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Mathematics Quotes (367 quotes)
Math Quotes, Maths Quotes

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

...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.
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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.
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Das Leben der Gotter ist Mathematik.
Mathematics is the Life of the Gods.
Attributed.
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L'analyse mathématique, n'est elle donc qu'un vain jeu d'esprit? Elle ne peut pas donner au physicien qu'un langage commode; n'est-ce pa là un médiocre service, dont on aurait pu se passer à la rigueur; et même n'est il pas à craindre que ce langage artificiel ne soit pas un voile interposé entre la réalité at l'oeil du physicien? Loin de là, sans ce langage, la pluspart des anaologies intimes des choses nous seraient demeurées à jamais inconnues; et nous aurions toujours ignoré l'harmonie interne du monde, qui est, nous le verrons, la seule véritable réalité objective.
So is not mathematical analysis then not just a vain game of the mind? To the physicist it can only give a convenient language; but isn't that a mediocre service, which after all we could have done without; and, it is not even to be feared that this artificial language be a veil, interposed between reality and the physicist's eye? Far from that, without this language most of the initimate analogies of things would forever have remained unknown to us; and we would never have had knowledge of the internal harmony of the world, which is, as we shall see, the only true objective reality.
La valeur de la science. In Anton Bovier, Statistical Mechanics of Disordered Systems (2006), 6.
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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.
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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.
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Mathematical Knowledge adds a manly Vigour to the Mind, frees it from Prejudice, Credulity, and Superstition.
On the Usefulness of Mathematical Learning, (3rd Ed., 1745), 7.
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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.
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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)
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Tous les effets de la nature ne sont que résultats mathématiques d'un petit noinbre de lois immuables.
All the effects of Nature are only the mathematical consequences of a small number of immutable laws.
From the original French in Oeuvres de Laplace, Vol. VII: Théorie des probabilités (1847), Introduction, cliv.
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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.
Everyone now agrees that a Physics where you banish all relationship with mathematics, to confine itself to a mere collection of experiences and observations, would be but an historical amusement, more fitting to entertain idle people, than to engage the mind of a true philosopher.
In Dictionnaire de Physique (1781), Vol. 8, 209. English version via Google Translate, tweaked by Webmaster. 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).
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A chemist who does not know mathematics is seriously handicapped.
Quoted in Albert Rosenfeld, Langmuir: The Man and the Scientist (1962), 293.
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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.
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A good deal of my research in physics has consisted in not setting out to solve some particular problem, but simply examining mathematical equations 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.
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A good mathematical joke is better, and better mathematics, than a dozen mediocre papers.
A Mathematician's Miscellany (1953). In Béla Bollobás, Littlewood's Miscellany (1986), 24.
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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.
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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) .
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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.
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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.'
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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)
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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.
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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.
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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.
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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.
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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.
The Complete Angler (1915), 7.
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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.
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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.
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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
Spoken by character Lazarus Long in Time Enough For Love (1973), 263. In Carl C. Gaither, Mathematically Speaking (1998), 206.

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.
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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.
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Arithmetically speaking, rabbits multiply faster than adders add.
Anonymous
In Evan Esar, 20,000 Quips and Quotes, 509.
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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.
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As history proves abundantly, mathematical achievement, whatever its intrinsic worth, is the most enduring of all.
In A Mathematician’s Apology (1940, 1967), 80.
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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.
Opticks, 2nd edition (1718), Book 3, Query 31, 380.
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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.
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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.
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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.

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.
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Before you generalize, formalize, and axiomatize there must be mathematical substance.
In Eberhard Zeidler, Applied Functional Analysis: main principles and their applications (1995), 282.
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Business, to be succcessful, must be based on science, for demand and supply are matters of mathematics, not guesswork.
The Book of Business (1913), 56.
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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.
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Chess problems are the hymn-tunes of mathematics.
'A Mathematician's Apology', in James Roy Newman, The World of Mathematics (2000), 2028.
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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...
'Russell's Mathematical Logic', in P. A. Schilpp (ed.), The Philosophy of Bertrand Russell (1944), Vol. 1, 137.
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Common integration is only the memory of differentiation...
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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.
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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.
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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.
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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.
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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.
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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 eory 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.
Variety of Men (1966), 100-1.
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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.
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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.
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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.
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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.
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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.
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For many parts of Nature can neither be invented with sufficient subtlety, nor demonstrated with sufficient perspicuity, nor accommodated unto use with sufficient dexterity, without the aid and intervening of the mathematics, of which sort are perspective, music, astronomy, cosmography, architecture, engineery, and divers others.
The Advancement of Learning (1605), Book 2. Reprinted in The Two Books of Francis Bacon: Of the Proficience and Advancement of Learning, Divine and Human (2009), 97.
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For scholars and laymen alike it is not philosophy but active experience in mathematics itself that can alone answer the question: What is mathematics?
In Richard Courant and Herbert Robbins, What Is Mathematics?: An Elementary Approach to Ideas and Methods (1941, 1996), xiii.
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From the age of 13, I was attracted to physics and mathematics. My interest in these subjects derived mostly from popular science books that I read avidly. Early on I was fascinated by theoretical physics and determined to become a theoretical physicist. I had no real idea what that meant, but it seemed incredibly exciting to spend one's life attempting to find the secrets of the universe by using one's mind.
From 'Autobiography', in Tore Frängsmyr (ed.) Les Prix Nobel. The Nobel Prizes 2004, (2005).
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Genetics is the first biological science which got in the position in which physics has been in for many years. One can justifiably speak about such a thing as theoretical mathematical genetics, and experimental genetics, just as in physics. There are some mathematical geniuses who work out what to an ordinary person seems a fantastic kind of theory. This fantastic kind of theory nevertheless leads to experimentally verifiable prediction, which an experimental physicist then has to test the validity of. Since the times of Wright, Haldane, and Fisher, evolutionary genetics has been in a similar position.
Oral history memoir. Columbia University, Oral History Research Office, New York, 1962. Quoted in William B. Provine, Sewall Wright and Evolutionary Biology (1989), 277.
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Geology has its peculiar difficulties, from which all other sciences are exempt. Questions in chemistry may be settled in the laboratory by experiment. Mathematical and philosophical questions may be discussed, while the materials for discussion are ready furnished by our own intellectual reflections. Plants, animals and minerals, may be arranged in the museum, and all questions relating to their intrinsic principles may be discussed with facility. But the relative positions, the shades of difference, the peculiar complexions, whether continuous or in subordinate beds, are subjects of enquiry in settling the character of rocks, which can be judged of while they are in situ only.
A Geological and Agricultural Survey of the District Adjoining the Erie Canal (1824), 8.
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Given for one instant an intelligence which could comprehend all the forces by which nature is animated and the respective positions of the beings which compose it, if moreover this intelligence were vast enough to submit these data to analysis, it would embrace in the same formula both the movements of the largest bodies in the universe and those of the lightest atom; to it nothing would be uncertain, and the future as the past would be present to its eyes.
expressing his belief in causal determinism Introduction to Oeuvres vol. VII, Theorie Analytique de Probabilites (1812-1820)
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God does not care about our mathematical difficulties. He integrates empirically.
Quoted, without citation, by Léopold Infeld in Quest (1942, 1980), 279. If you know the primary source, please contact Webmaster.
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God used beautiful mathematics in creating the world.
Quoted in Behram Kursunoglu and Eugene Paul Wigner, Paul Adrien Maurice Dirac (1990), Preface, xv.
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Gradually, at various points in our childhoods, we discover different forms of conviction. There’s the rock-hard certainty of personal experience (“I put my finger in the fire and it hurt,”), which is probably the earliest kind we learn. Then there’s the logically convincing, which we probably come to first through maths, in the context of Pythagoras’s theorem or something similar, and which, if we first encounter it at exactly the right moment, bursts on our minds like sunrise with the whole universe playing a great chord of C Major.
In short essay, 'Dawkins, Fairy Tales, and Evidence', 2.
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Greek mathematics is the real thing. The Greeks first spoke a language which modern mathematicians can understand… So Greek mathematics is ‘permanent’, more permanent even than Greek literature.
In A Mathematician’s Apology (1940, 1967), 81.
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Hardly a pure science, history is closer to animal husbandry than it is to mathematics, in that it involves selective breeding. The principal difference between the husbandryman and the historian is that the former breeds sheep or cows or such, and the latter breeds (assumed) facts. The husbandryman uses his skills to enrich the future; the historian uses his to enrich the past. Both are usually up to their ankles in bullshit.
Another Roadside Attraction (1990), 127.
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Having been the discoverer of many splendid things, he is said to have asked his friends and relations that, after his death, they should place on his tomb a cylinder enclosing a sphere, writing on it the proportion of the containing solid to that which is contained.
Plutarch, Life of Marcellus, 17.12. Trans. R. W. Sharples.
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Here I shall present, without using Analysis [mathematics], the principles and general results of the Théorie, applying them to the most important questions of life, which are indeed, for the most part, only problems in probability. One may even say, strictly speaking, that almost all our knowledge is only probable; and in the small number of things that we are able to know with certainty, in the mathematical sciences themselves, the principal means of arriving at the truth—induction and analogy—are based on probabilities, so that the whole system of human knowledge is tied up with the theory set out in this essay.
Philosophical Essay on Probabilities (1814), 5th edition (1825), trans. Andrew I. Dale (1995), 1.
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Here's to pure mathematics—may it never be of any use to anybody.
Anonymous
A toast, variously attributed as used of old at Cambridge University, or as used by G.N. Hardy (according to Arthur C. Clarke in 'The Joy of Maths', Greetings, Carbon-Based Bipeds!: Collected Essays, 1934-1998 (2001), 460).
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Histories make men wise; poets, witty; the mathematics, subtle; natural philosophy, deep; moral, grave; logic and rhetoric, able to contend.
'L. Of Studies,' Essays (1597). In Francis Bacon and Basil Montagu, The Works of Francis Bacon, Lord Chancellor of England (1852), 55.
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How can it be that mathematics, being after all a product of human thought which is independent of experience, is so admirably appropriate to the objects of reality?
From 'Geometry and Experience', an expanded form of an Address by Albert Einstein to the Prussian Academy of Sciences in Berlin (27 Jan 1921). In Albert Einstein, translated by G. B. Jeffery and W. Perrett, Sidelights on Relativity (1923).
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How is it that there are so many minds that are incapable of understanding mathematics? ... the skeleton of our understanding, ... and actually they are the majority. ... We have here a problem that is not easy of solution, but yet must engage the attention of all who wish to devote themselves to education.
Science and Method (1914, 2003), 117-118.
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I admit that mathematical science is a good thing. But excessive devotion to it is a bad thing.
Interview with J.W. N. Sullivan, Contemporary Mind (1934). In John De Pillis, 777 Mathematical Conversation Starters (2002), 198.
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I also ask you my friends not to condemn me entirely to the mill of mathematical calculations, and allow me time for philosophical speculations, my only pleasures.
Letter to Vincenzo Bianchi (17 Feb 1619). Johannes Kepler Gesammelte Werke (1937- ), Vol. 17, letter 827, l. 249-51, p. 327.
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I also require much time to ponder over the matters themselves, and particularly the principles of mechanics (as the very words: force, time, space, motion indicate) can occupy one severely enough; likewise, in mathematics, the meaning of imaginary quantities, of the infinitesimally small and infinitely large and similar matters.
In Davis Baird, R.I.G. Hughes and Alfred Nordmann, Heinrich Hertz: Classical Physicist, Modern Philosopher (1998), 159.
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I am ever more intrigued by the correspondence between mathematics and physical facts. The adaptability of mathematics to the description of physical phenomena is uncanny.
From Nobel Banquet Speech (10 Dec 1981), in Wilhelm Odelberg (ed.), Les Prix Nobel 1981 (1981), 59.
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I am interested in mathematics only as a creative art.
In A Mathematician’s Apology (1940, reprint with Foreward by C.P. Snow 1992), 115.

I believe that mathematical reality lies outside us, that our function is to discover or observe it, and that the theorems which we prove, and which we describe grandiloquently as our “creations,” are simply the notes of our observations.
In A Mathematician's Apology (1940, reprint with Foreward by C.P. Snow 1992), 113.
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I can see him now at the blackboard, chalk in one hand and rubber in the other, writing rapidly and erasing recklessly, pausing every few minutes to face the class and comment earnestly, perhaps on the results of an elaborate calculation, perhaps on the greatness of the Creator, perhaps on the beauty and grandeur of Mathematics, always with a capital M. To him mathematics was not the handmaid of philosophy. It was not a humanly devised instrument of investigation, it was Philosophy itself, the divine revealer of TRUTH.
Writing as a Professor Emeritus at Harvard University, a former student of Peirce, in 'Benjamin Peirce: II. Reminiscences', The American Mathematical Monthly (Jan 1925), 32, No. 1, 5.
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I confess, that very different from you, I do find sometimes scientific inspiration in mysticism … but this is counterbalanced by an immediate sense for mathematics.
Letter to Niels Bohr (1955). Quoted in Robert J. Scully, The Demon and the Quantum (2007), 7.
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I do not think the division of the subject into two parts - into applied mathematics and experimental physics a good one, for natural philosophy without experiment is merely mathematical exercise, while experiment without mathematics will neither sufficiently discipline the mind or sufficiently extend our knowledge in a subject like physics.
to Henry Roscoe, Professor of Chemistry at Owens College (2 Jun 1870), B.C.S Archive Quoted in R.H. Kargon, Science in Victorian Manchester (1977), 215.
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I had a dislike for [mathematics], and ... was hopelessly short in algebra. ... [One extraordinary teacher of mathematics] got the whole year's course into me in exactly six [after-school] lessons of half an hour each. And how? More accurately, why? Simply because he was an algebra fanatic—because he believed that algebra was not only a science of the utmost importance, but also one of the greatest fascination. ... [H]e convinced me in twenty minutes that ignorance of algebra was as calamitous, socially and intellectually, as ignorance of table manners—That acquiring its elements was as necessary as washing behind the ears. So I fell upon the book and gulped it voraciously. ... To this day I comprehend the binomial theorem.
In Prejudices: third series (1922), 261-262.
For a longer excerpt, see H. L. Mencken's Recollections of School Algebra.
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I had begun it, it will now be unnecessary for me to finish it.[At a late age, expressing his enthusiasm for mathematics had gone, as when informed of some other mathematician's current work.]
As quoted by Charles Hutton in A Philosophical and Mathematical Dictionary (1815), Vol. 1, 708.
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I have never done anything 'useful'. No discovery of mine has made, or is likely to make, directly or indirectly, for good or ill, the least difference to the amenity of the world... Judged by all practical standards, the value of my mathematical life is nil; and outside mathematics it is trivial anyhow. I have just one chance of escaping a verdict of complete triviality, that I may be judged to have created something worth creating. And that I have created something is undeniable: the question is about its value.
A Mathematician's Apology (1940), 90-1.
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I learnt to distrust all physical concepts as the basis for a theory. Instead one should put one's trust in a mathematical scheme, even if the scheme does not appear at first sight to be connected with physics. One should concentrate on getting interesting mathematics.
From a 1977 lecture. Quoted in Pesi Rustom Masani, Norbert Wiener, 1894-1964 (1990), 6.
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I liked science. I wasn’t mathematically oriented, so I became an organic chemist.
As quoted in Columbia University Press Release, On Campus (21 Feb 2013).
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I love mathematics not only because it is applicable to technology but also because it is beautiful.
In Eberhard Zeidler, Quantum Field Theory (2006), 955.
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I never could do anything with figures, never had any talent for mathematics, never accomplished anything in my efforts at that rugged study, and to-day the only mathematics I know is multiplication, and the minute I get away up in that, as soon as I reach nine times seven— [He lapsed into deep thought, trying to figure nine times seven. Mr. McKelway whispered the answer to him.] I've got it now. It's eighty-four. Well, I can get that far all right with a little hesitation. After that I am uncertain, and I can't manage a statistic.
Speech at the New York Association for Promoting the Interests of the Blind (29 Mar 1906). In Mark Twain and William Dean Howells (ed.), Mark Twain's Speeches? (1910), 323.
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I presume that few who have paid any attention to the history of the Mathematical Analysis, will doubt that it has been developed in a certain order, or that that order has been, to a great extent, necessary—being determined, either by steps of logical deduction, or by the successive introduction of new ideas and conceptions, when the time for their evolution had arrived. And these are the causes that operate in perfect harmony. Each new scientific conception gives occasion to new applications of deductive reasoning; but those applications may be only possible through the methods and the processes which belong to an earlier stage.
Explaining his choice for the exposition in historical order of the topics in A Treatise on Differential Equations (1859), Preface, v-vi.
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I started studying law, but this I could stand just for one semester. I couldn't stand more. Then I studied languages and literature for two years. After two years I passed an examination with the result I have a teaching certificate for Latin and Hungarian for the lower classes of the gymnasium, for kids from 10 to 14. I never made use of this teaching certificate. And then I came to philosophy, physics, and mathematics. In fact, I came to mathematics indirectly. I was really more interested in physics and philosophy and thought about those. It is a little shortened but not quite wrong to say: I thought I am not good enough for physics and I am too good for philosophy. Mathematics is in between.
From interview on his 90th birthday. In D J Albers and G L Alexanderson (eds.), Mathematical People: Profiles and Interviews (1985), 245-254.
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I tell them if they will occupy themselves with the study of mathematics they will find in it the best remedy against the lusts of the flesh.
The Magic Mountain (1924, 1965), 417.
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I think it is a peculiarity of myself that I like to play about with equations, just looking for beautiful mathematical relations which maybe don’t have any physical meaning at all. Sometimes they do.
At age 60.
"Interview with T. Kuhn (7 May 1963), Niels Bohr Library, American Intitute of Physics, New York. 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), 109.
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I was x years old in the year x2.
When asked about his age (43).
Quoted in H. Eves, In Mathematical Circles (1969).
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I'm sorry to say that the subject I most disliked was mathematics. I have thought about it. I think the reason was that mathematics leaves no room for argument.
Malcolm X
The Autobiography of Malcolm X (1965, 1999), 34.
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If a man's wit be wandering, let him study the mathematics; for in demonstrations, if his wit be called away never so little, he must begin again.
Translation in Francis Bacon, James Spedding (ed.) et al., Works of Francis Bacon (1858) Vol. 6, 498.

If others would but reflect on mathematical truths as deeply and continuously as I have, they would make my discoveries.
Quoted in J. R. Newman (ed.), The World of Mathematics (1956).
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If physics leads us today to a world view which is essentially mystical, it returns, in a way, to its beginning, 2,500 years ago. ... This time, however, it is not only based on intuition, but also on experiments of great precision and sophistication, and on a rigorous and consistent mathematical formalism.
In The Tao of Physics (1975), 19.
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If scientific reasoning were limited to the logical processes of arithmetic, we should not get very far in our understanding of the physical world. One might as well attempt to grasp the game of poker entirely by the use of the mathematics of probability.
Endless Horizons (1946), 27.
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If the proof starts from axioms, distinguishes several cases, and takes thirteen lines in the text book … it may give the youngsters the impression that mathematics consists in proving the most obvious things in the least obvious way.
Mathematical Discovery: on Understanding, Learning, and Teaching Problem Solving (1981), 129.
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If there is anything that can bind the heavenly mind of man to this dreary exile of our earthly home and can reconcile us with our fate so that one can enjoy living,—then it is verily the enjoyment of the mathematical sciences and astronomy.
In a letter to his son-in-law, Jakob Bartsch. Quoted in Norman Davidson, Sky Phenomena (2004), 131. Also see Johannes Kepler and Carola Baumgardt (ed.), Johannes Kepler: Life and Letters (1951), 190.
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If there is something very slightly wrong in our definition of the theories, then the full mathematical rigor may convert these errors into ridiculous conclusions.
Feynman Lectures on Gravitation, edited by Brian Hatfield (2002), 21.
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If we ever establish contact with intelligent aliens living on a planet around a distant star … They would be made of similar atoms to us. They could trace their origins back to the big bang 13.7 billion years ago, and they would share with us the universe's future. However, the surest common culture would be mathematics.
In 'Take Me to Your Mathematician', New Scientist (14 Feb 2009), 201, No. 2695.
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If we wish to foresee the future of mathematics, our proper course is to study the history and present condition of the science.
Science and Method (1914, 2003), 25.
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If you ask ... the man in the street ... the human significance of mathematics, the answer of the world will be, that mathematics has given mankind a metrical and computatory art essential to the effective conduct of daily life, that mathematics admits of countless applications in engineering and the natural sciences, and finally that mathematics is a most excellent instrumentality for giving mental discipline... [A mathematician will add] that mathematics is the exact science, the science of exact thought or of rigorous thinking.
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), 65-66.
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If you want to be a physicist, you must do three things—first, study mathematics, second, study more mathematics, and third, do the same.
Interview with Paul H. Kirkpatrick, in Daniel J. Kevles, The Physicists (1978), 200.
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Imagination is the Discovering Faculty, pre-eminently ... It is that which feels & discovers what is, the REAL which we see not, which exists not for our senses... Mathematical science shows what is. It is the language of unseen relations between things... Imagination too shows what is ... Hence she is or should be especially cultivated by the truly Scientific, those who wish to enter into the worlds around us!
In Time I Will Do All, I Dare Say. In Dorothy Stein (ed.), Ada: A Life and a Legacy (1985), 129.
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Imagine a person with a gift of ridicule [He might say] First that a negative quantity has no logarithm; secondly that a negative quantity has no square root; thirdly that the first non-existent is to the second as the circumference of a circle is to the diameter.
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In a lot of scientists, the ratio of wonder to skepticism declines in time. That may be connected with the fact that in some fields—mathematics, physics, some others—the great discoveries are almost entirely made by youngsters.
Quoted in interview with magazine staff, Psychology Today (Jan 1996).
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In fact, Gentlemen, no geometry without arithmetic, no mechanics without geometry... you cannot count upon success, if your mind is not sufficiently exercised on the forms and demonstrations of geometry, on the theories and calculations of arithmetic ... In a word, the theory of proportions is for industrial teaching, what algebra is for the most elevated mathematical teaching.
... a l'ouverture du cours de mechanique industrielle á Metz (1827), 2-3, trans. Ivor Grattan-Guinness.
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In his wretched life of less than twenty-seven years Abel accomplished so much of the highest order that one of the leading mathematicians of the Nineteenth Century (Hermite, 1822-1901) could say without exaggeration, “Abel has left mathematicians enough to keep them busy for five hundred years.” Asked how he had done all this in the six or seven years of his working life, Abel replied, “By studying the masters, not the pupils.”
The Queen of the Sciences (1931, 1938), 10.
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In mathematics there are no true controversies. (1811)
This quote is usually seen without any specific source citation. The sense of it is given, not in quotation marks, as “In 1811 Gauss stated that there are no true controversies in mathematics,” in G. Waldo Dunnington, Jeremy Gray and Fritz-Egbert Dohse, Carl Friedrich Gauss: Titan of Science (2003), 418. If you know the primary source, please contact Webmaster.
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In mathematics we find the primitive source of rationality; and to mathematics must the biologists resort for means to carry out their researches.
The Positive Philosophy, trans. Harriet Martineau (1853), Vol. 1, 388.

In mathematics, fractions speak louder than words.
Anonymous
In Evan Esar, 20,000 Quips and Quotes, 509.
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In order to translate a sentence from English into French two things are necessary. First, we must understand thoroughly the English sentence. Second, we must be familiar with the forms of expression peculiar to the French language. The situation is very similar when we attempt to express in mathematical symbols a condition proposed in words. First, we must understand thoroughly the condition. Second, we must be familiar with the forms of mathematical expression.
In How to Solve It: A New Aspect of Mathematical Method (2004), 174.
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In the beginning (if there was such a thing), God created Newton’s laws of motion together with the necessary masses and forces. This is all; everything beyond this follows from the development of appropriate mathematical methods by means of deduction.
Autobiographical Notes (1946), 19. In Albert Einstein, Alice Calaprice, Freeman Dyson , The Ultimate Quotable Einstein (2011), 397.
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In the company of friends, writers can discuss their books, economists the state of the economy, lawyers their latest cases, and businessmen their latest acquisitions, but mathematicians cannot discuss their mathematics at all. And the more profound their work, the less understandable it is.
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, 7.
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In the last two months I have been very busy with my own mathematical speculations, which have cost me much time, without my having reached my original goal. Again and again I was enticed by the frequently interesting prospects from one direction to the other, sometimes even by will-o'-the-wisps, as is not rare in mathematic speculations.
Letter to Ernst Weber (21 May 1843). Quoted in G. Waldo Dunnington, Carl Friedrich Gauss: Titan of Science (2004), 416.
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In the mathematics I can report no deficience, except it be that men do not sufficiently understand this excellent use of the pure mathematics, in that they do remedy and cure many defects in the wit and faculties intellectual. For if the wit be too dull, they sharpen it; if too wandering, they fix it; if too inherent in the sense, they abstract it. So that as tennis is a game of no use in itself, but of great use in respect it maketh a quick eye and a body ready to put itself into all postures, so in the mathematics that use which is collateral and intervenient is no less worthy than that which is principal and intended.
The Advancement of Learning (1605), Book 2. Reprinted in The Two Books of Francis Bacon: Of the Proficience and Advancement of Learning, Divine and Human (2009), 97.
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In [great mathematics] there is a very high degree of unexpectedness, combined with inevitability and economy.
In A Mathematician’s Apology (1940, reprint with Foreward by C.P. Snow 1992), 113.

Integers are the fountainhead of all mathematics.
In Diophantische Approximationen, Preface. As cited in Claudi Alsina and Roger B. Nelsen, Charming Proofs: A Journey into Elegant Mathematics (2011), 1.
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It always bothers me that according to the laws as we understand them today, it takes a computing machine an infinite number of logical operations to figure out what goes on in no matter how tiny a region of space and no matter how tiny a region of time ... I have often made the hypothesis that ultimately physics will not require a mathematical statement, that in the end the machinery will be revealed and the laws will turn out to be simple. ... But this speculation is of the same nature as those other people make - 'I like it','I don't like it' - and it is not good to be too prejudiced about these things.
The Character of Physical Law (1965), 57. Quoted in Brian Rotman, Mathematics as Sign (2000), 82.
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It becomes the urgent duty of mathematicians, therefore, to meditate about the essence of mathematics, its motivations and goals and the ideas that must bind divergent interests together.
In 'Mathematics in the Modern World', Scientific American (Sep 1964) 211, No. 3, 42. Collected in Ronald J. Comer and Morris Kline, Mathematics in the Modern World: Readings from Scientific American (1988), 20.
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It is a common observation that a science first begins to be exact when it is quantitatively treated. What are called the exact sciences are no others than the mathematical ones.
On The Doctrine of Chances, with Later Reflections (1878), 61.
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It is a mathematical fact that the casting of a pebble from my hand alters the centre of gravity of the universe.
In James Wood, Dictionary of Quotations from Ancient and Modern, English and Foreign Sources (1893), 190:1.
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It is always noteworthy that all those who seriously study this science [the theory of numbers] conceive a sort of passion for it.
Letter to Jonos Boyai (2 Sep 1808). Quoted in G. Waldo Dunnington, Carl Friedrich Gauss: Titan of Science (2004), 413.

It is by mathematical formulation of its observations and measurements that a science is able to form mathematically expressed hypotheses, and it is through its hypotheses that a natural science is able to make predictions.
The Nature of Science, and Other Essays (1971), 14.
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It is interesting thus to follow the intellectual truths of analysis in the phenomena of nature. This correspondence, of which the system of the world will offer us numerous examples, makes one of the greatest charms attached to mathematical speculations.
Exposition du système du monde (1799)
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It is not of the essence of mathematics to be conversant with the ideas of number and quantity. Whether as a general habit of mind it would be desirable to apply symbolic processes to moral argument, is another question.
An Investigation of the Laws of Thought (1854), 12.
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It is perplexing to see the flexibility of the so-called 'exact sciences' which by cast-iron laws of logic and by the infallible help of mathematics can lead to conclusions which are diametrically opposite to one another.
In The Nature of Light: an Historical Survey (1970), 229
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It is probably no exaggeration to suppose that in order to improve such an organ as the eye at all, it must be improved in ten different ways at once. And the improbability of any complex organ being produced and brought to perfection in any such way is an improbability of the same kind and degree as that of producing a poem or a mathematical demonstration by throwing letters at random on a table.
[Expressing his reservations about Darwin's proposed evolution of the eye by natural selection.]
Opening address to the Belfast Natural History Society, as given in the 'Belfast Northern Whig,' (19 Nov 1866). As cited by Charles Darwin in The Variation of Animals & Plants Under Domestication (1868), 222.
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It needs scarcely be pointed out that in placing Mathematics at the head of Positive Philosophy, we are only extending the application of the principle which has governed our whole Classification. We are simply carrying back our principle to its first manifestation. Geometrical and Mechanical phenomena are the most general, the most simple, the most abstract of all,—the most irreducible to others, the most independent of them; serving, in fact, as a basis to all others. It follows that the study of them is an indispensable preliminary to that of all others. Therefore must Mathematics hold the first place in the hierarchy of the sciences, and be the point of departure of all Education whether general or special.
The Positive Philosophy, trans. Harriet Martineau (1853), Vol.1, 33.

It often happens that men, even of the best understandings and greatest circumspection, are guilty of that fault in reasoning which the writers on logick call the insufficient, or imperfect enumeration of parts, or cases: insomuch that I will venture to assert, that this is the chief, and almost the only, source of the vast number of erroneous opinions, and those too very often in matters of great importance, which we are apt to form on all the subjects we reflect upon, whether they relate to the knowledge of nature, or the merits and motives of human actions. It must therefore be acknowledged, that the art which affords a cure to this weakness, or defect, of our understandings, and teaches us to enumerate all the possible ways in which a given number of things may be mixed and combined together, that we may be certain that we have not omitted anyone arrangement of them that can lead to the object of our inquiry, deserves to be considered as most eminently useful and worthy of our highest esteem and attention. And this is the business of the art, or doctrine of combinations ... It proceeds indeed upon mathematical principles in calculating the number of the combinations of the things proposed: but by the conclusions that are obtained by it, the sagacity of the natural philosopher, the exactness of the historian, the skill and judgement of the physician, and the prudence and foresight of the politician, may be assisted; because the business of all these important professions is but to form reasonable conjectures concerning the several objects which engage their attention, and all wise conjectures are the results of a just and careful examination of the several different effects that may possibly arise from the causes that are capable of producing them.
Ars conjectandi (1713). In F. Maseres, The Doctrine of Permutations and Combinations (1795), 36.
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It seems perfectly clear that Economy, if it is to be a science at all, must be a mathematical science. There exists much prejudice against attempts to introduce the methods and language of mathematics into any branch of the moral sciences. Most persons appear to hold that the physical sciences form the proper sphere of mathematical method, and that the moral sciences demand some other method—I know not what.
The Theory of Political Economy (1871), 3.
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Just as mathematics aims to study such entities as numbers, functions, spaces, etc., the subject matter of metamathematics is mathematics itself.
In 'Mathematics: A Non-Technical Exposition', American Scientist (3 Jul 1954), 42, No. 3, 490.
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Just by studying mathematics we can hope to make a guess at the kind of mathematics that will come into the physics of the future ... If someone can hit on the right lines along which to make this development, it m may lead to a future advance in which people will first discover the equations and then, after examining them, gradually learn how to apply the ... My own belief is that this is a more likely line of progress than trying to guess at physical pictures.
'The Evolution of the Physicist's Picture of Nature', Scientific American, May 1963, 208, 47. In Steve Adams, Frontiers (2000), 57.
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Kant, discussing the various modes of perception by which the human mind apprehends nature, concluded that it is specially prone to see nature through mathematical spectacles. Just as a man wearing blue spectacles would see only a blue world, so Kant thought that, with our mental bias, we tend to see only a mathematical world.
In The Mysterious Universe (1930), 115.
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Lagrange, in one of the later years of his life, imagined that he had overcome the difficulty (of the parallel axiom). He went so far as to write a paper, which he took with him to the Institute, and began to read it. But in the first paragraph something struck him that he had not observed: he muttered: 'Il faut que j'y songe encore', and put the paper in his pocket.' [I must think about it again]
Budget of Paradoxes (1872), 173.
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Laplace would have found it child's-play to fix a ratio of progression in mathematical science between Descartes, Leibnitz, Newton and himself
The Education of Henry Adams: An Autobiography? (1918), 491.
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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 'Mathematics as an Element in the History of Thought' in James R. Newman, The World of Mathematics (1957), Vol. 1, 403.
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Like the crest of a peacock, like the gem on the head of a snake, so is mathematics at the head of all knowledge.
Anonymous
From the oldest extant Indian astronomical text, Vedanga Jyotisa (c. 500 B.C.). Quoted, as cited by George Gheverghese Joseph, in Dick Teresi, Lost Discoveries (2003), 28. G. G. Joseph has written a book by the title Crest of the Peacock (1991).
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Logic issues in tautologies, mathematics in identities, philosophy in definitions; all trivial, but all part of the vital work of clarifying and organising our thought.
'Last Papers: Philosophy' (1929), in The Foundations of Mathematics and Other Logical Essays (1931), 264.
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Making mathematics accessible to the educated layman, while keeping high scientific standards, has always been considered a treacherous navigation between the Scylla of professional contempt and the Charybdis of public misunderstanding.
In Rota's 'Introduction' written (1980) to preface Philip J. Davis and Reuben Hersh, The Mathematical Experience (1981, 2012), xxiii.
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Making out an income tax is a lesson in mathematics: addition, division, multiplication and extraction.
Anonymous
In Evan Esar, 20,000 Quips and Quotes, 419.
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Many arts there are which beautify the mind of man; of all other none do more garnish and beautify it than those arts which are called mathematical.
The Elements of Geometric of the most ancient Philosopher Euclide of Megara (1570), Note to the Reader. In Robert Édouard Moritz, Memorabilia Mathematica; Or, The Philomath's Quotation-book (1914), 44.
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Many who have never had an opportunity of knowing any more about mathematics confound it with arithmetic, and consider it an arid science. In reality, however, it is a science which requires a great amount of imagination.
In a letter to Madame Schabelskoy, quoted in Sónya Kovalévsky: Her Recollections of Childhood, translated by Isabel F. Hapgood (1895), 316.
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Mathematical discoveries, like springtime violets in the woods, have their season which no human can hasten or retard.
Quoted in E.T. Bell, The Development of Mathematics (1945).
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Mathematics as an expression of the human mind reflects the active will, the contemplative reason, and the desire for aesthetic perfection. Its basic elements are logic and intuition, analysis and construction, generality and individuality. Though different traditions may emphasize different aspects, it is only the interplay of these antithetic forces and the struggle for their synthesis that constitute the life, usefulness, and supreme value of mathematical science.
In Richard Courant and Herbert Robbins, What Is Mathematics?: An Elementary Approach to Ideas and Methods (1941, 1996), x.
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Mathematics began to seem too much like puzzle solving. Physics is puzzle solving, too, but of puzzles created by nature, not by the mind of man.
Quoted in Joan Dash, 'Maria Goeppert-Mayer', A Life of One's Own, 252.
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Mathematics can remove no prejudices and soften no obduracy. It has no influence in sweetening the bitter strife of parties, and in the moral world generally its action is perfectly null.
In James Wood, Dictionary of Quotations from Ancient and Modern, English and Foreign Sources (1893), 271:3.

Mathematics education is much more complicated than you expected, even though you expected it to be more complicated than you expected.
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Mathematics had never had more than a secondary interest for him [her husband, George Boole]; and even logic he cared for chiefly as a means of clearing the ground of doctrines imagined to be proved, by showing that the evidence on which they were supposed to give rest had no tendency to prove them. But he had been endeavoring to give a more active and positive help than this to the cause of what he deemed pure religion.
In Eleanor Meredith Cobham, Mary Everest Boole: Collected Works (1931), 40.
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Mathematics has not a foot to stand upon which is not purely metaphysical.
'Kant in His Miscellaneous Essays', Blackwood's Magazine, 1830, 28, 244-68.

Mathematics is a dangerous profession; an appreciable proportion of us go mad.
A Mathematician's Miscellany (1953). In Béla Bollobás, Littlewood's Miscellany (1986), 104.
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Mathematics is a form of poetry which transcends poetry in that it proclaims a truth; a form of reasoning which transcends reasoning in that it wants to bring about the truth it proclaims; a form of action, of ritual behavior, which does not find fulfilment in the act but must proclaim and elaborate a poetic form of truth.
'Why Mathematics Grows', Journal of the History of Ideas (Jan-Mar 1965), 26, No. 1, 3. In Salomon Bochner and Robert Clifford Gunning (ed.) Collected Papers of Salomon Bochner (1992), Vol. 4, 191. Footnoted as restating about Mathematics what was written about Myth by Henri Frankfort, et al., in The Intellectual Adventures of Ancient Man (1946), 8.
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Mathematics is an interesting intellectual sport but it should not be allowed to stand in the way of obtaining sensible information about physical processes.
Quoted in N. Rose, Mathematical Maxims and Minims (1988).

Mathematics is an obscure field, an abstruse science, complicated and exact; yet so many have attained perfection in it that we might conclude almost anyone who seriously applied himself would achieve a measure of success.
In George Edward Martin, The Foundations of Geometry and the Non-Euclidean Plane (1982), 82.
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Mathematics is being lazy. Mathematics is letting the principles do the work for you so that you do not have to do the work for yourself
In Marion Walter and Tom O'Brien, 'Memories of George Polya', Mathematics Teaching (Sep 1986), 116, 4.
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Mathematics is concerned only with the enumeration and comparison of relations.
Quoted in E. T. Bell, The Development of Mathematics (1945).

Mathematics is like childhood diseases. The younger you get it, the better.
Quoted by Dudley Herschbach in 'Einstein as a Student', collected in Peter Galison (ed.), Gerald James Holton (ed.), Silvan S. Schweber (ed.), Einstein for the 21st Century: His Legacy in Science, Srt, and Modern Culture (2008), 236. The remark quoted was footnoted as heard by Dudley Herschbach many years earlier in a class by George Polya, but not found in print.
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Mathematics is much more than a language for dealing with the physical world. It is a source of models and abstractions which will enable us to obtain amazing new insights into the way in which nature operates. Indeed, the beauty and elegance of the physical laws themselves are only apparent when expressed in the appropriate mathematical framework.
In Principles of Electrodynamics (1972, 1987), 105.
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Mathematics is music for the mind; music is mathematics for the soul.
Anonymous
In Nat Shapiro (ed.) An Encyclopedia of Quotations About Music (1981), 3.
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Mathematics is not a careful march down a well-cleared highway, but a journey into a strange wilderness, where the explorers often get lost. Rigour should be a signal to the historian that the maps have been made, and the real explorers have gone elsewhere.
'Mathematics and History', Mathematical Intelligencer, 4, No. 4, 10.
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Mathematics is not a deductive science—that's a cliché. When you try to prove a theorem, you don't just list the hypotheses, and then start to reason. What you do is trial and error, experiment and guesswork.
I Want to be a Mathematician: an Automathography in Three Parts (1985), 321.
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Mathematics is not arithmetic. Though mathematics may have arisen from the practices of counting and measuring it really deals with logical reasoning in which theorems—general and specific statements—can be deduced from the starting assumptions. It is, perhaps, the purest and most rigorous of intellectual activities, and is often thought of as queen of the sciences.
Essay,'Private Games', in Lewis Wolpert, Alison Richards (eds.), A Passion for Science (1988), 53.
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Mathematics is not only one of the most valuable inventions—or discoveries—of the human mind, but can have an aesthetic appeal equal to that of anything in art. Perhaps even more so, according to the poetess who proclaimed, “Euclid alone hath looked at beauty bare.”
From 'The Joy of Maths'. Collected in Arthur C. Clarke, Greetings, Carbon-Based Bipeds!: Collected Essays, 1934-1998, 460.
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Mathematics is strange: many make thousands but not many make millions.
Anonymous
In Evan Esar, 20,000 Quips and Quotes, 250.
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Mathematics is the art of giving the same name to different things.
Henri Poincaré and George Bruce Halsted (trans.) The Foundations of Science: Science and Hypothesis (1921), 375.
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Mathematics is the cheapest science. Unlike physics or chemistry, it does not require any expensive equipment. All one needs for mathematics is a pencil and paper.
Quoted in 'And Sometimes the Mathematician Wants a Powerful Computer', in Donald J. Albers and Gerald L. Alexanderson (eds.), Mathematical People (1985). In John De Pillis, 777 Mathematical Conversation Starters (2002), 193.
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Mathematics is the key and door to the sciences.
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Mathematics is the only good metaphysics.
Quoted in E. T. Bell, Men of Mathematics, xvii.
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Mathematics is the only true metaphysics.
Silvanus Phillips Thompson, Life of Lord Kelvin (1910), 10. In Robert Édouard Moritz, Memorabilia Mathematica; Or, The Philomath's Quotation-book (1914)
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Mathematics is the queen of the sciences and arithmetic [number theory] is the queen of mathematics. She often condescends to render service to astronomy and other natural sciences, but in all relations, she is entitled to first rank.
I>Sartorius von Waltershausen: Gauss zum Gedächtniss (1856), 79. Quoted in Robert Edouard Moritz, Memorabilia Mathematica (1914), 271.
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Mathematics is the science which draws necessary conclusions.
Headline to opening paragraph, Linear Associative Algebra, read at National Academy of Sciences, Washington, D.C. (1870).
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Mathematics is the study of analogies between analogies. All science is. Scientists want to show that things that don’t look alike are really the same. That is one of their innermost Freudian motivations. In fact, that is what we mean by understanding.
In 'A Mathematician's Gossip', Indiscrete Thoughts (2008), 214.
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Mathematics is the tool specially suited for dealing with abstract concepts of any kind and there is no limit to its power in this field.
The Principles of Quantum Mechanics (1930, 1981), Preface, viii.

Mathematics is written for mathematicians.
'To His Holiness Pope Paul III, in Copernicus: On the Revolutions of the Heavenly Spheres (1543), trans. A. M. Duncan (1976), 27.

Mathematics may be compared to a mill of exquisite workmanship, which grinds you stuff of any degree of fineness; but, nevertheless, what you get out depends upon what you put in; and as the grandest mill in the world will not extract wheat-flour from peascods, so pages of formulae will not get a definite result out of loose data.
'Geological Reform' (1869). In Collected Essays (1894), Vol. 8, 333.
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Mathematics, as much as music or any other art, is one of the means by which we rise to a complete self-consciousness. The significance of mathematics resides precisely in the fact that it is an art; by informing us of the nature of our own minds it informs us of much that depends on our minds.
In Aspects of Science: Second Series (1926), 94.
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Mathematics, rightly viewed, possesses not only truth, but supreme beauty—a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the georgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show. The true spirit of delight, the exaltation, the sense of being more than Man, which is the touchstone of the highest excellence, is to be found in mathematics as surely as in poetry.
Essay, 'The Study of Mathematics' (1902), collected in Philosophical Essays (1910), 73-74. Also collected in Mysticism and Logic: And Other Essays (1919), 60.
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Mathematics—a wonderful science, but it hasn't yet come up with a way to divide one tricycle between three small boys.
In Bob Phillips, Phillips' Treasury of Humorous Quotations (2004), 171.
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Men can construct a science with very few instruments, or with very plain instruments; but no one on earth could construct a science with unreliable instruments. A man might work out the whole of mathematics with a handful of pebbles, but not with a handful of clay which was always falling apart into new fragments, and falling together into new combinations. A man might measure heaven and earth with a reed, but not with a growing reed.
Heretics (1905), 146-7.
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Modern bodybuilding is ritual, religion, sport, art, and science, awash in Western chemistry and mathematics. Defying nature, it surpasses it.
'Alice in Muscle Land,' Boston Globe (27 Jan 1991). Reprinted in Sex, Art, and American Culture (1992), 82.
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Modern mathematics, that most astounding of intellectual creations, has projected the mind's eye through infinite time and the mind's hand into boundless space.
'What Knowledge is of Most Worth?', Presidential address to the National Education Association, Denver, Colorado (9 Jul 1895). In Educational Review (Sep 1895), 10, 108.
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My two Jamaican cousins ... were studying engineering. 'That's where the money is,' Mom advised. ... I was to be an engineering major, despite my allergy to science and math. ... Those who preceded me at CCNY include the polio vaccine discoverer, Dr. Jonas Salk ... and eight Nobel Prize winners. ... In class, I stumbled through math, fumbled through physics, and did reasonably well in, and even enjoyed, geology. All I ever looked forward to was ROTC.
Autobiographical comments on his original reason for going to the City College of New York, where he shortly turned to his military career.
My American Journey (1996), 23-26. ROTC is the Reserve Officers' Training Corps (ROTC) school-based program of the U.S. military. From there, the self-described 'C-average student out of middling Morris High School' went on to become a four-star general.
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Nature's economy shall be the base for our own, for it is immutable, but ours is secondary. An economist without knowledge of nature is therefore like a physicist without knowledge of mathematics.
'Tankar om grunden til oeconomien', 1740, 406. Trans. Lisbet Koerner, Linnaeus: Nature and Nation (1999), 103.
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No discovery of mine has made, or is likely to make, directly or indirectly, for good or ill, the least difference to the amenity of the world.
In A Mathematician’s Apology (1941, reprint with Foreward by C.P. Snow 1992), 150.
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No old Men (excepting Dr. Wallis) love Mathematicks.
Comment made by Newton to William Whiston. Quoted in Richard Westfall, Never at Rest: A Biography of Isaac Newton (1980), 139.
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No one shall expel us from the paradise which Cantor has created for us.
Expressing the importance of Cantor's set theory in the development of mathematics.
In George Edward Martin, The Foundations of Geometry and the Non-Euclidean Plane (1982), 33.
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Non-standard analysis frequently simplifies substantially the proofs, not only of elementary theorems, but also of deep results. This is true, e.g., also for the proof of the existence of invariant subspaces for compact operators, disregarding the improvement of the result; and it is true in an even higher degree in other cases. This state of affairs should prevent a rather common misinterpretation of non-standard analysis, namely the idea that it is some kind of extravagance or fad of mathematical logicians. Nothing could be farther from the truth. Rather, there are good reasons to believe that non-standard analysis, in some version or other, will be the analysis of the future.
'Remark on Non-standard Analysis' (1974), in S. Feferman (ed.), Kurt Gödel Collected Works: Publications 1938-1974 (1990), Vol. 2, 311.
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Nothing enrages me more than when people criticize my criticism of school by telling me that schools are not just places to learn math and spelling, they are places where children learn a vaguely defined thing called socialization. I know. I think schools generally do an effective and terribly damaging job of teaching children to be infantile, dependent, intellectually dishonest, passive and disrespectful to their own developmental capacities. (1981)
Quoted in K.P. Yaday and Malti Sundram, Encyclopaedia Of Child And Primary Education Development, Vol. 2, 99.
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Now this supreme wisdom, united to goodness that is no less infinite, cannot but have chosen the best. For as a lesser evil is a kind of good, even so a lesser good is a kind of evil if it stands in the way of a greater good; and the would be something to correct in the actions of God if it were possible to the better. As in mathematics, when there is no maximum nor minimum, in short nothing distinguished, everything is done equally, or when that is not nothing at all is done: so it may be said likewise in respect of perfect wisdom, which is no less orderly than mathematics, that if there were not the best (optimum) among all possible worlds, God would not have produced any.
Theodicy: Essays on the Goodness of God and Freedom of Man and the Origin of Evil (1710), 128.
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Now, in the development of our knowledge of the workings of Nature out of the tremendously complex assemblage of phenomena presented to the scientific inquirer, mathematics plays in some respects a very limited, in others a very important part. As regards the limitations, it is merely necessary to refer to the sciences connected with living matter, and to the ologies generally, to see that the facts and their connections are too indistinctly known to render mathematical analysis practicable, to say nothing of the complexity.
From article 'Electro-magnetic Theory II', in The Electrician (16 Jan 1891), 26, No. 661, 331.
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Numbers written on restaurant checks [bills] within the confines of restaurants do not follow the same mathematical laws as numbers written on any other pieces of paper in any other parts of the Universe.
This single statement took the scientific world by storm. It completely revolutionized it. So many mathematical conferences got held in such good restaurants that many of the finest minds of a generation died of obesity and heart failure and the science of math was put back by years.
Life, the Universe and Everything (1982, 1995), 49.
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Of all regions of the earth none invites speculation more than that which lies beneath our feet, and in none is speculation more dangerous; yet, apart from speculation, it is little that we can say regarding the constitution of the interior of the earth. We know, with sufficient accuracy for most purposes, its size and shape: we know that its mean density is about 5½ times that of water, that the density must increase towards the centre, and that the temperature must be high, but beyond these facts little can be said to be known. Many theories of the earth have been propounded at different times: the central substance of the earth has been supposed to be fiery, fluid, solid, and gaseous in turn, till geologists have turned in despair from the subject, and become inclined to confine their attention to the outermost crust of the earth, leaving its centre as a playground for mathematicians.
'The Constitution of the Interior of the Earth, as Revealed by Earthquakes', Quarterly Journal of the Geological Society (1906), 62, 456.
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Of these austerer virtues the love of truth is the chief, and in mathematics, more than elsewhere, the love of truth may find encouragement for waning faith. Every great study is not only an end in itself, but also a means of creating and sustaining a lofty habit of mind; and this purpose should be kept always in view throughout the teaching and learning of mathematics.
Essay, 'The Study of Mathematics' (1902), collected in Philosophical Essays (1910), 73-74. Also collected in Mysticism and Logic: And Other Essays (1919), 73.
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On all levels primary, and secondary and undergraduate - mathematics is taught as an isolated subject with few, if any, ties to the real world. To students, mathematics appears to deal almost entirely with things whlch are of no concern at all to man.
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).
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On foundations we believe in the reality of mathematics, but of course, when philosophers attack us with their paradoxes, we rush to hide behind formalism and say 'mathematics is just a combination of meaningless symbols,'... Finally we are left in peace to go back to our mathematics and do it as we have always done, with the feeling each mathematician has that he is working with something real. The sensation is probably an illusion, but it is very convenient.
'The Work of Nicholas Bourbaki'American Mathematical Monthly (1970), 77, 134. In Carl C. Gaither, Alma E. Cavazos-Gaither, Mathematically Speaking: a Dictionary of Quotations (), 194.
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One of the chiefest triumphs of modern mathematics consists in having discovered what mathematics really is.
International Monthly (1901), 4, 84. In Robert Édoward Moritz, Memorabilia Mathematica (1914), 109.
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One of the first and foremost duties of the teacher is not to give his students the impression that mathematical problems have little connection with each other, and no connection at all with anything else. We have a natural opportunity to investigate the connections of a problem when looking back at its solution.
In How to Solve It: A New Aspect of Mathematical Method (2004), 15.
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Only dead mathematics can be taught where the attitude of competition prevails: living mathematics must always be a communal possession.
In Mary Everest Boole: Collected Works (1931), Vol. 3, 1008.
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Only mathematics and mathematical logic can say as little as the physicist means to say. (1931)
In The Scientific Outlook (1931, 2009), 57.
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Our present work sets forth mathematical principles of philosophy. For the basic problem of philosophy seems to be to discover the forces of nature from the phenomena of motions and then to demonstrate the other phenomena from these forces. It is to these ends that the general propositions in books 1 and 2 are directed, while in book 3 our explanation of the system of the world illustrates these propositions.
The Principia: Mathematical Principles of Natural Philosophy (1687), 3rd edition (1726), trans. I. B. Cohen and Anne Whitman (1999), Preface to the first edition, 382.
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Perhaps I can best describe my experience of doing mathematics in terms of a journey through a dark unexplored mansion. You enter the first room of the mansion and it's completely dark. You stumble around bumping into the furniture, but gradually you learn where each piece of furniture is. Finally, after six months or so, you find the light switch, you turn it on, and suddenly it''s all illuminated. You can see exactly where you were. Then you move into the next room and spend another six months in the dark. So each of these breakthroughs, while sometimes they're momentary, sometimes over a period of a day or two, they are the culmination of—and couldn't exist without—the many months of stumbling around in the dark that proceed them.
Quoted in interview for website for PBS TV Nova program, 'The Proof'.
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Perhaps the best reason for regarding mathematics as an art is not so much that it affords an outlet for creative activity as that it provides spiritual values. It puts man in touch with the highest aspirations and lofiest goals. It offers intellectual delight and the exultation of resolving the mysteries of the universe.
Mathematics: a Cultural Approach (1962), 671. Quoted in H. E. Hunter, The Divine Proportion (1970), 6.
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Perhaps we see equations as simple because they are easily expressed in terms of mathematical notation already invented at an earlier stage of development of the science, and thus what appears to us as elegance of description really reflects the interconnectedness of Nature's laws at different levels.
Nobel Banquet Speech (10 Dec 1969), in Wilhelm Odelberg (ed.),Les Prix Nobel en 1969 (1970).
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Perspective is a most subtle discovery in mathematical studies, for by means of lines it causes to appear distant that which is near, and large that which is small.
Attributed.
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Philosophy is a game with objectives and no rules. Mathematics is a game with rules and no objectives.
Anonymous
In Wieslaw Krawcewicz, Bindhyachal Rai, Calculus with Maple Labs (2003), 328. In this book, and also in Julian Havil, Nonplussed!: Mathematical Proof of Implausible Ideas? (2007), 68, the quote is attributed to Ian Ellis, but most sources vite it as Anonymous.
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Philosophy is written in this grand book, the universe, which stands continually open to our gaze. But the book cannot be understood unless one first learns to comprehend the language and read the letters in which it is composed. It is written in the language of mathematics, and its characters are triangles, circles, and other geometric figures without which it is humanly impossible to understand a single word of it; without these, one wanders about in a dark labyrinth.
'The Assayer' (1623), trans. Stillman Drake, Discoveries and Opinions of Galileo (1957), 237-8.
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Physics is NOT a body of indisputable and immutable Truth; it is a body of well-supported probable opinion only .... Physics can never prove things the way things are proved in mathematics, by eliminating ALL of the alternative possibilities. It is not possible to say what the alternative possibilities are.... Write down a number of 20 figures; if you multiply this by a number of, say, 30 figures, you would arrive at some enormous number (of either 49 or 50 figures). If you were to multiply the 30-figure number by the 20-figure number you would arrive at the same enormous 49- or 50-figure number, and you know this to be true without having to do the multiplying. This is the step you can never take in physics.
In Science is a Sacred Cow (1950), 68, 88, 179.
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Plenty of mathematicians, Hardy knew, could follow a step-by-step discursus unflaggingly—yet counted for nothing beside Ramanujan. Years later, he would contrive an informal scale of natural mathematical ability on which he assigned himself a 25 and Littlewood a 30. To David Hilbert, the most eminent mathematician of the day, he assigned an 80. To Ramanujan he gave 100.
In The Man who Knew Infinity: A Life of the Genius Ramanujan (1975), 226.
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Poincaré was a vigorous opponent of the theory that all mathematics can be rewritten in terms of the most elementary notions of classical logic; something more than logic, he believed, makes mathematics what it is.
Quoted in Thomson Gale, Online, 'Jules Henri Poincaré', World of Mathematics (2006).
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Professor [Max] Planck, of Berlin, the famous originator of the Quantum Theory, once remarked to me that in early life he had thought of studying economics, but had found it too difficult! Professor Planck could easily master the whole corpus of mathematical economics in a few days. He did not mean that! But the amalgam of logic and intuition and the wide knowledge of facts, most of which are not precise, which is required for economic interpretation in its highest form is, quite truly, overwhelmingly difficult for those whose gift mainly consists in the power to imagine and pursue to their furthest points the implications and prior conditions of comparatively simple facts which are known with a high degree of precision.
'Alfred Marshall: 1842-1924' (1924). In Geoffrey Keynes (ed.), Essays in Biography (1933), 191-2
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Pure mathematics consists entirely of such asseverations as that, if such and such is a proposition is true of anything, then such and such another propositions is true of that thing. It is essential not to discuss whether the first proposition is really true, and not to mention what the anything is of which it is supposed to be true. … If our hypothesis is about anything and not about some one or more particular things, then our deductions constititute mathematics. Thus mathematics may be defined as the the subject in which we never know what we are talking about, not whether what we are saying is true.
'Recent Work on the Principles of Mathematics', International Monthly (1901), 4, 84. In Robert Édouard Moritz, Memorabilia Mathematica (1914), 7.
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Putting together the mysteries of nature with the laws of mathematics, he dared to hope to be able to unlock the secrets of both with the same key.
Epitaph of René Descartes
Epitaph
In Peter Pešic, Labyrinth: A Search for the Hidden Meaning of Science (2001), 73.
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Questions that pertain to the foundations of mathematics, although treated by many in recent times, still lack a satisfactory solution. Ambiguity of language is philosophy's main source of problems. That is why it is of the utmost importance to examine attentively the very words we use.
Arithmetices Principia, (1889)
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Reductio ad absurdum, which Euclid loved so much, is one of a mathematician's finest weapons. It is a far finer gambit than any chess play: a chess player may offer the sacrifice of a pawn or even a piece, but a mathematician offers the game.
In A Mathematician's Apology (1940, reprint with Foreward by C.P. Snow 1992), 94.
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Remote from human passions, remote even from the pitiful facts of nature, the generations have gradually created an ordered cosmos [mathematics], where pure thought can dwell in its natural home...
'The Study of Mathematics', Philosophical Essays (1910), 73-74. In J. E. Creighton (Ed.), Evander Bradley McGilvary, 'Reviews of Books', The Philosophical Review (1911), Vol 20, 422.

Science and mathematics [are] much more compelling and exciting than the doctrines of pseudoscience, whose practitioners were condemned as early as the fifth century B.C. by the Ionian philosopher Heraclitus as 'night walkers, magicians, priests of Bacchus, priestesses of the wine-vat, mystery-mongers.' But science is more intricate and subtle, reveals a much richer universe, and powerfully evokes our sense of wonder. And it has the additional and important virtue—to whatever extent the word has any meaning—of being true.
Broca's Brain (1986), 76.
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Science is complex and chilling. The mathematical language of science is understood by very few. The vistas it presents are scary—an enormous universe ruled by chance and impersonal rules, empty and uncaring, ungraspable and vertiginous. How comfortable to turn instead to a small world, only a few thousand years old, and under God's personal; and immediate care; a world in which you are His peculiar concern.
The 'Threat' of Creationism. In Ashley Montagu (ed.), Science and Creationism (1984), 192.
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Since the seventeenth century, physical intuition has served as a vital source for mathematical porblems and methods. Recent trends and fashions have, however, weakened the connection between mathematics and physics; mathematicians, turning away from their roots of mathematics in intuition, have concentrated on refinement and emphasized the postulated side of mathematics, and at other times have overlooked the unity of their science with physics and other fields. In many cases, physicists have ceased to appreciate the attitudes of mathematicians. This rift is unquestionably a serious threat to science as a whole; the broad stream of scientific development may split into smaller and smaller rivulets and dry out. It seems therefore important to direct our efforts towards reuniting divergent trends by classifying the common features and interconnections of many distinct and diverse scientific facts.
In R. Courant and David Hilbert, Methods of Mathematical Physics (1937, 1989), Preface, v.
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Sir Arthur Eddington deduces religion from the fact that atoms do not obey the laws of mathematics. Sir James Jeans deduces it from the fact that they do.
In The Scientific Outlook (1931, 2009), 77.
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Some mathematics problems look simple, and you try them for a year or so, and then you try them for a hundred years, and it turns out that they're extremely hard to solve. There's no reason why these problems shouldn't be easy, and yet they turn out to be extremely intricate. [Fermat's] Last Theorem is the most beautiful example of this.
From interview for PBS website on the NOVA program, 'The Proof'.
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Some persons have contended that mathematics ought to be taught by making the illustrations obvious to the senses. Nothing can be more absurd or injurious: it ought to be our never-ceasing effort to make people think, not feel.
Seven Lectures on Shakespeare and Milton (1856) 24.
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Srinivasa Ramanujan was the strangest man in all of mathematics, probably in the entire history of science. He has been compared to a bursting supernova, illuminating the darkest, most profound corners of mathematics, before being tragically struck down by tuberculosis at the age of 33... Working in total isolation from the main currents of his field, he was able to rederive 100 years’ worth of Western mathematics on his own. The tragedy of his life is that much of his work was wasted rediscovering known mathematics.
In Hyperspace:A Scientific Odyssey through Parallel Universes, Time Warps, and the Tenth Dimension (1994), 172.
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Standard mathematics has recently been rendered obsolete by the discovery that for years we have been writing the numeral five backward. This has led to reevaluation of counting as a method of getting from one to ten. Students are taught advanced concepts of Boolean algebra, and formerly unsolvable equations are dealt with by threats of reprisals.
Getting Even (1978), 44.
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Suppose we loosely define a religion as any discipline whose foundations rest on an element of faith, irrespective of any element of reason which may be present. Quantum mechanics for example would be a religion under this definition. But mathematics would hold the unique position of being the only branch of theology possessing a rigorous demonstration of the fact that it should be so classified.
Concluding remark in 'Consistency and Completeness—A Résumé', The American Mathematical Monthly (May 1956), 63, No.5, 305.
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Taking mathematics from the beginning of the world to the time when Newton lived, what he had done was much the better half.
As quoted in Edmund Fillingham King, A Biographical Sketch of Sir Isaac Newton (1858), 97, stating this was Leibniz's reply “when asked at the royal table in Berlin his opinion of Newton.” No source citation was given, although all the next quotes that followed had footnotes. The lack of citation leaves the accuracy of the quote unverified. If you know a primary source, please contact the Webmaster.
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The Mathematics are Friends to Religion, inasmuch as they charm the Passions, restrain the Impetuosity of the Imagination, and purge the Mind from Error and Prejudice. Vice is Error, Confusion, and false Reasoning; and all Truth is more or less opposite to it. Besides, Mathematical Studies may serve for a pleasant Entertainment for those Hours which young Men are apt to throw away upon their Vices; the Delightfulness of them being such as to make Solitude not only easy, but desirable.
On the Usefulness of Mathematical Learning, (3rd Ed., 1745) 8.
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The advancement and perfection of mathematics are intimately connected with the prosperity of the State.
Correspondance de Napoléon, t. 24 (1868), 112. In Robert Édouard Moritz, Memorabilia Mathematica; Or, The Philomath's Quotation-book (1914), 42.

The average gambler will say “The player who stakes his whole fortune on a single play is a fool, and the science of mathematics can not prove him to be otherwise.” The reply is obvious: “The science of mathematics never attempts the impossible, it merely shows that other players are greater fools.”
[Concluding remarks to his mathematical proof, with certain assumptions, that a gambler's best strategy would be to always make his largest stake on his first play.]
In 'Gambler's Ruin', Annals of Mathematics (Jul 1909), 2nd Series, 10, No. 4, 189. This is also seen, without primary source, quoted as “It is true that a man who does this is a fool. I have only proved that a man who does anything else is an even bigger fool,” in Harold Eves, Return to Mathematical Circles (1988), 39.
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The British Mathematical Colloquium consists of three days of mathematics with no dogs and no wives.
Quoted in Des MacHale, Comic Sections (1993)
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The computer is important, but not to mathematics.
From interview (1981) with Donald J. Albers. In John H. Ewing and Frederick W. Gehring, Paul Halmos Celebrating 50 Years of Mathematics (1991), 3.
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The conception of objective reality ... has thus evaporated ... into the transparent clarity of mathematics that represents no longer the behavior of particles but rather our knowledge of this behavior.
In 'The Representation of Nature in Contemporary Physics', Daedalus (1958), 87, 95-108. As cited in Karl Popper, Quantum Theory and the Schism in Physics (1992), 85.
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The deep study of nature is the most fruitful source of mathematical discoveries. By offering to research a definite end, this study has the advantage of excluding vague questions and useless calculations; besides it is a sure means of forming analysis itself and of discovering the elements which it most concerns us to know, and which natural science ought always to conserve.
Théorie Analytique de la Chaleur, Discours Préliminaire. Translation as in Robert Édouard Moritz, Memorabilia Mathematica; Or, The Philomath's Quotation-book (1914), 89.
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The definition of a good mathematical problem is the mathematics it generates rather than the problem itself.
From interview for PBS website on the NOVA program, 'The Proof'.
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The development of mathematics toward greater precision has led, as is well known, to the formalization of large tracts of it, so that one can prove any theorem using nothing but a few mechanical rules... One might therefore conjecture that these axioms and rules of inference are sufficient to decide any mathematical question that can at all be formally expressed in these systems. It will be shown below that this is not the case, that on the contrary there are in the two systems mentioned relatively simple problems in the theory of integers that cannot be decided on the basis of the axioms.
'On Formally Undecidable Propositions of Principia Mathematica and Related Systems I' (193 1), in S. Feferman (ed.), Kurt Gödel Collected Works: Publications 1929-1936 (1986), Vol. I, 145.
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The elegance of a mathematical theorem is directly proportional to the number of independent ideas one can see in the theorem and inversely proportional to the effort it takes to see them.
In Mathematical Discovery: On Understanding, Learning, and Teaching Problem Solving (1981). As cited, with no more details, in Yi Ma, An Invitation to 3-D Vision (2004), 228.
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The end of the eighteenth and the beginning of the nineteenth century were remarkable for the small amount of scientific movement going on in this country, especially in its more exact departments. ... Mathematics were at the last gasp, and Astronomy nearly so—I mean in those members of its frame which depend upon precise measurement and systematic calculation. The chilling torpor of routine had begun to spread itself over all those branches of Science which wanted the excitement of experimental research.
Quoted in Sophia Elizabeth De Morgan, Memoir of Augustus De Morgan (1882), 41
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The essence of mathematics lies precisely in its freedom.
Gesammelte Abhandlungen (1932), 182, trans. Ivor Grattan-Guinness. Also, givened as: 'The essence of mathematics lies in its freedom,' in Mathematische Annalen, 21, 564. In Robert Édouard Moritz, Memorabilia Mathematica (1914), 12.

The essential fact is that all the pictures which science now draws of nature, and which alone seem capable of according with observational facts, are mathematical pictures.
In The Mysterious Universe (1930), 150.
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The fact is that there are few more “popular” subjects than mathematics. Most people have some appreciation of mathematics, just as most people can enjoy a pleasant tune; and there are probably more people really interested in mathematics than in music. Appearances may suggest the contrary, but there are easy explanations. Music can be used to stimulate mass emotion, while mathematics cannot; and musical incapacity is recognized (no doubt rightly) as mildly discreditable, whereas most people are so frightened of the name of mathematics that they are ready, quite unaffectedly, to exaggerate their own mathematical stupidity.
In A Mathematician's Apology (1940, reprint with Foreward by C.P. Snow 1992), 86.
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The fact that all Mathematics is Symbolic Logic is one of the greatest discoveries of our age; and when this fact has been established, the remainder of the principles of mathematics consists of the analysis of Symbolic Logic itself.
In Bertrand Russell, The Principles of Mathematics (1903), 5.
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The faith of scientists in the power and truth of mathematics is so implicit that their work has gradually become less and less observation, and more and more calculation. The promiscuous collection and tabulation of data have given way to a process of assigning possible meanings, merely supposed real entities, to mathematical terms, working out the logical results, and then staging certain crucial experiments to check the hypothesis against the actual empirical results. But the facts which are accepted by virtue of these tests are not actually observed at all. With the advance of mathematical technique in physics, the tangible results of experiment have become less and less spectacular; on the other hand, their significance has grown in inverse proportion. The men in the laboratory have departed so far from the old forms of experimentation—typified by Galileo's weights and Franklin's kite—that they cannot be said to observe the actual objects of their curiosity at all; instead, they are watching index needles, revolving drums, and sensitive plates. No psychology of 'association' of sense-experiences can relate these data to the objects they signify, for in most cases the objects have never been experienced. Observation has become almost entirely indirect; and readings take the place of genuine witness.
Philosophy in a New Key; A Study in Inverse the Symbolism of Reason, Rite, and Art (1942), 19-20.
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The first nonabsolute number is the number of people for whom the table is reserved. This will vary during the course of the first three telephone calls to the restaurant, and then bear no apparent relation to the number of people who actually turn up, or to the number of people who subsequently join them after the show/match/party/gig, or to the number of people who leave when they see who else has turned up.
The second nonabsolute number is the given time of arrival, which is now known to be one of the most bizarre of mathematical concepts, a recipriversexcluson, a number whose existence can only be defined as being anything other than itself. In other words, the given time of arrival is the one moment of time at which it is impossible that any member of the party will arrive. Recipriversexclusons now play a vital part in many branches of math, including statistics and accountancy and also form the basic equations used to engineer the Somebody Else's Problem field.
The third and most mysterious piece of nonabsoluteness of all lies in the relationship between the number of items on the check [bill], the cost of each item, the number of people at the table and what they are each prepared to pay for. (The number of people who have actually brought any money is only a subphenomenon of this field.)
Life, the Universe and Everything (1982, 1995), 47-48.
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The formulation of a problem is often more essential than its solution, which may be merely a matter of mathematical or experimental skill. To raise new questions, new possibilities, to regard old problems from a new angle requires creative imagination and marks real advances in science.
In Albert Einstein and Léopold Infeld, The Evolution of Physics: The Growth of Ideas from Early Concepts to Relativity and Quanta (1938, 1966), 92.
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The foundations of population genetics were laid chiefly by mathematical deduction from basic premises contained in the works of Mendel and Morgan and their followers. Haldane, Wright, and Fisher are the pioneers of population genetics whose main research equipment was paper and ink rather than microscopes, experimental fields, Drosophila bottles, or mouse cages. Theirs is theoretical biology at its best, and it has provided a guiding light for rigorous quantitative experimentation and observation.
'A Review of Some Fundamental Concepts and Problems of Population Genetics', Cold Spring Harbor Symposia on Quantitative Biology, 1955, 20, 13-14.
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The fundamental laws necessary for the mathematical treatment of a large part of physics and the whole of chemistry are thus completely known, and the difficulty lies only in the fact that application of these laws leads to equations that are too complex to be solved.
'Quantum Mechanics of Many-Electron Systems', Proceedings of the Royal Society (1929), A, 123, 714-733. Quoted in Steven M. Bachrach, Computational Organic Chemistry, Preface, xiii.
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The gambling reasoner is incorrigible; if he would but take to the squaring of the circle, what a load of misery would be saved.
Quoted in D. MacHale, Comic Sections (1993).

The general mental qualification necessary for scientific advancement is that which is usually denominated “common sense,” though added to this, imagination, induction, and trained logic, either of common language or of mathematics, are important adjuncts.
From presidential address (24 Nov 1877) to the Philosophical Society of Washington. As cited by L.A. Bauer in his retiring president address (5 Dec 1908), 'The Instruments and Methods of Research', published in Philosophical Society of Washington Bulletin, 15, 103. Reprinted in William Crookes (ed.) The Chemical News and Journal of Industrial Science (30 Jul 1909), 59.
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The genius of Laplace was a perfect sledge hammer in bursting purely mathematical obstacles; but, like that useful instrument, it gave neither finish nor beauty to the results. In truth, in truism if the reader please, Laplace was neither Lagrange nor Euler, as every student is made to feel. The second is power and symmetry, the third power and simplicity; the first is power without either symmetry or simplicity. But, nevertheless, Laplace never attempted investigation of a subject without leaving upon it the marks of difficulties conquered: sometimes clumsily, sometimes indirectly, always without minuteness of design or arrangement of detail; but still, his end is obtained and the difficulty is conquered.
'Review of "Théorie Analytique des Probabilites" par M. le Marquis de Laplace, 3eme edition. Paris. 1820', Dublin Review (1837), 2, 348.
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The Golden Gate Bridge is a giant moving math problem.
Quoted on web site for PBS American Experience episode for 'Golden Gate Bridge.'
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The Good Spirit never cared for the colleges, and though all men and boys were now drilled in Greek, Latin, and Mathematics, it had quite left these shells high on the beach, and was creating and feeding other matters [science] at other ends of the world.
The Prose Works of Ralph Waldo Emerson (1870), 553.
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The great problem of today is, how to subject all physical phenomena to dynamical laws. With all the experimental devices, and all the mathematical appliances of this generation, the human mind has been baffled in its attempts to construct a universal science of physics.
'President's Address', Proceedings of the American Association for the Advancement of Science (1874), 23, 34-5.
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The imaginary expression √(-a) and the negative expression -b, have this resemblance, that either of them occurring as the solution of a problem indicates some inconsistency or absurdity. As far as real meaning is concerned, both are imaginary, since 0 - a is as inconceivable as √(-a).

The intensity and quantity of polemical literature on scientific problems frequently varies inversely as the number of direct observations on which the discussions are based: the number and variety of theories concerning a subject thus often form a coefficient of our ignorance. Beyond the superficial observations, direct and indirect, made by geologists, not extending below about one two-hundredth of the Earth's radius, we have to trust to the deductions of mathematicians for our ideas regarding the interior of the Earth; and they have provided us successively with every permutation and combination possible of the three physical states of matter—solid, liquid, and gaseous.
'Address delivered by the President of Section [Geology] at Sydney (Friday, Aug 21), Report of the Eighty-Fourth Meeting of the British Association for the Advancement of Science: Australia 1914, 1915, 345.
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The life and soul of science is its practical application, and just as the great advances in mathematics have been made through the desire of discovering the solution of problems which were of a highly practical kind in mathematical science, so in physical science many of the greatest advances that have been made from the beginning of the world to the present time have been made in the earnest desire to turn the knowledge of the properties of matter to some purpose useful to mankind.
From 'Electrical Units of Measurement', a lecture delivered at the Institution of Civil Engineers, London (3 May 1883), Popular Lectures and Addresses Vol. 1 (1891), 86-87.
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The manuscript looks chaotic, even by mathematics standards.
[About newly-found late work of Srinivasa Ramanujan.]
Quoted in John Noble Wilford, 'Mathematician's Final Equations Praised', New York Times (9 Jun 1981), C1.
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The mathematical framework of quantum theory has passed countless successful tests and is now universally accepted as a consistent and accurate description of all atomic phenomena. The verbal interpretation, on the other hand – i.e., the metaphysics of quantum theory – is on far less solid ground. In fact, in more than forty years physicists have not been able to provide a clear metaphysical model.
In The Tao of Physics (1975), 132.
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The mathematician's patterns, like the painter's or the poet's must be beautiful; the ideas, like the colours or the words must fit together in a harmonious way. 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.

The mathematics clearly called for a set of underlying elementary objects—at that time we needed three types of them—elementary objects that could be combined three at a time in different ways to make all the heavy particles we knew. ... I needed a name for them and called them quarks, after the taunting cry of the gulls, “Three quarks for Muster mark,” from Finnegan's Wake by the Irish writer James Joyce.
From asppearance in the BBC-TV program written by Nigel Calder, 'The Key to the Universe,' (27 Jan 1977). As cited in Arthur Lewis Caso, 'The Production of New Scientific Terms', American Speech (Summer 1980), 55, No. 2, 101-102.
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The mathematics of cooperation of men and tools is interesting. Separated men trying their individual experiments contribute in proportion to their numbers and their work may be called mathematically additive. The effect of a single piece of apparatus given to one man is also additive only, but when a group of men are cooperating, as distinct from merely operating, their work raises with some higher power of the number than the first power. It approaches the square for two men and the cube for three. Two men cooperating with two different pieces of apparatus, say a special furnace and a pyrometer or a hydraulic press and new chemical substances, are more powerful than their arithmetical sum. These facts doubtless assist as assets of a research laboratory.
Quoted from a speech delivered at the fiftieth anniversary of granting of M.I.T's charter, in Guy Suits, 'Willis Rodney Whitney', National Academy of Sciences, Biographical Memoirs (1960), 352.
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The Mathematics, I say, which effectually exercises, not vainly deludes or vexatiously torments studious Minds with obscure Subtilties, perplexed Difficulties, or contentious Disquisitions; which overcomes without Opposition, triumphs without Pomp, compels without Force, and rules absolutely without Loss of Liberty; which does not privately over-reach a weak Faith, but openly assaults an armed Reason, obtains a total Victory, and puts on inevitable Chains; whose Words are so many Oracles, and Works as many Miracles; which blabs out nothing rashly, nor designs anything from the Purpose, but plainly demonstrates and readily performs all Things within its Verge; which obtrudes no false Shadow of Science, but the very Science itself, the Mind firmly adhering to it, as soon as possessed of it, and can never after desert it of its own Accord, or be deprived of it by any Force of others: Lastly the Mathematics, which depends upon Principles clear to the Mind, and agreeable to Experience; which draws certain Conclusions, instructs by profitable Rules, unfolds pleasant Questions; and produces wonderful Effects; which is the fruitful Parent of, I had almost said all, Arts, the unshaken Foundation of Sciences, and the plentiful Fountain of Advantage to human Affairs.
Address to the University of Cambridge upon being elected Lucasian Professor of Mathematics (14 Mar 1664). In Mathematical Lectures (1734), xxviii.
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The mere man of pleasure is miserable in old age, and the mere drudge in business is but little better, whereas, natural philosophy, mathematical and mechanical science, are a continual source of tranquil pleasure, and in spite of the gloomy dogmas of priests and of superstition, the study of these things is the true theology; it teaches man to know and admire the Creator, for the principles of science are in the creation, and are unchangeable and of divine origin.
Age of Reason (1818), 35.
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The more I study the things of the mind the more mathematical I find them. In them as in mathematics it is a question of quantities; they must be treated with precision. I have never had more satisfaction than in proving this in the realms of art, politics and history.
Notes made after the completion of the third chapter of Vol. 3 of La Rivolution, 22 April 1883. In E. Sparvel-Bayly (trans.), Life and Letters of H. Taine (1902-1908), Vol. 3, 239.
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The most difficult problem in mathematics is to make the date of a woman's birth agree with her present age.
Anonymous
In Evan Esar, 20,000 Quips and Quotes, 22.
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The moving power of mathematical invention is not reasoning but imagination.
Quoted in Robert Perceval Graves, Life of Sir W. R. Hamilton, Vol. 3 (1889), 219.
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The new mathematics is a sort of supplement to language, affording a means of thought about form and quantity and a means of expression, more exact, compact, and ready than ordinary language. The great body of physical science, a great deal of the essential facts of financial science, and endless social and political problems are only accessible and only thinkable to those who have had a sound training in mathematical analysis, and the time may not be very remote when it will be understood that for complete initiation as an efficient citizen of the great complex world-wide States that are now developing, it is as necessary to be able to compute, to think in averages and maxima and minima, as it is now to be able to read and write.
Mankind in the Making (1903), 204.
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The only place where a dollar is still worth one hundred cents today is in the problems in an arithmetic book.
Anonymous
In Evan Esar, 20,000 Quips and Quotes, 509.
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The power of mathematics rests on its evasion of all unnecessary thought and on its wonderful saving of mental operations.
Quoted in Freeman Dyson, 'Mathematic; in the Physical Sciences', Scientific American (Sep 1964), 211, No. 3, 133.
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The presentation of mathematics where you start with definitions, for example, is simply wrong. Definitions aren't the places where things start. Mathematics starts with ideas and general concepts, and then definitions are isolated from concepts. Definitions occur somewhere in the middle of a progression or the development of a mathematical concept. The same thing applies to theorems and other icons of mathematical progress. They occur in the middle of a progression of how we explore the unknown.
Interview for website of the Mathematical Association of America.
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The principles of logic and mathematics are true universally simply because we never allow them to be anything else. And the reason for this is that we cannot abandon them without contradicting ourselves, without sinning against the rules which govern the use of language, and so making our utterances self-stultifying. In other words, the truths of logic and mathematics are analytic propositions or tautologies.
Language, Truth and Logic (1960), 77.
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The progress of mathematics can be viewed as progress from the infinite to the finite.
In 'A Mathematician's Gossip', Indiscrete Thoughts, (2008), 214.
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The proof given by Wright, that non-adaptive differentiation will occur in small populations owing to “drift,” or the chance fixation of some new mutation or recombination, is one of the most important results of mathematical analysis applied to the facts of neo-mendelism. It gives accident as well as adaptation a place in evolution, and at one stroke explains many facts which puzzled earlier selectionists, notably the much greater degree of divergence shown by island than mainland forms, by forms in isolated lakes than in continuous river-systems.
Evolution: The Modern Synthesis (1942), 199-200.
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The Reader may here observe the Force of Numbers, which can be successfully applied, even to those things, which one would imagine are subject to no Rules. There are very few things which we know, which are not capable of being reduc'd to a Mathematical Reasoning, and when they cannot, it's a sign our Knowledge of them is very small and confus'd; and where a mathematical reasoning can be had, it's as great folly to make use of any other, as to grope for a thing in the dark when you have a Candle standing by you.
Of the Laws of Chance, or, a Method of the Hazards of Game (1692), Preface.
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The real trouble with this world of ours is not that it is an unreasonable world, nor even that it is a reasonable one. The commonest kind of trouble is that it is nearly reasonable, but not quite. … It looks just a little more mathematical and regular than it is; its exactitude is obvious, but its inexactitude is hidden; its wilderness lies in wait.
In Orthodoxy (1909), 148.
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The reasoning of mathematics is a type of perfect reasoning.
Common Sense in Education and Teaching (1905), 222.
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The results of mathematics are seldom directly applied; it is the definitions that are really useful. Once you learn the concept of a differential equation, you see differential equations all over, no matter what you do. This you cannot see unless you take a course in abstract differential equations. What applies is the cultural background you get from a course in differential equations, not the specific theorems. If you want to learn French, you have to live the life of France, not just memorize thousands of words. If you want to apply mathematics, you have to live the life of differential equations. When you live this life, you can then go back to molecular biology with a new set of eyes that will see things you could not otherwise see.
In 'A Mathematician's Gossip', Indiscrete Thoughts (2008), 213.
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The rudest numerical scales, such as that by which the mineralogists distinguish different degrees of hardness, are found useful. The mere counting of pistils and stamens sufficed to bring botany out of total chaos into some kind of form. It is not, however, so much from counting as from measuring, not so much from the conception of number as from that of continuous quantity, that the advantage of mathematical treatment comes. Number, after all, only serves to pin us down to a precision in our thoughts which, however beneficial, can seldom lead to lofty conceptions, and frequently descend to pettiness.
On the Doctrine of Chances, with Later Reflections (1878), 61-2.
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The science of government is my duty. ... I must study politics and war that my sons may have liberty to study mathematics and philosophy. My sons ought to study mathematics and philosophy, geography, natural history, naval architecture, navigation, commerce, and agriculture, in order to give their children a right to study painting, poetry, music, architecture, statuary, tapestry, and porcelain.
Letter to Abigail Adams, (1780). In John Adams and Charles Francis Adams, Letters of John Adams, Addressed to His Wife (1841), 68.
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The science of mathematics performs more than it promises, but the science of metaphysics promises more than it performs.
Reflection 342, Lacon: Many Things in Few Words (1820), 161-162.
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The science of mathematics presents the most brilliant example of how pure reason may successfully enlarge its domain without the aid of experience
In Joey Green, Philosophy on the Go (2007), 128
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The sciences do not try to explain, they hardly even try to interpret, they mainly make models. By a model is meant a mathematical construct which, with the addition of certain verbal interpretations, describes observed phenomena. The justification of such a mathematical construct is solely and precisely that it is expected to work—that is, correctly to describe phenomena from a reasonably wide area.
'Method in the Physical Sciences', in The Unity of Knowledge, editted by L. Leary (1955), 158. Reprinted in John Von Neumann, F. Bródy (ed.) and Tibor Vámos (ed.), The Neumann Compendium (2000), 628.
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The sciences, even the best,—mathematics and astronomy,—are like sportsmen, who seize whatever prey offers, even without being able to make any use of it.
Emerson's Complete Works (1883),62.
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The second [argument about motion] is the so-called Achilles, and it amounts to this, that in a race the quickest runner can never overtake the slowest, since the pursuer must first reach the point whence the pursued started, so that the slower must always hold a lead.
Statement of the Achilles and the Tortoise paradox in the relation of the discrete to the continuous.; perhaps the earliest example of the reductio ad absurdum method of proof.
Zeno
Aristotle, Physics, 239b, 14-6. In Jonathan Barnes (ed.), The Complete Works of Aristotle (1984), Vol. 1, 404.
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The smallest particles of matter were said [by Plato] to be right-angled triangles which, after combining in pairs, ... joined together into the regular bodies of solid geometry; cubes, tetrahedrons, octahedrons and icosahedrons. These four bodies were said to be the building blocks of the four elements, earth, fire, air and water ... [The] whole thing seemed to be wild speculation. ... Even so, I was enthralled by the idea that the smallest particles of matter must reduce to some mathematical form ... The most important result of it all, perhaps, was the conviction that, in order to interpret the material world we need to know something about its smallest parts.
[Recalling how as a teenager at school, he found Plato's Timaeus to be a memorable poetic and beautiful view of atoms.]
In Werner Heisenberg and A.J. Pomerans (trans.) The Physicist's Conception of Nature (1958), 58-59. Quoted in Jagdish Mehra and Helmut Rechenberg, The Historical Development of Quantum Theory (2001), Vol. 2, 12. Cited in Mauro Dardo, Nobel Laureates and Twentieth-Century Physics (2004), 178.
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The steady progress of physics requires for its theoretical formulation a mathematics which get continually more advanced. ... it was expected that mathematics would get more and more complicated, but would rest on a permanent basis of axioms and definitions, while actually the modern physical developments have required a mathematics that continually shifts its foundation and gets more abstract. Non-euclidean geometry and noncommutative algebra, which were at one time were considered to be purely fictions of the mind and pastimes of logical thinkers, have now been found to be very necessary for the description of general facts of the physical world. It seems likely that this process of increasing abstraction will continue in the future and the advance in physics is to be associated with continual modification and generalisation of the axioms at the base of mathematics rather than with a logical development of any one mathematical scheme on a fixed foundation.
Introduction to a paper on magnetic monopoles, 'Quantised singularities in the electromagnetic field', Proceedings of the Royal Society of Lonndon (1931), A, 133 60. In Helge Kragh, Dirac: a Scientific Biography (1990), 208.
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The study of geometry is a petty and idle exercise of the mind, if it is applied to no larger system than the starry one. Mathematics should be mixed not only with physics but with ethics; that is mixed mathematics.
A Week on the Concord and Merrimack Rivers (1873), 383.
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The study of mathematics is, if an unprofitable, a perfectly harmless and innocent occupation.
From Inaugural Lecture, Oxford (1920). Recalled in A Mathematician’s Apology (1940, 1967), 74.
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The study of the mathematics is like climbing up a steep and craggy mountain; when once you reach the top, it fully recompenses your trouble, by opening a fine, clear, and extensive prospect.
Anonymous
In Tryon Edwards (ed.), A Dictionary of Thoughts (1908), 337.
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The study of the mathematics, like the Nile, begins in minuteness, but ends in magnificence.
Reflection 342, in Lacon: or Many things in Few Words; Addressed to Those Who Think (1820), Vol. 1, 162.

The theory of probability is the only mathematical tool available to help map the unknown and the uncontrollable. It is fortunate that this tool, while tricky, is extraordinarily powerful and convenient.
The Fractal Geometry of Nature (1977), 201.
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The traditional mathematics professor of the popular legend is absentminded. He usually appears in public with a lost umbrella in each hand. He prefers to face a blackboard and to turn his back on the class. He writes a, he says b, he means c, but it should be d. Some of his sayings are handed down from generation to generation:
“In order to solve this differential equation you look at it till a solution occurs to you.”
“This principle is so perfectly general that no particular application of it is possible.”
“Geometry is the science of correct reasoning on incorrect figures.”
“My method to overcome a difficulty is to go round it.”
“What is the difference between method and device? A method is a device which you used twice.”
In How to Solve It: A New Aspect of Mathematical Method (2004), 208.
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The transfinite numbers are in a sense the new irrationalities [ ... they] stand or fall with the finite irrational numbers.
Gesammelte Abhandlungen (1932),395, trans. Ivor Grattan-Guinness.
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The true spirit of delight, the exaltation, the sense of being more than man, which is the touchstone of highest excellence, is to be found in mathematics as surely as in poetry.
Essay, 'The Study of Mathematics' (1902), collected in Philosophical Essays (1910), 73-74. Also collected in Mysticism and Logic: And Other Essays (1919), 60.
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The unreasonable efficiency of mathematics in science is a gift we neither understand nor deserve.
Quoted in Robert J. Scully, The Demon and the Quantum (2007), 191.
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The usefulness of mathematics in furthering the sciences is commonly acknowledged: but outside the ranks of the experts there is little inquiry into its nature and purpose as a deliberate human activity. Doubtless this is due to the inevitable drawback that mathematical study is saturated with technicalities from beginning to end.
Opening remark in Preface to The Great Mathmaticians (1929). Collected in James Newman, The World of Mathematics (1956), Vol. 1, 75.
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The validity of mathematical propositions is independent of the actual world—the world of existing subject-matters—is logically prior to it, and would remain unaffected were it to vanish from being. Mathematical propositions, if true, are eternal verities.
In The Pastures of Wonder: The Realm of Mathematics and the Realm of Science (1929), 99.
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The word “mathematics” is a Greek word and, by origin, it means “something that has been learned or understood,” or perhaps “acquired knowledge,” or perhaps even, somewhat against grammar, “acquirable knowledge,” that is, “learnable knowledge,” that is, “knowledge acquirable by learning.”
'Why Mathematics Grows', Journal of the History of Ideas (Jan-Mar 1965), 26, No. 1, 4. In Salomon Bochner and Robert Clifford Gunning (ed.) Collected Papers of Salomon Bochner (1992), Vol. 4, 192.
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The works of Lavoisier and his associates operated upon many of us at that time like the Sun's rising after a night of moonshine: but Chemistry is now betrothed to the Mathematics, and is in consequence grown somewhat shy of her former admirers.
In Luke Howard, Johann Wolfgang von Goethe and D.F.S. Scott (ed.), Luke Howard (1772-1864): His Correspondence with Goethe and his Continental Journey of 1816(1976), 2.
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The world is anxious to admire that apex and culmination of modern mathematics: a theorem so perfectly general that no particular application of it is feasible.
In 'A Story With a Moral', Mathematical Gazette (Jun 1973), 57, No. 400, 87.
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The world of mathematics and theoretical physics is hierarchical. That was my first exposure to it. There's a limit beyond which one cannot progress. The differences between the limiting abilities of those on successively higher steps of the pyramid are enormous. I have not seen described anywhere the shock a talented man experiences when he finds, late in his academic life, that there are others enormously more talented than he. I have personally seen more tears shed by grown men and women over this discovery than I would have believed possible. Most of those men and women shift to fields where they can compete on more equal terms
Alvarez: Adventures of a Physicist (1987), 20.
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The world of mathematics, which you condemn, is really a beautiful world; it has nothing to do with life and death and human sordidness, but is eternal, cold and passionless. To me, pure, mathematics is one of the highest forms of art; it has a sublimity quite special to itself, and an immense dignity derived, from the fact that its world is exempt I, from change and time. I am quite serious in this. The only difficulty is that none but mathematicians can enter this enchanted region, and they hardly ever have a sense of beauty. And mathematics is the only thing we know of that is capable of perfection; in thinking about it we become Gods.
Letter to Helen Thomas (30 Dec 1901). Quoted in Nicholas Griffin (ed.), The Selected Letters of Bertrand Russell (1992), Vol. 1, 224.
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Theorems are not to mathematics what successful courses are to a meal.
In Rota's 'Introduction' written (1980) to preface Philip J. Davis and Reuben Hersh, The Mathematical Experience (1981, 2012), xxii-xxiii.
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Theoretical physicists accept the need for mathematical beauty as an act of faith... For example, the main reason why the theory of relativity is so universally accepted is its mathematical beauty.
'Methods in Theoretical Physics', From A Life of Physics: Evening Lectures at the International Centre for Theoretical Physics, Trieste, Italy. A Special Supplement of the IAEA Bulletin (1968), 22.
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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 ... He who is ignorant of this [mathematics] cannot know the other sciences nor the affairs of this world.
Opus Majus [1266-1268], Part IV, distinction I, chapter I, trans. R. B. Burke, The Opus Majus of Roger Bacon (1928), Vol. I, 116.

There are many arts and sciences of which a miner should not be ignorant. First there is Philosophy, that he may discern the origin, cause, and nature of subterranean things; for then he will be able to dig out the veins easily and advantageously, and to obtain more abundant results from his mining. Secondly there is Medicine, that he may be able to look after his diggers and other workman ... Thirdly follows astronomy, that he may know the divisions of the heavens and from them judge the directions of the veins. Fourthly, there is the science of Surveying that he may be able to estimate how deep a shaft should be sunk ... Fifthly, his knowledge of Arithmetical Science should be such that he may calculate the cost to be incurred in the machinery and the working of the mine. Sixthly, his learning must comprise Architecture, that he himself may construct the various machines and timber work required underground ... Next, he must have knowledge of Drawing, that he can draw plans of his machinery. Lastly, there is the Law, especially that dealing with metals, that he may claim his own rights, that he may undertake the duty of giving others his opinion on legal matters, that he may not take another man's property and so make trouble for himself, and that he may fulfil his obligations to others according to the law.
De Re Metallica (1556), trans. H. C. and L. H. Hoover (1950), 3-4.
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There are several kinds of truths, and it is customary to place in the first order mathematical truths, which are, however, only truths of definition. These definitions rest upon simple, but abstract, suppositions, and all truths in this category are only constructed, but abstract, consequences of these definitions ... Physical truths, to the contrary, are in no way arbitrary, and do not depend on us.
'Premier Discours: De la Manière d'Étudier et de Traiter l'Histoire naturelle', Histoire Naturelle, Generale et Particulière, Avec la Description du Cabinet du Roi (1749), Vol. I, 53-4. Trans. Phillip R. Sloan.
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There are, at present, fundamental problems in theoretical physics … the solution of which … will presumably require a more drastic revision of our fundmental concepts than any that have gone before. Quite likely, these changes will be so great that it will be beyond the power of human intelligence to get the necessary new ideas by direct attempts to formulate the experimental data in mathematical terms. The theoretical worker in the future will, therefore, have to proceed in a more direct way. The most powerful method of advance that can be suggested at present is to employ all the resources of pure mathematics in attempts to perfect and generalize the mathematical formalism that forms the existing basis of theoretical physics, and after each success in this direction, to try to interpret the new mathematical features in terms of physical entities.
At age 28.
Proceedings of the Royal Society (1931), A133, 60. 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), 109.
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There have been many authorities who have asserted that the basis of science lies in counting or measuring, i.e. in the use of mathematics. Neither counting nor measuring can however be the most fundamental processes in our study of the material universe—before you can do either to any purpose you must first select what you propose to count or measure, which presupposes a classification.
Classification and Biology (1970), 2.
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There is a science which investigates being as being and the attributes which belong to this in virtue of its own nature. Now this is not the same as any of the so-called special sciences; for none of these treats universally of being as being. They cut off a part of being and investigate the attribute of this part; this is what the mathematical sciences for instance do. Now since we are seeking the first principles and the highest causes, clearly there must be some thing to which these belong in virtue of its own nature. If then those who sought the elements of existing things were seeking these same principles, it is necessary that the elements must be elements of being not by accident but just because it is being. Therefore it is of being as being that we also must grasp the first causes.
Aristotle
'Book Gamma (1003a17-1011b23' in Metaphysics, trans. W.D. Ross (1924). Excerpt 'Being Qua Being', in Joseph Margolis and Jacques Catudal, The Quarrel between Invariance and Flux (2001), 18-19.
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There is a strange disparity between the sciences of inert matter and those of life. Astronomy, mechanics, and physics are based on concepts which can be expressed, tersely and elegantly, in mathematical language. They have built up a universe as harmonious as the monuments of ancient Greece. They weave about it a magnificent texture of calculations and hypotheses. They search for reality beyond the realm of common thought up to unutterable abstractions consisting only of equations of symbols. Such is not the position of biological sciences. Those who investigate the phenomena of life are as if lost in an inextricable jungle, in the midst of a magic forest, whose countless trees unceasingly change their place and their shape. They are crushed under a mass of facts, which they can describe but are incapable of defining in algebraic equations.
Man the Unknown (1935), 1.
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There is an astonishing imagination, even in the science of mathematics. … We repeat, there was far more imagination in the head of Archimedes than in that of Homer.
In A Philosophical Dictionary: from the French (1824), 126.
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There is no branch of mathematics, however abstract, which may not some day be applied to phenomena of the real world.
From a page of quotations, without citations, in G.E. Martin The Foundations of Geometry and the Non-Euclidean Plane (1975), 225. If you know the primary source, please contact Webmaster.
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There is no national science, just as there is no national multiplication table; what is national is no longer science.
In Anton Chekhov, S. S. Koteliansky (trans.) and Leonard Woolf (trans.), Note-Book of Anton Chekhov (1921), 18.
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There was, I think, a feeling that the best science was that done in the simplest way. In experimental work, as in mathematics, there was “style” and a result obtained with simple equipment was more elegant than one obtained with complicated apparatus, just as a mathematical proof derived neatly was better than one involving laborious calculations. Rutherford's first disintegration experiment, and Chadwick's discovery of the neutron had a “style” that is different from that of experiments made with giant accelerators.
From 'Physics in a University Laboratory Before and After World War II', Proceedings of the Royal Society of London, Series A, (1975), 342, 463. As cited in Alan McComas, Galvani's Spark: The Story of the Nerve Impulse (2011), 107.
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These estimates may well be enhanced by one from F. Klein (1849-1925), the leading German mathematician of the last quarter of the nineteenth century. “Mathematics in general is fundamentally the science of self-evident things.” ... If mathematics is indeed the science of self-evident things, mathematicians are a phenomenally stupid lot to waste the tons of good paper they do in proving the fact. Mathematics is abstract and it is hard, and any assertion that it is simple is true only in a severely technical sense—that of the modern postulational method which, as a matter of fact, was exploited by Euclid. The assumptions from which mathematics starts are simple; the rest is not.
Mathematics: Queen and Servant of Science (1952),19-20.
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This splendid subject [mathematics], queen of all exact sciences, and the ideal and norm of all careful thinking...
Educational Problems (1911), Vol. 2, 393.

This therefore is Mathematics:
She reminds you of the invisible forms of the soul;
She gives life to her own discoveries;
She awakens the mind and purifies the intellect;
She brings light to our intrinsic ideas;
She abolishes oblivion and ignorance which are ours by birth...
Proclus
Quoted in Benjamin Franklin Finkel, Mathematical Association of America, The American Mathematical Monthly (1947), Vol. 54, 425.
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Throughout his life Newton must have devoted at least as much attention to chemistry and theology as to mathematics. …
History of Mathematics (3rd Ed., 1901), 335.
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To a mind of sufficient intellectual power, the whole of mathematics would appear trivial, as trivial as the statement that a four-footed animal is an animal. (1959)
My Philosophical Development (1995), 207.

To what purpose should People become fond of the Mathematicks and Natural Philosophy? ... People very readily call Useless what they do not understand. It is a sort of Revenge... One would think at first that if the Mathematicks were to be confin'd to what is useful in them, they ought only to be improv'd in those things which have an immediate and sensible Affinity with Arts, and the rest ought to be neglected as a Vain Theory. But this would be a very wrong Notion. As for Instance, the Art of Navigation hath a necessary Connection with Astronomy, and Astronomy can never be too much improv'd for the Benefit of Navigation. Astronomy cannot be without Optics by reason of Perspective Glasses: and both, as all parts of the Mathematicks are grounded upon Geometry ... .
Of the Usefulness of Mathematical Learning (1699)
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To-day, science has withdrawn into realms that are hardly understanded of the people. Biology means very largely histology, the study of the cell by difficult and elaborate microscopical processes. Chemistry has passed from the mixing of simple substances with ascertained reactions, to an experimentation of these processes under varying conditions of temperature, pressure, and electrification—all requiring complicated apparatus and the most delicate measurement and manipulation. Similarly, physics has outgrown the old formulas of gravity, magnetism, and pressure; has discarded the molecule and atom for the ion, and may in its recent generalizations be followed only by an expert in the higher, not to say the transcendental mathematics.
Anonymous
‘Exit the Amateur Scientist.’ Editorial, The Nation, 23 August 1906, 83, 160.
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Unless the chemist learns the language of mathematics, he will become a provincial and the higher branches of chemical work, that require reason as well as skill, will gradually pass out of his hands.
Quoted in Journal of the Chemical Society, 1929, 6, 254.
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Until now, physical theories have been regarded as merely models with approximately describe the reality of nature. As the models improve, so the fit between theory and reality gets closer. Some physicists are now claiming that supergravity is the reality, that the model and the real world are in mathematically perfect accord.
Superforce (1985), 149.
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We know the laws of trial and error, of large numbers and probabilities. We know that these laws are part of the mathematical and mechanical fabric of the universe, and that they are also at play in biological processes. But, in the name of the experimental method and out of our poor knowledge, are we really entitled to claim that everything happens by chance, to the exclusion of all other possibilities?
From Nobel Prize Lecture (Dec 1974), 'The Coming Age of the Cell'. Collected in Jan Lindsten (ed.) Nobel Lectures, Physiology or Medicine 1971-1980 (1992).
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We may always depend on it that algebra, which cannot be translated into good English and sound common sense, is bad algebra.
In Common Sense in the Exact Sciences (1885), 21.
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We may as well cut out group theory. That is a subject that will never be of any use in physics.
Discussing mathematics curriculum reform at Princeton University (1910), as quoted in Abraham P. Hillman, Gerald L. Alexanderson, Abstract Algebra: A First Undergraduate Course (1994), 94.
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We may discover resources on the moon or Mars that will boggle the imagination, that will test our limits to dream. And the fascination generated by further exploration will inspire our young people to study math, and science, and engineering and create a new generation of innovators and pioneers.
Speech, NASA Headquarters (14 Jan 2004). In Office of the Federal Register (U.S.) Staff (eds.), Public Papers of the Presidents of the United States, George W. Bush (2007), 58-59.
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We often hear that mathematics consists mainly of “proving theorems.” Is a writer's job mainly that of “writing sentences?”
In Rota's 'Introduction' written (1980) to preface Philip J. Davis and Reuben Hersh, The Mathematical Experience (1981, 2012), xxii.
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What binds us to space-time is our rest mass, which prevents us from flying at the speed of light, when time stops and space loses meaning. In a world of light there are neither points nor moments of time; beings woven from light would live “nowhere” and “nowhen”; only poetry and mathematics are capable of speaking meaningfully about such things.
In 'Mathematics and Physics', collected in Mathematics as Metaphor: Selected Essays of Yuri I. Manin (2007), 130.
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What is best in mathematics deserves not merely to be learnt as a task, but to assimilated as a part of daily thought, and brought again and again before the mind with ever-renewed encouragement.
Essay, 'The Study of Mathematics' (1902), collected in Philosophical Essays (1910), 73-74. Also collected in Mysticism and Logic: And Other Essays (1919), 60.
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What Is Mathematics? This question, if asked in earnest, has no answer.
'Why Mathematics Grows', Journal of the History of Ideas (Jan-Mar 1965), 26, No. 1, 3. In Salomon Bochner and Robert Clifford Gunning (ed.) Collected Papers of Salomon Bochner (1992), Vol. 4, 191.
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What's the best part of being a mathematician? I'm not a religious man, but it's almost like being in touch with God when you're thinking about mathematics. God is keeping secrets from us, and it's fun to try to learn some of the secrets.
From interview with Donald J. Albers. In John H. Ewing and Frederick W. Gehring, Paul Halmos Celebrating 50 Years of Mathematics (1991), 21.
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When ever we turn in these days of iron, steam and electricity we find that Mathematics has been the pioneer. Were its back bone removed, our material civilization would inevitably collapse. Modern thought and belief would have been altogether different, had Mathematics not made the various sciences exact.
The Teaching of Mathematics in the Elementary and the Secondary School (1907), 13.
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When the difficulty of a problem lies only in finding out what follows from certain fixed premises, mathematical methods furnish invaluable wings for flying over intermediate obstructions.
From The Economic Theory of the Location of Railways (1887, 1914), viii.
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Whereas, to borrow an illustration from mathematics, life was formerly an equation of, say, 100 unknown quantities, it is now one of 99 only, inasmuch as memory and heredity have been shown to be one and the same thing.
Samuel Butler, Henry Festing Jones (ed.), The Note-Books of Samuel Butler (1917), 57.
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Wherever it was, I did not come to know it through the bodily senses; the only things we know through the bodily senses are material objects, which we have found are not truly and simply one. Moreover, if we do not perceive one by the bodily sense, then we do not perceive any number by that sense, at least of those numbers that we grasp by understanding.
De Ubero Arbitrio (On Free Choice of the Will) [386], trans. T. Williams (1993), 45.
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While, on the one hand, the end of scientific investigation is the discovery of laws, on the other, science will have reached its highest goal when it shall have reduced ultimate laws to one or two, the necessity of which lies outside the sphere of our cognition. These ultimate laws—in the domain of physical science at least—will be the dynamical laws of the relations of matter to number, space, and time. The ultimate data will be number, matter, space, and time themselves. When these relations shall be known, all physical phenomena will be a branch of pure mathematics.
'Address to the section of Mathematical and Physical Science', Reports of the British Association for the Advancement of Science (1895), 595.
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With the exception of the geometrical series, there does not exist in all of mathematics a single infinite series the sum of which has been rigorously determined. In other words, the things which are the most important in mathematics are also those which have the least foundation.
From letter (Jan 1828) to his former teacher Berndt Holmböe. In Morris Kline, Mathematics: The Loss of Certainty (1982), 170.
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You can't go by mathematics: the dollar you borrow is never as big as the dollar you pay back.
Anonymous
In Evan Esar, 20,000 Quips and Quotes, 240.
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You know we’re constantly taking. We don’t make most of the food we eat, we don’t grow it, anyway. We wear clothes other people make, we speak a language other people developed, we use a mathematics other people evolved and spent their lives building. I mean we’re constantly taking things. It’s a wonderful ecstatic feeling to create something and put it into the pool of human experience and knowledge.
Expressing the driving force behind his passion. Interview with Rolling Stone writer, Steven Levy (late Nov 1983). As quoted in Nick Bilton, 'The 30-Year-Old Macintosh and a Lost Conversation With Steve Jobs' (24 Jan 2014), on New York Times blog web page. Levy appended a transcript of the interview to an updated Kindle version of his book, Insanely Great: The Life and Times of Macintosh, the Computer that Changed Everything.
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You may object that by speaking of simplicity and beauty I am introducing aesthetic criteria of truth, and I frankly admit that I am strongly attracted by the simplicity and beauty of mathematical schemes which nature presents us. You must have felt this too: the almost frightening simplicity and wholeness of the relationship, which nature suddenly spreads out before us.
Letter to Albert Einstein. In Ian Stewart, Why Beauty is Truth (), 278.
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You propound a complicated arithmetical problem: say cubing a number containing four digits. Give me a slate and half an hour's time, and I can produce a wrong answer.
Cashel Byron's Profession (1886, 1901), xxiii.
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[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 (1919), 80.

[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 E. Manuel, The Prophets of Paris (1962), 73.
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[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.
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[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.
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[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.
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[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.
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[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.
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[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.
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[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.
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[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.
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[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.

[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.
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[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.
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[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.
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[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).
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[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.
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[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.
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[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.
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[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.
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[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.
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[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.

[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).
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“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.

“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.
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Carl Sagan Thumbnail In science it often happens that scientists say, 'You know that's a really good argument; my position is mistaken,' and then they would actually change their minds and you never hear that old view from them again. They really do it. It doesn't happen as often as it should, because scientists are human and change is sometimes painful. But it happens every day. I cannot recall the last time something like that happened in politics or religion. (1987) -- Carl Sagan
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- 100 -
Sophie Germain
Gertrude Elion
Ernest Rutherford
James Chadwick
Marcel Proust
William Harvey
Johann Goethe
John Keynes
Carl Gauss
Paul Feyerabend
- 90 -
Antoine Lavoisier
Lise Meitner
Charles Babbage
Ibn Khaldun
Euclid
Ralph Emerson
Robert Bunsen
Frederick Banting
Andre Ampere
Winston Churchill
- 80 -
John Locke
Bronislaw Malinowski
Bible
Thomas Huxley
Alessandro Volta
Erwin Schrodinger
Wilhelm Roentgen
Louis Pasteur
Bertrand Russell
Jean Lamarck
- 70 -
Samuel Morse
John Wheeler
Nicolaus Copernicus
Robert Fulton
Pierre Laplace
Humphry Davy
Thomas Edison
Lord Kelvin
Theodore Roosevelt
Carolus Linnaeus
- 60 -
Francis Galton
Linus Pauling
Immanuel Kant
Martin Fischer
Robert Boyle
Karl Popper
Paul Dirac
Avicenna
James Watson
William Shakespeare
- 50 -
Stephen Hawking
Niels Bohr
Nikola Tesla
Rachel Carson
Max Planck
Henry Adams
Richard Dawkins
Werner Heisenberg
Alfred Wegener
John Dalton
- 40 -
Pierre Fermat
Edward Wilson
Johannes Kepler
Gustave Eiffel
Giordano Bruno
JJ Thomson
Thomas Kuhn
Leonardo DaVinci
Archimedes
David Hume
- 30 -
Andreas Vesalius
Rudolf Virchow
Richard Feynman
James Hutton
Alexander Fleming
Emile Durkheim
Benjamin Franklin
Robert Oppenheimer
Robert Hooke
Charles Kettering
- 20 -
Carl Sagan
James Maxwell
Marie Curie
Rene Descartes
Francis Crick
Hippocrates
Michael Faraday
Srinivasa Ramanujan
Francis Bacon
Galileo Galilei
- 10 -
Aristotle
John Watson
Rosalind Franklin
Michio Kaku
Isaac Asimov
Charles Darwin
Sigmund Freud
Albert Einstein
Florence Nightingale
Isaac Newton