Quotes about arithmetic (174 quotes, page 2)

“I refer, of course, to the debts our nation has amassed for itself over decades of indulgence. It is the new Red Menace, this time consisting of ink. We can debate its origins endlessly and search for villains on ideological grounds, but the reality is pure arithmetic.”

Mitch Daniels (1949) Governor of Indiana

Reported in Kathryn Jean Lopez, " Mitch Daniels Takes CPAC http://www.nationalreview.com/corner/259623/mitch-daniels-takes-cpac-kathryn-jean-lopez", National Review Online (February 11, 2011).

Search , Reality , Time

“Remember that algebra, with all its deep and intricate problems, is nothing but a development of the four fundamental operations of arithmetic. Everyone who understands the meaning of addition, subtraction, multiplication, and division holds the key to all algebraic problems.”

Richard von Mises (1883–1953) Austrian physicist and mathematician

Second Lecture, The Elements of the Theory of Probability, p. 38
Probability, Statistics And Truth - Second Revised English Edition - (1957)

Understanding , Problem

“A finished or even a competent reasoner is not the work of nature alone… education develops faculties which would otherwise never have manifested their existence. It is, therefore, as necessary to learn to reason before we can expect to be able to reason, as it is to learn to swim or fence, in order to attain either of those arts. Now, something must be reasoned upon, it matters not much what it is, provided that it can be reasoned upon with certainty. The properties of mind or matter, or the study of languages, mathematics, or natural history may be chosen for this purpose. Now, of all these, it is desirable to choose the one… in which we can find out by other means, such as measurement and ocular demonstration of all sorts, whether the results are true or not.
.. Now the mathematics are peculiarly well adapted for this purpose, on the following grounds:—
1. Every term is distinctly explained, and has but one meaning, and it is rarely that two words are employed to mean the same thing.
2. The first principles are self-evident, and, though derived from observation, do not require more of it than has been made by children in general.
3. The demonstration is strictly logical, taking nothing for granted except the self-evident first principles, resting nothing upon probability, and entirely independent of authority and opinion.
4. When the conclusion is attained by reasoning, its truth or falsehood can be ascertained, in geometry by actual measurement, in algebra by common arithmetical calculation. This gives confidence, and is absolutely necessary, if… reason is not to be the instructor, but the pupil.
5. There are no words whose meanings are so much alike that the ideas which they stand for may be confounded.
…These are the principal grounds on which… the utility of mathematical studies may be shewn to rest, as a discipline for the reasoning powers. But the habits of mind which these studies have a tendency to form are valuable in the highest degree. The most important of all is the power of concentrating the ideas which a successful study of them increases where it did exist, and creates where it did not. A difficult position or a new method of passing from one proposition to another, arrests all the attention, and forces the united faculties to use their utmost exertions. The habit of mind thus formed soon extends itself to other pursuits, and is beneficially felt in all the business of life.”

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

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

Life , Children , Truth , Education

“The uniformity and repeatability of print created the “political arithmetic” of the seventeenth century and the “hedonistic calculus” of the eighteenth.”

Marshall McLuhan (1911–1980) Canadian educator, philosopher, and scholar-- a professor of English literature, a literary critic, and a …

Source: 1960s, The Gutenberg Galaxy (1962), p. 237

“I have throughout introduced the Integral Calculus in connexion with the Differential Calculus…. Is it always proper to learn every branch of a direct subject before anything connected with the inverse relation is considered? If so why are not multiplication and involution in arithmetic made to follow addition and precede subtraction? The portion of the Integral Calculus, which properly belongs to any given portion of the Differential Calculus increases its power a hundred-fold…”

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

Advertisement, pp.3-4
The Differential and Integral Calculus (1836)

Learning

“In the early days of the computer revolution computer designers and numerical analysts worked closely together and indeed were often the same people. Now there is a regrettable tendency for numerical analysts to opt out of any responsibility for the design of the arithmetic facilities and a failure to influence the more basic features of software. It is often said that the use of computers for scientific work represents a small part of the market and numerical analysts have resigned themselves to accepting facilities "designed" for other purposes and making the best of them. […] One of the main virtues of an electronic computer from the point of view of the numerical analyst is its ability to "do arithmetic fast."”

James H. Wilkinson (1919–1986) English mathematician

Need the arithmetic be so bad!
Some Comments from a Numerical Analyst (1971)

People , Failure , Parting

“The bond price, and, as an arithmetical and rigid consequence, the interest rate is an inherently restless variable.”

G. L. S. Shackle (1903–1992) British economist

Source: Epistemics and Economics. (1972), p. 201

“Modern mind has become more and more calculating. The calculative exactness of practical life which the money economy has brought about corresponds to the ideal of natural science: to transform the world into an arithmetic problem, to fix every part of the world by mathematical formulas. Only money economy has filled the days of so many people with weighing, calculating, with numerical determinations, with a reduction of qualitative values to quantitative ones.”

Georg Simmel (1858–1918) German sociologist, philosopher, and critic

Source: The Metropolis and Modern Life (1903), p. 414

Life , Money , People , World

“The way in which the irrational numbers are usually introduced is based directly upon the conception of extensive magnitudes—which itself is nowhere carefully defined—and explains number as the result of measuring such a magnitude by another of the same kind. Instead of this I demand that arithmetic shall be developed out of itself.”

Richard Dedekind (1831–1916) German mathematician

Footnote: The apparent advantage of the generality of this definition of number disappears as soon as we consider complex numbers. According to my view, on the other hand, the notion of the ratio between two numbers of the same kind can be clearly developed only after the introduction of irrational numbers.
Stetigkeit und irrationale Zahlen (1872)

Way

“It is an arithmetic, moreover, which cannot be denied even though we nearly all try to deny it. The arithmetic is simply this: Any positive rate of growth whatever eventually carries a human population to an unacceptable magnitude, no matter how small the rate of growth may be unless the rate of population growth can be reduced to zero before the population reaches an unacceptable magnitude. There is a famous theorem in economics, one which I call the dismal theorem, which states that if the only thing which can check the growth of population is starvation and misery, then the population will grow until it is sufficiently miserable and starving to check its growth. There is a second, even worse theorem which I call the utterly dismal theorem. It says that if the only thing which can check the growth of population is starvation and misery, then the ultimate result of any technological improvement is to enable a larger number of people to live in misery than before and hence to increase the total sum of human misery.”

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

Source: 1960s, The meaning of the twentieth century: the great transition, 1964, p. 126

People

“I must tell you that mathematicians are not scientists, they are artists…. Apart from the most elementary mathematics, like arithmetic or high school algebra, the symbols, formulas and words of mathematics have no meaning at all. The entire structure of pure mathematics is a monstrous swindle, simply a game, a reckless prank. You may well ask: "Are there no renegades to reveal the truth?" Yes, of course. But the facts are so incredible that no one takes them seriously. So the secret is in no danger.”

Theodore Kaczynski (1942) American domestic terrorist, mathematician and anarchist

The Net: The Unabomber, LSD and the Internet https://www.youtube.com/watch?v=xLqrVCi3l6E
Interviews

Truth , Secret , School , Game

“The researches of the last thirty or forty years into the history of mathematics (I need only mention such names as those of [Carl Anton] Bretschneider, Hankel, Moritz Cantor, [Friedrich] Hultsch, Paul Tannery, Zeuthen, Loria, and Heiberg) have put the whole subject upon a different plane. I have endeavoured in this edition to take account of all the main results of these researches up to the present date. Thus, so far as the geometrical Books are concerned, my notes are intended to form a sort of dictionary of the history of elementary geometry, arranged according to subjects; while the notes on the arithmetical Books VII.-IX. and on Book X follow the same plan.”

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

p, 125
The Thirteen Books of Euclid's Elements (1908)

Books , History , Mathematics

“It is a far cry from the abacus to the modern keyboard accounting machine. It will be an equal step to the arithmetical machine of the future. But even this new machine will not take the scientist where he needs to go. Relief must be secured from laborious detailed manipulation of higher mathematics as well, if the users of it are to free their brains for something more than repetitive detailed transformations in accordance with established rules.”

Vannevar Bush book As We May Think

As We May Think (1945)

Mathematics , Future , Cry

“After having passed through the various phases of plastic creation [the phases of arrangement, composition, and construction] I have arrived at the creation of 'universal forms' through constructing upon an arithmetical basis with the pure elements of painting.”

Theo van Doesburg (1883–1931) Dutch architect, painter, draughtsman and writer

Quote in Van Doesburg's article 'From intuition towards certitude', 1930; as quoted in 'Réalités nouvelles', 1947, no. 1, p. 3
1926 – 1931

Universe , Painting

“The invention of the symbol ≡ by Gauss affords a striking example of the advantage which may be derived from an appropriate notation, and marks an epoch in the development of the science of arithmetic.”

George Ballard Mathews (1861–1922) British mathematician

Part 1, sect. 29.
Theory of Numbers, 1892

Science

“More evidently still astronomy attains through arithmetic the investigations that pertain to it, not alone because it is later than geometry in origin—for motion naturally comes after rest—nor because the motions of the stars have a perfectly melodious harmony, but also because risings, settings, progressions, retrogressions, increases, and all sorts of phases are governed by numerical cycles and quantities. So then we have rightly undertaken first the systematic treatment of this, as the science naturally prior, more honorable, and more venerable, and, as it were, mother and nurse of the rest; and here we will take our start for the sake of clearness.”

Nicomachus (60–120) Ancient Greek mathematician

Book I, Chapter V
Nicomachus of Gerasa: Introduction to Arithmetic (1926)

Star , Nature , Science

“Whether in number theory, or space-time cosmology, Gödel’s method was to advance the formalization of the system under consideration and then test it to destruction upon the ‘strange loops’ it generated (paradoxes of self-reference and time-travel). In each case, the system was shown to permit cases that it could not consistently absorb, opening it to an interminable process of revision, or technical improvement. It thus defined dynamic intelligence, or the logic of evolutionary imperfection, with an adequacy that was both sufficient and necessarily inconclusive. What it did not do was trash the very possibility of arithmetic, mathematical logic, or cosmic history — except insofar as these were falsely identified with idols of finality or closure.”

Nick Land (1962) British philosopher

"A Republic, If You Can Keep It" https://web.archive.org/web/20140327090001/http://www.thatsmags.com/shanghai/articles/12321 (2013)

Travel , History , Intelligence , Mathematics

“Somewhere about 1893 I started teaching Scouting to young soldiers in my regiment. When these young fellows joined the Army they had learned reading, writing, and arithmetic in school but as a rule not much else. They were nice lads and made very good parade soldiers, obeyed orders, kept themselves clean and smart and all that, but they had never been taught to be men, how to look after themselves, how to take responsibility, and so on. They had not had my chances of education outside the classroom.
They had been brought up in the herd at school, they were trained as a herd in the Army; they simply did as they were told and had no ideas or initiative of their own. In action they carried out orders, but if their officer was shot they were as helpless as a flock of sheep. Tell one of them to ride out alone with a message on a dark night and ten to one he would lose his way.
I wanted to make them feel that they were a match for any enemy, able to find their way by the stars or map, accustomed to notice all tracks and signs and to read their meaning, and able to fend for themselves away from regimental cooks and barracks.”

Robert Baden-Powell (1857–1941) lieutenant-general in the British Army, writer, founder and Chief Scout of the Scout Movement

"BE PREPARED" http://www.pinetreeweb.com/bp-listener.htm, Listener Magazine (1937)

Men , Education , School , Way

“When we speak of the early history of algebra it is necessary to consider… the meaning of the term. If… we mean the science that allows us to solve the equation ax^2 + bx + c = 0, expressed in these symbols, then the history begins in the 17th century; if we remove the restriction as to these particular signs, and allow for other and less convenient symbols, we might properly begin the history in the 3rd century; if we allow for the solution of the above equation by geometric methods, without algebraic symbols of any kind, we might say that algebra begins with the Alexandrian School or a little earlier; and if we say that we should class as algebra any problem that we should now solve with algebra (even though it was as first solved by mere guessing or by some cumbersome arithmetic process), the the science was known about 1800 B. C., and probably still earlier.<”

David Eugene Smith (1860–1944) American mathematician

Source: History of Mathematics (1925) Vol.2, Ch. 6: Algebra, p. 378

History , Problem , Science

“Mathematics says the sum value of a network increases as the square of the number of members. In other words, as the number of nodes in a network increases arithmetically, the value of the network increases exponentially. Adding a few more members can dramatically increase the value of the network.”

Kevin Kelly (1952) American author and editor

Out of Control: The New Biology of Machines, Social Systems and the Economic World (1995), New Rules for the New Economy: 10 Radical Strategies for a Connected World (1999)

Mathematics

“We know, and we feel, that the vast business of our redemption, arranged in the councils of the far-back eternity, and acted out amid the wonderings and throbbings of the universe, could not have been that stupendous transaction which gave God glory by giving sinners safety, if the inspired account brought its dimensions within the compass of a human arithmetic, or denned its issues by the lines of a human demarcation.”

Henry Melvill (1798–1871) British academic

Source: Dictionary of Burning Words of Brilliant Writers (1895), P. 423.

God , Feeling , Business , Universe

“Bill, why is it that some apparently-grown men never learn to do simple arithmetic?”

Robert A. Heinlein book Farmer in the Sky

Source: Farmer in the Sky (1950), Chapter 14, “Land of My Own” (p. 142)

Men , Learning

“The needs of business, and the extensive market obviously waiting, assured the advent of mass-produced arithmetical machines just as soon as production methods were sufficiently advanced.
With machines for advanced analysis no such situation existed; for there was and is no extensive market; the users of advanced methods of manipulating data are a very small part of the population.”

Vannevar Bush book As We May Think

As We May Think (1945)

Business , Waiting , Parting

“Others … are in the habit of teaching that religion and philosophy are really the same thing. Such a statement, however, appears to be true only in the sense in which Francis I is supposed to have said in a very conciliatory tone with reference to Charles V: ‘what my brother Charles wants is also what I want’, namely Milan. Others again do not stand on such ceremony, but talk bluntly of a Christian philosophy, which is much the same as if we were to speak of a Christian arithmetic, and this would be stretching a point. Moreover, epithets taken from such dogmas are obviously unbecoming of philosophy, for it is devoted to the attempt of the faculty of reason to solve by its own means and independently of all authority the problem of existence.”

Arthur Schopenhauer book Parerga and Paralipomena

Andere wieder, von diesen Wahrheitsforschern, schmelzen Philosophie und Religion zu einem Kentauren zusammen, den sie Religionsphilosophie nennen; Pflegen auch zu lehren, Religion und Philosophie seien eigentlich das Selbe;—welcher Sah jedoch nur in dem Sinne wahr zu seyn scheint, in welchem Franz I., in Beziehung auf Karl V., sehr versöhnlich gesagt haben soll: „was mein Bruder Karl will, das will ich auch,”—nämlich Mailand, Wieder andere machen nicht so viele Umstände, sondern reden geradezu von einer Christlichen Philosophie;—welches ungefähr so herauskommt, wie wenn man von einer Christlichen Arithmetik reden wollte, die fünf gerade seyn ließe. Dergleichen von Glaubenslehren entnommene Epitheta sind zudem der Philosophie offenbar unanständig, da sie sich für den Versuch der Vernunft giebt, aus eigenen Mitteln und unabhängig von aller Auktorität das Problem des Daseyns zu lösen.
Sämtliche Werke, Bd. 5, p. 155, E. Payne, trans. (1974) Vol. 1, pp. 142-143
Parerga and Paralipomena (1851), On Philosophy in the Universities

Religion , Problem , Sense , Appearance

“Men cannot be treated as units in operations of political arithmetic because they behave like the symbols for zero and the infinite, which dislocate all mathematical operations.”

Arthur Koestler (1905–1983) Hungarian-British author and journalist

Crossman (1949), p. 68.

Men , Mathematics

“The science of the age, in short, is physical, chemical, physiological; in all shapes mechanical. Our favourite Mathematics, the highly prized exponent of all these other sciences, has also become more and more mechanical. Excellence in what is called its higher departments depends less on natural genius than on acquired expertness in wielding its machinery. Without undervaluing the wonderful results which a Lagrange or Laplace educes by means of it, we may remark, that their calculus, differential and integral, is little else than a more cunningly-constructed arithmetical mill; where the factors, being put in, are, as it were, ground into the true product, under cover, and without other effort on our part than steady turning of the handle. We have more Mathematics than ever; but less Mathesis.”

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

1820s, Signs of the Times (1829)

Age , Education , Mathematics , Nature

“When, where, and how were you, Joseph Smith, first called? How old were you? and what were you qualifications? I was between fourteen and fifteen years of age. Had you been to college? No. Had you studied in any seminary of learning? No. Did you know how to read? Yes. How to write? Yes. Did you understand much about arithmetic? No. About grammar? No. Did you understand all the branches of education which are generally taught in our common schools? No. But yet you say the Lord called you when you were but fourteen or fifteen years of age? How did he call you? I will give you a brief history as it came from his own mouth. I have often heard him relate it. He was wrought upon by the Spirit of God, and felt the necessity of repenting of his sins and serving God. He retired from his father's house a little way, and bowed himself down in the wilderness, and called upon the name of the Lord. He was inexperienced, and in great anxiety and trouble of mind in regard to what church he should join. He had been solicited by many churches to join with them, and he was in great anxiety to know which was right. He pleaded with the Lord to give him wisdom on the subject; and while he was thus praying, he beheld a vision, and saw a light approaching him from the heavens; and as it came down and rested on the tops of the trees, it became more glorious; and as it surrounded him, his mind was immediately caught away from beholding surrounding objects. In this cloud of light he saw two glorious personages; and one, pointing to the other, said, "Behold my beloved son! hear ye him."”

Orson Pratt (1811–1881) Apostle of the LDS Church

Journal of Discourses 7:220 (August 14, 1859).
Joseph Smith Jr.'s First Vision

Wisdom , God , Age , Education

“The Essential Parts of the Simple or Pure Mathematicks are Arithmetick and Geometry, which mutually assist one another, and are independent on any other Sciences, except perhaps on Artificial Logick: But doubtless Natural Logick may be sufficient to a Man of Sense. The other parts are chiefly Physical Subjects explained by the Principles of Arithmetics or Geometry.”

Jacques Ozanam (1640–1718) French mathematician

Source: A Mathematical Dictionary: Or; A Compendious Explication of All Mathematical Terms, 1702, p. 2, The introduction

Nature , Sense , Science , Parting

“Simon, always a fool for simplicity, accepted. Punch took an envelope out of his pocket and scribbled on the back of it. He said, 'This has a simple arithmetical proof but no rational explanation of the paradox.' He gave it to Simon. Simon read it, looked at Punch with raised eyebrows, hunched his shoulders, shook his head sadly, and got up and left the room without a word.”

William Grey Walter (1910–1977) American-born British neuroscientist and roboticist

Source: The Curve of the Snowflake (1956), p. 72.

Fools , Reading

“That such comparisons with non-arithmetic notions have furnished the immediate occasion for the extension of the number-concept may, in a general way, be granted (though this was certainly not the case in the introduction of complex numbers); but this surely is no sufficient ground for introducing these foreign notions into arithmetic, the science of numbers.”

Richard Dedekind (1831–1916) German mathematician

p, 125
Stetigkeit und irrationale Zahlen (1872)

Way , Science

“The science of arithmetic may be called the science of exact limitation of matter and things in space, force, and time.”

Francis Wayland Parker (1837–1902) Union Army officer

Source: Talks on Pedagogics, (1894), p. 64. Reported in Robert Edouard Moritz. Memorabilia mathematica; or, The philomath's quotation-book https://archive.org/stream/memorabiliamathe00moriiala#page/81/mode/2up, (1914), p. 263

Science , Time

“Now as to what pertains to these Surd numbers (which, as it were by way of reproach and calumny, having no merit of their own are also styled Irrational, Irregular, and Inexplicable) they are by many denied to be numbers properly speaking, and are wont to be banished from arithmetic to another Science, (which yet is no science) viz. algebra.”

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

Source: Mathematical Lectures (1734), p. 44

Way , Science

“The particles of this cosmos rock and roll to their own self-generated beat. They defy the rules of arithmetic. Protons plus neutrons… equals music. Is this harmony… entropy? …No. …it's social behavior …riddled with form. And it's so antientropic that those in the scientific world who are trying desparately to rescue entropy… call it "negentropy."”

Howard Bloom (1943) American publicist and author

…Entropy is a very big assumption.
Heresy Number Three
The God Problem: How a Godless Cosmos Creates (2012)

World , Music , Behavior

“Legendre's Law of Quadratic Reciprocity' … is …the most important general truth in the science of integral numbers which has been discovered since the time of Fermat. It has been called by Gauss 'the gem of the higher arithmetic,' and is equally remarkable whether we consider the simplicity of its enunciation, the difficulties which for a long time attended its demonstration, or the number and variety of the results which have been obtained by its means. …[W]e find in the 'Opuscula Analytica' of Euler… a memoir… which contains a general and very elegant theorem from which the Law of Reciprocity is immediately deducible, and which is, vice versâ, deducible from that law. But Euler… expressly observes that the theorem is undemonstrated; and this would seem to be the only place in which he mentions it in connexion with the theory of the Residues of Powers; though in other researches he has frequently developed results which are consequences of the theorem, and which relate to the linear forms of the divisors of quadratic formulae. But here also his conclusions repose on induction only; though in one memoir he seems to have imagined… that he had obtained a satisfactory demonstration.”

Henry John Stephen Smith (1826–1883) mathematician

Report on the Theory of Numbers (1859) Part I, pp. 56-57.
The Collected Mathematical Papers of Henry John Stephen Smith (1894) Vol. 1

Truth , Law , Imagination , Science

“I have no great faith in political arithmetic, and I mean not to warrant the exactness of either of these computations.”

Adam Smith (1723–1790) Scottish moral philosopher and political economist

Source: (1776), Book IV, Chapter V, p. 577.

Faith

“The Reason of the present Animadversions. …How far Hevelius has proceeded. That his instruments do not much exceed Ticho. The bigness, Sights and Divisions, not considerably differing. Ticho not ignorant of his new way of Division. …That so great curiosity as Hevelius strives for is needless without the use of Telescopic Sights, the power of the naked eye being limited. That no one part of an Instrument should be more perfect than another. …
That if Hevelius could have been prevail'd on by the Author to have used Telescopic Sights, his observations might have been 40 times more exact than they are.
That Hevelius his Objections against Telescopic sights are of no validity; but the Sights without Telescopes cannot distinguish a less angle then half a Minute.
That an Instrument of 3 foot Radius with Telescopes, will do more then one of 3 score foot Radius with common Sights, the eye being unable to distinguish. This is proved by the undiscernableness of spots in the Moon, and by an Experiment with Lines on a paper, by which a Standard is made of the power of the eye. …
A Conclusion of the Animadversions. That the learn'd World is obllig'd to Hevelius for what he hath done, but would have more, if he had used other instruments.
That the Animadvertor both contrived some hundreds of Instruments, each of very great accurateness for taking Angles, Levels, &c.; and a particular Arithmetical lnstrument for performing all Operations in Arithmetick, with the greatest ease, swiftness and certainty imaginable.
That the Reader may be the more certain of this, the Author describes an Instrument for taking Angles in the Heavens…”

Robert Hooke (1635–1703) English natural philosopher, architect and polymath

Contents, Animadversions on the First Part of the Machina Coelestis of the Astronomer Johannes Hevelius https://books.google.com/books?id=KAtPAAAAcAAJ (1674)

World , Way , Imagination , Time

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

Nicomachus (60–120) Ancient Greek mathematician

Nicomachus of Gerasa: Introduction to Arithmetic (1926)

Life , God , Truth , Fear

“I hope I may claim in the present work to have made it probable that the laws of arithmetic are analytic judgments and consequently a priori. Arithmetic thus becomes simply a development of logic, and every proposition of arithmetic a law of logic, albeit a derivative one. To apply arithmetic in the physical sciences is to bring logic to bear on observed facts; calculation becomes deduction.”

Gottlob Frege book The Foundations of Arithmetic

Gottlob Frege (1950 [1884]). The Foundations of Arithmetic. p. 99.

Law , Hope , Science

“At the repeated solicitations of my son Charley to procure him a berth as a midshipman in the navy, I take the liberty to address you the following lines, requesting you will be pleased to interest yourself in my son's behalf with Mr. Hamilton, so as to procure him the midshipman's berth in some of our frigates. Mr. Hamilton, I am sure, is a gentleman of great merit and of superior discernment in his department; consequently will duly appreciate the merits of a youth stepping forward in defense of his country. Though he is only sixteen years old, as a parent I can with confidence say [that] my son is of an engaging, aspiring disposition by nature, regardless of those little difficulties and trials that generally retard and dishearten those of his age in the pursuit of their favorite object. He has been educated for some years in Georgetown College and, consequently, has studied the languages for a certain space of time and is as well versed in arithmetic as most of his age generally are.”

Charles Boarman (1795–1879) US Navy Rear Admiral

Charles Boarman, Sr. in a letter to Robert Brent, the mayor of Washington, D.C., asking for a letter of recommendation for his son's application to enlist in the United States Navy (1811)
A Gentlemanly and Honorable Profession: The Creation of the U.S. Naval Officer Corps, 1794-1815 (1991)

Youth , Age , Education , Confidence

“Your discovery of the contradiction caused me the greatest surprise and, I would almost say, consternation, since it has shaken the basis on which I intended to build arithmetic.”

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

"Letter to Bertrand Russel" (1902) in J. van Heijenoort, ed., From Frege to Godel: A Source Book in Mathematical Logic, 1879-1931 (1967)

“In the arithmetic of love, one plus one equals everything, and two minus one equals nothing.”

Mignon McLaughlin (1913–1983) American journalist

The Complete Neurotic's Notebook (1981), Love

Wedding , Love

“… your 90MHz Pentium won't have any trouble doing arithmetic (except for certain divisions).”

Paul DiLascia (1959–2008) American software developer

1995/3
About the Industry

“Some writers maintain arithmetic to be only the only sure guide in political economy; for my part, I see so many detestable systems built upon arithmetical statements, that I am rather inclined to regard that science as the instrument of national calamity.”

Jean-Baptiste Say (1767–1832) French economist and businessman

Source: A Treatise On Political Economy (Fourth Edition) (1832), Book I, On Production, Chapter XVII, Section III, p. 188

Science , Parting

“Strictly speaking, the theory of numbers has nothing to do with negative, or fractional, or irrational quantities, as such. No theorem which cannot be expressed without reference to these notions is purely arithmetical: and no proof of an arithmetical theorem, can be considered finally satisfactory if it intrinsically depends upon extraneous analytical theories.”

George Ballard Mathews (1861–1922) British mathematician

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

Dependency

“Suppose then I want to give myself a little training in the art of reasoning; suppose I want to get out of the region of conjecture and probability, free myself from the difficult task of weighing evidence, and putting instances together to arrive at general propositions, and simply desire to know how to deal with my general propositions when I get them, and how to deduce right inferences from them; it is clear that I shall obtain this sort of discipline best in those departments of thought in which the first principles are unquestionably true. For in all 59 our thinking, if we come to erroneous conclusions, we come to them either by accepting false premises to start with—in which case our reasoning, however good, will not save us from error; or by reasoning badly, in which case the data we start from may be perfectly sound, and yet our conclusions may be false. But in the mathematical or pure sciences,—geometry, arithmetic, algebra, trigonometry, the calculus of variations or of curves,—we know at least that there is not, and cannot be, error in our first principles, and we may therefore fasten our whole attention upon the processes. As mere exercises in logic, therefore, these sciences, based as they all are on primary truths relating to space and number, have always been supposed to furnish the most exact discipline. When Plato wrote over the portal of his school. “Let no one ignorant of geometry enter here,” he did not mean that questions relating to lines and surfaces would be discussed by his disciples. On the contrary, the topics to which he directed their attention were some of the deepest problems,—social, political, moral,—on which the mind could exercise itself. Plato and his followers tried to think out together conclusions respecting the being, the duty, and the destiny of man, and the relation in which he stood to the gods and to the unseen world. What had geometry to do with these things? Simply this: That a man whose mind has not undergone a rigorous training in systematic thinking, and in the art of drawing legitimate inferences from premises, was unfitted to enter on the discussion of these high topics; and that the sort of logical discipline which he needed was most likely to be obtained from geometry—the only mathematical science which in Plato’s time had been formulated and reduced to a system. And we in this country [England] have long acted on the same principle. Our future lawyers, clergy, and statesmen are expected at the University to learn a good deal about curves, and angles, and numbers and proportions; not because these subjects have the smallest relation to the needs of their lives, but because in the very act of learning them they are likely to acquire that habit of steadfast and accurate thinking, which is indispensable to success in all the pursuits of life.”

Joshua Girling Fitch (1824–1903) British educationalist

Source: Lectures on Teaching, (1906), pp. 291-292

Life , God , Truth , World

“Arithmetic is the science of the Evaluation of Functions, Algebra is the science of the Transformation of Functions.”

George Holmes Howison (1834–1916) American philosopher

Journal of Speculative Philosophy, Vol. 5, p. 175. Reported in: Memorabilia mathematica or, The philomath's quotation-book, by Robert Edouard Moritz. Published 1914
Journals

Science

“The free world, at least dimly aware of these facts, has naturally embarked on a large program of warning and defense systems. That program will be accelerated and extended. But let no one think that the expenditure of vast sums for weapons and systems of defense can guarantee absolute safety for the cities and citizens of any nation. The awful arithmetic of the atomic bomb does not permit of any such easy solution. Even against the most powerful defense, an aggressor in possession of the effective minimum number of atomic bombs for a surprise attack could probably place a sufficient number of his bombs on the chosen targets to cause hideous damage.”

Dwight D. Eisenhower (1890–1969) American general and politician, 34th president of the United States (in office from 1953 to 1961)

1950s, Atoms for Peace (1953)

World , Nature , Thinking

“Unfortunately, truth is neither a listable nor a decidable property; nor is the truth of a statement of arithmetic. The American logician John Myhill has used the term 'prospective' to characterize those attributes of the world that are neither listable nor decidable. They are properties that cannot be recognized by the application of some formula, made to conform to a rule, or generated by some computer program. They are characterized by incessant novelty that cannot be encompassed by any finite set of rules. 'Beauty', 'ugliness', 'truth', 'harmony', simplicity', and 'poetry' are names we give to some of the attributes of this sort. There is no way of listing all examples of beauty or ugliness, nor any procedure for saying whether or not something possesses either of those attributes, without redefining them in some more restrictive fashion that kills their prospective character.”

John D. Barrow (1952–2020) British scientist

Source: The Artful Universe (1995), Ch. 5, pp. 219-220

Truth , World , Beauty , Way

“If geometry exists, arithmetic must also needs be implied”

Nicomachus (60–120) Ancient Greek mathematician

Nicomachus of Gerasa: Introduction to Arithmetic (1926)
Context: If geometry exists, arithmetic must also needs be implied... But on the contrary 3, 4, and the rest might be 5 without the figures existing to which they give names. Hence arithmetic abolishes geometry along with itself, but is not abolished by it, and while it is implied by geometry, it does not itself imply geometry.<!--Book I, Chapter IV

“I believe mysticism is a very serious endeavor. One must be equipped for it. One doesn't study calculus before studying arithmetic.”

Elie Wiesel (1928–2016) writer, professor, political activist, Nobel Laureate, and Holocaust survivor

As quoted in "10 Questions for Elie Wiesel" by Jeff Chu in TIME (22 January 2006) http://www.time.com/time/magazine/article/0,9171,1151803,00.html
Context: I believe mysticism is a very serious endeavor. One must be equipped for it. One doesn't study calculus before studying arithmetic. In my tradition, one must wait until one has learned a lot of Bible and Talmud and the Prophets to handle mysticism. This isn't instant coffee. There is no instant mysticism.

Study

“Babbage, even with remarkably generous support for his time, could not produce his great arithmetical machine. His idea was sound enough, but construction and maintenance costs were then too heavy.”

Vannevar Bush book As We May Think

As We May Think (1945)
Context: Babbage, even with remarkably generous support for his time, could not produce his great arithmetical machine. His idea was sound enough, but construction and maintenance costs were then too heavy. Had a Pharaoh been given detailed and explicit designs of an automobile, and had he understood them completely, it would have taxed the resources of his kingdom to have fashioned the thousands of parts for a single car, and that car would have broken down on the first trip to Giza.

Idea , Support , Time

“There is scarcely any one who states purely arithmetical questions, scarcely any who understands them. Is this not because arithmetic has been treated up to this time geometrically rather than arithmetically?”

Pierre de Fermat (1601–1665) French mathematician and lawyer

Letter to Frénicle (1657) Oeuvres de Fermat Vol.II as quoted by Edward Everett Whitford, The Pell Equation http://books.google.com/books?id=L6QKAAAAYAAJ (1912)
Context: There is scarcely any one who states purely arithmetical questions, scarcely any who understands them. Is this not because arithmetic has been treated up to this time geometrically rather than arithmetically? This certainly is indicated by many works ancient and modern. Diophantus himself also indicates this. But he has freed himself from geometry a little more than others have, in that he limits his analysis to rational numbers only; nevertheless the Zetcica of Vieta, in which the methods of Diophantus are extended to continuous magnitude and therefore to geometry, witness the insufficient separation of arithmetic from geometry. Now arithmetic has a special domain of its own, the theory of numbers. This was touched upon but only to a slight degree by Euclid in his Elements, and by those who followed him it has not been sufficiently extended, unless perchance it lies hid in those books of Diophantus which the ravages of time have destroyed. Arithmeticians have now to develop or restore it. To these, that I may lead the way, I propose this theorem to be proved or problem to be solved. If they succeed in discovering the proof or solution, they will acknowledge that questions of this kind are not inferior to the more celebrated ones from geometry either for depth or difficulty or method of proof: Given any number which is not a square, there also exists an infinite number of squares such that when multiplied into the given number and unity is added to the product, the result is a square.

Understanding , Question , Time

“If scientific reasoning were limited to the logical processes of arithmetic, we should not get far in our understanding of the physical world. One might as well attempt to grasp the game of poker entirely by the use of the mathematics of probability.”

Vannevar Bush book As We May Think

As We May Think (1945)
Context: If scientific reasoning were limited to the logical processes of arithmetic, we should not get far in our understanding of the physical world. One might as well attempt to grasp the game of poker entirely by the use of the mathematics of probability. The abacus, with its beads strung on parallel wires, led the Arabs to positional numeration and the concept of zero many centuries before the rest of the world; and it was a useful tool — so useful that it still exists.

World , Game , Mathematics , Understanding

“Any one who considers arithmetical methods of producing random digits is, of course, in a state of sin.”

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

On mistaking pseudorandom number generators for being truly "random" — this quote is often erroneously interpreted to mean that von Neumann was against the use of pseudorandom numbers, when in reality he was cautioning about misunderstanding their true nature while advocating their use. From "Various techniques used in connection with random digits" by John von Neumann in Monte Carlo Method (1951) edited by A.S. Householder, G.E. Forsythe, and H.H. Germond 
Context: Any one who considers arithmetical methods of producing random digits is, of course, in a state of sin. For, as has been pointed out several times, there is no such thing as a random number — there are only methods to produce random numbers, and a strict arithmetic procedure of course is not such a method.

“But in my arithmetic, take one from one-”

Laxmi Prasad Devkota (1909–1959) Nepali poet

Lunatic. 3
पागल (The Lunatic)
Context: You're clever, quick with words, your exact equations are right forever and ever. But in my arithmetic, take one from one- and there's still one left. You get along with five senses, I with a sixth. You have a brain, friend, I have a heart. A rose is just a rose to you- to me it's Helen and Padmini. You are forceful prose I liquid verse. When you freeze I melt, When you're clear I get muddled and then it works the other way around. Your world is solid, mine vapor, yours coarse, mine subtle. You think a stone reality; harsh cruelty is real for you. I try to catch a dream, the way you grasp the rounded truth of cold, sweet coin.

“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

Mathematics , Science

“The more we progress the more we tend to progress. We advance not in arithmetical but in geometrical progression. We draw compound interest on the whole capital of knowledge and virtue which has been accumulated since the dawning of time.”

Arthur Conan Doyle book The Stark Munro Letters

The Stark Munro Letters (1894)
Context: The more we progress the more we tend to progress. We advance not in arithmetical but in geometrical progression. We draw compound interest on the whole capital of knowledge and virtue which has been accumulated since the dawning of time. Some eighty thousand years are supposed to have existed between paleolithic and neolithic man. Yet in all that time he only learned to grind his flint stones instead of chipping them. But within our father's lives what changes have there not been? The railway and the telegraph, chloroform and applied electricity. Ten years now go further than a thousand then, not so much on account of our finer intellects as because the light we have shows us the way to more. Primeval man stumbled along with peering eyes, and slow, uncertain footsteps. Now we walk briskly towards our unknown goal.

Knowledge , Time , Drawing

“The Fourier expansion of an elliptic modular form has been fruitfully utilized in various arithmetical problems as well as in the study of the analytic properties of the form itself. The same can be said also for the Hilbert and Siegel modular forms.”

Goro Shimura (1930–2019) Japanese mathematician

[The arithmetic of forms with respect to a unitary group, Annals of Mathematics, 107, 1978, 569–605, https://books.google.com/books?id=f8gB564cK68C&pg=PA38]

Problem , Study

“Knowledge is immense and life is short: therefore it is not obligatory to learn all the sciences, such as Astronomy and Medicine, and Arithmetic, etc., but only so much of each as bears upon the religious law: enough astronomy to know the times (of prayer) in the night, enough medicine to abstain from what is injurious, enough arithmetic to understand the division of inheritances and to calculate the duration of the Iddat.”

Ali Hujwiri book Kashf ul Mahjoob

Kashf ul Mahjoob, Chapter I, Affirmation of Knowledge, p. 80
Kashf ul Mahjoob

Time , Night , Law , Understanding

“We are amphibious creatures, weaponed for two elements, having two sets of faculties, the particular and the catholic. We adjust our instrument for general observation, and sweep the heavens as easily as we pick out a single figure in the terrestrial landscape. We are practically skilful in detecting elements, for which we have no place in our theory, and no name. Thus we are very sensible of an atmospheric influence in men and in bodies of men, not accounted for in an arithmetical addition of all their measurable properties. There is a genius of a nation, which is not to be found in the numerical citizens, but which characterizes the society. England, strong, punctual, practical, well-spoken England, I should not find, if I should go to the island to seek it. In the parliament, in the playhouse, at dinner-tables, I might see a great number of rich, ignorant, book-read, conventional, proud men, — many old women, — and not anywhere the Englishman who made the good speeches, combined the accurate engines, and did the bold and nervous deeds. It is even worse in America, where, from the intellectual quickness of the race, the genius of the country is more splendid in its promise, and more slight in its performance.”

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

1840s, Essays: Second Series (1844), Nominalist and Realist

Reading , Country , Strong , Books

“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 , Mathematics , Success

“I think any such analysis must take into account the intelligence of the people. People’s ability to understand the arithmetic and the logic behind the coming together of the Mahamilawat (great contamination) parties should never be underestimated. People know exactly why they are coming together, no matter how much these parties themselves try to mislead. There was a time when many different parties did come together to fight an election. This was the Emergency era when survival of our democracy was first priority. So, when many parties united, there was a clear issue that brought them together in national interest and people blessed them. What is the uniting factor for the Mahamilawat now? Personal survival and personal interest. People are aware of this and will reject such opportunism.”

Narendra Modi (1950) Prime Minister of India

2019, "2014 was a mandate for hope and aspiration, 2019 is about confidence and acceleration", 2019

Democracy , Time , Understanding , Personality

“I hope I may claim in the present work to have made it probable that the laws of arithmetic are analytic judgments and consequently a priori.”

Gottlob Frege book The Foundations of Arithmetic

Arithmetic thus becomes simply a development of logic, and every proposition of arithmetic a law of logic, albeit a derivative one. To apply arithmetic in the physical sciences is to bring logic to bear on observed facts; calculation becomes deduction.
Gottlob Frege (1950 [1884]). The Foundations of Arithmetic. p. 99.

Hope , Law

“I have been taking stock of my 50 years since I left Wichita. How I have existed fills me with horror for I failed in everything. Spelling, arithmetic, writing, swimming, tennis, golf, dancing, singing, acting, wife, mistress, whore, friend, even cooking. And I do not excuse myself with the usual escape of not trying. I tried with all my heart.”

Louise Brooks (1906–1985) American dancer and actress

Looking for Lulu (1998)

Singing , Whore , Heart , Dance

“I have throughout introduced the Integral Calculus in connexion with the Differential Calculus. …Is it always proper to learn every branch of a direct subject before anything connected with the inverse relation is considered? If so why are not multiplication and involution in arithmetic made to follow addition and precede subtraction?”

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

The portion of the Integral Calculus, which properly belongs to any given portion of the Differential Calculus increases its power a hundred-fold...
The Differential and Integral Calculus (1836)

Relation ships , Learning