Quotes about mathematics
page 11

“No mathematical idea has ever been published in the way it was discovered. Techniques have been developed and are used, if a problem has been solved, to turn the solution procedure upside down, or if it is a larger complex of statements and theories, to turn definitions into propositions, and propositions into definitions, the hot invention into icy beauty. This then if it has affected teaching matter, is the didactical inversion, which as it happens may be anti-didactical. Rather than behaving anti-didactically, one should recognise that the learner is entitled to recapitulate in a fashion of mankind. Not in the trivial matter of an abridged version, but equally we cannot require the new generation to start at the point where their predecessors left off.”

Hans Freudenthal (1905–1990) Dutch mathematician

Source: The Concept and the Role of the Model in Mathematics and Natural and Social Sciences (1961), p. ix

Beauty, Way, Idea, Fashion

“The science you enjoy now is the science we taught you. The medicine you treat yourselves with is the medicine we gave you. It’s the same with the astronomy you know, the mathematics, the literature, the art…”

Muammar Gaddafi (1942–2011) Libyan revolutionary, politician and political theorist

Interview with Oriana Fallaci (2 December 1979), Corriere della Sera
Interviews

Art, Literature, Science

“For we may remark generally of our mathematical researches, that these auxiliary quantities, these long and difficult calculations into which we are often drawn, are almost always proofs that we have not in the beginning considered the objects themselves so thoroughly and directly as their nature requires, since all is abridged and simplified, as soon as we place ourselves in a right point of view.”

Louis Poinsot (1777–1859) French mathematician and physicist

[Louis Poinsot, translated by Charles Thomas Whitley, Outlines of a new theory of rotatory motion, R. Newby (Cambridge), 1834, 4]

Nature

“The theory of perspective was taught in painting schools from the sixteenth century onward according to principles laid down by the masters… However, their treatises on perspective had on the whole been precept, rule, and ad hoc procedure; they lacked a solid mathematical basis. In the period from 1500 to 1600 artists and subsequently mathematicians put the subject on a satisfactory deductive basis, and it passed from quasi-empirical art to a true science. Definitive works on perspective were written much later by eighteenth-century mathematicians Brook Taylor and J. H. Lambert.”

Morris Kline (1908–1992) American mathematician

Source: Mathematical Thought from Ancient to Modern Times (1972), p. 183

School, Art, Science, Painting

“Human beings are incontestably capital from an abstract and mathematical point of view.”

Theodore Schultz (1902–1998) American economist

Source: "Investment in human capital," 1961, p. 3

“Hungary, severed from half of the population and most of the natural resources that it had once claimed, had now to practice a sort of economic acupuncture, striving to know the magic nodes in the global energy flow where a pinprick could alter the workings of a major organ. Mathematics was one of the few disciplines where it was possible to exert that degree of leverage, and so the Hungarians had become phenomenally good at teaching it to their children.”

Neal Stephenson book Reamde

Day 4
Reamde (2011), Part I: Nine Dragons

Children, Nature

“A model is a qualitative or quantitative representation of a process or endeavor that shows the effects of those factors which are significant for the purposes being considered. A model may be pictorial, descriptive, qualitative, or generally approximate in nature; or it may be mathematical and quantitative in nature and reasonably precise. It is important that effective means for modeling be understood such as analog, stochastic, procedural, scheduling, flow chart, schematic, and block diagrams.”

Harold Chestnut (1917–2001) American engineer

Source: Systems Engineering Tools, (1965), p. 108; As cited in: Alberto Ortiz (1992) Systems engineering concept demonstration, process model http://www.dtic.mil/dtic/tr/fulltext/u2/a265469.pdf. p. 12

Nature

“Mathematical and physiological researches have shown that the space of experience is simply an actual case of many conceivable cases, about whose peculiar properties experience alone can instruct us.”

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

Source: 20th century, Popular Scientific Lectures, (Chicago, 1910), p. 205; On the space of experience.

“It is a remarkable fact in the history of geometry, that the Elements of Euclid, written two thousand years ago, are still regarded by many as the best introduction to the mathematical sciences.”

Florian Cajori book A History of Mathematics

Source: A History of Mathematics (1893), p. 30 Reported in Memorabilia mathematica or, The philomath's quotation-book by Robert Edouard Moritz. Published 1914.

History, Science

“For reasons mathematical, psychological, and sociological, it is a good idea to use a money management system that is relatively forgiving of estimation errors.”

William Poundstone (1955) American writer

Part Six, Blowing Up, Survival Motive, p. 296-297
Fortune's Formula (2005)

Money, Error, Idea, Forgiveness

“The nice thing about mathematics is doing mathematics.”

Pierre Deligne (1944) mathematician

Pierre Deligne in: Philip Ball. "Mathematician wins award for shaping algebra: 2013 Abel Prize goes to Belgian Pierre Deligne, who proved a deep conjecture about algebra and geometry." in Nature, 20 March 2013

“In many cases, mathematics is an escape from reality. The mathematician finds his own monastic niche and happiness in pursuits that are disconnected from external affairs. Some practice it as if using a drug.”

Stanislaw Ulam (1909–1984) Polish-American mathematician

Source: Adventures of a Mathematician - Third Edition (1991), Chapter 6, Transition And Crisis, p. 120

Happiness, Reality

“Gradually… during the second half of the nineteenth century, the uncomfortable feeling of dislike of the action at a distance, which had been so strong in Huygens and other contemporaries of Newton, but had subsided during the eighteenth century, began to emerge again, and gained strength rapidly.
This was favoured by the purely mathematical transformation (which can be compared in a sense with that from the Ptolemaic to the Copernican system), replacing Newton's finite equations by the differential equations, the potential becoming the primary concept, instead of the force, which is only the gradient of the potential. These ideas, of course, arose first in the theory of electricity and magnetism or perhaps one should say in the brain of Faraday.”

Willem de Sitter (1872–1934) Dutch cosmologist

Kosmos (1932)

Feeling, Idea, Sense, Strong

“The modern scientific worldview with all its hope for clarity and precision, has a "flipside," a complementary set of views which it generates as its train. And this is its misology, sophistry, its abandonment of rationality in the world of human significance. There is thus a quite literal type of schizophrenia in the world bequeathed to us by the Cartesians. It is, on the one hand, hyper-rational; it seeks to extend the purview of mathematical physics throughout the universe. On the other hand, it relegates the world in which the physicist himself dwells, the unique world of humanity and its communities, to the junkpile of the irrational. We who know so much are prohibited from knowing ourselves.”

David Roochnik (1951) American philosopher

The Tragedy of Reason: Toward a Platonic Conception of Logos (Routledge: 1991), p. 74.

World, Hope, Communication, Universe

“The most powerful influence exercised by the Arabs on general natural physics was that directed to the advances of chemistry; a science for which this race created a new era.(…) Besides making laudatory mention of that which we owe to the natural science of the Arabs in both the terrestrial and celestial spheres, we must likewise allude to their contributions in separate paths of intellectual development to the general mass of mathematical science.”

Alexander von Humboldt book Kosmos

Cosmos, H.G. Bohn, 1860, v2, p.589,595.
Kosmos (1845 - 1847)

Exercises, Nature, Science

“At an unknown hour, from a source that is still sealed to us, but inexorable, the Work comes into the world. Cold calculations, splashes leaping up without plan, mathematically accurate construction (laid bare or concealed) silent, screaming drawing, scrupulous finish, colour in fanfares or played pianissimo on the strings, large, serene, cradling, fragmented planes. Isn’t that a Form? Aren't those the means?
Suffering, seeking, tormented souls with a deep fissure, caused by the collision of the spiritual with the material... Shame on him who turns his soul's ear away from the mouth of art. A human being speaks to human beings about the superhuman – the language of art.”

Wassily Kandinsky (1866–1944) Russian painter

Quote of Kandinsky, from the catalog of the second exhibition of the 'Neue Künstlervereinigung', München, August, 1910; as cited by de:Wolf-Dieter Dube, in Expressionism; Praeger Publishers, New York, 1973, p. 95
1910 - 1915

World, Art, Soul, Suffering

“[In] economics… you can disguise charlatanism under the weight of equations and nobody can catch you since there is no such thing as a controlled experiment. Now the spirit of such methods, called scientism by its detractors, continued into the discipline of finance as a few technicians thought their mathematical knowledge could lead them to understand markets. The practice of financial engineering came along with massive doses of pseudoscience.”

Nassim Nicholas Taleb book Fooled by Randomness

page 115
Fooled by Randomness (2001)

Understanding, Knowledge

“It’s true that my initial training was in mathematics. However, almost by accident, I happened to take a course from Kenneth Arrow on “Information Economics,” which was so inspiring that I decided to change direction. It seemed to me that economics combined the best of both worlds: the rigor of mathematics with the immediate relevance of a social science. As for how much math I would recommend, I’d say that basic analysis, including measure theory, is certainly very useful. Also, linear algebra and stochastic processes always helps. But beyond that, I don’t think a huge mathematical investment is necessary to do economic theory unless you are planning to work in an extremely technical area.”

Eric Maskin (1950) American Nobel laureate in economics

" Interview with Eric S. Maskin: Questions by TSE students http://www.tseconomist.com/all-publications/interview-with-nobel-prize-winner-eric-maskin" at tseconomist.com, 04/07/2013; In answer to the question of why he decided to become an economist.

World, Thinking, Science, Change

“In its conception the literature prize belongs to days when a writer could still be thought of as, by virtue of his or her occupation, a sage, someone with no institutional affiliations who could offer an authoritative word on our times as well as on our moral life. (It has always struck me as strange, by the way, that Alfred Nobel did not institute a philosophy prize, or for that matter that he instituted a physics prize but not a mathematics prize, to say nothing of a music prize - music is, after all, more universal than literature, which is bound to a particular language.) The idea of writer as sage is pretty much dead today. I would certainly feel very uncomfortable in the role.”

J.M. Coetzee (1940) South African writer

Dagens Nyheter http://www.dn.se/kultur-noje/an-exclusive-interview-with-j-m-coetzee interview with David Attwell (December 8, 2003)

Life, Music, Way, Feeling

“For the mathematician the important consideration is that the foundations of mathematics and a great portion of its content are Greek. The Greeks laid down the first principles, invented the methods ab initio, and fixed the terminology. Mathematics in short is a Greek science, whatever new developments modern analysis has brought or may bring.”

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

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

Science

“At present, when the prevailing forms of society have become hindrances to the free expression of human powers, it is precisely the abstract branches of science, mathematics and theoretical physics, which … offer a less distorted form of knowledge than other branches of science which are interwoven with the pattern of daily life, and the practicality of which seemingly testifies to their realistic character.”

Max Horkheimer (1895–1973) German philosopher and sociologist

Source: "The Latest Attack on Metaphysics" (1937), p. 133.

Life, Knowledge, Science

“My work is more varied than at any other point in my life. I am still carrying out research in pure mathematics. And I am working on an idea that I had several years ago on negative dimensions. … Negative dimensions are a way of measuring how empty something is. In mathematics, only one set is called empty. It contains nothing whatsoever. But I argued that some sets are emptier than others in a certain useful way. It is an idea that almost everyone greets with great suspicion, thinking I've gone soft in the brain in my old age. Then I explain it and people realise it is obvious. Now I'm developing the idea fully with a colleague. I have high hopes that once we write it down properly and give a few lectures about it at suitable places that negative dimensions will become standard in mathematics.”

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

New Scientist interview (2004)

Life, People, Age, Hope

“If I hadn't an absolute faith in the harmony of creation, I wouldn't have tried for thirty years to express it in a mathematical formula. It is only man's consciousness of what he does with his mind that elevates him above the animals, and enables him to become aware of himself and his relationship to the universe.”

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

Source: Attributed in posthumous publications, Einstein and the Poet (1983), p. 61

Faith, Universe, Animals

“To those who ask what the infinitely small quantity in mathematics is, we answer that it is actually zero. Hence there are not so many mysteries hidden in this concept as they are usually believed to be.”

Leonhard Euler (1707–1783) Swiss mathematician

As quoted in Fundamentals of Teaching Mathematics at University Level (2000) by Benjamin Baumslag, p. 214

“Mathematics was not just about keeping track of were the moon was going, but also where the money was going.”

David Orrell (1962) Canadian mathematician

Source: The Other Side Of The Coin (2008), Chapter 1, Limited Versus Unlimited, p. 30

Money

“There was no way, without full understanding, that one could have confidence that conditions the next time might not produce erosion three times more severe than the time before. Nevertheless, officials fooled themselves into thinking they had such understanding and confidence, in spite of the peculiar variations from case to case. A mathematical model was made to calculate erosion. This was a model based not on physical understanding but on empirical curve fitting.”

Richard Feynman (1918–1988) American theoretical physicist

Rogers Commission Report (1986)

Fools, Way, Confidence, Understanding

“The philosophy of the foundations of probability must be divorced from mathematics and statistics, exactly as the discussion of our intuitive space concept is now divorced from geometry.”

William Feller (1906–1970) Croatian-American mathematician

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

“The material which a scientist actually has at his disposal, his laws, his experimental results, his mathematical techniques, his epistemological prejudices, his attitude towards the absurd consequences of the theories which he accepts, is indeterminate in many ways, ambiguous, and never fully separated from the historical background. This material is always contaminated by principles which he does not know and which, if known, would be extremely hard to test.”

Paul Karl Feyerabend book Against Method

Pg. 66.
Against Method (1975)

Law, Way

“In April of 1959, ten of this country's leading scholars forgathered on the campus of Purdue University to discuss the nature of information and the nature of decision… What interests do these men have in common?… To answer these questions it is necessary to view the changing aspect of the scientific approach to epistemology, and the striking progress which has been wrought in the very recent past. The decade from 1940 to 1950 witnessed the operation of the first stored- program digital computer. The concept of information was quantified, and mathematical theories were developed for communication (Shannon) and decision (Wald). Known mathematical techniques were applied to new and important fields, as the techniques of complex- variable theory to the analysis of feedback systems and the techniques of matrix theory to the analysis of systems under multiple linear constraints. The word "cybernetics" was coined, and with it came the realization of the many analogies between control and communication in men and in automata. New terms like "operations research" and "system engineering" were introduced; despite their occasional use by charlatans, they have signified enormous progress in the solution of exceedingly complex problems, through the application of quantitative ness and objectivity.”

Robert E. Machol (1917–1998) American systems engineer

Source: Information and Decision Processes (1960), p. vii

Men, Communication, Question, Nature

“Computer Algebra Systems are NOT the Devil but the new MESSIAH that will take us out of the current utterly trivial phase of human-made mathematics into the much deeper semi-trivial computer-generated phase of future mathematics. Even more important, Computer Algebra Systems will turn out to be much more than just a `tool', since the methodology of computer-assisted and computer-generated research will rule in the future, and will make past mathematics seem like alchemy and astrology, or, at best, theology.”

Doron Zeilberger (1950) Israeli mathematician

Past, Future

“Although this book deals with the conflict between religion and science, I see this as only one battle in a wider war—a war between rationality and superstition. Religion is but a single brand of superstition (others include beliefs in astrology, paranormal phenomena, homeopathy, and spiritual healing), but it is the most widespread and harmful form of superstition. And science is but one form of rationality (philosophy and mathematics are others), but it is a highly developed form, and the only one capable of describing and understanding reality.”

Jerry Coyne book Faith vs. Fact: Why Science and Religion are Incompatible

Source: Faith vs. Fact (2015), p. xii

War, Books, Religion, Understanding

“The one intelligible theory of the universe is that of objective idealism, that matter is effete mind, inveterate habits becoming physical laws. But before this can be accepted it must show itself capable of explaining the tridimensionality of space, the laws of motion, and the general characteristics of the universe, with mathematical clearness and precision; for no less should be demanded of every Philosophy.”

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

The Architecture of Theories (1891)

Law, Intelligence, Universe

“There is no doubt of the need for an up-to-date, balanced, and comprehensive work on the history of atomism, drawing ideas, mathematics, and experiment together into a single story. When available, it should become required reading for all students of the exact sciences.”

Lancelot Law Whyte (1896–1972) Scottish industrial engineer

p, 125
Essay on Atomism: From Democritus to 1960 (1961)

History, Idea, Science, Reading

“I'm interested in the computer as a new idea, a new and fundamental philosophical concept that changes mathematics, that solves old problems better and suggests new problems, that changes our way of thinking and helps us to understand things better, that gives us radically new insights…”

Gregory Chaitin (1947) Argentinian mathematician and computer scientist

Meta Maths!: The Quest for Omega https://books.google.com/books?id=ZACLDQAAQBAJ&pg=PA11. Vintage Books (2006). p. 11

Way, Idea, Understanding, Problem

“God exists since mathematics is consistent, and the Devil exists since we cannot prove it.”

André Weil (1906–1998) French mathematician

As quoted in Mathematical Circles Adieu (Boston 1977) by H Eves

God

“The members of the Vienna Circle (Moritz Schlick, Rudolf Carnap, Philipp Frank, Hans Hahn, Herbert Feigl, Fritz Waismann, Kurt Godel, Otto Neurath and others) are working out a ‘Logical Empiricism’. Following Ernst Mach and Poincaré, but above all Russell and Wittgenstein, all the sciences are treated uniformly. Carnap’s Logischer Aufbau der Welt (1928) shows in which direction future systematic work will move. Wittgenstein’s Tractatus Logico-Philosophicus (1921) clarified, among other things, the position of logic and mathematics; besides the statements that make additions to what is meaningful, there are the ‘tautologies’ that show us which transformations are possible within language. By its syntax the language of science excludes anything that is meaningless from the very beginning.”

Otto Neurath (1882–1945) austrian economist, philosopher and sociologist

Source: 1930s, "Physicalism" (1931), p. 52

Science, Future

“What would geometry be without Gauss, mathematical logic without Boole, algebra without Hamilton, analysis without Cauchy?”

George Frederick James Temple (1901–1992) British mathematician

100 Years of Mathematics: a Personal Viewpoint (1981)

“It is only by intuition that the infinite can be apprehended. But why is this? Why cannot the infinite be apprehended by concepts? To see this we must understand that the word "infinite," in the religious sense, has nothing at all to do with that sense of the word in which it is applied to space, time, and the number series. We may call this latter the mathematical infinite to distinguish it from the religious infinite. And it is the confusion between these two which misled us into the false trail of supposing that the infinity of God's mind refers to the amount of His knowledge and that the finitude of man's mind refers to his ignorance. The religious infinite, or in other words the infinity of God, means that than 'which there is no other. In this sense neither space nor time could be infinite, since space is an "other" to time, and time is an "other" to space.”

Walter Terence Stace (1886–1967) British civil servant, educator and philosopher.

p. 46-47.

God, Understanding, Knowledge, Sense

“In the early 1990s, I came to suspect that the quest for a unified theory is religious rather than scientific. Physicists want to show that all things came from one thing: a force, or essence, or membrane wriggling in eleven dimensions, or something that manifests perfect mathematical symmetry. In their search for this primordial symmetry, however, physicists have gone off the deep end, postulating particles and energies and dimensions whose existence can never be experimentally verified.”

John Horgan (journalist) (1953) American science journalist

(2002 wager, 18 year duration) [Bet 12 (John Horgan vs. Michio Kaku), longbets.org, http://longbets.org/12/]

Search, Perfection

“In order for there to be a variable quantity in some mathematical study, the domain of its variability must strictly speaking be known beforehand through a definition. However, this domain cannot itself be something variable, since otherwise each fixed support for the study would collapse. Thus this domain is a definite, actually infinite set of values. Hence each potential infinite, if it is rigorously applicable mathematically, presupposes an actual infinite.”

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

"Über die verschiedenen Ansichten in Bezug auf die actualunendlichen Zahlen" ["Over the different views with regard to the actual infinite numbers"] - Bihand Till Koniglen Svenska Vetenskaps Akademiens Handigar (1886)

Support, Study

“What exactly is mathematics? Many have tried but nobody has really succeeded in defining mathematics; it is always something else.”

Stanislaw Ulam (1909–1984) Polish-American mathematician

Source: Adventures of a Mathematician - Third Edition (1991), Chapter 15, Random Reflections on Mathematics and Science, p. 273-274

“Mathematics is a presuppositionless science. To found it I do not need God, as does Kronecker, or the assumption of a special faculty of our understanding attuned to the principle of mathematical induction, as does Poincaré, or the primal intuition of Brouwer, or, finally, as do Russell and Whitehead, axioms of infinity, reducibility, or completeness, which in fact are actual, contentual assumptions that cannot be compensated for by consistency proofs.”

David Hilbert (1862–1943) German prominent mathematician

Quoted in Hilbert's Die Grundlagen der Mathematik (1927)

God, Understanding, Science

“The rest of Sitka’s homicides are so-called crimes of passion, which is a shorthand way of expressing the mathematical product of alcohol and firearms.”

Michael Chabon (1963) Novelist, short story writer, essayist

Source: The Yiddish Policemen’s Union (2007), Chapter 1

Alcohol, Way, Passion

“Even the simplest calculation in the purest mathematics can have terrible consequences. Without the invention of the infinitesimal calculus most of our technology would have been impossible. Should we say therefore that calculus is bad?”

Stanislaw Ulam (1909–1984) Polish-American mathematician

Source: Adventures of a Mathematician - Third Edition (1991), Chapter 11, The 'Super', p. 222

“So, ultimately, in order to understand nature it may be necessary to have a deeper understanding of mathematical relationships. But the real reason is that the subject is enjoyable, and although we humans cut nature up in different ways, and we have different courses in different departments, such compartmentalization is really artificial, and we should take our intellectual pleasures where we find them.”

Richard Feynman (1918–1988) American theoretical physicist

volume I; lecture 22, "Algebra"; section 22-1, "Addition and multiplication"; p. 22-1
The Feynman Lectures on Physics (1964)

Pleasure, Way, Understanding, Nature

“Natural science is the attempt to comprehend nature by precise concepts.
According to the concepts by which we comprehend nature not only are observations completed at every instant but also future observations are pre-determined as necessary, or, in so far as the concept-system is not quite adequate therefor, they are predetermined as probable; these concepts determine what is "possible" (accordingly also what is "necessary," or the opposite of which is impossible), and the degree of the possibility (the "probability") of every separate event that is possible according to them, can be mathematically determined, if the event is sufficiently precise.
If what is necessary or probable according to these concepts occurs, then the latter are thereby confirmed and upon this confirmation by experience rests our confidence in them. If, however, something happens which according to them is not expected and which is therefore according to them impossible or improbable, then arises the problem so to complete them, or if necessary, to transform them, that according to the completed or ameliorated concept-system, what is observed ceases to be impossible or improbable. The completion or amelioration of the concept-system forms the "explanation" of the unexpected observation. By this process our comprehension of nature becomes gradually always more complete and assured, but at the same time recedes even farther behind the surface of phenomena.”

Bernhard Riemann (1826–1866) German mathematician

Theory of Knowledge
Gesammelte Mathematische Werke (1876)

Confidence, Nature, Problem, Science

“[W]e shall be concerned with the general nature of pure mathematics, and how it is distinguished from other sciences. Here there are… two distinct categories of things of which an account must be given—the ideas or concepts of mathematics, and the propositions of mathematics. …the great majority of writers on the subject have concentrated their attention on the explanation of one or the other… and erroneously supposed that a satisfactory explanation of the other would immediately follow.”

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

Footnote: In the future by 'mathematics' will always be meant 'pure mathematics'.
The Foundations of Mathematics (1925)

Idea, Nature, Science

“The Great Conversation began before the beginnings of experimental science. But the birth of the Conversation and the birth of science were simultaneous. The earliest of the pre-Socratics were investigating and seeking to understand natural phenomena; among them were men who used mathematical notions for this purpose. Even experimentation is not new; it has been going on for hundreds of years. But faith in experimentation as an exclusive method is a modern manifestation. … It is now regarded in some quarters … as the sole method of obtaining knowledge of any kind.”

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

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

Men, Faith, Birth, Understanding

“War as a whole, and each operation taken separately, are first of all mathematics and calculations.”

Vasily Blyukher (1889–1938) Soviet military commander

Quoted in Jonathan Fenby; Chiang Kai Shek: China's Generalissimo and the Nation He Lost

War

“Only if mathematical rigor is adhered to, can systems problems be dealt with effectively, and so it is that the systems engineer must, at least, develop an appreciation for mathematical rigor if not also considerable mathematical competence.”

A. Wayne Wymore (1927–2011) American mathematician

Source: A Mathematical Theory of Systems Engineering (1967), p. 3.

Problem

“Oblivious of Democritus, the unwilling materialists of our day have generally been awkwardly intellectual and quite incapable of laughter. If they have felt anything, they have felt melancholy. Their allegiance and affection were still fixed on those mythical sentimental worlds which they saw to be illusory. The mechanical world they believed in could not please them, in spite of its extent and fertility. Giving rhetorical vent to their spleen and prejudice, they exaggerated nature's meagreness and mathematical dryness. When their imagination was chilled they spoke of nature, most unwarrantably, as dead, and when their judgment was heated they took the next step and called it unreal.”

George Santayana (1863–1952) 20th-century Spanish-American philosopher associated with Pragmatism

Source: The Life of Reason: The Phases of Human Progress (1905-1906), Vol. V, Reason in Science, Ch. 3 "Mechanism"

World, Imagination, Nature, Laughter

“Primary causes are unknown to us; but are subject to simple and constant laws, which may be discovered by observation, the study of them being the object of natural philosophy.
Heat, like gravity, penetrates every substance of the universe, its rays occupy all parts of space. The object of our work is to set forth the mathematical laws which this element obeys. The theory of heat will hereafter form one of the most important branches of general physics.”

Joseph Fourier book The Analytical Theory of Heat

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

Law, Nature, Study, Universe

“Many of the core ideas of cybernetics have been assimilated by other disciplines, where they continue to influence scientific developments. Other important cybernetic principles seem to have been forgotten, though, only to be periodically rediscovered or reinvented in different domains. Some examples are the rebirth of neural networks, first invented by cyberneticists in the 1940's, in the late 1960's and again in the late 1980's; the rediscovery of the importance of autonomous interaction by robotics and AI in the 1990's; and the significance of positive feedback effects in complex systems, rediscovered by economists in the 1990's. Perhaps the most significant recent development is the growth of the complex adaptive systems movement, which, in the work of authors such as John Holland, Stuart Kauffman and Brian Arthur and the subfield of artificial life, has used the power of modern computers to simulate and thus experiment with and develop many of the ideas of cybernetics. It thus seems to have taken over the cybernetics banner in its mathematical modelling of complex systems across disciplinary boundaries, however, while largely ignoring the issues of goal-directedness and control.”

Francis Heylighen (1960) Belgian cyberneticist

Source: Cybernetics and Second-Order Cybernetics (2001), p.5 : About the state of the art of contemporary cybenetics

Idea

“Like all of mathematics, game theory is a tautology whose conclusions are true because they are contained in the premises.”

Thomas Flanagan (political scientist) (1944) author, academic, and political activist

Source: Game Theory and Canadian Politics (1998), Chapter 10, What Have We Learned?, p. 164.

Game

“In the judgment of the most competent living mathematicians, Fräulein Noether was the most significant creative mathematical genius thus far produced since the higher education of women began.”

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

1930s, Obituary for Emmy Noether (1935)

Women, Education

“Mathematics is the only true metaphysics.”

William Thomson (1824–1907) British physicist and engineer

As quoted by Silvanus Phillips Thompson, The Life of William Thomson, Baron Kelvin of Largs (1910) Vol. 2 https://books.google.com/books?id=S_PPAAAAMAAJ, p. 1124 <br class="br">Variant: Mathematics is the only good metaphysics.

“The laws of classical mechanics represent a mathematical idealization and should not be assumed to correspond to the real laws of nature. … We now have to realize that errors are inevitable (..) a discovery that makes strict determinism impossible.”

Léon Brillouin (1889–1969) French physicist

[Léon Brillouin, Science and Information Theory, second edition, Academic Press, New York, 1962, 0-48643-918-6, 314]

Law, Error, Nature

“The Postulates of Mathematics Were Not on the Stone Tablets that Moses Brought Down from Mt. Sinai.”

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

Emphatic capitalization in original.
The Unreasonable Effectiveness of Mathematics (1980)

“Dave always arrives at the right conclusions, but his mathematics is terrible. I take it home and find all sorts of errors and then have to spend the night trying to develop the correct proof. But in the end, the result is always exactly the same as the one Dave saw directly.”

David Bohm (1917–1992) American theoretical physicist

Professor Basil Hiley http://www.bbk.ac.uk/tpru/BasilHiley.html, a colleague at Birkbeck College, University of London.

Error, Home, Night

“The chief innovator of symbolism in algebra was François Viète… an amateur in the sense that his professional life was devoted to the law… John Wallis… says that Viète, in denoting a class of numbers by a letter, followed the custom of lawyers who discussed legal cases by using arbitrary names [for the litigants]… and later the abbreviations… and still more briefly A, B, and C. Actually, letters had been used occasionally by the Greek Diophantus and by the Hindus. However, in these cases letters were confined to designating a fixed unknown number, powers of that number, and some operations. Viète recognized that a more extensive use of letters, and, in particular, the use of letters to denote classes of numbers, would permit the development of a new kind of mathematics; this he called logistica speciosa in distinction from logistica numerosa.”

Morris Kline (1908–1992) American mathematician

...the growth of symbolism was slow. Even simple ideas take hold slowly. Only in the last few centuries has the use of symbolism become widespread and effective.
Source: Mathematics and the Physical World (1959), p. 60

Life, Law, Idea, Sense

“I discovered the works of Euler and my perception of the nature of mathematics underwent a dramatic transformation. I was de-Bourbakized, stopped believing in sets, and was expelled from the Cantorian paradise. I still believe in abstraction, but now I know that one ends with abstraction, not starts with it. I learned that one has to adapt abstractions to reality and not the other way around. Mathematics stopped being a science of theories but reappeared to me as a science of numbers and shapes.”

Alexander Stepanov (1950) Russian programmer

Bjarne Stroustrup: Evolving a language in and for the real world: C++ 1991-2006. ACM HOPL-III. June 2007., 2008-04-25, http://web.archive.org/web/20071120015600/http://www.research.att.com/~bs/hopl-almost-final.pdf, 2007-11-20 http://www.research.att.com/~bs/hopl-almost-final.pdf,

Way, Nature, Reality, Learning

“Bertie Brookes and I shared a common view that, beyond the practical activities of information provision, there could be discerned a more general science of information. He tended towards a mathematical formulation of this: I was more interested in its social aspects.”

Brian Campbell Vickery (1918–2009) British information theorist

Source: A Long Search for Information (2004), p. 24-25.

Science

“Mathematical physics represents the purest image that the view of nature may generate in the human mind; this image presents all the character of the product of art; it begets some unity, it is true and has the quality of sublimity; this image is to physical nature what music is to the thousand noises of which the air is full…”

Théophile de Donder (1872–1957) Belgian physicist

as quoted by Ilya Prigogine in his Autobiography http://nobelprize.org/nobel_prizes/chemistry/laureates/1977/prigogine-autobio.html given at the occasion of Prigogine's 1977 Nobel Prize in Chemistry.

Art, Music, Nature

“The object of pure Physic[s] is the unfolding of the laws of the intelligible world; the object of pure Mathematic[s] that of unfolding the laws of human intelligence.”

James Joseph Sylvester (1814–1897) English mathematician

Reported in: Memorabilia Mathematica by Robert Edouard Moritz, quote #129.

World, Law, Intelligence

“At the time the book of Marquis de l'Hôpital had appeared, and almost all mathematicians began to turn to the new geometry of the infinite [that is, the new infinitesimal calculus], until then little known. The surprising universality of the methods, the elegant brevity of the proofs, the neatness and speed of the most difficult solutions, a singular and unexpected novelty, all attracted the mind and there was in the mathematical world a well marked revolution [une révolution bien marquée.”

Bernard Le Bovier de Fontenelle (1657–1757) French writer, satirist and philosopher of enlightenment

1792) as quoted by I. Bernard Cohen, Revolution in Science (1985

Books, World, Universe, Appearance

“I started with a kind of artistic approach… I visualized the best-looking shapes and sizes. I worked with the variables until it got to the point where, if I changed one of them, it didn't get any better… [only then I] figure out the mathematical formula to produce that effect.”

Arthur H. Robinson (1915–2004) American geographer

Robinson (1988) in The New York Times as cited in: John Noble Wilford (2004) " Arthur H. Robinson, 89, Geographer Who Reinterpreted World Map, Dies http://www.nytimes.com/2004/11/15/obituaries/15robinson.html?_r=0" in: The New York Times November 15, 2004: About the development of the Robinson projection.

Change

“Concept art was meant to replace all of mathematics with an endeavor which involved a Rorschach-blot semantics; and which did not claim to be cognitive, at least not in the inherited sense. Mathematics had already been disconnected from claims of realism; and I was extending that disavowal to a disconnection from claims of a priori truth. Concept art's value consisted in beauty, a beauty which was non-sentimental. Later I would say that its value consisted in "the invention of new mental abilities."”

Henry Flynt (1940) American musician

Popularity had nothing to do with whether this avenue was worth taking. <br class="br">Henry Flynt. " The Crystallization of Concept Art in 1961 http://www.henryflynt.org/meta_tech/crystal.html," at henryflynt.org, 1994.

Truth, Beauty, Art, Sense

“With the assistance of mathematics, I can see a property of the ninety-nine dimensional surfaces hidden from the naked eye. If an increase in the price of fertilizer alone always increases the amount the firm buys of caviar, from that fact alone I can predict the answer to the following experiment which I have never seen performed and upon which I have no observations: an increase in the price of caviar alone will increase the amount the firm buys of fertilizer. In thermodynamics such reciprocity or integrability conditions are known as Maxwell Conditions; in economics they are known as Hotelling conditions in honor of Harold Hotelling’s 1932 work.”

Paul A. Samuelson (1915–2009) American economist

Source: 1950s–1970s, Maximum Principles in Analytical Economics, 1970, p. 67

“Euclid … manages to obtain a rigorous proof without ever dealing with infinity, by reducing the problem [of the infinitude of primes] to the study of finite numbers. This is exactly what contemporary mathematical analysis does.”

Lucio Russo (1944) Italian historian and scientist

2.4, "Discrete Mathematics and the Notion of Infinity", p. 45
The Forgotten Revolution: How Science Was Born in 300 BC and Why It Had to Be Reborn (2004)

Problem, Study

“The spider-men came first, and presented her Majesty with a table full of mathematical points, lines and figures of all sorts of squares, circles, triangles, and the like; which the Empress, notwithstanding that she had a very ready wit, and quick apprehension, could not understand; but the more she endeavoured to learn, the more was she confounded”

Margaret Cavendish (1623–1673) English aristocrat, a prolific writer, and a scientist

Description of a New World, Called The Blazing World (1666)

Men, Understanding, Learning

“The beauty of mathematics often makes the subject matter much more attractive and easier to master.”

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

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

Beauty

“Analysis of the mathematical form and underlying principles of games was made by von Neumann " as early as 1928. In this early work von Neumann was not so much interested in executive-type problems as he was in the logical foundations of quantum mechanics. It was not until 1944, when von Neumann and Morgenstern published their now well known Theory of Games and Economic Behavior, that the mathematical treatment of games "took fire."”

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

Source: 1940s - 1950s, Introduction to Operations Research (1957), p. 519: Partly cited in: E. Roy Weintraub (1992) Toward a history of game theory. p. 235

Behavior, Game, Problem

“Many workers in the biological sciences — physiologists, psychologists, sociologists — are interested in cybernetics and would like to apply its methods and techniques to their own specialty. Many have, however, been prevented from taking up the subject by an impression that its use must be preceded by a long study of electronics and advanced pure mathematics; for they have formed the impression that cybernetics and these subjects are inseparable.
The author is convinced, however, that this impression is false. The basic ideas of cybernetics can be treated without reference to electronics, and they are fundamentally simple; so although advanced techniques may be necessary for advanced applications, a great deal can be done, especially in the biological sciences, by the use of quite simple techniques, provided they are used with a clear and deep understanding of the principles involved. It is the author’s belief that if the subject is founded in the common-place and well understood, and is then built up carefully, step by step, there is no reason why the worker with only elementary mathematical knowledge should not achieve a complete understanding of its basic principles. With such an understanding he will then be able to see exactly what further techniques he will have to learn if he is to proceed further; and, what is particularly useful, he will be able to see what techniques he can safely ignore as being irrelevant to his purpose.”

W. Ross Ashby (1903–1972) British psychiatrist

Preface
An Introduction to Cybernetics (1956)

Idea, Understanding, Knowledge, Study

“God does not care about our mathematical difficulties. He integrates empirically.”

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

Attributed to Einstein by his colleague Léopold Infeld in his book Quest: An Autobiography (1949), p. 279 http://books.google.com/books?id=fsvXYpOSowkC&lpg=PP1&pg=PA279#v=onepage&q&f=false <br class="br">Attributed in posthumous publications

God

“I hope that seeing the excitement of solving this problem will make young mathematicians realize that there are lots and lots of other problems in mathematics which are going to be just as challenging in the future.”

Andrew Wiles (1953) British mathematician

Nova Interview

Hope, Problem, Future

“The above interpretation has been introduced, not on account of its immediate value in the present system, but as an illustration of a significant fact in the philosophy of the intellectual powers, viz., that what has commonly been regarded as the fundamental axiom of metaphysics is but the consequence of a law of thought, mathematical in its form.”

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

Source: 1850s, An Investigation of the Laws of Thought (1854), p. 50

Law

“Historically, it was Euclidean geometry that, developed to a large extent as a votive offering to the God of Reason, opened men's eyes to the possibility of design and to the possibility of uncovering it by the pursuit of mathematics.”

Morris Kline (1908–1992) American mathematician

Source: Mathematics and the Physical World (1959), p. 89

Men, God

“Of the contemporaries of Newton one of the most prominent was John Wallis. …Wallis was a voluminous writer, and not only are his writings erudite, but they show a genius in mathematics… He was one of the first to recognize the significance of the generalization of exponents to include negative and fractional as well as positive and integral numbers. He recognized also the importance of Cavalieri's method of indivisibles, and employed it in the quadrature of such curves as y=xn, y=x1/n, and y=x0 + x1 + x2 +… He failed in his attempts at the approximate quadrature of the circle by means of series because he was not in possession of the general form of the binomial theorem. He reached the result, however, by another method. He also obtained the equivalent of ds = \! dx \sqrt{1+(\frac{dy}{dx})^2} for the length of an element of a curve, thus connecting the problem of rectification with that of quadrature.”

David Eugene Smith (1860–1944) American mathematician

History of Mathematics (1923) Vol.1

Problem

“In Plato's Republic, when the interlocutor of Socrates appears to bring certain plausible reasons to bear upon the mathematical sciences, to show that they are useful to human life, arithmetic for reckoning, distributions, contributions, exchanges, and partnerships, geometry for sieges, the founding of cities and sanctuaries, and the partition of land, music for festivals, entertainment, and the worship of the gods, and the doctrine of the spheres, or astronomy, for farming, navigation and other undertakings, revealing beforehand the proper procedure and suitable season, Socrates, reproaching him says: "You amuse me, because you seem to fear that these are useless studies that I recommend; but that is very difficult, nay, impossible. For the eye of the soul, blinded and buried by other pursuits, is rekindled and aroused again by these and these alone, and it is better that this be saved than thousands of bodily eyes, for by it alone is the truth of the universe beheld."”

Nicomachus (60–120) Ancient Greek mathematician

Nicomachus of Gerasa: Introduction to Arithmetic (1926)

Life, God, Truth, Fear

“Diophantos lived in a period when the Greek mathematicians of great original power had been succeeded by a number of learned commentators, who confined their investigations within the limits already reached, without attempting to further the development of the science. To this general rule there are two most striking exceptions, in different branches of mathematics, Diophantos and Pappos. These two mathematicians, who would have been an ornament to any age, were destined by fate to live and labour at a time when their work could not check the decay of mathematical learning. There is scarcely a passage in any Greek writer where either of the two is so much as mentioned. The neglect of their works by their countrymen and contemporaries can be explained only by the fact that they were not appreciated or understood. The reason why Diophantos was the earliest of the Greek mathematicians to be forgotten is also probably the reason why he was the last to be re-discovered after the Revival of Learning. The oblivion, in fact, into which his writings and methods fell is due to the circumstance that they were not understood. That being so, we are able to understand why there is so much obscurity concerning his personality and the time at which he lived. Indeed, when we consider how little he was understood, and in consequence how little esteemed, we can only congratulate ourselves that so much of his work has survived to the present day.”

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

Historical Introduction, p.17
Diophantos of Alexandria: A Study in the History of Greek Algebra (1885)

Fate, Age, Understanding, Learning

“Alexandre Grothendieck was very different from Weil in the way he approached mathematics: Grothendieck was not just a mathematician who could understand the discipline and prove important results— he was a man who could create mathematics. And he did it alone.”

André Weil (1906–1998) French mathematician

[Amir D. Aczel, The Artist and the Mathematician, http://books.google.com/books?id=fRCH-at7wgYC&pg=PA53, 29 April 2009, Basic Books, 978-0-7867-3288-3, 54]
Quote About

Way, Understanding

“I surmise that mathematical knowledge amounts to the crystallization of officially endorsed delusions in an intellectual quicksand”

Henry Flynt (1940) American musician

Henry Flynt " Is Mathematics a Scientific Discipline? http://www.henryflynt.org/studies_sci/mathsci.html," at henryflynt.org, 1996.

Knowledge

“All I'm trying to say to you is that our world hinges on moral foundations. God has made it so. God has made the universe to be based on a moral law. So long as man disobeys it he is revolting against God. That's what we need in the world today: people who will stand for right and goodness. It's not enough to know the intricacies of zoology and biology, but we must know the intricacies of law. It is not enough to know that two and two makes four, but we've got to know somehow that it's right to be honest and just with our brothers. It's not enough to know all about our philosophical and mathematical disciplines, but we've got to know the simple disciplines of being honest and loving and just with all humanity. If we don't learn it, we will destroy ourselves by the misuse of our own powers.”

Martin Luther King, Jr. (1929–1968) American clergyman, activist, and leader in the American Civil Rights Movement

1950s, Rediscovering Lost Values (1954)

Love, God, People, World

“Dalton, the mathematical tutor, following up the lead of Newton, combined the whole of the results of quantitative measurement which had accumulated up to his time, in a comprehensive theory, based on the concept of the chemical atom.”

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

Introduction
Higher Mathematics for Chemical Students (1911)

Time

“The Mathematical School; Although mathematical methods can be used by any school of management theory, and have been, I have chosen to group under a school those theorists who see management as a system of mathematical models and processes. Perhaps the most widely known group I arbitrarily so lump are the operations researchers or operations analysts, who have sometimes anointed themselves with the rather pretentious name of "management scientists." The abiding belief of this group is that, if management, or organization, or planning, or decision making is a logical process, it can be expressed in terms of mathematical symbols and relationships. The central approach of this school is the model, for it is through these devices that the problem is expressed in its basic relationships and in terms of selected goals or objectives.”

Harold Koontz (1909–1984)

Source: "The Management Theory Jungle," 1961, p. 181

School, Problem, Decision

“It is therefore not unreasonable to suppose that some portion of the neglect of science in England, may be attributed to the system of education we pursue. A young man passes from our public schools to the universities, ignorant of almost every branch of useful knowledge; and at these latter establishments … classical and mathematical pursuits are nearly the sole objects proposed to the student's ambition.”

Charles Babbage (1791–1871) mathematician, philosopher, inventor and mechanical engineer who originated the concept of a programmable c…

Source: Reflections on the Decline of Science in England, and on Some of its Causes (1830), p. 3

Education, School, Knowledge, Universe

“India was the motherland of our race, and Sanskrit the mother of Europe's languages: she was the mother of our philosophy; mother, through the Arabs, of much of our mathematics; mother, through the Buddha, of the ideals embodied in Christianity; mother, through the village community, of self-government and democracy. Mother India is in many ways the mother of us all.”

Will Durant (1885–1981) American historian, philosopher and writer

The Case for India (1931)

Way, Democracy, Communication

“One should not be deceived by philosophical works that pretend to be mathematical, but are merely dubious and murky metaphysics. Just because a philosopher can recite the words lemma, theorem and corollary doesn't mean that his work has the certainty of mathematics. That certainty does not derive from big words, or even from the method used by geometers, but rather from the utter simplicity of the objects considered by mathematics.”

Pierre Louis Maupertuis (1698–1759) French mathematician, philosopher and man of letters

Les Loix du Mouvement et du Repos, déduites d'un Principe Métaphysique (1746)

“\frac {dy}{dx} = \frac {\omega^2x}{g}…The first derivative, the result of the differentiation of y with respect to x, was written by Leibniz in the form
\frac {dy}{dx}…Leibniz's notation …is both extremely useful and dangerous. Today, as the concepts of limit and derivative are sufficiently clarified, the use of the notation… need not be dangerous. Yet, the situation was different in the 150 years between the discovery of calculus by Newton and Leibniz and the time of Cauchy. The derivative \frac {dy}{dx} was considered as the ratio of two "infinitely small quanitites", of the infinitesimals dy and dx. …it greatly facilitated the systematization of the rules of the calculus and gave intuitive meaning to its formulas. Yet this consideration was also obscure… it brought mathematics into disrepute… some of the best minds… such as… Berkeley, complained that calculus is incomprehensible. …\frac {dy}{dx} is the limit of a ratio of dy to dx… Once we have realized this sufficiently clearly, we may, under certain circumstances, treat \frac {dy}{dx} so as if it were a ratio… and multiply by dx to achieve the separation of variables. We get
{dy} = \frac {\omega^2x}{g}xdx”

George Pólya (1887–1985) Hungarian mathematician

Mathematical Methods in Science (1977)

Time

“Intermediate between mathematics, statistics, and economics, we find a new discipline which, for lack of a better name, may be called econometrics. Econometrics has as its aim to subject abstract laws of theoretical political economy or "pure" economics to experimental and numerical verification, and thus to turn pure economics, as far as possible, into a science in the strict sense of the word.”

Ragnar Frisch (1895–1973) Norwegian economist

Ragnar Frisch (1926) "On a Problem in Pure Economics: Translated by JS Chipman." Preferences, Utility, and Demand: A Minnesota Symposium. 1926."
Original in French:
Intermediaire entre les mathematiques, la statistique et l'economie politique, nous trouvons une discipline nouvelle que ion peut, faute de mieux, designer sous le nom de reconometrie. L'econometrie se pose le but de soumettre les lois abstraites de l'economie politique theorique ou l'economie 'pure' A une verification experimentale et numeriques, et ainsi de constituer, autant que cela est possible, l'economie pure en une science dans le sens restreint de ce mot.
1920

Law, Sense, Science

“Between sessions at a physics conference, I asked a number of attendees: Who is the smartest physicist of them all? …the name mentioned most often was Witten's. He seemed to evoke a special kind of awe, as though he belonged to a category unto himself. He is often likened to Einstein; one colleague reached even further back for a comparison, suggesting that Witten possessed the greatest mathematical mind since Newton.”

John Horgan (journalist) (1953) American science journalist

Source: The End of Science (1996), p. 60

“In other branches of science, where quick publication seems to be so much desired, there may possibly be some excuse for giving to the world slovenly or ill-digested work, but there is no such excuse in mathematics. The form ought to be as 102 perfect as the substance, and the demonstrations as rigorous as those of Euclid. The mathematician has to deal with the most exact facts of Nature, and he should spare no effort to render his interpretation worthy of his subject, and to give to his work its highest degree of perfection. “Pauca sed matura” was Gauss’s motto.”

James Whitbread Lee Glaisher (1848–1928) English mathematician and astronomer

Source: "Presidential Address British Association for the Advancement of Science," 1890, p. 467 : On the perfection of math. productions

World, Nature, Effort, Science

“The progress of mathematics has been most erratic, and… intuition has played a predominant rôle in it…. It was the function of intuition to create new forms; it was the acknowledged right of logic to accept or reject these new forms, in whose birth in had no part…. the children had to live, so while waiting for logic to sanctify their existence, they throve and multiplied.”

Tobias Dantzig (1884–1956) American mathematician

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

Children, Birth, Waiting, Parting

“In the interior of the atom, Bohr had tried the plan of retaining the particle-electron and modifying the classical mechanics. Heisenberg took the opposite course, his procedure amounting in effect to retaining the classical mechanics, at least in form, and modifying the electron. Actually, the electron dropped out all together, because it exists only as a matter of inference and not of direct observation. For the same reason, the new theory contains no mention of atoms, nuclei, protons, or of electricity in any shape or form. The existences of all these are matters of inference, and Heisenberg's purely mathematical theory could no more make contact with them than with the efficiency of a turbine or with the price of wheat.”

James Jeans (1877–1946) British mathematician and astronomer

Physics and Philosophy (1942)

“The 19th and first half of the 20th century conceived of the world as chaos. Chaos was the oft-quoted blind play of atoms, which, in mechanistic and positivistic philosophy, appeared to represent ultimate reality, with life as an accidental product of physical processes, and mind as an epi-phenomenon. It was chaos when, in the current theory of evolution, the living world appeared as a product of chance, the outcome of random mutations and survival in the mill of natural selection. In the same sense, human personality, in the theories of behaviorism as well as of psychoanalysis, was considered a chance product of nature and nurture, of a mixture of genes and an accidental sequence of events from early childhood to maturity.
Now we are looking for another basic outlook on the world -- the world as organization. Such a conception -- if it can be substantiated -- would indeed change the basic categories upon which scientific thought rests, and profoundly influence practical attitudes.
This trend is marked by the emergence of a bundle of new disciplines such as cybernetics, information theory, general system theory, theories of games, of decisions, of queuing and others; in practical applications, systems analysis, systems engineering, operations research, etc. They are different in basic assumptions, mathematical techniques and aims, and they are often unsatisfactory and sometimes contradictory. They agree, however, in being concerned, in one way or another, with "systems," "wholes" or "organizations"; and in their totality, they herald a new approach.”

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

Source: General System Theory (1968), 7. Some Aspects of System Theory in Biology, p. 166-167 as quoted in Lilienfeld (1978, pp. 7-8) and Alexander Laszlo and Stanley Krippner (1992) " Systems Theories: Their Origins, Foundations, and Development http://archive.syntonyquest.org/elcTree/resourcesPDFs/SystemsTheory.pdf" In: J.S. Jordan (Ed.), Systems Theories and A Priori Aspects of Perception. Amsterdam: Elsevier Science, 1998. Ch. 3, pp. 47-74.

Life, World, Way, Behavior

“Let us begin by observing that the word "system" is almost never used by itself; it is generally accompanied by an adjective or other modifier: physical system; biological system; social system; economic system; axiom system; religious system; and even "general" system. This usage suggests that, when confronted by a system of any kind, certain of its properties are to be subsumed under the adjective, and other properties are subsumed under the "system," while still others may depend essentially on both. The adjective describes what is special or particular; i. e., it refers to the specific "thinghood" of the system; the "system" describes those properties which are independent of this specific "thinghood."
This observation immediately suggests a close parallel between the concept of a system and the development of the mathematical concept of a set. Given any specific aggregate of things; e. g., five oranges, three sticks, five fingers, there are some properties of the aggregate which depend on the specific nature of the things of which the aggregate is composed. There are others which are totally independent of this and depend only on the "set-ness" of the aggregate. The most prominent of these is what we can call the cardinality of the aggregate…
It should now be clear that system hood is related to thinghood in much the same way as set-ness is related to thinghood. Likewise, what we generally call system properties are related to systemhood in the same way as cardinality is related to set-ness. But systemhood is different from both set-ness and from thinghood; it is an independent category.”

Robert Rosen (1934–1998) American theoretical biologist

Source: "Some comments on systems and system theory," (1986), p. 1-2 as quoted in George Klir (2001) Facets of Systems Science, p. 4

Way, Nature, Dependency

“… From the age of 13, I was attracted to physics and mathematics. My interest in these subjects derived mostly from popular science books that I read avidly. Early on I was fascinated by theoretical physics and determined to become a theoretical physicist. I had no real idea what that meant, but it seemed incredibly exciting to spend one's life attempting to find the secrets of the universe by using one's mind.”

David Gross (1941) American particle physicist and string theorist

"David J. Gross - Biographical" http://www.nobelprize.org/nobel_prizes/physics/laureates/2004/gross-bio.html, Nobel Prize in Physics, nobelprize.org (2004)

Life, Secret, Books, Age

“In the most general mathematical sense, a space is a set of elements which conform to certain postulates.”

James Grier Miller (1916–2002) biologist

Source: Living Systems: Basic Concepts (1969), p. 51

Sense

“Numbers instill a feeling for the lie of the land, and furnish grist for the mathematical mill that is the physicist's principal tool.”

Hans Christian von Baeyer (1938) American physicist

Source: Information, The New Language of Science (2003), Chapter 6, The Book of Life, Genetic information, p. 48

Lie, Feeling