Quotes about mathematics
page 6

“We especially need imagination in science. It is not all mathematics, nor all logic, but it is somewhat beauty and poetry. There will come with the greater love of science greater love to one another. Living more nearly to Nature is living farther from the world and from its follies, but nearer to the world's people; it is to be of them, with them, and for them, and especially for their improvement. We cannot see how impartially Nature gives of her riches to all, without loving all, and helping all; and if we cannot learn through Nature's laws the certainty of spiritual truths, we can at least learn to promote spiritual growth while we are together, and live in a trusting hope of a greater growth in the future.”

Maria Mitchell (1818–1889) American astronomer

Maria Mitchell: Life, Letters and Journals (illustrated) by Maria Mitchell, 1896, p. 186.

Love, People, Truth, World

“The word "Sanskrit" means "prepared, pure, refined or perfect". It was not for nothing that it was called the "devavani" (language of the Gods). It has an outstanding place in our culture and indeed was recognized as a language of rare sublimity by the whole world. Sanskrit was the language of our philosophers, our scientists, our mathematicians, our poets and playwrights, our grammarians, our jurists, etc. In grammar, Panini and Patanjali (authors of Ashtadhyayi and the Mahabhashya) have no equals in the world; in astronomy and mathematics the works of Aryabhatta, Brahmagupta and Bhaskar opened up new frontiers for mankind, as did the works of Charak and Sushrut in medicine.”

Markandey Katju (1946) Indian judge

On Sanskrit, as quoted in the transcript of a speech, titled "Sanskrit as a Language of Science" http://www.iisc.ernet.in/misc/bang_speech.html and delivered on 13 October 2009, published by Indian Institute of Science, Bangalore

God, World, Perfection

“There are problems to whose solution I would attach an infinitely greater importance than to those of mathematics, for example touching ethics, or our relation to God, or concerning our destiny and our future; but their solution lies wholly beyond us and completely outside the province of science.”

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

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

God, Problem, Science, Future

“A fractal is a mathematical set or concrete object that is irregular or fragmented at all scales…”

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

As quoted in a review of The Fractal Geometry of Nature by J. W. Cannon in The American Mathematical Monthly, Vol. 91, No. 9 (November 1984), p. 594

“Our family are an alternate stratification of poetry and mathematics.”

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

In a letter to Andrew Crosse, as quoted in Eugen Kölbing's Englische Studien, Volume 19 https://archive.org/stream/englischestudien19leipuoft#page/156/mode/1up (1894), Leipzig; O.R. Reisland, "Byron's Daughter", p. 156.

Family

“Wherever mathematics has entered it has never again been pushed out by other developments. The mathematization of an area of human endeavor is not a passing fad; it is the prime mover of scientific and technological progress.”

Oskar Morgenstern (1902–1977) austrian economist

Oskar Morgenstern (Mathematica/Mathematic Policy Research), (from "A Look Back at Some of Our Contributions Over Time")

“The so-called mysteries of quantum mechanics are in its philosophical interpretation, not in its mathematics.”

Victor J. Stenger (1935–2014) American philosopher

In God and the Folly of Faith: The Incompatibility of Science and Religion (2012)

“I thought fit to… explain in detail in the same book the peculiarity of a certain method, by which it will be possible… to investigate some of the problems in mathematics by means of mechanics. This procedure is… no less useful even for the proof of the theorems themselves; for certain things first became clear to me by a mechanical method, although they had to be demonstrated by geometry afterwards… But it is of course easier, when we have previously acquired, by the method, some knowledge of the questions, to supply the proof than it is to find it without any previous knowledge.”

Archimedes book The Method of Mechanical Theorems

The Method of Mechanical Theorems

Books, Question, Knowledge, Problem

“Today we preach that science is not science unless it is quantitative. We substitute correlations for causal studies, and physical equations for organic reasoning. Measurements and equations are supposed to sharpen thinking, but, in my observation, they more often tend to make the thinking noncausal and fuzzy. They tend to become the object of scientific manipulation instead of auxiliary tests of crucial inferences.
Many - perhaps most - of the great issues of science are qualitative, not quantitative, even in physics and chemistry. Equations and measurements are useful when and only when they are related to proof; but proof or disproof comes first and is in fact strongest when it is absolutely convincing without any quantitative measurement.
Or to say it another way, you can catch phenomena in a logical box or in a mathematical box. The logical box is coarse but strong. The mathematical box is fine-grained but flimsy. The mathematical box is a beautiful way of wrapping up a problem, but it will not hold the phenomena unless they have been caught in a logical box to begin with.”

John R. Platt (1918–1992) American physicist

John R. Platt (1964) " Science, Strong Inference -- Proper Scientific Method (The New Baconians) http://256.com/gray/docs/strong_inference.html. In: Science Magazine 16 October 1964, Volume 146, Number 3642. Cited in: Gerald Weinberg (1975) Introduction to General Systems Thinking. p. 1, and in multiple other sources.

Beauty, Way, Problem, Thinking

“One can prove or refute anything at all with words. Soon people will perfect language technology to such an extent that they’ll be proving with mathematical precision that twice two is seven.”

Anton Chekhov (1860–1904) Russian dramatist, author and physician

Lights (1888)

People, Perfection

“I shall often discuss mathematical discoveries… I shall try to make up a likely story how the discovery could have happened. I shall try to emphasize the motives underlying the discovery, the plausible inferences that led to it… everything that deserves imitation.”

George Pólya (1887–1985) Hungarian mathematician

Induction and Analogy in Mathematics (1954)

Motivation

“The golden age of mathematics - that was not the age of Euclid, it is ours.”

Cassius Jackson Keyser (1862–1947) American mathematician and journalist of pronounced philosophical inclinations

Source: The Human Worth of Rigorous Thinking: Essays and Addresses, p. 268

Age

“In the Cambridge mathematical tripods, they said, ’do six questions. Complete questions carry proportionately more marks than an equal number of fragments. Till you attempt half of the questions you won’t get full credit’. Upon declaration of result I found that I had scored 110 out of hundred in one paper, 140 in another and likewise in all the rest. I was confused. So my tutor explained to me that although they write ‘do six questions’, you can attempt as many as you want. They award you marks for whatever questions you answer correctly and the ranks are on basis of the score.”

Jayant Narlikar (1938) Indian physicist

An Exclusive Interview with Prof. Jayant Vishnu Narlikar

Question

“What we call objective reality is, in the last analysis, what is common to many thinking beings, and could be common to all; this common part, we shall see, can only be the harmony expressed by mathematical laws. It is this harmony then which is the sole objective reality, the only truth we can attain; and when I add that the universal harmony of the world is the source of all beauty, it will be understood what price we should attach to the slow and difficult progress which little by little enables us to know it better.”

Henri Poincaré book The Value of Science

Introduction, p. 14
The Value of Science (1905)

Truth, World, Law, Beauty

“A few programming is taking you away from mathematics; a lot will get you back in.”

Xavier Leroy (1968) French computer scientistand programmer

Sources Source: Xavier Leroy (2007) Conclusion of his seminar at Collège de France, 2009-03-13 http://www.college-de-france.fr/default/EN/all/inn_tec2007/seminaire_n3_xavier_leroy.htm,

“There is no foundational mathematical or physical reason the relationship between Pythagorean and tempered western music should exist. It just does. The rich flexibility of the tempered scale and the … bountiful archives of western music are a testimonial to this wonderful coincidence provided by nature.”

Robert J. Marks II (1950) American electrical engineering researcher and intelligent design advocate

"Handbook of Fourier Analysis and Its Applications" (Oxford University Press, 2009), p. 623, Robert J. Marks II, 2009, 2011-04-29 http://books.google.com/books?id=Sp7O4bocjPAC&printsec=frontcover&dq=Handbook+of+Fourier+Analysis+and+Its+Applications&hl=en&ei=wcm5TaPvJYba0QHYi7nRDw&sa=X&oi=book_result&ct=result&resnum=1&ved=0CDEQ6AEwAA#v=onepage&q&f=false,

Music, Nature

“Wallis was in sympathy with Greek mathematics and astronomy, editing parts of the works of Archimedes, Eutocius, Ptolemy, and Aristarchus; but at the same time he recognized the fact that the analytic method was to replace the synthetic, as when he defined a conic as a curve of the second degree instead of as a section of a cone, and treated it by the aid of coordinates.”

David Eugene Smith (1860–1944) American mathematician

History of Mathematics (1923) Vol.1

Time, Parting

“Factor analysis, i. e., isolation by way of mathematical analysis, of factors in multivariable phenomena in psychology and other fields”

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

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

Way

“If my painting depicts faithfully and without over-refinement the simple and true character of the place you have frequented, if I succeed.... in giving its own life to that world of vegetation, then you will hear the trees moaning under the winter wind, the birds that call their young and cry after their dispersion; you will feel the old chateau tremble; it will tell you that, as the wife you loved, it too will.... disappear and be reborn in multiple forms.. One does not copy with mathematical precision what one sees, but one feels and interprets a real world, all of whose fatalities hold you fast bound.”

Théodore Rousseau (1812–1867) French painter (1812-1867)

Quote in a letter to M. Guizot, c. 1839-41; as cited by Charles Sprague Smith, in Barbizon days, Millet-Corot-Rousseau-Barye publisher, A. Wessels Company, New York, July 1902, pp. 172-173
The Duke de Broglie had ordered of Rousseau a painting of the 'Chateau de Broglie', for his friend M. Guizot. Madame Guizot had died there, and The Duke de Broglie urged Rousseau to make the painting grave and sad.. The quote presents Rousseau’s responding
1830 - 1850

Love, Life, World, Feeling

“The mathematics is not there till we put it there.”

Arthur Stanley Eddington (1882–1944) British astrophysicist

The Philosophy of Physical Science (1938)

“In spite of all the foregoing, there is a science of economics, a true, and even exact, science, which reaches laws as universal as those of mathematics and mechanics. The greatest need for the development of economics as a growing body of thought and practice is an adequate appreciation of the meaning, and the limitations, of this body of accurate premises and rigorously established conclusions. It comes about in the same general way as all science, except perhaps in a higher degree, i. e., through abstraction. There are no laws regarding the content of economic behavior, but there are laws universally valid as to its form. There is an abstract rationale of all conduct which is rational at alt, and a rationale of all social relations arising through the organization of rational activity.”

Frank Knight (1885–1972) American economist

Source: "The limitations of scientific method in economics", 1924, p. 127 (2009 edition)

Law, Way, Behavior, Universe

“It is possible that the major collaboration between economists and engineers is still to come, in the greater use of physical analogues and computers of the analogue type to avoid the difficulties of calculation. Apart from their major use as possible tools for economic regulation, physical analogues have a subsidiary use, for there are students of economics, as there are many students of engineering, who can better understand the significance of the somewhat formidable mathematics that tends to be used in this field, if they can first acquaint themselves with the types of behaviour in question as exhibited by physical objects that can be seen, felt, handled and experimented with. It may also be suggested that economists may find that what they have to say about economic policy will be very readily assimilated by one group of attentive pupils, namely the scientists, engineers and technicians of industry, if explanatory notions can be drawn from the theory of automatic control, which is now part of a normal engineering education. The aim of this essay has been to give explanations of system behaviour, and some approaches to its analysis, using geometrical construction and physical analogy where possible to clarify the implications of the more usual formal algebraic approach.”

Arnold Tustin (1899–1994) British engineer

Source: The Mechanism of Economic Systems (1953), p. 2

Education, Understanding, Question, Parting

“Mathematics as an Element in the History of Thought.”

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

1920s, Science and the Modern World (1925)

History

“… treatises on mechanics do not clearly distinguish between what is experiment, what is mathematical reasoning, what is convention, and what is hypothesis.”

Henri Poincaré book Science and Hypothesis

... les traités de mécanique ne distinguent pas bien nettement ce qui est expérience, ce qui est raisonnement mathématique, ce qui est convention, ce qui est hypothèse.
Source: Science and Hypothesis (1901), Ch. VI: The Classical Mechanics, Tr. George Bruce Halsted (1913)

“Although mathematics originates in the human mind, its remarkable effectiveness in explaining the world does not extend to the mind itself. Psychology has proved unusually resistant to the mathematization that works so well in physics.”

Nick Herbert (1936) American physicist

Source: Quantum Reality - Beyond The New Physics, Chapter 1, The Quest For Reality, p. 2

World

“It is now known that the truth or falsity of the continuum hypothesis and other related conjectures cannot be determined by set theory as we know it today. This state of affairs regarding a classical and presumably well-posed problem must certainly appear rather unsatisfactory to the average mathematician. One is tempted to look more closely and perhaps more critically at the foundations of mathematics. Although our present "Cantorian" mathematics is highly successful in its treatment of abstractions, one must not overlook the fact that from the very beginning the use of infinite processes was regarded with suspicion by many people.”

Paul Cohen (1934–2007) American mathematician

Set theory and the continuum hypothesis, p. 1. https://books.google.com/books?id=Z4NCAwAAQBAJ&pg=PA1 Courier Corporation, 2008 (Dover reprint). Set Theory and the Continuum Hypothesis (1966)

People, Truth, Problem, Appearance

“I believe synthetic biology forces society into the chaos of ethics. And, I believe that every connotation of "chaos" applies to the ethics of synthetic biology. Researchers must decide if a particular endeavor is ethical because, of course, mistakes can result due to the unpredictability of these very complex systems. I also believe that the original Greek and the Genesis, Chapter 1, understanding of formlessness applies, since scientists are trying to move to the simplest creatures possible, so-called "chassis bacteria", onto which they can add desirable traits at their choosing. Finally, I believe that the mathematical connotation of chaos applies, since even the most simple living creature is a dynamical system that is highly sensitive to initial conditions. I sometimes wonder about the extent to which those most intimately involved and those strongly advocating synthetic biology think about this.”

Daniel Alan Vallero (1953) American scientist

Class notes from Vallero's optimization course at Duke University. 2017.

Understanding, Thinking

“Mathematics may be compared to a mill of exquisite workmanship, which grinds you stuff of any degree of fineness; but, nevertheless, what you get out depends upon what you put in; and as the grandest mill in the world will not extract wheat-flour from peascod, so pages of formulae will not get a definite result out of loose data.”

Thomas Henry Huxley (1825–1895) English biologist and comparative anatomist

"Geological Reform", Quarterly Journal of the Geological Society of London, Vol. 25 (1869); as reprinted in Huxley, Discourses, Biological and Geological essays (1909), pp. 335–336
1860s

World, Dependency

“I presume that few who have paid any attention to the history of the Mathematical Analysis, will doubt that it has been developed in a certain order, or that that order has been, to a great extent, necessary -- being determined, either by steps of logical deduction, or by the successive introduction of new ideas and conceptions, when the time for their evolution had arrived.”

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

Source: 1850s, A treatise on differential equations (1859), p. v; cited in: Quotations by George Boole http://www-history.mcs.st-and.ac.uk/Quotations/Boole.html, MacTutor History of Mathematics, August 2010.

History, Idea, Time, Success

“Mathematics is one of the most basic -- and most ancient -- types of knowledge. Yet the details of its historical development remain obscure to all but a few specialists.”

Ivor Grattan-Guinness (1941–2014) Historian of mathematics and logic

Text back cover.
Companion encyclopedia of the history and philosophy of the mathematical sciences (2003)

Knowledge

“I am accustomed, as a professional mathematician, to living in a sort of vacuum, surrounded by people who declare with an odd sort of pride that they are mathematically illiterate.”

David Mumford (1937) American mathematician

David Mumford, cited in: Michael Harris (2015), Mathematics without Apologies: Portrait of a Problematic Vocation. p. 5

People

“You will feel interested to know the fate of my mathematical speculations in Cambridge. One of the papers is already printed in the Mathematical Journal. Another, which I sent a short time ago, has been very favourably received, and will shortly be printed together with one I had previously sent.”

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

George Boole in letter to a friend, 1840, cited in: R. H. Hutton, " Professor Boole http://books.google.com/books?id=pfMEAAAAQAAJ&pg=PA147," in: The British Quarterly Review. (1866), p. 147; Cited in Des MacHale. George Boole: his life and work, Boole Press, 1985. p. 52 1840s

Fate, Feeling, Time

“Though a skilled mathematician, Alfred Marshall used mathematics sparingly. He saw that excessive reliance on this instrument might lead us astray in pursuit of intellectual toys, imaginary problems not conforming to the conditions of real life: and further, might distort our sense of proportion by causing us to neglect factors that could not easily be worked up in the mathematical machine.”

Arthur Cecil Pigou (1877–1959) British economist

Arthur Cecil Pigou ed. (1925). Memorials of Alfred Marshall, New York: Augustus M. Kelley, 1966. p. 84

Life, Problem, Sense

“The arithmetization of mathematics… which began with Weierstrass… had for its object the separation of purely mathematical concepts, such as number and correspondence and aggregate, from intuitional ideas, which mathematics had acquired from long association with geometry and mechanics.
These latter, in the opinion of the formalists, are so firmly entrenched in mathematical thought that in spite of the most careful circumspection in the choice of words, the meaning concealed behind these words, may influence our reasoning. For the trouble with human words is that they possess content, whereas the purpose of mathematics is to construct pure thought.”

Tobias Dantzig (1884–1956) American mathematician

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

Idea

“I know that most men — not only those considered clever, but even those who are very clever and capable of understanding most difficult scientific, mathematical, or philosophic, problems — can seldom discern even the simplest and most obvious truth if it be such as obliges them to admit the falsity of conclusions they have formed, perhaps with much difficulty — conclusions of which they are proud, which they have taught to others, and on which they have built their lives.”

Leo Tolstoy (1828–1910) Russian writer

Opening to Ch 14. Translation from: What Is Art and Essays on Art (Oxford University Press, 1930, trans. Aylmer Maude)
As quoted by physicist Joseph Ford in Chaotic Dynamics and Fractals (1985) edited by Michael Fielding Barnsley and Stephen G. Demko
What is Art? (1897)
Variant: I know that most men, including those at ease with problems of the greatest complexity, can seldom accept even the simplest and most obvious truth if it be such as would oblige them to admit the falsity of conclusions which they have delighted in explaining to colleagues, which they have proudly taught to others, and which they have woven, thread by thread, into the fabric of their lives.

Men, Truth, Understanding, Problem

“At each stage of in the advance of mathematical thought the outstanding characteristics are novelty and originality. That is why mathematics is such a delight to study, such a challenge to practise and such a puzzle to define.”

George Frederick James Temple (1901–1992) British mathematician

100 Years of Mathematics: a Personal Viewpoint (1981)

Study

“I would teach the world how the Greeks proved, more than 2,000 years ago, that there are infinitely many prime numbers. In my mind, this discovery is the beginning of mathematics – when humankind realised that, by pure thought alone, it could prove eternal truths of the universe.
Prime numbers are the indivisible numbers, numbers that can be divided only by themselves and one. They are the most important numbers in mathematics, because every number is built by multiplying prime numbers together – for example, 60 = 2 x 2 x 3 x 5. They are like the atoms of arithmetic, the hydrogen and oxygen of the world of numbers.”

Marcus du Sautoy (1965) British professor of mathematics

In "Life lessons" http://www.theguardian.com/science/2005/apr/07/science.highereducation?fb_ref=desktop The Guardian (7 April 2005)

Truth, World, Universe

“It must be admitted that among a certain number of advanced mathematical physicists whose work has lain mainly in the twentieth century, the ether is regarded with suspicion, or even with contempt. And some of the opponents go so far as to as to say that the nineteenth century idea of the ether has failed to establish itself, and that in consequence the whole idea of the ether is under a cloud, and that it is only upheld by a few antiquated supporters, who, though are willing to admit many modifications in the original nineteenth century notions of an ether, feel the need of a medium capable of performing the functions attributed to it.”

Oliver Lodge (1851–1940) British physicist

My Philosophy: Representing My Views on the Many Functions of the Ether of Space, p. 109 https://books.google.com/books?id=pC28TnExGEEC&pg=PA109 My Philosophy (1933)

Feeling, Idea, Support

“Magic is that which it is; it is by itself, like the mathematics; for it is the exact and absolute science of Nature and its laws.
Magic is the science of the Ancient Magi: and the Christian religion, which has imposed silence on the lying oracles, and put an end to the prestiges of the false Gods, itself reveres those Magi who came from the East, guided by a Star, to adore the Saviour of the world in His cradle.”

Albert Pike book Morals and Dogma of the Ancient and Accepted Scottish Rite of Freemasonry

Source: Morals and Dogma of the Ancient and Accepted Scottish Rite of Freemasonry (1871), Ch. XXXII : Sublime Prince of the Royal Secret, p. 841

God, World, Religion, Law

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

David Eugene Smith (1860–1944) American mathematician

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

Study, Universe, Time

“There are two suffixes in our language (and similar ones in other European languages) which suggest organized knowledge. One is the venerable, academic "ology," that reminds one of university curricula and scholarship. The other is the energetic and somewhat mysterious "ics," which has a connotative flavor of magic. Where "ology" suggests academic isolation (ichthyology, philology) "its" suggests a method of attack on life’s problems. It contains a faint throwback to the ancient dreams of the philosopher’s stone and of "keys" to the riddles of the universe. Ancient words ending in "ics" are mathematics and metaphysics. Of more recent origin are economics, statistics, semantics, and cybernetics.”

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

1950, p. 12 (1952, p. 123) lead paragraph
1950s, "What is Semantics?", 1950

Life, Dreams, Knowledge, Problem

“From the statements we have made, a stupendous perspective emerges, a vista towards a hitherto unsuspected unity of the conception of the world. Similar general principles have evolved everywhere, whether we are dealing with inanimate things, organisms, mental or social processes. What is the origin of these correspondences?
We answer this question by the claim for a new realm of science, which we call General System Theory. It is a logico-mathematical field, the subject matter of which is the formulation and derivation of those principles which hold for systems in general. A "system" can be defined as a complex of elements standing in interaction. There are general principles holding for systems, irrespective of the nature of the component elements and of the relations or forces between them.”

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

Source: 1950s, Problems of Life (1952, 1960), p. 199 as cited in: D.C. (1969) "Systems Theory — A Discredited Philosophy". in: Abacus V. p. 8

World, Question, Nature, Science

“Science and mathematics… have added little to our understanding of such things as Truth, Beauty, and Justice. There may be definite limits to the applicability of the scientific method.”

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

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

Truth, Justice, Beauty, Understanding

“Does anyone not have any problems with taking ideas seriously? I think I'm in this category because ideas like cryonics, the Singularity, [unfriendly artificial intelligence], and Tegmark's mathematical universe were all immediately obvious to me as ideas to take seriously, and I did so without much conscious effort or deliberation.”

Wei Dai Cryptocurrency pioneer and computer scientist

In a discussion thread https://www.lesswrong.com/posts/Q8jyAdRYbieK8PtfT/taking-ideas-seriously#Ym77AptKtD2h9bXXd on LessWrong, August 2010

Intelligence, Idea, Problem, Thinking

“There is no problem in all mathematics that cannot be solved by direct counting. But with the present implements of mathematics many operations can be performed in a few minutes which without mathematical methods would take a lifetime.”

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

Source: 19th century, Popular Scientific Lectures [McCormack] (Chicago, 1898), p. 197; On mathematics and counting.

Problem

“It is number which regulates everything and it is measure which establishes universal order…. A quiet peace, an inviolable order, an inflexible security amidst all change and turmoil characterize the world which mathematics discloses and whose depths it unlocks.”

Christian Heinrich von Dillmann (1829–1899) German educationist

Source: Die Mathematik die Fackelträgerin einer neuen Zeit (Stuttgart, 1889), p. 12.

World, Peace, Universe, Change

“"Is it agreeable?" somebody asked.
"Neither agreeable nor disagreeable," I answered. "it just is." Istigkeit - wasn't that the word Meister Eckhart liked to use? "Is-ness." The Being of Platonic philosophy - except that Plato seems to have made the enormous, the grotesque mistake of separating Being from becoming and identifying it with the mathematical abstraction of the Idea. He could never, poor fellow, have seen a bunch of flowers shining with their own inner light and all but quivering under the pressure of the significance with which they were charged; could never have perceived that what rose and iris and carnation so intensely signified was nothing more, and nothing less, than what they were - a transience that was yet eternal life, a perpetual perishing that was at the same time pure Being, a bundle of minute, unique particulars in which, by some unspeakable and yet self-evident paradox, was to be seen the divine source of all existence.”

Aldous Huxley book The Doors of Perception

page 4-5
The Doors of Perception (1954)

Life, Flowers, Idea, Light

“But we have higher mathematics, haven't we? This gives me freedom from my senses. The language of mathematics is even more inborn and universal than the language of music; a mathematical formula is crystal clear and independent of all sense organs. I therefore built a mathematical laboratory, set myself in it as if I were sitting in a car, and moved along with a beam of light.”

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. 11

Music, Cars, Sense, Freedom

“No matter how correct a mathematical theorem may appear to be, one ought never to be satisfied that there was not something imperfect about it until it gives the impression of also being beautiful.”

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

Attributed to George Boole in: Des MacHale (1993) Comic sections: the book of mathematical jokes, humour, and wisdom. p, 107
Attributed from posthumous publications

Beauty, Appearance

“In order to see the difference which exists between… studies,—for instance, history and geometry, it will be useful to ask how we come by knowledge in each. Suppose, for example, we feel certain of a fact related in history… if we apply the notions of evidence which every-day experience justifies us in entertaining, we feel that the improbability of the contrary compels us to take refuge in the belief of the fact; and, if we allow that there is still a possibility of its falsehood, it is because this supposition does not involve absolute absurdity, but only extreme improbability.
In mathematics the case is wholly different… and the difference consists in this—that, instead of showing the contrary of the proposition asserted to be only improbable, it proves it at once to be absurd and impossible. This is done by showing that the contrary of the proposition which is asserted is in direct contradiction to some extremely evident fact, of the truth of which our eyes and hands convince us. In geometry, of the principles alluded to, those which are most commonly used are—
I. If a magnitude is divided into parts, the whole is greater than either of those parts.
II. Two straight lines cannot inclose a space.
III. Through one point only one straight line can be drawn, which never meets another straight line, or which is parallel to it.
It is on such principles as these that the whole of geometry is founded, and the demonstration of every proposition consists in proving the contrary of it to be inconsistent with one of these.”

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

Source: On the Study and Difficulties of Mathematics (1831), Ch. I.

Truth, Feeling, History, Knowledge

“A distinguished agnostic once observed that in these days Christianity was not refuted, it was explained. Doubtless the difference between the two operations was, in his view, a matter rather of form than of substance. That which was once explained needed, he thought, no further refutation. And certainly we are all made happy when a belief, which seems to us obviously absurd, is shown nevertheless to be natural in those who hold it. But we must be careful. True beliefs are effects no less than false. In this respect magic and mathematics are on a level. Both demand scientific explanation; both are susceptible of it. Manifestly, then, we cannot admit that explanation may be treated as a kind of refutation. /…/ This way lies universal scepticism. Thus would all intellectual values be utterly destroyed.”

Arthur James Balfour (1848–1930) British Conservative politician and statesman

Theism and humanism

Happiness, Way, Nature, Universe

“We can… treat only the geometrical aspects of mathematics and shall be satisfied in having shown that there is no problem of the truth of geometrical axioms and that no special geometrical visualization exists in mathematics.”

Hans Reichenbach (1891–1953) American philosopher

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

Truth, Problem

“I had become interested in economics, an interest that was transformed into a lifetime dedication when I met with the mathematical theory of general economic equilibrium.”

Gérard Debreu (1921–2004) French economist and mathematician

" Gerard Debreu - Biographical http://www.nobelprize.org/nobel_prizes/economic-sciences/laureates/1983/debreu-bio.html". in: Les Prix Nobel. The Nobel Prizes 1983, Editor Wilhelm Odelberg, [Nobel Foundation], Stockholm, 1984; Republished at Nobelprize.org. Nobel Media AB 2014.

“We have defined the main task of economic analysis as the explanation of the magnitudes of economic quantities. The student will find also that the main part of this, as of most other works on the subject, is concerned with the theory of the determination of prices, wages, interest rates, incomes, and the like. He may well inquire, therefore, in the midst of so much mathematics, whether the first task of economics is not the investigation of wealth, or welfare. Some economists have endeavored to restrict the boundaries of the science to the investigation of those quantities which are numerically measurable. Well-being, under such a restriction, would not be part of economics at all.”

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

Source: 1940s, Economic Analysis, 1941, p. 7-8

Science, Parting, Wealth

“Mathematics is that form of intelligence in which we bring the objects of the phenomenal world under the control of the conception of quantity. [Provisional definition. ]”

George Holmes Howison (1834–1916) American philosopher

"The Departments of Mathematics, and their Mutual Relations," Journal of Speculative Philosophy, Vol. 5, p. 164. Reported in Moritz (1914)
Journals

World, Intelligence

“I will ask of you only the ability to read English and to think logically—no high school mathematics, and certainly no higher mathematics.”

Edmund Landau (1877–1938) German Jewish mathematician

Grundlagen der Analysis [Foundations of Analysis] (1930) Preface for the Student, as quoted by Eli Maor, Trigonometric Delights (2013)

School, Thinking, Reading

“About ancient mathematics (whether Greek or Mesopotamian) and medieval mathematics (Western or Oriental), the would-be historian must of necessity confine himself to the description of a comparatively small number of islands accidentally emerging from an ocean of ignorance, and to tenuous conjectural reconstructions of the submerged continents which at one time must have bridged the gaps between them.”

André Weil (1906–1998) French mathematician

Number Theory: An approach through history from Hammurapi to Legendre (Springer, 2006), p. 3 https://books.google.com/books?id=XSV0hDFj3loC&pg=PA3

Time

“We are approaching a more fluid state. I have talked about cultural boiling. The idea of the phase-transition period which, in fractal mathematics, is the chaotic flux between one state and another. …Culturally, and as a species, we are approaching a phase-transition. I don’t know quite what that means, on a human level.”

Alan Moore (1953) English writer primarily known for his work in comic books

De Abaitua interview (1998)

Idea

“I mused upon the ironic fate which had compelled a mathematical genius to make his sole confidant of a philistine lawyer, and induced that lawyer to repeat it confusedly to an ignoramus at twilight on a Scotch hill.”

John Buchan (1875–1940) British politician

Space (1912)

Fate, Confidence

“It is not true, out of geometry, that the mathematical sciences are, in all their parts those models of finished accuracy which many suppose. The extreme boundaries of analysis have always been as imperfectly understood as the tract beyond the boundaries was absolutely unknown. But the way to enlarge the settled country has not been by keeping within it, but by making voyages of discovery, and I am perfectly convinced that the student should be exercised in this manner; that is, that he should be taught how to examine the boundary, as well as how to cultivate the interior. …allowing all students whose capacity will let them read on the higher branches of applied mathematics, to have each his chance of being led to the cultivation of those parts of analysis on which rather depends its future progress than its present use in the sciences of matter.”

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

The Differential and Integral Calculus (1836)

Way, Exercises, Science, Chance

“The art of doing mathematics consists in finding that special case which contains all the germs of generality.”

David Hilbert (1862–1943) German prominent mathematician

Quoted in Constance Reid, "Hilbert" (1970)

Art

“…Once you entomb mathematics in an artificial language à la Hilbert, once you set up a completely formal axiomatic system, then you can forget that it has any meaning and just look at it as a game that you play with marks on paper that enable you to deduce theorems from axioms. You can forget about the meaning of the game, the game of mathematical reasoning, it's just combinatorial play with symbols! There are certain rules, and you can study these rules and forget that they have any meaning!”

Gregory Chaitin (1947) Argentinian mathematician and computer scientist

1999 Lecture—"A Century of Controversy over the Foundations of Mathematics" at U. Massachusetts at Lowell, quoted in [2012, Conversations with a Mathematician: Math, Art, Science and the Limits of Reason, Springer, https://books.google.com/books?id=DczTBwAAQBAJ&pg=PA15] p. 15

Game, Study, Forgetting

“In the last few years there has been a harvest of books and lectures about the "Mysterious Universe." The inconceivable magnitudes with which astronomy deals produce a sense of awe which lends itself to a poetic and philosophical treatment. "When I consider thy heavens, the work of thy hands, the moon and the starts, whuch thou hast ordained: what is man that thou art mindful of him? The literary skill with which this branch of science has been exploited compels one's admiration, but alos, a little, one's sense of the ridiculous. For other facts than those of astronomy, oother disciplines than of mathematics, can produce the same lively feelings of awe and reverence: the extraordinary finenness of their adjustments to the world outside: the amazing faculties of the human mind, of which we know neither whence it comes not whither it goes. In some fortunate people this reverence is produced by the natural bauty of a landscape, by the majesty of an ancient building, by the heroism of a rescue party, by poetry, or by music. God is doubtless a Mathematician, but he is also a Physiologist, an Engineer, a Mother, an Architect, a Coal Miner, a Poet, and a Gardener. Each of us views things in his own peculiar war, each clothes the Creator in a manner which fits into his own scheme. My God, for instance, among his other professions, is an Inventor: I picture him inventing water, carbon dioxide, and haemoglobin, crabs, frogs, and cuttle fish, whales and filterpassing organisms ( in the ratio of 100,000,000,000,000,000,000,000 to 1 in size), and rejoicing greatly over these weird and ingenious things, just as I rejoice greatly over some simple bit of apparatus. But I would nor urge that God is only an Inventor: for inventors are apt, as those who know them realize, to be very dull dogs. Indeed, I should be inclined rather to imagine God to be like a University, with all its teachers and professors together: not omittin the students, for he obviously possesses, judging from his inventions, that noblest human characteristic, a sense of humour.”

Archibald Hill (1886–1977) English physiologist and biophysicist

The Ethical Dilemma of Science and Other Writings https://books.google.com.mx/books?id=zaE1AAAAIAAJ&printsec=frontcover#v=onepage&q&f=false (1960, Cap 1. Scepticism and Faith, p. 41)

God, War, People, Books

“Aristotle would… by no means admit that mathematics was divorced from aesthetic; he could conceive, he said, of nothing more beautiful than the objects of mathematics.”

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

Beauty

“I do feel strongly that this is nonsense! … So perhaps I could entertain future historians by saying I think all this superstring stuff is crazy and is in the wrong direction. I think all this superstring stuff is crazy and is in the wrong direction. … I don’t like it that they’re not calculating anything. … why are the masses of the various particles such as quarks what they are? All these numbers … have no explanations in these string theories – absolutely none! … I don’t like that they don’t check their ideas. I don’t like that for anything that disagrees with an experiment, they cook up an explanation—a fix-up to say, “Well, it might be true.” For example, the theory requires ten dimensions. Well, maybe there’s a way of wrapping up six of the dimensions. Yes, that’s all possible mathematically, but why not seven? When they write their equation, the equation should decide how many of these things get wrapped up, not the desire to agree with experiment. In other words, there’s no reason whatsoever in superstring theory that it isn’t eight out of the ten dimensions that get wrapped up and that the result is only two dimensions, which would be completely in disagreement with experience. So the fact that it might disagree with experience is very tenuous, it doesn’t produce anything.”

Richard Feynman (1918–1988) American theoretical physicist

interview published in Superstrings: A Theory of Everything? (1988) edited by Paul C. W. Davies and Julian R. Brown, p. 193-194

Way, Feeling, Idea, Thinking

“The opinion appears to be gaining ground that this very general conception of functionality, born on mathematical ground, is destined to supersede the narrower notion of causation, traditional in connection with the natural sciences. As an abstract formulation of the idea of determination in its most general sense, the notion of functionality includes and transcends the more special notion of causation as a one-sided determination of future phenomena by means of present conditions; it can be used to express the fact of the subsumption under a general law of past, present, and future alike, in a sequence of phenomena. From this point of view the remark of Huxley that Mathematics "knows nothing of causation" could only be taken to express the whole truth, if by the term "causation" is understood "efficient causation." The latter notion has, however, in recent times been to an increasing extent regarded as just as irrelevant in the natural sciences as it is in Mathematics; the idea of thorough-going determinancy, in accordance with formal law, being thought to be alone significant in either domain.”

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

Source: Presidential Address British Association for the Advancement of Science, Section A (1910), p. 290 ; Cited in: Moritz (1914, 29): The Nature of Mathematics.

Truth, Law, Idea, Nature

“Vacuum stands and remains a mathematical space. A cube placed in a vacuum would not displace anything, as it would displace air or water in a space already containing those fluids.”

Robert Grosseteste (1175–1253) English bishop and philosopher

Commentarius in VIII Libros Physicorum Aristoteles (c. 1230-1235)

Water

“Universality is the distinguishing mark of genius. There is no such thing as a special genius, a genius for mathematics, or for music, or even for chess, but only a universal genius. … The theory of special genius, according to which for instance, it is supposed that a musical genius should be a fool at other subjects, confuses genius with talent. … There are many kinds of talent, but only one kind of genius, and that is able to choose any kind of talent and master it.”

Otto Weininger book Sex and Character

Source: Sex and Character (1903), p. 112.

Fools, Music, Universe

“I went to Cambridge to do higher mathematics, that was my first goal and appearing in the university exams in mathematics. You are given a menu of various branches of mathematics, pure as well as applied. So I found that applied aspects, especially application to astronomy were very interesting. And the speakers on both courses, that is the lecturers were also very good. At that time, I also read a book by Fred Hoyle called ‘Frontiers of Astronomy’, which gave a very readable account for a layman for what was happening in astronomy. So, all these factors made me go into the research field of astronomy. Because one is required to choose which branch of mathematics one takes as research field. In Cambridge, astronomy is treated as a branch of mathematics. So I choose that.”

Jayant Narlikar (1938) Indian physicist

An Exclusive Interview with Prof. Jayant Vishnu Narlikar

Books, Universe, Appearance, Time

“Arithmetic and geometry, according to Plato, are the two wings of the mathematician. The object indeed of all mathematical questions, is to determine the ratios of numbers, or of magnitudes; and it may even be said, to continue the comparison of the ancient philosopher, that arithmetic is the mathematician's right wing; for it is an incontestable truth, that geometrical determinations would, for the most part, present nothing satisfactory to the mind, if the ratios thus determined could not be reduced to numerical ratios. This justifies the common practice, which we shall here follow, of beginning with arithmetic.”

Jacques Ozanam (1640–1718) French mathematician

Source: Recreations in Mathematics and Natural Philosophy, (1803), p. 2

Truth, Question, Parting

“The use of practically any computing technique itself raises a number of mathematical problems. There is thus a very considerable impact of computation on mathematics itself, and this may be expected to influence mathematical research to an increasing degree.”

George Forsythe (1917–1972) Stanford University computer scientist

George Forsythe (1958) cited in: Computers and people Vol 23. (1974). p. 11 Pagina 11

Problem

“Georg Cantor claimed the essence of mathematics lies in its freedom. But mathematicians do not pick problems from thin air for the pleasure of solving them. To the contrary, a mark of greatness resides in the ability to identify the most interesting problems in the framework of what is already known.”

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

Source: The Fractalist (2012), Ch. 17, p. 178

Pleasure, Problem, Freedom

“Mechanical drawings and blueprints are not mere pictures, but a complete and rich language. In blueprint language, scientific, mathematical, and geometric formulations, notations, mensurations, and naming do not merely describe an object or process, they actually model it. Because of broad differences in subject, purpose, roles, and the needs of the people who use them, many forms of blueprint have evolved, but all rigorously present well structured information in understandable form.”

Douglas T. Ross (1929–2007) American computer scientist

Source: Structured analysis (SA): A language for communicating ideas (1977), p. 16.

People, Understanding, Drawing

“The present book is not a methodology of mathematics in the sense that I will systematically show how some teaching matter should taught; it is not even a systematic analysis of subject matter. I hardly ever refer to well-organized classroom experiments evaluated by statistical methods, nor do I cite experimental results of developmental psychology or the psychology of learning. Maybe the most striking feature is that this book contains few quotations. I will try to justify all these features.”

Hans Freudenthal (1905–1990) Dutch mathematician

Source: Mathematics as an Educational Task (1973), p. v;As cited in: Ben Wilbrink (2013)

Books, Sense, Learning

“At this time it is difficult to put one's finger on any single contribution in the decade 1950 - 1960 which is comparable to those above, and yet progress has probably been even greater. From the point of view of an educator, one cannot overlook the wide distribution which has been given to these ideas. There has been remarkable progress from analysis to synthesis, always a sign of maturity in a field of analytic endeavour. There has been consolidation, for example in the establishment of a more rigorous basis for information theory; there has been unification, for example in the demonstration of the formal similarity between game theory and linear programming; there has been application to mathematically more difficult situations, for example nonlinear servo systems and information channels with memory; there has been implementation, as in commercially available computers which by any reasonable measure are hun- dreds of times more powerful than the primitive devices of 1950; there has been de-limitation of the boundaries of many of these fields.”

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

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

Education, Game, Idea, Memory

“The sciences do not try to explain, they hardly even try to interpret, they mainly make models. By a model is meant a mathematical construct which, with the addition of certain verbal interpretations, describes observed phenomena. The justification of such a mathematical construct is solely and precisely that it is expected to work.”

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

"Method in the Physical Sciences", in The Unity of Knowledge (1955), ed. L. G. Leary (Doubleday & Co., New York), p. 157

Science

“Johann Herbart’s work on education and particularly mathematical psychology influenced me. I think mathematics is the pure instance of construct functioning—the model of human behaviour.”

George Kelly (psychologist) (1905–1967) American psychologist and therapist

Attributed to George A. Kelly in Hinkle (1970, p. 91), as cited in: Fay Fransella and Robert A. Neimeyer. "George Alexander Kelly: The man and his theory." International handbook of personal construct psychology (2003): 21-31.

Education, Thinking

“In the absolute universe all events can be regarded as absolutely deterministic, and if we can’t perceive the greater structures, it’s because our vision is faulty. If we had a real grasp of causality down to the molecular level, we wouldn’t need to rely on mathematical approximations, on statistics and probabilities, in making predictions. If our perceptions of cause and effect were only good enough, we’d be able to attain absolute knowledge of what is to come. We would make ourselves all-seeing.”

Robert Silverberg book The Stochastic Man

Source: The Stochastic Man (1975), Chapter 3 (p. 11)

Knowledge, Universe

“I was proud of my work with Yakov Isaevich, and he of me. Despite our good relationship, however, I kept my “other” mathematical life – my work with Fuchs and Feigin and all of that – secret from him as I did from most other people. It was as though applied mathematics was my spouse, and pure mathematics was my secret lover.”

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

Source: Love and Math, 2013, p. 139

Life, People, Secret, Lovers

“If one has really technically penetrated a subject, things that previously seemed in complete contrast, might be purely mathematical transformations of each other.”

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

As quoted in Proportions, Prices, and Planning (1970) by András Bródy

“Yet the complexity of all this pales in comparison to the patterns of information processing in your brain. Your roughly 100 billion neurons are constantly generating electrical signals (“firing”), which involves shuffling around billions of trillions of atoms, notably sodium, potassium, and calcium ions. The trajectories of these atoms form an extremely elaborate braid through spacetime, whose complex intertwining corresponds to storing and processing information in a way that somehow gives rise to our familiar sensation of self-awareness. There’s broad consensus in the scientific community that we still don’t understand how this works, so it’s fair to say that we humans don’t yet fully understand what we are. However, in broad brush, we might say this: You’re a pattern in spacetime. A mathematical pattern. Specifically, you’re a braid in spacetime—indeed, one of the most elaborate braids known.”

Max Tegmark (1967) Swedish-American cosmologist

Life Is A Braid In Spacetime http://nautil.us/issue/9/time/life-is-a-braid-in-spacetime

Way, Communication, Understanding

“The methods of mathematics are the main topic of the course, not a long list of finished mathematical results with such highly polished proofs that the poor student can only marvel at the results, with no hope of understanding how mathematics is actually created by practicing mathematicians.”

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

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

Hope, Understanding

“Whereas originally the hopes for string theory, and its descendants, were that some kind of uniqueness would be arrived at, whereby the theory would supply mathematical explanations for the measured numbers of experimental physics, the string theorists were driven to find refugee in the strong anthropic argument in an attempt to narrow down an absolutely vast number of alternatives. In my own view, this a very sad and unhelpful place for a theory to find itself.”

Roger Penrose book Fashion, Faith, and Fantasy in the New Physics of the Universe

Source: Fashion, Faith, and Fantasy in the New Physics of the Universe (2016), Ch. 3 Fantasy, p 322

Sadness, Hope, Strong

“A student of history, who cares little for Greek or mathematics in particular, but who likes to watch how things grow, will be able to extract from these pages a notion of the whole history of mathematical science down to Newton's time…”

James Gow (scholar) (1854–1923) scholar

Preface
A Short History of Greek Mathematics (1884)

History, Science, Time

“For many years I was the youngest among my mathematical friends. It makes me melancholy to realize that I now have become the oldest in most groups of scientists.”

Stanislaw Ulam (1909–1984) Polish-American mathematician

Source: Adventures of a Mathematician - Third Edition (1991), Chapter 2, Student Years, p. 37

“For all the sublimity of art, physics, music, mathematics, and other manifestations of human genius, everything depends on the mundane, frustrating, often debased vocation known as politics (and its most exacting subspecialty — statecraft). Because if we don't get politics right, everything else risks extinction.”

Charles Krauthammer (1950–2018) American journalist

2010s, 2011, Are we alone in the universe? (2011)

Art, Music, Dependency

“One voice in the Great Conversation itself announces this modern point of view. In the closing paragraph of his An Enquiry Concerning Human Understanding, David Hume writes: "When we run over libraries, persuaded of these principles, what havoc must we make? If we take in our hand any volume… let us ask, Does it contain any abstract reasoning concerning quantity or number? No. Does it contain any experimental reasoning concerning matter of fact and existence? No. Commit it then to flames: for it can contain nothing but sophistry and illusion."… the positivists of our own day, would commit to burning or, what is the same, to dismissal from serious consideration… Those books… argue the case against the kind of positivism that asserts that everything except mathematics and experimental science is sophistry and illusion…. The Great Conversation… contains both sides of the issue.”

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

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

Books, Understanding, Science

“Laplace created a number of new mathematical methods that were subsequently expanded into branches of mathematics, but he never cared for mathematics except as it helped him to study nature.”

Morris Kline (1908–1992) American mathematician

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

Nature, Study, Help

“As a mathematician, von Neumann was quick, brilliant, efficient, and enormously broad in scientific interests beyond mathematics itself. He knew his technical abilities; his virtuosity in following complicated reasoning and his insights were supreme; yet he lacked absolute self confidence.”

Stanislaw Ulam (1909–1984) Polish-American mathematician

Source: Adventures of a Mathematician - Third Edition (1991), Chapter 4, Princeton Days, p. 76

Confidence, Self-confidence

“.. if the objects will be mathematical values, the ambient in which they live will be a particular rhythm in the emotion which surrounds them. The graphic translation of this rhythm will be a state of form, a state of color, each of which will give back to the spectator the 'state of mind' which produced it..”

Umberto Boccioni (1882–1916) Italian painter and sculptor

in a letter of 12 Feb. 1912 from Paris, to his friend Nino Barbantini (director of the Ca' Pesaro in Venice); as cited in: Shannon N. Pritchard, Gino Severini and the symbolist aesthetics of his futurist dance imagery, 1910-1915 https://getd.libs.uga.edu/pdfs/pritchard_shannon_n_200305_ma.pdf Diss. uga, 2003, p. 67 1912

Color

“(…) the bottom line is that if you believe in an external reality independent of humans, then you must also believe that our physical reality is a mathematical structure. Nothing else has a baggage-free description. In other words, we all live in a gigantic mathematical object—one that’s more elaborate than a dodecahedron, and probably also more complex than objects with intimidating names such as Calabi-Yau manifolds, tensor bundles and Hilbert spaces, which appear in today’s most advanced physics theories. Everything in our world is purely mathematical—including you.”

Max Tegmark book Our Mathematical Universe

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

World, Reality, Appearance

“To emphasize only the beautiful seems to me to be like a mathematical system that only concerns itself with positive numbers.”

Paul Klee (1879–1940) German Swiss painter

Diary entry (March 1906), # 759, in The Diaries of Paul Klee, 1898-1918; University of California Press, 1968
1903 - 1910

Beauty

“I have been aware from the outset (end of January 1959, the birthdate of the second paper in the citation) that the deep analysis of something which is now called Kalman filtering were of major importance. But even with this immodesty I did not quite anticipate all the reactions to this work. Up to now there have been some 1000 related publications, at least two Citation Classics, etc. There is something to be explained.
To look for an explanation, let me suggest a historical analogy, at the risk of further immodesty. I am thinking of Newton, and specifically his most spectacular achievement, the law of Gravitation. Newton received very ample "recognition" (as it is called today) for this work. it astounded - really floored - all his contemporaries. But I am quite sure, having studied the matter and having added something to it, that nobody then (1700) really understood what Newton's contribution was. Indeed, it seemed an absolute miracle to his contemporaries that someone, an Englishman, actually a human being, in some magic and un-understandable way, could harness mathematics, an impractical and eternal something, and so use mathematics as to discover with it something fundamental about the universe.”

Rudolf E. Kálmán (1930–2016) Hungarian-born American electrical engineer

Kalman (1986) " Steele Prizes Awarded at the Annual Meeting in San Antonio http://www-history.mcs.st-and.ac.uk/Extras/Kalman_response.html", Notices Amer. Math. Soc. 34 (2) (1987), 228-229.

Law, Way, Understanding, Thinking

“Modern positivism…expresses criticism against the naïve use of certain terms… by the general postulate that the question whether a given sentence has any meaning… should always be thoroughly and critically examined. This… is derived from mathematical logic. The procedure of natural science is pictured as an attachment of symbols to the phenomena. The symbols can, as in mathematics, be combined according to certain rules… However, a combination of symbols that does not comply with the rules is not wrong but conveys no meaning.
The obvious difficulty in this argument is the lack of any general criterion as to when a sentence should be considered meaningless. A definite decision is possible only when the sentence belongs to a closed system of concepts and axioms, which in the development of natural science will be rather the exception than the rule. In some case the conjecture that a certain sentence is meaningless has historically led to important progress… new connections which would have been impossible if the sentence had a meaning. An example… sentence: "In which orbit does the electron move around the nucleus?"”

Werner Heisenberg (1901–1976) German theoretical physicist

But generally the positivistic scheme taken from mathematical logic is too narrow in a description of nature which necessarily uses words and concepts that are only vaguely defined.
Physics and Philosophy (1958)

Question, Nature, Decision, Science

“Mathematics is the fruitful Parent of, I had almost said all, Arts, the unshaken Foundation of Sciences, and the plentiful Fountain of Advantage to Human Affairs. In which last Respect, we may be said to receive from the Mathematics, the principal Delights of Life, Securities of Health, Increase of Fortune, and Conveniences of Labour: That we dwell elegantly and commodiously, build decent Houses for ourselves, erect stately Temples to God, and leave wonderful Monuments to Posterity: That we are protected by those Rampires from the Incursions of the Enemy; rightly use Arms, skillfully range an Army, and manage War by Art, and not by the Madness of wild Beasts: That we have safe Traffick through the deceitful Billows, pass in a direct Road through the tractless Ways of the Sea, and come to the designed Ports by the uncertain Impulse of the Winds: That we rightly cast up our Accounts, do Business expeditiously, dispose, tabulate, and calculate scattered 248 Ranks of Numbers, and easily compute them, though expressive of huge Heaps of Sand, nay immense Hills of Atoms: That we make pacifick Separations of the Bounds of Lands, examine the Moments of Weights in an equal Balance, and distribute every one his own by a just Measure: That with a light Touch we thrust forward vast Bodies which way we will, and stop a huge Resistance with a very small Force: That we accurately delineate the Face of this Earthly Orb, and subject the Oeconomy of the Universe to our Sight: That we aptly digest the flowing Series of Time, distinguish what is acted by due Intervals, rightly account and discern the various Returns of the Seasons, the stated Periods of Years and Months, the alternate Increments of Days and Nights, the doubtful Limits of Light and Shadow, and the exact Differences of Hours and Minutes: That we derive the subtle Virtue of the Solar Rays to our Uses, infinitely extend the Sphere of Sight, enlarge the near Appearances of Things, bring to Hand Things remote, discover Things hidden, search Nature out of her Concealments, and unfold her dark Mysteries: That we delight our Eyes with beautiful Images, cunningly imitate the Devices and portray the Works of Nature; imitate did I say? nay excel, while we form to ourselves Things not in being, exhibit Things absent, and represent Things past: That we recreate our Minds and delight our Ears with melodious Sounds, attemperate the inconstant Undulations of the Air to musical Tunes, add a pleasant Voice to a sapless Log and draw a sweet Eloquence from a rigid Metal; celebrate our Maker with an harmonious Praise, and not unaptly imitate the blessed Choirs of Heaven: That we approach and examine the inaccessible Seats of the Clouds, the distant Tracts of Land, unfrequented Paths of the Sea; lofty Tops of the Mountains, low Bottoms of the Valleys, and deep Gulphs of the Ocean: That in Heart we advance to the Saints themselves above, yea draw them to us, scale the etherial Towers, freely range through the celestial Fields, measure the Magnitudes, and determine the Interstices of the Stars, prescribe inviolable Laws to the Heavens themselves, and confine the wandering Circuits of the Stars within fixed Bounds: Lastly, that we comprehend the vast Fabrick of the Universe, admire and contemplate the wonderful Beauty of the Divine 249 Workmanship, and to learn the incredible Force and Sagacity of our own Minds, by certain Experiments, and to acknowledge the Blessings of Heaven with pious Affection.”

Isaac Barrow (1630–1677) English Christian theologian, and mathematician

Source: Mathematical Lectures (1734), p. 27-30

Life, God, War, Health

“Empirical evidence can never establish mathematical existence--nor can the mathematician's demand for existence be dismissed by the physicist as useless rigor. Only a mathematical existence proof can ensure that the mathematical description of a physical phenomenon is meaningful.”

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

Richard Courant in: The Parsimonious Universe, Stefan Hildebrandt & Anthony Tromba, Springer-Verlag, 1996, page 148

“You need only a sheet of paper and so mathematics starts.”

Guido Beck (1903–1988) German physicist

Interview of Guido Beck http://www.aip.org/history/ohilist/4500.html by John Heilbron on April 22, 1967, Niels Bohr Library & Archives, American Institute of Physics, College Park, MD USA