Quotes about mathematics
page 10

“The most general form of systems theory is a set of logical or mathematical statements about all conceptual systems. A subset of this concerns all concrete systems. A subsubset concerns the very special and very important living systems, i. e., general living systems theory.”

James Grier Miller (1916–2002) biologist

Source: Living systems, 1978, p. 41

“In theoretical science, the question is—What are we to think? and when a doubtful point arises, for the solution of which either experimental data are wanting, or mathematical methods are not sufficiently advanced, it is the duty of philosophic minds not to dispute about the probability of conflicting suppositions, but to labour for the advancement of experimental inquiry and of mathematics, and await patiently the time when they shall be adequate to solve the question.”

William John Macquorn Rankine (1820–1872) civil engineer

p, 125
"On the Harmony of Theory and Practice in Mechanics" (Jan. 3, 1856)

Question, Thinking, Science, Time

“Every decade or so, the business world invents another term for how it extracts managerial and decision-making value from computerized data. In the 1970s the favored term was “decision support systems,” accurately reflecting the importance of a decision-centered approach to data analysis. In the early 80s, “executive information systems” was the preferred nomenclature, which addressed the use of these systems by senior managers. Later in that decade, emphasis shifted to the more technical-sounding “online analytical processing,” or OLAP. The 90s saw the rise of “business intelligence” as a descriptor. In the middle of 2000’s first decade, “analytics” began to come into favor, at least for the more statistical and mathematical forms of data analysis.”

Thomas H. Davenport (1954) American academic

Thomas H. Davenport. The New World of “Business Analytics”. International Institute for Analytics. March 2010

World, Intelligence, Business, Support

“Without mathematics, we are blind.”

Alain Badiou (1937) French writer and philosopher

Original French: Hors les mathématiques, nous sommes aveugles.
From Court traité d'ontologie transitoire. Paris: Éditions du Seuil, 1998. ISBN 2020348853.

“Grasping spatial gestalts as figures is mathematizing of space. Arranging the properties of a parallelogram such that a particular one pops up to base the others on it in order to arrive at a definition of parallelogram, that is mathematizing the conceptual field of the parallelogram.”

Hans Freudenthal (1905–1990) Dutch mathematician

Source: Mathematics as an Educational Task (1973), p. 133

“The actual infinite arises in three contexts: first when it is realized in the most complete form, in a fully independent otherworldly being, in Deo, where I call it the Absolute Infinite or simply Absolute; second when it occurs in the contingent, created world; third when the mind grasps it in abstracto as a mathematical magnitude, number or order type.”

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

As quoted in Mind Tools: The Five Levels of Mathematical Reality (1988) by Rudy Rucker. ~ ISBN 0395468108

World

“Mathematics says the sum value of a network increases as the square of the number of members. In other words, as the number of nodes in a network increases arithmetically, the value of the network increases exponentially. Adding a few more members can dramatically increase the value of the network.”

Kevin Kelly (1952) American author and editor

Out of Control: The New Biology of Machines, Social Systems and the Economic World (1995), New Rules for the New Economy: 10 Radical Strategies for a Connected World (1999)

“Classical mechanics is mathematical not only in the sense that it makes use of mathematical terms and methods for abbreviating arguments which might, if necessary, also be expressed in the language of everyday speech; it is so also in the much more stringent sense that its basic concepts are mathematical concepts, that mechanics itself is a mathematics.”

Eduard Jan Dijksterhuis (1892–1965) Dutch historian

Source: The mechanization of the world picture, 1961, p. 499

Sense

“One rainy day in Cologne on the Rhine, the catalogue of a teaching aids company caught my attention. It was illustrated with models of all kind – mathematical, geometrical, anthropological, zoological, botanical, anatomical, mineralogical, paleontological, and so forth- elements of of such a diverse nature that the absurdity of the collection confused the eye and mind, producing hallucinations and lending the objects depicted new and rapidly changing meanings. I suddenly felt my 'visionary faculties' so intensified that I began seeing the newly emerged objects against a new background. To capture it, a little paint or a few lines were enough, a horizon, a sky, a wooden floor, that sort of things. My hallucination had been fixed. Now it was a matter of interpreting the hallucination in a few words or sentences Such as: 'Above the clouds midnight passes. Above midnight glides the invisible bird of day..”

Max Ernst (1891–1976) German painter, sculptor and graphic artist

Quote in 'Biographical Notes. Tissue of truth, Tissue of Lies', 1929; as cited in Max Ernst. A Retrospective, Munich, Prestel, 1991, p. 290
1910 - 1935

Nature, Bird, Change, Painting

“With respect to the mathematical part of optics, the curvature of the images, formed by lenses and mirrors, has been correctly investigated, and the inaccuracy of some former estimations has been demonstrated.”

Thomas Young (scientist) (1773–1829) English polymath

Preface
A Course of Lectures on Natural Philosophy and the Mechanical Arts (1807)

Parting

“There are, in fact, as I began to say above, not a few principles which are the special property of mathematics, such principles as are discovered by the common light of nature, require no demonstration, and which concern quantities primarily; then they are applied to other things, so far as the latter have something in common with quantities. Now there are more of these principles in mathematics than in the other theoretical sciences because of that very characteristic of the human understanding which seems to be such from the law of creation, that nothing can be known completely except quantities or by quantities. And so it happens that the conclusions of mathematics are most certain and indubitable.”

Johannes Kepler (1571–1630) German mathematician, astronomer and astrologer

Vol. VIII, p. 148
Joannis Kepleri Astronomi Opera Omnia, ed. Christian Frisch (1858)

Law, Understanding, Nature, Light

“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.”
Quaestiones, quae ad mathematicae fundamenta pertinent, etsi hisce temporibus a multis tractatae, satisfacienti solutione et adhuc carent. Hic difficultas maxime en sermonis ambiguitate oritur. Quare summi interest verba ipsa, quibus utimur attente perpendere.

Giuseppe Peano (1858–1932) Italian mathematician

Arithmetices principia, nova methodo exposita [The Principles of Arithmetic, presented by a new method] (1889)

Question, Problem, Time

“What appear to be the most valuable aspects of the theoretical physics we have are the mathematical descriptions which enable us to predict events. These equations are, we would argue, the only realities we can be certain of in physics; any other ways we have of thinking about the situation are visual aids or mnemonics which make it easier for beings with our sort of macroscopic experience to use and remember the equations.”

Celia Green (1935) British philosopher

The Lost Cause (2003)

Way, Thinking, Reality, Appearance

“The moving power of mathematical invention is not reasoning, but imagination.”

Augustus De Morgan (1806–1871) British mathematician, philosopher and university teacher (1806-1871)

Quoted in Robert Perceval Graves, The Life of Sir William Rowan Hamilton, Vol. 3 (1889), p. 219.

Imagination

“Off in his head, he solved extraordinary problems of mathematics, science, medicine, the humanities. He wrote poetry, saw music. … People have told him, with your IQ you should be a genius. I was a genius and have long since abused myself into a state of average and like it better that way. Now there comes a momentary clearing, a moment of satorial splendor, and then it goes oblique. It fades away like blown dust.”

Robert Olmstead (1954) American writer

America By Land (1993)

People, Music, Way, Problem

“I remember thinking of mathematics as a kind of omnipotent protector. You could prove things to people and they would have to believe you whether they liked you or not.”

John Allen Paulos (1945) American mathematician

Source: Innumeracy: Mathematical Illiteracy and its Consequences (1988), Chapter 4, “Whence Innumeracy?” (p. 99)

People, Thinking

“The following work is not a republication of a former treatise by the Author, entitled, "The Mathematical Analysis of Logic." Its earlier portion is indeed devoted to the same object, and it begins by establishing the same system of fundamental laws, but its methods are more general, and its range of applications far wider. It exhibits the results, matured by some years of study and reflection, of a principle of investigation relating to the intellectual operations, the previous exposition of which was written within a few weeks after its idea had been conceived.”

George Boole (1815–1864) English mathematician, philosopher and logician

Source: 1850s, An Investigation of the Laws of Thought (1854), p. i; Preface, lead paragraph

Law, Idea, Study

“The idea of an isolated American painting, so popular in this country during the thirties, seems absurd to me, just as the idea of a purely American mathematics or physics would seem absurd... And in another sense, the problem doesn't exist at all; or, if it did, would solve itself: An American is an American and his painting would naturally be qualified by the fact, whether he wills or not. But the basic problems of contemporary painting are independent of any one country.”

Jackson Pollock (1912–1956) American artist

As quoted in Abstract Expressionism: Creators and Critics, ed. Clifford Ross, Abrahams Publishers, New York, 1990, p. 138
1940's, Art and Architecture (1944)

Idea, Nature, Problem, Sense

“Guided only by their feeling for symmetry, simplicity, and generality, and an indefinable sense of the fitness of things, creative mathematicians now, as in the past, are inspired by the art of mathematics rather than by any prospect of ultimate usefulness.”

Eric Temple Bell (1883–1960) mathematician and science fiction author born in Scotland who lived in the United States for most of his li…

As quoted in 777 Mathematical Conversation Starters http://books.google.co.in/books?id=JNbKURWmODkC&pg=PA172 (2002) by John de Pillis, p. 172

Art, Feeling, Sense, Past

“Modernism began in the mid-nineteenth century as a response not only to the restrictions and hypocrisies of everyday life, but also as a reaction to the Enlightenment's emphasis on the rationality of human behavior. …The founders of the Royal Society thought of God as a mathematician who had designed the universe according to logical and mathematical principles. The role of the scientist—the natural philosopher was to… decipher the codebook that God had used in creating the cosmos.”

Eric R. Kandel (1929) American neuropsychiatrist

The Age of Insight (2012)

Life, God, Behavior, Nature

“It was not until my second year as a doctoral student that I began to understand that mathematics was an ever-expanding universe. My thesis advisor at Princeton was Emil Artin, one of the great algebraists of the century. Unfortunately, or perhaps fortunately, he offered me no advice in the selection of a thesis topic. I think it was a fluke that I got started at all. But once I did, a whole new world opened up, to which I would devote a vast amount of time and energy for over thirty years.”

O. Timothy O'Meara (1928–2018) American mathematician

[The Idea of a Catholic University: A Personal Perspective, Marquette Law Review, Winter 1995: Symposium on Religiously Affiliated Law Schools, 78, 2, 389–396, http://scholarship.law.marquette.edu/cgi/viewcontent.cgi?article=1579&context=mulr]

World, Understanding, Thinking, Universe

“This idea is that laws which purport to be statements of what actually occurs are statistical in character as distinct from so-called dynamic laws that are abstract and mathematical, and disguised definitions. Recognition of the statistical nature of physical laws was first effected in the case of gases when it became evident that generalizations regarding the behavior of swarms of molecules were not descriptions or predictions of the behavior of any individual particle. A single molecule is not and cannot be a gas. It is consequently absurd to suppose that a scientific law is about the elementary constituents of a gas. It is a statement of what happens when a large number of such constituents interact with one another under certain conditions.”

John Dewey (1859–1952) American philosopher, psychologist, and educational reformer

Time and Individuality (1940)

Law, Behavior, Idea, Nature

“The history of Greek mathematics is, for the most part, only the history of such mathematics as are learnt daily in all our public schools. …If it was not wanted, as it ought to have been, by our classical professors and our mathematicians, it would have served at any rate to quicken, with some human interest, the melancholy labours of our schoolboys.”

James Gow (scholar) (1854–1923) scholar

Preface
A Short History of Greek Mathematics (1884)

School, History, Parting

“Like all great churches, that are not mere store-houses of theology, Chartres expressed, besides whatever else it meant, an emotion, the deepest man ever felt,— the struggle of his own littleness to grasp the infinite. You may, if you like, figure in it a mathematic formula of infinity,— the broken arch, our finite idea of space; the spire, pointing, with its converging lines, to Unity beyond space; the sleepless, restless thrust of the vaults, telling the unsatisfied, incomplete, overstrained effort of man to rival the energy, intelligence and purpose of God. Thomas Aquinas and the schoolmen tried to put it in words, but their church is another chapter. In act, all man's work ends there;— mathematics, physics, chemistry, dynamics, optics, every sort of machinery science may invent,— to this favor come at last, as religion and philosophy did before science was born.”

Henry Adams (1838–1918) journalist, historian, academic, novelist

Mont Saint Michel and Chartres (1904)

God, Religion, Intelligence, Idea

“[Scottish accent] "We don't need your English bastard pounds! We're our own country, we'll have our own bloody money, eh?!" "Would you like your own currency?" "Ah, it's complicated mathematically. Let's just have yours with our photos, I think that's the best way!"”

Michael McIntyre (1976) British comedian

Source: Live at the Apollo (November 26, 2007)

Money, Way, Photo, Thinking

“I scarcely believe that in psychology data are present which can be mathematically evaluated. But one cannot know this with certainty, without having made the experiment. God alone is in possession of the mathematical bases of psychic phenomena.”

Carl Friedrich Gauss (1777–1855) German mathematician and physical scientist

As quoted in Carl Friedrich Gauss: Titan of Science (1955) by Guy Waldo Dunnington. p. 306

God

“Simulation is the use of models and/or the actual conditions of either the thing being modeled or the environment in which it operates, with the models or conditions in physical, mathematical, or some other form. The purpose of simulation is to explore the various results which might be obtained from the real system by subjecting the model to representative environments which are equivalent to, or in some way representative of, the situations it is desired to understand or investigate. Simulation may involve system hardware and the actual physical environment, or it may involve mathematical models subjected to mathematical forcing or disturbance functions representative of the systems conditions to be studied.”

Harold Chestnut (1917–2001) American engineer

Source: Systems Engineering Tools, (1965), p. 111 as cited in

Way, Understanding, Study

“When this book was first conceived (more than 25 years ago) few mathematicians outside the Soviet Union recognized probability as a legitimate branch of mathematics.”

William Feller (1906–1970) Croatian-American mathematician

Preface to the Third Edition, p. vii.
An Introduction To Probability Theory And Its Applications (Third Edition)

Books

“The physicist who states a law of nature with the aid of a mathematical formula is abstracting a real feature of a real material world, even if he has to speak of numbers, vectors, tensors, state-functions, or whatever to make the abstraction.”

Hilary Putnam (1926–2016) American philosopher

in What is Mathematics, in [Hilary Putnam, Mathematics, matter, and method, Cambridge University Press, 1979, 0521295505, 60]

World, Law, Nature

“Those skilled in mathematical analysis know that its object is not simply to calculate numbers, but that it is also employed to find the relations between magnitudes which cannot be expressed in numbers and between functions whose law is not capable of algebraic expression.”

Antoine Augustin Cournot Researches into the Mathematical Principles of the Theory of Wealth

Source: Researches into the Mathematical Principles of the Theory of Wealth, 1897, p. 3 ; Cited in: Robert Edouard Moritz. Memorabilia mathematica; or, The philomath's quotation-book https://archive.org/stream/memorabiliamathe00moriiala#page/198/mode/2up, (1914) p. 33: About the nature of mathematics

Law

“Men cannot be treated as units in operations of political arithmetic because they behave like the symbols for zero and the infinite, which dislocate all mathematical operations.”

Arthur Koestler (1905–1983) Hungarian-British author and journalist

Crossman (1949), p. 68.

Men

“Bohr's investigation had typified what had become a standard procedure in problems of theoretical physics. The first step was to discover the mathematical laws governing certain groups of phenomena; the second was to devise hypothetical models or pictures to interpret these laws in terms of motion or mechanism; the third was to examine in what way these models would behave in other respects, and this would lead to prediction of other phenomena-predictions which might or might not be confirmed when put to the test of experiment. For instance, Newton had explained the phenomena of gravitation in terms of a force of gravitation; a later age had seen the luminiferous ether introduced to explain the propagation of light and, subsequently, the general phenomena of electricity and magnetism; finally Bohr had introduced electronic jumps in an attempt to explain atomic spectra. In each case the models had fulfilled their primary purpose, but had failed to predict further phenomena with accuracy.”

James Jeans (1877–1946) British mathematician and astronomer

Physics and Philosophy (1942)

Age, Law, Way, Problem

“He that could teach mathematics well, would not be a bad teacher in any of the rest [physics, chemistry, biology, psychology] unless by the accident of total inaptitude for experimental illustration; while the mere experimentalist is likely to fall into the error of missing the essential condition of science as reasoned truth; not to speak of the danger of making the instruction an affair of sensation, glitter, or pyrotechnic show.”

Alexander Bain (1818–1903) Scottish philosopher and educationalist

Source: Education as a Science, 1898, p. 298.

Truth, Error, Teachers, Science

“The subject matter of mathematics is the expressions themselves together with the rules for manipulating them—nothing more.”

Edward Nelson (1932–2014) American mathematical physicist and logician

[Nelson, E., Predicative Arithmetic, 1986, Princeton University Press, Princeton, 0-691-08455-6, https://books.google.co.uk/books?id=pvr_AwAAQBAJ&printsec=frontcover#v=onepage&q&f=false, 86018730, 14001745, 173, harv]

“The world operates by number, by physical laws, expressed mathematically. If you know these, you will have a better grasp of things. And some possible job skills.”

Kim Stanley Robinson book The Years of Rice and Salt

Book 9: "Nsara", § 5
The Years of Rice and Salt (2002)

World, Law

“The science of the age, in short, is physical, chemical, physiological; in all shapes mechanical. Our favourite Mathematics, the highly prized exponent of all these other sciences, has also become more and more mechanical. Excellence in what is called its higher departments depends less on natural genius than on acquired expertness in wielding its machinery. Without undervaluing the wonderful results which a Lagrange or Laplace educes by means of it, we may remark, that their calculus, differential and integral, is little else than a more cunningly-constructed arithmetical mill; where the factors, being put in, are, as it were, ground into the true product, under cover, and without other effort on our part than steady turning of the handle. We have more Mathematics than ever; but less Mathesis.”

Thomas Carlyle (1795–1881) Scottish philosopher, satirical writer, essayist, historian and teacher

1820s, Signs of the Times (1829)

Age, Education, Nature, Effort

“The applications of knowledge, especially mathematics, reveal the unity of all knowledge. In a new situation almost anything and everything you ever learned might be applicable, and the artificial divisions seem to vanish.”

Richard Hamming (1915–1998) American mathematician and information theorist

Methods of Mathematics Applied to Calculus, Probability, and Statistics (1985)

Knowledge, Learning

“In accordance with the foregoing investigations on mathematical principles, let bronze vessels be made, proportionate to the size of the theatre, and let them be so fashioned that, when touched, they may produce with one another the notes of the fourth, the fifth, and so on up the double octave.”

Vitruvius book De architectura

Source: De architectura (The Ten Books On Architecture) (~ 15BC), Book V, Chapter V, Sec. 1

Fashion

“It was a dramatic moment in the history of our civilisation when, in about the year 240 B. C., Archimedes leapt out of his bath and ran naked into the street of Syracuse shouting "Eureka" - "I have found it!" He had found the theoretical answer to a practical problem, that of finding the specific gravity of solids, known ever since as "Archimedes' Principle". We know this because Archimedes had written and published many books, on mathematics and mechanics, and some of these have survived and are preserved in libraries for us to use today. He made a major contribution to the progress of humanity, and over the centuries, that progress has been continually stimulated and accelerated by the invention of new theories, new tools and machines, which explain our world, and lighten the burden of securing the basic necessities of life.”

Douglas John Foskett (1918–2004)

Librarians and Information Systems (1995)

Life, Books, World, History

“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? Is human reason, then, without experience, merely by taking thought, able to fathom the properties of real things?”

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

1920s, Sidelights on Relativity (1922)

Reality

“Objections… inspired Kronecker and others to attack Weierstrass' "sequential" definition of irrationals. Nevertheless, right or wrong, Weierstrass and his school made the theory work. The most useful results they obtained have not yet been questioned, at least on the ground of their great utility in mathematical analysis and its implications, by any competent judge in his right mind. This does not mean that objections cannot be well taken: it merely calls attention to the fact that in mathematics, as in everything else, this earth is not yet to be confused with the Kingdom of Heaven, that perfection is a chimaera, and that, in the words of Crelle, we can only hope for closer and closer approximations to mathematical truth—whatever that may be, if anything—precisely as in the Weierstrassian theory of convergent sequences of rationals defining irrationals.”

Eric Temple Bell book Men of Mathematics

Men of Mathematics (1937)

Truth, School, Hope, Question

“Abstractness, sometimes hurled as a reproach at mathematics, is its chief glory and its surest title to practical usefulness. It is also the source of such beauty as may spring from mathematics.”

Eric Temple Bell (1883–1960) mathematician and science fiction author born in Scotland who lived in the United States for most of his li…

Source: The Development of Mathematics (1940), p. 9

Beauty, Spring

“Basically there’s a gulf between truth and untruth, I don’t want to be too mathematical about it, because I’m not very good at maths. But it’s a divide between, I dunno, a film and a cartoon. I’ve just become this cartoon character. I try not to follow it, but when you see pictures of yourself that have been photo-shopped to show you doing something you didn’t…. that’s wrong. It’s my worst nightmare – and being misquoted, too, especially as I’m so precious about words.”

Pete Doherty (1979) English musician, writer, actor, poet and artist

Speaking out on the media's caricature of him
People

Truth, Photo

“Out of fifty mathematical papers presented in brief at such a meeting, it is a rare mathematician indeed who really understands what more than half a dozen are about.”

Eric Temple Bell (1883–1960) mathematician and science fiction author born in Scotland who lived in the United States for most of his li…

Source: Mathematics: Queen and Servant of Science (1938), p. 7

Understanding

“The subject matter of mathematics has increased so rapidly and extensively that there is some element of truth in maintaining that mathematics is not so much a subject as a way of studying any subject, not so much a science as a way of life. We turn, then, from the attempt to characterize the material object of mathematics to an attempt to determine its formal object, i. e., its methodology.”

George Frederick James Temple (1901–1992) British mathematician

100 Years of Mathematics: a Personal Viewpoint (1981)

Life, Truth, Way, Study

“To investigate the essence of a natural phenomenon, three conditions are necessary: We must first study and know the phenomenon itself, from all sides; we must then determine in what relation it stands to other natural phenomena; and lastly, when we have ascertained all these relations, we have to solve the problem of measuring these relations and the laws of mutual dependence—that is, of expressing them in numbers. In the first period of chemistry, all the powers of men's minds were devoted to acquiring a knowledge of the properties of bodies; it was necessary to discover, observe, and ascertain their peculiarities. This is the alchemistical period. The second period embraces the determination of the mutual relations or connections of these properties; this is the period of phlogistic chemistry. In the third period, in which we now are, we ascertain by weight and measure and express in numbers the degree in which the properties of bodies are mutually dependent. The inductive sciences begin with the substance itself, then come just ideas, and lastly, mathematics are called in, and with the aid of numbers, complete the work.”

Justus von Liebig (1803–1873) German chemist

Familiar Letters on Chemistry, Tr. Blythe, 4th ed., London, 1859, p. 60 as quoted by John Theodore Merz, A History of European Thought in the Nineteenth Century Vol.1 http://books.google.com/books?id=xqwQAAAAYAAJ (1903).

Men, Law, Idea, Nature

“We can now return to the distinction between language and symbolism. A symbol is language and yet not language. A mathematical or logical or any other kind of symbol is invented to serve a purpose purely scientific; it is supposed to have no emotional expressiveness whatever. But when once a particular symbolism has been taken into use and mastered, it reacquires the emotional expressiveness of language proper. Every mathematician knows this. At the same time, the emotions which mathematicians find expressed in their symbols are not emotions in general, they are the peculiar emotions belonging to mathematical thinking.”

R. G. Collingwood (1889–1943) British historian and philosopher

Source: The Principles of Art (1938), p. 268

Thinking, Time

“Actually, even with tested missiles, results of attacks are not really mathematically predictable. The probability of extreme variations in performance, the upper and lower limits, cannot be calculated accurately. But laymen or narrow professionals persist in regarding the matter as a simple problem in engineering and physics.”

Herman Kahn (1922–1983) American futurist

The Magnum Opus; On Thermonuclear War

Problem

“The history of mathematics may be instructive as well as agreeable; it may not only remind us of what we have, but may also teach us to increase our store. Says De Morgan, "The early history of the mind of men with regards to mathematics leads us to point out our own errors; and in this respect it is well to pay attention to the history of mathematics." It warns us against hasty conclusions; it points out the importance of a good notation upon the progress of the science; it discourages excessive specialization on the part of the investigator, by showing how apparently distinct branches have been found to possess unexpected connecting links; it saves the student from wasting time and energy upon problems which were, perhaps, solved long since; it discourages him from attacking an unsolved problem by the same method which has led other mathematicians to failure; it teaches that fortifications can be taken by other ways than by direct attack, that when repulsed from a direct assault it is well to reconnoitre and occupy the surrounding ground and to discover the secret paths by which the apparently unconquerable position can be taken.”

Florian Cajori book A History of Mathematics

Source: A History of Mathematics (1893), pp. 1-2; Cited in: Robert Edouard Moritz. Memorabilia mathematica; or, The philomath's quotation-book https://archive.org/stream/memorabiliamathe00moriiala#page/198/mode/2up, (1914) p. 90; Study and research in mathematics

Men, Secret, Way, Error

“Galileo had provided the methodology for the analysis of motions on and near the earth and had applied it successfully. Copernicus and Kepler had previously obtained the laws of motion of the planets and their satellites. …But Galileo had succeeded in deriving numerous laws from a few physical principles and… the axioms and theorems of mathematics. …The Keplerian laws …were not logically related to each other. Each was an independent inference from observations. …They seemed to be suspended in the same vacuum in which the planets moved.
Galileo's laws had the additional advantage of supplying physical insight. The first law of motion and the law that the force of graviation gives… a downward acceleration of 32 ft/sec2… explain the vertrical rise and fall of bodies, motion on slopes, and projectile motion. Kepler's laws… had no physical basis. …Kepler tried to introduce the idea of a magnetic force which the sun exerted… But he failed to related the behavior of the planets to the precise laws of planetary motion. …
The new astronomical theory was completely isolated from the theory of motion on earth. …it bothered mathematicians and scientists who believed that all the phenomena of the universe were governed by one master plan instituted by the master planner—God.”

Morris Kline (1908–1992) American mathematician

Source: Mathematics and the Physical World (1959), pp. 224-225

God, Law, Behavior, Idea

“We do not aim at « mathematical rigour » of exposition, which in theoretical physics often amounts to self-deception.”

Evgeny Lifshitz (1915–1985) Soviet physicist

"Quantum Mechanics: Non-relativistic Theory", together with L. D. Landau, translated by John Menzies (third edition 1991), p. xi

“After logic we must proceed to philosophy proper. Here too we have to learn from our predecessors, just as in mathematics and law. Thus it is wrong to forbid the study of ancient philosophy. Harm from it is accidental, like harm from taking medicine, drinking water, or studying law.”

Averroes (1126–1198) Medieval Arab scholar and philosopher

Ch 1.9
The Decisive Treatise

Law, Drink, Study, Learning

“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.”

Richard Courant (1888–1972) German American mathematician (1888-1972)

Richard Courant in: The Australian Mathematics Teacher, Volumes 39-40 http://books.google.co.in/books?id=CofxAAAAMAAJ, Australian Association of Mathematics Teachers, 1983, p. 3

Life, Science, Perfection

“I shall now address you on the subject of the present situation in research in the foundations of mathematics. Since there remain open questions in this field, I am not in a position to paint a definitive picture of it for you. But it must be pointed out that the situation is not so critical as one could think from listening to those who speak of a foundational crisis. From certain points of view, this expression can be justified; but it could give rise to the opinion that mathematical science is shaken at its roots.”

Paul Bernays (1888–1977) Swiss mathematician

Paul Bernays, Platonism in mathematics http://sites.google.com/site/ancientaroma2/book_platonism.pdf (1935) Lecture delivered June 18, 1934, in the cycle of Conferences internationales des Sciences mathematiques organized by the University of Geneva, in the series on Mathematical Logic.) Translation by: Charles Parsons

Question, Thinking, Science, Painting

“Homeric mind is ingenuity, practical intelligence. There is no Rodin-like deep thinking, no mathematical or philosophical speculation. Odysseus thinks with his hands.”

Camille Paglia (1947) American writer

Source: Sexual Personae: Art and Decadence from Nefertiti to Emily Dickinson (1990), p. 85

Intelligence, Thinking

“Now what are we looking at right here? What’s bang in the middle? Some people think that’s mathematics there. Fine. But if maths is the study that best allows you to think your way to the centre, what’re the forces you’re investigating? Maths is totally abstract, at one level, square roots of minus one and the like, but the world is nothing if not rigorously mathematical. So this is a way of looking at the world which unifies all the forces: mental, social, physical.”

China Miéville book Perdido Street Station

Source: Perdido Street Station (2000), pp. 145-146

People, World, Way, Thinking

“1. We are the contemporaries of a third epoch of science after the Greek and Galilean. The caesura which opens this third epoch is not (a with the Greek) an invention - that of demonstrative mathematics - nor is it (like the Galilean) a break - that which mathematized the discourse of physics. It is a split, through, which the very nature of the base of mathematical rationality reveals itself, as does the character of hte dision of thought which establishes it.”

Alain Badiou (1937) French writer and philosopher

Introduction
Being and Event (1988)

Nature, Science

“We want the mathematical genius - there is work for him. We want the literary genius - there is work for him, especially in my office. We want the scientific brain - there is more than enough work for him. We want the man of brains and we want the man of common sense and little brain. We want the man of initiative and the man of action, the methodical man and even the crank. We open our ranks widely to all.”

Hugh Trenchard, 1st Viscount Trenchard (1873–1956) Royal Flying Corps commander and first Royal Air Force Chief of the Air Staff

Speech to the Cambridge University Aeronautical Society, April 1925 in Trenchard, Man of Vision (1962) p. 519

Sense

“[The books of Euclid pass on to us] something admirable and very necessary to see and to read, namely the order in the method of writing on mathematics in that aforementioned time of the wise age.”

Simon Stevin (1548–1620) Flemish scientist, mathematician and military engineer

Géographie, in Les Oeuvres Mathématiques de Simon Stevin de Bruges (1634) ed. Girard, p. 109, as quoted by Jacob Klein]], Greek Mathematical Thought and the Origin of Algebra (1968)

Books, Age, Time, Reading

“And last of all we have the secondary forms of crystals bursting in upon us, and sparkling in the rigidity of mathematical necessity and telling us, neither of harmony of design, usefulness or moral significance, — nothing but spherical trigonometry and Napier's analogies. It is because we have blindly excluded the lessons of these angular bodies from the domain of human knowledge that we are still in doubt about the great doctrine that the only laws of matter are those which our minds must fabricate, and the only laws of mind are fabricated for it by matter.”

James Clerk Maxwell (1831–1879) Scottish physicist

Essay "Analogies in Nature" (February 1856), reprinted in The Scientific Letters and Papers of James Clerk Maxwell: 1846-1862 edited by P.M. Harman, p. 376 (the quote appears on p. 383 http://books.google.com/books?id=zfM8AAAAIAAJ&lpg=PP1&pg=PA383#v=onepage&q&f=false)

Law, Knowledge

“Whether game theory leads to clear-cut solutions, to vague solutions, or to impasses, it does achieve one thing. In bringing techniques of logical and mathematical analysis gives men an opportunity to bring conflicts up from the level of fights, where the intellect is beclouded by passions, to the level of games, where the intellect has a chance to operate.”

Anatol Rapoport (1911–2007) Russian-born American mathematical psychologist

Source: 1960s, "The Use and Misuse of Game Theory," 1962, p. 114

Men, Game, Passion, Chance

“Moreover, according to Alexander, the “troubled mathematical martyr,” exemplified not only by Galois but also by Abel, János Bolyai, Riemann, Cantor, Gödel, Turing, John Nash, Grothendieck, Perelman, and even, in a certain sense, Cauchy, remains to this day the dominant image of the “ideal mathematician,” long after the romantic paradigm was exhausted in the arts.”

Michael Harris (1954) American mathematician

Source: Mathematics without Apologies: Portrait of a Problematic Vocation, 2015, pp. 147-148

Art, Sense

“Unfortunately, the critics of economics have had a tendency to discuss the whole structure as a tissue of misconceptions. It is a critique that fails. The strength of economics is its considerable, if far from complete, understanding of the flows and comparative advantages that underlie trade, jobs, capital and incomes, and the logic of optimising behaviour, all backed by glittering accomplishment in mathematics. That makes it a powerful analytical instrument, so that just a few misconceptions – such as a failure to understand the informal economy or resource depletion – have leverage: like a baby monkey at the controls of a Ferrari, they can turn it into an instrument with extraordinarily destructive potential. If it were a tissue of errors, it would not be dangerous: it is its 90 percent brilliance which makes it so.”

David Fleming (1940–2010) British activist

Lean Logic, (2016), p. 123, entry on Economics http://www.flemingpolicycentre.org.uk/lean-logic-surviving-the-future/

Error, Failure, Understanding, Baby

“Three principles — the conformability of nature to herself, the applicability of the criterion of simplicity, and the "unreasonable effectiveness" of certain parts of mathematics in describing physical reality — are thus consequences of the underlying law of the elementary particles and their interactions. Those three principles need not be assumed as separate metaphysical postulates. Instead, they are emergent properties of the fundamental laws of physics.”

Murray Gell-Mann (1929–2019) American physicist

TED talk on beauty and truth in physics —video timecode 14m28s (March 2007) http://ted.com/index.php/talks/murray_gell_mann_on_beauty_and_truth_in_physics.html.

Law, Nature, Reality, Parting

“The illiterate expression ‘given data’ constantly recurs in Lange. It appears to have an irresistable attraction to mathematical economists because it doubly assures them that they know what they do not know. It seems to bewitch them into making assertions about the real world for which they have no empirical justification whatever.”

Friedrich Hayek (1899–1992) Austrian and British economist and Nobel Prize for Economics laureate

1980s and later, "Two Pages of Fiction" (1982)

World, Appearance

“Newton… (after having remarked that geometry only requires two of the mechanical actions which it postulates, namely, to describe a straight line and a circle) says: geometry is proud of being able to achieve so much while taking so little from extraneous sources. One might say of metaphysics, on the other hand: it stands astonished, that with so much offered it by pure mathematics it can effect so little. In the meantime, this little is something which mathematics indispensably requires in its application to natural science, which, inasmuch as it must here necessarily borrow from metaphysics, need not be ashamed to allow itself to be seen in company with the latter.”

Immanuel Kant (1724–1804) German philosopher

Preface, Tr. Bax (1883) citing Isaac Newton's Principia
(1786)

Nature, Science

“The word 'definition' has come to have a dangerously reassuring sound, owing no doubt to its frequent occurrence in logical and mathematical writings.”

Willard van Orman Quine (1908–2000) American philosopher and logician

"Two dogmas of Empiricism", p. 26
From a Logical Point of View: Nine Logico-Philosophical Essays (1953)

“The field of mathematics is now so extensive that no one can [any] longer pretend to cover it, least of all the specialist in any one department. Furthermore it takes a century or more to weigh men and their discoveries, thus making the judgment of contemporaries often quite worthless.”

David Eugene Smith (1860–1944) American mathematician

David Eugene Smith, History of Modern Mathematics https://books.google.com/books?id=EwcCAAAAYAAJ, 1896; 1904

Men

“All natural philosophers, who wished to proceed mathematically in their work, have hence invariably (although unknown to themselves) made use of metaphysical principles, and must make use of such, it matters not how energetically they may otherwise repudiate any claim of metaphysics on their science.”

Immanuel Kant (1724–1804) German philosopher

Preface, Tr. Bax (1883)
(1786)

Nature, Science

“Mathematics, in the development of its ideas, has only to take account of the immanent reality of its concepts and has absolutely no obligation to examine their transient reality.”

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

From Kant to Hilbert (1996)

Idea, Reality

“This was the might of France, the pride of France, the tactic of the world's first conscript army, and this column, Clausel's counter-attack, ignored cold mathematical logic. It was not defeated by the line.”

Bernard Cornwell (1944) British writer

Narrator, p. 253
Sharpe (Novel Series), Sharpe's Sword (1983)

World

“We shed as we pick up, like travellers who must carry everything in their arms, and what we let fall will be picked up by those behind. The procession is very long and life is very short. We die on the march. But there is nothing outside the march so nothing can be lost to it. The missing plays of Sophocles will turn up piece by piece, or be written again in another language. Ancient cures for diseases will reveal themselves once more. Mathematical discoveries glimpsed and lost to view will have their time again. You do not suppose, my lady, that if all of Archimedes had been hiding in the great library of Alexandria, we would be at a loss for a corkscrew?”

Tom Stoppard Arcadia

Septimus, Act I
Arcadia (1993)

Life, Disease, Travel, Loss

“You live in an age that is dominated by science and engineering. …Thus if you wish to be effective in the world and to achieve the things that you want, it is necessary to understand both science and engineering”

Richard Hamming (1915–1998) American mathematician and information theorist

and those require mathematics
Methods of Mathematics Applied to Calculus, Probability, and Statistics (1985)

Age, World, Understanding, Science

“We must… maintain that mathematical geometry is not a science of space insofar as we understand by space a visual structure that can be filled with objects - it is a pure theory of manifolds.”

Hans Reichenbach (1891–1953) American philosopher

The Philosophy of Space and Time (1928, tr. 1957)

Understanding, Science

“As mathematical and absolute certainty is seldom to be attained in human affairs, reason and public utility require that judges and all mankind in forming their opinions of the truth of facts should be regulated by the superior number of the probabilities on the one side or the other whether the amount of these probabilities be expressed in words and arguments or by figures and numbers.”

William Murray, 1st Earl of Mansfield (1705–1793) British judge

Reported in Andrew Stuart, Letters to the Right Honorable Lord Mansfield (1773), p. 29.

Truth

“In the last 30 years, the prestige of mathematics has declined in all countries. I think that mathematicians are partially to be blamed as well—foremost, Hilbert and Bourbaki—the ones who proclaimed that the goal of their science was investigation of all corollaries of arbitrary systems of axioms.”

Vladimir I. Arnold (1937–2010) Russian mathematician

Interview translated from the Russian into English and republished in the book Boris A. Khesin; Serge L. Tabachnikov (editors), Arnold: Swimming Against the Tide (2014) Google Books preview http://books.google.com/books?id=aBWHBAAAQBAJ&pg=PA4 pages 4–5.

Thinking, Science, Country

“I came into history from a primary concern with mathematics and science. This has been a tremendous help to me as a person and as a historian, although it must be admitted it has served to make my historical interpretations less conventional than may be acceptable of many of my colleagues in the field.”

Carroll Quigley (1910–1977) American historian

Preface to the First Edition, p. 27
The Evolution of Civilizations (1961) (Second Edition 1979)

History, Science, Help, Personality

“[I]nterest, say in mathematics, has usually been killed by routine teaching, exactly as the literary interest… has been killed…”

Jacob Bronowski (1908–1974) Polish-born British mathematician

"Sense and Sensibility"
The Common Sense of Science (1951)

“Mathematics my foot! Algorithms are mathematics too, and often more interesting and definitely more useful.”

Doron Zeilberger (1950) Israeli mathematician

The Narrow-Minded and Ignorant Referee's Report [and Zeilberger's Response] of Zeilberger's Paper "Automaric CounTilings" that was rejected by Helene Barcelo and the Members of the Advisory Board [that includes(!) Enumeration Expert Mireille Bousquet-Melou] of the Journal of Combinatorial Theory-Series A. http://www.math.rutgers.edu/~zeilberg/RefTipesh.html

“[Describing the bazar held in Delhi near the Red Fort, Francois Bernier (seventeenth century) says that] “Hither, likewise, the astrologers resort, both Mahometan and Gentile. These wise doctors remain seated in the sun, on a dusty piece of carpet, handling some old mathematical instruments, and having open before them a large book which represents the sign of the Zodiac. In this way they attract the attention of the passenger… by whom they are considered as so many infallible oracles. They tell a poor person his fortune for a payssa… Silly women, wrapping themselves in a white cloth from head to foot, flock to the astrologers, whisper to them all the transaction of their lives, and disclose every secret with no more reserve than is practised by a penitent in the presence of her confessor. The ignorant and infatuated people really believe that the stars have an influence (on their lives) which the astrologers can control.””

François Bernier (1620–1688) French physician and traveller

Quoted from Lal, K. S. (1992). The legacy of Muslim rule in India. New Delhi: Aditya Prakashan. Chapter 1
Travels in the Mogul Empire (1656-1668)

Women, People, Secret, Books

“We are mainly interested in the processes… not… in presenting mathematics in its most abstract form. …we will often begin with concrete forms and then exhibit the process of abstraction.”

Richard Hamming (1915–1998) American mathematician and information theorist

Methods of Mathematics Applied to Calculus, Probability, and Statistics (1985)

“Mathematicians are extremely lucky, they are paid for doing what they would by nature have to do anyway. One should not have a non-teaching fellowship too long, there comes a time when one must make a contribution to society. Great mathematics is achieved by solving difficult problems not by fabricating elaborate theories in search of a problem.”

Harold Davenport (1907–1969) English Mathematician

Harold Davenport, ‎Bryan John Birch, ‎Heini Halberstam (1977), The collected works of Harold Davenport. p. xviii

Search, Nature, Problem, Time

“It is clear that the building of models is not a purely mechanical process but requires skill of a high order – not merely mathematical skill but a sensitivity to the relative importance of different factors and a critical, almost an artistic, faculty in the selection of behaviour equations which are reasonable, tentative hypotheses in explaining the behaviour of actual economies.”

Kenneth E. Boulding (1910–1993) British-American economist

Source: 1950s, The Skills of the Economist, 1958, p. 16-17 as cited in Andrew Mearman (2011).

“The calculus was the first achievement of modern mathematics and it is difficult to overestimate its importance. I think it defines more unequivocally than anything else the inception of modern mathematics; and the system of mathematical analysis, which is its logical development, still constitutes the greatest technical advance in exact thinking.”

John Von Neumann (1903–1957) Hungarian-American mathematician and polymath

As quoted in Bigeometric Calculus: A System with a Scale-Free Derivative (1983) by Michael Grossman, and in Single Variable Calculus (1994) by James Stewart.

Thinking

“Our children grow up in a culture permeated with the idea that there are "smart people" and "dumb people". The social construction of the individual is as a bundle of aptitudes. There are people who are "good at math" and people who "can't do math." Everything is set up for children to attribute their first unsuccessful or unpleasant learning experiences to their own disabilities. As a result, children perceive failure as relegating them either to the group of "dumb people" or, more often, to a group of people "dumb at x" (where, as we have pointed out, x often equals mathematics). Within this framework children will define themselves in terms of their limitations, and this definition will be consolidated and reinforced throughout their lives. Only rarely does some exceptional event lead people to reorganize their intellectual self-image in such a way as to open up new perspectives on what is learnable.”

Seymour Papert book Mindstorms: Children, Computers, and Powerful Ideas

Source: Mindstorms: Children, Computers, and Powerful Ideas (1980), Chapter 2, Mathophobia: The Fear of Learning

Children, People, Way, Idea

“There are three intellectual pursuits, and, so far as I am aware, only three, in which human beings have performed major feats before the age of puberty. They are music, mathematics, and chess.”

George Steiner (1929–2020) American writer

"A Death of Kings"
Extraterritorial (1971)

Age, Music

“The last thirty years [1925 to 1955] have seen an enormous improvement in the position of geometry as a branch of mathematics, or, rather, have seen the re-integration of geometry into the main fabric of mathematics. Indeed, one can go further and say that with the restoration of geometry to its rightful place in the mathematical scheme the process of fragmentation which had been doing so much harm to mathematics has been reversed, and we may look forward to the day in which there are no longer analysts, algebraists, geometers and so on, but simply mathematicians. Mathematical research has two aspects, motivation and technique, and when the latter gains control the result is apt to be excessive specialisation. The revolution of geometrical thought, and the reinstatement of geometry as one of the major mathematical disciplines, have helped to bring about a unification of mathematics which we may justly regard as one of the major contributions of the last quarter century to the subject.”

W. V. D. Hodge (1903–1975) British mathematician

W. V. D. Hodge, Changing Views of Geometry. Presidential Address to the Mathematical Association, 14th April, 1955, The Mathematical Gazette 39 (329) (1955), 177-183.

Motivation, Help