Quotes about mathematics
page 5

“What I have achieved has been largely a matter of chance. Many problems I thought about at length with no success. With other problems, there was the inspiration—indeed, some that astound me today. Certainly the best times were when I was alone with mathematics, free of ambition and of pretense, and indifferent to the world.”

Robert Langlands (1936) Canadian mathematician

Source: "Mathematical retrospections," 2013, p. 142

World, Problem, Chance, Time

“As in Mathematics in general, the really great advances, embodying new ideas of far-reaching fruitfullness, have been due to an exceedingly small number of great men… there are periods when for a long series of centuries no advance was made; when the results obtained in a more enlightened age have been forgotten. We observe the times of revival, when the older learning has been rediscovered, and when the results of the progress made in distinct countries have been made available as the starting points of new efforts and a fresh period of activity.”

E. W. Hobson (1856–1933) British mathematician

Source: Squaring the Circle (1913), p. 10

Men, Age, Idea, Effort

“The study of mathematics is apt to commence in disappointment… We are told that by its aid the stars are weighed and the billions of molecules in a drop of water are counted. Yet, like the ghost of Hamlet's father, this great science eludes the efforts of our mental weapons to grasp it.”

Alfred North Whitehead (1861–1947) English mathematician and philosopher

Source: 1910s, An Introduction to Mathematics (1911), ch. 1.

Disappointment, Star, Effort, Study

“When, as a result of what was called Enlightenment in the eighteenth century, the priests had in fact almost entirely lost this function of guidance. Their place was taken by writers and scientists. In both cases it is equally absurd. Mathematics, physics, and biology are as remote from spiritual guidance as the art of arranging words. When that function is usurped by literature and science it proves there is no longer any spiritual life.”

Simone Weil (1909–1943) French philosopher, Christian mystic, and social activist

“Morality and literature,” pp. 164-165
On Science, Necessity, and the Love of God (1968)

Life, Art, Literature, Science

“Mathematics is the science which draws necessary conclusions.”

Benjamin Peirce Linear Associative Algebra

§ 1.
Linear Associative Algebra (1882)

Science, Drawing

“A work of genius is a complex object and there is light to be shed about what went into the making of it. Even in the case of scientific and mathematical achievement we can say a great deal about the existing state of knowledge, the problems it failed to address, why those problems were or had become interesting, the particular capacities which the solver brought to their solution. At the other extreme is a work of art which seems to have much more immediate roots in the personal life of the artist or writer. In between is the area in which Keynes worked, which was partly scientific, partly artistic. This gives a wide justification for a biographical approach. As I put it in the introduction to my first volume: 'If underlying Keynesian theory was Keynes's vision of his age, knowledge of his state of mind and the circumstances which formed it is essential, not only in order to understand how he came to see the world as he did, but also in order to pass judgement on the theory itself.”

Robert Skidelsky (1939) Economist and author

John Maynard Keynes: 1883-1946: Economist, Philosopher, Statesman (2003), Introduction

Life, Age, World, Art

“Nonsense,” Weston denied, with an involuntary nervous shiver. “That’s completely ridiculous. We had a long discussion at the time we bought Robbie about the First Law of Robotics. You know that it is impossible for a robot to harm a human being; that long before enough can go wrong to alter that First Law, a robot would be completely inoperable. It’s a mathematical impossibility. Besides I have an engineer from U. S. Robots here twice a year to give the poor gadget a complete overhaul. Why, there’s no more chance of anything at all going wrong with Robbie than there is of you or I suddenly going looney—considerably less, in fact. Besides, how are you going to take him away from Gloria?”

Isaac Asimov book I, Robot

“Robbie”, p. 17
I, Robot (1950)

Law, Chance, Time

“I discovered that a whole range of problems of the most diverse character relating to the scientiﬁc organization of production (questions of the optimum distribution of the work of machines and mechanisms, the minimization of scrap, the best utilization of raw materials and local materials, fuel, transportation, and so on) lead to the formulation of a single group of mathematical problems (extremal problems). These problems are not directly comparable to problems considered in mathematical analysis. It is more correct to say that they are formally similar, and even turn out to be formally very simple, but the process of solving them with which one is faced [i. e., by mathematical analysis] is practically completely unusable, since it requires the solution of tens of thousands or even millions of systems of equations for completion.
I have succeeded in ﬁnding a comparatively simple general method of solving this group of problems which is applicable to all the problems I have mentioned, and is sufﬁciently simple and effective for their solution to be made completely achievable under practical conditions.”

Leonid Kantorovich (1912–1986) Russian mathematician

Kantorovich (1960) "Mathematical Methods of Organizing and Planning Production." Management Science, 6(4):366–422, 1960, p. 368); As cited in: Cockshott, W. Paul. " Mises, Kantorovich and economic computation http://www.dcs.gla.ac.uk/publications/PAPERS/8707/standalonearticle.pdf." (2007).

Question, Problem

“Regardless of whether or not God exists, God has no place in mathematics, at least in my book.”

Doron Zeilberger (1950) Israeli mathematician

An Enquiry Concerning Human (and Computer!) [Mathematical] Understanding C.S. Calude, ed., "Randomness & Complexity, from Leibniz to Chaitin", World Scientific, Singapore, (October 2007)

God, Books

“Mathematical knowledge may, just like our scientific knowledge, be deep and broad, it may be subtle and wonderfully explanatory, it may be uncontroversially accepted; but it cannot be certain.”

David Deutsch book The Fabric of Reality

Source: The Fabric of Reality (1997), Ch. 10

Knowledge

“One reason why mathematics enjoys special esteem, above all other sciences, is that its laws are absolutely certain and indisputable, while those of other sciences are to some extent debatable and in constant danger of being overthrown by newly discovered facts.”

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

1920s, Sidelights on Relativity (1922)

Law, Science

“Mathematics is one of the essential emanations of the human spirit, a thing to be valued in and for itself, like art or poetry.”

Oswald Veblen (1880–1960) American mathematician

Bulletin of the American Mathematical Society, Volume 30 (1924), p. 289.

Art

“In mathematics, logic, linguistics, and other abstract disciplines, the systems are not assigned to objects. They are defined by an enumeration of the variables, their admissible values, and their algebraic, topological, grammatical, and other properties which, in the given case, determine the relations between the variables under consideration.”

George Klir (1932–2016) American computer scientist

Source: An approach to general systems theory (1969), p. 40.

“The truth is that the mathematical sciences are growing in complete security and harmony. The ideas of Dedekind, Poincare, and Hilbert have been systematically developed with great success, without any conflict in the results. It is only from the philosophical point of view that objections have been raised. They bear on certain ways of reasoning peculiar to analysis and set theory. These modes of reasoning were first systematically applied in giving a rigorous form to the methods of the calculus. [According to them, ] the objects of a theory are viewed as elements of a totality such that one can reason as follows: For each property expressible using the notions of the theory, it is [an] objectively determinate [fact] whether there is or there is not an element of the totality which possesses this property. Similarly, it follows from this point of view that either all the elements of a set possess a given property, or there is at least one element which does not possess it”

Paul Bernays (1888–1977) Swiss mathematician

Paul Bernays, Platonism in mathematics http://sites.google.com/site/ancientaroma2/book_platonism.pdf (1935)

Truth, Way, Idea, Science

“Despite the vociferous claims of the Platonists and Neoplatonists, Plato was not a mathematician. To Plato and his followers mathematics was largely a means to an end… they viewed the technical aspects of mathematics as a mere device for sharpening one's wits, or at most a course of training peparatory to handling the larger issues of philosophy. This is reflected in the very name "mathematics,"… a course of studies or… a curriculum. …in the Dialogues… such topics as harmony, triangular numbers, figurate numbers… which we view today as more or less irrelevant, if not trivial, were taken up at length. …the guiding motive behind the… Pythagoreans and Platonists was… metaphysical …which for the nonprofessional have all the earmarks of the occult.”

Tobias Dantzig (1884–1956) American mathematician

...We also discover in the Pythagorean speculations more than a mere germ of... the scientific attitude.
The Bequest of the Greeks (1955)

Motivation, Study, Training

“However, in all honesty, I must say that one must essentially forget that all proofs are transcribed in this formal language. In order to think productively, one must use all the intuitive and informal methods of reasoning at one's disposal. At the very end one must check that no errors have been committed; but in practice set theory is treated as any other branch of mathematics. The reason that we can do this is that we will never speak about proofs but only about models.”

Paul Cohen (1934–2007) American mathematician

p. 1078 of "The discovery of forcing." http://www.logic.univie.ac.at/~ykhomski/ST2013/The%20Discovery%20of%20Forcing.pdf Rocky Mountain Journal of Mathematics 32, no. 4 (2002): 1071–1100.

Error, Thinking, Forgetting

“In electromagnetism… the law of the inverse square had been supreme, but, as a consequence of the work of Faraday and Maxwell, it was superseded by the field. And the same change took place in the theory of gravitation. By and by the material particles, electrically charged bodies, and magnets which are the things that we actually observe come to be looked upon only as "singularities" in the field. So far this transformation from the force to the potential, from the action at a distance to the field, is only a purely mathematical operation.”

Willem de Sitter (1872–1934) Dutch cosmologist

Kosmos (1932), Above is Beginning Quote of the Last Chapter: Relativity and Modern Theories of the Universe -->

Law, Change

“Thanks to my memory, which enabled me to quote Latin and to discuss Greek and Roman civilization, it became obvious to some of my colleagues in other fields that I was interested in things outside mathematics. This lead quickly to very pleasant relationships.”

Stanislaw Ulam (1909–1984) Polish-American mathematician

Source: Adventures of a Mathematician - Third Edition (1991), Chapter 7, The University of Wisconsin, p. 125

Quotes, Memory

“One of the first applications of the simplex algorithm was to the determination of an adequate diet that was of least cost. In the fall of 1947, Jack Laderman of the Mathematical Tables Project of the National Bureau of Standards undertook, as a test of the newly proposed simplex method, the first large-scale computation in this field. It was a system with nine equations in seventy-seven unknowns. Using hand-operated desk calculators, approximately 120 man-days were required to obtain a solution. … The particular problem solved was one which had been studied earlier by George Stigler (who later became a Nobel Laureate) who proposed a solution based on the substitution of certain foods by others which gave more nutrition per dollar. He then examined a "handful" of the possible 510 ways to combine the selected foods. He did not claim the solution to be the cheapest but gave his reasons for believing that the cost per annum could not be reduced by more than a few dollars. Indeed, it turned out that Stigler's solution (expressed in 1945 dollars) was only 24 cents higher than the true minimum per year $39.69.”

George Dantzig (1914–2005) American mathematician

cited in: John J. O'Connor & Edmund F.; Robertson (2003) " George Dantzig http://www-history.mcs.st-and.ac.uk/Biographies/Dantzig_George.html". in: MacTutor History of Mathematics archive, University of St Andrews. Linear programming and extensions (1963)

Food, Way, Problem, Study

“The confidence placed in physical theory owes much to its possessing the same kind of excellence from which pure geometry and pure mathematics in general derive their interest, and for the sake of which they are cultivated. … We cannot truly account for our acceptance of such theories without endorsing our acknowledgement of a beauty that exhilarates and a profundity that entrances us.”

Michael Polanyi (1891–1976) Hungarian-British polymath

Source: Personal Knowledge (1958), p. 15

Beauty, Confidence

“The nude gains its enduring value from the fact that it reconciles several contrary states. It takes the most sensual and immediately interesting object, the human body, and puts it out of reach of time and desire; it takes the most purely rational concept of which mankind is capable, mathematical order, and makes it a delight to the senses; and it takes the vague fears of the unknown and sweetens them by showing that the gods are like men and may be worshiped for their life-giving beauty rather than their death-dealing powers.”

Kenneth Clark (1903–1983) Art historian, broadcaster and museum director

Source: The Nude: A Study in Ideal Form (1951), Ch. I: The Naked and the Nude

Life, Death, Men, God

“I want to do kiddie movies now. I'm fed up with adult movies — most of them stink. At a certain point with movies it becomes all about mathematics: this has to lead up to this, this has to lead up to that — you're always bound by some kind of formula. But since having kids and watching lots of animated cartoons and all those great old Disney films, I think they're better, they're much better. They're more fun and they take more risks.”

Johnny Depp (1963) American actor, film producer, and musician

Quoted in Jessamy Calkin, "One That Got Away," http://www.johnnydeppfan.com/interviews/telegraphmag.htm The Daily Telegraph Magazine (2002-08-03)

Thinking, Animals

“Motion with respect to the universal ocean of aether eludes us. We say, "Let V be the velocity of a body through the aether", and form the various electromagnetic equations in which V is scattered liberally. Then we insert the observed values, and try to eliminate everything which is unknown except V. The solution goes on famously; but just as we have got rid of all the other unknowns, behold! V disappears as well, and we are left with the indisputable but irritating conclusion —
::: 0 = 0
This is a favourite device that mathematical equations resort to, when we propound stupid questions.”

Arthur Stanley Eddington (1882–1944) British astrophysicist

Source: The Nature of the Physical World (1928), Ch. 2 Relativity

Stupidity, Question, Universe

“That all men are created equal does not mean that human beings are the same, or equal, in size, strength, beauty, virtue, or intelligence. There are obviously great differences in individual aptitudes and talents in sports, music, mathematics, speaking, and writing. They are also unequal in the virtues, among them courage, temperance, and justice. But as Jefferson once said, the fact that Sir Isaac Newton may be the most intelligent of living human beings does not give him any right whatever to my person or my property.”

Harry V. Jaffa (1918–2015) American historian and collegiate professor

2000s, God Bless America (2008), The American Proposition

Men, Justice, Beauty, Music

“The young mathematical disciple 'topology' might be of some help in making psychology a real science.”

Kurt Lewin (1890–1947) German-American psychologist

Source: 1930s, Principles of topological psychology, 1936, p. vii.

Science, Help

“Chaotic theory is mathematically based on non-linear propositions, "meaning that they expressed relationships that were not strictly proportional. Linear relationships can be captured with a straight line on a graph"”

James Gleick book Chaos: Making a New Science

Source: Chaos: Making a New Science, 1987, p. 23 as cited in John A. Rush (1996), Clinical Anthropology: An Application of Anthropological Concepts, p. 75

“Pure mathematics is a collection of hypothetical, deductive theories, each consisting of a definite system of primitive, undefined, concepts or symbols and primitive, unproved, but self-consistent assumptions (commonly called axioms) together with their logically deducible consequences following by rigidly deductive processes without appeal to intuition.”

Joshua Girling Fitch (1824–1903) British educationalist

The Fourth Dimension simply Explained. (New York, 1910), p. 58. Reported in Moritz (1914); Also cited in: Howard Eves (2012), Foundations and Fundamental Concepts of Mathematics, p. 167

“The university immediately published my pamphlet, and it was sent to ﬁfty People’s Commissariats. It was distributed only in the Soviet Union, since in the days just before the start of the World War it came out in an edition of one thousand copies in all.
Soviet Union, since in the days just before the start of the World War it came out in an edition of one thousand copies in all. The number of responses was not very large. There was quite an interesting reference from the People’s Commissariat of Transportation in which some optimization problems directed at decreasing the mileage of wagons was considered, and a good review of the pamphlet appeared in the journal "The Timber Industry."
At the beginning of 1940 I published a purely mathematical version of this work in Doklady Akad. Nauk [76], expressed in terms of functional analysis and algebra. However, I did not even put in it a reference to my published pamphlet—taking into account the circumstances I did not want my practical work to be used outside the country
In the spring of 1939 I gave some more reports—at the Polytechnic Institute and the House of Scientists, but several times met with the objection that the work used mathematical methods, and in the West the mathematical school in economics was an anti-Marxist school and mathematics in economics was a means for apologists of capitalism. This forced me when writing a pamphlet to avoid the term "economic" as much as possible and talk about the organization and planning of production; the role and meaning of the Lagrange multipliers had to be given somewhere in the outskirts of the second appendix and in the semi Aesopian language.”

Leonid Kantorovich (1912–1986) Russian mathematician

L.V. Kantorovich (1996) Descriptive Theory of Sets and Functions. p. 41; As cited in: K. Aardal, ‎George L. Nemhauser, ‎R. Weismantel (2005) Handbooks in Operations Research and Management Science, p. 19-20

War, People, World, School

“It has never been in my power to study anything, — mathematics, ethics, metaphysics, gravitation, thermodynamics, optics, chemistry, comparative anatomy, astronomy, psychology, phonetics, economics, the history of science, whist, men and women, wine, metrology, except as a study of semeiotic.”

Charles Sanders Peirce (1839–1914) American philosopher, logician, mathematician, and scientist

Letter to Victoria (23 December 1908)

Women, Men, History, Study

“What mathematics, therefore are expected to do for the advanced student at the university, Arithmetic, if taught demonstratively, is capable of doing for the children even of the humblest school. It furnishes training in reasoning, and particularly in deductive reasoning. It is a discipline in closeness and continuity of thought. It reveals the nature of fallacies, and refuses to avail itself of unverified assumptions. It is the one department of school-study in which the sceptical and inquisitive spirit has the most legitimate scope; in which authority goes for nothing. In other departments of instruction you have a right to ask for the scholar’s confidence, and to expect many things to be received on your testimony with the understanding that they will be explained and verified afterwards. But here you are justified in saying to your pupil “Believe nothing which you cannot understand. Take nothing for granted.” In short, the proper office of arithmetic is to serve as elementary 268 training in logic. All through your work as teachers you will bear in mind the fundamental difference between knowing and thinking; and will feel how much more important relatively to the health of the intellectual life the habit of thinking is than the power of knowing, or even facility of achieving visible results. But here this principle has special significance. It is by Arithmetic more than by any other subject in the school course that the art of thinking—consecutively, closely, logically—can be effectually taught.”

Joshua Girling Fitch (1824–1903) British educationalist

Source: Lectures on Teaching, (1906), pp. 292-293.

Life, Children, Health, School

“The history of arithmetic and algebra illustrates one of the striking and curious features of the history of mathematics. Ideas that seem remarkably simple once explained were thousands of years in the making.”

Morris Kline (1908–1992) American mathematician

Morris Kline, p.22.
Mathematics for the Nonmathematician (1967)

History, Idea

“Our first objective in undertaking the experimental program described here has been to gain some understanding of what goes on in long sequences of plays of Prisoner's Dilemma. We have attempted to gain this understanding by postulating a system going through a sequence of states and by attempting to formulate some mathematical models from which the dynamics of the system could be deduced. Once such a model is found, its parameters, properly interpreted, become the key terms in the emerging psychological theory. This strategy can be deemed successful, if the parameters so discovered are independent of the process itself, if they suggest further investigation, and if the further investigations, in turn, lead to a more inclusive theory.”

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

Source: 1960s, Prisoner's dilemma: A study in conflict and cooperation (1965), p. 185

Understanding, Success

“In mathematics he was greater
Than Tycho Brahe, or Erra Pater:
For he, by geometric scale,
Could take the size of pots of ale;
Resolve, by sines and tangents straight,
If bread and butter wanted weight;
And wisely tell what hour o' th' day
The clock doth strike, by algebra.”

Samuel Butler (poet) (1612–1680) poet and satirist

Canto I, line 119
Source: Hudibras, Part I (1663–1664)

“It is hard for you to appreciate that modern mathematics has become so extensive and so complex that it is essential, if mathematics is to stay as a whole and not become a pile of little bits of research, to provide a unification, which absorbs in some simple and general theories all the common substrata of the diverse branches if the science, suppressing what is not so useful and necessary, and leaving intact what is truly the specific detail of each big problem. This is the good one can achieve with axiomatics (and this is no small achievement). This is what Bourbaki is up to.”

André Weil (1906–1998) French mathematician

as translated by Martin H. Krieger "A 1940 letter of André Weil on analogy in mathematics." http://www.ams.org/notices/200503/fea-weil.pdf Notices of the AMS 52, no. 3 (2005) pp. 334–341, quote on p. 341

Problem, Science

“But to return to the Newtonian Philosophy: Tho' its Truth is supported by Mathematicks, yet its Physical Discoveries may be communicated without. The great Mr. Locke was the first who became a Newtonian Philosopher without the help of Geometry; for having asked Mr. Huygens, whether all the mathematical Propositions in Sir Isaac's Principia were true, and being told he might depend upon their Certainty; he took them for granted, and carefully examined the Reasonings and Corollaries drawn from them, became Master of all the Physics, and was fully convinc'd of the great Discoveries contained in that Book.”

John Theophilus Desaguliers (1683–1744) French-born British natural philosopher and clergyman

Source: Course of Experimental Philosophy, 1745, p. viii: Preface; Cited in Joseph Schwartz (1992), The creative moment: how science made itself alien to modern culture, p. 20

Truth, Books, Communication, Support

“Angling may be said to be so like the mathematics that it can never be fully learnt.”

Izaak Walton book The Compleat Angler

Epistle to the Reader.
The Compleat Angler (1653-1655)

“(Game theory is) essentially a structural theory. It uncovers the logical structure of a great variety of conflict situations and describes this structure in mathematical terms. Sometimes the logical structure of a conflict situation admits rational decisions; sometimes it does not.”

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

Source: 1960s, Prisoner's dilemma: A study in conflict and cooperation (1965), p. 196

Game, Decision

“Mathematics brought rigor to Economics. Unfortunately, it also brought mortis”

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

Attributed to Kenneth Boulding in: Peter J. Dougherty (2002) Who's afraid of Adam Smith?: how the market got its soul. p. 110
1990s and attributed

“I have asked many teachers and parents what they thought mathematics to be and why it was important to learn it. Few held a view of mathematics that was sufficiently coherent to justify devoting several thousand hours of a child's life to learning it, and children sense this. When a teacher tells a student that the reason for those many hours of arithmetic is to be able to check the change at the supermarket, the teacher is simply not believed. Children see such "reasons" as one more example of adult double talk. The same effect is produced when children are told school math is "fun" when they are pretty sure that teachers who say so spend their leisure hours on anything except this allegedly fun-filled activity. Nor does it help to tell them that they need math to become scientists---most children don't have such a plan. The children can see perfectly well that the teacher does not enjoy math any more than they do and that the reason for doing it is simply that it has been inscribed into the curriculum. All of this erodes children's confidence in the adult world and the process of education. And I think it introduces a deep element of dishonestly into the education relationship.”

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

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

Life, Children, World, Education

“Clearly the present organization of the scientific community, cutting across the lines of nation states, bureaus, and almost all previously existing institutions, cannot be the result of conscious planning. There is, today, a good deal of organizational planning, but all of the instrumentalities which engage in this activity were founded after the development of science was well under way. Further, most of these organizations are parochial in nature, concerning themselves with only some special part of the scientific community like mathematical biophysics or Russian science. There is no general institution which has shaped or now can shape the development of science, only a mass of institutions which provide little more than liaison (and sometimes funds) for the scientific “producers.””

Gordon Tullock (1922–2014) American economist

The Organization of Inquiry (1966) Ch 1. The Social Organization of Science

Way, Communication, Nature, Science

“Today we face a problem that involves two difficult to satisfy conditions. On the one hand we have to find a way for computer assisted verification of mathematical proofs. This is necessary, first of all, because we have to stop the dissolution of the concept of proof in mathematics. On the other hand, we have to preserve the intimate connection between mathematics and the world of human intuition. This connection is what moves mathematics forward and what we often experience as the beauty of mathematics.”

Vladimir Voevodsky (1966–2017) Russian mathematician

UniMath by Vladimir Voevodsky, Heidelberg Laureate Forum, Sept. 22, 2016, Heidelberg https://www.math.ias.edu/vladimir/sites/math.ias.edu.vladimir/files/2016_09_22_HLF_Heidelberg.pdf p. 3

World, Beauty, Way, Problem

“Being a language, mathematics may be used not only to inform but also, among other things, to seduce.”

Benoît Mandelbrot (1924–2010) Polish-born, French and American mathematician

Fractals : Form, chance and dimension (1977)

“A painter should never speak because words are not the means at his command. Words cannot express visually dimension at a glance they can only establish their own relationship in time.
However, it is possible for a painter, at certain moments of his development to formulate some of the problems he is facing in the growth of his work. A painting cannot be explained. Words can only stimulate the act of looking.
A visual problem is never put a priori as a mathematical problem but is born in the process of painting and evolves in a state of unawareness of the painter.”

Fritz Glarner (1899–1972) Swiss artist

Source: George L. K. Morris, Willem De Kooning, Alexander Calder, Fritz Glarner, Robert Motherwell, Stuart Davis. " What Abstract Art Means to Me http://www.jstor.org/stable/4058250," in: The Bulletin of the Museum of Modern Art, Vol. 18, No. 3, (Spring, 1951), pp. 2-15

Problem, Time, Painting

“In recent years, many men of science have come to realize that the scientific picture of the world is a partial one — the product of their special competence in mathematics and their special incompetence to deal systematically with aesthetic and moral values, religous experiences and intuitions of significance. Unhappily, novel ideas become acceptable to the less intelligent members of society only with a very considerable time-lag. Sixty or seventy years ago the majority of scientists believed — and the belief caused them considerable distress — that the product of their special incompetence was identical with reality as a whole. Today this belief has begun to give way, in scientific circles, to a different and obviously truer conception of the relation between science and total experience.”

Aldous Huxley book Ends and Means

Ends and Means (1937)

Men, World, Way, Intelligence

“But how can we avoid the use of human language? The… symbol. Only by using a symbolic language not yet usurped by those vague ideas of space, time, continuity which have their origin in intuition and tend to obscure pure reason—only thus may we hope to build mathematics on the solid foundation of logic.”

Tobias Dantzig (1884–1956) American mathematician

p, 125
Number: The Language of Science (1930)

Hope, Idea, Time

“Women have been discouraged from genres such as sculpture that require studio training or expensive materials. But in philosophy, mathematics, and poetry, the only materials are pen and paper. Male conspiracy cannot explain all female failures. I am convinced that, even without restrictions, there still would have been no female Pascal, Milton, or Kant. Genius is not checked by social obstacles: it will overcome. Men's egotism, so disgusting in the talentless, is the source of their greatness as a sex. […] Even now, with all vocations open, I marvel at the rarity of the woman driven by artistic or intellectual obsession, that self-mutilating derangement of social relationship which, in its alternate forms of crime and ideation, is the disgrace and glory of the human species.”

Camille Paglia (1947) American writer

Source: Sexual Personae: Art and Decadence from Nefertiti to Emily Dickinson (1990), p. 653

Women, Men, Sex, Failure

“Research in statistical theory and techniques is necessarily mathematical, scholarly, and abstract in character, requiring some degree of leisure, detachment, and access to a good mathematical and historical library. The importance of continuing such research is very great, though it is not always obvious to those whose interest is entirely in practical applications of already existing theory. Excepting in the presence of active research in pure science, the application of the science tend to drop into a deadly rut of unthinking routine, incapable of progress beyond a limited range predetermined by the accomplishments of pure science, and are in constant danger of falling into the hands of people who do not really understand the tools that they are working with and who are out of touch with those that do. … It is in fact rather absurd, though quite in line with the precedents of earlier centuries, that scientific men of the highest talents can live only by doing work that could be done by others of lesser special abilities, while the real worth of their most important work receives no official recognition.”

Harold Hotelling (1895–1973) American economist and statistician

Men, People, Understanding, Science

“If physics leads us today to a world view which is essentially mystical, it returns, in a way, to its beginning, 2,500 years ago. […] This time, however, it is not only based on intuition, but also on experiments of great precision and sophistication, and on a rigorous and consistent mathematical formalism.”

Fritjof Capra book The Tao of Physics

Source: The Tao of Physics (1975), Ch. 1, Modern Physics, p. 19.

World, Way, Time

“We now come to a decisive step of mathematical abstraction: we forget about what the symbols stand for… [The mathematician] need not be idle; there are many operations which he may carry out with these symbols, without ever having to look at the things they stand for.”

Hermann Weyl (1885–1955) German mathematician

The Mathematical Way of Thinking (1941)

Decision, Forgetting

“The properties of executability and universality associated with programming languages can be combined, in a single language, with the well-known properties of mathematical notation which make it such an effective tool of thought.”

Kenneth E. Iverson (1920–2004) Canadian computer scientist

§5
Notation as a Tool of Thought (1979)

Universe

“Recognition of the subjectivity of the qualities of sense is found in Galilei (and also in Descartes and Hobbes) in a form closely related to the principle underlying the constructive mathematical method of our modern physics which repudiates" qualities."”

Hermann Weyl (1885–1955) German mathematician

Introduction
Space—Time—Matter (1952)

Sense

“The manner in which mathematical theories are applied does not depend on preconceived ideas; it is a purposeful technique depending on, and changing with, experience.”

William Feller (1906–1970) Croatian-American mathematician

Introduction, The Nature of Probability Theory, p. 2 - 3.
An Introduction To Probability Theory And Its Applications (Third Edition)

Idea, Change, Dependency

“As far as the use of mathematics in economics is concerned, there is an abundance of formulas where such are not needed. They are frequently introduced, one fears, in order to show off. The more difficult the mathematical theorem, the more esoteric the name of the mathematician quoted, the better.”

Oskar Morgenstern (1902–1977) austrian economist

Oskar Morgenstern, " Limits of the Use of Mathematics in Economics https://www.princeton.edu/~erp/ERParchives/archivepdfs/M49.pdf," in: James C. Charlesworth (Hg.), Mathematics and the Social Science. The Utility and Inutility of Mathematics in the Study of Economics, Political Sciences and Sociology, Philadelphia 1963, S. 12-29, hier S. 18.

Fear, Quotes

“The object of mathematics is to discover "true" theorems. We shall use the term "valid" to describe statements formed according to certain rules and then shall discuss how this notion compares with the intuitive idea of "true."”

Paul Cohen (1934–2007) American mathematician

Set theory and the continuum hypothesis, p. 8. https://books.google.com/books?id=Z4NCAwAAQBAJ&pg=PA8 Set Theory and the Continuum Hypothesis (1966)

Idea

“Geometry is that of mathematical science which is devoted to consideration of form and size, and may be said to be the best and surest guide to study of all sciences in which ideas of dimension or space are involved. Almost all the knowledge required by navigators, architects, surveyors, engineers, and opticians, in their respective occupations, is deduced from geometry and branches of mathematics. All works of art constructed according to the rules which geometry involves; and we find the same laws observed in the works of nature. The study of mathematics, generally, is also of great importance in cultivating habits of exact reasoning; and in this respect it forms a useful auxiliary to logic.”

Robert Chambers (publisher, born 1802) (1802–1871) Scottish publisher and writer

Robert Chambers, Chambers's Information for the People (1875) Vol. 2 https://books.google.com/books?id=vNpTAAAAYAAJ

Law, Art, Idea, Nature

“Let me also remind you that zero, like all of mathematics, is fictional and an idealization. It is impossible to reach absolute zero temperature or to get perfect vacuum. Luckily, mathematics is a fairyland where ideal and fictional objects are possible.”

Doron Zeilberger (1950) Israeli mathematician

" " (nothing) published in the Personal Journal of Shalosh B. Ekhad and Doron Zeilberger

Perfection

“When I had the honour of his conversation, I endeavoured to learn his thoughts upon mathematical subjects, and something historical concerning his inventions, that I had not been before acquainted with. I found, he had read fewer of the modern mathematicians, than one could have expected; but his own prodigious invention readily supplied him with what he might have an occasion for in the pursuit of any subject he undertook. I have often heard him censure the handling geometrical subjects by algebraic calculations; and his book of Algebra he called by the name of Universal Arithmetic, in opposition to the injudicious title of Geometry, which Des Cartes had given to the treatise, wherein he shews, how the geometer may assist his invention by such kind of computations. He frequently praised Slusius, Barrow and Huygens for not being influenced by the false taste, which then began to prevail. He used to commend the laudable attempt of Hugo de Omerique to restore the ancient analysis, and very much esteemed Apollonius's book De sectione rationis for giving us a clearer notion of that analysis than we had before.”

Henry Pemberton (1694–1771) British doctor

Preface; The bold passage is subject of the 1809 article " Remarks on a Passage in Castillione's Life' of Sir Isaac Newton http://books.google.com/books?id=BS1WAAAAYAAJ&pg=PA519." By John Winthrop, in: The Philosophical Transactions of the Royal Society of London, from Their Commencement, in 1665, to the Year 1800: 1770-1776: 1770-1776. Charles Hutton et al. eds. (1809) p. 519. Preface to View of Newton's Philosophy, (1728)

Books, Learning, Universe, Reading

“Mathematicians tells us that it is easy to invent mathematical theorems which are true, but that it is hard to find interesting ones. In analyzing music or writing its history, we meet the same difficulty, and it is compounded by another. For whom is it interesting? To paraphrase a famous remark of Barnett Newman, musicology is for musicians what ornithology is for the birds.”

Charles Rosen (1927–2012) American pianist and writer on music

Source: The Frontiers of Meaning: Three Informal Lectures on Music (1994), Ch. 3 : Explaining the Obvious

Music, History, Bird

“I may remark that the curious transformations many formulae can undergo, the unsuspected and to a beginner apparently impossible identity of forms exceedingly dissimilar at first sight, is I think one of the chief difficulties in the early part of mathematical studies. I am often reminded of certain sprites and fairies one reads of, who are at one's elbows in one shape now, and the next minute in a form most dissimilar.”

Ada Lovelace (1815–1852) English mathematician, considered the first computer programmer

As quoted in Toole, Betty Alexandra (1998), Ada, the Enchantress of Numbers: Prophet of the Computer Age, Strawberry Press, ISBN 0912647183. p. 99

Thinking, Study, Reading, Parting

“Game theory, analyzing in a novel mathematical framework, rational competition between two or more antagonists for maximum gain and minimum loss.”

Ludwig von Bertalanffy (1901–1972) austrian biologist and philosopher

General System Theory (1968), 4. Advances in General Systems Theory

Game, Loss

“A model may be pictorial, descriptive, qualitative, or generally approximate in nature, or it may be mathematical and quantitative in nature and reasonably precise.”

Harold Chestnut (1917–2001) American engineer

Source: Systems Engineering Tools, (1965), p. 108

Nature

“In many schools today, the phrase "computer-aided instruction" means making the computer teach the child. One might say the computer is being used to program the child. In my vision, the child programs the computer and, in doing so, both acquires a sense of master over a piece of the most modern and powerful technology and establishes an intimate contact with some of the deepest ideas from science, from mathematics, and from the art of intellectual model building.”

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

Introduction
Mindstorms: Children, Computers, and Powerful Ideas (1980)

School, Art, Idea, Sense

“The only generally agreed upon definition of mathematics is "Mathematics is what mathematicians do." […]
In the face of this difficulty [of defining "computer science"] many people, including myself at times, feel that we should ignore the discussion and get on with doing it. But as George Forsythe points out so well in a recent article*, it does matter what people in Washington D. C. think computer science is. According to him, they tend to feel that it is a part of applied mathematics and therefore turn to the mathematicians for advice in the granting of funds. And it is not greatly different elsewhere; in both industry and the universities you can often still see traces of where computing first started, whether in electrical engineering, physics, mathematics, or even business. Evidently the picture which people have of a subject can significantly affect its subsequent development. Therefore, although we cannot hope to settle the question definitively, we need frequently to examine and to air our views on what our subject is and should become.”

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

Hamming cites Forsythe, G.E., "What to do until the computer scientist comes", Am. Math. Monthly 75 (5), May 1968, p. 454-461.
One Man's View of Computer Science (1969)

People, Hope, Feeling, Question

“It is not uncommon for intelligent adults to turn into passive observers of their own incompetence in anything but the most elementary mathematics. Individuals may see the direct consequences of this intellectual paralysis in terms of limiting job possibilities. But the indirect, secondary consequences are even more serious. One of the main lessons learned by most people in math class is a sense of having rigid limitations. They learn a balkanized image of human knowledge which they come to see as a patchwork of territories separated by impassable iron curtains.”

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

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

People, Intelligence, Knowledge, Sense

“In the realm of physics it is perhaps only the theory of relativity which has made it quite clear that the two essences, space and time, entering into our intuition, have no place in the world constructed by mathematical physics. Colours are thus "really" not even æther-vibrations, but merely a series of values of mathematical functions in which occur four independent parameters corresponding to the three dimensions of space, and the one of time.”

Hermann Weyl (1885–1955) German mathematician

Introduction
Space—Time—Matter (1952)

World, Time

“I cannot refrain… from expressing my surprise that, according to the report in The Times there should be so much complaint about the difficulty of understanding the new theory. It is evident that Einstein's little book "About the Special and the General Theory of Relativity in Plain Terms," did not find its way into England during wartime. Any one reading it will, in my opinion, come to the conclusion that the basic ideas of the theory are really clear and simple; it is only to be regretted that it was impossible to avoid clothing them in pretty involved mathematical terms, but we must not worry about that. …
The Newtonian theory remains in its full value as the first great step, without which one cannot imagine the development of astronomy and without which the second step, that has now been made, would hardly have been possible. It remains, moreover, as the first, and in most cases, sufficient, approximation. It is true that, according to Einstein's theory, because it leaves us entirely free as to the way in which we wish to represent the phenomena, we can imagine an idea of the solar system in which the planets follow paths of peculiar form and the rays of light shine along sharply bent lines—think of a twisted and distorted planetarium—but in every case where we apply it to concrete questions we shall so arrange it that the planets describe almost exact ellipses and the rays of light almost straight lines.
It is not necessary to give up entirely even the ether. …according to the Einstein theory, gravitation itself does not spread instantaneously, but with a velocity that at the first estimate may be compared with that of light. …In my opinion it is not impossible that in the future this road, indeed abandoned at present, will once more be followed with good results, if only because it can lead to the thinking out of new experimental tests. Einstein's theory need not keep us from so doing; only the ideas about the ether must accord with it.”

Hendrik Lorentz (1853–1928) Dutch physicist

Theory of Relativity: A Concise Statement (1920)

Books, Way, Imagination, Idea

“[T]here is a tremendous egotism and conceit in those popular articles and lectures entitled,"My Idea of Religion," or "My Idea of God." An individual religion can be as misleading and uninformed as an individual astronomy or an individual mathematics.”

Fulton J. Sheen (1895–1979) Catholic bishop and television presenter

Source: Peace of Soul (1949), Ch. 4, p. 59

God, Religion, Idea

“The lesson here is that it is insufficient to protect ourselves with laws; we need to protect ourselves with mathematics. Encryption is too important to be left solely to governments.”

Bruce Schneier (1963) American computer scientist

[John Wiley & Sons, 1996, Applied Cryptography 2nd edition Source Code in C, Bruce Schneier, http://www.schneier.com/book-applied.html]
Cryptography

Law

“Two important features in the modern development of economics are the application of mathematics to abstract economic reasoning… and the attempt at placing economics on a numerical and experimental basis by an intensive study of economic statistics.
Both these developments have a common characteristic: they emphasize the quantitative character of economics. This quantitative movement in our estimation is one of the most promising developments in modern economics. We also consider it important that the two aspects of the quantitative method referred to should be furthered, developed, and studied jointly as two integrating parts of economics.
We therefore venture to propose the establishment of an international periodical devoted to the advancement of the quantitative study of economic phenomena, and especially to the development of a closer relation between pure economics and economic statistics.
We believe that the scope of the new journal would be happily suggested if it is called "Oekonometrika."”

Ragnar Frisch (1895–1973) Norwegian economist

Accordingly, the quantitative study of economic phenomena here considered may be termed econometrics. Frisch (1927) as quoted in Divisia 1953, pp.24-25; Cited in: Bjerkholt, Olav. " Ragnar Frisch and the foundation of the Econometric Society and Econometrica http://www.ssb.no/a/histstat/doc/doc_199509.pdf." ECONOMETRIC SOCIETY MONOGRAPHS 31 (1998): 26-57. Lead paragraph of a memorandum on the importance of establishing the journal "Oekonometrika" 1920

Study, Parting

“A mathematical formula should never be "owned" by anybody! Mathematics belong to God.”

Donald Ervin Knuth Digital Typography

Digital Typography, ch. 1, p. 8 (1999)

God

“Of the splendid constellation of great names… we admire the living and revere dead far too warmly and too deeply to suffer us sit in judgment on their respective claims to in this or that particular discovery; to balance mathematical skill of one against the experimental dexterity of another, or the philosophical acumen a third. So long as "one star differs from another in glory,"—so long as there shall exist varieties, or even incompatibilities of excellence,—so long will the admiration of mankind be found sufficient for all who merit it.”

John Herschel (1792–1871) English mathematician, astronomer, chemist and photographer

On the Theory of Light https://books.google.com/books?id=Lo4_AAAAcAAJ (1828) p.494

Star, Suffering, Dead

“Now my humble fear is that this double training, in language as well as in thought, imposes somewhat too heavy a burden upon the young, especially when, at the age of three years old, they are taken from the maternal care and taught to unlearn the old language — except for the purpose of repeating it in the presence of their Mothers and Nurses — and to learn the vocabulary and idiom of science. Already methinks I discern a weakness in the grasp of mathematical truth at the present time as compared with the more robust intellect of our ancestors three hundred years ago.”

Edwin Abbott Abbott book Flatland

Source: Flatland: A Romance of Many Dimensions (1884), PART I: THIS WORLD, Chapter 12. Of the Doctrine of our Priests

Truth, Age, Fear, Learning

“The work of Renoir, after that of Cézanne whose great influence had been manifested among artists, save us from whatever drying effect there is in pure abstraction. The rules that one might deduce in considering the work of these two masters appear to be more difficult to discover in the work of Renoir, who hides his efforts better. Whereas the continuous tension of the mind of Cézanne, his lack of self-confidence, prevent him from giving himself to us entirely even though he shows the evidence of his corrections, from which are easily (too easily) deduced rules that have a mathematical precision.”

Henri Matisse (1869–1954) French artist

critical quote on Cubism
In a short text of Matisse, 1918, written for the catalogue of 'Den Franske Utstilling', 1918, Copenhagen; as quoted in Matisse on Art, Jack Flam, University of California Press 1995 p. 272, note 2
1910 - 1920

Quotes, Confidence, Effort, Appearance

“Historians of the future will amuse themselves by coming up with theories to explain why European civilization, at the height of its powers, rich with unparalleled achievements in science, music, art, literature, mathematics, and technology, gave up its lands and its treasures to people for whom those achievements were mere hated tokens of oppression or the impious and superfluous productions of infidels.”

John Derbyshire (1945) writer

Multiculturalism: When Will the Sleeper Wake? http://takimag.com/article/multiculturalism_when_will_the_sleeper_wake_john_derbyshire/print#ixzz3xOopVxdb, Taki's Magazine, March 29, 2012.

Hate, People, Art, Music

“In June of 1964 the research group and academic program moved to Penn bringing with it most of the faculty, students, and research projects. Our activities flourished in the very supportive environment that Penn and Wharton provided. The wide variety of faculty members that we were able to involve in our activities significantly enhanced our capabilities. By the mid-1960s I had become uncomfortable with the direction, or rather, the lack of direction, of professional Operations Research. I had four major complaints.
First, it had become addicted to its mathematical tools and had lost sight of the problems of management. As a result it was looking for problems to which to apply its tools rather than looking for tools that were suitable for solving the changing problems of management. Second, it failed to take into account the fact that problems are abstractions extracted from reality by analysis. Reality consists of systems of problems, problems that are strongly interactive, messes. I believed that we had to develop ways of dealing with these systems of problems as wholes. Third, Operations Research had become a discipline and had lost its commitment to interdisciplinarity. Most of it was being carried out by professionals who had been trained in the subject, its mathematical techniques. There was little interaction with the other sciences professions and humanities. Finally, Operations Research was ignoring the developments in systems thinking — the methodology, concepts, and theories being developed by systems thinkers.”

Russell L. Ackoff (1919–2009) Scientist

Preface, cited in Gharajedaghi, Jamshid. Systems thinking: Managing chaos and complexity: A platform for designing business architecture http://booksite.elsevier.com/samplechapters/9780123859150/Front_Matter.pdf. Elsevier, 2011. p. xiii Towards a Systems Theory of Organization, 1985

Way, Support, Problem, Thinking

“The next day, I called one of them and asked if I could come to his office to discuss the briefing. Fine with him, so off I went. I sat down and asked him a question: “Norm, how did you decide so exactly where the bomb would explode?” He looked at me as if I were a country bumpkin and explained how SAC calculated its bombing accuracy and he had gotten the accuracy of this particular drop straight from the horse’s mouth. Now I don’t want to bore you with how SAC arrived at planning estimates for the delivery accuracy of its nuclear bombers, except to say that it was a statistical process based on thousands of practice sorties, whose results would be mathematically analyzed to allow estimates to be made of the results of a large bombing campaign; not one bomber flying over one city and one bombardier, with the lives of perhaps millions of Muscovites at his fingertips, dropping one bomb. This I pointed out to Norm, implying that he had gone to all that fuss and bother for naught. His response was that in doing his calculations as a mathematician, he was going by the accepted ground rules. The bombing accuracy he had assumed had been provided to him by others. His was not to reason why.”

Samuel T. Cohen (1921–2010) American physicist

F*** You! Mr. President: Confessions of the Father of the Neutron Bomb (2006)

Question, Horse, Flying, Country

“More than any of his predecessors Plato appreciated the scientific possibilities of geometry... By his teaching he laid the foundations of the science, insisting upon accurate definitions, clear assumptions, and logical proof. His opposition to the materialists, who saw in geometry only what was immediately useful to the artisan and the mechanic is… clear. …That Plato should hold the view… is not a cause for surprise. The world's thinkers have always held it. No man has ever created a mathematical theory for practical purposes alone. The applications of mathematics have generally been an afterthought.”

David Eugene Smith (1860–1944) American mathematician

Source: History of Mathematics (1923) Vol.1, p. 90

World, Science

“I can now report that the mathematics of the multiple focus are a most improbable thicket, and the useful service I enforced upon what I must call an absurd set of contradictions is one of my secrets. I know that thousands of scientists, at home and abroad, are attempting to duplicate my work; they are welcome to the effort. None will succeed. Why do I speak so positively? That is my other secret.”

Jack Vance (1916–2013) American mystery and speculative fiction writer

Section 3 (p. 169)
Short fiction, Rumfuddle (1973)

Secret, Home, Effort

“1. The human mind is so constructed that it must see every perception in a time-relation—in an order—and every perception of an object in a space-relation—as outside or beside our perceiving selves.
2. These necessary time-relations are reducible to Number, and they are studied in the theory of number, arithmetic and algebra.
3. These necessary space-relations are reducible to Position and Form, and they are studied in geometry.
Mathematics, therefore, studies an aspect of all knowing, and reveals to us the universe as it presents itself, in one form, to mind. To apprehend this and to be conversant with the higher developments of mathematical reasoning, are to have at hand the means of vitalizing all teaching of elementary mathematics.”

Nicholas Murray Butler (1862–1947) American philosopher, diplomat, and educator

Editor's Introduction, The Teaching of Elementary Mathematics https://books.google.com/books?id=NKoAAAAAMAAJ (1906) by David Eugene Smith

Study, Universe, Time

“Mathematics have a triple aim. They must furnish an instrument for the study of nature. But that is not all: they have a philosophic aim and, I dare maintain, an esthetic aim. They must aid the philosopher to fathom the notions of number, of space, of time. And above all, their adepts find therein delights analogous to those given by painting and music. They admire the delicate harmony of numbers and forms; they marvel when a new discovery opens to them an unexpected perspective; and has not the joy they thus feel the esthetic character, even though the senses take no part therein? Only a privileged few are called to enjoy it fully, it is true, but is not this the case for all the noblest arts?
This is why I do not hesitate to say that mathematics deserve to be cultivated for their own sake, and the theories inapplicable to physics as well as the others. Even if the physical aim and the esthetic aim were not united, we ought not to sacrifice either.”

Henri Poincaré book The Value of Science

Source: The Value of Science (1905), Ch. 5: Analysis and Physics

Art, Music, Feeling, Nature

“Teachers, not students, are the ones who are failing. As Albert Einstein said that “all religions, arts and sciences are branches of the same tree,” the detachment of philosophy – the forefather of all knowledge and academic disciplines – from mathematics, sciences, and technology is the fundamental reason for failure in modern-day K-12 and higher education.”

Newton Lee American computer scientist

Google It: Total Information Awareness, 2016

Education, Religion, Art, Failure

“A great many individuals ever since the rise of the mathematical method, have, each for himself, attacked its direct and indirect consequences…. I shall call each of these persons a paradoxer, and his system a paradox. I use the word in the old sense:… something which is apart from general opinion, either in subject-matter, method, or conclusion…. Thus in the sixteenth century many spoke of the earth's motion as the paradox of Copernicus, who held the ingenuity of that theory in very high esteem, and some, I think, who even inclined towards it. In the seventeenth century, the depravation of meaning took place… Phillips says paradox is "a thing which seemeth strange"—here is the old meaning…—"and absurd, and is contrary to common opinion," which is an addition due to his own time.”

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

A Budget of Paradoxes (1872)

Thinking, Sense, Time, Personality

“Mathematical Reasoning is not only exact; it has its own criteria of reality”

Paul Karl Feyerabend book Science in a Free Society

pg 52.
Science in a Free Society (1978)

Reality

“Nothing is more impressive than the fact that as mathematics withdrew increasingly into the upper regions of ever greater extremes of abstract thought, it returned back to earth with a corresponding growth of importance for the analysis of concrete fact. …The paradox is now fully established that the utmost abstractions are the true weapons with which to control our thought of concrete fact.”

Alfred North Whitehead (1861–1947) English mathematician and philosopher

1920s, Science and the Modern World (1925)

“I find enough mystery in mathematics to satisfy my spiritual needs. I think, for example, that pi is mysterious enough (don't get me started!) without having to worry about God. Or if pi isn't enough, how about fractals? or quantum mechanics?”

Tom Lehrer (1928) American singer-songwriter and mathematician

Interview at celebathiests.com (June 1996)

God, Thinking

“The accounting methods based on mathematical models, the use of computers for computations and information data processing make up only one part of the control mechanism, another part is the control structure.”

Leonid Kantorovich (1912–1986) Russian mathematician

"Mathematics in Economics: Achievements, Difficulties, Perspectives," 1975

Parting

“If others would but reflect on mathematical truths as deeply and as continuously as I have, they would make my discoveries.”

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

The World of Mathematics (1956) Edited by J. R. Newman

Truth

“First of all, why a descent into novelty rather than an ascent? It was my thing to do as I wanted to do it, and it seemed to me—the way I thought of time was I thought of it like a river. And so I thought of it as flowing toward its lowest level. And I thought of history as a river and Eternity as the ocean. So naturally history flows downhill to reach Eternity. I also like the fact that when the descent in elevation is rapid, the river runs faster, and when the landscape is almost flat, the river broadens out and meanders. So it was to preserve this idea of time as a fluid. The other reason is a mathematical reason. It has to do with the fact that if we have novelty moving downward, then the maximum of novelty is zero.”

Terence McKenna (1946–2000) American ethnobotanist

Talk at the Cathedral Church of Saint John the Divine, NYC https://web.archive.org/web/20120429183018/http://www.abrupt.org/abruptlog/logos/terence-mckenna-at-saint-johns-2785/ 25 April 1996

Way, History, Idea, Nature

“I cannot see why it is necessary that every deduction from algebra should be bound to certain conventions incident to an earlier stage of mathematical learning, even supposing them to have been consistently used up to the point in question. I should not care if any one thought this treatise unalgebraical, but should only ask whether the premises were admissible and the conclusions logical.”

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

Advertisement, p.3
The Differential and Integral Calculus (1836)

Question, Learning

“A great deal of study has been directed to denning 'best decisions,' particularly since the pioneering work of mathematical statisticians (such as Wald), of mathematicians (such as von Neumann), of economists (such as Arrow)… The main effect of this development on the practice of OR has been the growing realization that there are decision objectives other than maximizing expected return and minimizing maximum loss. That is, in many practical situations there are criteria of optimality that are more appropriate than these two mentioned.”

Russell L. Ackoff (1919–2009) Scientist

Source: 1950s, The development of operations research as a science, 1956, p. 270.

Optimism, Decision, Study, Loss

“We are told … statements that are not mathematical or logical formulae may look as if they were necessarily or certainly true, but they only look like that. They cannot really be either necessary or certain.”

Robert Maynard Hutchins (1899–1977) philosopher and university president

Great Books: The Foundation of a Liberal Education (1954)

“Too often, this concern for the big picture is simply obscurantist and is put forward by people who prefer vagueness and mystery to (partial) answers. Vagueness is at times necessary and mystery is never in short supply, but I don’t think they’re anything to worship. Genuine science and mathematical precision are more intriguing than are the “facts” published in supermarket tabloids or a romantic innumeracy which fosters credulity, stunts skepticism, and dulls one to real imponderables.”

John Allen Paulos (1945) American mathematician

Source: Innumeracy: Mathematical Illiteracy and its Consequences (1988), Chapter 4, “Whence Innumeracy?” (pp. 126-127)

People, Thinking, Science, Time

“Money, having freed itself from the physical universe, has become number itself, and finance a strange form of mathematical alchemy.”

David Orrell (1962) Canadian mathematician

Source: The Other Side Of The Coin (2008), Chapter 3, One Versus Plurality, p. 89

Money, Universe

“Often we were joined by Maurice Princet. Although very young, he held an important post in an insurance company which he owed to his knowledge of mathematics. But outside his profession it was as an artist that he thought of mathematics, as a specialist in aesthetics that he evoked continuities in n dimensions. He liked to interest painters in the new visions of space that had been opened up by Victor Schlegel and several others. He succeeded. After having heard him by chance, Henri Matisse was caught reading an essay on hyperspace. Oh! it was only a potboiler [un ouvrage de vulgarisation]! but at least that shows that for the great 'fauve' the days of the painter who knows nothing, who runs towards a pretty subject with his beard blowing in the wind, was passed.”

Jean Metzinger (1883–1956) French painter

Cubism was born

Knowledge, Chance, Reading

“It is less than four years since cohomological methods (i. e. methods of Homological Algebra) were introduced into Algebraic Geometry in Serre's fundamental paper[11], and it seems certain that they are to overflow the part of mathematics in the coming years, from the foundations up to the most advanced parts. … [11] Serre, J. P. Faisceaux algébriques cohérents. Ann. Math. (2), 6, 197–278”

Alexander Grothendieck (1928–2014) French mathematician

1955
[1960, Cambridge University Press, The cohomology theory of abstract algebraic varieties, Proc. Internat. Congress Math.(Edinburgh, 1958), 103–118, https://webusers.imj-prg.fr/~leila.schneps/grothendieckcircle/CohomologyVarieties.pdf] (p. 103)

Parting

“In all the rest of my life’s wanderings, I never met another person who spoke words to rival the beauty of mathematics.”

Lawrence M. Schoen (1959) American writer and klingonist

Source: Barsk: The Elephants' Graveyard (2015), Chapter 17, “Dead Voices” (p. 171)

Life, Beauty, Personality

“I am afraid that mathematics will perish by the end of this century if the present trend for senseless abstractin — as I call it: theory of the empty set — cannot be blocked up.”

Carl Ludwig Siegel (1896–1981) German mathematician

in a 1964 letter to L. J. Mordell as quoted by [C. S. Yogananda, The Life and Times of Bourbaki, June 2015, Resonance, 556–559, http://www.ias.ac.in/article/fulltext/reso/020/06/0556-0559] (quote from p. 558)