Quotes about mathematics (1323 quotes, page 4)

“The only progress I can see is progress in organization. The ordinary human being does not live long enough to draw any substantial benefit from his own experience. And no one, it seems, can benefit by the experiences of others. Being both a father and teacher, I know we can teach our children nothing. We can transmit to them neither our knowledge of life nor of mathematics. Each must learn its lesson anew.”

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

1920s, Viereck interview (1929)

Life , Children , Knowledge , Learning

“From Pythagoras to Boethius, when pure mathematics consisted of arithmetic and geometry while applied mathematics consisted of music and astronomy, mathematics could be characterized as the deductive study of 'such abstractions as quantities and their consequences, namely figures and so forth' (Acquinas ca. 1260). But since the emergence of abstract algebra it has become increasingly difficult to formulate a definition to cover the whole of the rich, complex and expanding domain of mathematics.”

George Frederick James Temple (1901–1992) British mathematician

100 Years of Mathematics: a Personal Viewpoint (1981)

Music , Study

“Mr. Stone was satisfied, being sure in his heart that any person skilled with mathematical tools could learn anything else he needed to know, with or without a master.”

Robert A. Heinlein book The Rolling Stones

Source: The Rolling Stones (1952), Chapter 4, “Aspects of Domestic Engineering” (p. 61)

Learning , Heart , Personality

“To the average mathematician who merely wants to know that his work is securely based, the most appealing choice is to avoid difficulties by means of Hilbert's program. Here one regards mathematics as a formal game and one is only concerned with the question of consistency.”

Paul Cohen (1934–2007) American mathematician

p. 11 of "Comments on the foundations of set theory." https://books.google.com/books?id=TVi2AwAAQBAJ&pg=PA11 In Axiomatic set theory, pp. 9-15. Providence (RI). American Mathematical Society, 1971.

Game , Question

“With the computer and programming languages, mathematics has newly-acquired tools, and its notation should be reviewed in the light of them. The computer may, in effect, be used as a patient, precise, and knowledgeable "native speaker" of mathematical notation.”

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

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

Knowledge , Light

“It wasn’t so long ago that complexity thinking was synonymous with bottom-up computer simulation. However, in the past 5-10 years we have seen other threads emerge from this mathematically focused starting point that acknowledge the profound philosophical implications of complexity.”

Gerald Midgley (1960) New Zealand acaedmic

Kurt A. Richardson and Gerald Midgley (2007) " Systems theory and complexity: Part 4 http://kurtrichardson.com/publications/richardson_midgley.pdf" in: E:CO Issue Vol. 9 Nos. 1-2 2007 pp. xx–xx.

Thinking , Past

“Eudoxes… not only based the method [of exhaustion] on rigorous demonstration… but he actually applied the method to find the volumes (1) of any pyramid, (2) of the cone, proving (1) that any pyramid is one third part of the prism which has the same base and equal height, and (2) that any cone is one third part of the cylinder which has the same base and equal height. Archimedes, however, tells us the remarkable fact that these two theorems were first discovered by Democritus, though he was not able to prove them (which no doubt means, not that he gave no sort of proof, but that he was not able to establish the propositions by the rigorous methods of Eudoxes. Archimedes adds that we must give no small share of the credit for these theorems to Democritus… another testimony to the marvellous powers, in mathematics as well as in other subjects, of the great man who, in the words of Aristotle, "seems to have thought of everything"…. Democritus wrote on irrationals; he is also said to have discussed the question of two parallel sections of a cone (which were evidently supposed to be indefinitely close together), asking whether we are to regard them as equal or unequal… Democritus was already close on the track of infinitesimals.”

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

Achimedes (1920)

Question , Parting

“Laplace made many important discoveries in mathematical physics… Indeed, he was interested in anything that helped to interpret nature. He worked on hydrodynamics, the wave propagation of sound, and the tides. In the field of chemistry, his work on the liquid state of matter is classic. His studies of the tension in the surface layer of water, which accounts for the rise of liquids inside a capillary tube, and of the cohesive forces in liquids, are fundamental. Laplace and Lavoisier designed an ice calorimeter (1784) to measure heat and measured the specific heat of numerous substances; heat, to them, was still a special kind of matter. Most of Laplace's life was, however, devoted to celestial mechanics.”

Morris Kline (1908–1992) American mathematician

Source: Mathematical Thought from Ancient to Modern Times (1972), p. 495

Life , Nature , Study , Water

“The result of the mathematician's creative work is demonstrative reasoning, a proof; but the proof is discovered by plausible reasoning, by guessing. If the learning of mathematics reflects to any degree the invention of mathematics, it must have a place for guessing, for plausible inference.”

George Pólya (1887–1985) Hungarian mathematician

Induction and Analogy in Mathematics (1954)

Learning

“The best history of Greek mathematics which exists at present is undoubtedly that of Gino Loria under the title Le scienze esatte nell' antica Grecia (second edition 1914…) …the arrangement is chronological …they raise the question whether in a history of this kind it is best to follow chronological order or to arrange the material according to subjects…
I have adopted a new arrangement, mainly according to subjects…”

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

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

History , Question

“They [the critics] forget that while the artist never works outside his time yet his art will go on to be merged gradually into the new art of a new age. There will be no short stop. We shall not, contrary to the expectation of these people, hear of the sudden death of Cubism, abstraction, so-called modern art.... if they could but realize that energy is a spiritual movement and that they must conceive of working under a law of universal aesthetic progress, as we do in science, in mathematics, in physics.”

Arshile Gorky (1904–1948) Armenian-American painter

Quote from: 'Stuart Davis', Arshile Gorky, in 'Creative Art 9', September 1931
1930 - 1941

Death , People , Age , Law

“I was an atheist, finding no reason to postulate the existence of any truths outside of mathematics, physics and chemistry. But then I went to medical school, and encountered life and death issues at the bedsides of my patients. Challenged by one of those patients, who asked "What do you believe, doctor?", I began searching for answers.”

Francis S. Collins (1950) Geneticist; Director of the National Institutes of Health

cnn.com http://edition.cnn.com/2007/US/04/03/collins.commentary/

Life , Death , Truth , School

“Courage is not a quality one normally associates with mathematicians. Yet it should apply to people who work in their attics in secret for seven years without cease on a problem that has eluded the greatest mathematical minds since first proposed in 1637.”

Charles Krauthammer (1950–2018) American journalist

1990s, 1993, Eureka! A Modest Genius (1993)

People , Secret , Courage , Problem

“Calculus is the mathematics of change. …Change is characteristic of the world.”

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

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

World , Change

“This implication of the Mathematical Universe Hypothesis is pretty radical, so please pause your reading for a moment to take it in and think about it. What you’re aware of right at this moment feels not like a photo but like a movie clip. This movie isn’t reality—it exists only in your head, as part of your brain’s reality model. It contains lots of information about the actual external physical reality—as long as you aren’t dreaming or hallucinating—but still constitutes only a very heavily edited version of reality, akin to the evening news on TV, mainly featuring certain highlights of patterns nearby in space and time that your brain thinks are useful for you to be aware of.”

Max Tegmark book Our Mathematical Universe

Our Mathematical Universe: My Quest for the Ultimate Nature of Reality (2014)

Dreams , Feeling , Photo , Thinking

“Right now, at my institute in the US we are working on gene-caused failures in brain development that frequently lead to autism and schizophrenia. We may also find that differences in these respective brain development genes also lead to differences in our abilities to carry out different mental tasks.
In some cases, how these genes function may help us to understand variations in IQ, or why some people excel at poetry but are terrible at mathematics. All too often people with high mathematical abilities have autistic traits. The same gene that gives some people such great mathematical abilities may also lead to autistic behaviour. This is why, in studying autism and schizophrenia, we believe that we shall come very close to a better understanding of intelligence and, therefore, of the differences in intelligence.”

James D. Watson (1928) American molecular biologist, geneticist, and zoologist.

To question genetic intelligence is not racism (2007)

People , Intelligence , Failure , Understanding

“An artist who sees that the imitation of natural appearances, however artistic, is not for him – the kind of creative artist who wants to, and has to express his own inner world – sees with envy how naturally and easily such goals can be attained in music, the least material of the arts today. Understandably, he may turn toward it and try to find the same means in his own art. Hence the current search for rhythm in painting, for mathematical abstract construction.... the way colors are set in motion, etc.”

Wassily Kandinsky (1866–1944) Russian painter

1910 - 1915
Source: On the Spiritual in Art, 1911; as quoted in Schönberg and Kandinsky: An Historic Encounter, by Klaus Kropfinger; edited by Konrad Boehmer; published by Routledge (imprint of Taylor & Francis, an informal company), 2003, p. 15

World , Art , Music , Color

“To the scientists of 1850, Hamilton's principle was the realization of a dream. …from the time of Galileo scientists had been striving to deduce as many phenomena of nature as possible from a few fundamental physical principles. …they made striking progress …But even before these successes were achieved Descartes had already expressed the hope and expectation that all the laws of science would be derivable from a single basic law of the universe. This hope became a driving force in the late eighteenth century after Maupertuis's and Euler's work showed that optics and mechanics could very likely be unified under one principle. Hamilton's achievement in encompassing the most developed and largest branches of physical science, mechanics, optics, electricity, and magnetism under one principle was therefore regarded as the pinnacle of mathematical physics.”

Morris Kline (1908–1992) American mathematician

Source: Mathematical Thought from Ancient to Modern Times (1972), p. 441.

Law , Dreams , Hope , Nature

“I trust the clergy are not involved in this problem. They‘re notoriously unscientific. One can‘t attack them with mathematics.”

S. S. Van Dine book The Bishop Murder Case

Ch 5
The Bishop Murder Case (1929)

Problem

“This book… is primarily and essentially an account of the discovery or invention of mathematical concepts, and the historical material is… divided and classified, neither chronologically nor biographically, but philosophically.”

George Frederick James Temple (1901–1992) British mathematician

100 Years of Mathematics: a Personal Viewpoint (1981)

Books

“How did we recover our liberty? By fraud and violence. We tried to overcome the thirty thousand majority by honest methods, which was a mathematical impossibility. After we had borne these indignities for eight years life became worthless under such conditions.”

Benjamin Tillman (1847–1918) American politician

As quoted in "The Question of Race in the South Carolina Constitutional Convention of 1895" (July 1952), by George B. Tindall. The Journal of Negro History, 37 (3): 277–303. JSTOR 2715494., p. 94.

Life , Fraud

“An explanation of a phenomenon is regarded, apparently instinctively, as the most general possible when it is a mechanical explanation. The "mechanism" of the process is the ultimate goal of experiment. Now this mechanism in general lies beyond the range of the senses; either by reason of their limitations, as in the case of the atomic structure of matter, or by the very nature of the supposed mechanism, as in the theory of the ether. The only way to bridge the gap between the machinery of the physical process and the world of sense-impressions is to think out some consequence of that mechanism. This we will call the hypothesis. The hypothesis, resting still on the mechanical basis, is yet beyond the range of direct experimental investigation; but if, by mathematical reasoning, a consequence of the hypothesis can be deduced, this will often lie within the range of experimental inquiry, and thus a test of the soundness of the original mechanical conception may be instituted.”

J. R. Partington (1886–1965) British chemist

Introduction
Higher Mathematics for Chemical Students (1911)

Lie , World , Way , Nature

“The word science of administration has been used. There are many who object to the term. Now if by science is meant a conceptual scheme of things in which every particularity coveted may be assigned a mathematical value, then administration is not a science. In this sense only astro-physics may be called a science and it is well to remember that mechanical laws of the heavens tell us nothing about the color and composition of the stars and as yet cannot account for some of the disturbances and explosions which seem accidental. If, on the other hand, we may rightly use the term science in connection with a body of exact knowledge derived from experience and observation, and a body of rules or axioms which experience has demonstrated to be applicable in concrete practice, and to work out in practice approximately as forecast, then we may, if we please, appropriately and for convenience, speak of a science of administration. Once, when the great French mathematician, Poincaré, was asked whether Euclidean geometry is true, he replied that the question had no sense but that Euclidean geometry is and still remains the most convenient. The Oxford English Dictionary tells us that a science is, among other things, a particular branch of knowledge or study; a recognized department of learning.”

Charles A. Beard (1874–1948) American historian

Source: Philosophy, Science and Art of Public Administration (1939), p. 660-1

Law , Color , Star , Question

“In the natural sciences, language (mathematics) is a useful tool: like the microscope or telescope, it enables us to see what is otherwise invisible. In the social sciences, language (literalized metaphor) is an impediment: like a distorting mirror, it prevents us from seeing the obvious.
That is why in the natural sciences, knowledge can be gained only with the mastery of their special languages; whereas in human affairs, knowledge can be gained only by rejecting the pretentious jargons of the social sciences.”

Thomas Szasz (1920–2012) Hungarian psychiatrist

The Untamed Tongue: A Dissenting Dictionary (1990)

Nature , Knowledge , Science

“A scientific theory is a concise and coherent set of concepts, claims, and laws (frequently expressed mathematically) that can be used to precisely and accurately explain and predict natural phenomena.
A theory should include a mechanism that explains how its concepts, claims, and laws arise from lower-level theories.”

Mordechai Ben-Ari (1948) Israeli computer scientist

Source: Just a Theory: Exploring the Nature of Science (2005), Chapter 2, “Just a Theory: What Scientists Do” (p. 24)

Law , Nature

“The mathematical forms of order which the mind of a physicist manipulates coincides "miraculously" with experimental measurements.”

Marie-Louise von Franz (1915–1998) Swiss psychologist and scholar

Source: Psyche and Matter (1992), p. 269

“There are numerous theorems in economics that rely upon mathematically fallacious propositions.”

Steve Keen (1953) Australian economist

Source: Debunking Economics - The Naked Emperor Of The Social Sciences (2001), Chapter 12, Don't Shoot Me, I'm Only The Piano, p. 259

“.. [by making his work 'Onement', in 1948].. from then on I had to give up any relation to nature, as seen [by himself till then]. That doesn't mean that I think my things are mathematical or removed from life. By 'nature' I mean something very specific. I think that some abstractions - for example Kandinsky's - are really nature paintings. The triangles and the spheres or circles could be bottles. They could be trees, or buildings. I think that in 'Euclydean Abyss' and 'Onement' I removed myself from nature. But I did not remove myself from life.”

Barnett Newman (1905–1970) American artist

interview, April 1965, edited for broadcasting by the BBC first published in 'The Listener', Aug. 1972; as quoted in Interviews with American Artists, by David Sylvester; Chatto & Windus, London 2001, p. 37
1960 - 1970, Interview with David Sylvester 1. Spring 1965

Life , Nature , Thinking , Tree

“We must not be enticed by mathematically attractive assumptions into pretending that the contingencies of men's social positions and the asymmetries of their situations somehow even out in the end. Rather we must choose our conception of justice fully recognizing that this is not and cannot be the case.”

John Rawls book A Theory of Justice

Source: A Theory of Justice (1971; 1975; 1999), Chapter III, Section 28, pg. 171

Men , Justice

“Surely, after 62 years, we should have an exact formulation of some serious part of quantum mechanics? By 'exact' I do not of course mean 'exactly true'. I mean only that the theory should be fully formulated in mathematical terms, with nothing left to the discretion of the theoretical physicist… until workable approximations are needed in applications. By 'serious' I mean that some substantial fragment of physics should be covered. Nonrelativistic 'particle' quantum mechanics, perhaps with the inclusion of the electromagnetic field and a cut-off interaction, is serious enough.”

John S. Bell (1928–1990) Northern Irish physicist

Against 'measurement' (1990)

Parting

“Superficially at least the forces of electricity and magnetism seem to present the same kind of problem as gravitation. Experiment shows that two electrically charged bodies attract one another (or repel if their charges are of the same kind) with a force which conforms to the same mathematical law as the force of gravitation - both forces fall off inversely as the inverse square of the distance. The same is true of the magnetic force also; two magnetic poles attract or repel one another with a force which again follows the law of the inverse square of the distance.”

James Jeans (1877–1946) British mathematician and astronomer

Physics and Philosophy (1942)

Law , Problem

“The concept of 'number' in its most elementary sense as the signless integer appears to be an immediate abstraction from quantitative reality subjected to processes of counting and measurement. Vulgar fractions arise from division of a quantity into equal parts. But in what sense is zero a number? Are there negative numbers? Are there numbers corresponding to incommensurable ratios? Each question requires for its solution a fresh exercise of that kind of creative imagination which we call mathematical abstraction.”

George Frederick James Temple (1901–1992) British mathematician

100 Years of Mathematics: a Personal Viewpoint (1981)

Exercises , Imagination , Question , Reality

“Music trains the mind, like mathematics, or logic, to precision of mind.”

Guy Gavriel Kay book Tigana

Source: Tigana (1990), Chapter 4 (p. 77)

Music , Training

“The symbol is the tool which gives man his power, and it is the same tool whether the symbols are images or words, mathematical signs or mesons.”

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

"The Reach of Imagination" (1967)

“All true metaphysics is taken from the essential nature of the thinking faculty itself, and therefore in nowise invented, since it is not borrowed from experience, but contains the pure operations of thought, that is, conceptions and principles à priori, which the manifold of empirical presentations first of all brings into legitimate connection, by which it can become empirical knowledge, i. e. experience. …mathematical physicists were thus quite unable to dispense with such metaphysical principles…”

Immanuel Kant book Metaphysical Foundations of Natural Science

Preface, Tr. Bax (1883)
Metaphysical Foundations of Natural Science (1786)

Nature , Knowledge , Thinking

“Science does not speak of the world in the language of words alone, and in many cases it simply cannot do so. The natural language of science is a synergistic integration of words, diagrams, pictures, graphs, maps, equations, tables, charts, and other forms of visual and mathematical expression… [Science thus consists of] the languages of visual representation, the languages of mathematical symbolism, and the languages of experimental operations.”

Jay Lemke (1946) American academic

Jay Lemke (2003), "Teaching all the languages of science: Words , symbols, images and actions," p. 3; as cited in: Scott, Phil, Hilary Asoko, and John Leach. "Student conceptions and conceptual learning in science." Handbook of research on science education (2007): 31-56.

World , Nature , Science

“It is true that M. Fourier had the opinion that the principal end of mathematics was the public utility and the explanation of natural phenomena; but such a philosopher as he is should have known that the unique end of science is the honor of the human mind, and that from this point of view a question of number is as important as a question of the system of the world.”

Carl Gustav Jacob Jacobi (1804–1851) German mathematician

Letter to Legendre (July 2, 1830) in response to Fourier's report to the Paris Academy Science that mathematics should be applied to the natural sciences, as quoted in Science (March 10, 1911) Vol. 33 https://books.google.com/books?id=4LU7AQAAMAAJ&pg=PA359, p.359, with additional citations and dates from H. Pieper, "Carl Gustav Jacob Jacobi," Mathematics in Berlin (2012) p.46

World , Question , Nature , Science

“Look at the budget that was just proposed in Washington. It is an attack of unimaginable cruelty on the most vulnerable among us, the youngest, the oldest, the poorest, and hard-working people who need a little help to gain or hang on to a decent middle-class life. It grossly underfunds public education, mental health, and efforts even to combat the opioid epidemic. And in reversing our commitment to fight climate change, it puts the future of our nation and our world at risk. And to top it off, it is shrouded in a trillion-dollar mathematical lie. Let's call it what it is. It's a con. They don't even try to hide it.”

Hillary Clinton (1947) American politician, senator, Secretary of State, First Lady

Post Presidential Election, Wellesley Commencement Speech (2017)

Life , People , Health , Lie

“[The mathematical character of Descartes' physics lies in its methodological nature, namely, the] axiomatic structure of the whole system, in the establishment of indubitable foundations and the deduction of the phenomena.”

Eduard Jan Dijksterhuis (1892–1965) Dutch historian

Source: The mechanization of the world picture, 1961, p. 414; as cited in: ‎Marleen Rozemond (2009), Descartes's Dualism. p. 235

Nature

“The fundamental doctrines of motion have [herein]… been more immediately referred to axioms simply mathematical than has hitherto been usual; and the application of these doctrines to practical purposes has [herein]… been facilitated.”

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

Preface (March 30, 1807)
A Course of Lectures on Natural Philosophy and the Mechanical Arts (1807)

“My specific… object has been to contain, within the prescribed limits, the whole of the student's course, from the confines of elementary algebra and trigonometry, to the entrance of the highest works on mathematical physics. A learner who has a good knowledge of the subjects just named, and who can master the present treatise, taking up elementary works on conic sections, application of algebra to geometry, and the theory of equations, as he wants them, will, I am perfectly sure, find himself able to conquer the difficulties of anything he may meet with; and need not close any book of Laplace, Lagrange, Legendre, Poisson, Fourier, Cauchy, Gauss, Abel, Hindenburgh and his followers. or of any one of our English mathematicians, under the idea that it is too hard for him.”

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

Preface, p. iii
The Differential and Integral Calculus (1836)

Books , Idea , Knowledge

“When a man counts one, two, three, he is not only doing mathematics, he is on the path to the mysticism of numbers in Pythagoras and Vitruvius and Kepler, to the Trinity and the signs of the Zodiac.”

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

"The Reach of Imagination" (1967)

“Charles Francis Adams's memory was hardly above the average; his mind was not bold like his grandfather's or restless like his father's, or imaginative or oratorical — still less mathematical; but it worked with singular perfection, admirable self-restraint, and instinctive mastery of form. Within its range it was a model.”

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

The Education of Henry Adams (1907)

Imagination , Memory , Perfection

“Because of mathematical indeterminancy and the uncertainty principle, it may be a law of nature that no nervous system is capable of acquiring enough knowledge to significantly predict the future of any other intelligent system in detail. Nor can intelligent minds gain enough self-knowledge to know their own future, capture fate, and in this sense eliminate free will.”

Edward O. Wilson book On Human Nature

On Human Nature (1978), Ch.4 Emergence

Fate , Law , Intelligence , Nature

“What had already been done for music by the end of the eighteenth century has at last been begun for the pictorial arts. Mathematics and physics furnished the means in the form of rules to be followed and to be broken. In the beginning it is wholesome to be concerned with the functions and to disregard the finished form. Studies in algebra, in geometry, in mechanics characterize teaching directed towards the essential and the functional, in contrast to apparent. One learns to look behind the façade, to grasp the root of things. One learns to recognize the undercurrents, the antecedents of the visible. One learns to dig down, to uncover, to find the cause, to analyze.”

Paul Klee (1879–1940) German Swiss painter

1921 - 1930
Source: 'Bauhaus prospectus 1929'; as quoted in Artists on Art, from the 14th – 20th centuries, ed. by Robert Goldwater and Marco Treves; Pantheon Books, 1972, London, p. 444

Art , Music , Study , Learning

“It is known that the mathematics prescribed for the high school [Gymnasien] is essentially Euclidean, while it is modern mathematics, the theory of functions and the infinitesimal calculus, which has secured for us an insight into the mechanism and laws of nature. Euclidean mathematics is indeed, a prerequisite for the theory of functions, but just as one, though he has learned the inflections of Latin nouns and verbs, will not thereby be enabled to read a Latin author much less to appreciate the beauties of a Horace, so Euclidean mathematics, that is the mathematics of the high school, is unable to unlock nature and her laws. Euclidean mathematics assumes the completeness and invariability of mathematical forms; these forms it describes with appropriate accuracy and enumerates their inherent and related properties with perfect clearness, order, and completeness, that is, Euclidean mathematics operates on forms after the manner that anatomy operates on the dead body and its members.
On the other hand, the mathematics of variable magnitudes—function theory or analysis—considers mathematical forms in their genesis. By writing the equation of the parabola, we express its law of generation, the law according to which the variable point moves. The path, produced before the eyes of the 113 student by a point moving in accordance to this law, is the parabola.
If, then, Euclidean mathematics treats space and number forms after the manner in which anatomy treats the dead body, modern mathematics deals, as it were, with the living body, with growing and changing forms, and thus furnishes an insight, not only into nature as she is and appears, but also into nature as she generates and creates,—reveals her transition steps and in so doing creates a mind for and understanding of the laws of becoming. Thus modern mathematics bears the same relation to Euclidean mathematics that physiology or biology … bears to anatomy. But it is exactly in this respect that our view of nature is so far above that of the ancients; that we no longer look on nature as a quiescent complete whole, which compels admiration by its sublimity and wealth of forms, but that we conceive of her as a vigorous growing organism, unfolding according to definite, as delicate as far-reaching, laws; that we are able to lay hold of the permanent amidst the transitory, of law amidst fleeting phenomena, and to be able to give these their simplest and truest expression through the mathematical formulas”

Christian Heinrich von Dillmann (1829–1899) German educationist

Source: Die Mathematik die Fackelträgerin einer neuen Zeit (Stuttgart, 1889), p. 37.

School , Law , Beauty , Understanding

“The eulogy of the non-lover in the speech of Lysias … stresses the fact that the non-lover … does not commit extreme acts under the influence of passions. Since he acts from calculation, he never has occasion for remorse. … The non-lover demonstrates his superiority through prudence and objectivity. … We must now observe how these points of superiority correspond to those of “semantically purified” speech, … the kind of speech approaching pure notation in the respect that it communicates abstract intelligence without impulsion. It is a simple instrumentality, showing no affection for the object of its symbolizing and incapable of inducing bias in the hearer. In its ideal conception, it would have less power to move than 2 + 2 = 4, since it is generally admitted that mathematical equations may have the beauty of elegance, and hence are not above suspicion where beauty is suspect. But this neuter language will be an unqualified medium of transmission of meanings from mind to mind, and by virtue of it minds can remain in an unprejudiced relationship to the world and also to other minds. … Instead of passion, it offers the serviceability of objectivity … It distrusts any departure from the literal and prosaic.”

Richard M. Weaver (1910–1963) American scholar

“The Phaedrus and the Nature of Rhetoric,” pp. 6-7.
The Ethics of Rhetoric (1953)

World , Beauty , Intelligence , Communication

“Mathematics allows you to see the invisible.”

Edward Frenkel (1968) mathematician working in representation theory, algebraic geometry, and mathematical physics

[Contenta, Sandro, The Canadian who reinvented mathematics, http://projects.thestar.com/math-the-canadian-who-reinvented-mathematics/, Toronto Star]

“The answer is in the problem, not away from the problem. I go through the searching, analysing, dissecting process, in order to escape from the problem. But, if I do not escape from the problem and try to look at the problem without any fear or anxiety, if I merely look at the problem — mathematical, political, religious, or any other — and not look to an answer, then the problem will begin to tell me. Surely, this is what happens. We go through this process and eventually throw it aside because there is no way out of it. So, why can’t we start right from the beginning, that is, not seek an answer to a problem? — which is extremely arduous, isn’t it? Because, the more I understand the problem, the more significance there is in it. To understand, I must approach it quietly, not impose on the problem my ideas, my feelings of like and dislike. Then the problem will reveal its significance. Why is it not possible to have tranquillity of the mind right from the beginning?”

Jiddu Krishnamurti (1895–1986) Indian spiritual philosopher

"Eighth Talk in The Oak Grove, 7 August 1949" http://www.jkrishnamurti.org/krishnamurti-teachings/view-text.php?tid=320&chid=4643&w=%22The+answer+is+in+the+problem%2C+not+away+from+the+problem%22, J.Krishnamurti Online, JKO Serial No. 490807, Vol. V, p. 283
Posthumous publications, The Collected Works

Fear , Way , Feeling , Search

“Everyone knows that mathematics offers an excellent opportunity to learn demonstrative reasoning, but I contend also that there is no other subject in the usual curricula of the schools that affords a comparable opportunity to learn plausible reasoning. …let us learn proving, but also let us learn guessing.”

George Pólya (1887–1985) Hungarian mathematician

Induction and Analogy in Mathematics (1954)

School , Learning

“Mathematics may be a way of developing physically, that is anatomically, new connections in the brain.”

Stanislaw Ulam (1909–1984) Polish-American mathematician

Source: Adventures of a Mathematician - Third Edition (1991), Chapter 15, Random Reflections on Mathematics and Science, p. 277

Way

“No.3 Commando was very anxious to be chums with Lord Glasgow, so they offered to blow up an old tree stump for him and he was very grateful and said don't spoil the plantation of young trees near it because that is the apple of my eye and they said no of course not we can blow a tree down so it falls on a sixpence and Lord Glasgow said goodness you are clever and he asked them all to luncheon for the great explosion.
So Col. Durnford-Slater DSO said to his subaltern, have you put enough explosive in the tree?. Yes, sir, 75lbs. Is that enough? Yes sir I worked it out by mathematics it is exactly right. Well better put a bit more. Very good sir.
And when Col. D Slater DSO had had his port he sent for the subaltern and said subaltern better put a bit more explosive in that tree. I don't want to disappoint Lord Glasgow. Very good sir.
Then they all went out to see the explosion and Col. DS DSO said you will see that tree fall flat at just the angle where it will hurt no young trees and Lord Glasgow said goodness you are clever.
So soon they lit the fuse and waited for the explosion and presently the tree, instead of falling quietly sideways, rose 50 feet into the air taking with it ½ acre of soil and the whole young plantation.
And the subaltern said Sir, I made a mistake, it should have been 7½ not 75. Lord Glasgow was so upset he walked in dead silence back to his castle and when they came to the turn of the drive in sight of his castle what should they find but that every pane of glass in the building was broken.
So Lord Glasgow gave a little cry and ran to hide his emotions in the lavatory and there when he pulled the plug the entire ceiling, loosened by the explosion, fell on his head.
This is quite true.”

Evelyn Waugh (1903–1966) British writer

Letter to his wife (31 May 1942)

Disappointment , Tree , Silence , Waiting

“We are so accustomed to hear arithmetic spoken of as one of the three fundamental ingredients in all schemes of instruction, that it seems like inquiring too curiously to ask why this should be. Reading, Writing, and Arithmetic—these three are assumed to be of co-ordinate rank. Are they indeed co-ordinate, and if so on what grounds?
In this modern “trivium” the art of reading is put first. Well, there is no doubt as to its right to the foremost place. For reading is the instrument of all our acquisition. It is indispensable. There is not an hour in our lives in which it does not make a great difference to us whether we can read or not. And the art of Writing, too; that is the instrument of all communication, and it becomes, in one form or other, useful to us every day. But Counting—doing sums,—how often in life does this accomplishment come into exercise? Beyond the simplest additions, and the power to check the items of a bill, the arithmetical knowledge required of any well-informed person in private life is very limited. For all practical purposes, whatever I may have learned at school of fractions, or proportion, or decimals, is, unless I happen to be in business, far less available to me in life than a knowledge, say, of history of my own country, or the elementary truths of physics. The truth is, that regarded as practical arts, reading, writing, and arithmetic have no right to be classed together as co-ordinate elements of education; for the last of these is considerably less useful to the average man or woman not only than the other two, but than 267 many others that might be named. But reading, writing, and such mathematical or logical exercise as may be gained in connection with the manifestation of numbers, have a right to constitute the primary elements of instruction. And I believe that arithmetic, if it deserves the high place that it conventionally holds in our educational system, deserves it mainly on the ground that it is to be treated as a logical exercise. It is the only branch of mathematics which has found its way into primary and early education; other departments of pure science being reserved for what is called higher or university instruction. But all the arguments in favor of teaching algebra and trigonometry to advanced students, apply equally to the teaching of the principles or theory of arithmetic to schoolboys. It is calculated to do for them exactly the same kind of service, to educate one side of their minds, to bring into play one set of faculties which cannot be so severely or properly exercised in any other department of learning. In short, relatively to the needs of a beginner, Arithmetic, as a science, is just as valuable—it is certainly quite as intelligible—as the higher mathematics to a university student.”

Joshua Girling Fitch (1824–1903) British educationalist

Source: Lectures on Teaching, (1906), pp. 267-268.

Life , Truth , Education , School

“Ideality is preëminently the foundation of Mathematics.”

Benjamin Peirce (1809–1880) American mathematician

As quoted by Arnold B. Chace, in Benjamin Peirce, 1809-1880 : Biographical Sketch and Bibliography (1925) by R. C. Archibald.

“Computer science research is different from these more traditional disciplines. Philosophically it differs from the physical sciences because it seeks not to discover, explain, or exploit the natural world, but instead to study the properties of machines of human creation. In this it as analogous to mathematics, and indeed the "science" part of computer science is, for the most part mathematical in spirit. But an inevitable aspect of computer science is the creation of computer programs: objects that, though intangible, are subject to commercial exchange.”

Dennis M. Ritchie (1941–2011) American computer scientist

Reflections on Software Research (1984)

World , Nature , Study , Science

“Even though principles of rationality seem as often violated as followed, we still cling to the notion that human thought should be rational, logical, and orderly. Much of law is based upon the concept of rational thought and behavior. Much of economic theory is based upon the model of the rational human who attempts to optimize personal benefit, utility, or comfort. Many scientists who study artificial intelligence use the mathematics of formal logic—the predicate calculus—as their major tool to simulate thought. […] Human thought is not like logic; it is fundamentally different in kind and spirit. The difference is neither worse nor better. But it is the difference that leads to creative discovery and to great robustness of behavior.”

Donald A. Norman book The Design of Everyday Things

Source: The Design of Everyday Things (1988, 2002), Ch. 5, pp. 114–115.

Law , Behavior , Intelligence , Optimism

“Nonmathematical people sometimes ask me, “You know math, huh? Tell me something I’ve always wondered, What is infinity divided by infinity?” I can only reply, “The words you just uttered do not make sense. That was not a mathematical sentence. You spoke of ‘infinity’ as if it were a number. It’s not. You may as well ask, 'What is truth divided by beauty?’ I have no clue. I only know how to divide numbers. ‘Infinity,’ ‘truth,’ ‘beauty’—those are not numbers.””

John Derbyshire book Prime Obsession

Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics (2003)

People , Truth , Beauty , Sense

“A central problem in teaching mathematics is to communicate a reasonable sense of taste—meaning often when to, or not to, generalize, abstract, or extend something you have just done.”

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

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

Communication , Problem , Sense

“It is easy to create an interstellar radio message which can be recognized as emanating unambiguously from intelligent beings. A modulated signal (‘beep,’ ‘beep-beep,’. . . ) comprising the numbers 1, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, for example, consists exclusively of the first 12 prime numbers—that is, numbers that can be divided only by 1, or by themselves. A signal of this kind, based on a simple mathematical concept, could only have a biological origin.. . . But by far the most promising method is to send pictures.”

Carl Sagan (1934–1996) American astrophysicist, cosmologist, author and science educator

Smithsonian magazine, May 1978, pp. 43, 44. Quoted in Awake! magazine, 1978, 8/22.

Intelligence

“If we believe that the task of physics is the discovery of a timeless mathematical equation that captures every aspect of the universe, then we believe that the truth about the universe lies outside the universe.”

Lee Smolin (1955) American cosmologist

Time Reborn: From the Crisis in Physics to the Future of the Universe (2013)

Truth , Universe

“Classification is thought of by many librarians as either a fearful complication of a very simply act, or an outmoded, almost prehistoric, method of doing a very complex mathematical task.”

Douglas John Foskett (1918–2004)

Source: The Classification Research Group 1952—1962 (1962), p. 138

Fear

“In brief, one can view organizing essentially as either an economic, behavioral, adaptive, mathematical, or decisional entity. In the aggregate most of these theories are concerned with structure, behavior, and strategy under conditions of change and complexity brought about by technology, environment, and human behavior.”

George R. Terry (1909–1979)

Source: Principles of Management, 1960, p. 314 (6th ed. 1971)

Behavior , Change

“Modern science was born in the period beginning with Copernicus's work De Revolutionibus Orbium Coelestium (l543) and ending with Newton's Philosophia Naturalis Philosophiae Mathematica.
For reasons not to be entered into here, mediaeval scholasticism had not succeeded in finding an effective method for the investigation of natural phenomena. And Humanism, though indirectly instrumental in the creation of natural science through the promotion of knowledge of Greek works on mathematics, mechanics, and astronomy, had not been able to find the new paths that science would have to follow. The conviction shared by the two movements, viz. that science was something which mankind had once possessed, but had since lost, turned men's eyes towards the past instead of to the future — to ancient books instead of to new investigations and experiments.
The creation of modern science required a different philosophy. Man had to realize that if science is to grow, each generation must make its own contribution; and that the accumulated wisdom of antiquity is useful only as a starting-point for new research.”

Eduard Jan Dijksterhuis (1892–1965) Dutch historian

Source: Simon Stevin: Science in the Netherlands around 1600, 1970, p. 1; Lead paragraph

Wisdom , Men , Books , Nature

“See how involved it gets? Clover, bees, nitrogen, escape speed, power, plant-animal balance, gas laws, compound interest laws, meteorology—a mathematical ecologist has to think of everything and think of it ahead of time. Ecology is explosive; what seems like a minor and harmless invasion can change the whole balance.”

Robert A. Heinlein book Farmer in the Sky

Source: Farmer in the Sky (1950), Chapter 12, “Bees and Zeroes” (p. 125)

Law , Thinking , Bee , Change

“It is exceptional that one should be able to acquire the understanding of a process without having previously acquired a deep familiarity with running it, with using it, before one has assimilated it in an instinctive and empirical way… Thus any discussion of the nature of intellectual effort in any field is difficult, unless it presupposes an easy, routine familiarity with that field. In mathematics this limitation becomes very severe.”

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

As quoted in "The Mathematician" in The World of Mathematics (1956), by James Roy Newman

Way , Understanding , Nature , Effort

“The viewpoint of the formalist must lead to the conviction that if other symbolic formulas should be substituted for the ones that now represent the fundamental mathematical relations and the mathematical-logical laws, the absence of the sensation of delight, called "consciousness of legitimacy," which might be the result of such substitution would not in the least invalidate its mathematical exactness. To the philosopher or to the anthropologist, but not to the mathematician, belongs the task of investigating why certain systems of symbolic logic rather than others may be effectively projected upon nature. Not to the mathematician, but to the psychologist, belongs the task of explaining why we believe in certain systems of symbolic logic and not in others, in particular why we are averse to the so-called contradictory systems in which the negative as well as the positive of certain propositions are valid.”

L. E. J. Brouwer (1881–1966) Dutch mathematician and logician

as translated by Arnold Dresden from: Brouwer, L. E. J. (1913). Intuitionism and formalism. Bulletin of the American Mathematical Society, 20(2), 81–96. (quote on p. 84)

Law , Nature

“We know by actual observation only a comparatively small part of the whole universe. I will call this "our neighborhood." Even within the confines of this province our knowledge decreases very rapidly as we get away from our own particular position in space and time. It is only within the solar system that our empirical knowledge extends to the second order of small quantities (and that only for g44 and not for the other gαβ), the first order corresponding to about 10-8. How the gαβ outside our neighborhood are, we do not know, and how they are at infinity of space or time we shall never know. Infinity is not a physical but a mathematical concept, introduced to make our equations more symmetrical and elegant. From the physical point of view everything that is outside our neighborhood is pure extrapolation, and we are entirely free to make this extrapolation as we please to suit our philosophical or aesthetical predilections—or prejudices. It is true that some of these prejudices are so deeply rooted that we can hardly avoid believing them to be above any possible suspicion of doubt, but this belief is not founded on any physical basis. One of these convictions, on which extrapolation is naturally based, is that the particular part of the universe where we happen to be, is in no way exceptional or privileged; in other words, that the universe, when considered on a large enough scale, is isotropic and homogeneous.”

Willem de Sitter (1872–1934) Dutch cosmologist

"The Astronomical Aspect of the Theory of Relativity" (1933)

Way , Nature , Knowledge , Universe

“When you are into mathematics, you have been so high on the scale of complexity of reasoning that you are living in some kind of altered reality. You think everybody on the street is able to understand complicated reasoning […]. And you get very frustrated, when you discover that's not the case.”

Cédric Villani (1973) French mathematician

"Mathematics: Beauty vs Utility - Numberphile". youtube.com. January 19, 2017. Retrieved September 16, 2018.

Understanding , Thinking , Reality

“Mathematical physics is in the first place physics and it could not exist without experimental investigations.”

Peter Debye (1884–1966) Dutch-American physicist and physical chemist

Inaugural lecture for his professorship of mathematical physics at the University of Utrecht (1913), as quoted by Davies, Mansel. Peter Joseph Wilhelm Debye: 1884-1966. Biographical Memoirs of Fellows of The Royal Society, Vol. 16 (1970).

“Pure mathematics is much more than an armoury of tools and techniques for the applied mathematician. On the other hand, the pure mathematician has ever been grateful to applied mathematics for stimulus and inspiration. From the vibrations of the violin string they have drawn enchanting harmonies of Fourier Series, and to study the triode valve they have invented a whole theory of non-linear oscillations.”

George Frederick James Temple (1901–1992) British mathematician

100 Years of Mathematics: a Personal Viewpoint (1981)

Study

“It is true that in recent years a number of attractive histories of mathematics have been published in England and America, but these have only dealt with Greek mathematics as part of the larger subject, and in consequence the writers have been precluded… from presenting the work of the Greeks in suflicient detail. The same remark applies to the German histories of mathematics, even to the great work of Moritz Cantor…”

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

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

History , Parting

“If you expect to continue learning all your life, you will be teaching yourself much of the time. You must learn to learn, especially the difficult topic of mathematics.”

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

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

Life , Learning , Time

“Another feature of Alexandrian algebra is the absence of any explicit deductive structure. The various types of numbers… were not defined. Nor was there any axiomatic basis on which a deductive structure could be erected. The work of Heron, Nichomachus, and Diophantus, and of Archimedes as far as his arithmetic is concerned, reads like the procedural texts of the Egyptians and Babylonians… The deductive, orderly proof of Euclid and Apollonius, and of Archimedes' geometry is gone. The problems are inductive in spirit, in that they show methods for concrete problems that presumably apply to general classes whose extent is not specified. In view of the fact that as a consequence of the work of the classical Greeks mathematical results were supposed to be derived deductively from an explicit axiomatic basis, the emergence of an independent arithmetic and algebra with no logical structure of its own raised what became one of the great problems of the history of mathematics. This approach to arithmetic and algebra is the clearest indication of the Egyptian and Babylonian influences… Though the Alexandrian Greek algebraists did not seem to be concerned about this deficiency… it did trouble deeply the European mathematicians.”

Morris Kline (1908–1992) American mathematician

Source: Mathematical Thought from Ancient to Modern Times (1972), p.144

History , Problem , Reading

“There are two general Methods made use of in the Mathematicks, viz. Synthesis and Analysis, which we shall explain, after having acquainted the Reader, that the Method we make use of to resolve a Mathematical Problem, is called Zetetick; and that that Method which determines when, and by what way, and how many different ways a Problem may be resolved, is called Poristick. But in treating of Methods, we will first premise, that in general, a Method is the Art of disposing a Train of Arguments or Consequences in a right Order, either to discover the Truth of a Theorem, which we would find out, or to demonstrate it to others, when found.”

Jacques Ozanam (1640–1718) French mathematician

Source: A Mathematical Dictionary: Or; A Compendious Explication of All Mathematical Terms, 1702, p. 26

Truth , Art , Way , Problem

“A technique succeeds in mathematical physics, not by a clever trick, or a happy accident, but because it expresses some aspect of a physical truth.”

Graham Sutton (1903–1977) Welsh scientist

Mathematics in Action (1954) page 3

Truth , Happiness

““What is the Vienna Circle?” is a question which is neither rhetorical nor trivial. It is perhaps an attempt to ‘square the circle’ – which is, meanwhile, mathematically possible, as Karl Menger described as early as 1934.
This question might be also a problem of how the whole relates logically to the parts or the parts to the whole, which was already addressed by mereology (whole-part theory) according to Stanislaw (1916).
Of course, we are all familiar with the irritating fact that one and the same phenomenon can be described consistently by more than one theory”

Friedrich Stadler (1951) Austrian historian

underdetermination of a theory by observation
Source: "What is the Vienna Circle?" 2006, p. xi

Question , Problem , Parting

“I beg to introduce myself to you as a clerk in the Accounts Department of the Port Trust Office at Madras… I have no University education but I have undergone the ordinary school course. After leaving school I have been employing the spare time at my disposal to work at Mathematics. I have not trodden through the conventional regular course which is followed in a University course, but I am striking out a new path for myself. I have made a special investigation of divergent series in general and the results I get are termed by the local mathematicians as "startling"…. Very recently I came across a tract published by you styled Orders of Infinity in page 36 of which I find a statement that no definite expression has been as yet found for the number of prime numbers less than any given number. I have found an expression which very nearly approximates to the real result, the error being negligible. I would request that you go through the enclosed papers. Being poor, if you are convinced that there is anything of value I would like to have my theorems published. I have not given the actual investigations nor the expressons that I get but I have indicated the lines on which I proceed. Being inexperienced I would very highly value any advice you give me. Requesting to be excused for the trouble I give you. I remain, Dear Sir, Yours truly…”

Srinivasa Ramanujan (1887–1920) Indian mathematician

Letter to G. H. Hardy, (16 January 1913), published in Ramanujan: Letters and Commentary American Mathematical Society (1995) History of Mathematics, Vol. 9

Education , School , Error , Universe

“To put it bluntly, the discipline of economics has yet to get over its childish passion for mathematics.”

Thomas Piketty book Capital in the Twenty-First Century

Source: Capital in the Twenty-First Century (2013), p. 32.

Passion

“What expressions we used – in part taken over and in part newly invented! — above all, the famous 'wholly other' breaking in upon us ‘perpendicularly from above,’ the not less famous 'infinite qualitative distinction' between God and man, the vacuum, the mathematical point, and the tangent in which alone they must meet.”

Karl Barth (1886–1968) Swiss Protestant theologian

The Humanity of God (1960), p. 42.

God , Parting

“Pure mathematics proves itself a royal science both through its content and form, which contains within itself the cause of its being and its methods of proof. For in complete independence mathematics creates for itself the object of which it treats, its magnitudes and laws, its formulas and symbols.”

Christian Heinrich von Dillmann (1829–1899) German educationist

Source: Die Mathematik die Fackelträgerin einer neuen Zeit (Stuttgart, 1889), p. 94.

Law , Science

“He was a man of immense intellectual capabilities and most of us were marvelling at his doctorate thesis which focused on the mathematical solutions using digit zero. As a politician he was a man who choose his words carefully. As a personal and trusted friend whom I had assigned key responsibilities, the loss of Saitoti has been a painful experience to me”

Mwai Kibaki (1931) Former president of Kenya

Speaking at the George Saitoti burial (16 June 2012) at All Africa

Pain , Loss , Personality

“What renders a problem definite, and what leaves it indefinite, may best be understood from mathematics. The very important idea of solving a problem within limits of error is an element of rational culture, coming from the same source. The art of totalizing fluctuations by curves is capable of being carried, in conception, far beyond the mathematical domain, 65 where it is first learned. The distinction between laws and coefficients applies in every department of causation. The theory of Probable Evidence is the mathematical contribution to Logic, and is of paramount importance.”

Alexander Bain (1818–1903) Scottish philosopher and educationalist

Source: Education as a Science, 1898, pp. 151-152.

Law , Art , Error , Idea

“I maintain that in every special natural doctrine only so much science proper is to be met with as mathematics; for… science proper, especially of nature, requires a pure portion, lying at the foundation of the empirical, and based upon à priori knowledge of natural things. …the conception should be constructed. But the cognition of the reason through construction of conceptions is mathematical. A pure philosophy of nature in general, namely, one that only investigates what constitutes a nature in general, may thus be possible without mathematics; but a pure doctrine of nature respecting determinate natural things (corporeal doctrine and mental doctrine), is only possible by means of mathematics; and as in every natural doctrine only so much science proper is to be met with therein as there is cognition à priori, a doctrine of nature can only contain so much science proper as there is in it of applied mathematics.”

Immanuel Kant book Metaphysical Foundations of Natural Science

Preface, Tr. https://books.google.com/books?id=OCJLAAAAMAAJ Ernest Belfort Bax (1883)
Metaphysical Foundations of Natural Science (1786)

Lying , Nature , Knowledge , Science

“A psychological space is established for any set of stimuli by determining metric distances between the stimuli such that the probability that a response learned to any stimulus will generalize to any other is an invariant monotonic function of the distance between them. To a good approximation, this probability of generalization (i) decays exponentially with this distance, and (ii) does so in accordance with one of two metrics, depending on the relation between the dimensions along which the stimuli vary. These empirical regularities are mathematically derivable from universal principles of natural kinds and probabilistic geometry that may, through evolutionary internalization, tend to govern the behaviors of all sentient organisms.”

Roger Shepard (1929) American psychologist

Source: "Toward a universal law of generalization for psychological science," 1987, p. 1317

Behavior , Nature , Learning , Universe

“General Systems Theory is a name which has come into use to describe a level of theoretical model-building which lies somewhere between the highly generalized constructions of pure mathematics and the specific theories of the specialized disciplines. Mathematics attempts to organize highly general relationships into a coherent system, a system however which does not have any necessary connections with the "real" world around us. It studies all thinkable relationships abstracted from any concrete situation or body of empirical knowledge.”

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

Source: 1950s, General Systems Theory - The Skeleton of Science, 1956, p. 197: Opening sentences

World , Knowledge , Study

“I had a feeling once about Mathematics, that I saw it all—Depth beyond depth was revealed to me—the Byss and the Abyss. I saw, as one might see the transit of Venus—or even the Lord Mayor's Show, a quantity passing through infinity and changing its sign from plus to minus. I saw exactly how it happened and why the tergiversation was inevitable: and how the one step involved all the others. It was like politics. But it was after dinner and I let it go!”

Winston S. Churchill (1874–1965) Prime Minister of the United Kingdom

Source: My Early Life: A Roving Commission (1930), Chapter 3 (Examinations), p. 27.

Feeling , Change

“The work Was begun in 1913, but the bulk of it was written, as a distraction, during the first three years of the war, the hideous course of which seemed day by day to enforce the profound truth conveyed in the answer of Plato to the Delians. When they consulted him on the problem set them by the Oracle, namely that of duplicating the cube, he replied, 'It must be supposed, not that the god specially wished this problem solved, but that he would have the Greeks desist from war and wickedness and cultivate the Muses, so that, their passions being assuaged by philosophy and mathematics, they might live in innocent and mutually helpful intercourse with one another'.
Truly,Greece and her foundations are
Built below the tide of war,
Based on the crystàlline sea
Of thought and its eternity.</center”

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

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

God , War , Truth , Problem

“If the establishment of the thesis 'mathematics is ontology' is the basis of this book, it is in no way its goal.”

Alain Badiou (1937) French writer and philosopher

Introduction
Being and Event (1988)

Books , Way

“My conviction was justified: art, that which lasts, is based on mathematics.”

Jean Metzinger (1883–1956) French painter

Cubism was born

Art

“The 'physical' does not mean any particular kind of reality, but a particular kind of denoting reality, namely a system of concepts in the natural sciences which is necessary for the cognition of reality. 'The physical' should not be interpreted wrongly as an attribute of one part of reality, but not of the other; it is rather a word denoting a kind of conceptual construction, as, e. g., the markers 'geographical' or 'mathematical', which denote not any distinct properties of real things, but always merely a manner of presenting them by means of ideas.”

Moritz Schlick book Théorie générale de la connaissance

Source: Allgemeine Erkenntnislehre, 1925, p. 27 i. ; as cited in: Adam Schaff (1962). Introduction to semantics, p. 83

Idea , Nature , Reality , Science

“The success of mathematical physics led the social scientist to be jealous of its power without quite understanding the intellectual attitudes that had contributed to this power. The use of mathematical formulae had accompanied the development of the natural sciences and become the mode in the social sciences. Just as primitive peoples adopt the Western modes of denationalized clothing and of parliamentatism out of a vague feeling that these magic rites and vestments will at once put them abreast of modern culture and technique, so the economists have developed the habit of dressing up their rather imprecise ideas in the language of the infinitesimal calculus.”

Norbert Wiener (1894–1964) American mathematician

Source: Ex-Prodigy: My Childhood and Youth (1964), p. 89; partly cited in: Herman E. Daly. Steady-State Economics: Second Edition With New Essays. 1977/1991 p. 4

People , Feeling , Idea , Understanding

“It is clear that economics, if it is to be a science at all, must be a mathematical science.”

William Stanley Jevons The Theory of Political Economy

Source: The Theory of Political Economy (1871), Chapter I, Introduction, p. 38.

Science

“The formalists neglected the content altogether and made mathematics meaningless, the logicians neglected the form and made mathematics consist of any true generalizations; only by taking account of both sides and regarding it as composed of tautologous generalizations can we obtain an adequate theory.”

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

The Foundations of Mathematics (1925)

“Needless to say, there is no formal or mathematical content to ID—a “theory” that explains everything, explains nothing, and predicts nothing. ID cannot explain why millions of species were created and then became extinct. Even more importantly, it cannot explain “mistakes” in the design of living organisms such as us.”

Mordechai Ben-Ari (1948) Israeli computer scientist

Source: Just a Theory: Exploring the Nature of Science (2005), Chapter 2, “Just a Theory: What Scientists Do” (p. 39)

“Origami helps in the study of mathematics and science in many ways. … Using origami anyone can become a scientific experimenter with no fuss.”

Martin David Kruskal (1925–2006) American mathematician

at the AAAS meeting: Mathematics and Science of Origami: Visualize the Possibilities, February 15, 2002, as quoted by Science Daily Origami Helps Scientists Solve Problems http://www.sciencedaily.com/releases/2002/02/020219080203.htm, February 21, 2002.

Way , Study , Science , Help

“I was appalled to find that the mathematical notation on which I had been raised failed to fill the needs of the courses I was assigned, and I began work on extensions to notation that might serve. In particular, I adopted the matrix algebra used in my thesis work, the systematic use of matrices and higher-dimensional arrays (almost) learned in a course in Tensor Analysis rashly taken in my third year at Queen’s, and (eventually) the notion of Operators in the sense introduced by Heaviside in his treatment of Maxwell’s equations.”

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

"Kenneth E. Iverson" http://keiapl.info/rhui/autobio.htm, autobiographical sketch from an unfinished work (ca. 2004), on his experience at Harvard with "a Masters program in Automatic Data Processing in 1955; in effect, the first computer science program."

Sense , Learning

“Flaming torches arching from hand to hand, the silken rolling of flesh on flesh, tautened wire vibrating to the human word, ideogrammatic gestures of fear, love, and rage, the mathematical grace of bodies moving through space—all seemed revealed as shadows on the void, the pauvre panoply of man’s attempt to transcend the universe of space and time through the transmaterial purity of abstract form.
Yet beyond this noble dance of human art, the highest expression of our spirit’s striving to transcend the realm of time and form, lay that which could not be encompassed by the artifice of man. From nothing are we born, to nothing do we go; the universe we know is but the void looped back upon itself, and form is but illusion’s final veil.
We touch that which lies beyond only in those fleeting rare moments when the reality of form dissolves—through molecule and charge, the perfection of the meditative trance, orgasmic ego-loss, transcendent peaks of art, mayhap the instant of our death.
Vraiment, is not the history of man from pigments smeared on the walls of caves to our present starflung age, our sciences and arts, our religions and our philosophies, our cultures and our noble dreams, our heroics and our darkest deeds, but the dance of spirit round this central void, the striving to transcend, and the deadly fear of same?”

Norman Spinrad book The Void Captain's Tale

Source: The Void Captain's Tale (1983), Chapter 10 (p. 117)

Love , Death , Age , Fear

“A great department of thought must have its own inner life, however transcendent may be the importance of its relations to the outside. No department of science, least of all one requiring so high a degree of mental concentration as Mathematics, can be developed entirely, or even mainly, with a view to applications outside its own range. The increased complexity and specialisation of all branches of knowledge makes it true in the present, however it may have been in former times, that important advances in such a department as Mathematics can be expected only from men who are interested in the subject for its own sake, and who, whilst keeping an open mind for suggestions from outside, allow their thought to range freely in those lines of advance which are indicated by the present state of their subject, untrammelled by any preoccupation as to applications to other departments of science. Even with a view to applications, if Mathematics is to be adequately equipped for the purpose of coping with the intricate problems which will be presented to it in the future by Physics, Chemistry and other branches of physical science, many of these problems probably of a character which we cannot at present forecast, it is essential that Mathematics should be allowed to develop freely on its own lines.”

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

Source: Presidential Address British Association for the Advancement of Science, Section A (1910), p. 286; Cited in: Moritz (1914, 106): Modern mathematics.

Life , Men , Knowledge , Problem

“It is possible to relegate the epoch of the starting of the expansion to minus infinity, e. g. by using instead of the ordinary time the logarithm of the time elapsed since the beginning. But this is only a mathematical trick. We call zero minus infinity, but that only means that we allow the universe an infinite time to get well started on its course of expansion, but it does not make the time during which anything really happens any longer.”

Willem de Sitter (1872–1934) Dutch cosmologist

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

Universe , Time

“The essential meaning of such an assertion is this: events a and b are necessarily dependent moments of a single unified occurrence. The mathematical formula states the quantitative relations involved in the occurrence. Already in such cases the dependent moment of the occurrence are moments that obtain temporally by side.”

Kurt Lewin (1890–1947) German-American psychologist

Kurt Lewin (1927, p. 305) as cited in: K. Mulligan & B. Smith (1988) " Mach and Ehrenfels: Foundations of Gestalt Theory http://ontology.buffalo.edu/smith/articles/mach/mach.pdf". p. 149.
1920s

Dependency

Quotes about mathematics page 4

Quotes about mathematics
page 4