Quotes about mathematics
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“There is a connected set of events (light-waves) travelling outward from a centre… there are some respects in which all events are alike, and others in which they differ… We must not think of a light-wave as a 'thing', but as a connected group of rhythmical events. The mathematical characteristics of such a group can be inferred by physics, but the intrinsic character of the component events cannot be inferred.”

Bertrand Russell (1872–1970) logician, one of the first analytic philosophers and political activist

An Outline of Philosophy Ch.15 The Nature of our Knowledge of Physics (1927)
1920s

Travel, Thinking, Light

“It is not to be supposed that young men and women who are busy acquiring valuable specialized knowledge can spare a great deal of time for the study of philosophy, but even in the time that can easily be spared without injury to the learning of technical skills, philosophy can give certain things that will greatly increase the student’s value as a human being and as a citizen. It can give a habit of exact and careful thought, not only in mathematics and science, but in questions of large practical import. It can give an impersonal breadth and scope to the conception of the ends of life. It can give to the individual a just measure of himself in relation to society, of man in the present to man in the past and in the future, and of the whole history of man in relation to the astronomical cosmos. By enlarging the objects of his thoughts it supplies an antidote to the anxieties and anguish of the present, and makes possible the nearest approach to serenity that is available to a sensitive mind in our tortured and uncertain world.”

Bertrand Russell (1872–1970) logician, one of the first analytic philosophers and political activist

1940s, Philosophy for Laymen (1946)

Life, Women, Men, World

“The object of this paper is to give a satisfactory account of the Foundations of Mathematics in accordance with the general method of Frege, Whitehead and Russell. Following these authorities, I hold that mathematics is part of logic, and so belong to what may be called the logical school as opposed to the formalist and intuitionist schools. I have therefore taken Principia Mathematica as a basis for discussion and ammendment; and believe myself to have discovered how, by using the work of Mr Ludwig Wittgenstein, it can be rendered free from the serious objections which have caused its rejection by the majority of German authorities, who have deserted altogether its line of approach.”

Frank P. Ramsey (1903–1930) British mathematician, philosopher

Preface
The Foundations of Mathematics (1925)

School, Parting

“If a "religion" is defined to be a system of ideas that contains unprovable statements, then Gödel taught us that mathematics is not only a religion, it is the only religion that can prove itself to be one.”

Bertrand Russell (1872–1970) logician, one of the first analytic philosophers and political activist

John D. Barrow, Between Inner and Outer Space: Essays on Science, Art and Philosophy (Oxford University Press, 2000, ISBN 0-192-88041-1, Part 4, ch. 13: Why is the Universe Mathematical? (p. 88). Also found in Barrow's "The Mathematical Universe" http://www.lasalle.edu/~didio/courses/hon462/hon462_assets/mathematical_universe.htm (1989) and The Artful Universe Expanded (Oxford University Press, 2005, ISBN 0-192-80569-X, ch. 5, Player Piano: Hearing by Numbers, p. 250 Misattributed

Religion, Idea

“I thought we should require physical determinations, and not abstract integrations. A pernicious taste begins to infiltrate, from which real science will suffer far more than it will progress, and it would be often better for the true physics if there were no mathematics in the world.”

Daniel Bernoulli

Ich vermeinte, man verlange physische Determinationen und nicht abstracte integrationes. Es fängt sich ein verderblicher goût an einzuschleichen, durch welchen die wahren Wissenschaften viel mehr leiden, als sie avancirt werden, und wäre es oft besser für die realem physicam, wenn keine Mathematik auf der Welt wäre.
Letter to Leonhard Euler, 26 January 1750, published in [Correspondance mathématique et physique de quelques célèbres géomètres du XVIIIème siècle, P. H. Fuss, Saint Petersburg, 1843, 650]

World, Suffering, Science

“Every kind of science, if it has only reached a certain degree of maturity, automatically becomes a part of mathematics.”

David Hilbert (1862–1943) German prominent mathematician

"Axiomatic Thought" (1918), printed in From Kant to Hilbert, Vol. 2 by William Bragg Ewald

Science, Parting

“Econometrics is the name for a field of science in which mathematical-economic and mathematical-statistical research are applied in combination. Econometrics, therefore, forms a borderland between two branches of science, with the advantages and disadvantages thereof; advantages, because new combinations are introduced which often open up new perspectives; disadvantages, because the work in this field requires skill in two domains, which either takes up too much time or leads to insufficient training of its students in one of the two respects.”

Jan Tinbergen (1903–1994) Dutch economist

Source: Econometrics, 1951, p. 3; Cited in: Econometrica: journal of the Econometric Society. (1953) p. 36

Science, Time, Training

“We had principles in mathematics that were granted to be absolute in mathematics for over 800 years, but new science has gotten rid of those absolutism, gotten — forward other different logics of looking at mathematics, and sort of turned the way we look at it as a science altogether after 800 years.”

Mahmoud Ahmadinejad (1956) 6th President of the Islamic Republic of Iran

Columbia University speech, 24 September 2007
[24 September 2007, http://www.azstarnet.com/sn/hourlyupdate/202820.php, "Iran's president at Columbia University - a transcript", azstarnet.com, 2007-09-25]
2007

Way, Science

“Mathematics knows no races or geographic boundaries; for mathematics, the cultural world is one country.”

David Hilbert (1862–1943) German prominent mathematician

Quoted in Mathematical Circles Revisited (1971) by Howard Whitley Eves

World, Country

“The man who blames the supreme certainty of mathematics feeds on confusion, and can never silence the contradictions of sophistical sciences which lead to an eternal quackery.”

Leonardo Da Vinci (1452–1519) Italian Renaissance polymath

The Notebooks of Leonardo da Vinci (1883), XIX Philosophical Maxims. Morals. Polemics and Speculations.

Silence, Science

“I feel entitled to hope that I have found in this new analysis the only natural method according to which mathematics should be applied to nature, and according to which geometry may also be treated, whenever it leads to general and to fruitful results.”

Hermann Grassmann (1809–1877) German polymath, linguist and mathematician

Ausdehnungslehre (1844)

Hope, Feeling, Nature

“So much of modern mathematical work is obviously on the border-line of logic, so much of modern logic is symbolic and formal, that the very close relationship of logic and mathematics has become obvious to every instructed student. The proof of their identity is, of course, a matter of detail: starting with premisses which would be universally admitted to belong to logic, and arriving by deduction at results which as obviously belong to mathematics, we find that there is no point at which a sharp line can be drawn, with logic to the left and mathematics to the right. If there are still those who do not admit the identity of logic and mathematics, we may challenge them to indicate at what point, in the successive definitions and deductions of Principia Mathematica, they consider that logic ends and mathematics begins. It will then be obvious that any answer must be quite arbitrary.”

Bertrand Russell (1872–1970) logician, one of the first analytic philosophers and political activist

Source: 1910s, Introduction to Mathematical Philosophy (1919), Ch. 18: Mathematics and Logic

Universe, Success

“The ideal of strictly scientific method in mathematics which I have tried to realise here, and which perhaps might be named after Euclid I should like to describe in the following way… The novelty of this book does not lie in the content of the theorems but in the development of the proofs and the foundations on which they are based… With this book I accomplish an object which I had in view in my Begriffsschrift of 1879 and which I announced in my Grundlagen der Arithmetik. I am here trying to prove the opinion on the concept of number that I expressed in the book last mentioned.”

Gottlob Frege (1848–1925) mathematician, logician, philosopher

Vol. 1. pp. 137-140, as cited in: Ralph H. Johnson (2012), Manifest Rationality: A Pragmatic Theory of Argument, p. 87
Grundgesetze der Arithmetik, 1893 and 1903

Lie, Books, Way

“I wanted certainty in the kind of way in which people want religious faith. I thought that certainty is more likely to be found in mathematics than elsewhere. But I discovered that many mathematical demonstrations, which my teachers expected me to accept, were full of fallacies, and that, if certainty were indeed discoverable in mathematics, it would be in a new field of mathematics, with more solid foundations than those that had hitherto been thought secure. But as the work proceeded, I was continually reminded of the fable about the elephant and the tortoise. having constructed an elephant upon which the mathematical world could rest, I found the elephant tottering, and proceeded to construct a tortoise to keep the elephant from falling. But the tortoise was no more secure than the elephant, and after some twenty years of very arduous toil, I came to the conclusion that there was nothing more that I could do in the way of making mathematical knowledge indubitable.”

Bertrand Russell (1872–1970) logician, one of the first analytic philosophers and political activist

Source: 1950s, Portraits from Memory and Other Essays (1956), p. 53

People, World, Faith, Way

“I don't suppose that applies so much to other physicists; I think it’s 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.”

Paul Dirac (1902–1984) theoretical physicist

Interview with Dr. P. A. M. Dirac by Thomas S. Kuhn at Dirac's home, Cambridge, England, May 7, 1963 http://www.aip.org/history/ohilist/4575_3.html

Beauty, Thinking

“I doubt not, but from self-evident Propositions, by necessary Consequences, as incontestable as those in Mathematics, the measures of right and wrong might be made out.”

John Locke book An Essay Concerning Human Understanding

Book IV, Ch. 3, sec. 18
An Essay Concerning Human Understanding (1689)

“all the standard equations of mathematical physics can be separated and solved in Kerr geometry.”

Subrahmanyan Chandrasekhar (1910–1995) physicist

From Chandrasekhar's Nobel lecture, in his summary of his work on black holes; Republished in: D. G. Caldi, ‎George D. Mostow (1989) Proceedings of the Gibbs Symposium: Yale University, May 15-17, 1989 p. 230

“Pure 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.”

G. H. Hardy book A Mathematician's Apology

A Mathematician's Apology (1941)

“It remains to discuss briefly what general requirements may be justly laid down for the solution of a mathematical problem. I should say first of all, this: that it shall be possible to establish the correctness of the solution by means of a finite number of steps based upon a finite number of hypotheses which are implied in the statement of the problem and which must always be exactly formulated. This requirement of logical deduction by means of a finite number of processes is simply the requirement of rigor in reasoning.”

David Hilbert Mathematical Problems

Mathematical Problems (1900)

Problem

“By the time I was a student in high school I was reading the classic Men of Mathematics by E. T. Bell and I remember succeeding in proving the classic Fermat theorem about an integer multiplied by itself p times where p is a prime.”

John Nash (1928–2015) American mathematician and Nobel Prize laureate

Autobiographical essay (1994)

Men, School, Time, Reading

“But my understanding is there are many villages you go to where there’s really no schools, as a practical matter, and many of the schools still teach just how to memorize certain things rather than how to think critically about problems. And every country at this point, if it wants to succeed, needs to put in place free, compulsory education for its young people -- because they just can’t succeed unless they have some basic skills. They have to be able to read. They have to be able to do mathematics. They have to have some familiarity with computers. They have to be able to understand basic principles of science. If you don’t have those basic tools, then it’s very hard to find a decent job in today’s economy.”

Barack Obama (1961) 44th President of the United States of America

2014, Young Southeast Asian Leaders Initiative Town Hall Speech (November 2014)

People, Education, School, Understanding

“Mathematical Analysis is… the true rational basis of the whole system of our positive knowledge.”

Auguste Comte (1798–1857) French philosopher

Bk. 1, chap. 1; as cited in: Robert Edouard Moritz. Memorabilia mathematica; or, The philomath's quotation-book https://archive.org/stream/memorabiliamathe00moriiala#page/81/mode/2up, (1914), p. 224 System of positive polity (1852)

Knowledge

“Oh! human stupidity, do you not perceive that, though you have been with yourself all your life, you are not yet aware of the thing you possess most of, that is of your folly? and then, with the crowd of sophists, you deceive yourselves and others, despising the mathematical sciences, in which truth dwells and the knowledge of the things included in them. And then you occupy yourself with miracles, and write that you possess information of those things of which the human mind is incapable and which cannot be proved by any instance from nature. And you fancy you have wrought miracles when you spoil a work of some speculative mind, and do not perceive that you are falling into the same error as that of a man who strips a tree of the ornament of its branches covered with leaves mingled with the scented blossoms or fruit.”

Leonardo Da Vinci (1452–1519) Italian Renaissance polymath

The Notebooks of Leonardo da Vinci (1883), XIX Philosophical Maxims. Morals. Polemics and Speculations.

Life, Truth, Stupidity, Error

“Mathematics would certainly have not come into existence if one had known from the beginning that there was in nature no exactly straight line, no actual circle, no absolute magnitude.”

Friedrich Nietzsche (1844–1900) German philosopher, poet, composer, cultural critic, and classical philologist

As quoted in The Puzzle Instinct : The Meaning of Puzzles in Human Life‎ (2004) by Marcel Danesi, p. 71 from Human All-Too-Human

Nature

“The first impulse came from the consideration of negatives in geometry; I was accustomed to viewing the distances AB and BA as opposite magnitudes. Arising from this idea was the conclusion that if A, B, and C are points of a straight line, then in all cases AB + BC = AC, this being true whether AB and BC are directed in the same direction or in opposite directions (where C lies between A and B). In the latter case AB and BC were not viewed as merely lengths, but simultaneously their considered since they were oppositely directed, Thus dawned the distinction between the sum of lengths and the sum of distances which were fixed in direction. From this resulted the requirement for establishing this latter concept of sum, not simply for the case where the distances were directed in the same or opposite directions, but also for any other case. This could be done in the most simple manner, since the law that AB + BC = AC remains valid when A, B, and C do not lie on a straight line.
This then was the first step which led to a new branch of mathematics… I did not however realize how fruitful and how rich was the field that I had opened up; rather that result seemed scarcely worthy of note until it was combined with a related idea.”

Hermann Grassmann (1809–1877) German polymath, linguist and mathematician

Ausdehnungslehre (1844)

Lie, Law, Idea

“A good deal of my research work in physics has consisted in not setting out to solve some particular problems, but simply examining mathematical quantities of a kind that physicists use and trying to get 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 that the work does have an application. Then one has had good luck.”

Paul Dirac (1902–1984) theoretical physicist

P.A.M. Dirac, "Pretty Mathematics," International Journal of Theoretical Physics, Vol. 21, Issue 8–9, August 1982, p. 603 http://link.springer.com/article/10.1007/BF02650229#page-1

Way, Search, Problem

“I am probably the least sensuous of all living beings; being almost exclusively visual and quasi-abstract in imagination, and tending to view and enjoy all things as a passive, detached, and sometimes remote spectator. Those arts which appeal most to the ideational imagination—the sense of drama, pageantry, historic flux, collective organisation, or escape from the natural limitations of time, space, and natural law—are undoubtedly those which appeal chiefly to me. Even my strong love of architectural and decorative beauty is probably largely dependent upon the historical bearings of the forms and motifs in which I delight. I am not wholly insensible to abstract form, but seem to relish the associative element in art more instantly and acutely than the lyrical or mathematical element... I don't really revel in anything unless it reminds me of something else either real or visionary—unless it opens up visual avenues of linked pseudo-recollections leading to sensations of ego-expansion and liberation... usually bringing in the element of time, somehow based on the past, and harbouring hints of an elusive, intangible kind of adventurous expectancy.”

H.P. Lovecraft (1890–1937) American author

Letter to Virgil Finlay (25 September 1936), in Selected Letters V, 1934-1937 edited by August Derleth and Donald Wandrei, p. 310
Non-Fiction, Letters

Love, Law, Beauty, Art

“No man can visualize four dimensions, except mathematically … I think in four dimensions, but only abstractly. The human mind can picture these dimensions no more than it can envisage electricity. Nevertheless, they are no less real than electro-magnetism, the force which controls our universe, within, and by which we have our being.”

Albert Einstein (1879–1955) German-born physicist and founder of the theory of relativity

1920s, Viereck interview (1929)

Thinking, Universe

“Profound study of nature is the most fertile source of mathematical discoveries.”

Joseph Fourier book The Analytical Theory of Heat

Source: The Analytical Theory of Heat (1878), Ch. 1, p. 7

Nature, Study

“God used beautiful mathematics in creating the world.”

Paul Dirac (1902–1984) theoretical physicist

As quoted in The Cosmic Code : Quantum Physics As The Language Of Nature (1982) by Heinz R. Pagels, p. 295; also in Paul Adrien Maurice Dirac : Reminiscences about a Great Physicist (1990) edited by Behram N. Kursunoglu and Eugene Paul Wigner, p. xv

God, World, Beauty

“Mechanics is the paradise of the mathematical sciences because by means of it one comes to the fruits of mathematics.”

Leonardo Da Vinci (1452–1519) Italian Renaissance polymath

The Notebooks of Leonardo da Vinci (1883), I Prolegomena and General Introduction to the Book on Painting

Science

“Wherever Mathematics is mixed up with anything, which is outside its field, you will find attempts to demonstrate these merely conventional propositions a priori, and it will be your task to find out the false deduction in each case.”

Carl Gustav Jacob Jacobi (1804–1851) German mathematician

Vorlesungen über analytische Mechanik [Lectures on Analytical Mechanics] (1847/48; edited by Helmut Pulte in 1996).

Falseness

“Among all the studies of natural causes and reasons Light chiefly delights the beholder; and among the great features of Mathematics the certainty of its demonstrations is what preeminently (tends to) elevate the mind of the investigator. Perspective, therefore, must be preferred to all the discourses and systems of human learning. In this branch [of science] the beam of light is explained on those methods of demonstration which form the glory not so much of Mathematics as of Physics and are graced with the flowers of both.”

Leonardo Da Vinci (1452–1519) Italian Renaissance polymath

The Notebooks of Leonardo da Vinci (1883), I Prolegomena and General Introduction to the Book on Painting

Flowers, Nature, Study, Light

“In mathematics there is a certain way of seeking the truth, a way which Plato is said first to have discovered and which was called "analysis" by Theon and was defined by him as "taking the thing sought as granted and proceeding by means of what follows to a truth which is uncontested"; so, on the other hand, "synthesis" is "taking the thing that is granted and proceeding by means of what follows to the conclusion and comprehension of the thing sought." And although the ancients set forth a twofold analysis, the zetetic and the poristic, to which Theon's definition particularly refers, it is nevertheless fitting that there be established also a third kind, which may be called rhetic or exegetic, so that there is a zetetic art by which is found the equation or proportion between the magnitude that is being sought and those that are given, a poristic art by which from the equation or proportion the truth of the theorem set up is investigated, and an exegetic art by which from the equation set up or the proportion, there is produced the magnitude itself which is being sought. And thus, the whole threefold analytic art, claiming for itself this office, may be defined as the science of right finding in mathematics…. the zetetic art does not employ its logic on numbers—which was the tediousness of the ancient analysts—but uses its logic through a logistic which in a new way has to do with species [of number]…”

François Viète (1540–1603) French mathematician

Source: In artem analyticem Isagoge (1591), Ch. 1 as quoted by Jacob Klein, Greek Mathematical Thought and the Origin of Algebra (1934-1936) Appendix.

Truth, Art, Way, Science

“I am obliged to interpolate some remarks on a very difficult subject: proof and its importance in mathematics. All physicists, and a good many quite respectable mathematicians, are contemptuous about proof. I have heard Professor Eddington, for example, maintain that proof, as pure mathematicians understand it, is really quite uninteresting and unimportant, and that no one who is really certain that he has found something good should waste his time looking for proof.”

G. H. Hardy (1877–1947) British mathematician

Source: Ramanujan (1940), Ch. I : The Indian mathematician Ramanujan.

Understanding, Time

“Of such are the mathematical sciences alone; that is, geometry and arithmetic, in which the Divine intellect indeed knows infinitely more propositions, since it knows all. But with regard to those few which the human intellect does understand, I believe its knowledge equals the Divine in objective certainty, for here it succeeds in understanding necessity, beyond which there can be no greater sureness.”

Galileo Galilei book Dialogue Concerning the Two Chief World Systems

In the 1661 translation by Thomas Salusbury: … such are the pure Mathematical sciences, to wit, Geometry and Arithmetick: in which Divine Wisdom knows infinite more propositions, because it knows them all; but I believe that the knowledge of those few comprehended by humane understanding, equalleth the divine, as to the certainty objectivè, for that it arriveth to comprehend the necessity thereof, than which there can be no greater certainty." p. 92 (from the Archimedes Project http://archimedes.mpiwg-berlin.mpg.de/cgi-bin/toc/toc.cgi?page=92;dir=galil_syste_065_en_1661;step=textonly) In the original Italian: … tali sono le scienze matematiche pure, cioè la geometria e l’aritmetica, delle quali l’intelletto divino ne sa bene infinite proposizioni di piú, perché le sa tutte, ma di quelle poche intese dall’intelletto umano credo che la cognizione agguagli la divina nella certezza obiettiva, poiché arriva a comprenderne la necessità, sopra la quale non par che possa esser sicurezza maggiore." (from the copy at the Italian Wikisource). Dialogue Concerning the Two Chief World Systems (1632)

Understanding, Knowledge, Science

“To those who inquire as to the purpose of mathematics, the usual answer will be that it facilitates the making of machines, the travelling from place to place, and the victory over foreign nations, whether in war or commerce. … The reasoning faculty itself is generally conceived, by those who urge its cultivation, as merely a means for the avoidance of pitfalls and a help in the discovery of rules for the guidance of practical life.”

Bertrand Russell (1872–1970) logician, one of the first analytic philosophers and political activist

1900s, "The Study of Mathematics" (November 1907)

Life, War, Travel, Victory

“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.”

G. H. Hardy book A Mathematician's Apology

A Mathematician's Apology (1941)

Idea, Chance

“In my youth it was said that what was too silly to be said may be sung. In modern economics it may be put into mathematics.”

Ronald H. Coase (1910–2013) British economist and author

Source: 1960s-1980s, "Note on the problem of social costs", 1988, p. 185

Youth

“The smallest natural point is larger than all mathematical points, and this is proved because the natural point has continuity, and any thing that is continuous is infinitely divisible; but the mathematical point is indivisible because it has no size.”

Leonardo Da Vinci (1452–1519) Italian Renaissance polymath

The Notebooks of Leonardo da Vinci (1883), II Linear Perspective

Nature

“I think the first thing that led me toward philosophy (though at that time the word 'philosophy' was still unknown to me) occurred at the age of eleven. My childhood was mainly solitary as my only brother was seven years older than I was. No doubt as a result of much solitude I became rather solemn, with a great deal of time for thinking but not much knowledge for my thoughtfulness to exercise itself upon. I had, though I was not yet aware of it, the pleasure in demonstrations which is typical of the mathematical mind. After I grew up I found others who felt as I did on this matter. My friend G. H. Hardy, who was professor of pure mathematics, enjoyed this pleasure in a very high degree. He told me once that if he could find a proof that I was going to die in five minutes he would of course be sorry to lose me, but this sorrow would be quite outweighed by pleasure in the proof. I entirely sympathized with him and was not at all offended. Before I began the study of geometry somebody had told me that it proved things and this caused me to feel delight when my brother said he would teach it to me. Geometry in those days was still 'Euclid.' My brother began at the beginning with the definitions. These I accepted readily enough. But he came next to the axioms. 'These,' he said, 'can't be proved, but they have to be assumed before the rest can be proved.' At these words my hopes crumbled. I had thought it would be wonderful to find something that one could prove, and then it turned out that this could only be done by means of assumptions of which there was no proof. I looked at my brother with a sort of indignation and said: 'But why should I admit these things if they can't be proved?”

Bertrand Russell (1872–1970) logician, one of the first analytic philosophers and political activist

He replied, 'Well, if you won't, we can't go on.'
Source: 1950s, Portraits from Memory and Other Essays (1956), p. 19

Pleasure, Age, Hope, Feeling

“The books to be read should not be limited to those written in English…. Instead it should be devoted to the great works of history, biography, philosophy, theology, natural science, social science, and mathematics, as well as the… tradition of Western literature -- in English translation… Its aim should not be a survey of Western civilization, but an effort to understand the basic ideas and issues in Western thought.”

Mortimer J. Adler (1902–2001) American philosopher and educator

Source: Reforming Education: The Opening of the American Mind (1990), p. 316

Books, History, Literature, Idea

“The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve.”

Eugene Paul Wigner The Unreasonable Effectiveness of Mathematics in the Natural Sciences

"The Unreasonable Effectiveness of Mathematics in the Natural Sciences," Communications in Pure and Applied Mathematics, February 1960, final sentence.

Law, Understanding

“Would you like to know, for instance, to what extent higher mathematics is used in our two great industries? Well, it is never used at all. Having found this to be the case in my own experience, after quite a long career, I wondered whether I was not an exception; so I made enquiries, and I found that it was a general rule that neither engineers nor managers used higher mathematics in carrying out their duties. We must, of course, learn mathematics that goes without saying but the question is how much must we learn? Up to the present this point has nearly always been decided simply by professors, but it seems to me to be a question in which professors do not count very much, and in which they count less as they become more learned and more devoted to their work. They would like to pass on all their scientific knowledge and they find that their pupils always leave them too soon.”

Henri Fayol (1841–1925) Developer of Fayolism

Source: Henri Fayol addressed his colleagues in the mineral industry, 1900, p. 909

Question, Knowledge, Learning

“I would not dare to say that there is a direct relation between mathematics and madness, but there is no doubt that great mathematicians suffer from maniacal characteristics, delirium and symptoms of schizophrenia.”

John Nash (1928–2015) American mathematician and Nobel Prize laureate

Statement of 1996, as quoted in Dr. Riemann's Zeros (2003) by Karl Sabbagh, p. 88
1990s

Suffering, Madness

“Mathematics is in its development entirely free and is only bound in the self-evident respect that its concepts must both be consistent with each other, and also stand in exact relationships, ordered by definitions, to those concepts which have previously been introduced and are already at hand and established.”

Georg Cantor (1845–1918) mathematician, inventor of set theory

From Kant to Hilbert (1996)
Context: Mathematics is in its development entirely free and is only bound in the self-evident respect that its concepts must both be consistent with each other, and also stand in exact relationships, ordered by definitions, to those concepts which have previously been introduced and are already at hand and established. In particular, in the introduction of new numbers, it is only obligated to give definitions of them which will bestow such a determinacy and, in certain circumstances, such a relationship to the other numbers that they can in any given instance be precisely distinguished. As soon as a number satisfies all these conditions, it can and must be regarded in mathematics as existent and real.

“The fact that all Mathematics is Symbolic Logic is one of the greatest discoveries of our age”

Bertrand Russell (1872–1970) logician, one of the first analytic philosophers and political activist

Principles of Mathematics (1903), Ch. I: Definition of Pure Mathematics, p. 5
1900s
Context: 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 in the analysis of Symbolic Logic itself.

Age

“In modern quantum theory there can be no doubt that the elementary particles will finally also be mathematical forms”

Werner Heisenberg (1901–1976) German theoretical physicist

Physics and Philosophy (1958)
Context: But the resemblance of the modern views to those of Plato and the Pythagoreans can be carried somewhat further. The elementary particles in Plato's Timaeus are finally not substance but mathematical forms. "All things are numbers" is a sentence attributed to Pythagoras. The only mathematical forms available at that time were such geometric forms as the regular solids or the triangles which form their surface. In modern quantum theory there can be no doubt that the elementary particles will finally also be mathematical forms but of a much more complicated nature.

“Mathematics is not just a language. Mathematics is a language plus reasoning.”

Richard Feynman book The Character of Physical Law

Source: The Character of Physical Law (1965), chapter 2, “The Relation of Mathematics to Physics”
Context: Mathematics is not just a language. Mathematics is a language plus reasoning. It's like a language plus logic. Mathematics is a tool for reasoning. It's, in fact, a big collection of the results of some person's careful thought and reasoning. By mathematics, it is possible to connect one statement to another.

“Arrange all these results as described, and you will produce a Table, certainly the most excellent of all Mathematical tables, and prepared for the most important uses.”

John Napier (1550–1617) Scottish mathematician

The Construction of the Wonderful Canon of Logarithms (1889)
Context: From the Radical table completed in this way, you will find with great exactness the logarithms of all sines between radius and the sine 45 degrees; from the arc of 45 degrees doubled, you will find the logarithm of half radius; having obtained all these, you will find the other logarithms. Arrange all these results as described, and you will produce a Table, certainly the most excellent of all Mathematical tables, and prepared for the most important uses.

“It seems to be one of the fundamental features of nature that fundamental physical laws are described in terms of a mathematical theory of great beauty and power”

Paul Dirac (1902–1984) theoretical physicist

The Evolution of the Physicist's Picture of Nature (1963)
Context: It seems to be one of the fundamental features of nature that fundamental physical laws are described in terms of a mathematical theory of great beauty and power, needing quite a high standard of mathematics for one to understand it. You may wonder: Why is nature constructed along these lines? One can only answer that our present knowledge seems to show that nature is so constructed. We simply have to accept it. One could perhaps describe the situation by saying that God is a mathematician of a very high order, and He used very advanced mathematics in constructing the universe. Our feeble attempts at mathematics enable us to understand a bit of the universe, and as we proceed to develop higher and higher mathematics we can hope to understand the universe better.

Law, Beauty, Nature

“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.”

Paul Dirac (1902–1984) theoretical physicist

The Evolution of the Physicist's Picture of Nature (1963)
Context: 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. A good many people are working on the mathematical basis of quantum theory, trying to understand the theory better and to make it more powerful and more beautiful. If someone can hit on the right lines along which to make this development, it may lead to a future advance in which people will first discover the equations and then, after examining them, gradually learn how to apply them.

Hope, Study, Future

“It is interesting that the term mystic is used in this derogatory sense to mean anything we cannot segmentize and count. The odd belief prevails in our culture that a thing or experience is not real if we cannot make it mathematical, and that somehow it must be real if we can reduce it to numbers.”

Rollo May (1909–1994) US psychiatrist

Existence (1956) p. 39; also published in The Discovery of Being : Writings in Existential Psychology (1983), Part III : Contributions to Therapy, Ch. 6 : To Be and Not to Be, p. 94
Existence (1958)
Context: It is interesting that the term mystic is used in this derogatory sense to mean anything we cannot segmentize and count. The odd belief prevails in our culture that a thing or experience is not real if we cannot make it mathematical, and that somehow it must be real if we can reduce it to numbers. But this means making an abstraction out of it … Modern Western man thus finds himself in the strange situation, after reducing something to an abstraction, of having then to persuade himself it is real. … the only experience we let ourselves believe in as real, is that which precisely is not.

Sense

“It was an All-in-One and One-in-All of limitless being and self — not merely a thing of one Space-Time continuum, but allied to the ultimate animating essence of existence's whole unbounded sweep — the last, utter sweep which has no confines and which outreaches fancy and mathematics alike.”

H.P. Lovecraft book Through the Gates of the Silver Key

"Through the Gates of the Silver Key " - written with E. Hoffman Price, October 1932 - Apr 1933; first published in Weird Tales, Vol. 24, No. 1 (July 1934)
Fiction
Context: It was an All-in-One and One-in-All of limitless being and self — not merely a thing of one Space-Time continuum, but allied to the ultimate animating essence of existence's whole unbounded sweep — the last, utter sweep which has no confines and which outreaches fancy and mathematics alike. It was perhaps that which certain secret cults of earth have whispered of as YOG-SOTHOTH, and which has been a deity under other names; that which the crustaceans of Yuggoth worship as the Beyond-One, and which the vaporous brains of the spiral nebulae know by an untranslatable Sign...

Time, Animals

“Only mathematics and mathematical logic can say as little as the physicist means to say.”

Bertrand Russell (1872–1970) logician, one of the first analytic philosophers and political activist

The Scientific Outlook (1931)
1930s
Context: Ordinary language is totally unsuited for expressing what physics really asserts, since the words of everyday life are not sufficiently abstract. Only mathematics and mathematical logic can say as little as the physicist means to say.

“You will perceive that economy, scientifically speaking, is a very contracted science; it is in fact a sort of vague mathematics which calculates the causes and effects of man's industry, and shows how it may be best applied.”

William Stanley Jevons (1835–1882) English economist and logician

Letter to Henrietta Jevons (28 February 1858), published in Letters and Journal of W. Stanley Jevons (1886), edited by Harriet A. Jevons, his wife, p. 101.
Context: You will perceive that economy, scientifically speaking, is a very contracted science; it is in fact a sort of vague mathematics which calculates the causes and effects of man's industry, and shows how it may be best applied. There are a multitude of allied branches of knowledge connected with mans condition; the relation of these to political economy is analogous to the connexion of mechanics, astronomy, optics, sound, heat, and every other branch more or less of physical science, with pure mathematics.

Science

“For truly on countless occasions throughout my life I have had this experience; persons for a time talk pleasantly with me because of my work among the sick, in which they think me very well trained, but when they learn later on that I am also trained in mathematics, they avoid me for the most part and are no longer at all glad to be with me.”

Galén (129–216) Roman physician, surgeon and philosopher

Galen. Margaret Tallmadge May (trans.) On the Usefulness of the Parts of the Body, Ithaca, New York: Cornell U. Press, 1968. p. 502.
Context: A god, as I have said, commanded me to tell the first use also, and he himself knows that I have shrunk from its obscurity. He knows too that not only here but also in many other places in these commentaries, if it depended on me, I would omit demonstrations requiring astronomy, geometry, music, or any other logical discipline, lest my books should be held in utter detestation by physicians. For truly on countless occasions throughout my life I have had this experience; persons for a time talk pleasantly with me because of my work among the sick, in which they think me very well trained, but when they learn later on that I am also trained in mathematics, they avoid me for the most part and are no longer at all glad to be with me. Accordingly, I am always wary of touching on such subjects, and in this case it is only in obedience to the command of a divinity, as I have said, that I have used the theorems of geometry

Life, Thinking, Learning, Time

“Pure mathematics consists entirely of assertions to the effect that, if such and such a proposition is true of anything, then such and such another proposition is true of that thing.”

Bertrand Russell (1872–1970) logician, one of the first analytic philosophers and political activist

Recent Work on the Principles of Mathematics, published in International Monthly, Vol. 4 (1901), later published as "Mathematics and the Metaphysicians" in Mysticism and Logic and Other Essays (1917)
1900s
Context: Pure mathematics consists entirely of assertions to the effect that, if such and such a proposition is true of anything, then such and such another proposition 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. Both these points would belong to applied mathematics. We start, in pure mathematics, from certain rules of inference, by which we can infer that if one proposition is true, then so is some other proposition. These rules of inference constitute the major part of the principles of formal logic. We then take any hypothesis that seems amusing, and deduce its consequences. If our hypothesis is about anything, and not about some one or more particular things, then our deductions constitute mathematics. Thus mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true. People who have been puzzled by the beginnings of mathematics will, I hope, find comfort in this definition, and will probably agree that it is accurate.

“If nature leads us to mathematical forms of great simplicity and beauty”

Werner Heisenberg (1901–1976) German theoretical physicist

Conversation with Einstein, as quoted in Bittersweet Destiny: The Stormy Evolution of Human Behavior by Del Thiessen
Context: If nature leads us to mathematical forms of great simplicity and beauty—by forms I am referring to coherent systems of hypothesis, axioms, etc.—to forms that no one has previously encountered, we cannot help thinking that they are "true," that they reveal a genuine feature of nature... You must have felt this too: The almost frightening simplicity and wholeness of relationships which nature suddenly spreads out before us and for which none of us was in the least prepared.

Beauty, Nature

“All acts of mathematical reasoning may… be considered but as applications of a corresponding axiom of quantity”

William Stanley Jevons (1835–1882) English economist and logician

The Substitution of Similars, The True Principles of Reasoning (1869)
Context: Aristotle's dictim... may then be formulated somewhat as follows:—Whatever is known of a term may be stated of its equal or equivalent. Or, in other words, Whatever is true of a thing is true of its like.... the value of the formula must be judged by its results;... it not only brings into harmony all the branches of logical doctrine, but... unites them in close analogy to the corresponding parts of mathematical method. All acts of mathematical reasoning may... be considered but as applications of a corresponding axiom of quantity...

“Mathematical science is in my opinion an indivisible whole, an organism whose vitality is conditioned upon the connection of its parts.”

David Hilbert Mathematical Problems

Mathematical Problems (1900)
Context: Mathematical science is in my opinion an indivisible whole, an organism whose vitality is conditioned upon the connection of its parts. For with all the variety of mathematical knowledge, we are still clearly conscious of the similarity of the logical devices, the relationship of the ideas in mathematics as a whole and the numerous analogies in its different departments. We also notice that, the farther a mathematical theory is developed, the more harmoniously and uniformly does its construction proceed, and unsuspected relations are disclosed between hitherto separate branches of the science. So it happens that, with the extension of mathematics, its organic character is not lost but only manifests itself the more clearly.

Science, Parting

“A mathematical problem should be difficult in order to entice us, yet not completely inaccessible, lest it mock at our efforts.”

David Hilbert Mathematical Problems

Mathematical Problems (1900)
Context: A mathematical problem should be difficult in order to entice us, yet not completely inaccessible, lest it mock at our efforts. It should be to us a guide post on the mazy paths to hidden truths, and ultimately a reminder of our pleasure in the successful solution.

Problem, Effort

“[B]ecause that which is finite is always bounded with reference to something… it is necessary that there should be no end… [N]umber also appears to be infinite, and mathematical magnitudes, and that which is beyond the heavens. And since that which is beyond is infinite, body also appears to be infinite, and it would seem that there are infinite worlds; for why is there rather void here than there? …If also there is a vacuum, and an infinite place, it is necessary that there should be an infinite body: for in things which have a perpetual subsistence, capacity differs nothing from being. The speculation of the infinite is, however, attended with doubt: for many impossibilities happen both to those who do not admit that it has a subsistence, and to those who do. …It is …especially the province of a natural philosopher to consider if there be a sensible infinite magnitude.”

Aristotle book Physics

Book III, Ch. V, p. 156.
Physics

World, Nature, Appearance

“My suggestion is that at each state the proper order of operation of the mind requires an overall grasp of what is generally known, not only in formal logical, mathematical terms, but also intuitively, in images, feelings, poetic usage of language, etc.”

David Bohm book Wholeness and the Implicate Order

Wholeness and the Implicate Order (1980)
Context: My suggestion is that at each state the proper order of operation of the mind requires an overall grasp of what is generally known, not only in formal logical, mathematical terms, but also intuitively, in images, feelings, poetic usage of language, etc. (Perhaps we could say that this is what is involved in harmony between the 'left brain' and the 'right brain'). This kind of overall way of thinking is not only a fertile source of new theoretical ideas: it is needed for the human mind to function in a generally harmonious way, which could in turn help to make possible an orderly and stable society. <!-- p. xi

Feeling

“The underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are”

Paul Dirac (1902–1984) theoretical physicist

Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character, Vol. 123, No. 792 http://doi.org/10.1098/rspa.1929.0094 (6 April 1929) Context: The underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known, and the difficulty is only that the exact application of these laws leads to equations much too complicated to be soluble. It therefore becomes desirable that approximate practical methods of applying quantum mechanics should be developed, which can lead to an explanation of the main features of complex atomic systems without too much computation.

Law, Parting

“A mathematical theory is not to be considered complete until you have made it so clear that you can explain it to the first man whom you meet on the street.”

David Hilbert Mathematical Problems

Eine mathematische Theorie ist nicht eher als vollkommen anzusehen, als bis du sie so klar gemacht hast, daß du sie dem ersten Manne erklären könntest, den du auf der Straße triffst.
Mathematical Problems (1900)
Context: An old French mathematician said: A mathematical theory is not to be considered complete until you have made it so clear that you can explain it to the first man whom you meet on the street. This clearness and ease of comprehension, here insisted on for a mathematical theory, I should still more demand for a mathematical problem if it is to be perfect; for what is clear and easily comprehended attracts, the complicated repels us.

“Physics is mathematical not because we know so much about the physical world, but because we know so little”

Bertrand Russell (1872–1970) logician, one of the first analytic philosophers and political activist

An Outline of Philosophy Ch.15 The Nature of our Knowledge of Physics (1927)
1920s
Context: Physics is mathematical not because we know so much about the physical world, but because we know so little: it is only its mathematical properties that we can discover.

World

“In mathematics we can take our inner distance from the content of our statements. In the final analysis mathematics is a mental game that we can play or not play as we choose. Religion, on the other hand, deals with ourselves, with our life and death; its promises are meant to govern our actions and thus, at least indirectly, our very existence. We cannot just look at them impassively from the outside. Moreover, our attitude to religious questions cannot be separated from our attitude to society.”

Niels Bohr (1885–1962) Danish physicist

Remarks after the Solvay Conference (1927)
Context: In mathematics we can take our inner distance from the content of our statements. In the final analysis mathematics is a mental game that we can play or not play as we choose. Religion, on the other hand, deals with ourselves, with our life and death; its promises are meant to govern our actions and thus, at least indirectly, our very existence. We cannot just look at them impassively from the outside. Moreover, our attitude to religious questions cannot be separated from our attitude to society. Even if religion arose as the spiritual structure of a particular human society, it is arguable whether it has remained the strongest social molding force through history, or whether society, once formed, develops new spiritual structures and adapts them to its particular level of knowledge. Nowadays, the individual seems to be able to choose the spiritual framework of his thoughts and actions quite freely, and this freedom reflects the fact that the boundaries between the various cultures and societies are beginning to become more fluid. But even when an individual tries to attain the greatest possible degree of independence, he will still be swayed by the existing spiritual structures — consciously or unconsciously. For he, too, must be able to speak of life and death and the human condition to other members of the society in which he's chosen to live; he must educate his children according to the norms of that society, fit into its life. Epistemological sophistries cannot possibly help him attain these ends. Here, too, the relationship between critical thought about the spiritual content of a given religion and action based on the deliberate acceptance of that content is complementary. And such acceptance, if consciously arrived at, fills the individual with strength of purpose, helps him to overcome doubts and, if he has to suffer, provides him with the kind of solace that only a sense of being sheltered under an all-embracing roof can grant. In that sense, religion helps to make social life more harmonious; its most important task is to remind us, in the language of pictures and parables, of the wider framework within which our life is set.

Life, Death, Religion, Game

“In order to understand the Freudian concepts, one must set out on the basis that it is the subject who is called—the subject of Cartesian origin. This basis gives its true function to what, in analysis, is called recollection or remembering. Recollection is not Platonic reminiscence —it is not the return of a form, an imprint, a eidos of beauty and good,a supreme truth, coming to us from the beyond. It is something that comes to us from the structural necessities, something humble, born at the level of the lowest encounters and of all the talking crowd that precedes us, at the level of the structure of the signifier, of the languages spoken in a stuttering, stumbling way, but which cannot elude constraints whose echoes, model, style can be found, curiously enough, in contemporary mathematics.”

Jacques Lacan (1901–1981) French psychoanalyst and psychiatrist

Of the Network of Signifiers
The Four Fundamental Concepts of Psycho Analysis (1978)
Context: It is on this step that depends the fact that one can call upon the subject to re-enter himself in the unconscious—for, after all, it is important to know who one is calling. It is not the soul, either mortal or immortal, which has been with us for so long, nor some shade, some double, some phantom, nor even some supposed psycho-spherical shell, the locus of the defences and other such simplified notions. It is the subject who is called— there is only he, therefore, who can be chosen. There may be, as in the parable, many called and few chosen, but there will certainly not be any others except those who are called. In order to understand the Freudian concepts, one must set out on the basis that it is the subject who is called—the subject of Cartesian origin. This basis gives its true function to what, in analysis, is called recollection or remembering. Recollection is not Platonic reminiscence —it is not the return of a form, an imprint, a eidos of beauty and good, a supreme truth, coming to us from the beyond. It is something that comes to us from the structural necessities, something humble, born at the level of the lowest encounters and of all the talking crowd that precedes us, at the level of the structure of the signifier, of the languages spoken in a stuttering, stumbling way, but which cannot elude constraints whose echoes, model, style can be found, curiously enough, in contemporary mathematics.

Truth, Beauty, Way, Understanding

“Experimenters are the shocktroops of science… An experiment is a question which science poses to Nature, and a measurement is the recording of Nature’s answer. But before an experiment can be performed, it must be planned – the question to nature must be formulated before being posed. Before the result of a measurement can be used, it must be interpreted – Nature’s answer must be understood properly. These two tasks are those of theorists, who find himself always more and more dependent on the tools of abstract mathematics.”

Max Planck (1858–1947) German theoretical physicist

Scientific Autobiography and Other Papers (1949)

Question, Nature, Science, Dependency

“He lived in a period of competitive patronage of the sciences, especially mathematics and astronomy, in the Middle East and Central Asia. He is said to have been a high administrative official in a small principality made up of Basra, in what is now Iraq, and the adjacent region of Ahwâz.”

Alhazen (965–1038) Arab physicist, mathematician and astronomer

Abdelhamid I. Sabra, in “Ibn al-Haytham Brief life of an Arab mathematician: died circa 1040 (September-October 2003)”

Science

“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 gorgeous 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 highest excellence, is to be found in mathematics as surely as in poetry. What is best in mathematics deserves not merely to be learnt as a task, but to be assimilated as a part of daily thought, and brought again and again before the mind with ever-renewed encouragement.”

Bertrand Russell (1872–1970) logician, one of the first analytic philosophers and political activist

1900s, "The Study of Mathematics" (November 1907)

Truth, Beauty, Art, Music

“He has made contributions to many areas of science; among them are agronomy, anthropology, astronomy, bacteriology, botany, economics, forestry, meteorology, psychology, public health, and-above all-genetics, in which he is recognized as one of the leaders. Out of this varied scientific research and his skill in mathematics, he has evolved systematic principles for the interpretation of empirical data; and he has founded a science of experimental design. On the foundations he has laid down, there has been erected a structure of statistical techniques that are used whenever people attempt to learn about nature from experiment and observation.”

Ronald Fisher (1890–1962) English statistician, evolutionary biologist, geneticist, and eugenicist

W. Allen Wallis (1952) at the University of Chicago while honoring Fisher with the Honorary degree of Doctor of Science; cited in: George E. P. Box (1976) " Science and Statistics http://www-sop.inria.fr/members/Ian.Jermyn/philosophy/writings/Boxonmaths.pdf" Journal of the American Statistical Association, Vol. 71, No. 356. (Dec., 1976), pp. 791-799.

People, Health, Nature, Learning

“It is striking that one of the most consequential representatives of this abstract scientific orientation of the seventeenth century became so personalistic. This is because as a juristic thinker he wanted to grasp the reality of societal life just as much as he, as a philosopher and natural scientist, wanted to grasp the reality of nature. He did not discover that there is a juristic reality and life that need not be reality in the sense of the natural sciences. Mathematical relativism and nominalism also operate concurrently. Often he seemed to be able to construct the unity of the state from any arbitrary given point. But juristic thought in those days had not yet become so overpowered by the natural sciences that he, in the intensity of his scientific approach, should unsuspectingly have overlooked the specific reality of legal life inherent in the legal form. The form that he sought lies in the concrete decision, one that emanates from a particular authority. In the independent meaning of the decision, the subject of the decision has an independent meaning, apart from the question of content. What matters for the reality of legal life is who decides.”

Carl Schmitt (1888–1985) German jurist, political theorist and professor of law

Political Theology (1922), Ch. 2 : The Problem of Sovereignty as the Problem of the Legal Form and of the Decision

Life, Question, Nature, Reality

“His work, called Aryabhatiya, is composed of three parts, in only the first of which use is made of a special notation of numbers. It is an alphabetical system in which the twenty-five consonants represent 1-25, respectively; other letters stand for 30, 40, …., 100 etc. The other mathematical parts of Aryabhatiya consists of rules without examples. Another alphabetic system prevailed in Southern India, the numbers 1-19 being designated by consonants, etc.”

Aryabhata (476–550) Indian mathematician-astronomer

Florian Cajori in: A History of Mathematical Notations http://books.google.co.in/books?id=_byqAAAAQBAJ&pg=PT961&dq=Notations&hl=en&sa=X&ei=Wz65U5WYDIKulAW1qIGYDA&ved=0CBwQ6AEwAA#v=onepage&q=Notation&f=false, Courier Dover Publications, 26 September 2013, p. 47.

Parting

“Mathematicians have various ways of judging the merits of new theorems and constructions. One very important criterion is esthetic — some developments just “feel” right, fitting, and beautiful. Just as in other venues where beauty or esthetics are discussed, taste plays an important role in this, but I think I am not alone in being especially excited when apparently different fields suddenly meet in a new concept, a new understanding. It is often of the sparks of such encounters that our esthetic enjoyment of mathematics is born.
Another important criterion for according merit to some particular piece of mathematics is the extent to which it can be useful in applications; this is the criterion almost exclusively used by nonmathematicians.”

Ingrid Daubechies (1954) Belgian physicist and mathematician

(1995) Wavelets and Other Phase Space Localization Methods. In: Chatterji, S.D. (ed.). Proceedings of the International Congress of Mathematicians. Birkhäuser, Basel. [10.1007/978-3-0348-9078-6_8]

Beauty, Way, Feeling, Understanding

“I think I have already said somewhere that mathematics is the art of giving the same name to different things.”

Henri Poincaré book Science and Method

Original: (fr) Je ne sais si je n’ai déjà dit quelque part que la Mathématique est l’art de donner le même nom à des choses différentes.
Source: Science and Method (1908), Part I. Ch. 2 : The Future of Mathematics, p. 31

Art, Thinking

“Maybe that is what beauty was, for humans. Accidents, imperfections, placed inside a pretty pattern. Asymmetry. The defiance of mathematics.”

Matt Haig (1975) British writer

Source: The Humans

Beauty

“The core of science is not a mathematical modeling--it is intellectual honesty. It is a willingness to have our certainties about the world constrained by good evidence and good argument.”

Sam Harris (1967) American author, philosopher and neuroscientist

World, Science

“The 3-legged stool of understanding is held up by history, languages, and mathematics. Equipped with those three you can learn anything you want to learn. But if you lack any one of them you are just another ignorant peasant with dung on your boots.”

Robert A. Heinlein book Expanded Universe

Source: "The Happy Days Ahead" in Expanded Universe (1980)
Context: I started clipping and filing by categories on trends as early as 1930 and my "youngest" file was started in 1945.
Span of time is important; the 3-legged stool of understanding is held up by history, languages, and mathematics. Equipped with these three you can learn anything you want to learn. But if you lack any one of them you are just another ignorant peasant with dung on your boots.

History, Understanding, Learning

“Life has puffed and blown itself into a summer day, and clouds and spring billow over the heavens as if calendars were a listing of mathematical errors.”

Zelda Fitzgerald (1900–1948) Novelist, wife of F. Scott Fitzgerald

Source: Dear Scott, Dearest Zelda: The Love Letters of F. Scott and Zelda Fitzgerald

Life, Error, Spring, Summer

“A mathematical formula for happiness:Reality divided by Expectations. There were two ways to be happy:improve your reality or lower your expectations.”

Jodi Picoult book Nineteen Minutes

Source: Nineteen Minutes

Way, Reality

“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.”

Albert Einstein (1879–1955) German-born physicist and founder of the theory of relativity

Insofern sich die Sätze der Mathematik auf die Wirklichkeit beziehen, sind sie nicht sicher, und insofern sie sicher sind, beziehen sie sich nicht auf die Wirklichkeit. http://books.google.com/books?id=QF0ON71WuxEC&q=%22Insofern+sich+die+S%C3%A4tze+der+Mathematik+auf+die%22&pg=PA3#v=onepage Geometrie and Erfahrung (1921) pp. 3-4 link.springer.com http://link.springer.com/chapter/10.1007%2F978-3-642-49903-6_1#page-1 as cited by Karl Popper, The Two Fundamental Problems of the Theory of Knowledge (2014) Tr. Andreas Pickel, Ed. Troels Eggers Hansen. Ref: en.wikiquote.org - Albert Einstein / Quotes / 1920s http://books.google.com/books?id=QF0ON71WuxEC&q=%22beziehen+sind+sie+nicht+sicher+und+insofern+sie+sicher+sind+beziehen+sie+sich+nicht+auf+die+Wirklichkeit%22&pg=PA4#v=onepage 1920s, Sidelights on Relativity (1922)

Law, Reality

“If two people loved, they slept together; it was a mathematical formula, tested and proved by human experience.”

Graham Greene book The End of the Affair

Source: The End of the Affair

Love, People

“I think that modern physics has definitely decided in favor of Plato. In fact the smallest units of matter are not physical objects in the ordinary sense; they are forms, ideas which can be expressed unambiguously only in mathematical language.”

Werner Heisenberg (1901–1976) German theoretical physicist

Das Naturgesetz und die Struktur der Materie (1967), as translated in Natural Law and the Structure of Matter (1981), p. 34

Idea, Thinking, Sense

“Eternity is said not to be an extension of time but an absence of time, and sometimes it seemed to me that her abandonment touched that strange mathematical point of endlessness, a point with no width, occupying no space.”

Graham Greene (1904–1991) English writer, playwright and literary critic

Time

“What do we call visible light? We call it color. But the electromagnetic spectrum runs to zero in one direction and infinity in the other, so really, children, mathematically, all of light is invisible.”

Anthony Doerr book All the Light We Cannot See

Source: All the Light We Cannot See

Children, Color, Light

“With me, everything turns into mathematics.”

René Descartes (1596–1650) French philosopher, mathematician, and scientist

Mais apud me omnia fiunt Mathematicè in Natura More closely translated as: but in my opinion, all things in nature occur mathematically. Note: "Mais" is French for "but" and the "but in my opinion" comes from the context of the original conversation. apud me omnia fiunt Mathematicè in Natura is in latin. Sometimes the Latin version is incorrectly quoted as Omnia apud me mathematica fiunt. Sources: Correspondence with Mersenne http://fr.wikisource.org/wiki/Page%3aDescartes_-_%C5%92uvres,_%C3%A9d._Adam_et_Tannery,_III.djvu/48 note for line 7 (1640), page 36, Die Wiener Zeit http://books.google.com/books?id=9Xh3fVZLCycC&pg=PA532&lpg=PA532&dq=%22Omnia+apud+me+mathematica+fiunt%22+original+zitat&source=bl&ots=CgQOrveRiM&sig=WFHwIK20r5vRZ66FwCaxo857LCU&hl=de&sa=X&ei=_Wf2UcHlJYbfsgaf1IHABg#v=onepage&q=%22Omnia%20apud%20me%20mathematica%20fiunt%22%20original%20zitat&f=false page 532 (2008); StackExchange Math Q/A Where did Descartes write... http://math.stackexchange.com/questions/454599/where-did-descartes-write-with-me-everything-turns-into-mathematics?noredirect=1#comment978229_454599

“I merely say that all reading for pleasure is escape, whether it be Greek, mathematics, astronomy, Benedetto Croce, or The Diary of the Forgotten Man. To say otherwise is to be an intellectual snob, and a juvenile at the art of living.”

Raymond Chandler book The Simple Art of Murder

Source: The Simple Art of Murder

Pleasure, Art, Reading

“Histories make men wise; poets, witty; the mathematics, subtile; natural philosophy, deep; moral, grave; logic and rhetoric, able to contend.”

Francis Bacon (1561–1626) English philosopher, statesman, scientist, jurist, and author

Of Studies
Essays (1625)
Source: The Collected Works of Sir Francis Bacon

Men, History, Nature

“Now what is history? It is the centuries of systematic explorations of the riddle of death, with a view to overcoming death. That’s why people discover mathematical infinity and electromagnetic waves, that’s why they write symphonies..”

Jon Krakauer book Into the Wild

Source: Into the Wild

Death, People, History

“JOY goes against the foundations of mathematics: it multiplies when we divide.”

Paulo Coelho (1947) Brazilian lyricist and novelist

Joy

“… while the individual man is an insoluble puzzle, in the aggregate he becomes a mathematical certainty.”

Arthur Conan Doyle book The Sign of the Four

Source: The Sign of Four

“politics were as bad as mathematics, and that the mission of politicians seemed to be calling each other names”

Louisa May Alcott book Little Women

Source: Little Women

“Cyberspace. A consensual hallucination experienced daily by billions of legitimate operators, in every nation, by children being taught mathematical concepts…”

William Gibson book Neuromancer

Source: Neuromancer (1984)
Context: Cyberspace. A consensual hallucination experienced daily by billions of legitimate operators, in every nation, by children being taught mathematical concepts… A graphic representation of data abstracted from banks of every computer in the human system. Unthinkable complexity. Lines of light ranged in the nonspace of the mind, clusters and constellations of data. Like city lights, receding...

Children

“Sexually progressive cultures gave us mathematics, literature, philosophy, civilization and the rest, while sexually restrictive cultures gave us the Dark Ages and the Holocaust.”

Alan Moore (1953) English writer primarily known for his work in comic books

"BOG VENUS VERSUS NAZI COCK-RING: Some Thoughts Concerning Pornography" in Arthur magazine, Vol. 1, No. 25 (November 2006) http://www.arthurmag.com/magpie/?p=1685 Source: 25,000 Years of Erotic Freedom Context: Sexually progressive cultures gave us mathematics, literature, philosophy, civilization and the rest, while sexually restrictive cultures gave us the Dark Ages and the Holocaust. Not that I’m trying to load my argument, of course.

Age, Literature