Quotes about mathematics (1323 quotes, page 7)

“Do not imagine that mathematics is hard and crabbed, and repulsive to common sense. It is merely the etherealization of common sense.”

William Thomson (1824–1907) British physicist and engineer

Quoted in Life of Lord Kelvin (1910) by Silvanus Phillips Thompson

Imagination , Sense

“I always saw a close kinship between the needs of "pure" mathematics and a certain hero of Greek mythology, Antaeus. The son of Earth, he had to touch the ground every so often in order to reestablish contact with his Mother; otherwise his strength waned. To strangle him, Hercules simply held him off the ground. Back to mathematics. Separation from any down-to-earth input could safely be complete for long periods — but not forever. In particular, the mathematical study of Brownian motion deserved a fresh contact with reality.”

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

A Theory of Roughness (2004)

Reality , Study

“The world can no longer accept, the world can no longer accept that basic education is enough. Why do leaders accept that for children in developing countries, only basic literacy is sufficient, when their own children do homework in Algebra, Mathematics, Science and Physics?”

Malala Yousafzai (1997) Pakistani children's education activist

Nobel Peace Prize Lecture (December 10, 2014)

Children , World , Education , Science

“Rate of change is this mathematics known as Calculus. … Now I hope you understand this, because I've never been able to make head nor tail of it. It must be some sort of a Black Magic operation, started out by the Luce cult — some immoral people who are operating up in New York City, Rockefeller Plaza — been thoroughly condemned by the whole society. Anyway, their rate-of-change theory — I've never seen any use for that mathematics, by the way — I love that mathematics, because it — I asked an engineer, one time, who was in his 6th year of engineering, if he'd ever used Calculus, and he told me yeah, once, once I did, he said. When did you use it? And he said I used it once. Let me see, what did you use it on? Oh yeah. Something on the rate-of-change of steam particles in boilers. And then we went out and tested it and found the answer was wrong.”

L. Ron Hubbard (1911–1986) American science fiction author, philosopher, cult leader, and the founder of the Church of Scientology

Philadelphia Doctorate Course, Tape #58 (November 1952) http://www.rr.cistron.nl/xenu/quotes.htm.

Love , People , Hope , Way

“For four or five centuries, Islam was the most brilliant civilization in the Old World. (…) At its higher level the golden age of Muslim civilization was both an immense scientific success and a exceptional revival of ancient philosophy. These was not its only triumphs; literature was another: but they eclipse the rest. First, science: it was there that the Saracens (…) made the most original contributions. These, in brief, were nothing less than trigonometry and algebra (with its significantly Arab name). (…) Equally distinguished were Islam's mathematical geographers, its astronomical observatories and instruments (…). The Muslims also deserve high marks for optics, for chemistry (…) and for pharmacy. More than half the remedies and healing aids used by the West came from Islam (…). Muslim medical skill was incontestable. (…) In the field of philosophy, what took place was rediscovery - a return, essentially of the peripatetic philosophy. The scope of this rediscovery, however, was not limited to copying and handling on, valuable as that undoubtely was. It also involved continuing, elucidating and creating.”

Fernand Braudel (1902–1985) French historian and a leader of the Annales School

A History of Civilizations , Penguin, 1995, p. 73-81

Age , World , Literature , Science

“The fact that randomness requires a physical rather than a mathematical source is noted by almost everyone who writes on the subject, and yet the oddity of this situation is not much remarked.”

Brian Hayes (scientist) (1900) American scientist, columnist and author

Source: Group Theory in the Bedroom (2008), Chapter 2, Random Resources, p. 35

“The bases of music are rhythm and harmony. Rhythm is ordered recurrence in time… As the planets move around the sun, they repeat their orbits periodically; thus there is already a primitive kind of rhythm in their motion…. Harmony… can be considered a special kind of rhythm…. pure musical tones are produced when the vibrations are… periodic or… repeat themselves regularly in time. Two tones harmonize if their intervals of repetition are in rhythm—or, in mathematical language, if their periods are in proportion. Kepler… in the third book of Harmonice mundi… attempted to make other… related, connections between musical harmony and mathematical proportion.”

Frank Wilczek (1951) physicist

Longing for the Harmonies: Themes and Variations from Modern Physics (1987)

Books , Music , Time , Sun

“The history of Alexandrian mathematics begins with the Elements of Euclid and closes with the Algebra of Diophantus, both of which are founded on the discoveries of several preceding centuries.”

James Gow (scholar) (1854–1923) scholar

Preface
A Short History of Greek Mathematics (1884)

History

“The reason is not to glorify "bit chasing"; a more fundamental issue is at stake here: Numerical subroutines should deliver results that satisfy simple, useful mathematical laws whenever possible. […] Without any underlying symmetry properties, the job of proving interesting results becomes extremely unpleasant. The enjoyment of one's tools is an essential ingredient of successful work.”

Donald Ervin Knuth book The Art of Computer Programming

Vol. II, Seminumerical Algorithms, Section 4.2.2 part A, final paragraph [Italics in source]
The Art of Computer Programming (1968–2011)

Law , Success

“The first demonstration (Disq. Arith., Arts. 125-145) which is presented by Gauss in a form very repulsive to any but the most laborious students, has been resumed by Lejeune Dirichlet in a memoir in Crelle's Journal… and has been developed by him with that luminous perspicuity by which his mathematical writings are distinguished.”

Henry John Stephen Smith (1826–1883) mathematician

Report on the Theory of Numbers (1859) Part I, p. 59.
The Collected Mathematical Papers of Henry John Stephen Smith (1894) Vol. 1

Art

“The final truth about a phenomenon resides in the mathematical description of it; so long as there is no imperfection in this, our knowledge of the phenomenon is complete. We go beyond the mathematical formula at our own risk; we may find a model or a picture which helps us understand it, but we have no right to expect this, and our failure to find such a model or picture need not indicate that either our reasoning or our knowledge is at fault. The making of models or pictures to explain mathematical formulas and the phenomena they describe is not a step towards, but a step away from reality; it is like making a graven image of a spirit.”

James Jeans book The Mysterious Universe

The Mysterious Universe (1930)

Truth , Failure , Understanding , Knowledge

“What seems certain is that Pythagoras developed the idea of mathematical logic… He realized that numbers exist independently of the tangible world and therefore their study was untainted by inaccuracies of perception. This meant he could discover truths which were independent of opinion of prejudice and which were more absolute then any previous knowledge.”

Simon Singh (1964) British author

Fermat's Last Theorem (1997)

Truth , World , Idea , Knowledge

“I have said that mathematics is the oldest of the sciences; a glance at its more recent history will show that it has the energy of perpetual youth. The output of contributions to the advance of the science during the last century and more has been so enormous that it is difficult to say whether pride in the greatness of achievement in this subject, or despair at his inability to cope with the multiplicity of its detailed developments, should be the dominant feeling of the mathematician. Few people outside of the small circle of mathematical specialists have any idea of the vast growth of mathematical literature. The Royal Society Catalogue contains a list of nearly thirty-nine thousand papers on subjects of Pure Mathematics alone, which have appeared in seven hundred serials during the nineteenth century. This represents only a portion of the total output, the very large number of treatises, dissertations, and monographs published during the century being omitted.”

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

Source: Presidential Address British Association for the Advancement of Science, Section A (1910), p. 283; Cited in: Moritz (1914, 108-9): Modern mathematics.

People , Youth , Feeling , History

“The 'language theory' is inadequate as a description of the nature of mathematics.”

George Frederick James Temple (1901–1992) British mathematician

100 Years of Mathematics: a Personal Viewpoint (1981)

Nature

“In the original discovery of a proposition of practical utility, by deduction from general principles and from experimental data, a complex algebraical investigation is often not merely useful, but indispensable; but in expounding such a proposition as a part of practical science, and applying it to practical purposes, simplicity is of the importance:—and… the more thoroughly a scientific man has studied higher mathematics, the more fully does he become aware of this truth—and… the better qualified does he become to free the exposition and application of principles from mathematical intricacy.”

William John Macquorn Rankine (1820–1872) civil engineer

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

Truth , Study , Science , Parting

“To discover something in mathematics is to overcome an inhibition and a tradition. You cannot move forward if you are not subversive.”

Laurent Schwartz book A Mathematician Grappling with His Century

Un mathematicien aux prises avec le siecle (1997)

“.. and to think now that great mathematicians find my work interesting because I am able to illustrate their theories. They can not imagine that I was such a bad pupil in mathematics. I don't understand it myself neither. I never could understand why it was necessary to prove something that everyone already sees. I saw it, I knew it, so it is how it is… But yes, you had to prove it. I did overcome it when I realized I can make something else - I thought I was a good-for-nothing. In my family there were no other artists to find... I was just a weird duck, right?”

M. C. Escher (1898–1972) Dutch graphic artist

version in original Dutch (origineel citaat van M.C. Escher, in het Nederlands): En als je nu bedenkt dat grote wiskundigen mijn werk interessant vinden, omdat ik in staat ben hun theorieën te illustreren. Ze kunnen zich helemaal niet voorstellen dat ik zo slecht was in wiskunde. Ik snap er zelf ook niets van. Ik begreep niet dat je iets moest bewijzen wat iedereen ziet. Ik zag het, ik wist, het is toch zo.. .Maar jawel hoor, je moest het bewijzen. Ik ben er bovenuit gekomen toen ik me realiseerde, dat ik wat anders kon. Ik dacht, dat ik een nietsnut was. Ik kom uit een milieu waar geen artiesten in waren.. ..Ik was een rare eend in de bijt, he?
1960's, M.C. Escher, interviewed by Bibeb', 1968

Family , Imagination , Understanding , Thinking

“Mathematical research currently relies on a complex system of mutual trust based on reputations. By the time Simpson's paper appeared, both Kapranov and I had strong reputations. Simpson's paper created doubts in our result, which led to it being unused by other researchers, but no one came forward and challenged us on it.”

Vladimir Voevodsky (1966–2017) Russian mathematician

Univalent Foundations, Vladimir Voevodsky, IAS, March 26, 2014 http://www.math.ias.edu/vladimir/files/2014_IAS.pdf p. 10

Strong , Appearance , Time

“Among the various paradigmatic changes in science and mathematics in this century, one such change concerns the concept of uncertainty. In science, this change has been manifested by a gradual transition from the traditional view, which insists that uncertainty is undesirable in science and should be avoided by all possible means, to an alternative view, which is tolerant of uncertainty and insists that science cannot avoid it. According to the traditional view, science should strive for certainty in all its manifestations (precision, specificity, sharpness, consistency, etc.); hence, uncertainty (imprecision, nonspecificity, vagueness, inconsistency, etc.) is regarded as unscientific. According to the alternative (or modem) view, uncertainty is considered essential to science; it is not only an unavoidable plague, but it has, in fact, a great utility.”

George Klir (1932–2016) American computer scientist

Source: Fuzzy sets and fuzzy logic (1995), p. 1.

Science , Change

“Although there are at present many occupations that require a good deal of skill and training in advanced mathematics, mathematics itself is still often regarded as a curious profession demanding singular talents and a singular personality.”

Robert Langlands (1936) Canadian mathematician

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

Personality , Training

“Mathematics fluourished as long as freedom of thought prevailed; it decayed when creative joy gave way to blind faith and fanatical frenzy.”

Tobias Dantzig (1884–1956) American mathematician

The Bequest of the Greeks (1955)

Faith , Way , Freedom , Joy

“The technical literature on structural engineering abounds with theoretical works of a mathematical nature, but few publications are concerned with the various kinds of structures or the fundamental reasons for their existence.”

Eduardo Torroja (1899–1961) Spanish architect

p, 125
Philosophy of Structures (1958)

Literature , Nature

“Mass psychology is not simply a summation of individual psychologies; that is a prime theorem of social psychodynamics—not just my opinion; no exception has ever been found to this theorem. It is the social mass-action rule, the mob-hysteria law, known and used by military, political, and religious leaders, by advertising men and prophets and propagandists, by rabble rousers and actors and gang leaders, for generations before it was formulated in mathematical symbols. It works.”

Robert A. Heinlein book Methuselah's Children

Methuselah’s Children (p. 535)
Short fiction, The Past Through Tomorrow (1967)

Men , Law

“The two great conceptual revolutions of twentieth-century science, the overturning of classical physics by Werner Heisenberg and the overturning of the foundations of mathematics by Kurt Gödel, occurred within six years of each other within the narrow boundaries of German-speaking Europe. … A study of the historical background of German intellectual life in the 1920s reveals strong links between them. Physicists and mathematicians were exposed simultaneously to external influences that pushed them along parallel paths. … Two people who came early and strongly under the influence of Spengler's philosophy were the mathematician Hermann Weyl and the physicist Erwin Schrödinger. … Weyl and Schrödinger agreed with Spengler that the coming revolution would sweep away the principle of physical causality. The erstwhile revolutionaries David Hilbert and Albert Einstein found themselves in the unaccustomed role of defenders of the status quo, Hilbert defending the primacy of formal logic in the foundations of mathematics, Einstein defending the primacy of causality in physics. In the short run, Hilbert and Einstein were defeated and the Spenglerian ideology of revolution triumphed, both in physics and in mathematics. Heisenberg discovered the true limits of causality in atomic processes, and Gödel discovered the limits of formal deduction and proof in mathematics. And, as often happens in the history of intellectual revolutions, the achievement of revolutionary goals destroyed the revolutionary ideology that gave them birth. The visions of Spengler, having served their purpose, rapidly became irrelevant.”

Freeman Dyson (1923) theoretical physicist and mathematician

The Scientist As Rebel (2006)

Life , People , History , Birth

“I have tried to make clear in this brief sketch of the nature of deductively formulated theories, at least as they occur in economics, that, although their origin lies in common observation and introspection, nevertheless they are capable of a purely mathematical development through the exact statement of a of a set of postulates which, however, since they must inevitably contain assumptions about human behaviour, require to be tested by reference to actual events.”

Richard Stone (1913–1991) British economist, Nobel Memorial Prize winner

Source: The Role of Measurement in Economics. 1951, p. 14-15

Nature

“In the winter of 2008, Jenifer and I visited Chennai Mathematical Institute. This remarkable Institute is the creation of Seshadri. It is a unique blend of an American style liberal arts college with traditional Indian guru one-on-one teaching, adding physics, computer science, history and music to its maths curriculum. Only in India could an intellectual with no business or management experience, who spends all his spare time singing classical south Indian music, have been the catalyst for such a unique educational experiment.”

David Mumford (1937) American mathematician

[David Mumford, Passages to India, Mathematics Intelligencer, 2010, Hyderabad edition, 51-55, http://www.dam.brown.edu/people/mumford/beyond/papers/2010c--PassagesIndia-journal.pdf]

Education , Art , Music , History

“My real speciality is the mathematical analysis of Hilbert Space operators. But this was no time to come on like an ivory-tower idealist.”

Rudy Rucker (1946) American mathematician, computer scientist, science fiction author and philosopher

Source: The Sex Sphere (1983), p. 18

Time

“His [Sarpi's] career soon revealed another cause of his return; he evidently felt the same impulse which stirred his contemporaries, Lord Bacon and Galileo, for he began devoting himself to the whole range of scientific and philosophical studies, especially to mathematics, physics, astronomy, anatomy, and physiology. In these he became known as an authority, and before long was recognized as such throughout Europe. It is claimed, and it is not improbable, that he anticipated Harvey in discovering the circulation of the blood, and that he was the forerunner of noted discoverers in magnetism. Unfortunately the loss of the great mass of his papers by the fire which destroyed his convent in 1769 forbids any full estimate of his work; but it is certain that among those who sought his opinion and advice were such great discoverers as Acquapendente, Galileo, Torricelli, and Gilbert of Colchester, and that every one of these referred to him as an equal, and indeed as a master.”

Andrew Dickson White (1832–1918) American politician

Source: Seven Great Statesmen in the Warfare of Humanity with Unreason (1915), p. 4-5

Study , Loss

“The nature of the instability of an unregulated free-enterprise system is only now beginning to be clearly understood. Perhaps the degree of understanding already attained ensures that the grosser shortcomings have gone for ever, and to that extent the conflict between Capitalism and Communism is about issues that belong to the past. It may now be too late. The gods must smile to note how different the state of the world might have been if the progress of economic thought of the last twenty years had been advanced by even ten years.
The possibility of a stable economic life with full utilization of our resources is still not sufficiently assured, and it is extremely important that it should be so assured, and that the whole world should accept this as a fact. The work that is being done in econometrics is massive, and undaunted by mathematical difficulties, but it appears, at any rate as viewed from outside, to be unclear as to its aim.”

Arnold Tustin (1899–1994) British engineer

Source: The Mechanism of Economic Systems (1953), p. 1

Life , God , World , Smile

“Conventions of generality and mathematical elegance may be just as much barriers to the attainment and diffusion of knowledge as may contentment with particularity and literary vagueness… It may well be that the slovenly and literary borderland between economics and sociology will be the most fruitful building ground during the years to come and that mathematical economics will remain too flawless in its perfection to be very fruitful.”

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

Kenneth Boulding (1948) "Samuelson's Foundations: The Role of Mathematics in Economics," In: Journal of Political Economy, Vol 56 (June). as cited in: Peter J. Boettke (1998) " James M. Buchanan and the Rebirth of Political Economy http://publicchoice.info/Buchanan/files/boettke.htm". Boettke further explains "Boulding's words are even more telling today than they were then as we have seen the fruits of the formalist revolution in economic theory and how it has cut economics off from the social theoretic discourse on the human condition."
1940s

Knowledge , Perfection

“The laws of nature, as analyzed mathematically and descriptively by Ptolemy and Galen, bore an interesting, and perhaps not entirely accidental similarity to the law of nations and of nature, as discerned by a long succession of Roman jurists. …The concept of an objective law applicable to human affairs, yet operating in accord with Nature and Reason and apart both from divine revelation and from human whim or passion, was peculiar to Rome and societies descended from Rome.”

William H. McNeill book The Rise of the West: A History of the Human Community

The Rise of the West: A History of the Human Community (1963)

Law , Nature , Passion , Success

“[The process of system design is]… consisting of the development of a sequence of mathematical models of systems, each one more detailed than the last.”

A. Wayne Wymore (1927–2011) American mathematician

A. Wayne Wymore (1970) Systems Engineering Methodology. Department of Systems Engineering, The University of Arizona, p. 14/2; As cited in: J.C. Heckman (1973) Locating traveler support facilities along the interstate system--a simulation using general systems theory. p. 43.

“That classical theory had catastrophic implications for the constitution of matter was barely appreciated by Planck and others at the turn of the century, and played no substantial role in the development of quantum theory until Bohr's work 13 years later.
The lesson that could be drawn from this, and also from the development of general relativity, is that a crisis will only become creative if it is formulated in a mathematically precise manner. This conclusion has an echo in Bell's insistence on having quantum mechanics "fully formulated in mathematical terms, with nothing left to the discretion of the theoretical physicist", but what it really calls for is a critique of the "orthodox" theory fully formulated in mathematical terms with nothing left to the discretion of the critic.”

Kurt Gottfried (1929) American physicist

Does quantum mechanics carry the seeds of its own destruction? (1991)

“Understanding the methods of calculus is vital to the creative use of mathematics… Without this mastery the average scientist or engineer, or any other user of mathematics, will be perpetually stunted in development, and will at best be able to follow only what the textbooks say; with mastery, new things can be done, even in old, well-established fields.”

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

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

Understanding

“Philosophical knowledge is the knowledge gained by reason from concepts; mathematical knowledge is the knowledge gained by reason from the construction of concepts.”

Immanuel Kant book Critique of Pure Reason

A 713, B 741
Critique of Pure Reason (1781; 1787)

Knowledge

“The real is the present, conceived not as a mathematical point between the present and the past, but as the union of present and past in a duration or permanence that is at the same time change. Thus the past as past and the future as future do not exist at all, but are purely ideal; the past as living in the present and the future as germinating in the present are wholly real and indeed are just the present itself. It is because of the presence of these two elements in the present… that the present is a concrete and changing reality and not an empty mathematical point.”

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

Source: "Some Perplexities about time: with an attempted solution" (1925), p. 149. as cited in: Jonathan Gorman, "The transmission of our understanding of historical time." Historia Social y de la Educación 1.2 (2012): 129-152.

Reality , Change , Time , Past

“Data is transformed into graphics to understand. A map, a diagram are documents to be interrogated. But understanding means integrating all of the data. In order to do this it’s necessary to reduce it to a small number of elementary data. This is the objective of the “data treatment” be it graphic or mathematic.”

Jacques Bertin (1918–2010) French geographer and cartographer

About the true value of graphics
Interview with Jacques Bertin (2003)

Understanding

“But a kind of genius can come of this deprivation of sensation, of experience. It has been mistaken as naïve intelligence, when in fact it is empty intelligence, pure intelligence. The composition of the mind is altered. Its previous cultivation is disintegrated and it has greater access to the brain, the body: it is Supersanity. Learning is turned inside out. You have to start from the top and work your way down. You must study mathematical theory before simple arithmetic; theoretical physics before applied physics; anatomy, you might say, before you can walk.”

Jack Abbott book In the Belly of the Beast

In the Belly of the Beast (1981)

Way , Intelligence , Study , Learning

“The analytical equations, unknown to the ancient geometers, which Descartes was the first to introduce into the study of curves and surfaces, are not restricted to the properties of figures, and to those properties which are the object of rational mechanics; they extend to all general phenomena. There cannot be a language more universal and more simple, more free from errors and from obscurities, that is to say more worthy to express the invariable relations of natural things.
Considered from this point of view, mathematical analysis is as extensive as nature itself; it defines all perceptible relations, measures times, spaces, forces, temperatures; this difficult science is formed slowly, but it preserves every principle which it has once acquired; it grows and strengthens itself incessantly in the midst of the many variations and errors of the human mind.
Its chief attribute is clearness; it has no marks to express confused notions. It brings together phenomena the most diverse, and discovers the hidden analogies which unite them.”

Joseph Fourier book The Analytical Theory of Heat

Preliminary Discourse, p.7 Note: often quoted as Mathematics [or mathematical analysis] compares the most diverse phenomena and discovers the secret analogies that unite them.
The Analytical Theory of Heat (1878)

Error , Nature , Study , Universe

“The next grand extensions of mathematical physics will, in all likelihood, be furnished by quaternions.”

Peter Guthrie Tait (1831–1901) British mathematician

in Note on a Quaternion Transformation , Communication read on Monday, 6th April, 1863, Proceedings of the Royal Society of Edinburgh (1866), p. 117.

“During the 1950s and 1960s most of the work which was called cybernetics tended to focus on control systems in engineering or on applications of the concept of feedback in fields ranging from mathematics to sociology. At the 1970 meeting of the American Society for Cybernetics in Philadelphia Heinz von Foerster sought to redirect attention to the original interests which had led to the founding of the field of cybernetics. In a paper titled "Cybernetics of Cybernetics" he made a distinction between first order cybernetics, the cybernetics of observed systems, and second order cybernetics, the cybernetics of observing systems.”

Stuart A. Umpleby (1944) American scientist

Stuart A. Umpleby (1991) "Strategies for Winning Acceptance of Second Order Cybernetics." In George E. Lasker, et al. (eds.) Advances in Human Systems and Information Technologies. Windsor, Canada: International Institute for Advanced Studies in Systems Research and Cybernetics, 1992. pp. 97-196. (paper)

“To those who do not know mathematics it is difficult to get across a real feeling as to the beauty, the deepest beauty, of nature. … If you want to learn about nature, to appreciate nature, it is necessary to understand the language that she speaks in.”

Richard Feynman book The Character of Physical Law

Source: The Character of Physical Law (1965), chapter 2, “The Relation of Mathematics to Physics,” p. 58

Beauty , Feeling , Understanding , Nature

“I found myself obliged, through perhaps unique circumstances, to devote myself to my mathematical research, almost without help, advice or even books… Endlessly occupied by a thousand different matters and constrained my state duties, it is the work of an engineer that I herewith present and not the fruit of the meditations of a savant.”

Charles Dupin (1784–1873) French mathematician

Charles Dupin (1808) in: Hacette (1813; 86-87); as cited in Margaret Bradley, Charles Dupin (1784-1873) and His Influence on France, Cambria Press. p. 69

Books , Help

“Mathematics doesn’t care about those beyond the numbers.”

Dejan Stojanovic (1959) poet, writer, and businessman

"I and I," p. 30
The Shape (2000), Sequence: “Happiness of Atoms”

“What makes a piece of mathematical economics not only mathematics but also economics is, I believe, this: When we set up a system of theoretical relationships and use economic names for the otherwise purely theoretical variables involved, we have in mind some actual experiment, or some design of an experiment, which we could at least imagine arranging, in order to measure those quantities in real economic life that we think might obey the laws imposed on their theoretical namesakes.”

Trygve Haavelmo (1911–1999) Norwegian economist and econometrician

Trygve Haavelmo, "The probability approach in econometrics" in: Supplement to Econometrica. 12 91944), p. 5; Cited in Pearl (2012, 1-2)

Life , Law , Imagination , Thinking

“Newton and Descartes started to try and prove that God existed in the same way as they would try and prove something in the laboratory or with their mathematics … And when you try and mix science and religion you get bad science and bad religion. The two are doing two different things. … Science can give you a diagnosis of cancer. It can even cure your disease, but it cannot touch your grief and disappointment, nor can it help you to die well.”

Karen Armstrong (1944) author and comparative religion scholar from Great Britain

Ode interview (2009)

God , Disease , Religion , Disappointment

“It seems to be expected of every pilgrim up the slopes of the mathematical Parnassus, that he will at some point or other of his journey sit down and invent a definite integral or two towards the increase of the common stock.”

James Joseph Sylvester (1814–1897) English mathematician

James Joseph Sylvester, Collected Mathematical Papers, Vol. 2 (1908), p. 214.
Bigeometric Calculus: A System with a Scale-Free Derivative by Michael Grossman, p. 31.

“General systems theory, in a sense, is no news at all, as Von Foerster found out when he attempted to organize a conference of general systems people and anthropologists. In a sense, the situation is comparable to that found by the Committee for the Study of Mankind, in which a committee that included Robert Redfidd tried to get each discipline to consider its relationship to the concept of Mankind. Anthropologists replied, "we are related already," and so they were. Something similar may be said of attempts to date in mathematical anthropology. The kind of information that a computer program can finally provide, on a level of a particular culture, is simply a reflection of how detailed field work has been done, and to the careful field worker, on kinship, for example, it provides no illumination.”

Margaret Mead (1901–1978) American anthropologist

Source: 1970s, Changing Styles of Anthropological Work, 1973, p. 8

People , Sense , Study

“[Oddly enough, Putnam believes part of the attraction of some of these formalisms is their obscurity]. "I think part of the appeal of mathematical logic is that the formulas look mysterious - you write backward Es!"”

Hilary Putnam (1926–2016) American philosopher

Putnam as quoted in: Julian Baggini, Jeremy Stangroom (2005) What Philosophers Think. p. 233

Thinking , Parting

“We distinguish diagrammatic from sentential paper-and-pencil representations of information by developing alternative models of information-processing systems that are informationally equivalent and that can be characterized as sentential or diagrammatic. Sentential representations are sequential, like the propositions in a text. Diagrammatic representations are indexed by location in a plane. Diagrammatic representations also typically display information that is only implicit in sentential representations and that therefore has to be computed, sometimes at great cost, to make it explicit for use. We then contrast the computational efficiency of these representations for solving several. illustrative problems in mathematics and physics.”

Herbert A. Simon (1916–2001) American political scientist, economist, sociologist, and psychologist

Source: 1980s and later, "Why a diagram is (sometimes) worth ten thousand words," (1987), p. 65

Problem

“As a science mathematics has been adapted to the description of natural phenomena, and the great practitioners in this field… have never concerned themselves with the logical foundations of mathematics, but have boldly taken a pragmatic view of mathematics as an intellectual machine which works successfully. Description has been verified by further observation, still more strikingly be prediction, and sometimes, more ominously, by control of natural forces. Happily, unresolved problems… still remain as challenges.”

George Frederick James Temple (1901–1992) British mathematician

100 Years of Mathematics: a Personal Viewpoint (1981)

Nature , Problem , Science

“The sciences we are familiar with have been installed in a number of great 'continents'. Before Marx, two such continents had been opened up to scientific knowledge: the continent of Mathematics and the continent of Physics. The first by the Greeks (Thales), the second by Galileo. Marx opened up a third continent to scientific knowledge: the continent of History.”

Louis Althusser book Lenin and Philosophy and Other Essays

Source: Lenin and Philosophy and Other Essays (1968), "Philosophy as a Revolutionary Weapon", p. 4

History , Knowledge , Science

“Do not lose your faith. A mighty fortress is our mathematics. Mathematics will rise to the challenge, as it always has.”

Stanislaw Ulam (1909–1984) Polish-American mathematician

In Heinz R. Pagels, The Dreams of Reason: The Computer and the Rise of the Sciences of Complexity, Ch. 3, p. 94; as quoted in Gaither's Dictionary of Scientific Quotations (Springer, 2008), p. 861

Faith

“The young people who now proposed to devote themselves to intellectual studies no longer took the term to mean attending a university and taking a nibble of this or that from the dainties offered by celebrated and loquacious professors who without authority offered them the crumbs of what had once been higher education. Now they had to study just as stringently and methodically as the engineers and technicians of the past, if not more so. They had a steep path to climb, had to purify and strengthen their minds by dint of mathematics and scholastic exercises in Aristotelian philosophy. Moreover, they had to learn to renounce all those benefits which previous generations of scholars had considered worth striving for: rapid and easy money-making, celebrity and public honors, the homage of the newspapers, marriages with daughters of bankers and industrialists, a pampered and luxurious style of life.”

Hermann Hesse book The Glass Bead Game

The Glass Bead Game (1943)

Life , Money , Marriage , People

“There is this great danger in student life. Now, we rest all upon what Socrates said, or what Copernicus taught; how can we dispute authority which has come down to us, all established, for ages? We must at least question it; we cannot accept anything as granted, beyond the first mathematical formulae. Question everything else.”

Maria Mitchell (1818–1889) American astronomer

Maria Mitchell: Life, Letters and Journals (illustrated) by Maria Mitchell, 1896, p. 188.

Life , Age , Question

“Economic history matters. Students of economics should read Charles MacKay and Charles Kindleberger, and should study the history of the Wall Street Crash as well as the theory and the mathematics required to formalize it.”

Adair Turner, Baron Turner of Ecchinswell (1955) British businessman

Source: Economics after the crisis : objectives and means (2012), Ch. 2 : Financial Markets: Efficiency, Stability, and Income Distribution

History , Study , Reading

“Mrs. May knows that with her current, slim majority in Parliament, she is at the behest of the hard Brexiteers who could easily mount a coup against her, defying the party whip, and may be even worse, if she demurs from her short-lived mantra of “Brexit means Brexit”. And you don’t need to fall for it or worry about it either. Look at the mathematics of a parliamentary election, before you head down into the comments and start screaming, “But Corbyn doesn’t want ANY Brexit! She’s our only hope!” The fact is you’re going to end up with her anyway. There’s no chance Corbyn and this fantasy “coalition of chaos” comes to pass, given the situation up and down the country. It’s a magnificent, though somewhat predictable, piece of Tory electioneering. But that’s all it is.”

Raheem Kassam (1986) British journalist and politician

Brexit Voters Cannot Afford to Give Theresa May a Massive Majority as She Plans Compromise on Free Movement http://www.breitbart.com/london/2017/05/17/kassam-theresa-may-preparing-screw-brexit-voters-hard-youre-still-voting/ (May 17, 2017)

Hope , Chance , Country

“All such problems can be formulated as mathematical programming problems. Naturally, we can propose many sophisticated algorithms and a theory but the final test of a theory is its capacity to solve the problems which originated it.”

George Dantzig (1914–2005) American mathematician

Source: Linear programming and extensions (1963), p. vii.

Nature , Problem

“The objection made to the numerical method, to wit, the difficulty or impossibility of forming classes of similar facts, is alike applicable to all the methods that might be substituted; that it is impossible to appreciate each case with mathematical exactness, and it is precisely on this account that enumeration becomes necessary; by so doing, the errors, (which are inevitable,) being the same in two groups of patients subjected to different treatment, mutually compensate each other, and they may be disregarded without sensibly affecting the exactness of the results.”

Pierre Charles Alexandre Louis (1787–1872) French physician

p, 125
Researches on the effects of bloodletting... (1836)

Error

“The outstanding personalities of Euclid and Archimedes demand chapters to themselves. Euclid, the author of the incomparable Elements, wrote on almost all the other branches of mathematics known in his day. Archimedes's work, all original and set forth in treatises which are models of scientific exposition, perfect in form and style, was even wider in its range of subjects. The imperishable and unique monuments of the genius of these two men must be detached from their surroundings and seen as a whole if we would appreciate to the full the pre-eminent place which they occupy, and will hold for all time, in the history of science.”

Thomas Little Heath (1861–1940) British civil servant and academic

Preface p. viii
A History of Greek Mathematics (1921) Vol. 1. From Thales to Euclid

Men , History , Science , Time

“…a detailed mathematical discussion shows that whatever kind of wave-packet we select to represent the electron, the product of the two uncertainties of position and momentum can never be less than h, which is precisely what Heisenberg found…”

James Jeans (1877–1946) British mathematician and astronomer

Physics and Philosophy (1942)

“It becomes the urgent duty of mathematicians, therefore, to meditate about the essence of mathematics, its motivations and goals and the ideas that must bind divergent interests together.”

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

Richard Courant, "Mathematics in the Modern World", Scientific American, Vol 211, (Sep 1964), p. 42

Motivation , Idea

“Mathematics and philosophy are cultivated by two different classes of men: some make them an object of pursuit, either in consequence of their situation, or through a desire to render themselves illustrious, by extending their limits; while others pursue them for mere amusement, or by a natural taste which inclines them to that branch of knowledge. It is for the latter class of mathematicians and philosophers that this work is chiefly intended j and yet, at the same time, we entertain a hope that some parts of it will prove interesting to the former. In a word, it may serve to stimulate the ardour of those who begin to study these sciences; and it is for this reason that in most elementary books the authors endeavour to simplify the questions designed for exercising beginners, by proposing them in a less abstract manner than is employed in the pure mathematics, and so as to interest and excite the reader's curiosity. Thus, for example, if it were proposed simply to divide a triangle into three, four, or five equal parts, by lines drawn from a determinate point within it, in this form the problem could be interesting to none but those really possessed of a taste for geometry. But if, instead of proposing it in this abstract manner, we should say: "A father on his death-bed bequeathed to his three sons a triangular field, to be equally divided among them: and as there is a well in the field, which must be common to the three co-heirs, and from which the lines of division must necessarily proceed, how is the field to be divided so as to fulfill the intention of the testator?"”

Jean-Étienne Montucla (1725–1799) French mathematician

This way of stating it will, no doubt, create a desire in most minds to discover the method of solving the problem; and however little taste people may possess for real science, they will be tempted to try iheir ingenuity in finding the answer to such a question at this.
Source: Preface to Recreations in Mathematics and Natural Philosophy. (1803), p. ii; As cited in: Tobias George Smollett. The Critical Review: Or, Annals of Literature http://books.google.com/books?id=T8APAAAAQAAJ&pg=PA410, Volume 38, (1803), p. 410

Death , Men , People , Books

“The precision provided (or enforced) by programming languages and their execution can identify lacunas, ambiguities, and other areas of potential confusion in conventional [mathematical] notation.”

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

Source: Math for the Layman (1999), Ch. 10, §D

“It is common to think of statistical graphics and data visualization as relatively modern developments in statistics. In fact, the graphic representation of quantitative information has deep roots. These roots reach into the histories of the earliest map-making and visual depiction, and later into thematic cartography, statistics and statistical graphics, medicine, and other fields. Along the way, developments in technologies (printing, reproduction) mathematical theory and practice, and empirical observation and recording, enabled the wider use of graphics and new advances in form and content.”

Michael Friendly (1945) American psychologist

Michael Friendly. " A brief history of data visualization http://www.datavis.ca/papers/hbook.pdf at datavis.ca, March 21, 2006.

Way , History , Thinking

“As the prerogative of Natural Science is to cultivate a taste for observation, so that of Mathematics is, almost from the starting point, to stimulate the faculty of invention.”

James Joseph Sylvester (1814–1897) English mathematician

"A plea for the mathematician", Nature, Vol. 1, p. 261.

Nature , Science

“Grégoire de Saint-Vincent… was a Jesuit, taught mathematics in Rome and Prag (1629-1631), and was afterwards called to Spain by Phillip IV as tutor to his son… He wrote two works on geometry [Principia Matheseos Univerales (1651); Exercitationum Mathematicarum Libri quinque (1657)], giving in one of them the quadrature of the hyperbola referred to its asymptotes, and showing that as the area increased in arithmetic series the abscissas increased in geometric series.”

David Eugene Smith (1860–1944) American mathematician

p, 125
History of Mathematics (1923) Vol.1

“Once some engineers from the veneer trust laboratory came to me for consultation with a quite skilful presentation of their problems. Diﬀerent productivity is obtained for veneer-cutting machines for diﬀerent types of materials; linked to this the output of production of this group of machines depended, it would seem, on the chance factor of which group of raw materials to which machine was assigned. How could this fact be used rationally?
This question interested me, but nevertheless appeared to be quite particular and elementary, so I did not begin to study it by giving up everything else. I put this question for discussion at a meeting of the mathematics department, where there were such great specialists as Gyunter, Smirnov himself, Kuz’min, and Tartakovskii. Everyone listened but no one proposed a solution; they had already turned to someone earlier in individual order, apparently to Kuz’min. However, this question nevertheless kept me in suspense. This was the year of my marriage, so I was also distracted by this. In the summer or after the vacation concrete, to some extent similar, economic, engineering, and managerial situations started to come into my head, that also required the solving of a maximization problem in the presence of a series of linear constraints.
In the simplest case of one or two variables such problems are easily solved—by going through all the possible extreme points and choosing the best. But, let us say in the veneer trust problem for ﬁve machines and eight types of materials such a search would already have required solving about a billion systems of linear equations and it was evident that this was not a realistic method. I constructed particular devices and was probably the ﬁrst to report on this problem in 1938 at the October scientiﬁc session of the Herzen Institute, where in the main a number of problems were posed with some ideas for their solution.
The universality of this class of problems, in conjunction with their difficulty, made me study them seriously and bring in my mathematical knowledge, in particular, some ideas from functional analysis.
What became clear was both the solubility of these problems and the fact that they were widespread, so representatives of industry were invited to a discussion of my report at the university.”

Leonid Kantorovich (1912–1986) Russian mathematician

L.V. Kantorovich (1996) Descriptive Theory of Sets and Functions. p. 39; As cited in: K. Aardal, ‎George L. Nemhauser, ‎R. Weismantel (2005) Handbooks in Operations Research and Management Science, p. 15-26

Marriage , Search , Idea , Question

“A treatise called Astadhyayi (or Astaka) is Panini's major work. It consists of eight chapters, each subdivided into quarter chapters. In this work Panini distinguishes between the language of sacred texts and the usual language of communication. Panini gives formal production rules and definitions to describe Sanskrit grammar. Starting with about 1700 basic elements like nouns, verbs, vowels, consonants he put them into classes. The construction of sentences, compound nouns etc., is explained as ordered rules operating on underlying structures in a manner similar to modern theory. In many ways Panini's constructions are similar to the way that a mathematical function is defined today.”

Pāṇini ancient Sanskrit grammarian

An Analytical Study of 'Sanskrit' and 'Panini' as Foundation of Speech Communication in India and the World

Way , Communication

“In our theory we have multiple “mini bangs” and these mini bangs are not mysterious like the big bang but they come because of the concentration of what we call negative energy fields. Whenever there are pockets of negative energy, they explode and produce a mini bang. We believe there’s a mini bang producing energy in the centre of quasars, which are very bright star like objects. Gamma ray bursts, which are the explosive creations of gamma rays, are another example of a mini bang. These are actually happening. These can be described by normal physics, but the big bang cannot. The big bang theory does not use any mathematical formulation so they can’t say why the bang occurred.”

Jayant Narlikar (1938) Indian physicist

Jayant Narlikar's Cosmology

Star

“I have often been surprised that Mathematics, the quintessence of Truth, should have found admirers so few and so languid. Frequent consideration and minute scrutiny have at length unravelled the cause: viz. that though Reason is feasted, Imagination is starved; whilst Reason is luxuriating in its proper Paradise, Imagination is wearily travelling on a dreary desert.”

Samuel Taylor Coleridge (1772–1834) English poet, literary critic and philosopher

Letter to his brother (1791).
Letters

Truth , Travel , Imagination

“The calculus is probably the most useful single branch of mathematics. …I have found the ability to do simple calculus, easily and reliably, was the most valuable part of mathematics I ever learned.”

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

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

Learning , Parting

“As knowledge of the alphabet and its permutations and combinations is to traditional literacy, and as a knowledge of numbers and their permutations and combinations is to mathematics, so a knowledge of the biological and conceptual alphabets of the brain and its apparently infinite permutations and combinations is to mental literacy.”

Tony Buzan (1942–2019) British psychologist

The Mind Map Book, Buzan and Buzan (1991)

Knowledge

“Thought-economy is most highly developed in mathematics, that science which has reached the highest formal development, and on which natural science so frequently calls for assistance. Strange as it may seem, the strength of mathematics lies in the avoidance of all unnecessary thoughts, in the utmost economy of thought-operations. The symbols of order, which we call numbers, form already a system of wonderful simplicity and economy. When in the multiplication of a number with several digits we employ the multiplication table and thus make use of previously accomplished results rather than to repeat them each time, when by the use of tables of logarithms we avoid new numerical calculations by replacing them by others long since performed, when we employ determinants instead of carrying through from the beginning the solution of a system of equations, when we decompose new integral expressions into others that are familiar,—we see in all this but a faint reflection of the intellectual activity of a Lagrange or Cauchy, who with the keen discernment of a military commander marshalls a whole troop of completed operations in the execution of a new one.”

Ernst Mach (1838–1916) Austrian physicist and university educator

Source: 20th century, "Populär-wissenschafliche Vorlesungen" (1908), pp. 224-225: On thought-economy in m., 203.

Nature , Science , Time

“Humanism is only another name for spiritual laziness, or a vague half-creed adopted by men of science and logicians whose heads are too occupied with the world of mathematics and physics to worry about religious categories.”

Colin Wilson book The Outsider

Source: The Outsider (1956), Chapter Nine, Breaking the Circuit

Men , World , Science

“Although it is always somewhat dangerous to look for the cause of a complex sociological phenomenon in a single event, nonetheless, I believe that a cause can be made for the proposition that the present widespread interest in the quantum theory can be traced to a single paper with the nontransparent title “On the Einstein-Podolsky-Rosen Paradox,” which was written in 1964 by the then thirty-four-year-old Irish physicist John Bell. It was published in the obscure journal Physics, which expired after a few issues. Bell’s paper was, as it happens, published in its first issue. Bell, who has worked since 1960 at CERN, the gigantic elementary-particle physics laboratory near Geneva, has been known to claim that his paper involves only the use of “high school mathematics”; however, its six pages are dense with an extremely abstract set of arguments, which even professionals in the field must work hard at to understand. In fact, for several years after its publication, few if any professional physicists bothered to try.”

Jeremy Bernstein (1929) American physicist

Quantum Profiles (1991), John Stewart Bell: Quantum Engineer

School , Understanding

“It’s magic,” the chief cook concluded, in awe.
”No, not magic,” the ship’s doctor replied. “It’s much more. It’s mathematics.”

David Brin book Glory Season

Source: Glory Season (1993), Chapter 24 (p. 466)

Cooking

“Topology or relational mathematics, including non-metrical fields such as network and graph theory.”

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

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

“In the period 1965 - 69, the U. S. Arms Control and Disarmament Agency employed a group of about ten young game theorists as consultants. It was as a member of this group that I developed the simpler approach, already mentioned, to the analysis of I-games.
I realized that a major problem in arms control negotiations is the fact that each side is relatively well informed about its own position with respect to various variables relevant to arms control negotiations, such as its own policy objectives, its peaceful or bellicose attitudes toward the other side, its military strength, its own ability to introduce new military technologies, and so on - but may be rather poorly informed about the other side's position in terms of such variables.
I came to the conclusion that finding a suitable mathematical representation for this particular problem may very well be a crucial key to a better theory of arms control negotiations, and indeed to a better theory of all I-games.”

John Harsanyi (1920–2000) hungarian economist

Source: "Games with Incomplete Information," 1997, p. 138

Peace , Game , Problem

“The practice of first developing a clear and precise definition of a process without regard for efficiency, and then using it as a guide and a test in exploring equivalent processes possessing other characteristics, such as greater efficiency, is very common in mathematics. It is a very fruitful practice which should not be blighted by premature emphasis on efficiency in computer execution.”

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

§5.4
Notation as a Tool of Thought (1979)

“The mathematical activity of Ancient Greece reached its peak during the glorious era of Euclid, Eratosthenes, Archimedes and Apollonius, a time when Greek letters, art and philosophy were already on the decline. …it was not Greece proper but its outposts in Asia Minor, in Lower Italy, in Africa that had contributed most to the development of mathematics.”

Tobias Dantzig (1884–1956) American mathematician

The Bequest of the Greeks (1955)

Art , Time

“The future mathematical statistician needs early contacts with experimental sciences. He needs them because, at this stage of the development of statistics, the expeimental sciences are sources of theoretical problems. Also, he needs them because in almost any imaginable job which he may get after graduation, he will be called upon to apply his theory to experimental or observational problems.”

Jerzy Neyman (1894–1981) Polish statistician

Proceedings of the Berkeley Symposium on Mathematical Statistics and Probability. Vol. 1. http://books.google.com/books?id=p2T2bxyDSLMC&pg=PA48 University of California Press, 1949, p. 48.

Imagination , Problem , Science , Future

“Although mathematical notation undoubtedly possesses parsing rules, they are rather loose, sometimes contradictory, and seldom clearly stated. […] The proliferation of programming languages shows no more uniformity than mathematics. Nevertheless, programming languages do bring a different perspective. […] Because of their application to a broad range of topics, their strict grammar, and their strict interpretation, programming languages can provide new insights into mathematical notation.”

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

Source: Math for the Layman (1999), Ch. 10, §D

“Are there mathematical propositions for which there is a considerable amount of computational evidence, evidence that is so persuasive that a physicist would regard them as experimentally verified?”

Gregory Chaitin (1947) Argentinian mathematician and computer scientist

Thoughts on the Riemann hypothesis http://link.springer.com/article/10.1007/BF02985392 The Mathematical Intelligencer (December 2004) vol. 26, issue 1, pp. 4–7, quote on p. 4

“In the history of mathematics, the "how" always preceded the "why," the technique of the subject preceded its philosophy.”

Tobias Dantzig (1884–1956) American mathematician

Number: The Language of Science (1930)

History

“I may here refer to a curious mathematical calculation by Dr. Thomas Young, to the effect, that if three words coincide in two different languages, it is ten to one they must be derived in both cases from some parent language, or introduced in some other manner.”

Robert Chambers (publisher, born 1802) book Vestiges of the Natural History of Creation

Source: Vestiges of the Natural History of Creation (1844), p. 293

Parent

“One cannot help but be struck by the diversity that characterizes efforts to study the management process. If it is true that psychologists like to study personality traits in terms of a person's reactions to objects and events, they could not choose a better stimulus than management science. Some feel it is a technique, some feel it is a branch of mathematics, or of mathematical economics, or of the "behavioral sciences," or of consultation services, or just so much nonsense. Some feel it is for management (vs. labor), some feel it ought to be for the good of mankind — or for the good of underpaid professors.
But this diversity of attitude, which is really characteristic of all fields of endeavor, is matched by another and more serious kind of diversity. In the management sciences, we have become used to talking about game theory, inventory theory, waiting line theory. What we mean by "theory" in this context is that if certain assumptions are valid, then such-and-such conclusions follow. Thus inventory theory is not a set of statements that predict how inventories will behave, or even how they should behave in actual situations, but is rather a deductive system which becomes useful if the assumptions happen to hold. The diversity of attitude on this point is reflected in two opposing points of view: that the important problems of management science are theoretical, and that the important problems are factual.”

C. West Churchman (1913–2004) American philosopher and systems scientist

quote in: Fremont A. Shull (ed.), Selected readings in management https://archive.org/stream/selectedreadings00shul#page/n13/mode/2up, , 1957. p. 7-8
1940s - 1950s, "Management Science — Fact or Theory?" 1956

Behavior , Feeling , Game , Problem

“[T]he formalist school, of whom the most eminent representative is Hilbert, have concentrated on the propositions of mathematics, such as '2 + 2 = 4'. They have pronounced these to be meaningless formulae to be manipulated according to arbitrary rules, and they hold that mathematical knowledge consists in knowing what formulae can be derived from what others consistently with the rules…. for example…'2' is a meaningless mark occurring in these meaningless formulae. But… '2' occurs not only in '2 + 2 = 4', but also in 'It is 2 miles to the station', which is not a meaningless formulae, but a significant proposition, in which '2' cannot conceivably be a meaningless mark.”

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

The Foundations of Mathematics (1925)

School , Knowledge

“Cybernetics is a young discipline which, like applied mathematics, cuts across the entrenched departments of natural science; the sky, the earth, the animals and the plants. Its interdisciplinary character emerges when it considers economy not as an economist, biology not as a biologist, engines not as an engineer. In each case its theme remains the same, namely, how systems regulate themselves, reproduce themselves, evolve and learn. Its high spot is the question of how they organize themselves.
A cybernetic laboratory has a varied worksheet - concept formation in organized groups, teaching machines, brain models, and chemical computers for use in a cybernetic factory. As pure scientists we are concerned with brain-like artifacts, with evolution, growth and development; with the process of thinking and getting to know about the world. Wearing the hat of applied science, we aim to create what Boulanger,' in his presidential address to the International Association of Cybernetics, called the instruments of a new industrial revolution - control mechanisms that lay their own plans.”

Gordon Pask (1928–1996) British psychologist

Source: An Approach to Cybernetics (1961), p. 11. Partly cited in: A.M.E. Salazar, A. Espinosa, J. Walker (2011) A Complexity Approach to Sustainability: Theory and Application. p. 11.

World , Question , Nature , Thinking

“Those that can readily master the difficulties of Mathematics find a considerable charm in the study, sometimes amounting to fascination. This is far from universal; but the subject contains elements of strong interest of a kind that constitutes the pleasures of knowledge. The marvellous devices for solving problems elate the mind with the feeling of intellectual power; and the innumerable constructions of the science leave us lost in wonder.”

Alexander Bain (1818–1903) Scottish philosopher and educationalist

Source: Education as a Science, 1898, p. 153.

Pleasure , Feeling , Knowledge , Problem

“Personal computers will make our future adult population simultaneously more mathematically able and more visually literate. Ten years from now, teenagers are likely to enjoy a much richer panorama of options because the pursuit of intellectual achievement will not be tilted so much in favor of the bookworm, but instead cater to a wider range of cognitive styles, learning patterns, and expressive behaviors.”

Nicholas Negroponte book Being Digital

Being Digital (1995)

Behavior , Learning , Future , Personality

“The mathematical framework of quantum theory has passed countless successful tests and is now universally accepted as a consistent and accurate description of all atomic phenomena. The verbal interpretation, on the other hand – i. e., the metaphysics of quantum theory – is on far less solid ground. In fact, in more than forty years physicists have not been able to provide a clear metaphysical model.”

Fritjof Capra book The Tao of Physics

Source: The Tao of Physics (1975), Ch. 10, The Unity of All Things, p. 132.

Universe , Success

“My poem (Love -Kisses XL1) is like a nugget of gold or clump of earth it contains within itself the seeds rhetoric, ethics, philosophy, mathematics, geography, music, astronomy, medicine, nature, law, writing”

Quirinus Kuhlmann (1651–1689) German poet and mystic

Love-Kiss XL1

Love , Law , Music , Kiss

“As a boy of six I could understand the proof of a mathematical theorem more readily than that meat had to be cut with one's knife, not one's fork.”

Ferdinand Eisenstein (1823–1852) German mathematician

Curriculum Vitae - an autobiographical statement written when Eisenstein was 20, often referred to as his "Autobiography" (1843)

Boy , Understanding

“A scratch is nothing but the back-cueing that you hear in your ear before you push it out to the crowd. All you have to know is mathematically how many times to scratch it and when to let it go — when certain things will enhance the record you're listening to. For instance, if you're playing a record with drums--horns would sound nice to enhance it so you get a record with horns and slip it in at certain times.”

Grandmaster Flash (1958) musician

Quoted in Rap Attack 2 (1991) by David Toop, p. 65

Time

“If we do not succeed in solving a mathematical problem, the reason frequently consists in our failure to recognize the more general standpoint from which the problem before us appears only as a single link in a chain of related problems. After finding this standpoint, not only is this problem frequently more accessible to our investigation, but at the same time we come into possession of a method which is applicable also to related problems.”

David Hilbert Mathematical Problems

Mathematical Problems (1900)

Failure , Problem , Appearance , Time

“The only two good things in life are doing mathematics and teaching it.”

Siméon Denis Poisson (1781–1840) French mathematician, mechanician and physicist

La vie n'est bonne qu'à deux choses : à faire des mathématiques et à les professer.
quoted by François Arago in Notices biographiques, Volume 2 http://books.google.fr/books?pg=PA662&id=ZzNLAAAAYAAJ#v=onepage&q&f=false, 1854, p. 662.

Life

“The mathematician always proceeds from thought to being or things. Consequently, critical idealism was born the day Descartes decided that the mathematical method must henceforth be the method for metaphysics.”

Étienne Gilson (1884–1978) French historian and philosopher

Methodical Realism