Quotes about mathematics (1323 quotes, page 13)

“Science and mathematics
Run parallel to reality, they symbolize it, they squint at it,
They never touch it”

Robinson Jeffers (1887–1962) American poet

"The Silent Shepherds" (1958)
Context: Science and mathematics
Run parallel to reality, they symbolize it, they squint at it,
They never touch it: consider what an explosion
Would rock the bones of men into little white fragments and unsky the world
If any mind for a moment touch truth.

Reality , Science

“Those who really solve mathematical puzzles are the physicists.”

Carlo Beenakker (1960) Dutch physicist

In Interview with Professor Carlo Beenakker. Interviewers: Ramy El-Dardiry and Roderick Knuiman (February 1, 2006).
Context: … mathematicians are much more concerned for example with the structure behind something or with the whole edifice. Mathematicians are not really puzzlers. Those who really solve mathematical puzzles are the physicists. If you like to solve mathematical puzzles, you should not study mathematics but physics!

“Mathematics, under this definition, belongs to every enquiry, moral as well as physical.”

Benjamin Peirce Linear Associative Algebra

§ 1.
Linear Associative Algebra (1882)
Context: The sphere of mathematics is here extended, in accordance with the derivation of its name, to all demonstrative research, so as to include all knowledge strictly capable of dogmatic teaching. Mathematics is not the discoverer of laws, for it is not induction; neither is it the framer of theories, for it is not hypothesis; but it is the judge over both, and it is the arbiter to which each must refer its claims; and neither law can rule nor theory explain without the sanction of mathematics. It deduces from a law all its consequences, and develops them into the suitable form for comparison with observation, and thereby measures the strength of the argument from observation in favor of a proposed law or of a proposed form of application of a law.
Mathematics, under this definition, belongs to every enquiry, moral as well as physical. Even the rules of logic, by which it is rigidly bound, could not be deduced without its aid. The laws of argument admit of simple statement, but they must be curiously transposed before they can be applied to the living speech and verified by, observation.

“The sphere of mathematics is here extended, in accordance with the derivation of its name, to all demonstrative research, so as to include all knowledge strictly capable of dogmatic teaching.”

Benjamin Peirce Linear Associative Algebra

§ 1.
Linear Associative Algebra (1882)
Context: The sphere of mathematics is here extended, in accordance with the derivation of its name, to all demonstrative research, so as to include all knowledge strictly capable of dogmatic teaching. Mathematics is not the discoverer of laws, for it is not induction; neither is it the framer of theories, for it is not hypothesis; but it is the judge over both, and it is the arbiter to which each must refer its claims; and neither law can rule nor theory explain without the sanction of mathematics. It deduces from a law all its consequences, and develops them into the suitable form for comparison with observation, and thereby measures the strength of the argument from observation in favor of a proposed law or of a proposed form of application of a law.
Mathematics, under this definition, belongs to every enquiry, moral as well as physical. Even the rules of logic, by which it is rigidly bound, could not be deduced without its aid. The laws of argument admit of simple statement, but they must be curiously transposed before they can be applied to the living speech and verified by, observation.

Knowledge

“In this domain of arithmetic,.. the mathematical infinite already plays a preponderant rôle, and without it there would be no science, because there would be nothing general.”

Henri Poincaré book Science and Hypothesis

Source: Science and Hypothesis (1901), Ch. I. (1905) Tr. George Bruce Halstead
Context: This procedure is the demonstration by recurrence. We first establish a theorem for n = 1; then we show that if it is true of n - 1, it is true of n, and thence conclude that it is true for all the whole numbers... Here then we have the mathematical reasoning par excellence, and we must examine it more closely.
... The essential characteristic of reasoning by recurrence is that it contains, condensed, so to speak, in a single formula, an infinity of syllogisms.
... to arrive at the smallest theorem [we] can not dispense with the aid of reasoning by recurrence, for this is an instrument which enables us to pass from the finite to the infinite.
This instrument is always useful, for, allowing us to overleap at a bound as many stages as we wish, it spares us verifications, long, irksome and monotonous, which would quickly become impracticable. But it becomes indispensable as soon as we aim at the general theorem...
In this domain of arithmetic,.. the mathematical infinite already plays a preponderant rôle, and without it there would be no science, because there would be nothing general.<!--pp.10-12

Science

“The idealistic tinge in my conception of the physical world arose out of mathematical researches on the relativity theory. In so far as I had any earlier philosophical views, they were of an entirely different complexion.”

Arthur Stanley Eddington (1882–1944) British astrophysicist

The Nature of the Physical World (1928)
Context: The idealistic tinge in my conception of the physical world arose out of mathematical researches on the relativity theory. In so far as I had any earlier philosophical views, they were of an entirely different complexion.
From the beginning I have been doubtful whether it was desirable for a scientist to venture so far into extra-scientific territory. The primary justification for such an expedition is that it may afford a better view of his own scientific domain.

Preface http://www-groups.dcs.st-and.ac.uk/~history/Extras/Eddington_Gifford.html

World

“Those who assert that the mathematical sciences say nothing of the beautiful or the good are in error. For these sciences say and prove a great deal about them; if they do not expressly mention them, but prove attributes which are their results or definitions, it is not true that they tell us nothing about them. The chief forms of beauty are order and symmetry and definiteness, which the mathematical sciences demonstrate in a special degree.”

Aristotle book Metaphysics

Book XIII, 1078a.33
Metaphysics

Beauty , Error , Science

“Any impatient student of mathematics or science or engineering who is irked by having algebraic symbolism thrust on him should try to get on without it for a week.”

Eric Temple Bell (1883–1960) mathematician and science fiction author born in Scotland who lived in the United States for most of his li…

Source: Mathematics: Queen and Servant of Science (1938), p. 226
Context: Some of his deepest discoveries were reasoned out verbally with very few if any symbols, and those for the most part mere abbreviations of words. Any impatient student of mathematics or science or engineering who is irked by having algebraic symbolism thrust on him should try to get on without it for a week.

Science

“Therefore, the mathematical forms that represent the elementary particles will be solutions of some eternal law of motion for matter.”

Werner Heisenberg (1901–1976) German theoretical physicist

Physics and Philosophy (1958)
Context: The equation of motion holds at all times, it is in this sense eternal, whereas the geometrical forms, like the orbits, are changing. Therefore, the mathematical forms that represent the elementary particles will be solutions of some eternal law of motion for matter. Actually this is a problem which has not yet been solved.<!-- p. 72

Law

“The Modernist reaction to the Enlightenment came in the aftermath of the Industrial Revolution, whose brutalizing effects revealed that modern life had not become… mathematically perfect…”

Eric R. Kandel (1929) American neuropsychiatrist

The Age of Insight (2012)

Life , Perfection

“The results of a scrutiny of the materials of chemical science from a mathematical standpoint are pronounced in two directions. In the first we observe crude, qualitative notions”

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

Introduction
Higher Mathematics for Chemical Students (1911)
Context: The results of a scrutiny of the materials of chemical science from a mathematical standpoint are pronounced in two directions. In the first we observe crude, qualitative notions, such as fire-stuff, or phlogiston, destroyed; and at the same time we perceive definite measurable quantities such as fixed air, or oxygen, taking their place. In the second direction we notice the establishment of generalizations, laws, or theories, in which a mass of quantitative data is reduced to order and made intelligible. Such are the law of conservation of matter, the laws of chemical combination, and the atomic theory.

Science

“The elementary particles in Plato's Timaeus are finally not substance but mathematical forms. "All things are numbers" is a sentence attributed to Pythagoras.”

Werner Heisenberg (1901–1976) German theoretical physicist

Physics and Philosophy (1958)
Context: But the resemblance of the modern views to those of Plato and the Pythagoreans can be carried somewhat further. The elementary particles in Plato's Timaeus are finally not substance but mathematical forms. "All things are numbers" is a sentence attributed to Pythagoras. The only mathematical forms available at that time were such geometric forms as the regular solids or the triangles which form their surface. In modern quantum theory there can be no doubt that the elementary particles will finally also be mathematical forms but of a much more complicated nature.

“Galileo had raised the concepts of space and time to the status of fundamental categories by directing attention to the mathematical description of motion.”

Gerald James Whitrow (1912–2000) British mathematician

The Structure of the Universe: An Introduction to Cosmology (1949)
Context: Galileo had raised the concepts of space and time to the status of fundamental categories by directing attention to the mathematical description of motion. The midiaevel qualitative method had made these concepts relatively unimportant, but in the new mathematical philosophy the external world became a world of bodies moving in space and time. In the Timaeus Plato had expounded a theory that outside the universe, which he regarded as bounded and spherical, there was an infinite empty space. The ideas of Plato were much discussed in the middle of the seventeenth century by the Cambridge Platonists, and Newton's views were greatly influenced thereby. He regarded space as the 'sensorium of God' and hence endowed it with objective existence, although he confessed that it could not be observed. Similarly, he believed that time had an objective existence independent of the particular processes which can be used for measuring it.<!--p.46

Time

“After Pythagoras, Anaxagoras the Clazomenian succeeded, who undertook many things pertaining to geometry. And Oenopides the Chian, was somewhat junior to Anaxagoras, and whom Plato mentions in his Rivals, as one who obtained mathematical glory. To these succeeded Hippocrates, the Chian, who invented the quadrature of the lunula, and Theodorus the Cyrenean, both of them eminent in geometrical knowledge. For the first of these, Hippocrates composed geometrical elements: but Plato, who was posterior to these, caused as well geometry itself, as the other mathematical disciplines, to receive a remarkable addition, on account of the great study he bestowed in their investigation. This he himself manifests, and his books, replete with mathematical discourses, evince: to which we may add, that he every where excites whatever in them is wonderful, and extends to philosophy. But in his time also lived Leodamas the Thasian, Architas the Tarentine, and Theætetus the Athenian; by whom theorems were increased, and advanced to a more skilful constitution. But Neoclides was junior to Leodamas, and his disciple was Leon; who added many things to those thought of by former geometricians. So that Leon also constructed elements more accurate, both on account of their multitude, and on account of the use which they exhibit: and besides this, he discovered a method of determining when a problem, whose investigation is sought for, is possible, and when it is impossible.”

Proclus (412–485) Greek philosopher

Source: The Philosophical and Mathematical Commentaries of Proclus on the First Book of Euclid's Elements Vol. 1 (1788), Ch. IV.

Books , Knowledge , Problem , Study

“The mistakes and unresolved difficulties of the past in mathematics have always been the opportunities of its future;”

Eric Temple Bell (1883–1960) mathematician and science fiction author born in Scotland who lived in the United States for most of his li…

Source: The Development of Mathematics (1940), p. 283
Context: The mistakes and unresolved difficulties of the past in mathematics have always been the opportunities of its future; and should analysis ever appear to be without or blemish, its perfection might only be that of death.

Past , Future

“The branches of mathematics are as various as the sciences to which they belong, and each subject of physical enquiry has its appropriate mathematics.”

Benjamin Peirce Linear Associative Algebra

§ 2.
Linear Associative Algebra (1882)
Context: The branches of mathematics are as various as the sciences to which they belong, and each subject of physical enquiry has its appropriate mathematics. In every form of material manifestation, there is a corresponding form of human thought, so that the human mind is as wide in its range of thought as the physical universe in which it thinks.

Science

“They first began teaching their ugly scholastic pseudo-mathematics to their students, then to schoolchildren (forgetting Hardy's warning that ugly mathematics has no permanent place under the Sun).”

Vladimir I. Arnold (1937–2010) Russian mathematician

"On teaching mathematics", as translated by A. V. Goryunov, in Russian Mathematical Surveys Vol. 53, no. 1 (1998), p. 229–236.
Context: In the middle of the twentieth century it was attempted to divide physics and mathematics. The consequences turned out to be catastrophic. Whole generations of mathematicians grew up without knowing half of their science and, of course, in total ignorance of any other sciences. They first began teaching their ugly scholastic pseudo-mathematics to their students, then to schoolchildren (forgetting Hardy's warning that ugly mathematics has no permanent place under the Sun).

Sun , Forgetting

“Most persons of a deeply religious nature would tell you emphatically that nine churches out of ten actually were dead-born, after the thirteenth century, and that church architecture became a pure matter of mechanism or mathematics”

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

Mont Saint Michel and Chartres (1904)
Context: Every day, as the work went on, the Virgin was present, directing the architects, and it is this direction that we are going to study, if you have now got a realising sense of what it meant. Without this sense, the church is dead. Most persons of a deeply religious nature would tell you emphatically that nine churches out of ten actually were dead-born, after the thirteenth century, and that church architecture became a pure matter of mechanism or mathematics; but that is a question for you to decide when you come to it; and the pleasure consists not in seeing the death, but in feeling the life.

Nature , Personality , Dead

“Men are constantly attracted and deluded by two opposite charms: the charm of competence which is engendered by mathematics and everything akin to mathematics, and the charm of humble awe, which is engendered by meditation on the human soul and its experiences. Philosophy is characterized by the gentle, if firm, refusal to succumb to either charm.”

Leo Strauss (1899–1973) Classical philosophy specialist and father of neoconservativism

Source: What is Political Philosophy (1959), p. 40
Context: Men are constantly attracted and deluded by two opposite charms: the charm of competence which is engendered by mathematics and everything akin to mathematics, and the charm of humble awe, which is engendered by meditation on the human soul and its experiences. Philosophy is characterized by the gentle, if firm, refusal to succumb to either charm. It is the highest form of the mating of courage and moderation. In spite of its highness or nobility, it could appear as Sisyphean or ugly, when one contrasts its achievement with its goal. Yet it is necessarily accompanied, sustained and elevated by eros. It is graced by nature's grace.

Men , Soul

“This, therefore, is mathematics: she reminds you of the invisible form of the soul; she gives life to her own discoveries; she awakens the mind and purifies the intellect; she brings light to our intrinsic ideas; she abolishes oblivion and ignorance which are ours by birth.”

Proclus (412–485) Greek philosopher

As quoted by Morris Kline, Mathematical Thought from Ancient to Modern Times (1972)

Life , Soul , Idea , Birth

“Wisdom is not mathematical, nor astronomical, nor zoological; when it talks too much of any one thing it ceases to be itself.”

George Sarton (1884–1956) American historian of science

Preface.
A History of Science Vol.1 Ancient Science Through the Golden Age of Greece (1952)
Context: Wisdom is not mathematical, nor astronomical, nor zoological; when it talks too much of any one thing it ceases to be itself. There are wise physicists, but wisdom is not physical; there are wise physicians, but wisdom is not medical.

Wisdom

“A large part of mathematics which becomes useful developed with absolutely no desire to be useful, and in a situation where nobody could possibly know in what area it would become useful; and there were no general indications that it ever would be so.”

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

"The Role of Mathematics in the Sciences and in Society" (1954) an address to Princeton alumni, published in John von Neumann : Collected Works (1963) edited by A. H. Taub ; also quoted in Out of the Mouths of Mathematicians : A Quotation Book for Philomaths (1993) by R. Schmalz
Context: A large part of mathematics which becomes useful developed with absolutely no desire to be useful, and in a situation where nobody could possibly know in what area it would become useful; and there were no general indications that it ever would be so. By and large it is uniformly true in mathematics that there is a time lapse between a mathematical discovery and the moment when it is useful; and that this lapse of time can be anything from 30 to 100 years, in some cases even more; and that the whole system seems to function without any direction, without any reference to usefulness, and without any desire to do things which are useful.

Parting

“There is a logic of language and a logic of mathematics.”

Thomas Merton (1915–1968) Priest and author

The Secular Journal of Thomas Merton (1959)
Context: There is a logic of language and a logic of mathematics. The former is supple and lifelike, it follows our experience. The latter is abstract and rigid, more ideal. The latter is perfectly necessary, perfectly reliable: the former is only sometimes reliable and hardly ever systematic. But the logic of mathematics achieves necessity at the expense of living truth, it is less real than the other, although more certain. It achieves certainty by a flight from the concrete into abstraction. Doubtless, to an idealist, this would seem to be a more perfect reality. I am not an idealist. The logic of the poet — that is, the logic of language or the experience itself — develops the way a living organism grows: it spreads out towards what it loves, and is heliotropic, like a plant.

“I… present also examples of historic interest, examples of real mathematical beauty”

George Pólya (1887–1985) Hungarian mathematician

Induction and Analogy in Mathematics (1954)
Context: I... present also examples of historic interest, examples of real mathematical beauty, and examples illustrating the parallelism of the procedures in other sciences, or in everyday life.

Beauty

“There could not be a non-mathematical Universe containing living observers.”

John D. Barrow (1952–2020) British scientist

The Artful Universe (1995)
Context: Where there is life there is a pattern, and where there is a pattern there is mathematics. Once that germ of rationality and order exists to turn a chaos into a cosmos, then so does mathematics. There could not be a non-mathematical Universe containing living observers.<!-- Ch. 5, p. 230

Universe

“Supposing you knew — not by sight or by instinct, but by sheer intellectual knowledge, as I know the truth of a mathematical proposition — that what we call empty space was full, crammed.”

John Buchan (1875–1940) British politician

Space (1912)
Context: Supposing you knew — not by sight or by instinct, but by sheer intellectual knowledge, as I know the truth of a mathematical proposition — that what we call empty space was full, crammed. Not with lumps of what we call matter like hills and houses, but with things as real — as real to the mind.

Truth , Knowledge

“If perchance there should be foolish speakers who, together with those ignorant of all mathematics, will take it upon themselves to decide concerning these things, and because of some place in the Scriptures wickedly distorted to their purpose, should dare to assail this my work, they are of no importance to me, to such an extent do I despise their judgment as rash.”

Nicolaus Copernicus (1473–1543) Renaissance mathematician, Polish astronomer, physician

Translation as quoted in The Gradual Acceptance of the Copernican Theory of the Universe (1917) by Dorothy Stimson, p. 115
Context: If perchance there should be foolish speakers who, together with those ignorant of all mathematics, will take it upon themselves to decide concerning these things, and because of some place in the Scriptures wickedly distorted to their purpose, should dare to assail this my work, they are of no importance to me, to such an extent do I despise their judgment as rash. For it is not unknown that Lactantius, the writer celebrated in other ways but very little in mathematics, spoke somewhat childishly of the shape of the earth when he derided those who declared the earth had the shape of a ball. So it ought not to surprise students if such should laugh at us also. Mathematics is written for mathematicians to whom these our labors, if I am not mistaken, will appear to contribute something even to the ecclesiastical state the headship of which your Holiness now occupies. (Author's preface to de revolutionibus) http://la.wikisource.org/wiki/Pagina:Nicolai_Copernici_torinensis_De_revolutionibus_orbium_coelestium.djvu/8

“For the great majority of mathematicians, mathematics is”

George Frederick James Temple (1901–1992) British mathematician

100 Years of Mathematics: a Personal Viewpoint (1981)
Context: For the great majority of mathematicians, mathematics is... a whole world of invention and discovery—an art. The construction of a new theorem, the intuition of some new principle, or the creation of a new branch of mathematics is the triumph of the creative imagination of the mathematician, which can be compared to that of a poet, the painter and the sculptor.

“The league between virtue and nature engages all things to assume a hostile front to vice. The beautiful laws and substances of the world persecute and whip the traitor. He finds that things are arranged for truth and benefit, but there is no den in the wide world to hide a rogue. Commit a crime, and the earth is made of glass. Commit a crime, and it seems as if a coat of snow fell on the ground, such as reveals in the woods the track of every partridge and fox and squirrel and mole. You cannot recall the spoken word, you cannot wipe out the foot-track, you cannot draw up the ladder, so as to leave no inlet or clew. Some damning circumstance always transpires. The laws and substances of nature — water, snow, wind, gravitation — become penalties to the thief.
On the other hand, the law holds with equal sureness for all right action. Love, and you shall be loved. All love is mathematically just, as much as the two sides of an algebraic equation. The good man has absolute good, which like fire turns every thing to its own nature, so that you cannot do him any harm; but as the royal armies sent against Napoleon, when he approached, cast down their colors and from enemies became friends, so disasters of all kinds, as sickness, offence, poverty, prove benefactors: —
::"Winds blow and waters roll
Strength to the brave, and power and deity,
Yet in themselves are nothing."”

Ralph Waldo Emerson (1803–1882) American philosopher, essayist, and poet

The good are befriended even by weakness and defect. As no man had ever a point of pride that was not injurious to him, so no man had ever a defect that was not somewhere made useful to him.
1840s, Essays: First Series (1841), Compensation

Love , Hostility , Truth , World

“Does the mathematical method proceed from particular to the general, and, if so, how can it be called deductive? …If we refuse to admit these consequences, it must be conceded that mathematical reasoning has of itself a sort of creative virtue and consequently differs from a syllogism.”

Henri Poincaré book Science and Hypothesis

Source: Science and Hypothesis (1901), Ch. I: On the Nature of Mathematical Reasoning (1905) Tr. https://books.google.com/books?id=5nQSAAAAYAAJ George Bruce Halstead
Context: The very possibility of the science of mathematics seems an insoluble contradiction. If this science is deductive only in appearance, whence does it derive that perfect rigor no one dreams of doubting? If, on the contrary, all the propositions it enunciates can be deduced one from another by the rules of formal logic, why is not mathematics reduced to an immense tautology? The syllogism can teach us nothing essentially new, and, if everything is to spring from the principle of identity, everything should be capable of being reduced to it. Shall we then admit that the enunciations of all those theorems which fill so many volumes are nothing but devious ways of saying A is A!... Does the mathematical method proceed from particular to the general, and, if so, how can it be called deductive?... If we refuse to admit these consequences, it must be conceded that mathematical reasoning has of itself a sort of creative virtue and consequently differs from a syllogism.<!--pp.5-6

“Mathematics is written for mathematicians, to whom these my labours”

Nicolaus Copernicus book De revolutionibus orbium coelestium

Preface Letter to Pope Paul III as quoted by Edwin Arthur Burtt in The Metaphysical Foundations of Modern Physical Science (1925)
De revolutionibus orbium coelestium (1543)
Context: Nor do I doubt that skilled and scholarly mathematicians will agree with me if, what philosophy requires from the beginning, they will examine and judge, not casually but deeply, what I have gathered together in this book to prove these things.... Mathematics is written for mathematicians, to whom these my labours, if I am not mistaken, will appear to contribute something.... What... I may have achieved in this, I leave to the decision of your Holiness especially, and to all other learned mathematicians.... If perchance there should be foolish speakers who, together with those ignorant of all mathematics, will take it upon themselves to decide concerning these things, and because of some place in the Scriptures wickedly distorted to their purpose, should dare to assail this my work, they are of no importance to me, to such an extent do I despise their judgment as rash.

“The science of probability gives mathematical expression to our ignorance, not to our wisdom.”

Samuel R. Delany Time Considered as a Helix of Semi-Precious Stones

Time Considered as a Helix of Semi-Precious Stones (1968)
Context: If everything, everything were known, statistical estimates would be unnecessary. The science of probability gives mathematical expression to our ignorance, not to our wisdom.

Wisdom , Science

“If a 'religion' is defined to be a system of ideas that contains unprovable statements, then Gödel taught us that mathematics is not only a religion, it is the only religion that can prove itself to be one.”

John D. Barrow (1952–2020) British scientist

The Artful Universe (1995)
Context: If a 'religion' is defined to be a system of ideas that contains unprovable statements, then Gödel taught us that mathematics is not only a religion, it is the only religion that can prove itself to be one.<!-- Ch. 5, p. 211

Religion , Idea

“I must study politics and war, that our sons may have liberty to study mathematics and philosophy. Our sons ought to study mathematics and philosophy, geography, natural history and naval architecture, navigation, commerce and agriculture in order to give their children a right to study painting, poetry, music, architecture, statuary, tapestry and porcelain.”

John Quincy Adams (1767–1848) American politician, 6th president of the United States (in office from 1825 to 1829)

War , Children , Music , History

“The last two years have seen the emergence of a beautiful new subject in mathematical physics. It manages to combine a most exotic range of disciplines: two-dimensional quantum field theory, intersection theory on the moduli space of Riemann surfaces, integrable hierarchies, matrix integrals, random surfaces, and many more. The common denominator of all these fields is two-dimensional quantum gravity or, more general, low-dimensional string theory.”

Robbert Dijkgraaf (1960) Dutch mathematical physicist and string theorist

[1992, Intersection Theory, Integrable Hierarchies and Topological Field Theory by Robbert Dijkgraaf, Fröhlich J., ’t Hooft G., Jaffe A., Mack G., Mitter P.K., Stora R. (eds.), New Symmetry Principles in Quantum Field Theory, NATO ASI Series (Series B: Physics), vol. 295, 95–158, Springer, Boston, MA, 10.1007/978-1-4615-3472-3_4]

Beauty

“I always want to try to understand why things work. I’m not interested in getting a formula without knowing what it means. I always try to dig behind the scenes, so if I have a formula, I understand why it’s there. And understanding is a very difficult notion.People think mathematics begins when you write down a theorem followed by a proof. That’s not the beginning, that’s the end. For me the creative place in mathematics comes before you start to put things down on paper, before you try to write a formula. You picture various things, you turn them over in your mind. You’re trying to create, just as a musician is trying to create music, or a poet. There are no rules laid down. You have to do it your own way. But at the end, just as a composer has to put it down on paper, you have to write things down. But the most important stage is understanding. A proof by itself doesn’t give you understanding. You can have a long proof and no idea at the end of why it works. But to understand why it works, you have to have a kind of gut reaction to the thing. You’ve got to feel it.”

Michael Atiyah (1929–2019) British mathematician

On an article by Quanta magazine(when asked: Is there one big question that has always guided you?) https://www.quantamagazine.org/michael-atiyahs-mathematical-dreams-20160303

Understanding , Music , Thinking , Way

“The purpose of the mathematical theory of statistics is to deal with the relationship between 2 or more variable quantities without assuming that one is a single-valued mathematical function of the rest. The statistician does not think a certain x will produce a single-valued y; not a causative relation but a correlation. The relationship between x and y will be somewhere within a zone and we have to work out the probability that the point (x,y) will lie in different parts of that zone. The physicist is limited and shrinks the zone into a line. Our treatment will fit all the vagueness of biology, sociology, etc. A very wide science.”

Karl Pearson (1857–1936) English mathematician and biometrician

As quoted by E.S. Pearson, Karl Pearson: An Appreciation of Some Aspects of his Life and Work (1938) and cited in Bernard J. Norton, "Karl Pearson and Statistics: The Social Origins of Scientific Innovation" in Social Studies of Science, Vol. 8, No. 1, Theme Issue: Sociology of Mathematics (Feb.,1978), pp. 3-34.

Lie , Thinking , Relation ships , Science

“Heredity. Given any organ in a parent and the same or any other organ in its offspring, the mathematical measure of heredity is the correlation of these organs for pairs of parent and offspring... The word organ here must be taken to include any characteristic which can be quantitatively measured.”

Karl Pearson (1857–1936) English mathematician and biometrician

"Mathematical Contributions to the Theory of Evolution III: Regression, Heredity and Panmixia", Philosophical Transactions of the Royal Society, Series A, Vol. 187 (1896) p. 259.

Parent

“It remains a real world if there is a background to the symbols — an unknown quantity which the mathematical symbol x stands for. We think we are not wholly cut off from this background. It is to this background that our own personality and consciousness belong, and those spiritual aspects of our nature not to be described by any symbolism … to which mathematical physics has hitherto restricted itself.”

Arthur Stanley Eddington (1882–1944) British astrophysicist

Science and the Unseen World (1929)

Personality , Nature , Thinking , World

“If to-day you ask a physicist what he has finally made out the æther or the electron to be, the answer will not be a description in terms of billiard balls or fly-wheels or anything concrete; he will point instead to a number of symbols and a set of mathematical equations which they satisfy. What do the symbols stand for? The mysterious reply is given that physics is indifferent to that; it has no means of probing beneath the symbolism. To understand the phenomena of the physical world it is necessary to know the equations which the symbols obey but not the nature of that which is being symbolised. … this newer outlook has modified the challenge from the material to the spiritual world.”

Arthur Stanley Eddington (1882–1944) British astrophysicist

Science and the Unseen World (1929)

Understanding , Indifference , Nature , Flying

“He calls it the university of the Real One, and it teaches only things that are known to be true, which means it is largely devoted to mathematics and sciences.”

Sheri S. Tepper (1929–2016) American fiction writer

Elnith in Ch. 46 : nell latimer’s journal, p. 498
The Visitor (2002)

Universe , Science

“It can have no analogy in fact with the descriptions of those who know nothing of the subject; furthermore, it is not to be represented as this or that by any person whomsoever: it is that which it is, drawing from itself only, even as mathematics do, for it is the exact and absolute science of Nature and her laws.”

Eliphas Levi (1810–1875) French writer

Introduction
The History Of Magic (1860)

Drawing , Law , Personality , Nature

“The real problem in speech is not precise language. The problem is clear language. The desire is to have the idea clearly communicated to the other person. It is only necessary to be precise when there is some doubt as to the meaning of a phrase, and then the precision should be put in the place where the doubt exists. It is really quite impossible to say anything with absolute precision, unless that thing is so abstracted from the real world as to not represent any real thing.Pure mathematics is just such an abstraction from the real world, and pure mathematics does have a special precise language for dealing with its own special and technical subjects. But this precise language is not precise in any sense if you deal with real objects of the world, and it is only pedantic and quite confusing to use it unless there are some special subtleties which have to be carefully distinguished.”

Richard Feynman (1918–1988) American theoretical physicist

" New Textbooks for the "New" Mathematics http://calteches.library.caltech.edu/2362/1/feynman.pdf", Engineering and Science volume 28, number 6 (March 1965) p. 9-15 at p. 14
Paraphrased as "Precise language is not the problem. Clear language is the problem."

Personality , Communication , Problem , World

“To an increasing number of practitioners, computer simulations rooted in mathematics represent a third way of doing science, alongside theory and experiment.”

Ivars Peterson (1948) Canadian mathematician

Source: The Mathematical Tourist: New and Updated Snapshots of Modern Mathematics (1998), Chapter 1, “Explorations” (p. 10)

Way , Science

“Indeed, mathematics is full of conjectures—questions waiting for answers—with no assurance that the answers even exist.”

Ivars Peterson (1948) Canadian mathematician

Source: The Jungles of Randomness: A Mathematical Safari (1997), Chapter 10, “Lifetimes of Chance” (p. 199)

Waiting , Question

“In mathematics, in science, and in life, we constantly face the delicate, tricky task of separating design from happenstance.”

Ivars Peterson (1948) Canadian mathematician

Source: The Jungles of Randomness: A Mathematical Safari (1997), Chapter 2, “Sea of Life” (p. 43)

Science , Life

“Intriguingly, the mathematics of randomness, chaos, and order also furnishes what may be a vital escape from absolute certainty—an opportunity to exercise free will in a deterministic universe. Indeed, in the interplay of order and disorder that makes life interesting, we appear perpetually poised in a state of enticingly precarious perplexity. The universe is neither so crazy that we can’t understand it at all nor so predictable that there’s nothing left for us to discover.”

Ivars Peterson (1948) Canadian mathematician

Preface, “Infinite Possibility” (p. xiii)
The Jungles of Randomness: A Mathematical Safari (1997)

Understanding , Universe , Appearance , Life

“It is a safe rule to apply that, when a mathematical or philosophical author writes with a misty profundity, he is talking nonsense.”

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

Source: 1910s, An Introduction to Mathematics (1911), ch. 15.

“While its true that masculinity must be forced and fostered, this is also true of any human potentiality. One must be forced, or force oneself to learn a language or play an instrument or solve mathematical equations. No one calls an accomplished dancer, painter, athlete or singer a phony because it took years of disciplined practice and some kind of nurturing environment to become what they are- for them to develop their talents to their full potential. On the contrary, to ignore these talents is considered a tragedy.”

Jack Donovan (1974) American activist, editor and writer

pg 31
A More Complete Beast (2018)

Learning

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

Frank Wilczek (1951) physicist

Kepler... in the third book of Harmonice mundi... attempted to make other... related, connections between musical harmony and mathematical proportion.
Longing for the Harmonies: Themes and Variations from Modern Physics (1987)

Music , Sun , Time

“The reduction of a society and culture to dependence on mathematical abstraction has infantilised a grown-up civilisation and is well on the way to destroying it. Civilisations self-destruct anyway, but it is reasonable to ask whether they have done so before with such enthusiasm, in obedience to such an acutely absurd superstition, while claiming with such insistence that they were beyond being seduced by the irrational promises of religion.”

David Fleming (1940–2010) British activist

Surviving the Future, (2016), p. 180, Epilogue http://www.flemingpolicycentre.org.uk/lean-logic-surviving-the-future/

Religion , Dependency , Self-confidence , Way

“I know all about Thor and Balder and Mjolnir, the hammer. Nobody ever bothered with that stuff except me. I loved it in high school and I loved it in my pre-high school days. It was the thing that kept my mind off the general poverty in the area. When I went to school that’s what kept me in school — it wasn’t mathematics and it wasn’t geography; it was history.”

Jack Kirby (1917–1994) American comic book artist, writer and editor

"Jack Kirby Interview" http://www.tcj.com/jack-kirby-interview/6/, Gary Groth, The Comics Journal, #134, (February 1990, posted May 23, 2011).

School , History , Love , Making love

“The Other Place, the astral plane, the land of dreams—it’s not a real place like, say, Walsall. But it’s a metaphor for a mathematical abstraction, a manifold containing an n-dimensional space where everything is the product of geometrical transformations, including mass and energy and time. Leakage between dimensions occurs there: it’s how we summon demons from the vasty deep, communicate with aliens, and try to extract our tax codes from the Inland Revenue.”

Charles Stross The Laundry Files

Source: The Laundry Files, The Apocalypse Codex (2012), Chapter 14, “Appointment in Samarra” (p. 283)

Dreams , Time , Communication

“Who, that has pity, is disposed to censure a child for rebelling against the useless and absurd rumination, thru painful years, of mummified languages and fearful mathematical formulae, which have no more real bearing, and which to the average human being never will have any more real bearing, on the great, living, performing universe around him than the esoteric nonsense of the Five Kings?”

J. Howard Moore (1862–1916)

Source: Better-World Philosophy: A Sociological Synthesis (1899), Individual Culture, p. 266

Universe , Fear , Pain

“Pure mathematics is, in its way, the poetry of logical ideas. One seeks the most general ideas of operation which will bring together in simple, logical and unified form the largest possible circle of formal relationships. In this effort toward logical beauty spiritual formulas are discovered necessary for the deeper penetration into the laws of nature.”

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

1930s, Obituary for Emmy Noether (1935)

Law , Beauty , Nature , Way

“It’s about learning to listen, much like in music. You can train your ears to history. You can train your ears to the earth. You can train your ears to the wind. It’s important to listen and then to study the world, like astronomy or geology or the names of birds. A lot of poets can be semihistorians. Poetry is very mathematical. There’s a lot in the theoretical parts that is similar. Quantum physicists remind me of mystics. They are aware of what happens in timelessness, though they speak of it through theories and equations.”

Joy Harjo (1951) American writer

On her advice to poets in “The First Native American U.S. Poet Laureate on How Poetry Can Counter Hate” https://time.com/5658443/joy-harjo-poet-interview/ in Time Magazine (2019 Aug 22)

Study , History , Learning , Music

“Newton… (after having remarked that geometry only requires two of the mechanical actions which it postulates, namely, to describe a straight line and a circle) says: geometry is proud of being able to achieve so much while taking so little from extraneous sources. One might say of metaphysics, on the other hand: it stands astonished, that with so much offered it by pure mathematics it can effect so little.”

Immanuel Kant book Metaphysical Foundations of Natural Science

In the meantime, this little is something which mathematics indispensably requires in its application to natural science, which, inasmuch as it must here necessarily borrow from metaphysics, need not be ashamed to allow itself to be seen in company with the latter.
Preface, Tr. Bax (1883) citing Isaac Newton's Principia
Metaphysical Foundations of Natural Science (1786)

“Phenomena of the external sense are examined and set forth in physics; those of the internal sense in empirical psychology. Pure mathematics considers space in geometry and time in pure mechanics. To these is to be added a certain concept, intellectual to be sure in itself, but whose becoming actual in the concrete requires the auxiliary notions of time and space in the successive addition and simultaneous juxtaposition of separate units, which is the concept of number treated in arithmetic.”

Immanuel Kant (1724–1804) German philosopher

Kant's Inaugural Dissertation (1770), Section II On The Distinction Between The Sensible And The Intelligible Generally

Sense , Time , Success

“He thought of a way to state it: Ego is the point of consciousness, the latest term in a continuously expanding series along the line of memory duration. That sounded like a general statement, but he was not sure; he would have to try to formulate it mathematically before he could trust it. Verbal language had such queer booby traps in it.”

Robert A. Heinlein book By His Bootstraps

By His Bootstraps (p. 257)
Short fiction, Off the Main Sequence (2005)

Way , Memory

“He moved to Egypt and supported himself by teaching and by copying Arabic translations of Greek mathematical classics such as Euclid’s Elements and Ptolemy’s Almagest.”

Alhazen (965–1038) Arab physicist, mathematician and astronomer

Abdelhamid I. Sabra, in “Ibn al-Haytham Brief life of an Arab mathematician: died circa 1040 (September-October 2003)”

Support

“I have often been surprised that Mathematics, the quintessence of Truth, should have found admirers so few and so languid. Frequent consideration and minute scrutiny have at length unravelled the cause: viz.”

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

that though Reason is feasted, Imagination is starved; whilst Reason is luxuriating in its proper Paradise, Imagination is wearily travelling on a dreary desert.
Letter to his brother (1791)
Letters

Truth

“The answer is in the problem, not away from the problem. I go through the searching, analysing, dissecting process, in order to escape from the problem. But, if I do not escape from the problem and try to look at the problem without any fear or anxiety, if I merely look at the problem — mathematical, political, religious, or any other — and not look to an answer, then the problem will begin to tell me. Surely, this is what happens. We go through this process and eventually throw it aside because there is no way out of it. So, why can’t we start right from the beginning, that is, not seek an answer to a problem?”

Jiddu Krishnamurti (1895–1986) Indian spiritual philosopher

which is extremely arduous, isn’t it? Because, the more I understand the problem, the more significance there is in it. To understand, I must approach it quietly, not impose on the problem my ideas, my feelings of like and dislike. Then the problem will reveal its significance. Why is it not possible to have tranquillity of the mind right from the beginning?
"Eighth Talk in The Oak Grove, 7 August 1949" http://www.jkrishnamurti.org/krishnamurti-teachings/view-text.php?tid=320&chid=4643&w=%22The+answer+is+in+the+problem%2C+not+away+from+the+problem%22, J.Krishnamurti Online, JKO Serial No. 490807, Vol. V, p. 283
Posthumous publications, The Collected Works

Search , Problem , Way , Fear

“A grievous crime indeed against religion has been committed by the man who imagines that Islam is defended by the denial of the mathematical sciences.”

Abu Hamid al-Ghazali (1058–1111) Persian Muslim theologian, jurist, philosopher, and mystic

The Deliverance from Error https://www.amazon.com/Al-Ghazalis-Path-Sufism-Deliverance-al-Munqidh/dp/1887752307, p: 34

Imagination , Science , Religion

“Formal proofs, where there is deliberately no meaning, can convince only formalists, and of the results they themselves seem to deny any meaning. Is that to be the mathematics we are to use to understand the world we live in?”

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

Source: Mathematics on a Distant Planet (1998), p. 645

World , Understanding

“It is easy to measure your mastery of the results via a conventional examination; it is less easy to measure your mastery of doing mathematics, of creating new (to you) results, and of your ability to surmount the almost infinite details to see the general situation.”

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

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

“If we compare. e. g. the systems of classical mathematics and of intuitionistic mathematics, we find that the first is much simpler and technically more efficient, while the second is more safe from surprising occurences, e. g. contradictions. At the present time, any estimation of the degree of safety of the system of classical mathematics, in other words, the degree of plausibility of its principles, is rather subjective. The majority of mathematicians seem to regard this degree as sufficiently high for all practical purposes and therefore prefer the application of classical mathematics to that of intuitionistic mathematics. The latter has not, so far as I know, been seriously applied in physics by anybody.”

Rudolf Carnap (1891–1970) German philosopher

Rudolf Carnap (1939; 51), as cited in: Paul van Ulsen. Wetenschapsfilosofie http://www.illc.uva.nl/Research/Publications/Inaugurals/IV-10-Arend-Heyting.text.pdf, 6 november 2017.

Time

“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.”

Paul Karl Feyerabend book Against Method

This material is always contaminated by principles which he does not know and which, if known, would be extremely hard to test.
Pg. 66.
Against Method (1975)

Way , Law

“The shortcoming thus acknowledged to attach to the content turns out at the same time to be a shortcoming in respect of form. Spinoza puts substance at the head of his system, and defines it to be the unity of thought and extension, without demonstrating how he gets to this distinction, or how he traces it back to the unity of substance. The further treatment of the subject proceeds in what is called the mathematical method. Definitions and axioms are first laid down: after them comes a series of theorems, which are proved by an analytical reduction of them to these unproved postulates. Although the system of Spinoza, and that even by those who altogether reject its contents and results, is praised for the strict sequence of its method, such unqualified praise of the form is as little justified as an unqualified rejection of the content. The defect of the content is that the form is not known as immanent in it, and therefore only approaches it as an outer and subjective form. As intuitively accepted by Spinoza without a previous mediation by dialectic, Substance, as the universal negative power, is as it were a dark shapeless abyss which engulfs all definite content as radically null, and produces from itself nothing that has a positive subsistence of its own.”

Baruch Spinoza (1632–1677) Dutch philosopher

Georg Wilhelm Friedrich Hegel, Encyclopedia of Philosophical Sciences: The Logic
G - L, Georg Wilhelm Friedrich Hegel

Universe , Time

“Eratosthenes of Cyrene, employing mathematical theories and geometrical methods, discovered from the course of the sun the shadows cast by an equinoctial gnomon, and the inclination of the heaven that the circumference of the earth is two hundred and fifty-two thousand stadia, that is, thirty-one million five hundred thousand paces.”

Eratosthenes (-276–-194 BC) ancient Greek scientist

Vitruvius, De Architectura, Book 1, Chap 6, Sec. 9; as translated in Morris Hicky Morgan (trans.), Vitruvius: The Ten Books on Architecture (1914), 27-28.
About

Sun

“Dharampal, the noted Gandhian, used British data during the colonial period to show that in the ninetheenth century, the shudras comprised a larger student body than any other community did. … Besides the large number of schools at that time, there were also approximately a hundred institutions of higher learning in each district of Bengal and Bihar. Unfortunately, these numbers rapidly dwindled all across India during the nineteenth century under British rule. The British also noted that Sanskrit books were being widely used to teach grammar, lexicology, mathematics, medical science, logic, law and philosophy. …. Furthermore, in the early British period in India, British officials noted that education for the masses was more advanced and widespread in India than it was in England. …. According to Dharampal, the British later replaced this Sanskrit-based system with their own English-based one, the goal being to produce low-level clerks for the British administration.”

Dharampal (1922–2006) Indian historian

Rajiv Malhotra, The Battle for Sanskrit

School , Time , Law , Learning

“As a mathematical object, the constitution is maximally simple, consistent, necessarily incomplete, and interpretable as a model of natural law. Political authority is allocated solely to serve the constitution.”

Nick Land (1962) British philosopher

There are no authorities which are not overseen, within nonlinear structures. Constitutional language is formally constructed to eliminate all ambiguity and to be processed algorithmically. Democratic elements, along with official discretion, and legal judgment, is incorporated reluctantly, minimized in principle, and gradually eliminated through incremental formal improvement. Argument defers to mathematical expertise. Politics is a disease that the constitution is designed to cure.
"A Republic, If You Can Keep It" https://web.archive.org/web/20140327090001/http://www.thatsmags.com/shanghai/articles/12321 (2013) (original emphasis)

Law , Nature

“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.”

Tobias Dantzig (1884–1956) American mathematician

...the children had to live, so while waiting for logic to sanctify their existence, they throve and multiplied.
Number: The Language of Science (1930)

Birth , Parting

“It is on this step that depends the fact that one can call upon the subject to re-enter himself in the unconscious—for, after all, it is important to know who one is calling. It is not the soul, either mortal or immortal, which has been with us for so long, nor some shade, some double, some phantom, nor even some supposed psycho-spherical shell, the locus of the defences and other such simplified notions. It is the subject who is called— there is only he, therefore, who can be chosen. There may be, as in the parable, many called and few chosen, but there will certainly not be any others except those who are called. In order to understand the Freudian concepts, one must set out on the basis that it is the subject who is called—the subject of Cartesian origin. This basis gives its true function to what, in analysis, is called recollection or remembering. Recollection is not Platonic reminiscence —it is not the return of a form, an imprint, a eidos of beauty and good, a supreme truth, coming to us from the beyond. It is something that comes to us from the structural necessities, something humble, born at the level of the lowest encounters and of all the talking crowd that precedes us, at the level of the structure of the signifier, of the languages spoken in a stuttering, stumbling way, but which cannot elude constraints whose echoes, model, style can be found, curiously enough, in contemporary mathematics.”

Jacques Lacan (1901–1981) French psychoanalyst and psychiatrist

Of the Network of Signifiers
The Four Fundamental Concepts of Psycho Analysis (1978)

Soul , Understanding , Dependency , Beauty

“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.”

Frank Knight (1885–1972) American economist

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

Law , Universe , Way , Science

“As you can well imagine, any nuclear bombing study that neglected to target Moscow would be laughed out of the room. (That is, no study at that time; 10 or 15 years later senior policy officials were debating how good an idea this might be. If you wiped out the political leadership of the Soviet Union in the process, who would you deal with in arranging for a truce and who would be left to run the country after the war?) Consequently, two of RAND’s brightest mathematicians were assigned the task of determining, with the help of computers, in great detail, precisely what would happen to the city were a bomb of so many megatons dropped on it. It was truly a daunting task and called for devising a mathematical model unimaginably complex; one that would deal with the exact population distribution, the precise location of various industries and government agencies, the vulnerability of all the important structures to the bomb’s effects, etc., etc. However, these two guys were up to the task and toiled in the vineyards for some months, finally coming up with the results. Naturally, they were horrendous.”

Samuel T. Cohen (1921–2010) American physicist

Harold Mitchell, a medical doctor, an expert on human vulnerability to the H-bomb’s effects, told me when the study first began: “Why are they wasting their time going through all this shit? You know goddamned well that a bomb this big is going to blow the fucking city into the next county. What more do you have to know?” I had to agree with him.
F*** You! Mr. President: Confessions of the Father of the Neutron Bomb (2006)

Study , Time , Imagination , Country

“Swati Tirunal, now thirteen… took up a book of mathematics and selecting the forty seventh proposition of Euclid sketched the figure on a country slate but what astonished me most was his telling us in English that Geometry was derived from the Sanskrit, which as Jaw metor (Jyamiti) to measure the earth and that many of our mathematical terms were also derived from the same source such as hexagon, heptagon, octagon… This promising boy is now, I conclude, sovereign of the finest country in India for he was to succeed to the Musnud (throne) the moment he had attained his 16th year.”

Swathi Thirunal Rama Varma (1813–1846) Maharajah of Travencore

Colonel Welsh, in "The Monarch musician"
About Swathi Thirunal

Books , Boy , Country

“P. Bernays has pointed out on several occasions that, since the consistency of a system cannot be proved using means of proof weaker than those of the system itself, it is necessary to go beyond the framework of what is, in Hilbert’s sense, finitary mathematics if one wants to prove the consistency of classical mathematics, or even that of classical number theory. Consequently, since finitary mathematics is defined as the mathematics in which evidence rests on what is intuitive, certain abstract notions are required for the proof of the consistency of number theory…. In the absence of a precise notion of what it means to be evident, either in the intuitive or in the abstract realm, we have no strict proof of Bernays’ assertion; practically speaking, however, there can be no doubt that it is correct…”

Paul Bernays (1888–1977) Swiss mathematician

Kurt Gödel (1958, CW II, p. 241) as cited in: Feferman, Solomon. " Lieber Herr Bernays!, Lieber Herr Gödel! Gödel on finitism, constructivity and Hilbert's program* http://math.stanford.edu/~feferman/papers/bernays.pdf." dialectica 62.2 (2008): 179-203.

Sense

“Physics by its very nature requires extreme specialization on the part of its students. Its conclusions, which must eventually predict numbers for the results of actual measurements, are best expressed in mathematical formulae. This has the disadvantage of making the subject well-nigh unintelligible to the layman. There are unfortunately few teachers who are able to surmount this handicap. Professor Raman has written a book which avoids this pitfall and thus should give the lay reader an opportunity of penetrating at least part of the way into the mysteries of this interesting and important science.”

C. V. Raman (1888–1970) Indian physicist

Francis Low, a distinguished theoretical physicist then working at the Institute for Advanced Studies, Princeton, wrote in the introduction to this book quoted in Chandrasekhara Venkata Raman:A Legend of Modern Indian Science, 22 November 2013, Official Government of India's website Vigyan Prasar http://www.vigyanprasar.gov.in/scientists/cvraman/raman1.htm,

Nature , Teachers , Books , Way

“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.”

Hans Freudenthal (1905–1990) Dutch mathematician

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

Beauty , Problem , Way , Idea

“Among the various paradigmatic changes in science and mathematics in this century, one such change concerns the concept of uncertainty.”

George Klir (1932–2016) American computer scientist

In science, this change has been manifested by a gradual transition from the traditional view, which insists that uncertainty is undesirable in science and should be avoided by all possible means, to an alternative view, which is tolerant of uncertainty and insists that science cannot avoid it. According to the traditional view, science should strive for certainty in all its manifestations (precision, specificity, sharpness, consistency, etc.); hence, uncertainty (imprecision, nonspecificity, vagueness, inconsistency,etc.) is regarded as unscientific. According to the alternative (or modem) view, uncertainty is considered essential to science; it is not only an unavoidable plague, but it has, in fact, a great utility.
Source: Fuzzy sets and fuzzy logic (1995), p. 1.

Science , Change

“I shall here present the view that numbers, even whole numbers, are words, parts of speech, and that mathematics is their grammar.”

Carl Eckart (1902–1973) American physicist

Numbers were therefore invented by people in the same sense that language, both written and spoken, was invented. Grammar is also an invention. Words and numbers have no existence separate from the people who use them. Knowledge of mathematics is transmitted from one generation to another, and it changes in the same slow way that language changes. Continuity is provided by the process of oral or written transmission.
Source: Our Modern Idol: Mathematical Science (1984), p. 95.

Parting

“If we could be any mathematician in the history of the world (besides ourselves), who would we rather be? …we narrowed the choice down to Euler and Pólya, and finally settled on George Pólya because of the sheer enjoyment of mathematics that he has conveyed by so many examples.”

George Pólya (1887–1985) Hungarian mathematician

Donald E. Knuth, comments at Pólya's 90th birthday celebration quoted by Gerald L. Alexanderson, The Random Walks of George Polya (2000)

World , History

“The picture of the economic revolution as the final step to freedom was false as soon as I asked myself that question. For, in actual fact, The State, The Government, cannot exist. They are abstract concepts, useful enough in their place, as the theory of minus numbers is useful in mathematics. In actual living experience, however, it is impossible to subtract anything from nothing; when a purse is empty, it is empty, it cannot contain a minus ten dollars. On this same plane of actuality, no State, no Government, exists. What does in fact exist is a man, or a few men, in power over many men.”

Rose Wilder Lane (1886–1968) American journalist

Give Me Liberty (1936)

Question , Freedom , Falseness , Men

“A great many individuals ever since the rise of the mathematical method, have, each for himself, attacked its direct and indirect consequences. …I shall call each of these persons a paradoxer, and his system a paradox.”

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

I use the word in the old sense: ...something which is apart from general opinion, either in subject-matter, method, or conclusion. ...Thus in the sixteenth century many spoke of the earth's motion as the paradox of Copernicus, who held the ingenuity of that theory in very high esteem, and some, I think, who even inclined towards it. In the seventeenth century, the depravation of meaning took place... Phillips says paradox is "a thing which seemeth strange"—here is the old meaning...—"and absurd, and is contrary to common opinion," which is an addition due to his own time.
A Budget of Paradoxes (1872)

Personality

“He left in his Optics, the earliest surviving table of angles of refraction from air to water. …This table, quoted and requoted until modern times, has been admired… A closer glance at it, however, suggests that there was less experimentation involved in it than originally was thought, for the values of the angles of refraction form an arithmetic progression of second order… As in other portions of Greek Science, confidence in mathematics was here greater than that in the evidence of the senses, although the value corresponding to 60° agrees remarkably well with experience.”

Ptolemy (100–170) Greco-Egyptian writer and astronomer of Alexandria

Carl B. Boyer, in The Rainbow: From Myth to Mathematics (1959)

Confidence , Time , Self-confidence , Water

“We are… bound to attach the greatest importance to the preliminary step taken by Lavoisier, who is even more justly called the father of modern chemistry than Kepler is called the father of modern astronomy. The exact claims of Lavoisier to this important place in the history of chemistry have been variously stated: …since his time, and greatly through his labours, the quantitative method has been established as the ultimate test of chemical facts; the principle of this method being the rule that in all changes of combination and reaction, the total weight of the various ingredients—be they elementary bodies or compounds—remains unchanged. The science of chemistry was thus established upon an exact, a mathematical basis. By means of this method Lavoisier, utilising and analysing the results gained by himself and others before him, notably those of Priestley, Cavendish, and Black, succeeded in destroying the older theory of combustion, the so-called phlogistic theory.”

Antoine Lavoisier (1743–1794) French chemist

John Theodore Merz, A History of European Thought in the Nineteenth Century Vol.1 http://books.google.com/books?id=xqwQAAAAYAAJ (1903)

Time , History , Science , Change

“The reason is not to glorify "bit chasing"; a more fundamental issue is at stake here: Numerical subroutines should deliver results that satisfy simple, useful mathematical laws whenever possible.”

Donald Ervin Knuth book The Art of Computer Programming

[...] Without any underlying symmetry properties, the job of proving interesting results becomes extremely unpleasant. The enjoyment of one's tools is an essential ingredient of successful work.
Vol. II, Seminumerical Algorithms, Section 4.2.2 part A, final paragraph [Italics in source]
The Art of Computer Programming (1968–2011)

Law

“The laws of nature are but the mathematical thoughts of God.”

Johannes Kepler (1571–1630) German mathematician, astronomer and astrologer

Attributed to Kepler in some sources (though more recent sources often attribute it to Euclid), such as Mathematically Speaking: A Dictionary of Quotations edited by Carl C. Gaither and Alma E. Cavazos-Gaither (1998), p. 214 http://books.google.com/books?id=4abygoxLdwQC&lpg=PP1&pg=PA214#v=onepage&q&f=false. The earliest publication located that attributes the quote to Kepler is the piece "The Mathematics of Elementary Chemistry" by Principal J. McIntosh of Fowler Union High School in California, which appeared in School Science and Mathematics, Volume VII ( 1907 http://books.google.com/books?id=kAEUAAAAIAAJ&pg=PR3#v=onepage&q&f=false), p. 383 http://books.google.com/books?id=kAEUAAAAIAAJ&pg=PA383#v=onepage&q&f=false. Neither this nor any other source located gives a source in Kepler's writings, however, and in an earlier source, the 1888 Notes and Queries, Vol V., it is attributed on p. 165 http://books.google.com/books?id=0qYXAQAAMAAJ&pg=PA165#v=onepage&q&f=false to Plato. Expressions that relate geometry to the divine "mind of God" include comments in the Harmonices Mundi, e.g., "Geometry is one and eternal shining in the mind of God", and "Since geometry is co-eternal with the divine mind before the birth of things, God himself served as his own model in creating the world".
Disputed quotes

God , Law , Nature

“Science has always tried to understand nature by breaking things into their parts. Now it is overwhelmingly clear that wholes cannot he understood by analysis. This is one of those logical boomerangs, like the mathematical proof that no mathematical system can be truly coherent in itself.”

Marilyn Ferguson (1938–2008) American writer

The Aquarian Conspiracy (1980), Chapter Six, Liberating Knowledge: News from the Frontiers of Science

Understanding , Nature , Science , Parting

“[M]athematics is not a popular subject... The reason for this is to be found in the common superstition that [it] is but a continuation... of the fine art of arithmetic, of juggling with numbers. [We] combat that superstition, by offering, instead of formulas, figures that may be looked at and that may easily be supplemented by models which the reader may construct. This book... bring[s] about a greater enjoyment of mathematics, by making it easier... to penetrate the essence of mathematics without... a laborious course of studies.”

David Hilbert Geometry and the Imagination

Preface
Geometry and the Imagination (1952)

Study , Art , Books , Making love

“[O]ur purpose is to give a presentation of geometry... in its visual, intuitive aspects. With the aid of visual imagination we can illuminate the manifold facts and problems... beyond this, it is possible... to depict the geometric outline of the methods of investigation and proof, without... entering into the details... In this manner, geometry being as many-faceted as it is and being related to the most diverse branches of mathematics, we may even obtain a summarizing survey of mathematics as a whole, and a valid idea of the variety of problems and the wealth of ideas it contains. Thus a presentation of geometry in large brushstrokes... and based on the approach through visual intuition, should contribute to a more just appreciation of mathematics by a wider range of people than just the specialists.”

David Hilbert Geometry and the Imagination

Preface
Geometry and the Imagination (1952)

Imagination , Wealth , People , Problem

“In mathematics, as in any scientific research, we find two tendencies... [T]he tendency toward abstraction seeks to crystallize the logical relations inherent in the maze of material in a systematic and orderly manner. On the other hand, the tendency toward intuitive understanding fosters a more immediate grasp of the objects... a live rapport with them... which stresses the concrete meaning of their relations. ...[I]ntuitive understanding plays a major role in geometry. ...[S]uch concrete intuition is of great value not only for the research worker, but... for anyone who wishes to study and appreciate the results of research in geometry.”

David Hilbert Geometry and the Imagination

Preface
Geometry and the Imagination (1952)

Relation ships , Study , Understanding

“The theory of the nature of mathematics is extremely reactionary. We do not subscribe to the fairly recent notion that mathematics is an abstract language based, say, on set theory. In many ways, it is unfortunate that philosophers and mathematicians like Russell and Hilbert were able to tell such a convincing story about the meaning-free formalism of mathematics. In Greek, mathematics simply meant learning, and we have adapted this... to define the term as "learing to decide."”

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

Mathematics is a way of preparing for decisions through thinking. Sets and classes provide one way to subdivide a problem for decision preparation; a set derives its meaning from decision making, and not vice versa.

C. West Churchman, Leonard Auerbach, Simcha Sadan, Thinking for Decisions: Deductive Quantitative Methods (1975) Preface.
1960s - 1970s

Way , Learning , Nature

“The universe acts on us, we adapt to it, and the notions that we develop as a result, including the mathematical ones, are in a sense taught us by the universe. Evolution has selected those of our ancestors (both human and not) whose behavior and thought were consistent with the workings of the universe.”

John Allen Paulos (1945) American mathematician

Part 3 “Four Psycho-Mathematical Arguments”, Chapter 4 “The Universality Argument (and the Relevance of Morality and Mathematics)” (p. 131)
Irreligion: A Mathematician Explains Why the Arguments for God Just Don’t Add Up (2008)

Sense , Behavior , Universe

“It’s time to let the secret out: mathematics is not primarily a matter of plugging numbers into formulas and performing rote computations. It is a way of thinking and questioning that may be unfamiliar to many of us, but is available to almost all of us.”

John Allen Paulos book A Mathematician Reads the Newspaper

Introduction (p. 3)
A Mathematician Reads the Newspaper (1995)

Time , Question , Thinking , Secret

“I think a lot of people may take some heart from this: a lot of the math, the heavy math in projective geometry, [...] took me a long time. There were many many years, a decade, when I was considered this graphics guru genius, when I really couldn't do from scratch [...] the mathematics that underpins a lot of that. But slowly, eventually, with a couple decades of experience, most of it did eventually sink in on me.”

John D. Carmack (1970) American computer programmer, engineer, and businessman

Speaking about mathematics in engineering, Quoted in https://www.youtube.com/watch?v=dSCBCk4xVa0&t=1271s

Time , People , Heart , Thinking

“In mathematics we agree that clear thinking is very important, but fuzzy thinking is just as important.”

Harish-Chandra (1923–1983) Indian American mathematician and physicist (1923–1983)

Harish-Chandra, cited in: Robert Langlands, "Harish-Chandra. 11 October 1923-16 October 1983." http://rsbm.royalsocietypublishing.org/content/31/198, in: Biographical Memoirs of Fellows of the Royal Society of London, 31 (1985), 199-225. (quote from p. 211).

Thinking