Quotes about arithmetic

A collection of quotes on the topic of arithmetic, number, mathematics, geometry.

Quotes about arithmetic

“I obtained scientific guidance and stimulation mainly through personal mathematical contacts in Erlangen and in Göttingen. Above all I am indebted to Mr. E. Fischer from whom I received the decisive impulse to study abstract algebra from an arithmetical viewpoint, and this remained the governing idea for all my later work.”

Emmy Noether (1882–1935) German mathematician

Mathematics, Idea, Decision, Study

“They learn to speak, write, and do arithmetic. They have a phenomenal memory. If one read them the Encyclopedia Britannica they could repeat everything back in order, but they never think up anything original. They'd make fine university professors.”

Karel Čapek R.U.R.

R.U.R. (1920)

Thinking, Learning, Universe, Memory

“An ancient writer said that arithmetic and geometry are the wings of mathematics; I believe one can say without speaking metaphorically that these two sciences are the foundation and essence of all the sciences which deal with quantity. Not only are they the foundation, they are also, as it were, the capstones; for, whenever a result has been arrived at, in order to use that result, it is necessary to translate it into numbers or into lines; to translate it into numbers requires the aid of arithmetic, to translate it into lines necessitates the use of geometry.”

Joseph Louis Lagrange (1736–1813) Italian mathematician and mathematical physicist

Dans Les Leçons Élémentaires sur les Mathématiques (1795) Leçon cinquiéme, Tr. McCormack, cited in Moritz, Memorabilia mathematica or, The philomath's quotation-book (1914) Ch. 15 Arithmetic, p. 261. https://archive.org/stream/memorabiliamathe00moriiala#page/260/mode/2up

Mathematics, Science

“Reeling and Writhing of course, to begin with,' the Mock Turtle replied, 'and the different branches of arithmetic-ambition, distraction, uglification, and derision.”

Lewis Carroll (1832–1898) English writer, logician, Anglican deacon and photographer

Source: Alice In Wonderland: Including Alice's Adventures In Wonderland And Through The Looking Glass

“There are millions of chords. There are millions of numbers. And everyone forgets the one that is a zero. But without the zero, numbers are just arithmetic. Without the empty chord, music is just noise.”

Terry Pratchett book Soul Music

Source: Soul Music

Music, Forgetting

“All the geography, trigonometry, and arithmetic in the world are useless unless you learn to think for yourself. No school teaches you that. It's not on the curriculum.”

Carlos Ruiz Zafón book Marina

Source: Marina

World, School, Thinking, Learning

“The hardest arithmetic to master is that which enables us to count our blessings.”

Eric Hoffer (1898–1983) American philosopher

Section 172
Reflections on the Human Condition (1973)

Teachers' Day

“Arithmetic must be discovered in just the same sense in which Columbus discovered the West Indies, and we no more create numbers than he created the Indians.”

Bertrand Russell (1872–1970) logician, one of the first analytic philosophers and political activist

Principles of Mathematics (1903), p. 451
1900s

Sense

“I was born undisciplined. Never, even as a child, could I be made to obey a set rule. What little I know I learned at home. School was always like a prison to me, I could never bring myself to stay there, even four hours a day, when the sun was shining and the sea was so tempting, and it was such fun scrambling over cliffs and paddling in the shallows. Such, to the great despair of my parents, was the unruly but healthy life I lived until I was fourteen or fifteen. In the meantime I somehow picked up the rudiments of reading, writing and arithmetic, with a smattering of spelling. And there my schooling ended. It never worried me very much because I always had plenty of amusements on the side. I doodled in the margins of my books, I decorated our blue copy paper with ultra-fantastic drawings, and I drew the faces and profiles of my schoolmasters as outrageously as I could, distorting them out of all recognition.”

Claude Monet (1840–1926) French impressionist painter

in Denis Rouart (1972) Claude Monet, p. 21 : About his youth
after Monet's death

Life, Books, School, Home

“One day he said: For the soul there is a satisfaction of a higher type; the material is not at all necessary. Whether I apply mathematics to a couple of clods of dirt, which we call planets, or to purely arithmetical problems, it s just the same; the latter have only a higher charm for me.”

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

Carl Friedrich Gauss: Titan of Science (1955) by Guy Waldo Dunnington. p. 348

Soul, Mathematics, Problem

“But where is the antidote for lucid despair, perfectly articulated, proud, and sure? All of us are miserable, but how many know it? The consciousness of misery is too serious a disease to figure in an arithmetic of agonies or in the catalogues of the Incurable. It belittles the prestige of hell, and converts the slaughterhouses of time into idyls. What sin have you committed to be born, what crime to exist? Your suffering like your fate is without motive. To suffer, truly to suffer, is to accept the invasion of ills without the excuse of causality, as a favor of demented nature, as a negative miracle...”

Emil M. Cioran book A Short History of Decay

A Short History of Decay (1949)

Fate, Disease, Motivation, Hell

“The analysis of variance is not a mathematical theorem, but rather a convenient method of arranging the arithmetic.”

Ronald Fisher (1890–1962) English statistician, evolutionary biologist, geneticist, and eugenicist

Discussion to ‘Statistics in agricultural research’ by J.Wishart, Journal of the Royal Statistical Society, Supplement, 1, 26-61, 1934.
1930s

Mathematics

“That fondness for science, … that affability and condescension which God shows to the learned, that promptitude with which he protects and supports them in the elucidation of obscurities and in the removal of difficulties, has encouraged me to compose a short work on calculating by al-jabr and al-muqabala, confining it to what is easiest and most useful in arithmetic.”

Muhammad ibn Mūsā al-Khwārizmī

As quoted in: Victor J. Katz (2009) A history of mathematics: an introduction. p. 271

God, Support, Learning, Science

“Authoritarian, paralyzing, circular, occasionally elliptical, stock phrases, also jocularly referred to as nuggets of wisdom, are malignant plague, one of the very worst ever to ravage the earth. We say to the confused, Know thyself, as if knowing yourself was not the fifth and most difficult of human arithmetical operations, we say to the apathetic, Where there’s a will, there’s a way, as if the brute realities of the world did not amuse themselves each day by turning that phrase on its head, we say to the indecisive, Begin at the beginning, as if that beginning were the clearly visible point of a loosely wound thread and that all we had to do was to keep pulling until we reached the other end, and as if, between the former and the latter, we had held in our hands a smooth, continuous thread with no knots to untie, no snarled to untangle, a complete impossibility in the life of a skien, or indeed, if we may be permitted on more stock phrase, in the skien of life. … These are the delusions of the pure and unprepared, the beginning is never the clear, precise end of a thread, the beginning is a long, painfully slow process that requires time and patience in order to find out in which direction it is heading, a process that feels its way along the path ahead like a blind man the beginning is just the beginning, what came before is nigh on worthless.”

José Saramago book The Cave

Source: The Cave (2000), p. 54 (Vintage 2003)

Life, Wisdom, World, Way

“If I compare arithmetic with a tree that unfolds upward into a multitude of techniques and theorems while its root drives into the depths, then it seems to me that the impetus of the root.”

Gottlob Frege (1848–1925) mathematician, logician, philosopher

Gottlob Frege, Montgomery Furth (1964). The Basic Laws of Arithmetic: Exposition of the System. p. 10

Tree

“This ideography is a "formula language", that is, a lingua characterica, a language written with special symbols, "for pure thought", that is, free from rhetorical embellishments, "modeled upon that of arithmetic", that is, constructed from specific symbols that are manipulated according to definite rules.”

Gottlob Frege (1848–1925) mathematician, logician, philosopher

paraphrasing Frege's Begriffsschrift, a formula language, modeled upon that of arithmetic, for pure thought (1879) in Jean Van Heijenoort ed., in From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931 (1967)

“We started them to school. They learned to read. They learned to work simple arithmetic problems. Now some of our plantation owners can't figure the poor devils out of everything at the close of each year.”

Huey Long (1893–1935) American politician, Governor of Louisiana, and United States Senator

Huey Long on African American Education (Williams p. 524)

School, Problem, Learning, Reading

“India is industrially more developed than many less fortunate countries and is reckoned as the seventh or eighth among the world's industrial nations. But this arithmetical distinction cannot conceal the poverty of the great majority of our people. To remove this poverty by greater production, more equitable distribution, better education and better health, is the paramount need and the most pressing task before us and we are determined to accomplish this task. We realize that self-help is the first condition of success for a nation, no less than for an individual. We are conscious that ours must be the primary effort and we shall seek succour from none to escape from any part of our own responsibility. But though our economic potential is great, its conversion into finished wealth will need much mechanical and technological aid. We shall, therefore, gladly welcome such aid and co-operation on terms that are of mutual benefit. We believe that this may well help in the solution of the larger problems that confront the world. But we do not seek any material advantage in exchange for any part of our hard-won freedom.”

Jawaharlal Nehru (1889–1964) Indian lawyer, statesman, and writer, first Prime Minister of India

Speech to the US Congress (13 October 1949)

People, Health, World, Education

“Of such are the mathematical sciences alone; that is, geometry and arithmetic, in which the Divine intellect indeed knows infinitely more propositions, since it knows all. But with regard to those few which the human intellect does understand, I believe its knowledge equals the Divine in objective certainty, for here it succeeds in understanding necessity, beyond which there can be no greater sureness.”

Galileo Galilei book Dialogue Concerning the Two Chief World Systems

In the 1661 translation by Thomas Salusbury: … such are the pure Mathematical sciences, to wit, Geometry and Arithmetick: in which Divine Wisdom knows infinite more propositions, because it knows them all; but I believe that the knowledge of those few comprehended by humane understanding, equalleth the divine, as to the certainty objectivè, for that it arriveth to comprehend the necessity thereof, than which there can be no greater certainty." p. 92 (from the Archimedes Project http://archimedes.mpiwg-berlin.mpg.de/cgi-bin/toc/toc.cgi?page=92;dir=galil_syste_065_en_1661;step=textonly) <br class="br">In the original Italian: … tali sono le scienze matematiche pure, cioè la geometria e l’aritmetica, delle quali l’intelletto divino ne sa bene infinite proposizioni di piú, perché le sa tutte, ma di quelle poche intese dall’intelletto umano credo che la cognizione agguagli la divina nella certezza obiettiva, poiché arriva a comprenderne la necessità, sopra la quale non par che possa esser sicurezza maggiore." (from the copy at the Italian Wikisource). <br class="br">Dialogue Concerning the Two Chief World Systems (1632)

Mathematics, Understanding, Knowledge, Science

“The devil take this wrong arithmetic.”

Karl Marx book Grundrisse

Grundrisse (1857/58)
Variant: The devil take this wrong arithmetic. But never mind.
Source: Notebook IV, The Chapter on Capital, p. 297.

“We all have such stories. It is a brutal arithmetic. But I - I am alive. You are alive. As long as we breathe, we can see and hear. As long as we can remember, all those gone before are alive inside us.”

Jane Yolen book The Devil's Arithmetic

Source: The Devil's Arithmetic

“Part of her revolted against the insanity of the rules. Part of her was grateful. In a world of chaos, any guidelines helped. And she knew that each day she remained alive,. One plus one plus one. The Devil's arithmetic…”

Jane Yolen (1939) American speculative fiction and children's writer

World, Help, Parting

“Very possible! Possible, indeed. Maybe even probable, which, as you know if you study your arithmetic, can happen more often than possible. In other words, probable is more possible than possible. - Bubo”

Kathryn Lasky (1944) American children's writer

Source: The Rescue

Study

“The loony legacy of money was that the arithmetic by which things were measured had become more valuable than the things themselves.”

Tom Robbins Skinny Legs and All

Source: Skinny Legs and All

Money

“He who refuses to do arithmetic is doomed to talk nonsense.”

John McCarthy (1927–2011) American computer scientist and cognitive scientist

PROGRESS AND ITS SUSTAINABILITY http://www-formal.stanford.edu/jmc/progress/ (1995 – ) <br class="br">1990s

“The first period embraces the time between the first records of empirical determinations of the ratio of the circumference to the diameter of a circle until the invention of the Differential and Integral Calculus, in the middle of the seventeenth century. This period, in which the ideal of an exact construction was never entirely lost sight of, and was occasionally supposed to have been attained, was the geometrical period, in which the main activity consisted in the approximate determination of π by the calculation of the sides or the areas of regular polygons in- and circum-scribed to the circle. The theoretical groundwork of the method was the Greek method of Exhaustions. In the earlier part of the period the work of approximation was much hampered by the backward condition of arithmetic due to the fact that our present system of numerical notation had not yet been invented; but the closeness of the approximations obtained in spite of this great obstacle are truly surprising. In the later part of this first period methods were devised by which the approximations to the value of π were obtained which required only a fraction of the labour involved in the earlier calculations. At the end of the period the method was developed to so high a degree of perfection that no further advance could be hoped for on the lines laid down by the Greek Mathematicians; for further progress more powerful methods were required.”

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

Source: Squaring the Circle (1913), pp. 10-11

Hope, Time, Parting, Perfection

“Most, if not all, of the great ideas of modern mathematics have had their origin in observation. Take, for instance, the arithmetical theory of forms, of which the foundation was laid in the diophantine theorems of Fermat, left without proof by their author, which resisted all efforts of the myriad-minded Euler to reduce to demonstration, and only yielded up their cause of being when turned over in the blow-pipe flame of Gauss’s transcendent genius; or the doctrine of double periodicity, which resulted from the observation of Jacobi of a purely analytical fact of transformation; or Legendre’s law of reciprocity; or Sturm’s theorem about the roots of equations, which, as he informed me with his own lips, stared him in the face in the midst of some mechanical investigations connected (if my memory serves me right) with the motion of compound pendulums; or Huyghen’s method of continued fractions, characterized by Lagrange as one of the principal discoveries of that great mathematician, and to which he appears to have been led by the construction of his Planetary Automaton; or the new algebra, speaking of which one of my predecessors (Mr. Spottiswoode) has said, not without just reason and authority, from this chair, “that it reaches out and indissolubly connects itself each year with fresh branches of mathematics, that the theory of equations has become almost new through it, algebraic 31 geometry transfigured in its light, that the calculus of variations, molecular physics, and mechanics” (he might, if speaking at the present moment, go on to add the theory of elasticity and the development of the integral calculus) “have all felt its influence.”

James Joseph Sylvester (1814–1897) English mathematician

James Joseph Sylvester. "A Plea for the Mathematician, Nature," Vol. 1, p. 238; Collected Mathematical Papers, Vol. 2 (1908), pp. 655, 656.

Law, Mathematics, Idea, Effort

“As professor in the Polytechnic School [autumn of 1858] in Zurich I found myself for the first time obliged to lecture upon the elements of the differential calculus and felt, more keenly than ever before, the lack of a really scientific foundation for arithmetic. In discussing the notion of the approach of a variable magnitude to a fixed limiting value, and especially in proving the theorem that every magnitude which grows continually, but not beyond all limits, must certainly approach a limiting value, I had recourse to geometric evidences. Even now such resort to geometric intuition in a first presentation of the differential calculus, I regard as exceedingly useful, from the didactic standpoint, and indeed indispensable, if one does not wish to lose too much time. But that this form of introduction into the differential calculus can make no claim to being scientific, no one will deny. For myself this feeling of dissatisfaction was so overpowering that I made the fixed resolve to keep meditating on the question till I should find a purely arithmetic and perfectly rigorous foundation for the principles of infinitesimal analysis.”

Richard Dedekind (1831–1916) German mathematician

p, 125
Stetigkeit und irrationale Zahlen (1872)

School, Feeling, Question, Autumn

“For to pass by those Ancients, the wonderful Pythagoras, the sagacious Democritus, the divine Plato, the most subtle and very learned Aristotle, Men whom every Age has hitherto acknowledged as deservedly honored, as the greatest Philosophers, the Ring-leaders of Arts; in whose Judgments how much these Studies [mathematics] were esteemed, is abundantly proclaimed in History and confirmed by their famous Monuments, which are everywhere interspersed and bespangled with Mathematical Reasonings and Examples, as with so many Stars; and consequently anyone not in some Degree conversant in these Studies will in vain expect to understand, or unlock their hidden Meanings, without the Help of a Mathematical Key: For who can play well on Aristotle’s Instrument but with a Mathematical Quill; or not be altogether deaf to the Lessons of natural Philosophy, while ignorant of Geometry? Who void of (Geometry shall I say, or) Arithmetic can comprehend Plato’s 218 Socrates lisping with Children concerning Square Numbers; or can conceive Plato himself treating not only of the Universe, but the Polity of Commonwealths regulated by the Laws of Geometry, and formed according to a Mathematical Plan?”

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

Source: Mathematical Lectures (1734), pp. 26-27

Men, Children, Age, Law

“Experience has convinced me that the proper way of teaching is to bring together that which is simple from all quarters, and, if I may use such a phrase, to draw upon the surface of the subject a proper mean between the line of closest connexion and the line of easiest deduction. This was the method followed by Euclid, who, fortunately for us, never dreamed of a geometry of triangles, as distinguished from a geometry of circles, or a separate application of the arithmetics of addition and subtraction; but made one help out the other as he best could.”

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

Advertisement, p.4
The Differential and Integral Calculus (1836)

Dreams, Way, Help, Drawing

“There is only one other survey, Datta and Singh’s 1938 History of Hindu Mathematics, recently reprinted but very hard to obtain in the West (I found a copy in a small specialized bookstore in Chennai). They describe in some detail the Indian work in arithmetic and algebra and, supplemented by the equally hard to find Geometry in Ancient and Medieval India by Sarasvati Amma (1979), one can get an overview of most topics.”

David Mumford (1937) American mathematician

[David Mumford, Book Review, Notices of the AMS, March 2010, 57, 3, http://www.ams.org/notices/201003/rtx100300385p.pdf]

History, Mathematics

“Here's what I've started thinking: All the wonder of God happens right above our arithmetic and formula. The more I climb outside my pat answers, the more invigorating the view, the more my heart enters into worship.”

Donald Miller book Blue Like Jazz: nonreligious thoughts on Christian spirituality

Blue Like Jazz (2003, Nelson Books)

God, Thinking, Heart

“The most probable view is that adopted by Nesselmann, that the works which we know under the three titles formed part of one arithmetical work, which was, according to the author's own words, to consist of thirteen Books. The proportion of the lost parts to the whole is probably less than it might be supposed to be. The Porisms form the part the loss of which is most to be regretted, for from the references to them it is clear that they contained propositions in the Theory of Numbers most wonderful for the time.”

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

Source: Diophantos of Alexandria: A Study in the History of Greek Algebra (1885), Ch. II, p.37

Books, Loss, Time, Parting

“We alone can be wracked with doubt, and we alone have been provoked by that epistemic itch to seek a remedy: better truth-seeking methods. Wanting to keep better track of our food supplies, our territories, our families, our enemies, we discovered the benefits of talking it over with others, asking questions, passing on lore. We invented culture. Then we invented measuring, and arithmetic, and maps, and writing. These communicative and recording innovations come with a built-in ideal: truth. The point of asking questions is to find true answers; the point of measuring is to measure accurately; the point of making maps is to find your way to your destination. … In short, the goal of truth goes without saying, in every human culture.”

Daniel Dennett (1942) American philosopher

Postmodernism and truth (1998)

Truth, Family, Food, Way

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

George Frederick James Temple (1901–1992) British mathematician

100 Years of Mathematics: a Personal Viewpoint (1981)

Music, Mathematics, Study

“An important point is that the p-adic field, or respectively the real or complex field, corresponding to a prime ideal, plays exactly the role, in arithmetic, that the field of power series in the neighborhood of a point plays in the theory of functions: that is why one calls it a local field.”

André Weil (1906–1998) French mathematician

as translated by Martin H. Krieger "A 1940 letter of André Weil on analogy in mathematics." http://www.ams.org/notices/200503/fea-weil.pdf Notices of the AMS 52, no. 3 (2005) pp. 334–341, quote on p. 340

“We assert that it is possible to describe analytically any human function which can be reasonably defined in objective terms and we specifically include in such functions "thinking" insofar as that term is definable. If by "thinking" one means being able to do arithmetic, or play a good game of chess, or learn from experience, or make optimal decisions in exceedingly complex situations, then we assert that thinking can be described analytically. And there are two important corollaries: if It can be described analytically, it can be simulated; and if it can be simulated, it can be performed mechanically.”

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

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

Game, Optimism, Thinking, Decision

“Suppose I think, after doing my accounts, that I have a large balance at the bank. And suppose you want to find out whether this belief of mine is “wishful thinking.” You can never come to any conclusion by examining my psychological condition. Your only chance of finding out is to sit down and work through the sum yourself. When you have checked my figures, then, and then only, will you know whether I have that balance or not. If you find my arithmetic correct, then no amount of vapouring about my psychological condition can be anything but a waste of time. If you find my arithmetic wrong, then it may be relevant to explain psychologically how I came to be so bad at my arithmetic, and the doctrine of the concealed wish will become relevant—but only after you have yourself done the sum and discovered me to be wrong on purely arithmetical grounds. It is the same with all thinking and all systems of thought. If you try to find out which are tainted by speculating about the wishes of the thinkers, you are merely making a fool of yourself. You must first find out on purely logical grounds which of them do, in fact, break down as arguments. Afterwards, if you like, go on and discover the psychological causes of the error.”

Clive Staples Lewis (1898–1963) Christian apologist, novelist, and Medievalist

"Bulverism" (1941)

Fools, Error, Thinking, Chance

“This art is music. It stands quite apart from all the others. In it we do not recognize the copy, the repetition, of any Idea of the inner nature of the world. Yet it is such a great and exceedingly fine art, its effect on man's innermost nature is so powerful, and it is so completely and profoundly understood by him in his innermost being as an entirely universal language, whose distinctness surpasses even that of the world of perception itself, that in it we certainly have to look for more than that exercitium arithmeticae occultum nescientis se numerare animi [exercise in arithmetic in which the mind does not know it is counting] which Leibniz took it to be.”

Arthur Schopenhauer book The World as Will and Representation

Vol. I, Ch. III, The World As Representation: Second Aspect, as translated by Eric F. J. Payne (1958)
The World as Will and Representation (1819; 1844; 1859)

World, Art, Music, Exercises

“I have endeavoured… to present the principles and applications of Symbolical, in immediate sequence to those of Arithmetical, Algebra, and at the same time to preserve that strict logical order and simplicity of form and statement which is essential to an elementary work. This is a task of no ordinary difficulty, more particularly when the great generality of the language of Symbolical Algebra and the wide range of its applications are considered, and this difficulty has not been a little increased, in the present instance, by the wide departure of my own views of its principles from those which have been commonly entertained.”

George Peacock (1791–1858) Scottish mathematician

Vol. II: On Symbolical Algebra and its Applications to the Geometry of Position (1845) Preface, p. iii
A Treatise on Algebra (1842)

Departure, Time

“For, among the world's incertitudes, this thing called arithmetic is established by a sure reasoning that we comprehend as we do the heavenly bodies. It is an intelligible pattern, a beautiful system, that both binds the heavens and preserves the earth. For is there anything that lacks measure, or transcends weight? It includes all, it rules all, and all things have their beauty because they are perceived under its standard.”
Haec enim quae appellatur arithmetica inter ambigua mundi certissima ratione consistit, quam cum caelestibus aequaliter novimus: evidens ordo, pulchra dispositio, cognitio simplex, immobilis scientia, quae et superna continet et terrena custodit. quid est enim quod aut mensuram non habeat aut pondus excedat? omnia complectitur, cuncta moderatur et universa hinc pulchritudinem capiunt, quia sub modo ipsius esse noscuntur.

Cassiodorus Variae

Bk. 1, no. 10; p. 12.
Variae

World, Beauty, Intelligence

“The system R forms a well-arranged domain of one dimension extending to infinity on two opposite sides. What is meant by this is sufficiently indicated by my use of expressions borrowed from geometric ideas; but just for this reason it will be necessary to bring out clearly the corresponding purely arithmetic properties in order to avoid even the appearance as if arithmetic were in need of ideas foreign to it.”

Richard Dedekind (1831–1916) German mathematician

p, 125
Stetigkeit und irrationale Zahlen (1872)

Idea, Appearance

“No superior can supervise directly the work of more than five or, at the most, six subordinates whose work interlocks. The reason for this is simple. What is supervised is not only the individuals, but the permutations and combinations of the relationships between them. And while the former increase in arithmetical progression with the addition of each fresh subordinate, the latter increase by geometrical progression. If a superior adds a sixth to five immediate subordinates he Increases his opportunity of delegation by 20 per cent, but he adds over 100 per cent to the number of relationships he has to take into account. Because ultimately it is based on the limitations imposed by the human span of attention, this principle is called The Span of Control.”

Lyndall Urwick (1891–1983) British management consultant

Source: 1940s, The Elements of Business Administration, 1943, p. 53

“It may fairly be said that the germs of the modern algebra of linear substitutions and concomitants are to be found in the fifth section of the Disquisitiones Arithmeticae; and inversely, every advance in the algebraic theory of forms is an acquisition to the arithmetical theory.”

George Ballard Mathews (1861–1922) British mathematician

Part 1, sect. 48.
Theory of Numbers, 1892

“It is picked out from numbers progressing in continuous proportion. Of continuous progressions, an arithmetical is one which proceeds by equal intervals; a geometrical one which advances by unequal and proportionally increasing or decreasing intervals. Arithmetical progressions: 1, 2, 3, 4, 5, 6, 7, &c.; or 2, 4, 6, 8, 10, 12, 14, 16, &c, Geometrical progressions: 1, 2, 4, 8, 16, 32, 64, &c.; or 243, 81, 27, 9, 3, 1.”

John Napier (1550–1617) Scottish mathematician

The Construction of the Wonderful Canon of Logarithms (1889)

“You see, some mathematicians have clear and far-ranging. "programs". For instance, Grothendieck had such a program for algebraic geometry; now Langlands has one for representation theory, in relation to modular forms and arithmetic. I never had such a program, not even a small size one.”

Jean-Pierre Serre (1926) French mathematician

An Interview with Jean-Pierre Serre - Singapore Mathematical Society https://sms.math.nus.edu.sg/smsmedley/Vol-13-1/An%20interview%20with%20Jean-Pierre%20Serre(CT%20Chong%20&%20YK%20Leong).pdf

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

Joshua Girling Fitch (1824–1903) British educationalist

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

Life, Truth, Education, School

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

Morris Kline (1908–1992) American mathematician

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

History, Mathematics, Problem, Reading

“Of course the intellectual class did not notice this for many decades, as an intellectual is a highly educated man who can't do arithmetic with his shoes on, and is proud of the lack.”

Robert A. Heinlein book The Cat Who Walks Through Walls

Source: The Cat Who Walks Through Walls (1985), p. 564

Education, Shoe

“The whole of arithmetic now appeared within the grasp of mechanism.”

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

Passages from the Life of a Philosopher (1864), ch. 8 "Of the Analytical Engine"
Passages from the Life of a Philosopher (1864)

Appearance

“I am coming more and more to the conviction that the necessity of our geometry cannot be demonstrated, at least neither by, nor for, the human intellect... Geometry should be ranked, not with arithmetic, which is purely aprioristic, but with mechanics.”

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

As quoted in Solid Shape (1990) by Jan J. Koenderink

“The method of arithmetical teaching]] is perhaps the best understood of any of the methods concerned with elementary studies.”

Alexander Bain (1818–1903) Scottish philosopher and educationalist

Source: Education as a Science, 1898, p. 288.

Study

“Writing wasn't easy for her. It wasn't like arithmetic, not like a column of numbers to be added or subtracted from the top of the blackboard to the bottom, neat and organized; or science, figuring out the why of things.”

Patricia Reilly Giff (1935) American children's writer

Source: Water Street (2006), Chapters 11-20, p. 65

Science

“The advanced arithmetical machines of the future will be electrical in nature, and they will perform at 100 times present speeds, or more.
Moreover, they will be far more versatile than present commercial machines, so that they may readily be adapted for a wide variety of operations. They will be controlled by a control card or film, they will select their own data and manipulate it in accordance with the instructions thus inserted, they will perform complex arithmetical computations at exceedingly high speeds, and they will record results in such form as to be readily available for distribution or for later further manipulation.”

Vannevar Bush book As We May Think

As We May Think (1945)

Nature, Time, Future, Speed

“What mathematics, therefore are expected to do for the advanced student at the university, Arithmetic, if taught demonstratively, is capable of doing for the children even of the humblest school. It furnishes training in reasoning, and particularly in deductive reasoning. It is a discipline in closeness and continuity of thought. It reveals the nature of fallacies, and refuses to avail itself of unverified assumptions. It is the one department of school-study in which the sceptical and inquisitive spirit has the most legitimate scope; in which authority goes for nothing. In other departments of instruction you have a right to ask for the scholar’s confidence, and to expect many things to be received on your testimony with the understanding that they will be explained and verified afterwards. But here you are justified in saying to your pupil “Believe nothing which you cannot understand. Take nothing for granted.” In short, the proper office of arithmetic is to serve as elementary 268 training in logic. All through your work as teachers you will bear in mind the fundamental difference between knowing and thinking; and will feel how much more important relatively to the health of the intellectual life the habit of thinking is than the power of knowing, or even facility of achieving visible results. But here this principle has special significance. It is by Arithmetic more than by any other subject in the school course that the art of thinking—consecutively, closely, logically—can be effectually taught.”

Joshua Girling Fitch (1824–1903) British educationalist

Source: Lectures on Teaching, (1906), pp. 292-293.

Life, Children, Health, School

“The history of arithmetic and algebra illustrates one of the striking and curious features of the history of mathematics. Ideas that seem remarkably simple once explained were thousands of years in the making.”

Morris Kline (1908–1992) American mathematician

Morris Kline, p.22.
Mathematics for the Nonmathematician (1967)

History, Mathematics, Idea

“I have asked many teachers and parents what they thought mathematics to be and why it was important to learn it. Few held a view of mathematics that was sufficiently coherent to justify devoting several thousand hours of a child's life to learning it, and children sense this. When a teacher tells a student that the reason for those many hours of arithmetic is to be able to check the change at the supermarket, the teacher is simply not believed. Children see such "reasons" as one more example of adult double talk. The same effect is produced when children are told school math is "fun" when they are pretty sure that teachers who say so spend their leisure hours on anything except this allegedly fun-filled activity. Nor does it help to tell them that they need math to become scientists---most children don't have such a plan. The children can see perfectly well that the teacher does not enjoy math any more than they do and that the reason for doing it is simply that it has been inscribed into the curriculum. All of this erodes children's confidence in the adult world and the process of education. And I think it introduces a deep element of dishonestly into the education relationship.”

Seymour Papert book Mindstorms: Children, Computers, and Powerful Ideas

Source: Mindstorms: Children, Computers, and Powerful Ideas (1980), Chapter 2, Mathophobia: The Fear of Learning

Life, Children, World, Education

“When I had the honour of his conversation, I endeavoured to learn his thoughts upon mathematical subjects, and something historical concerning his inventions, that I had not been before acquainted with. I found, he had read fewer of the modern mathematicians, than one could have expected; but his own prodigious invention readily supplied him with what he might have an occasion for in the pursuit of any subject he undertook. I have often heard him censure the handling geometrical subjects by algebraic calculations; and his book of Algebra he called by the name of Universal Arithmetic, in opposition to the injudicious title of Geometry, which Des Cartes had given to the treatise, wherein he shews, how the geometer may assist his invention by such kind of computations. He frequently praised Slusius, Barrow and Huygens for not being influenced by the false taste, which then began to prevail. He used to commend the laudable attempt of Hugo de Omerique to restore the ancient analysis, and very much esteemed Apollonius's book De sectione rationis for giving us a clearer notion of that analysis than we had before.”

Henry Pemberton (1694–1771) British doctor

Preface; The bold passage is subject of the 1809 article " Remarks on a Passage in Castillione's Life' of Sir Isaac Newton http://books.google.com/books?id=BS1WAAAAYAAJ&pg=PA519." By John Winthrop, in: The Philosophical Transactions of the Royal Society of London, from Their Commencement, in 1665, to the Year 1800: 1770-1776: 1770-1776. Charles Hutton et al. eds. (1809) p. 519. <br class="br">Preface to View of Newton's Philosophy, (1728)

Books, Mathematics, Learning, Universe

“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.”

Nicholas Murray Butler (1862–1947) American philosopher, diplomat, and educator

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

Mathematics, Study, Universe, Time

“I wasn't motivated to do it [re-record Bach's Goldberg Variations] until rather recently, when it occurred to me, on one of my rare relistenings to that early recording, that it was very nice, but that it was perhaps a little bit like thirty very interesting but somewhat independent-minded pieces, going their own way, and all making a comment on the ground bass on which they are all formed and to which they all conform. And I suddenly felt, not having played it in, well, since I stopped playing concerts, about 20 years, having not played it in all that time, that maybe I wasn't savaged by any over-exposure to it, and that if I looked at it again, I could find a way of making some sort of almost arithmetical correspondence between the theme and the subsequent variations, so that there would be some sort of temporal relationship, I don't want to say just exactly 2, 4, 8, 16, 32, that kind of correspondence, but, you know what I mean, there would be a sense in which, substituting for the fact that Bach had absolutely no melodic design that is continuous but rather a base harmonic design that is continuous, there would be at least a rhythmic design that is continuous, and the sense of pulse that went through it. And that seemed to me sufficient justification […] to do it all over again.”

Glenn Gould (1932–1982) Canadian pianist

transcribed from The Glenn Gould Collection vol 13 (Sony laserdisc).

Way, Motivation, Sense, Time

“I had a running compiler and nobody would touch it. … they carefully told me, computers could only do arithmetic; they could not do programs.”

Grace Hopper (1906–1992) American computer scientist and United States Navy officer

As quoted in Grace Hopper : Navy Admiral and Computer Pioneer (1989) by Charlene W. Billings, p. 74 ISBN 089490194X

“This method of mine takes its beginnings where Cavalieri ends his Method of indivisibles…. for as his was the Geometry of indivisibles, so I have chosen to call my method the Arithmetic of infinitesimals.”

John Wallis (1616–1703) English mathematician

Arithmetica Infinitorum (1656)

“Any desired geometrical mean between two sines has for its Logarithm the corresponding arithmetical mean between the Logarithms of the sines.”

John Napier (1550–1617) Scottish mathematician

Appendix, The relations of Logarithms & their natural numbers to each other
The Construction of the Wonderful Canon of Logarithms (1889)

“It is India that gave us the ingenious method of expressing all numbers by means of ten symbols, each symbol receiving a value of position as well as an absolute value; a profound and important idea which appears so simple to us now that we ignore its true merit. But its very simplicity and the great ease which it has lent to computations put our arithmetic in the first rank of useful inventions; and we shall appreciate the grandeur of the achievement the more when we remember that it escaped the genius of Archimedes and Apollonius, two of the greatest men produced by antiquity.”

Pierre-Simon Laplace (1749–1827) French mathematician and astronomer

Quoted in H Eves Return to Mathematical Circles (Boston 1988). http://www-groups.dcs.st-and.ac.uk/history/Quotations/Laplace.html

Men, Idea, Appearance

“As Gauss first pointed out, the problem of cyclotomy, or division of the circle into a number of equal parts, depends in a very remarkable way upon arithmetical considerations. We have here the earliest and simplest example of those relations of the theory of numbers to transcendental analysis, and even to pure geometry, which so often unexpectedly present themselves, and which, at first sight, are so mysterious.”

George Ballard Mathews (1861–1922) British mathematician

Part 1, sect. 167.
Theory of Numbers, 1892

Way, Problem, Parting, Dependency

“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)

Mathematics, Idea

“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, Mathematics, Universe

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)

Mathematics, Study, Universe, Time

“In arithmetic the Arabs took one step backward. Though they were familiar with negative numbers and the rules for operating with them through the work of the Hindus, they rejected negative numbers.”

Morris Kline (1908–1992) American mathematician

Source: Mathematical Thought from Ancient to Modern Times (1972), p. 192.

“Population, when unchecked, increases in a geometrical ratio, Subsistence, increases only in an arithmetical ratio.”

Thomas Robert Malthus (1766–1834) British political economist

Source: An Essay on The Principle of Population (First Edition 1798, unrevised), Chapter I, paragraph 18, lines 1-2

“My father's education was altogether of the worst and most limited. I believe he was never more than three months at any school. What he learned there showed what he might have learned. A solid knowledge of arithmetic, a fine antique handwriting — these, with other limited practical etceteras, were all the things he ever heard mentioned as excellent. He had no room to strive for more.”

Thomas Carlyle (1795–1881) Scottish philosopher, satirical writer, essayist, historian and teacher

1880s, Reminiscences (1881)

Education, School, Knowledge, Learning

“The statement is so frequently made that the differential calculus deals with continuous magnitude, and yet an explanation of this continuity is nowhere given; even the most rigorous expositions of the differential calculus do not base their proofs upon continuity but, with more or less consciousness of the fact, they either appeal to geometric notions or those suggested by geometry, or depend upon theorems which are never established in a purely arithmetic manner. Among these, for example, belongs the above mentioned theorem, and a more careful investigation convinced me that this theorem, or any one equivalent to it, can be regarded in some way as a sufficient basis for infinitesimal analysis. It then only remained to discover its true origin in the elements of arithmetic and thus at the same time to secure a real definition of the essence of continuity. I succeeded Nov. 24, 1858.”

Richard Dedekind (1831–1916) German mathematician

p, 125
Stetigkeit und irrationale Zahlen (1872)

Way, Time, Dependency

“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, Mathematics, Question, Parting

“The desire to economize time and mental effort in arithmetical computations, and to eliminate human liability to error is probably as old as the science of arithmetic itself.”

Howard H. Aiken (1900–1973) pioneer in computing, original conceptual designer behind IBM's Harvard Mark I computer

"Proposed Automatic Calculating Machine" (1937)

Error, Effort, Science, Time

“All the constructions of this first phase of the youth of the group bear witness to its constant characteristics. The epic imagination is a proof of its esemplastic power. Everything is formal, inventive and arithmetical, therefore rhythmic.”

Albert Gleizes (1881–1953) French painter

after 1920, The Epic, From immobile form to mobile form (1925)

Youth, Imagination

“Imagination is perhaps the most decisive characteristic of mankind. My dream is the imagination of space – to change the optical impression of the world of objects by a transcendental arithmetic progression of the inner being. That is the precept. In principal any alteration of the object is allowed which has a sufficiently strong creative power behind it. Whether such alteration causes excitement or boredom in the spectator is for you to decide.”

Max Beckmann (1884–1950) German painter, draftsman, printmaker, sculptor and writer

Source: 1930s, On my Painting (1938), p. 16

World, Dreams, Imagination, Decision

“Anyone who understands algebraic notation, reads at a glance in an equation results reached arithmetically only with great labour and pains.”

Antoine Augustin Cournot Researches into the Mathematical Principles of the Theory of Wealth

Source: Researches into the Mathematical Principles of the Theory of Wealth, 1897, p. 4; Cited in: Moritz (1914, 197): About mathematics as language

Pain, Understanding, Reading

“The fact that arithmetic and geometry took such a notable step forward… was due in no small measure to the introduction of Egyptian papyrus into Greece. This event occurred about 650 B. C., and the invention of printing in the 15th century did not more surely effect a revolution in thought than did this introduction of writing material on the northern shores of the Mediterranean Sea just before the time of Thales.”

David Eugene Smith (1860–1944) American mathematician

History of Mathematics (1923) Vol.1

Time, Sea

“With Diophantus the history of Greek arithmetic comes to an end. No original work, that we know of, was done afterwards.”

James Gow (scholar) (1854–1923) scholar

p, 125
A Short History of Greek Mathematics (1884)

History

“Science is turning into a monastery for the Order of Capitulant Friars. Logical calculus is supposed to supersede man as moralist. We submit to the blackmail of the "superior knowledge" that has the temerity to assert that nuclear war can be, by derivation, a good thing, because this follows from simple arithmetic.”

Stanisław Lem book His Master's Voice

Source: His Master's Voice (1968), Ch. 9

War, Knowledge, Science

“As in geometry, the oblique must be known, as well as the right; and in arithmetic, the odd as well as the even; so in actions of life, who seeth not the filthiness of evil, wanteth a great foil to perceive the beauty of virtue.”

Philip Sidney (1554–1586) English diplomat

Aphorisms of Sir Philip Sidney; with remarks, by Miss Porter (1807), p. 23. London: Longman, Hurst, Rees and Orme https://babel.hathitrust.org/cgi/pt?id=uc1.aa0000617332;view=1up;seq=53

Life, Beauty, Evil

“Probably Greek logistic, or calculation, extended to more difficult operations… and… probably Greek arithmetic, or theory of numbers, owed much more to induction than is permitted to appear by its first and chief professors.”

James Gow (scholar) (1854–1923) scholar

p, 125
A Short History of Greek Mathematics (1884)

Appearance

“The problem of distinguishing prime numbers from composite numbers and of resolving the latter into their prime factors is known to be one of the most important and useful in arithmetic. It has engaged the industry and wisdom of ancient and modern geometers to such an extent that it would be superfluous to discuss the problem at length. … Further, the dignity of the science itself seems to require that every possible means be explored for the solution of a problem so elegant and so celebrated.”

Carl Friedrich Gauss book Disquisitiones Arithmeticae

Problema, numeros primos a compositis dignoscendi, hosque in factores suos primos resolvendi, ad gravissima ac utilissima totius arithmeticae pertinere, et geometrarum tum veterum tum recentiorum industriam ac sagacitatem occupavisse, tam notum est, ut de hac re copiose loqui superfluum foret. … [P]raetereaque scientiae dignitas requirere videtur, ut omnia subsidia ad solutionem problematis tam elegantis ac celebris sedulo excolantur.
Disquisitiones Arithmeticae (1801): Article 329

Wisdom, Problem, Science

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

Jack Abbott book In the Belly of the Beast

In the Belly of the Beast (1981)

Way, Intelligence, Mathematics, Study

“No doubt once it was real progress when developers and teachers offered learners tangible material in order to teach them arithmetic of whole number… The best palpable material you can give the child is its own body.”

Hans Freudenthal (1905–1990) Dutch mathematician

Source: Mathematics as an Educational Task (1973), p. 75-76; As cited in: Anne Birgitte Fyhn (2007, p. 6)

Teachers

“A community conceived only as the arithmetical sum of its component individuals; a community is, in fact, no more than this once it ceases to be attached to any principle higher than the individuals.”

René Guénon (1886–1951) French metaphysician

Source: The Crisis of the Modern World (1927), p. 96

Communication

“"Any graduate of the ___ Business School should be able to beat an index fund over the course of a market cycle."
Statements such as these are made with alarming frequency by investment professionals. In some cases, subtle and sophisticated reasoning may be involved. More often (alas), the conclusions can only be justified by assuming that the laws of arithmetic have been suspended for the convenience of those who choose to pursue careers as active managers.”

William F. Sharpe (1934) American economist

William F Sharpe, "The arithmetic of active management." Financial Analysts Journal 47.1 (1991): 7-9.

School, Law, Business

“I have separated arithmetical from symbolical algebra, and I have devoted the present volume entirely to the exposition of the principles of the former science and their application to the theory of numbers and of arithmetical processes: the second volume, which is now in the press, will embrace the principles of symbolical algebra: it will be followed, if other and higher duties should allow me the leisure to complete them, by other works, embracing all the more important departments of analysis, with the view of presenting their principles in such a form, as may make them component parts of one uniform and connected system.”

George Peacock (1791–1858) Scottish mathematician

Vol. I: Arithmetical Algebra Preface, p. iii
A Treatise on Algebra (1842)

Science, Parting

“Symbolical algebra adopts the rules of arithmetical algebra, but removes altogether their restrictions: thus symbolical subtraction differs from the same operation in arithmetical algebra in being possible for all relations of value of the symbols or expressions employed… all the results of arithmetical algebra which are deduced by the application of its rules, and which are general in form, though particular in value, are results likewise of symbolical algebra, where they are general in value as well as in form: thus the product of a^{m} and a^{n}, which is a^{m+n} when m and n are whole numbers, and therefore general in form though particular in value, will be their product likewise when m and n are general in value as well as in form: the series for (a+b)^{n}, determined by the principles of arithmetical algebra, when n is any whole number, if it be exhibited in a general form, without reference to a final term, may be shewn, upon the same principle, to the equivalent series for (a+b)^n, when n is general both in form and value.”

George Peacock (1791–1858) Scottish mathematician

Vol. I: Arithmetical Algebra Preface, p. vi-vii
A Treatise on Algebra (1842)

“Brook Taylor… in his Methodus Incrementorum Directa et Inversa (1715), sought to clarify the ideas of the calculus but limited himself to algebraic functions and algebraic differential equations. …Taylor's exposition, based on what we would call finite differences, failed to obtain many backers because it was arithmetical in nature when the British were trying to tie the calculus to geometry or to the physical notion of velocity.”

Morris Kline (1908–1992) American mathematician

Source: Mathematical Thought from Ancient to Modern Times (1972), p. 427

Idea, Nature

“Vladimir Voevodsky and his collaborators have provided us with a very interesting candidate-category of motives: a category (of sheaves relative to an extraordinarily fine Grothendieck-style topology on the category of schemes) which in some intuitive sense “softens algebraic geometry” so as to allow for a good notion of homotopy in an algebro-geometric setup and is sufficiently directly connected to concrete algebraic geometry to have already yielded some extraordinary applications. The quest for a full theory of motives is a potent driving force in complex analysis, algebraic geometry, automorphic representation theory, the study of L functions, and arithmetic. It will continue to be so throughout the current century.”

Barry Mazur (1937) American mathematician

"WHAT IS...a Motive?" 2004

Motivation, Sense, Study

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

David Eugene Smith (1860–1944) American mathematician

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

Mathematics

“If you have observed nature, you would have proved that the question of the numbers of mates is certainly not a question of arithmetic. With gnats, ten females are born to one male. Now gnats are not polygamous. Nine out of those females dies spinsters. It is only the old maids who bite us, from which one sees that celibacy engenders ferocity among insects as well as among women.”

André Maurois (1885–1967) French writer

Les silences du colonel Bramble (The Silence of Colonel Bramble)

Women, Question, Nature

“Music is a hidden arithmetic exercise of the soul, which does not know that it is counting.”
Musica est exercitium arithmeticae occultum nescientis se numerare animi.

Gottfried Leibniz (1646–1716) German mathematician and philosopher

Letter to Christian Goldbach, April 17, 1712.
Arthur Schopenhauer paraphrased this quotation in the first book of Die Welt als Wille und Vorstellung: Musica est exercitium metaphysices occultum nescientis se philosophari animi. (Music is a hidden metaphysical exercise of the soul, which does not know that it is philosophizing.)

Music, Exercises, Soul

“The Hindus saw clearly that if the arithmetic operations… were properly defined for negative numbers, these numbers could be employed to as good advantage as people had previously derived from positive numbers. …To people to whom the word number had always meant positive whole numbers and positive fractions, the very idea that there could be other numbers came hard. For many centuries negative numbers were either rejected or treated as second-class citizens.
What was especially difficult for mathematicians to swallow was that negative numbers could be acceptable roots of equations.”

Morris Kline (1908–1992) American mathematician

Source: Mathematics and the Physical World (1959), p. 51.

People, Idea

“In arithmetical algebra we consider symbols as representing numbers, and the operations to which they are submitted as included in the same definitions as in common arithmetic; the signs + and - denote the operations of addition and subtraction in their ordinary meaning only, and those operations are considered as impossible in all cases where the symbols subjected to them possess values which would render them so in case they were replaced by digital numbers; thus in expressions such as a + b we must suppose a and b to be quantities of the same kind; in others, like a - b, we must suppose a greater than b and therefore homogeneous with it; in products and quotients, like ab and \frac{a}{b} we must suppose the multiplier and divisor to be abstract numbers; all results whatsoever, including negative quantities, which are not strictly deducible as legitimate conclusions from the definitions of the several operations must be rejected as impossible, or as foreign to the science.”

George Peacock (1791–1858) Scottish mathematician

Vol. I: Arithmetical Algebra Preface, p. iv
A Treatise on Algebra (1842)

Science