  |  Thinking (425)  |  Thought (995)

Another diversity of Methods is according to the subject or matter which is handled; for there is a great difference in delivery of the Mathematics, which are the most abstracted of knowledges, and Policy, which is the most immersed…, yet we see how that opinion, besides the weakness of it, hath been of ill desert towards learning, as that which taketh the way to reduce learning to certain empty and barren generalities; being but the very husks and shells of sciences, all the kernel being forced out and expulsed with the torture and press of the method.
Advancement of Learning, Book 2. In James Spedding, The Works of Francis Bacon (1863), Vol. 6, 292-293. Peter Pešić, explains that 'By Mathematics, he had in mind a sterile and rigid scheme of logical classifications, called dichotomies in his time,' inLabyrinth: A Search for the Hidden Meaning of Science (2001), 73.
Science quotes on:  |  Abstract (141)  |  According (236)  |  Barren (33)  |  Being (1276)  |  Certain (557)  |  Delivery (7)  |  Desert (59)  |  Difference (355)  |  Diversity (75)  |  Empty (82)  |  Generality (45)  |  Great (1610)  |  Husk (4)  |  Kernel (4)  |  Knowledge (1647)  |  Learning (291)  |  Matter (821)  |  Method (531)  |  Most (1728)  |  Opinion (291)  |  Policy (27)  |  Reduce (100)  |  See (1094)  |  Shell (69)  |  Subject (543)  |  Torture (30)  |  Way (1214)  |  Weakness (50)

Another great and special excellence of mathematics is that it demands earnest voluntary exertion. It is simply impossible for a person to become a good mathematician by the happy accident of having been sent to a good school; this may give him a preparation and a start, but by his own individual efforts alone can he reach an eminent position.
In Conflict of Studies (1873), 2.
Science quotes on:  |  Accident (92)  |  Alone (324)  |  Become (821)  |  Demand (131)  |  Earnest (3)  |  Effort (243)  |  Eminent (20)  |  Excellence (40)  |  Exertion (17)  |  Give (208)  |  Good (906)  |  Great (1610)  |  Happy (108)  |  Impossible (263)  |  Individual (420)  |  Mathematician (407)  |  Person (366)  |  Position (83)  |  Preparation (60)  |  Reach (286)  |  School (227)  |  Send (23)  |  Simply (53)  |  Special (188)  |  Start (237)  |  Value Of Mathematics (60)  |  Voluntary (6)

Any conception which is definitely and completely determined by means of a finite number of specifications, say by assigning a finite number of elements, is a mathematical conception. Mathematics has for its function to develop the consequences involved in the definition of a group of mathematical conceptions. Interdependence and mutual logical consistency among the members of the group are postulated, otherwise the group would either have to be treated as several distinct groups, or would lie beyond the sphere of mathematics.
In 'Mathematics', Encyclopedia Britannica (9th ed.).
Science quotes on:  |  Assign (15)  |  Beyond (316)  |  Complete (209)  |  Completely (137)  |  Conception (160)  |  Consequence (220)  |  Consistency (31)  |  Consistent (50)  |  Definite (114)  |  Definition (238)  |  Definitions and Objects of Mathematics (33)  |  Determine (152)  |  Develop (278)  |  Distinct (98)  |  Element (322)  |  Finite (60)  |  Function (235)  |  Group (83)  |  Interdependence (4)  |  Involve (93)  |  Involved (90)  |  Lie (370)  |  Logical (57)  |  Mean (810)  |  Means (587)  |  Member (42)  |  Mutual (54)  |  Number (710)  |  Otherwise (26)  |  Postulate (42)  |  Say (989)  |  Several (33)  |  Specification (7)  |  Sphere (118)  |  Treat (38)

Anyone who cannot cope with mathematics is not fully human. At best he is a tolerable subhuman who has learned to wear shoes, bathe and not make messes in the house
In Time Enough for Love: The Lives of Lazarus Long (1973), 265.
Science quotes on:  |  Best (467)  |  House (143)  |  Human (1512)  |  Learn (672)  |  Learned (235)  |  Shoe (12)  |  Subhuman (2)

Anyone who has had actual contact with the making of the inventions that built the radio art knows that these inventions have been the product of experiment and work based on physical reasoning, rather than on the mathematicians' calculations and formulae. Precisely the opposite impression is obtained from many of our present day text books and publications.
Attributed.
Science quotes on:  |  Actual (118)  |  Art (680)  |  Book (413)  |  Calculation (134)  |  Contact (66)  |  Experiment (736)  |  Impression (118)  |  Invention (400)  |  Know (1538)  |  Logic (311)  |  Making (300)  |  Obtain (164)  |  Opposite (110)  |  Physical (518)  |  Precisely (93)  |  Present (630)  |  Product (166)  |  Publication (102)  |  Radio (60)  |  Reasoning (212)  |  Work (1402)

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.
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:  |  Axiom (65)  |  Become (821)  |  Conscious (46)  |  Deeper (4)  |  Dependent (26)  |  Formation (100)  |  Indirect (18)  |  Knowledge (1647)  |  Layer (41)  |  Leading (17)  |  Method (531)  |  More (2558)  |  Object (438)  |  Ripe (5)  |  Role (86)  |  Scientific (955)  |  Scientific Thought (17)  |  Sign (63)  |  Soon (187)  |  Summon (11)  |  Theory (1015)  |  Thought (995)  |  Unity (81)

Archimedes will be remembered when Aeschylus is forgotten, because languages die and mathematical ideas do not. “Immortality” may be a silly word, but probably a mathematician has the best chance of whatever it may mean.
In A Mathematician's Apology (1940, reprint with Foreward by C.P. Snow 1992), 81.
Science quotes on:  |  Aeschylus (5)  |  Archimedes (63)  |  Best (467)  |  Chance (244)  |  Do (1905)  |  Forgotten (53)  |  Idea (881)  |  Language (308)  |  Mean (810)  |  Remember (189)  |  Silly (17)  |  Whatever (234)  |  Will (2350)  |  Word (650)

Archimedes, who combined a genius for mathematics with a physical insight, must rank with Newton, who lived nearly two thousand years later, as one of the founders of mathematical physics. … The day (when having discovered his famous principle of hydrostatics he ran through the streets shouting Eureka! Eureka!) ought to be celebrated as the birthday of mathematical physics; the science came of age when Newton sat in his orchard.
In An Introduction to Mathematics (1911), 37.
Science quotes on:  |  Age (509)  |  Archimedes (63)  |  Birthday (9)  |  Celebrate (21)  |  Discover (571)  |  Eureka (13)  |  Famous (12)  |  Founder (26)  |  Genius (301)  |  Insight (107)  |  Later (18)  |  Lived (3)  |  Mathematical Physics (12)  |  Mathematicians and Anecdotes (141)  |  Must (1525)  |  Nearly (137)  |  Sir Isaac Newton (363)  |  Orchard (4)  |  Physic (515)  |  Physical (518)  |  Physics (564)  |  Principle (530)  |  Rank (69)  |  Run (158)  |  Shout (25)  |  Sit (51)  |  Street (25)  |  Thousand (340)  |  Through (846)  |  Two (936)  |  Year (963)

Arithmetically speaking, rabbits multiply faster than adders add.
Anonymous
In Evan Esar, 20,000 Quips and Quotes, 509.
Science quotes on:  |  Adder (3)  |  Faster (50)  |  Joke (90)  |  Multiply (40)  |  Speaking (118)

Art is an expression of the world order and is, therefore, orderly, organic, subject to mathematical law, and susceptible to mathematical analysis.
In 'The Theosophic View of the Art of Architecture', The Beautiful Necessity, Seven Essays on Theosophy and Architecture (2nd ed., 1922), Preface to the Second Edition, 11.
Science quotes on:  |  Analysis (244)  |  Art (680)  |  Expression (181)  |  Law (913)  |  Mathematical Analysis (23)  |  Order (638)  |  Orderly (38)  |  Organic (161)  |  Subject (543)  |  Susceptible (8)  |  World (1850)

Art is usually considered to be not of the highest quality if the desired object is exhibited in the midst of unnecessary lumber.
In Mathematics: Queen and Servant of Sciences (1938), 20. Bell is writing about the postulational method and the art of pruning a set of postulates to bare essentials without internal duplication.
Science quotes on:  |  Art (680)  |  Consider (428)  |  Desired (5)  |  Exhibit (21)  |  High (370)  |  Lumber (5)  |  Midst (8)  |  Object (438)  |  Quality (139)  |  Unnecessary (23)  |  Usually (176)

As a little boy, I showed an abnormal aptitude for mathematics this gift played a horrible part in tussles with quinsy or scarlet fever, when I felt enormous spheres and huge numbers swell relentlessly in my aching brain.
In Speak, Memory: An Autobiography Revisited (1999), 2
Science quotes on:  |  Abnormal (6)  |  Ache (7)  |  Aptitude (19)  |  Boy (100)  |  Brain (281)  |  Enormous (44)  |  Feel (371)  |  Fever (34)  |  Gift (105)  |  Horrible (10)  |  Huge (30)  |  Little (717)  |  Number (710)  |  Relentless (9)  |  Scarlet Fever (2)  |  Show (353)  |  Sphere (118)  |  Swell (4)

As an Art, Mathematics has its own standard of beauty and elegance which can vie with the more decorative arts. In this it is diametrically opposed to a Baroque art which relies on a wealth of ornamental additions. Bereft of superfluous addenda, Mathematics may appear, on first acquaintance, austere and severe. But longer contemplation reveals the classic attributes of simplicity relative to its significance and depth of meaning.
In The Skeleton Key of Mathematics (1949), 12.
Science quotes on:  |  Acquaintance (38)  |  Addition (70)  |  Appear (122)  |  Art (680)  |  Attribute (65)  |  Austere (7)  |  Beauty (313)  |  Bereft (2)  |  Classic (13)  |  Contemplation (75)  |  Depth (97)  |  Diametrical (2)  |  Diametrically (6)  |  Elegance (40)  |  First (1302)  |  Longer (10)  |  Meaning (244)  |  More (2558)  |  Opposed (3)  |  Ornament (20)  |  Relative (42)  |  Reveal (152)  |  Severe (17)  |  Significance (114)  |  Simplicity (175)  |  Standard (64)  |  Superfluous (21)  |  Wealth (100)

As an exercise of the reasoning faculty, pure mathematics is an admirable exercise, because it consists of reasoning alone, and does not encumber the student with an exercise of judgment: and it is well to begin with learning one thing at a time, and to defer a combination of mental exercises to a later period.
In Annotations to Bacon’s Essays (1873), Essay 1, 493.
Science quotes on:  |  Admirable (20)  |  Alone (324)  |  Begin (275)  |  Combination (150)  |  Consist (223)  |  Encumber (4)  |  Exercise (113)  |  Faculty (76)  |  Judgment (140)  |  Late (119)  |  Learn (672)  |  Learning (291)  |  Mental (179)  |  Period (200)  |  Pure (299)  |  Pure Mathematics (72)  |  Reason (766)  |  Reasoning (212)  |  Student (317)  |  Thing (1914)  |  Time (1911)  |  Value Of Mathematics (60)

As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.
Sidelights on Relativity (1920), 28.
Science quotes on:  |  Certain (557)  |  Do (1905)  |  Law (913)  |  Reality (274)

As for everything else, so for a mathematical theory: beauty can be perceived but not explained.
President’s address (1883) to the British Association for the Advancement of Science, in The Collected Mathematical Papers (1895), Vol. 8, xxii.
Science quotes on:  |  Beauty (313)  |  Everything (489)  |  Explain (334)  |  Perceive (46)  |  Theory (1015)

As for the place of mathematics in relation to other sciences, mathematics can be seen as a big warehouse full of shelves. Mathematicians put things on the shelves and guarantee that they are true. They also explain how to use them and how to reconstruct them. Other sciences come and help themselves from the shelves; mathematicians are not concerned with what they do with what they have taken. This metaphor is rather coarse, but it reflects the situation well enough.
From interview with Marc Kirch, 'My First Fifty years at the Collège de France', collected in Helge Holden and Ragni Piene, The Abel Prize: 2003-2007 The First Five Years (2009), 15-29.
Science quotes on:  |  Coarse (4)  |  Concern (239)  |  Do (1905)  |  Enough (341)  |  Explain (334)  |  Guarantee (30)  |  Help (116)  |  Mathematician (407)  |  Metaphor (37)  |  Other (2233)  |  Reconstruct (5)  |  Reflect (39)  |  Relation (166)  |  Shelf (8)  |  Situation (117)  |  Themselves (433)  |  Thing (1914)  |  True (239)  |  Use (771)

As history proves abundantly, mathematical achievement, whatever its intrinsic worth, is the most enduring of all.
In A Mathematician’s Apology (1940, 1967), 80.
Science quotes on:  |  Achievement (187)  |  Enduring (6)  |  History (716)  |  Intrinsic (18)  |  Most (1728)  |  Proof (304)  |  Prove (261)  |  Whatever (234)  |  Worth (172)

As in Mathematicks, so in Natural Philosophy, the Investigation of difficult Things by the Method of Analysis, ought ever to precede the Method of Composition. This Analysis consists in making Experiments and Observations, and in drawing general Conclusions from them by Induction, and admitting of no Objections against the Conclusions, but such as are taken from Experiments, or other certain Truths. For Hypotheses are not to be regarded in experimental Philosophy.
From Opticks, (1704, 2nd ed. 1718), Book 3, Query 31, 380.
Science quotes on:  |  Against (332)  |  Analysis (244)  |  Certain (557)  |  Composition (86)  |  Conclusion (266)  |  Consist (223)  |  Difficult (263)  |  Drawing (56)  |  Experiment (736)  |  Experimental (193)  |  General (521)  |  Hypothesis (314)  |  Induction (81)  |  Investigation (250)  |  Making (300)  |  Method (531)  |  Natural (810)  |  Natural Philosophy (52)  |  Objection (34)  |  Observation (593)  |  Other (2233)  |  Philosophy (409)  |  Regard (312)  |  Thing (1914)  |  Truth (1109)

As in the domains of practical life so likewise in science there has come about a division of labor. The individual can no longer control the whole field of mathematics: it is only possible for him to master separate parts of it in such a manner as to enable him to extend the boundaries of knowledge by creative research.
In Die reine Mathematik in den Jahren 1884-99, 10. As quoted, cited and translated in Robert Édouard Moritz, Memorabilia Mathematica; Or, The Philomath’s Quotation-Book (1914), 94.
Science quotes on:  |  Boundary (55)  |  Control (182)  |  Creative (144)  |  Division (67)  |  Domain (72)  |  Enable (122)  |  Extend (129)  |  Field (378)  |  Individual (420)  |  Knowledge (1647)  |  Labor (200)  |  Life (1870)  |  Manner (62)  |  Master (182)  |  Part (235)  |  Possible (560)  |  Practical (225)  |  Research (753)  |  Separate (151)  |  Study And Research In Mathematics (61)  |  Whole (756)

As pure truth is the polar star of our science [mathematics], so it is the great advantage of our science over others that it awakens more easily the love of truth in our pupils. … If Hegel justly said, “Whoever does not know the works of the ancients, has lived without knowing beauty,” Schellbach responds with equal right, “Who does not know mathematics, and the results of recent scientific investigation, dies without knowing truth.”
Max Simon
From Didaktik und Methodik des Rechnens und der Mathematik (1908), 37. As quoted and translated in J.W.A. Young, Teaching of Mathematics in the Elementary and the Secondary School (1907), 44. From the original German, “Wenn Hegel mit Recht sagt: ‘Wer die Werke der Alten nicht kennt, der hat gelebt, ohne die Schönheit gekannt zu haben’, so erwidert Schellbach mit nicht minderem Recht: ‘Wer die Math. und die Resultate der neueren Naturforschung nicht gekannt hat, der stirbt, ohne die Wahrheit zu kennen.’”
Science quotes on:  |  Advantage (144)  |  Ancient (198)  |  Awake (19)  |  Beauty (313)  |  Die (94)  |  Equal (88)  |  Great (1610)  |  Georg Wilhelm Friedrich Hegel (7)  |  Investigation (250)  |  Know (1538)  |  Knowing (137)  |  Live (650)  |  Love (328)  |  More (2558)  |  Other (2233)  |  Polar (13)  |  Pole Star (2)  |  Pupil (62)  |  Pure (299)  |  Recent (78)  |  Respond (14)  |  Result (700)  |  Right (473)  |  Karl Heinrich Schellbach (2)  |  Scientific (955)  |  Star (460)  |  Truth (1109)  |  Whoever (42)  |  Work (1402)

As regards authority I so proceed. Boetius says in the second prologue to his Arithmetic, “If an inquirer lacks the four parts of mathematics, he has very little ability to discover truth.” And again, “Without this theory no one can have a correct insight into truth.” And he says also, “I warn the man who spurns these paths of knowledge that he cannot philosophize correctly.” And Again, “It is clear that whosoever passes these by, has lost the knowledge of all learning.”
Opus Majus [1266-1268], Part IV, distinction I, chapter I, trans. R. B. Burke, The Opus Majus of Roger Bacon (1928), Vol. I, 117.
Science quotes on:  |  Ability (162)  |  Arithmetic (144)  |  Authority (99)  |  Discover (571)  |  Inquirer (9)  |  Insight (107)  |  Knowledge (1647)  |  Lack (127)  |  Learning (291)  |  Little (717)  |  Man (2252)  |  Path (159)  |  Proceed (134)  |  Regard (312)  |  Say (989)  |  Theory (1015)  |  Truth (1109)

As the prerogative of Natural Science is to cultivate a taste for observation, so that of Mathematics is, almost from the starting point, to stimulate the faculty of invention.
In 'A Plea for the Mathematician', Nature, 1, 261 in Collected Mathematical Papers, Vol. 2 (1908), 717.
Science quotes on:  |  Cultivate (24)  |  Faculty (76)  |  Invention (400)  |  Natural (810)  |  Natural Science (133)  |  Nature Of Mathematics (80)  |  Observation (593)  |  Point (584)  |  Prerogative (3)  |  Starting Point (16)  |  Stimulate (21)  |  Taste (93)

As there is no study which may be so advantageously entered upon with a less stock of preparatory knowledge than mathematics, so there is none in which a greater number of uneducated men have raised themselves, by their own exertions, to distinction and eminence. … Many of the intellectual defects which, in such cases, are commonly placed to the account of mathematical studies, ought to be ascribed to the want of a liberal education in early youth.
In Elements of the Philosophy of the Human Mind (1827), Vol. 3, Chap. 1, Sec. 3, 183.
Science quotes on:  |  Account (195)  |  Advantage (144)  |  Defect (31)  |  Distinction (72)  |  Early (196)  |  Education (423)  |  Eminence (25)  |  Enter (145)  |  Exertion (17)  |  Greater (288)  |  Intellectual (258)  |  Knowledge (1647)  |  Liberal Education (2)  |  Number (710)  |  Preparation (60)  |  Study (701)  |  Themselves (433)  |  Uneducated (9)  |  Want (504)  |  Youth (109)

As to the need of improvement there can be no question whilst the reign of Euclid continues. My own idea of a useful course is to begin with arithmetic, and then not Euclid but algebra. Next, not Euclid, but practical geometry, solid as well as plane; not demonstration, but to make acquaintance. Then not Euclid, but elementary vectors, conjoined with algebra, and applied to geometry. Addition first; then the scalar product. Elementary calculus should go on simultaneously, and come into vector algebraic geometry after a bit. Euclid might be an extra course for learned men, like Homer. But Euclid for children is barbarous.
Electro-Magnetic Theory (1893), Vol. 1, 148. In George Edward Martin, The Foundations of Geometry and the Non-Euclidean Plane (1982), 130.
Science quotes on:  |  Acquaintance (38)  |  Addition (70)  |  Algebra (117)  |  Applied (176)  |  Arithmetic (144)  |  Barbarous (4)  |  Begin (275)  |  Calculus (65)  |  Child (333)  |  Children (201)  |  Continue (179)  |  Course (413)  |  Demonstration (120)  |  Education (423)  |  Elementary (98)  |  Euclid (60)  |  First (1302)  |  Geometry (271)  |  Idea (881)  |  Improvement (117)  |  Learn (672)  |  Learned (235)  |  Next (238)  |  Practical (225)  |  Product (166)  |  Question (649)  |  Reign (24)  |  Solid (119)  |  Useful (260)  |  Vector (6)

Astronomy and Pure Mathematics are the magnetic poles toward which the compass of my mind ever turns.
In Letter to Bolyai (30 Jun 1803), in Franz Schmidt and Paul Stäckel, Briefwechsel zwischen Carl Friedrich Gauss und Wolfgang Bolyai, (1899), Letter XXIII , 55.
Science quotes on:  |  Astronomy (251)  |  Compass (37)  |  Magnetic (44)  |  Mathematicians and Anecdotes (141)  |  Mind (1377)  |  Pole (49)  |  Pure (299)  |  Pure Mathematics (72)  |  Toward (45)  |  Turn (454)

At every major step physics has required, and frequently stimulated, the introduction of new mathematical tools and concepts. Our present understanding of the laws of physics, with their extreme precision and universality, is only possible in mathematical terms.
In Book Review 'Pulling the Strings,' of Lawrence Krauss's Hiding in the Mirror: The Mysterious Lure of Extra Dimensions, from Plato to String Theory and Beyond in Nature (22 Dec 2005), 438, 1081.
Science quotes on:  |  Concept (242)  |  Extreme (78)  |  Introduction (37)  |  Law (913)  |  Major (88)  |  New (1273)  |  Physic (515)  |  Physics (564)  |  Possible (560)  |  Precision (72)  |  Present (630)  |  Require (229)  |  Required (108)  |  Step (234)  |  Stimulate (21)  |  Term (357)  |  Terms (184)  |  Tool (129)  |  Understanding (527)  |  Universal (198)  |  Universality (22)

At the present time it is of course quite customary for physicists to trespass on chemical ground, for mathematicians to do excellent work in physics, and for physicists to develop new mathematical procedures. … Trespassing is one of the most successful techniques in science.
In Dynamics in Psychology (1940, 1973), 116.
Science quotes on:  |  Chemical (303)  |  Chemistry (376)  |  Course (413)  |  Custom (44)  |  Customary (18)  |  Develop (278)  |  Do (1905)  |  Ground (222)  |  Most (1728)  |  New (1273)  |  Physic (515)  |  Physicist (270)  |  Physics (564)  |  Present (630)  |  Procedure (48)  |  Success (327)  |  Successful (134)  |  Technique (84)  |  Time (1911)  |  Trespass (5)  |  Trespassing (2)  |  Work (1402)

Bacon himself was very ignorant of all that had been done by mathematics; and, strange to say, he especially objected to astronomy being handed over to the mathematicians. Leverrier and Adams, calculating an unknown planet into a visible existence by enormous heaps of algebra, furnish the last comment of note on this specimen of the goodness of Bacon’s view… . Mathematics was beginning to be the great instrument of exact inquiry: Bacon threw the science aside, from ignorance, just at the time when his enormous sagacity, applied to knowledge, would have made him see the part it was to play. If Newton had taken Bacon for his master, not he, but somebody else, would have been Newton.
In Budget of Paradoxes (1872), 53-54.
Science quotes on:  |  Algebra (117)  |  Applied (176)  |  Apply (170)  |  Astronomy (251)  |  Sir Francis Bacon (188)  |  Begin (275)  |  Beginning (312)  |  Being (1276)  |  Calculate (58)  |  Comment (12)  |  Enormous (44)  |  Exact (75)  |  Existence (481)  |  Furnish (97)  |  Goodness (26)  |  Great (1610)  |  Heap (15)  |  Himself (461)  |  Ignorance (254)  |  Ignorant (91)  |  Inquiry (88)  |  Instrument (158)  |  Knowledge (1647)  |  Last (425)  |  LeVerrier_Urbain (3)  |  Master (182)  |  Mathematician (407)  |  Mathematicians and Anecdotes (141)  |  Sir Isaac Newton (363)  |  Note (39)  |  Object (438)  |  Part (235)  |  Planet (402)  |  Play (116)  |  Sagacity (11)  |  Say (989)  |  See (1094)  |  Specimen (32)  |  Strange (160)  |  Throw (45)  |  Time (1911)  |  Unknown (195)  |  View (496)  |  Visible (87)

Be very vigilent over thy Childe … If he chuse the profession of a Scholler, advise him to study the most profitable Arts: Poetry, and the Mathematichs, take up too great a latitude of the Soule, and moderately used, are good Recreations, but bad Callings, bring nothing but their owne Reward.
From Alexander B. Grosart (ed), 'Enchyridion: The Fourth Book' (1641), The Complete Works in Prose and Verse of Francis Quarles (1880), Vol. 1, 48, Cap. XCIX.
Science quotes on:  |  Poetry (150)  |  Profession (108)  |  Recreation (23)  |  Scholar (52)  |  Unprofitable (7)

Beauty is the first test: there is no permanent place in the world for ugly mathematics.
In A Mathematician’s Apology (1940, reprint with Foreward by C.P. Snow 1992), 85.
Science quotes on:  |  Beauty (313)  |  First (1302)  |  Mathematical Beauty (19)  |  Permanent (67)  |  Test (221)  |  World (1850)

Before an experiment can be performed, it must be planned—the question to nature must be formulated before being posed. Before the result of a measurement can be used, it must be interpreted—nature's answer must be understood properly. These two tasks are those of the theorist, who finds himself always more and more dependent on the tools of abstract mathematics. Of course, this does not mean that the experimenter does not also engage in theoretical deliberations. The foremost classical example of a major achievement produced by such a division of labor is the creation of spectrum analysis by the joint efforts of Robert Bunsen, the experimenter, and Gustav Kirchoff, the theorist. Since then, spectrum analysis has been continually developing and bearing ever richer fruit.
'The Meaning and Limits of Exact Science', Science (30 Sep 1949), 110, No. 2857, 325. Advance reprinting of chapter from book Max Planck, Scientific Autobiography (1949), 110.
Science quotes on:  |  Abstract (141)  |  Abstract Mathematics (9)  |  Achievement (187)  |  Analysis (244)  |  Answer (389)  |  Bearing (10)  |  Being (1276)  |  Robert Bunsen (8)  |  Classical (49)  |  Collaboration (16)  |  Continuing (4)  |  Course (413)  |  Creation (350)  |  Deliberation (5)  |  Dependence (46)  |  Development (441)  |  Division (67)  |  Effort (243)  |  Engage (41)  |  Example (98)  |  Experiment (736)  |  Experimenter (40)  |  Find (1014)  |  Formulation (37)  |  Fruit (108)  |  Himself (461)  |  Interpretation (89)  |  Joint (31)  |  Kirchoff_Gustav (3)  |  Labor (200)  |  Major (88)  |  Mean (810)  |  Measurement (178)  |  More (2558)  |  Must (1525)  |  Nature (2017)  |  Perform (123)  |  Performance (51)  |  Plan (122)  |  Produced (187)  |  Properly (21)  |  Question (649)  |  Result (700)  |  Richness (15)  |  Spectral Analysis (4)  |  Spectrum (35)  |  Task (152)  |  Theorist (44)  |  Tool (129)  |  Two (936)  |  Understanding (527)  |  Understood (155)  |  Use (771)

Before the introduction of the Arabic notation, multiplication was difficult, and the division even of integers called into play the highest mathematical faculties. Probably nothing in the modern world could have more astonished a Greek mathematician than to learn that, under the influence of compulsory education, the whole population of Western Europe, from the highest to the lowest, could perform the operation of division for the largest numbers. This fact would have seemed to him a sheer impossibility. … Our modern power of easy reckoning with decimal fractions is the most miraculous result of a perfect notation.
In Introduction to Mathematics (1911), 59.
Science quotes on:  |  Arabic (4)  |  Astonish (39)  |  Astonished (10)  |  Call (781)  |  Compulsory (8)  |  Decimal (21)  |  Difficult (263)  |  Division (67)  |  Easy (213)  |  Education (423)  |  Europe (50)  |  Fact (1257)  |  Fraction (16)  |  Greek (109)  |  Impossibility (60)  |  Influence (231)  |  Integer (12)  |  Introduction (37)  |  Largest (39)  |  Learn (672)  |  Mathematician (407)  |  Mathematics As A Language (20)  |  Miraculous (11)  |  Modern (402)  |  Modern World (5)  |  More (2558)  |  Most (1728)  |  Multiplication (46)  |  Notation (28)  |  Nothing (1000)  |  Number (710)  |  Operation (221)  |  Perfect (223)  |  Perform (123)  |  Population (115)  |  Power (771)  |  Probably (50)  |  Reckon (31)  |  Reckoning (19)  |  Result (700)  |  Western (45)  |  Whole (756)  |  World (1850)

Before you generalize, formalize, and axiomatize there must be mathematical substance.
In Eberhard Zeidler, Applied Functional Analysis: main principles and their applications (1995), 282.
Science quotes on:  |  Axiom (65)  |  Generalize (19)  |  Must (1525)  |  Substance (253)

Being a language, mathematics may be used not only to inform but also, among other things, to seduce.
From Fractals: Form, Chance and Dimension (1977), 20.
Science quotes on:  |  Being (1276)  |  Inform (50)  |  Language (308)  |  Other (2233)  |  Seduce (4)  |  Thing (1914)

Bertrand, Darboux, and Glaisher have compared Cayley to Euler, alike for his range, his analytical power, and, not least, for his prolific production of new views and fertile theories. There is hardly a subject in the whole of pure mathematics at which he has not worked.
In Proceedings of London Royal Society (1895), 58, 21.
Science quotes on:  |  Alike (60)  |  Analysis (244)  |  Joseph Bertrand (6)  |  Arthur Cayley (17)  |  Compare (76)  |  Leonhard Euler (35)  |  Fertile (30)  |   James Whitbread Lee Glaisher (3)  |  Mathematicians and Anecdotes (141)  |  New (1273)  |  Power (771)  |  Production (190)  |  Prolific (5)  |  Pure (299)  |  Pure Mathematics (72)  |  Range (104)  |  Subject (543)  |  Theory (1015)  |  View (496)  |  Whole (756)  |  Work (1402)

Besides a mathematical inclination, an exceptionally good mastery of one’s native tongue is the most vital asset of a competent programmer.
…...
Science quotes on:  |  Asset (6)  |  Competent (20)  |  Exceptionally (3)  |  Good (906)  |  Inclination (36)  |  Mastery (36)  |  Most (1728)  |  Native (41)  |  Programmer (5)  |  Tongue (44)  |  Vital (89)

Besides accustoming the student to demand, complete proof, and to know when he has not obtained it, mathematical studies are of immense benefit to his education by habituating him to precision. It is one of the peculiar excellencies of mathematical discipline, that the mathematician is never satisfied with à peu près. He requires the exact truth. Hardly any of the non-mathematical sciences, except chemistry, has this advantage. One of the commonest modes of loose thought, and sources of error both in opinion and in practice, is to overlook the importance of quantities. Mathematicians and chemists are taught by the whole course of their studies, that the most fundamental difference of quality depends on some very slight difference in proportional quantity; and that from the qualities of the influencing elements, without careful attention to their quantities, false expectation would constantly be formed as to the very nature and essential character of the result produced.
In An Examination of Sir William Hamilton’s Philosophy (1878), 611. [The French phrase, à peu près means “approximately”. —Webmaster]
Science quotes on:  |  Accustom (52)  |  Advantage (144)  |  Approximate (25)  |  Attention (196)  |  Benefit (123)  |  Both (496)  |  Careful (28)  |  Character (259)  |  Chemist (169)  |  Chemistry (376)  |  Complete (209)  |  Constantly (27)  |  Course (413)  |  Demand (131)  |  Depend (238)  |  Difference (355)  |  Discipline (85)  |  Education (423)  |  Element (322)  |  Error (339)  |  Essential (210)  |  Exact (75)  |  Excellence (40)  |  Expectation (67)  |  False (105)  |  Form (976)  |  Fundamental (264)  |  Habituate (3)  |  Hardly (19)  |  Immense (89)  |  Importance (299)  |  Influence (231)  |  Know (1538)  |  Loose (14)  |  Mathematician (407)  |  Mode (43)  |  Most (1728)  |  Nature (2017)  |  Never (1089)  |  Obtain (164)  |  Opinion (291)  |  Overlook (33)  |  Peculiar (115)  |  Practice (212)  |  Precision (72)  |  Produce (117)  |  Produced (187)  |  Proof (304)  |  Proportional (5)  |  Quality (139)  |  Quantity (136)  |  Require (229)  |  Result (700)  |  Satisfied (23)  |  Slight (32)  |  Source Of Error (2)  |  Student (317)  |  Study (701)  |  Teach (299)  |  Thought (995)  |  Truth (1109)  |  Value Of Mathematics (60)  |  Whole (756)

Boltzmann was both a wizard of a mathematician and a physicist of international renown. The magnitude of his output of scientific papers was positively unnerving. He would publish two, three, sometimes four monographs a year; each one was forbiddingly dense, festooned with mathematics, and as much as a hundred pages in length.
In 'The Bulldog: A Profile of Ludwig Boltzmann', The American Scholar (1 Jan 1999), 99.
Science quotes on:  |  Ludwig Eduard Boltzmann (25)  |  Both (496)  |  Dense (5)  |  Festoon (3)  |  Hundred (240)  |  International (40)  |  Magnitude (88)  |  Mathematician (407)  |  Monograph (5)  |  Output (12)  |  Paper (192)  |  Physicist (270)  |  Publish (42)  |  Renown (3)  |  Scientific (955)  |  Two (936)  |  Wizard (4)  |  Year (963)

Bolyai [Janos] projected a universal language for speech as we have it for music and mathematics.
In János Bolyai, Science Absolute of Space, translated from the Latin by George Bruce Halsted (1896), Translator's Introduction, xxix.
Science quotes on:  |  János Bolyai (6)  |  Language (308)  |  Mathematicians and Anecdotes (141)  |  Music (133)  |  Project (77)  |  Speech (66)  |  Universal (198)

Büchsel in his reminiscences from the life of a country parson relates that he sought his recreation in Lacroix’s Differential Calculus and thus found intellectual refreshment for his calling. Instances like this make manifest the great advantage which occupation with mathematics affords to one who lives remote from the city and is compelled to forego the pleasures of art. The entrancing charm of mathematics, which captivates every one who devotes himself to it, and which is comparable to the fine frenzy under whose ban the poet completes his work, has ever been incomprehensible to the spectator and has often caused the enthusiastic mathematician to be held in derision. A classic illustration is the example of Archimedes….
From Die Entwickelung der Mathematik im Zusammenhange mit der Ausbreitung der Kultur (1893), 22. As translated in Robert Édouard Moritz, Memorabilia Mathematica; Or, The Philomath’s Quotation-Book (1914), 186. From the original German, “Wenn Büchsel in seinen Erinnerungen aus dem Leben eines Landgeistlichen erzählt, dass er in der Differentialrechnung von Lacroix Erholung gesucht und geistige Erfrischung ftir seinen Beruf gefunden habe, so erkennen wir darin den grossen Vorzug, den die Beschaftigung mit der Mathematik für jemanden hat, der fern von einer Stadt lebt und auf ihre Kunstgenüsse verzichten muss. Der berückende Zauber der Mathematik, dem jeder unterliegt, der sich ihr ergiebt, und der dem holden Wahnsinn vergleichbar ist, unter dessen Bann der Dichter sein Work vollendet, ist dem betrachtenden Mitmenschen immer unbegreiflich gewesen und hat den begeisterten Mathematiker oft zum Gespött werden lassen. Als klassisches Beispiel wird jedem Schüler Archimedes…”
Science quotes on:  |  Advantage (144)  |  Archimedes (63)  |  Art (680)  |  Calculus (65)  |  Captivate (5)  |  Cause (561)  |  Charm (54)  |  City (87)  |  Classic (13)  |  Compel (31)  |  Complete (209)  |  Country (269)  |  Derision (8)  |  Devote (45)  |  Differential Calculus (11)  |  Enthusiastic (7)  |  Entrance (16)  |  Example (98)  |  Forego (4)  |  Frenzy (6)  |  Great (1610)  |  Himself (461)  |  Hold (96)  |  Illustration (51)  |  Incomprehensible (31)  |  Intellect (251)  |  Intellectual (258)  |  Life (1870)  |  Live (650)  |  Mathematician (407)  |  Occupation (51)  |  Parson (3)  |  Pleasure (191)  |  Poet (97)  |  Recreation (23)  |  Refreshment (3)  |  Reminiscence (4)  |  Remote (86)  |  Spectator (11)  |  Work (1402)

Elbert (Green) Hubbard quote: Business, to be successful, must be based on science, for demand and supply are matters
Business, to be successful, must be based on science, for demand and supply are matters of mathematics, not guesswork.
The Book of Business (1913), 56.
Science quotes on:  |  Business (156)  |  Demand (131)  |  Guesswork (4)  |  Matter (821)  |  Must (1525)  |  Success (327)  |  Successful (134)  |  Supply (100)  |  Supply And Demand (4)

But indeed, the English generally have been very stationary in latter times, and the French, on the contrary, so active and successful, particularly in preparing elementary books, in the mathematical and natural sciences, that those who wish for instruction, without caring from what nation they get it, resort universally to the latter language.
Letter (29 Jan 1824) to Patrick K. Rodgers. Collected in Andrew A. Lipscomb (ed.), The Writings of Thomas Jefferson (1904), Vol. 16, 2.
Science quotes on:  |  Active (80)  |  Book (413)  |  Caring (6)  |  Contrary (143)  |  Elementary (98)  |  English (35)  |  French (21)  |  Indeed (323)  |  Instruction (101)  |  Language (308)  |  Nation (208)  |  Natural (810)  |  Natural Science (133)  |  Preparing (21)  |  Stationary (11)  |  Successful (134)  |  Time (1911)  |  Wish (216)

But it is precisely mathematics, and the pure science generally, from which the general educated public and independent students have been debarred, and into which they have only rarely attained more than a very meagre insight. The reason of this is twofold. In the first place, the ascendant and consecutive character of mathematical knowledge renders its results absolutely insusceptible of presentation to persons who are unacquainted with what has gone before, and so necessitates on the part of its devotees a thorough and patient exploration of the field from the very beginning, as distinguished from those sciences which may, so to speak, be begun at the end, and which are consequently cultivated with the greatest zeal. The second reason is that, partly through the exigencies of academic instruction, but mainly through the martinet traditions of antiquity and the influence of mediaeval logic-mongers, the great bulk of the elementary text-books of mathematics have unconsciously assumed a very repellant form,—something similar to what is termed in the theory of protective mimicry in biology “the terrifying form.” And it is mainly to this formidableness and touch-me-not character of exterior, concealing withal a harmless body, that the undue neglect of typical mathematical studies is to be attributed.
In Editor’s Preface to Augustus De Morgan and Thomas J. McCormack (ed.), Elementary Illustrations of the Differential and Integral Calculus (1899), v.
Science quotes on:  |  Absolute (153)  |  Academic (20)  |  Antiquity (34)  |  Ascendant (2)  |  Assume (43)  |  Attain (126)  |  Attribute (65)  |  Begin (275)  |  Beginning (312)  |  Biology (232)  |  Body (557)  |  Book (413)  |  Bulk (24)  |  Character (259)  |  Conceal (19)  |  Consecutive (2)  |  Consequent (19)  |  Cultivate (24)  |  Debar (2)  |  Devotee (7)  |  Distinguish (168)  |  Distinguished (84)  |  Educated (12)  |  Elementary (98)  |  End (603)  |  Exigency (3)  |  Exploration (161)  |  Exterior (7)  |  Field (378)  |  First (1302)  |  Form (976)  |  Formidable (8)  |  General (521)  |  Great (1610)  |  Greatest (330)  |  Harmless (9)  |  Independent (74)  |  Influence (231)  |  Insight (107)  |  Instruction (101)  |  Knowledge (1647)  |  Logic (311)  |  Meager (2)  |  Medieval (12)  |  More (2558)  |  Nature Of Mathematics (80)  |  Necessity (197)  |  Neglect (63)  |  Part (235)  |  Patient (209)  |  Person (366)  |  Precisely (93)  |  Presentation (24)  |  Protective (5)  |  Public (100)  |  Pure (299)  |  Pure Science (30)  |  Rarely (21)  |  Reason (766)  |  Render (96)  |  Repellent (4)  |  Result (700)  |  Something (718)  |  Speak (240)  |  Student (317)  |  Study (701)  |  Term (357)  |  Terrify (12)  |  Textbook (39)  |  Theory (1015)  |  Thorough (40)  |  Through (846)  |  Touch (146)  |  Tradition (76)  |  Typical (16)  |  Unacquainted (3)  |  Unconscious (24)  |  Undue (4)  |  Zeal (12)

But of this I can assure you that there is not a movement of any body of Men however small whether on Horse-back or on foot, nor an operation or March of any description nor any Service in the field that is not formed upon some mathematical principle, and in the performance of which the knowledge and practical application of the mathematicks will be found not only useful but necessary. The application of the Mathematicks to Gunnery, Fortification, Tactics, the survey and knowledge of formal Castrenantion etc. cannot be acquired without study.
Duke of Wellington to his son Douro (1826). Quoted in A Selection of the Private Correspondence of the First Duke of Wellington (1952), 44.
Science quotes on:  |  Acquired (77)  |  Application (257)  |  Back (395)  |  Body (557)  |  Field (378)  |  Form (976)  |  Fortification (6)  |  Horse (78)  |  Horseback (3)  |  Knowledge (1647)  |  March (48)  |  Movement (162)  |  Necessary (370)  |  Operation (221)  |  Performance (51)  |  Practical (225)  |  Principle (530)  |  Service (110)  |  Small (489)  |  Study (701)  |  Survey (36)  |  Tactic (9)  |  Useful (260)  |  Will (2350)

But the creative principle resides in mathematics. In a certain sense, therefore, I hold it true that pure thought can grasp reality, as the ancients dreamed.
From Herbert Spencer Lecture, at University of Oxford (10 Jun 1933), 'On the Methods of Theoretical Physics'. Printed in Philosophy of Science, (Apr 1934), 1, No. 2. Quoted and cited in epigraph, A. H. Louie, More Than Life Itself: A Synthetic Continuation in Relational Biology (2013), 81.
Science quotes on:  |  Ancient (198)  |  Creative (144)  |  Dream (222)  |  Grasp (65)  |  Principle (530)  |  Pure (299)  |  Reality (274)  |  Thought (995)

Buttercups do not think, yet they are also built of mathematics. If buttercups do not cogitate, but we do, yet are built of the same ultimate stuff, then the difference must lie in the complexity of our structures that has emerged from the process of evolution.
In Creation Revisited: The Origin of Space, Time and the Universe (1992), 119.
Science quotes on:  |  Buttercup (2)  |  Complexity (121)  |  Evolution (635)

By and large it is uniformly true in mathematics that there is a time lapse between a mathematical discovery and the moment when it is useful; and that this lapse of time can be anything from 30 to 100 years, in some cases even more.
From Address (1954) to Princeton Alumni, 'The Role of Mathematics in the Sciences and in Society', published in A.H. Taub (ed.), John von Neumann: Collected Works (1963), Vol. 6, 489. As quoted and cited in Rosemary Schmalz,Out of the Mouths of Mathematicians: A Quotation Book for Philomaths (1993), 123.
Science quotes on:  |  Discovery (837)  |  Lapse (2)  |  Large (398)  |  Moment (260)  |  More (2558)  |  Time (1911)  |  True (239)  |  Uniformly (2)  |  Useful (260)  |  Year (963)

By keenly confronting the enigmas that surround us, and by considering and analyzing the observations that I had made I ended up in the domain of mathematics.
In M.C. Escher: The Graphic Work (1978), 8.
Science quotes on:  |  Analyze (12)  |  Confront (18)  |  Consider (428)  |  Domain (72)  |  End (603)  |  Enigma (16)  |  Observation (593)  |  Surround (33)

Can science ever be immune from experiments conceived out of prejudices and stereotypes, conscious or not? (Which is not to suggest that it cannot in discrete areas identify and locate verifiable phenomena in nature.) I await the study that says lesbians have a region of the hypothalamus that resembles straight men and I would not be surprised if, at this very moment, some scientist somewhere is studying brains of deceased Asians to see if they have an enlarged ‘math region’ of the brain.
…...
Science quotes on:  |  Area (33)  |  Asian (3)  |  Await (6)  |  Brain (281)  |  Conceive (100)  |  Conscious (46)  |  Discrete (11)  |  Enlarge (37)  |  Experiment (736)  |  Identify (13)  |  Immune (3)  |  Locate (7)  |  Moment (260)  |  Nature (2017)  |  Phenomenon (334)  |  Prejudice (96)  |  Region (40)  |  Resemble (65)  |  Say (989)  |  Scientist (881)  |  See (1094)  |  Stereotype (4)  |  Straight (75)  |  Study (701)  |  Studying (70)  |  Suggest (38)  |  Surprise (91)  |  Verifiable (6)

Catastrophe Theory is a new mathematical method for describing the evolution of forms in nature. … It is particularly applicable where gradually changing forces produce sudden effects. We often call such effects catastrophes, because our intuition about the underlying continuity of the forces makes the very discontinuity of the effects so unexpected, and this has given rise to the name.
From Catastrophe Theory: Selected Papers, 1972-1977 (1977), 1. As quoted and cited in a Review by: Hector J. Sussmann, SIAM Review (Apr 1979), 21, No. 2, 269.
Science quotes on:  |  Catastrophe (35)  |  Catastrophe Theory (3)  |  Change (639)  |  Continuity (39)  |  Describe (132)  |  Discontinuity (4)  |  Effect (414)  |  Evolution (635)  |  Force (497)  |  Form (976)  |  Gradual (30)  |  Intuition (82)  |  Method (531)  |  Name (359)  |  Nature (2017)  |  New (1273)  |  Nomenclature (159)  |  Sudden (70)  |  Unexpected (55)

Cauchy is mad, and there is no way of being on good terms with him, although at present he is the only man who knows how mathematics should be treated. What he does is excellent, but very confused…
In Oeuvres (1826), Vol. 2, 259. As quoted and cited in Ernst Hairer and Gerhard Wanner Analysis by Its History (2008), 188. From the original French, “Cauchy est fou, et avec lui il n’y a pas moyen de s’entendre, bien que pour le moment il soit celui qui sait comment les mathématiques doivent être traitées. Ce qu’il fait est excellent, mais très brouillé….”
Science quotes on:  |  Being (1276)  |  Baron Augustin-Louis Cauchy (11)  |  Confused (13)  |  Excellent (29)  |  Good (906)  |  Know (1538)  |  Mad (54)  |  Man (2252)  |  Present (630)  |  Term (357)  |  Terms (184)  |  Treat (38)  |  Way (1214)

Cayley was singularly learned in the work of other men, and catholic in his range of knowledge. Yet he did not read a memoir completely through: his custom was to read only so much as would enable him to grasp the meaning of the symbols and understand its scope. The main result would then become to him a subject of investigation: he would establish it (or test it) by algebraic analysis and, not infrequently, develop it so to obtain other results. This faculty of grasping and testing rapidly the work of others, together with his great knowledge, made him an invaluable referee; his services in this capacity were used through a long series of years by a number of societies to which he was almost in the position of standing mathematical advisor.
In Proceedings of London Royal Society (1895), 58, 11-12.
Science quotes on:  |  Advisor (3)  |  Algebraic (5)  |  Analysis (244)  |  Become (821)  |  Capacity (105)  |  Catholic (18)  |  Arthur Cayley (17)  |  Completely (137)  |  Custom (44)  |  Develop (278)  |  Enable (122)  |  Establish (63)  |  Faculty (76)  |  Grasp (65)  |  Great (1610)  |  Infrequent (2)  |  Invaluable (11)  |  Investigation (250)  |  Knowledge (1647)  |  Learn (672)  |  Learned (235)  |  Long (778)  |  Main (29)  |  Mathematicians and Anecdotes (141)  |  Mean (810)  |  Meaning (244)  |  Memoir (13)  |  Number (710)  |  Obtain (164)  |  Other (2233)  |  Position (83)  |  Range (104)  |  Rapid (37)  |  Rapidly (67)  |  Read (308)  |  Referee (8)  |  Result (700)  |  Scope (44)  |  Series (153)  |  Service (110)  |  Society (350)  |  Stand (284)  |  Subject (543)  |  Symbol (100)  |  Test (221)  |  Through (846)  |  Together (392)  |  Understand (648)  |  Work (1402)  |  Year (963)

Characteristically skeptical of the idea that living things would faithfully follow mathematical formulas, [Robert Harper] seized upon factors in corn which seemed to blend in the hybrid—rather than be represented by plus or minus signs, and put several seasons into throwing doubt upon the concept of immutable hypothetical units of inheritance concocted to account for selected results.
In 'Robert Almer Harper', National Academy Biographical Memoirs (1948), 25, 233-234.
Science quotes on:  |  Account (195)  |  Blend (9)  |  Concept (242)  |  Concoct (3)  |  Corn (20)  |  Doubt (314)  |  Factor (47)  |  Follow (389)  |  Formula (102)  |  Robert Harper (2)  |  Hybrid (14)  |  Hypothesis (314)  |  Idea (881)  |  Immutable (26)  |  Inheritance (35)  |  Life (1870)  |  Living (492)  |  Minus (7)  |  Plus (43)  |  Represent (157)  |  Representation (55)  |  Result (700)  |  Season (47)  |  Seize (18)  |  Select (45)  |  Selection (130)  |  Skeptic (8)  |  Skeptical (21)  |  Thing (1914)  |  Throwing (17)

Chemical engineering is the profession in which a knowledge of mathematics, chemistry and other natural sciences gained by study, experience and practice is applied with judgment to develop economic ways of using materials and energy for the benefit of mankind.
AIChE
In Article III, 'Definition of the Profession', Constitution of the American Institute of Chemical Engineers (as amended 17 Jan 2003). The same wording is found in the 1983 Constitution, as quoted in Nicholas A. Peppas (ed.), One Hundred Years of Chemical Engineering: From Lewis M. Norton (M.I.T. 1888) to Present (2012), 334.
Science quotes on:  |  Applied (176)  |  Benefit (123)  |  Chemical (303)  |  Chemical Engineering (4)  |  Chemistry (376)  |  Develop (278)  |  Economic (84)  |  Economics (44)  |  Energy (373)  |  Engineering (188)  |  Experience (494)  |  Gain (146)  |  Judgment (140)  |  Knowledge (1647)  |  Mankind (356)  |  Material (366)  |  Natural (810)  |  Natural Science (133)  |  Other (2233)  |  Practice (212)  |  Profession (108)  |  Study (701)  |  Use (771)  |  Way (1214)

Chess combines the beauty of mathematical structure with the recreational delights of a competitive game.
In 'Preface', Mathematics, Magic, and Mystery (1956), ix.
Science quotes on:  |  Beauty (313)  |  Chess (27)  |  Combine (58)  |  Competitive (8)  |  Delight (111)  |  Game (104)  |  Recreation (23)  |  Structure (365)

Chess problems are the hymn-tunes of mathematics.
'A Mathematician's Apology', in James Roy Newman, The World of Mathematics (2000), 2028.
Science quotes on:  |  Chess (27)  |  Hymn (6)  |  Problem (731)  |  Tune (20)

Children are told that an apple fell on Isaac Newton’s head and he was led to state the law of gravity. This, of course, is pure foolishness. What Newton discovered was that any two particles in the universe attract each other with a force that is proportional to the product of their masses and inversely proportional to the square of the distance between them. This is not learned from a falling apple, but by observing quantities of data and developing a mathematical theory that can be verified by additional data. Data gathered by Galileo on falling bodies and by Johannes Kepler on motions of the planets were invaluable aids to Newton. Unfortunately, such false impressions about science are not universally outgrown like the Santa Claus myth, and some people who don’t study much science go to their graves thinking that the human race took until the mid-seventeenth century to notice that objects fall.
In How to Tell the Liars from the Statisticians (1983), 127.
Science quotes on:  |  17th Century (20)  |  Additional (6)  |  Aid (101)  |  Apple (46)  |  Attract (25)  |  Body (557)  |  Century (319)  |  Child (333)  |  Children (201)  |  Course (413)  |  Data (162)  |  Discover (571)  |  Distance (171)  |  Fall (243)  |  False (105)  |  Foolish (41)  |  Foolishness (10)  |  Force (497)  |  Galileo Galilei (134)  |  Gather (76)  |  Grave (52)  |  Gravity (140)  |  Head (87)  |  Human (1512)  |  Human Race (104)  |  Impression (118)  |  Invaluable (11)  |  Inversely Proportional (7)  |  Johannes Kepler (95)  |  Law (913)  |  Law Of Gravity (16)  |  Learn (672)  |  Learned (235)  |  Mass (160)  |  Motion (320)  |  Myth (58)  |  Sir Isaac Newton (363)  |  Notice (81)  |  Object (438)  |  Observe (179)  |  Other (2233)  |  Particle (200)  |  People (1031)  |  Planet (402)  |  Product (166)  |  Proportional (5)  |  Pure (299)  |  Quantity (136)  |  Race (278)  |  Santa Claus (2)  |  Square (73)  |  State (505)  |  Study (701)  |  Theory (1015)  |  Think (1122)  |  Thinking (425)  |  Two (936)  |  Unfortunately (40)  |  Universe (900)  |  Verify (24)

Classes and concepts may, however, also be conceived as real objects, namely classes as “pluralities of things” or as structures consisting of a plurality of things and concepts as the properties and relations of things existing independently of our definitions and constructions. It seems to me that the assumption of such objects is quite as legitimate as the assumption of physical bodies and there is quite as much reason to believe in their existence. They are in the same sense necessary to obtain a satisfactory system of mathematics as physical bodies are necessary for a satisfactory theory of our sense perceptions…
In 'Russell's Mathematical Logic', in P.A. Schilpp (ed.), The Philosophy of Bertrand Russell (1944), Vol. 1, 137.
Science quotes on:  |  Assumption (96)  |  Class (168)  |  Concept (242)  |  Construction (114)  |  Definition (238)  |  Existence (481)  |  Independently (24)  |  Legitimate (26)  |  Necessary (370)  |  Object (438)  |  Obtain (164)  |  Perception (97)  |  Physical (518)  |  Reason (766)  |  Sense (785)  |  Structure (365)  |  System (545)  |  Theory (1015)  |  Thing (1914)

Common integration is only the memory of differentiation...
Science quotes on:  |  Common (447)  |  Differentiation (28)  |  Integration (21)  |  Memory (144)

Confined to its true domain, mathematical reasoning is admirably adapted to perform the universal office of sound logic: to induce in order to deduce, in order to construct. … It contents itself to furnish, in the most favorable domain, a model of clearness, of precision, and consistency, the close contemplation of which is alone able to prepare the mind to render other conceptions also as perfect as their nature permits. Its general reaction, more negative than positive, must consist, above all, in inspiring us everywhere with an invincible aversion for vagueness, inconsistency, and obscurity, which may always be really avoided in any reasoning whatsoever, if we make sufficient effort.
In Synthèse Subjective (1856), 98. As translated in Robert Édouard Moritz, Memorabilia Mathematica; Or, The Philomath’s Quotation-Book (1914), 202-203. From the original French, “Bornée à son vrai domaine, la raison mathématique y peut admirablement remplir l’office universel de la saine logique: induire pour déduire, afin de construire. … Elle se contente de former, dans le domaine le plus favorable, un type de clarté, de précision, et de consistance, dont la contemplation familière peut seule disposer l’esprit à rendre les autres conceptions aussi parfaites que le comporte leur nature. Sa réaction générale, plus négative que positive, doit surtout consister à nous inspirer partout une invincible répugnance pour le vague, l’incohérence, et l’obscurité, que nous pouvons réellement éviter envers des pensées quelconques, si nous y faisons assez d’efforts.”
Science quotes on:  |  Adapt (70)  |  Alone (324)  |  Aversion (9)  |  Avoid (123)  |  Clearness (11)  |  Close (77)  |  Conception (160)  |  Confine (26)  |  Consist (223)  |  Consistency (31)  |  Consistent (50)  |  Construct (129)  |  Contemplation (75)  |  Content (75)  |  Deduce (27)  |  Domain (72)  |  Effort (243)  |  Everywhere (98)  |  Favorable (24)  |  Furnish (97)  |  General (521)  |  Inconsistent (9)  |  Induce (24)  |  Inspire (58)  |  Invincible (6)  |  Logic (311)  |  Mathematics And Logic (27)  |  Mind (1377)  |  Model (106)  |  More (2558)  |  Most (1728)  |  Must (1525)  |  Nature (2017)  |  Negative (66)  |  Obscurity (28)  |  Office (71)  |  Order (638)  |  Other (2233)  |  Perfect (223)  |  Perform (123)  |  Permit (61)  |  Positive (98)  |  Precision (72)  |  Prepare (44)  |  Reaction (106)  |  Reasoning (212)  |  Render (96)  |  Sound (187)  |  Sufficient (133)  |  True (239)  |  Universal (198)  |  Vagueness (15)  |  Whatsoever (41)

Coterminous with space and coeval with time is the kingdom of Mathematics; within this range her dominion is supreme; otherwise than according to her order nothing can exist; in contradiction to her laws nothing takes place. On her mysterious scroll is to be found written for those who can read it that which has been, that which is, and that which is to come.
From Presidential Address (Aug 1878) to the British Association, Dublin, published in the Report of the 48th Meeting of the British Association for the Advancement of Science (1878), 31.
Science quotes on:  |  According (236)  |  Contradiction (69)  |  Dominion (11)  |  Exist (458)  |  Kingdom (80)  |  Law (913)  |  Mysterious (83)  |  Nature Of Mathematics (80)  |  Nothing (1000)  |  Order (638)  |  Place (192)  |  Range (104)  |  Read (308)  |  Space (523)  |  Supreme (73)  |  Time (1911)  |  Written (6)

Deductivism in mathematical literature and inductivism in scientific papers are simply the postures we choose to be seen in when the curtain goes up and the public sees us. The theatrical illusion is shattered if we ask what goes on behind the scenes. In real life discovery and justification are almost always different processes.
Induction and Intuition in Scientific Thought (1969), 26.
Science quotes on:  |  Ask (420)  |  Behind (139)  |  Choice (114)  |  Choose (116)  |  Curtain (4)  |  Difference (355)  |  Different (595)  |  Discovery (837)  |  Illusion (68)  |  Justification (52)  |  Life (1870)  |  Literature (116)  |  Paper (192)  |  Posture (7)  |  Process (439)  |  Public (100)  |  Publication (102)  |  Real Life (8)  |  Scene (36)  |  Scientific (955)  |  See (1094)  |  Shatter (8)  |  Shattered (8)  |  Theatre (5)

Definition of Mathematics.—It has now become apparent that the traditional field of mathematics in the province of discrete and continuous number can only be separated from the general abstract theory of classes and relations by a wavering and indeterminate line. Of course a discussion as to the mere application of a word easily degenerates into the most fruitless logomachy. It is open to any one to use any word in any sense. But on the assumption that “mathematics” is to denote a science well marked out by its subject matter and its methods from other topics of thought, and that at least it is to include all topics habitually assigned to it, there is now no option but to employ “mathematics” in the general sense of the “science concerned with the logical deduction of consequences from the general premisses of all reasoning.”
In article 'Mathematics', Encyclopedia Britannica (1911, 11th ed.), Vol. 17, 880. In the 2006 DVD edition of the encyclopedia, the definition of mathematics is given as “The science of structure, order, and relation that has evolved from elemental practices of counting, measuring, and describing the shapes of objects.” [Premiss is a variant form of “premise”. —Webmaster]
Science quotes on:  |  Abstract (141)  |  Apparent (85)  |  Application (257)  |  Assign (15)  |  Assumption (96)  |  Become (821)  |  Class (168)  |  Concern (239)  |  Consequence (220)  |  Continuous (83)  |  Course (413)  |  Deduction (90)  |  Definition (238)  |  Degenerate (14)  |  Denote (6)  |  Discrete (11)  |  Discussion (78)  |  Employ (115)  |  Field (378)  |  Fruitless (9)  |  General (521)  |  Habitual (5)  |  Include (93)  |  Indeterminate (4)  |  Logic (311)  |  Mark (47)  |  Marked (55)  |  Matter (821)  |  Method (531)  |  Most (1728)  |  Number (710)  |  Open (277)  |  Option (10)  |  Other (2233)  |  Premise (40)  |  Province (37)  |  Reason (766)  |  Reasoning (212)  |  Relation (166)  |  Sense (785)  |  Separate (151)  |  Subject (543)  |  Subject Matter (4)  |  Theory (1015)  |  Thought (995)  |  Topic (23)  |  Traditional (16)  |  Use (771)  |  Waver (2)  |  Word (650)

Descartes is the completest type which history presents of the purely mathematical type of mind—that in which the tendencies produced by mathematical cultivation reign unbalanced and supreme.
In An Examination of Sir William Hamilton’s Philosophy (1878), 626.
Science quotes on:  |  Complete (209)  |  Cultivation (36)  |  René Descartes (83)  |  History (716)  |  Mathematicians and Anecdotes (141)  |  Mind (1377)  |  Present (630)  |  Produce (117)  |  Produced (187)  |  Purely (111)  |  Reign (24)  |  Supreme (73)  |  Tendency (110)  |  Type (171)  |  Unbalanced (2)

Difficulties [in defining mathematics with full generality, yet simplicity] are but consequences of our refusal to see that mathematics cannot be defined without acknowledging its most obvious feature: namely, that it is interesting. Nowhere is intellectual beauty so deeply felt and fastidiously appreciated.
In Personal Knowledge (1958, 2012), 200,
Science quotes on:  |  Acknowledge (33)  |  Appreciate (67)  |  Beauty (313)  |  Consequence (220)  |  Define (53)  |  Difficulty (201)  |  Fastidious (2)  |  Feature (49)  |  Generality (45)  |  Intellectual (258)  |  Interesting (153)  |  Most (1728)  |  Obvious (128)  |  Refusal (23)  |  See (1094)  |  Simplicity (175)

Dirichlet was not satisfied to study Gauss’ Disquisitiones arithmetical once or several times, but continued throughout life to keep in close touch with the wealth of deep mathematical thoughts which it contains by perusing it again and again. For this reason the book was never placed on the shelf but had an abiding place on the table at which he worked. … Dirichlet was the first one, who not only fully understood this work, but made it also accessible to others.
In Dirichlet, Werke, Bd. 2, 315. As translated in Robert Édouard Moritz, Memorabilia Mathematica; Or, The Philomath’s Quotation-book (1914), 159.
Science quotes on:  |  Abide (12)  |  Accessible (27)  |  Book (413)  |  Close (77)  |  Contain (68)  |  Continue (179)  |  Deep (241)  |  Peter Gustav Lejeune Dirichlet (3)  |  Disquisitiones Arithmeticae (2)  |  First (1302)  |  Fully (20)  |  Carl Friedrich Gauss (79)  |  Keep (104)  |  Life (1870)  |  Mathematicians and Anecdotes (141)  |  Never (1089)  |  Other (2233)  |  Peruse (2)  |  Place (192)  |  Reason (766)  |  Satisfied (23)  |  Shelf (8)  |  Study (701)  |  Table (105)  |  Thought (995)  |  Throughout (98)  |  Time (1911)  |  Touch (146)  |  Understand (648)  |  Understood (155)  |  Wealth (100)  |  Work (1402)

Distrust even Mathematics; albeit so sublime and highly perfected, we have here a machine of such delicacy it can only work in vacuo, and one grain of sand in the wheels is enough to put everything out of gear. One shudders to think to what disaster such a grain of sand may bring a Mathematical brain. Remember Pascal.
The Garden of Epicurus (1894) translated by Alfred Allinson, in The Works of Anatole France in an English Translation (1920), 187.
Science quotes on:  |  Brain (281)  |  Delicacy (8)  |  Disaster (58)  |  Distrust (11)  |  Enough (341)  |  Everything (489)  |  Grain (50)  |  Machine (271)  |  Blaise Pascal (81)  |  Perfect (223)  |  Remember (189)  |  Sand (63)  |  Sublime (50)  |  Think (1122)  |  Wheel (51)  |  Work (1402)

Do not imagine that mathematics is harsh and crabbed, and repulsive to common sense. It is merely the etherealisation of common sense.
'The Six Gateways of Knowledge', Presidential Address to the Birmingham and Midland Institute, Birmingham (3 Oct 1883). In Popular Lectures and Addresses (1891), Vol. 1, 280.
Science quotes on:  |  Common (447)  |  Common Sense (136)  |  Do (1905)  |  Harsh (9)  |  Imagine (176)  |  Merely (315)  |  Repulsive (7)  |  Sense (785)

Do not worry about your difficulties in Mathematics. I can assure you mine are still greater.
In letter (7 Jan 1943) to Barbara Wilson, a junior high school student, who had difficulties in school with mathematics. In Einstein Archives, 42-606. Quoted in Alice Calaprice, Dear Professor Einstein: Albert Einstein's Letters to and from Children (2002), 140.
Science quotes on:  |  Assure (16)  |  Difficulty (201)  |  Do (1905)  |  Greater (288)  |  Mine (78)  |  Still (614)  |  Worry (34)

Do not worry about your problems in mathematics. I assure you, my problems with mathematics are much greater than yours.
…...
Science quotes on:  |  Assure (16)  |  Do (1905)  |  Great (1610)  |  Greater (288)  |  Problem (731)  |  Worry (34)

Don’t talk to me of your Archimedes’ lever. He was an absent-minded person with a mathematical imagination. Mathematics commands all my respect, but I have no use for engines. Give me the right word and the right accent and I will move the world.
In 'Preface', A Personal Record (1912), 2.
Science quotes on:  |  Absent-Minded (4)  |  Accent (5)  |  Archimedes Lever (3)  |  Command (60)  |  Engine (99)  |  Imagination (349)  |  Lever (13)  |  Mind (1377)  |  Move (223)  |  Person (366)  |  Respect (212)  |  Right (473)  |  Talk (108)  |  Use (771)  |  Will (2350)  |  Word (650)  |  World (1850)

Doubtless the reasoning faculty, the mind, is the leading and characteristic attribute of the human race. By the exercise of this, man arrives at the properties of the natural bodies. This is science, properly and emphatically so called. It is the science of pure mathematics; and in the high branches of this science lies the truly sublime of human acquisition. If any attainment deserves that epithet, it is the knowledge, which, from the mensuration of the minutest dust of the balance, proceeds on the rising scale of material bodies, everywhere weighing, everywhere measuring, everywhere detecting and explaining the laws of force and motion, penetrating into the secret principles which hold the universe of God together, and balancing worlds against worlds, and system against system. When we seek to accompany those who pursue studies at once so high, so vast, and so exact; when we arrive at the discoveries of Newton, which pour in day on the works of God, as if a second fiat had gone forth from his own mouth; when, further, we attempt to follow those who set out where Newton paused, making his goal their starting-place, and, proceeding with demonstration upon demonstration, and discovery upon discovery, bring new worlds and new systems of worlds within the limits of the known universe, failing to learn all only because all is infinite; however we may say of man, in admiration of his physical structure, that “in form and moving he is express and admirable,” it is here, and here without irreverence, we may exclaim, “In apprehension how like a god!” The study of the pure mathematics will of course not be extensively pursued in an institution, which, like this [Boston Mechanics’ Institute], has a direct practical tendency and aim. But it is still to be remembered, that pure mathematics lie at the foundation of mechanical philosophy, and that it is ignorance only which can speak or think of that sublime science as useless research or barren speculation.
In Works (1872), Vol. 1, 180.
Science quotes on:  |  Accompany (22)  |  Acquisition (46)  |  Admirable (20)  |  Admiration (61)  |  Against (332)  |  Aim (175)  |  Apprehension (26)  |  Arrive (40)  |  Attainment (48)  |  Attempt (266)  |  Attribute (65)  |  Balance (82)  |  Barren (33)  |  Body (557)  |  Boston (7)  |  Branch (155)  |  Bring (95)  |  Call (781)  |  Characteristic (154)  |  Course (413)  |  Demonstration (120)  |  Deserve (65)  |  Detect (45)  |  Direct (228)  |  Discovery (837)  |  Doubtless (8)  |  Dust (68)  |  Emphatically (8)  |  Epithet (3)  |  Estimates of Mathematics (30)  |  Everywhere (98)  |  Exact (75)  |  Exclaim (15)  |  Exercise (113)  |  Explain (334)  |  Express (192)  |  Extensive (34)  |  Faculty (76)  |  Fail (191)  |  Far (158)  |  Fiat (7)  |  Follow (389)  |  Force (497)  |  Form (976)  |  Forth (14)  |  Foundation (177)  |  Goal (155)  |  God (776)  |  High (370)  |  Hold (96)  |  Human (1512)  |  Human Race (104)  |  Ignorance (254)  |  Infinite (243)  |  Institution (73)  |  Irreverence (3)  |  Know (1538)  |  Knowledge (1647)  |  Known (453)  |  Law (913)  |  Lead (391)  |  Learn (672)  |  Lie (370)  |  Limit (294)  |  Making (300)  |  Man (2252)  |  Material (366)  |  Measure (241)  |  Mechanic (120)  |  Mechanical (145)  |  Mechanics (137)  |  Mensuration (2)  |  Mind (1377)  |  Minute (129)  |  Motion (320)  |  Mouth (54)  |  Move (223)  |  Natural (810)  |  New (1273)  |  New Worlds (5)  |  Sir Isaac Newton (363)  |  Of Course (22)  |  Pause (6)  |  Penetrate (68)  |  Philosophy (409)  |  Physical (518)  |  Pour (9)  |  Practical (225)  |  Principle (530)  |  Proceed (134)  |  Proceeding (38)  |  Properly (21)  |  Property (177)  |  Pure (299)  |  Pure Mathematics (72)  |  Pursue (63)  |  Race (278)  |  Reason (766)  |  Reasoning (212)  |  Remember (189)  |  Research (753)  |  Rise (169)  |  Rising (44)  |  Say (989)  |  Scale (122)  |  Second (66)  |  Secret (216)  |  Seek (218)  |  Set (400)  |  Speak (240)  |  Speculation (137)  |  Starting Point (16)  |  Still (614)  |  Structure (365)  |  Study (701)  |  Sublime (50)  |  System (545)  |  Tendency (110)  |  Think (1122)  |  Together (392)  |  Truly (118)  |  Universe (900)  |  Useless (38)  |  Vast (188)  |  Weigh (51)  |  Will (2350)  |  Work (1402)  |  World (1850)

During the last two centuries and a half, physical knowledge has been gradually made to rest upon a basis which it had not before. It has become mathematical. The question now is, not whether this or that hypothesis is better or worse to the pure thought, but whether it accords with observed phenomena in those consequences which can be shown necessarily to follow from it, if it be true
In Augustus De Morgan and Sophia Elizabeth De Morgan (ed.), A Budget of Paradoxes (1872), 2.
Science quotes on:  |  Accord (36)  |  Basis (180)  |  Become (821)  |  Better (493)  |  Century (319)  |  Consequence (220)  |  Follow (389)  |  Gradually (102)  |  Hypothesis (314)  |  Knowledge (1647)  |  Last (425)  |  Necessarily (137)  |  Necessity (197)  |  Observation (593)  |  Observed (149)  |  Phenomenon (334)  |  Physical (518)  |  Physical Science (104)  |  Pure (299)  |  Question (649)  |  Rest (287)  |  Thought (995)  |  Truth (1109)  |  Two (936)  |  Worse (25)

During the three years which I spent at Cambridge my time was wasted, as far as the academical studies were concerned…. I attempted mathematics, … but I got on very slowly. The work was repugnant to me, chiefly from my not being able to see any meaning in the early steps in algebra. This impatience was very foolish…
In Charles Darwin and Francis Darwin (ed.), 'Autobiography', The Life and Letters of Charles Darwin (1887, 1896), Vol. 1, 40.
Science quotes on:  |  Academic (20)  |  Algebra (117)  |  Attempt (266)  |  Being (1276)  |  Cambridge University (2)  |  Chiefly (47)  |  Concern (239)  |  Early (196)  |  Foolish (41)  |  Impatience (13)  |  Meaning (244)  |  Repugnant (8)  |  See (1094)  |  Spent (85)  |  Step (234)  |  Study (701)  |  Time (1911)  |  Waste (109)  |  Work (1402)  |  Year (963)

Each generation has its few great mathematicians, and mathematics would not even notice the absence of the others. They are useful as teachers, and their research harms no one, but it is of no importance at all. A mathematician is great or he is nothing.
Reflections: Mathematics and Creativity', New Yorker (1972), 47, No. 53, 39-45. In Douglas M. Campbell, John C. Higgins (eds.), Mathematics: People, Problems, Results (1984), Vol. 2, 3.
Science quotes on:  |  Generation (256)  |  Great (1610)  |  Harm (43)  |  Importance (299)  |  Mathematician (407)  |  Nothing (1000)  |  Notice (81)  |  Other (2233)  |  Research (753)  |  Teacher (154)  |  Useful (260)

Each thing in the world has names or unnamed relations to everything else. Relations are infinite in number and kind. To be is to be related. It is evident that the understanding of relations is a major concern of all men and women. Are relations a concern of mathematics? They are so much its concern that mathematics is sometimes defined to be the science of relations.
In Mole Philosophy and Other Essays (1927), 94-95.
Science quotes on:  |  Concern (239)  |  Define (53)  |  Everything (489)  |  Evident (92)  |  Infinite (243)  |  It Is Evident (6)  |  Kind (564)  |  Major (88)  |  Name (359)  |  Number (710)  |  Relation (166)  |  Sometimes (46)  |  Thing (1914)  |  Understand (648)  |  Understanding (527)  |  Woman (160)  |  World (1850)

Education is like a diamond with many facets: It includes the basic mastery of numbers and letters that give us access to the treasury of human knowledge, accumulated and refined through the ages; it includes technical and vocational training as well as instruction in science, higher mathematics, and humane letters.
In Proclamation 5463, for Education Day (19 Apr 1986). Collected in Public Papers of the Presidents of the United States: Ronald Reagan, 1986 (1988), 490.
Science quotes on:  |  Access (21)  |  Accumulation (51)  |  Age (509)  |  Basic (144)  |  Diamond (21)  |  Education (423)  |  Facet (9)  |  Human (1512)  |  Humane (19)  |  Humanities (21)  |  Include (93)  |  Instruction (101)  |  Knowledge (1647)  |  Letter (117)  |  Mastery (36)  |  Number (710)  |  Refinement (19)  |  Technical (53)  |  Through (846)  |  Training (92)  |  Treasury (3)

Einstein, twenty-six years old, only three years away from crude privation, still a patent examiner, published in the Annalen der Physik in 1905 five papers on entirely different subjects. Three of them were among the greatest in the history of physics. One, very simple, gave the quantum explanation of the photoelectric effect—it was this work for which, sixteen years later, he was awarded the Nobel prize. Another dealt with the phenomenon of Brownian motion, the apparently erratic movement of tiny particles suspended in a liquid: Einstein showed that these movements satisfied a clear statistical law. This was like a conjuring trick, easy when explained: before it, decent scientists could still doubt the concrete existence of atoms and molecules: this paper was as near to a direct proof of their concreteness as a theoretician could give. The third paper was the special theory of relativity, which quietly amalgamated space, time, and matter into one fundamental unity.
This last paper contains no references and quotes no authority. All of them are written in a style unlike any other theoretical physicist’s. They contain very little mathematics. There is a good deal of verbal commentary. The conclusions, the bizarre conclusions, emerge as though with the greatest of ease: the reasoning is unbreakable. It looks as though he had reached the conclusions by pure thought, unaided, without listening to the opinions of others. To a surprisingly large extent, that is precisely what he had done.
In Variety of Men (1966), 100-101. First published in Commentary magazine.
Science quotes on:  |  Atom (381)  |  Authority (99)  |  Award (13)  |  Bizarre (6)  |  Brownian Motion (2)  |  Commentary (3)  |  Conclusion (266)  |  Concrete (55)  |  Concreteness (5)  |  Conjuring (3)  |  Crude (32)  |  Deal (192)  |  Decent (12)  |  Difference (355)  |  Different (595)  |  Direct (228)  |  Doubt (314)  |  Ease (40)  |  Easy (213)  |  Effect (414)  |  Einstein (101)  |  Albert Einstein (624)  |  Emergence (35)  |  Erratic (4)  |  Examiner (5)  |  Existence (481)  |  Explain (334)  |  Explanation (246)  |  Extent (142)  |  Fundamental (264)  |  Good (906)  |  Greatest (330)  |  History (716)  |  History Of Physics (3)  |  Large (398)  |  Last (425)  |  Law (913)  |  Liquid (50)  |  Listening (26)  |  Little (717)  |  Look (584)  |  Matter (821)  |  Molecule (185)  |  Motion (320)  |  Movement (162)  |  Nobel Prize (42)  |  Old (499)  |  Opinion (291)  |  Other (2233)  |  Paper (192)  |  Particle (200)  |  Patent (34)  |  Phenomenon (334)  |  Photoelectric Effect (2)  |  Physic (515)  |  Physicist (270)  |  Physics (564)  |  Precisely (93)  |  Privation (5)  |  Proof (304)  |  Publication (102)  |  Pure (299)  |  Quantum (118)  |  Quote (46)  |  Reach (286)  |  Reasoning (212)  |  Reference (33)  |  Relativity (91)  |  Scientist (881)  |  Show (353)  |  Simple (426)  |  Space (523)  |  Special (188)  |  Statistics (170)  |  Still (614)  |  Subject (543)  |  Suspension (7)  |  Theoretical Physicist (21)  |  Theorist (44)  |  Thought (995)  |  Time (1911)  |  Tiny (74)  |  Trick (36)  |  Unbreakable (3)  |  Unity (81)  |  Work (1402)  |  Year (963)

ENGINEER, in the military art, an able expert man, who, by a perfect knowledge in mathematics, delineates upon paper, or marks upon the ground, all sorts of forts, and other works proper for offence and defence. He should understand the art of fortification, so as to be able, not only to discover the defects of a place, but to find a remedy proper for them; as also how to make an attack upon, as well as to defend, the place. Engineers are extremely necessary for these purposes: wherefore it is requisite that, besides being ingenious, they should be brave in proportion. When at a siege the engineers have narrowly surveyed the place, they are to make their report to the general, by acquainting him which part they judge the weakest, and where approaches may be made with most success. Their business is also to delineate the lines of circumvallation and contravallation, taking all the advantages of the ground; to mark out the trenches, places of arms, batteries, and lodgments, taking care that none of their works be flanked or discovered from the place. After making a faithful report to the general of what is a-doing, the engineers are to demand a sufficient number of workmen and utensils, and whatever else is necessary.
In Encyclopaedia Britannica or a Dictionary of Arts and Sciences (1771), Vol. 2, 497.
Science quotes on:  |  Advantage (144)  |  Arm (82)  |  Arms (37)  |  Art (680)  |  Attack (86)  |  Being (1276)  |  Brave (16)  |  Business (156)  |  Care (203)  |  Defect (31)  |  Defence (16)  |  Delineate (2)  |  Demand (131)  |  Discover (571)  |  Doing (277)  |  Engineer (136)  |  Expert (67)  |  Find (1014)  |  Fort (2)  |  Fortification (6)  |  General (521)  |  Ground (222)  |  Ingenious (55)  |  Judge (114)  |  Knowledge (1647)  |  Making (300)  |  Man (2252)  |  Military (45)  |  Most (1728)  |  Necessary (370)  |  Number (710)  |  Offence (4)  |  Other (2233)  |  Paper (192)  |  Perfect (223)  |  Proper (150)  |  Proportion (140)  |  Purpose (336)  |  Remedy (63)  |  Success (327)  |  Sufficient (133)  |  Survey (36)  |  Trench (6)  |  Understand (648)  |  Utensil (3)  |  Whatever (234)  |  Work (1402)  |  Workman (13)

Engineering is quite different from science. Scientists try to understand nature. Engineers try to make things that do not exist in nature. Engineers stress invention. To embody an invention the engineer must put his idea in concrete terms, and design something that people can use. That something can be a device, a gadget, a material, a method, a computing program, an innovative experiment, a new solution to a problem, or an improvement on what is existing. Since a design has to be concrete, it must have its geometry, dimensions, and characteristic numbers. Almost all engineers working on new designs find that they do not have all the needed information. Most often, they are limited by insufficient scientific knowledge. Thus they study mathematics, physics, chemistry, biology and mechanics. Often they have to add to the sciences relevant to their profession. Thus engineering sciences are born.
Y.C. Fung and P. Tong, Classical and Computational Solid Mechanics (2001), 1.
Science quotes on:  |  Biology (232)  |  Characteristic (154)  |  Chemistry (376)  |  Concrete (55)  |  Design (203)  |  Device (71)  |  Different (595)  |  Dimension (64)  |  Do (1905)  |  Engineer (136)  |  Engineering (188)  |  Exist (458)  |  Experiment (736)  |  Find (1014)  |  Geometry (271)  |  Idea (881)  |  Improvement (117)  |  Information (173)  |  Invention (400)  |  Knowledge (1647)  |  Limit (294)  |  Limited (102)  |  Material (366)  |  Mechanic (120)  |  Mechanics (137)  |  Method (531)  |  Most (1728)  |  Must (1525)  |  Nature (2017)  |  New (1273)  |  Number (710)  |  People (1031)  |  Physic (515)  |  Physics (564)  |  Problem (731)  |  Profession (108)  |  Science And Engineering (16)  |  Scientific (955)  |  Scientist (881)  |  Solution (282)  |  Something (718)  |  Stress (22)  |  Study (701)  |  Term (357)  |  Terms (184)  |  Thing (1914)  |  Try (296)  |  Understand (648)  |  Use (771)

Engineering is the application of scientific and mathematical principles to practical ends such as the design, manufacture, and operation of efficient and economical structures, machines, processes, and systems.
In Bernice Zeldin Schacter, Issues and Dilemmas of Biotechnology: A Reference Guide (1999), 1, citing the American Heritage Dictionary, 2nd College Edition.
Science quotes on:  |  Application (257)  |  Design (203)  |  Economical (11)  |  Efficient (34)  |  End (603)  |  Engineering (188)  |  Machine (271)  |  Manufacture (30)  |  Operation (221)  |  Practical (225)  |  Principle (530)  |  Process (439)  |  Scientific (955)  |  Structure (365)  |  System (545)

Engineering is the profession in which a knowledge of the mathematical and natural sciences gained by study, experience, and practice is applied with judgment to develop ways to utilize, economically, the materials and forces of nature for the benefit of mankind.
ABET
In EAC Criteria for 1999-2000 as cited in Charles R. Lord, Guide to Information Sources in Engineering (2000), 5. Found in many sources, and earlier, for example, Otis E. Lancaster, American Society for Engineering Education, Engineers' Council for Professional Development, Achieve Learning Objectives (1962), 8.
Science quotes on:  |  Applied (176)  |  Benefit (123)  |  Develop (278)  |  Economical (11)  |  Engineering (188)  |  Experience (494)  |  Force (497)  |  Force Of Nature (9)  |  Gain (146)  |  Judgment (140)  |  Knowledge (1647)  |  Mankind (356)  |  Material (366)  |  Natural (810)  |  Natural Science (133)  |  Nature (2017)  |  Practice (212)  |  Profession (108)  |  Study (701)  |  Utilize (10)  |  Way (1214)

Engineering…is both an art and a science, and as a science it consists for the most part of mathematics applied to physics and mechanics. It is of necessity, therefore, a measuring science, and a congress of engineers ought, in the nature of things, to be interested in anything relating to progress in metrology.
From Address to the International Engineering Congress of the Columbia Exposition, Chicago, 1893. Published in Transactions of the American Society of Civil Engineers (Oct 1893), 120. Reprinted in 'Fundamental Units of Measure', Smithsonian Report for 1893 (1894), 135.
Science quotes on:  |  Apply (170)  |  Art (680)  |  Congress (20)  |  Consist (223)  |  Engineer (136)  |  Engineering (188)  |  Interest (416)  |  Measure (241)  |  Mechanics (137)  |  Nature Of Things (30)  |  Necessity (197)  |  Physics (564)  |  Progress (492)  |  Relate (26)  |  Science (39)

Engineers apply the theories and principles of science and mathematics to research and develop economical solutions to practical technical problems. Their work is the link between scientific discoveries and commercial applications. Engineers design products, the machinery to build those products, the factories in which those products are made, and the systems that ensure the quality of the product and efficiency of the workforce and manufacturing process. They design, plan, and supervise the construction of buildings, highways, and transit systems. They develop and implement improved ways to extract, process, and use raw materials, such as petroleum and natural gas. They develop new materials that both improve the performance of products, and make implementing advances in technology possible. They harness the power of the sun, the earth, atoms, and electricity for use in supplying the Nation’s power needs, and create millions of products using power. Their knowledge is applied to improving many things, including the quality of health care, the safety of food products, and the efficient operation of financial systems.
Bureau of Labor Statistics, Occupational Outlook Handbook (2000) as quoted in Charles R. Lord. Guide to Information Sources in Engineering (2000), 5. This definition has been revised and expanded over time in different issues of the Handbook.
Science quotes on:  |  Advance (298)  |  Application (257)  |  Applied (176)  |  Apply (170)  |  Atom (381)  |  Both (496)  |  Build (211)  |  Building (158)  |  Care (203)  |  Commercial (28)  |  Construction (114)  |  Create (245)  |  Design (203)  |  Develop (278)  |  Discovery (837)  |  Earth (1076)  |  Economical (11)  |  Efficiency (46)  |  Efficient (34)  |  Electricity (168)  |  Engineer (136)  |  Ensure (27)  |  Extract (40)  |  Factory (20)  |  Finance (4)  |  Food (213)  |  Gas (89)  |  Harness (25)  |  Health (210)  |  Health Care (10)  |  Highway (15)  |  Implement (13)  |  Improvement (117)  |  Knowledge (1647)  |  Machinery (59)  |  Manufacturing (29)  |  Material (366)  |  Million (124)  |  Nation (208)  |  Natural (810)  |  Natural Gas (2)  |  Need (320)  |  New (1273)  |  Operation (221)  |  Performance (51)  |  Petroleum (8)  |  Plan (122)  |  Possible (560)  |  Power (771)  |  Practical (225)  |  Principle (530)  |  Problem (731)  |  Process (439)  |  Product (166)  |  Quality (139)  |  Raw (28)  |  Research (753)  |  Safety (58)  |  Scientific (955)  |  Solution (282)  |  Solution. (53)  |  Sun (407)  |  Supervise (2)  |  System (545)  |  Technical (53)  |  Technology (281)  |  Theory (1015)  |  Thing (1914)  |  Transit (2)  |  Use (771)  |  Using (6)  |  Way (1214)  |  Work (1402)

Eratosthenes of Cyrene, employing mathematical theories and geometrical methods, discovered from the course of the sun, the shadows cast by an equinoctial gnomon, and the inclination of the heaven that the circumference of the earth is two hundred and fifty-two thousand stadia, that is, thirty-one million five hundred thousand paces.
Vitruvius
In De Architectura, Book 1, Chap 6, Sec. 9. As translated in Morris Hicky Morgan (trans.), Vitruvius: The Ten Books on Architecture (1914), 27-28.
Science quotes on:  |  Cast (69)  |  Circumference (23)  |  Course (413)  |  Discover (571)  |  Earth (1076)  |  Eratosthenes (6)  |  Geometry (271)  |  Heaven (266)  |  Hundred (240)  |  Inclination (36)  |  Method (531)  |  Pace (18)  |  Shadow (73)  |  Sun (407)  |  Thousand (340)  |  Two (936)

Essentially all civilizations that rose to the level of possessing an urban culture had need for two forms of science-related technology, namely, mathematics for land measurements and commerce and astronomy for time-keeping in agriculture and aspects of religious rituals.
From The Science Matrix: The Journey, Travails, Triumphs (1992, 1998), Preface, x.
Science quotes on:  |  Agriculture (78)  |  Aspect (129)  |  Astronomy (251)  |  Civilization (220)  |  Commerce (23)  |  Culture (157)  |  Form (976)  |  Land (131)  |  Measurement (178)  |  Need (320)  |  Religion (369)  |  Religious (134)  |  Ritual (9)  |  Rose (36)  |  Science And Religion (337)  |  Technology (281)  |  Time (1911)  |  Timekeeping (2)  |  Two (936)  |  Urban (12)

Euclid and Archimedes are allowed to be knowing, and to have demonstrated what they say: and yet whosoever shall read over their writings without perceiving the connection of their proofs, and seeing what they show, though he may understand all their words, yet he is not the more knowing. He may believe, indeed, but does not know what they say, and so is not advanced one jot in mathematical knowledge by all his reading of those approved mathematicians.
In Conduct of the Understanding, sect. 24.
Science quotes on:  |  Advance (298)  |  Allow (51)  |  Approve (6)  |  Archimedes (63)  |  Belief (615)  |  Connection (171)  |  Demonstrate (79)  |  Euclid (60)  |  Indeed (323)  |  Jot (3)  |  Know (1538)  |  Knowing (137)  |  Knowledge (1647)  |  Mathematician (407)  |  More (2558)  |  Perceive (46)  |  Proof (304)  |  Read (308)  |  Reading (136)  |  Say (989)  |  See (1094)  |  Seeing (143)  |  Show (353)  |  Study And Research In Mathematics (61)  |  Understand (648)  |  Word (650)  |  Writing (192)

Euclid avoids it [the treatment of the infinite]; in modern mathematics it is systematically introduced, for only then is generality obtained.
'Geometry', Encyclopedia Britannica, 9th edition. In George Edward Martin, The Foundations of Geometry and the Non-Euclidean Plane (1982), 130. This is part of a longer quote, which begins “In Euclid each proposition…”, on the Arthur Cayley Quotes page of this website.
Science quotes on:  |  Avoid (123)  |  Euclid (60)  |  Generality (45)  |  Infinite (243)  |  Introduce (63)  |  Modern (402)  |  Modern Mathematics (50)  |  Obtain (164)  |  Treatment (135)

Euclidean mathematics assumes the completeness and invariability of mathematical forms; these forms it describes with appropriate accuracy and enumerates their inherent and related properties with perfect clearness, order, and completeness, that is, Euclidean mathematics operates on forms after the manner that anatomy operates on the dead body and its members. On the other hand, the mathematics of variable magnitudes—function theory or analysis—considers mathematical forms in their genesis. By writing the equation of the parabola, we express its law of generation, the law according to which the variable point moves. The path, produced before the eyes of the student by a point moving in accordance to this law, is the parabola.
If, then, Euclidean mathematics treats space and number forms after the manner in which anatomy treats the dead body, modern mathematics deals, as it were, with the living body, with growing and changing forms, and thus furnishes an insight, not only into nature as she is and appears, but also into nature as she generates and creates,—reveals her transition steps and in so doing creates a mind for and understanding of the laws of becoming. Thus modern mathematics bears the same relation to Euclidean mathematics that physiology or biology … bears to anatomy.
In Die Mathematik die Fackelträgerin einer neuen Zeit (1889), 38. As translated in Robert Édouard Moritz, Memorabilia Mathematica; Or, The Philomath’s Quotation-book (1914), 112-113.
Science quotes on:  |  Accord (36)  |  Accordance (10)  |  According (236)  |  Accuracy (81)  |  Analysis (244)  |  Anatomy (75)  |  Appear (122)  |  Appropriate (61)  |  Bear (162)  |  Become (821)  |  Becoming (96)  |  Biology (232)  |  Body (557)  |  Change (639)  |  Clearness (11)  |  Completeness (19)  |  Consider (428)  |  Create (245)  |  Dead (65)  |  Deal (192)  |  Describe (132)  |  Doing (277)  |  Enumerate (3)  |  Equation (138)  |  Euclid (60)  |  Express (192)  |  Eye (440)  |  Form (976)  |  Function (235)  |  Furnish (97)  |  Generate (16)  |  Generation (256)  |   Genesis (26)  |  Grow (247)  |  Growing (99)  |  Inherent (43)  |  Insight (107)  |  Invariability (6)  |  Law (913)  |  Living (492)  |  Living Body (3)  |  Magnitude (88)  |  Manner (62)  |  Member (42)  |  Mind (1377)  |  Modern (402)  |  Modern Mathematics (50)  |  Move (223)  |  Nature (2017)  |  Number (710)  |  On The Other Hand (40)  |  Operate (19)  |  Order (638)  |  Other (2233)  |  Parabola (2)  |  Path (159)  |  Perfect (223)  |  Physiology (101)  |  Point (584)  |  Produce (117)  |  Produced (187)  |  Property (177)  |  Relate (26)  |  Relation (166)  |  Reveal (152)  |  Same (166)  |  Space (523)  |  Step (234)  |  Student (317)  |  Theory (1015)  |  Transition (28)  |  Treat (38)  |  Understand (648)  |  Understanding (527)  |  Variable (37)  |  Write (250)  |  Writing (192)

Even if there is only one possible unified theory, it is just a set of rules and equations. What is it that breathes fire into the equations and makes a universe for them to describe? The usual approach of science of constructing a mathematical model cannot answer the questions of why there should be a universe for the model to describe. Why does the universe go to all the bother of existing?
A Brief History of Time (1998), 190.
Science quotes on:  |  Answer (389)  |  Approach (112)  |  Breathe (49)  |  Describe (132)  |  Description (89)  |  Equation (138)  |  Existence (481)  |  Fire (203)  |  Model (106)  |  Possibility (172)  |  Possible (560)  |  Question (649)  |  Rule (307)  |  Set (400)  |  Theory (1015)  |  Unified Theory (7)  |  Universe (900)  |  Why (491)

Even now there is a very wavering grasp of the true position of mathematics as an element in the history of thought. I will not go so far as to say that to construct a history of thought without profound study of the mathematical ideas of successive epochs is like omitting Hamlet from the play which is named after him That would be claiming too much. But it is certainly analogous to cutting out the part of Ophelia. This simile is singularly exact. For Ophelia is quite essential to the play, she is very charming—and a little mad. Let us grant that the pursuit of mathematics is a divine madness of the human spirit, a refuge from the goading urgency of contingent happenings.
From Lecture to the Mathematical Society, Brown University, 'Mathematics as an Element in the History of Thought', collected as Chap. 2 in Science and the Modern World: Lowell Lectures, 1925 (1925), 31.
Science quotes on:  |  Charming (4)  |  Hamlet (10)  |  History (716)  |  Mad (54)  |  Ophelia (2)  |  Simile (8)  |  Thought (995)

Every new body of discovery is mathematical in form, because there is no other guidance we can have.
(1931). As quoted, without citation, in Eric Temple Bell, 'They Say, What They Say, Let Them Say', Men of Mathematics (1937, 2014), Vol. 2, xvii. Webmaster has searched, but not yet found a primary source. Can you help?
Science quotes on:  |  Body (557)  |  Discovery (837)  |  Form (976)  |  Guidance (30)  |  New (1273)  |  Other (2233)

Every discipline must be honored for reason other than its utility, otherwise it yields no enthusiasm for industry.
For both reasons, I consider mathematics the chief subject for the common school. No more highly honored exercise for the mind can be found; the buoyancy [Spannkraft] which it produces is even greater than that produced by the ancient languages, while its utility is unquestioned.
In 'Mathematischer Lehrplan für Realschulen' Werke [Kehrbach] (1890), Bd. 5, 167. (Mathematics Curriculum for Secondary Schools). As quoted, cited and translated in Robert Édouard Moritz, Memorabilia Mathematica; Or, The Philomath’s Quotation-Book (1914), 61.
Science quotes on:  |  Ancient (198)  |  Both (496)  |  Buoyancy (7)  |  Chief (99)  |  Common (447)  |  Consider (428)  |  Discipline (85)  |  Enthusiasm (59)  |  Exercise (113)  |  Greater (288)  |  Honor (57)  |  Honored (3)  |  Industry (159)  |  Language (308)  |  Mind (1377)  |  More (2558)  |  Must (1525)  |  Other (2233)  |  Produce (117)  |  Produced (187)  |  Reason (766)  |  School (227)  |  Subject (543)  |  Unquestioned (7)  |  Utility (52)  |  Value Of Mathematics (60)  |  Yield (86)

Every human activity, good or bad, except mathematics, must come to an end.
Quoted as a favorite saying of Paul Erdös, by Béla Bollobás, 'The Life and Work of Paul Erdos', in Shiing-Shen Chern and Friedrich Hirzebruch (eds.) Wolf Prize in Mathematics (2000), Vol. 1, 292.
Science quotes on:  |  Activity (218)  |  Bad (185)  |  End (603)  |  Good (906)  |  Human (1512)  |  Must (1525)

Every mathematical book that is worth reading must be read “backwards and forwards”, if I may use the expression. I would modify Lagrange’s advice a little and say, “Go on, but often return to strengthen your faith.” When you come on a hard or dreary passage, pass it over; and come back to it after you have seen its importance or found the need for it further on.
In Algebra, Part 2 (1889), Preface, viii.
Science quotes on:  |  Advice (57)  |  Back (395)  |  Backwards (18)  |  Book (413)  |  Dreary (6)  |  Expression (181)  |  Faith (209)  |  Far (158)  |  Find (1014)  |  Forward (104)  |  Hard (246)  |  Importance (299)  |  Count Joseph-Louis de Lagrange (26)  |  Little (717)  |  Modify (15)  |  Must (1525)  |  Need (320)  |  Often (109)  |  Pass (241)  |  Passage (52)  |  Read (308)  |  Reading (136)  |  Return (133)  |  Say (989)  |  See (1094)  |  Strengthen (25)  |  Study And Research In Mathematics (61)  |  Use (771)  |  Worth (172)

Every mathematical discipline goes through three periods of development: the naive, the formal, and the critical.
Quoted in R Remmert, Theory of complex functions (New York, 1989).
Science quotes on:  |  Critical (73)  |  Development (441)  |  Discipline (85)  |  Formal (37)  |  Naive (13)  |  Period (200)  |  Through (846)

Every new theory as it arises believes in the flush of youth that it has the long sought goal; it sees no limits to its applicability, and believes that at last it is the fortunate theory to achieve the 'right' answer. This was true of electron theory—perhaps some readers will remember a book called The Electrical Theory of the Universe by de Tunzelman. It is true of general relativity theory with its belief that we can formulate a mathematical scheme that will extrapolate to all past and future time and the unfathomed depths of space. It has been true of wave mechanics, with its first enthusiastic claim a brief ten years ago that no problem had successfully resisted its attack provided the attack was properly made, and now the disillusionment of age when confronted by the problems of the proton and the neutron. When will we learn that logic, mathematics, physical theory, are all only inventions for formulating in compact and manageable form what we already know, like all inventions do not achieve complete success in accomplishing what they were designed to do, much less complete success in fields beyond the scope of the original design, and that our only justification for hoping to penetrate at all into the unknown with these inventions is our past experience that sometimes we have been fortunate enough to be able to push on a short distance by acquired momentum.
The Nature of Physical Theory (1936), 136.
Science quotes on:  |  Acquired (77)  |  Age (509)  |  Already (226)  |  Answer (389)  |  Arise (162)  |  Attack (86)  |  Belief (615)  |  Beyond (316)  |  Book (413)  |  Brief (37)  |  Call (781)  |  Claim (154)  |  Compact (13)  |  Complete (209)  |  Depth (97)  |  Design (203)  |  Disillusionment (2)  |  Distance (171)  |  Do (1905)  |  Electrical (57)  |  Electron (96)  |  Enough (341)  |  Experience (494)  |  Field (378)  |  First (1302)  |  Form (976)  |  Fortunate (31)  |  Future (467)  |  General (521)  |  General Relativity (10)  |  Goal (155)  |  Invention (400)  |  Justification (52)  |  Know (1538)  |  Last (425)  |  Learn (672)  |  Limit (294)  |  Logic (311)  |  Long (778)  |  Mechanic (120)  |  Mechanics (137)  |  Momentum (10)  |  Neutron (23)  |  New (1273)  |  Past (355)  |  Penetrate (68)  |  Physical (518)  |  Problem (731)  |  Proton (23)  |  Push (66)  |  Quantum Theory (67)  |  Relativity (91)  |  Remember (189)  |  Right (473)  |  Scheme (62)  |  Scope (44)  |  See (1094)  |  Short (200)  |  Space (523)  |  Success (327)  |  Theory (1015)  |  Time (1911)  |  Universe (900)  |  Unknown (195)  |  Wave (112)  |  Will (2350)  |  Year (963)  |  Youth (109)

Everybody firmly believes in it [Nomal Law of Errors] because the mathematicians imagine it is a fact of observation, and observers that it is a theory of mathematics.
…...
Science quotes on:  |  Belief (615)  |  Error (339)  |  Everybody (72)  |  Fact (1257)  |  Firmly (6)  |  Imagine (176)  |  Law (913)  |  Mathematician (407)  |  Observation (593)  |  Observer (48)  |  Theory (1015)

Everybody is to some small extent a philosopher of mathematics. Let him only exclaim on occasion: “But figures can’t lie!” and he joins the ranks of Plato and of Lakatos.
Co-author with Reuben Hersh, in The Mathematical Experience (1981), xi.
Science quotes on:  |  Everybody (72)  |  Exclaim (15)  |  Figure (162)  |  Join (32)  |  Lie (370)  |  Philosopher (269)  |  Plato (80)  |  Rank (69)

Everybody praises the incomparable power of the mathematical method, but so is everybody aware of its incomparable unpopularity.
In Jahresbericht der Deutschen Mathematiker Vereinigung, Bd. 13, 17.
Science quotes on:  |  Aware (36)  |  Everybody (72)  |  Incomparable (14)  |  Method (531)  |  Modern Mathematics (50)  |  Power (771)  |  Praise (28)  |  Unpopular (4)

Everyone makes for himself a clear idea of the motion of a point, that is to say, of the motion of a corpuscle which one supposes to be infinitely small, and which one reduces by thought in some way to a mathematical point.
Théorie Nouvelle de la Rotation des Corps (1834). As translated by Charles Thomas Whitley in Outlines of a New Theory of Rotatory Motion (1834), 1.
Science quotes on:  |  Clear (111)  |  Corpuscle (14)  |  Himself (461)  |  Idea (881)  |  Infinitely (13)  |  Motion (320)  |  Point (584)  |  Reduce (100)  |  Say (989)  |  Small (489)  |  Suppose (158)  |  Thought (995)  |  Way (1214)

Everyone now agrees that a Physics where you banish all relationship with mathematics, to confine itself to a mere collection of observations and experiences, would be but an historical amusement, more fitting to entertain idle people, than to engage the mind of a true philosopher.
In 'Préface Contenant l’Exposition du Système', Dictionnaire de Physique (1761), Vol. 1, iii. English version via Google Translate, tweaked by Webmaster. From the original French, “Tout le monde convient maintenant qu’une Physique d’où l'on banniroit tout ce qui peut avoir quelque rapport avec les mathématiques, pour se borner à un simple recueil d’observations & d’experiences, ne seroit qu’un amusement historique, plus propre à récréer un cercle de personnes oisives, qu’à occuper un esprit véritablement philosophique.” Also seen translated as—“Everyone now agrees that a physics lacking all connection with mathematics…would only be an historical amusement, fitter for entertaining the idle than for occupying the mind of a philosopher,” in John L. Heilbron, Electricity in the 17th and 18th centuries: A Study of Early Modern Physics (1979), 74. In the latter source, the subject quote immediately follows a different one by Franz Karl Achard. An editor misreading that paragraph is the likely reason the subject quote will be found in Oxford Dictionary of Science Quotations attributed to Achard. Webmaster checked the original footnoted source, and corrected the author of this entry to Paulian (16 May 2014).
Science quotes on:  |  Amusement (37)  |  Banish (11)  |  Collection (68)  |  Engage (41)  |  Entertain (27)  |  Experience (494)  |  Historical (70)  |  Idle (34)  |  Mind (1377)  |  More (2558)  |  Observation (593)  |  People (1031)  |  Philosopher (269)  |  Philosophy (409)  |  Physic (515)  |  Physics (564)  |  Plus (43)  |  Relationship (114)  |  Simple (426)

Everything is controlled by immutable mathematical laws, from which there is, and can be, no deviation whatsoever. We learn the complex from the simple. We arrive at the abstract by way of the concrete.
In The Science of Poetry and the Philosophy of Language (1910), xi.
Science quotes on:  |  Abstract (141)  |  Complex (202)  |  Concrete (55)  |  Deviation (21)  |  Everything (489)  |  Immutable (26)  |  Law (913)  |  Learning (291)  |  Logic (311)  |  Simple (426)

Everything that the greatest minds of all times have accomplished toward the comprehension of forms by means of concepts is gathered into one great science, mathematics.
In 'Pestalozzi's Idee eines A B C der Anschauung', Werke[Kehrbach] (1890), Bd.l, 163. As quoted, cited and translated in Robert Édouard Moritz, Memorabilia Mathematica; Or, The Philomath’s Quotation-Book (1914), 5.
Science quotes on:  |  Accomplishment (102)  |  Comprehension (69)  |  Concept (242)  |  Definitions and Objects of Mathematics (33)  |  Everything (489)  |  Form (976)  |  Gather (76)  |  Great (1610)  |  Greatest (330)  |  Mean (810)  |  Means (587)  |  Mind (1377)  |  Time (1911)  |  Toward (45)

Everywhere in nature we seek some certainty, but all this is nothing more than an arrangement of the dark feeling of our own. All the mathematical laws that we find in Nature are always suspicious to me, despite their beauty. They give me no pleasure. They are merely expedients. Everything is not true at close range.
From the original German, in Physikalische und Mathematische Schriften (1806), Vol. 4, 145, “Wir suchen in der Natur überall eine gewisse Bestimmtheit, aber das Alles ist weiter nichts, als Anordnung des dunkeln Gefühls unserer eigenen. Alle mathematischen Gesetze, die wir in der Natur finden, sind mir trotz ihrer Schönheit immer verdächtig. Sie Freuen mich nicht. Sie sind bloss Hülfsmittel. In der Nähe ist Alles nicht wahr.” English version by Webmaster using Google translate.
Science quotes on:  |  Arrangement (93)  |  Auxiliary (11)  |  Beauty (313)  |  Certainty (180)  |  Close (77)  |  Dark (145)  |  Everything (489)  |  Everywhere (98)  |  Feeling (259)  |  Find (1014)  |  Law (913)  |  Merely (315)  |  More (2558)  |  Nature (2017)  |  Nothing (1000)  |  Pleasure (191)  |  Range (104)  |  Seek (218)  |  Suspect (18)  |  True (239)

Examples ... show how difficult it often is for an experimenter to interpret his results without the aid of mathematics.
Quoted in E. T. Bell, Men of Mathematics, xvi.
Science quotes on:  |  Aid (101)  |  Difficult (263)  |  Difficulty (201)  |  Example (98)  |  Experiment (736)  |  Experimenter (40)  |  Interpretation (89)  |  Result (700)  |  Show (353)

Experimenters are the shock troops of science … An experiment is a question which science poses to Nature, and a measurement is the recording of Nature’s answer. But before an experiment can be performed, it must be planned–the question to nature must be formulated before being posed. Before the result of a measurement can be used, it must be interpreted–Nature’s answer must be understood properly. These two tasks are those of theorists, who find himself always more and more dependent on the tools of abstract mathematics.
'The Meaning and Limits of Exact Science', Science (30 Sep 1949), 110, No. 2857, 325. Advance reprinting of chapter from book Max Planck, Scientific Autobiography (1949), 110.
Science quotes on:  |  Abstract Mathematics (9)  |  Answer (389)  |  Experiment (736)  |  Formulation (37)  |  Interpretation (89)  |  Nature (2017)  |  Planning (21)  |  Question (649)

Experiments may be of two kinds: experiments of simple fact, and experiments of quantity. ...[In the latter] the conditions will ... vary, not in quality, but quantity, and the effect will also vary in quantity, so that the result of quantitative induction is also to arrive at some mathematical expression involving the quantity of each condition, and expressing the quantity of the result. In other words, we wish to know what function the effect is of its conditions. We shall find that it is one thing to obtain the numerical results, and quite another thing to detect the law obeyed by those results, the latter being an operation of an inverse and tentative character.
Principles of Science: A Treatise on Logic and Scientific Method (1874, 1892), 439.
Science quotes on:  |  Being (1276)  |  Character (259)  |  Condition (362)  |  Detect (45)  |  Effect (414)  |  Experiment (736)  |  Expression (181)  |  Fact (1257)  |  Find (1014)  |  Function (235)  |  Induction (81)  |  Kind (564)  |  Know (1538)  |  Law (913)  |  Numerical (39)  |  Obey (46)  |  Obtain (164)  |  Operation (221)  |  Other (2233)  |  Quality (139)  |  Quantitative (31)  |  Quantity (136)  |  Result (700)  |  Simple (426)  |  Tentative (18)  |  Thing (1914)  |  Two (936)  |  Variation (93)  |  Will (2350)  |  Wish (216)  |  Word (650)

Few will deny that even in the first scientific instruction in mathematics the most rigorous method is to be given preference over all others. Especially will every teacher prefer a consistent proof to one which is based on fallacies or proceeds in a vicious circle, indeed it will be morally impossible for the teacher to present a proof of the latter kind consciously and thus in a sense deceive his pupils. Notwithstanding these objectionable so-called proofs, so far as the foundation and the development of the system is concerned, predominate in our textbooks to the present time. Perhaps it will be answered, that rigorous proof is found too difficult for the pupil’s power of comprehension. Should this be anywhere the case,—which would only indicate some defect in the plan or treatment of the whole,—the only remedy would be to merely state the theorem in a historic way, and forego a proof with the frank confession that no proof has been found which could be comprehended by the pupil; a remedy which is ever doubtful and should only be applied in the case of extreme necessity. But this remedy is to be preferred to a proof which is no proof, and is therefore either wholly unintelligible to the pupil, or deceives him with an appearance of knowledge which opens the door to all superficiality and lack of scientific method.
In 'Stücke aus dem Lehrbuche der Arithmetik', Werke, Bd. 2 (1904), 296.
Science quotes on:  |  Answer (389)  |  Anywhere (16)  |  Appearance (145)  |  Applied (176)  |  Apply (170)  |  Base (120)  |  Call (781)  |  Case (102)  |  Circle (117)  |  Comprehend (44)  |  Comprehension (69)  |  Concern (239)  |  Confession (9)  |  Consciously (6)  |  Consistent (50)  |  Deceive (26)  |  Defect (31)  |  Deny (71)  |  Development (441)  |  Difficult (263)  |  Door (94)  |  Doubtful (30)  |  Especially (31)  |  Extreme (78)  |  Fallacy (31)  |  Far (158)  |  Find (1014)  |  First (1302)  |  Forego (4)  |  Foundation (177)  |  Frank (4)  |  Give (208)  |  Historic (7)  |  Impossible (263)  |  Indeed (323)  |  Indicate (62)  |  Instruction (101)  |  Kind (564)  |  Knowledge (1647)  |  Lack (127)  |  Latter (21)  |  Merely (315)  |  Method (531)  |  Morally (2)  |  Most (1728)  |  Necessity (197)  |  Open (277)  |  Other (2233)  |  Plan (122)  |  Power (771)  |  Predominate (7)  |  Prefer (27)  |  Preference (28)  |  Present (630)  |  Proceed (134)  |  Proof (304)  |  Pupil (62)  |  Remedy (63)  |  Rigorous (50)  |  Scientific (955)  |  Scientific Method (200)  |  Sense (785)  |  So-Called (71)  |  State (505)  |  Superficiality (4)  |  System (545)  |  Teacher (154)  |  Teaching of Mathematics (39)  |  Textbook (39)  |  Theorem (116)  |  Time (1911)  |  Treatment (135)  |  Unintelligible (17)  |  Vicious Circle (4)  |  Way (1214)  |  Whole (756)  |  Wholly (88)  |  Will (2350)

Finite systems of deterministic ordinary nonlinear differential equations may be designed to represent forced dissipative hydrodynamic flow. Solutions of these equations can be identified with trajectories in phase space. For those systems with bounded solutions, it is found that nonperiodic solutions are ordinarily unstable with respect to small modifications, so that slightly differing initial states can evolve into considerably different states. Systems with bounded solutions are shown to possess bounded numerical solutions.
A simple system representing cellular convection is solved numerically. All of the solutions are found to be unstable, and almost all of them are nonperiodic.
The feasibility of very-long-range weather prediction is examined in the light of these results
Abstract from his landmark paper introducing Chaos Theory in relation to weather prediction, 'Deterministic Nonperiodic Flow', Journal of the Atmospheric Science (Mar 1963), 20, 130.
Science quotes on:  |  Bound (120)  |  Chaos Theory (4)  |  Convection (3)  |  Design (203)  |  Different (595)  |  Differential Equation (18)  |  Equation (138)  |  Feasibility (4)  |  Finite (60)  |  Flow (89)  |  Hydrodynamics (5)  |  Light (635)  |  Long (778)  |  Modification (57)  |  Nonlinear (4)  |  Numerical (39)  |  Ordinary (167)  |  Phase (37)  |  Phase Space (2)  |  Possess (157)  |  Prediction (89)  |  Range (104)  |  Represent (157)  |  Respect (212)  |  Result (700)  |  Simple (426)  |  Small (489)  |  Solution (282)  |  Solution. (53)  |  Space (523)  |  State (505)  |  System (545)  |  Unstable (9)  |  Weather (49)  |  Weather Prediction (2)

First of all, we ought to observe, that mathematical propositions, properly so called, are always judgments a priori, and not empirical, because they carry along with them necessity, which can never be deduced from experience. If people should object to this, I am quite willing to confine my statements to pure mathematics, the very concept of which implies that it does not contain empirical, but only pure knowledge a priori.
In Critique of Pure Reason (1900), 720.
Science quotes on:  |  A Priori (26)  |  Call (781)  |  Carry (130)  |  Concept (242)  |  Confine (26)  |  Contain (68)  |  Deduce (27)  |  Definitions and Objects of Mathematics (33)  |  Empirical (58)  |  Experience (494)  |  First (1302)  |  Imply (20)  |  Judgment (140)  |  Knowledge (1647)  |  Necessity (197)  |  Never (1089)  |  Object (438)  |  Observe (179)  |  People (1031)  |  Properly (21)  |  Proposition (126)  |  Pure (299)  |  Pure Mathematics (72)  |  Statement (148)  |  Willing (44)

First, [Newton’s Law of Universal Gravitation] is mathematical in its expression…. Second, it is not exact; Einstein had to modify it…. There is always an edge of mystery, always a place where we have some fiddling around to do yet…. But the most impressive fact is that gravity is simple…. It is simple, and therefore it is beautiful…. Finally, comes the universality of the gravitational law and the fact that it extends over such enormous distances…
In The Character of Physical Law (1965, 2001), 33.
Science quotes on:  |  Beautiful (271)  |  Distance (171)  |  Do (1905)  |  Edge (51)  |  Einstein (101)  |  Albert Einstein (624)  |  Enormous (44)  |  Expression (181)  |  Extend (129)  |  Fact (1257)  |  First (1302)  |  Gravitation (72)  |  Gravity (140)  |  Impressive (27)  |  Law (913)  |  Law Of Gravitation (23)  |  Law Of Universal Gravitation (3)  |  Modify (15)  |  Most (1728)  |  Mystery (188)  |  Sir Isaac Newton (363)  |  Simple (426)  |  Universal (198)  |  Universality (22)

First, as concerns the success of teaching mathematics. No instruction in the high schools is as difficult as that of mathematics, since the large majority of students are at first decidedly disinclined to be harnessed into the rigid framework of logical conclusions. The interest of young people is won much more easily, if sense-objects are made the starting point and the transition to abstract formulation is brought about gradually. For this reason it is psychologically quite correct to follow this course.
Not less to be recommended is this course if we inquire into the essential purpose of mathematical instruction. Formerly it was too exclusively held that this purpose is to sharpen the understanding. Surely another important end is to implant in the student the conviction that correct thinking based on true premises secures mastery over the outer world. To accomplish this the outer world must receive its share of attention from the very beginning.
Doubtless this is true but there is a danger which needs pointing out. It is as in the case of language teaching where the modern tendency is to secure in addition to grammar also an understanding of the authors. The danger lies in grammar being completely set aside leaving the subject without its indispensable solid basis. Just so in Teaching of Mathematics it is possible to accumulate interesting applications to such an extent as to stunt the essential logical development. This should in no wise be permitted, for thus the kernel of the whole matter is lost. Therefore: We do want throughout a quickening of mathematical instruction by the introduction of applications, but we do not want that the pendulum, which in former decades may have inclined too much toward the abstract side, should now swing to the other extreme; we would rather pursue the proper middle course.
In Ueber den Mathematischen Unterricht an den hoheren Schulen; Jahresbericht der Deutschen Mathematiker Vereinigung, Bd. 11, 131.
Science quotes on:  |  Abstract (141)  |  Accomplishment (102)  |  Accumulate (30)  |  Addition (70)  |  Application (257)  |  Attention (196)  |  Author (175)  |  Base (120)  |  Basis (180)  |  Begin (275)  |  Beginning (312)  |  Being (1276)  |  Bring (95)  |  Case (102)  |  Completely (137)  |  Concern (239)  |  Conclusion (266)  |  Conviction (100)  |  Correct (95)  |  Course (413)  |  Danger (127)  |  Decade (66)  |  Development (441)  |  Difficult (263)  |  Do (1905)  |  End (603)  |  Essential (210)  |  Exclusive (29)  |  Extent (142)  |  Extreme (78)  |  First (1302)  |  Follow (389)  |  Former (138)  |  Formerly (5)  |  Formulation (37)  |  Framework (33)  |  Gradual (30)  |  Gradually (102)  |  Grammar (15)  |  Harness (25)  |  High (370)  |  High School (15)  |  Hold (96)  |  Implant (5)  |  Important (229)  |  Inclined (41)  |  Indispensable (31)  |  Inquire (26)  |  Instruction (101)  |  Interest (416)  |  Interesting (153)  |  Introduction (37)  |  Kernel (4)  |  Language (308)  |  Large (398)  |  Leave (138)  |  Lie (370)  |  Logic (311)  |  Lose (165)  |  Majority (68)  |  Mastery (36)  |  Matter (821)  |  Middle (19)  |  Modern (402)  |  More (2558)  |  Must (1525)  |  Need (320)  |  Object (438)  |  Other (2233)  |  Outer (13)  |  Pendulum (17)  |  People (1031)  |  Permit (61)  |  Point (584)  |  Possible (560)  |  Premise (40)  |  Proper (150)  |  Psychological (42)  |  Purpose (336)  |  Pursue (63)  |  Quicken (7)  |  Quickening (4)  |  Reason (766)  |  Receive (117)  |  Recommend (27)  |  Rigid (24)  |  School (227)  |  Secure (23)  |  Sense (785)  |  Set (400)  |  Set Aside (4)  |  Share (82)  |  Sharpen (22)  |  Side (236)  |  Solid (119)  |  Starting Point (16)  |  Student (317)  |  Stunt (7)  |  Subject (543)  |  Success (327)  |  Surely (101)  |  Swing (12)  |  Teach (299)  |  Teaching (190)  |  Teaching of Mathematics (39)  |  Tendency (110)  |  Think (1122)  |  Thinking (425)  |  Throughout (98)  |  Transition (28)  |  True (239)  |  Understand (648)  |  Understanding (527)  |  Want (504)  |  Whole (756)  |  Wise (143)  |  World (1850)  |  Young (253)

For a physicist mathematics is not just a tool by means of which phenomena can be calculated, it is the main source of concepts and principles by means of which new theories can be created.
In 'Mathematics in the Physical Sciences', Scientific American (Sep 1964), 211, No. 3, 129.
Science quotes on:  |  Calculate (58)  |  Concept (242)  |  Create (245)  |  Mean (810)  |  Means (587)  |  New (1273)  |  Phenomenon (334)  |  Physicist (270)  |  Principle (530)  |  Source (101)  |  Theory (1015)  |  Tool (129)

For all their wealth of content, for all the sum of history and social institution invested in them, music, mathematics, and chess are resplendently useless (applied mathematics is a higher plumbing, a kind of music for the police band). They are metaphysically trivial, irresponsible. They refuse to relate outward, to take reality for arbiter. This is the source of their witchery.
In 'A Death of Kings', George Steiner at The New Yorker (2009), 209.
Science quotes on:  |  Applied (176)  |  Applied Mathematics (15)  |  Arbiter (5)  |  Band (9)  |  Chess (27)  |  Content (75)  |  History (716)  |  Institution (73)  |  Invest (20)  |  Irresponsible (5)  |  Kind (564)  |  Metaphysical (38)  |  Music (133)  |  Outward (7)  |  Plumbing (5)  |  Police (5)  |  Reality (274)  |  Refuse (45)  |  Relate (26)  |  Resplendent (3)  |  Social (261)  |  Source (101)  |  Sum (103)  |  Trivial (59)  |  Useless (38)  |  Wealth (100)

For he who knows not mathematics cannot know any other science; what is more, he cannot discover his own ignorance, or find its proper remedy.
Science quotes on:  |  Discover (571)  |  Find (1014)  |  Ignorance (254)  |  Know (1538)  |  More (2558)  |  Other (2233)  |  Proper (150)  |  Remedy (63)  |  Science And Mathematics (10)

For it being the nature of the mind of man (to the extreme prejudice of knowledge) to delight in the spacious liberty of generalities, as in a champion region, and not in the enclosures of particularity; the Mathematics were the goodliest fields to satisfy that appetite.
In De Augmentis, Bk. 8; Advancement of Learning, Bk. 2.
Science quotes on:  |  Appetite (20)  |  Being (1276)  |  Champion (6)  |  Delight (111)  |  Enclosure (4)  |  Estimates of Mathematics (30)  |  Extreme (78)  |  Field (378)  |  Generality (45)  |  Good (906)  |  Knowledge (1647)  |  Liberty (29)  |  Man (2252)  |  Mind (1377)  |  Mind Of Man (7)  |  Nature (2017)  |  Prejudice (96)  |  Region (40)  |  Satisfy (29)  |  Spacious (2)

For many parts of Nature can neither be invented with sufficient subtlety, nor demonstrated with sufficient perspicuity, nor accommodated to use with sufficient dexterity, without the aid and intervention of Mathematic: of which sort are Perspective, Music, Astronomy, cosmography, Architecture, Machinery, and some others.
In De Augmentis, Bk. 3; The Advancement of Learning (1605), Book 3. As translated in Francis Bacon, ‎James Spedding and ‎Robert Leslie Ellis, 'Of the great Appendix of Natural Philosophy, both Speculative and Operative, namely Mathematic; and that it ought rather to be placed among Appendices than among Substantive Sciences. Division of Mathematic into Pure and Mixed', The Works of Francis Bacon (1858), Vol. 4, Chap. 6, 371.
Science quotes on:  |  Accommodation (9)  |  Aid (101)  |  Architecture (50)  |  Astronomy (251)  |  Cosmography (4)  |  Demonstration (120)  |  Dexterity (8)  |  Diversity (75)  |  Engineering (188)  |  Intervention (18)  |  Invention (400)  |  Machinery (59)  |  Music (133)  |  Nature (2017)  |  Other (2233)  |  Perspective (28)  |  Perspicuity (2)  |  Subtlety (19)  |  Sufficiency (16)  |  Sufficient (133)  |  Use (771)  |  Value Of Mathematics (60)

For mathematics, in a wilderness of tragedy and change, is a creature of the mind, born to the cry of humanity in search of an invariant reality, immutable in substance, unalterable with time.
In The American Mathematical Monthly (1949), 56, 19. Excerpted in John Ewing (ed,), A Century of Mathematics: Through the Eyes of the Monthly (1996), 186.
Science quotes on:  |  Bear (162)  |  Change (639)  |  Creature (242)  |  Cry (30)  |  Humanity (186)  |  Immutable (26)  |  Invariant (10)  |  Mind (1377)  |  Reality (274)  |  Search (175)  |  Substance (253)  |  Time (1911)  |  Tragedy (31)  |  Unalterable (7)  |  Wilderness (57)

For scholars and laymen alike it is not philosophy but active experience in mathematics itself that can alone answer the question: What is mathematics?
As co-author with Herbert Robbins, in What Is Mathematics?: An Elementary Approach to Ideas and Methods (1941, 1996), xiii.
Science quotes on:  |  Active (80)  |  Activity (218)  |  Alike (60)  |  Alone (324)  |  Answer (389)  |  Experience (494)  |  Itself (7)  |  Layman (21)  |  Philosophy (409)  |  Question (649)  |  Scholar (52)

For the harmony of the world is made manifest in Form and Number, and the heart and soul and all the poetry of Natural Philosophy are embodied in the concept of mathematical beauty.
In 'Epilogue', On Growth and Form (1917), 778-9.
Science quotes on:  |  Beauty (313)  |  Concept (242)  |  Form (976)  |  Harmony (105)  |  Heart (243)  |  Mathematical Beauty (19)  |  Natural (810)  |  Natural Philosophy (52)  |  Number (710)  |  Philosophy (409)  |  Poetry (150)  |  Soul (235)  |  World (1850)

For the things of this world cannot be made known without a knowledge of mathematics.
(Sent to the Pope in 1267). As translated in Opus Majus (1928), Vol. 1, 128.
Science quotes on:  |  Knowledge (1647)  |  Known (453)  |  Thing (1914)  |  World (1850)

For twenty pages perhaps, he read slowly, carefully, dutifully, with pauses for self-examination and working out examples. Then, just as it was working up and the pauses should have been more scrupulous than ever, a kind of swoon and ecstasy would fall on him, and he read ravening on, sitting up till dawn to finish the book, as though it were a novel. After that his passion was stayed; the book went back to the Library and he was done with mathematics till the next bout. Not much remained with him after these orgies, but something remained: a sensation in the mind, a worshiping acknowledgment of something isolated and unassailable, or a remembered mental joy at the rightness of thoughts coming together to a conclusion, accurate thoughts, thoughts in just intonation, coming together like unaccompanied voices coming to a close.
In Mr. Fortune’s Maggot (1927), 161.
Science quotes on:  |  Accurate (88)  |  Acknowledgment (13)  |  Back (395)  |  Book (413)  |  Carefully (65)  |  Coming (114)  |  Conclusion (266)  |  Dawn (31)  |  Ecstasy (9)  |  Examination (102)  |  Fall (243)  |  Finish (62)  |  Joy (117)  |  Kind (564)  |  Library (53)  |  Mental (179)  |  Mind (1377)  |  More (2558)  |  Next (238)  |  Novel (35)  |  Passion (121)  |  Read (308)  |  Remain (355)  |  Remember (189)  |  Scrupulous (7)  |  Self (268)  |  Sensation (60)  |  Sitting (44)  |  Something (718)  |  Thought (995)  |  Together (392)

For we may remark generally of our mathematical researches, that these auxiliary quantities, these long and difficult calculations into which we are often drawn, are almost always proofs that we have not in the beginning considered the objects themselves so thoroughly and directly as their nature requires, since all is abridged and simplified, as soon as we place ourselves in a right point of view.
In Théorie Nouvelle de la Rotation des Corps (1834). As translated by Charles Thomas Whitley in Outlines of a New Theory of Rotatory Motion (1834), 4.
Science quotes on:  |  Abridge (3)  |  Auxiliary (11)  |  Beginning (312)  |  Calculation (134)  |  Consider (428)  |  Difficult (263)  |  Direct (228)  |  Long (778)  |  Nature (2017)  |  Object (438)  |  Ourselves (247)  |  Point (584)  |  Point Of View (85)  |  Proof (304)  |  Quantity (136)  |  Require (229)  |  Research (753)  |  Right (473)  |  Simplify (14)  |  Soon (187)  |  Themselves (433)  |  Thorough (40)  |  Thoroughly (67)  |  View (496)

For, in mathematics or symbolic logic, reason can crank out the answer from the symboled equations—even a calculating machine can often do so—but it cannot alone set up the equations. Imagination resides in the words which define and connect the symbols—subtract them from the most aridly rigorous mathematical treatise and all meaning vanishes. Was it Eddington who said that we once thought if we understood 1 we understood 2, for 1 and 1 are 2, but we have since found we must learn a good deal more about “and”?
In 'The Biological Basis of Imagination', American Thought: 1947 (1947), 81.
Science quotes on:  |  Alone (324)  |  Answer (389)  |  Arid (6)  |  Calculating Machine (3)  |  Connect (126)  |  Crank (18)  |  Deal (192)  |  Define (53)  |  Do (1905)  |  Sir Arthur Stanley Eddington (135)  |  Equation (138)  |  Good (906)  |  Imagination (349)  |  Learn (672)  |  Logic (311)  |  Machine (271)  |  Meaning (244)  |  More (2558)  |  Most (1728)  |  Must (1525)  |  Reason (766)  |  Reside (25)  |  Rigorous (50)  |  Set (400)  |  Set Up (3)  |  Subtract (2)  |  Symbol (100)  |  Symbolic (16)  |  Thought (995)  |  Treatise (46)  |  Understand (648)  |  Understood (155)  |  Vanish (19)  |  Word (650)

For, Mathematical Demonstrations being built upon the impregnable Foundations of Geometry and Arithmetick, are the only Truths, that can sink into the Mind of Man, void of all Uncertainty; and all other Discourses participate more or less of Truth, according as their Subjects are more or less capable of Mathematical Demonstration.
Inaugural lecture of Christopher Wren in his chair of astronomy at Gresham College (1657). From Parentelia (1741, 1951), 200-201.
Science quotes on:  |  According (236)  |  Arithmetic (144)  |  Being (1276)  |  Capable (174)  |  Demonstration (120)  |  Discourse (19)  |  Foundation (177)  |  Geometry (271)  |  Man (2252)  |  Mind (1377)  |  More (2558)  |  More Or Less (71)  |  Other (2233)  |  Participation (15)  |  Sink (38)  |  Subject (543)  |  Truth (1109)  |  Uncertainty (58)  |  Void (31)

Formal thought, consciously recognized as such, is the means of all exact knowledge; and a correct understanding of the main formal sciences, Logic and Mathematics, is the proper and only safe foundation for a scientific education.
In Number and its Algebra (1896), 134.
Science quotes on:  |  Conscious (46)  |  Correct (95)  |  Education (423)  |  Exact (75)  |  Formal (37)  |  Foundation (177)  |  Knowledge (1647)  |  Logic (311)  |  Main (29)  |  Mathematics And Logic (27)  |  Mean (810)  |  Means (587)  |  Proper (150)  |  Recognize (136)  |  Safe (61)  |  Science Education (16)  |  Scientific (955)  |  Thought (995)  |  Understand (648)  |  Understanding (527)

Fractal is a word invented by Mandelbrot to bring together under one heading a large class of objects that have [played] … an historical role … in the development of pure mathematics. A great revolution of ideas separates the classical mathematics of the 19th century from the modern mathematics of the 20th. Classical mathematics had its roots in the regular geometric structures of Euclid and the continuously evolving dynamics of Newton. Modern mathematics began with Cantor’s set theory and Peano’s space-filling curve. Historically, the revolution was forced by the discovery of mathematical structures that did not fit the patterns of Euclid and Newton. These new structures were regarded … as “pathological,” .… as a “gallery of monsters,” akin to the cubist paintings and atonal music that were upsetting established standards of taste in the arts at about the same time. The mathematicians who created the monsters regarded them as important in showing that the world of pure mathematics contains a richness of possibilities going far beyond the simple structures that they saw in Nature. Twentieth-century mathematics flowered in the belief that it had transcended completely the limitations imposed by its natural origins.
Now, as Mandelbrot points out, … Nature has played a joke on the mathematicians. The 19th-century mathematicians may not have been lacking in imagination, but Nature was not. The same pathological structures that the mathematicians invented to break loose from 19th-century naturalism turn out to be inherent in familiar objects all around us.
From 'Characterizing Irregularity', Science (12 May 1978), 200, No. 4342, 677-678. Quoted in Benoit Mandelbrot, The Fractal Geometry of Nature (1977, 1983), 3-4.
Science quotes on:  |  19th Century (41)  |  Art (680)  |  Belief (615)  |  Beyond (316)  |  Break (109)  |  Century (319)  |  Class (168)  |  Classical (49)  |  Completely (137)  |  Curve (49)  |  Development (441)  |  Discovery (837)  |  Euclid (60)  |  Fit (139)  |  Flower (112)  |  Fractal (11)  |  Gallery (7)  |  Great (1610)  |  Historical (70)  |  Idea (881)  |  Imagination (349)  |  Inherent (43)  |  Joke (90)  |  Large (398)  |  Limitation (52)  |  Benoît Mandelbrot (15)  |  Mathematician (407)  |  Modern (402)  |  Modern Mathematics (50)  |  Monster (