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Leibniz binary system, 1697.

Arithmetic or arithmetics (from the Greek word ἀριθμός, arithmos "number") is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple day-to-day counting to advanced science and business calculations.


  • 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.
  • A hieratic papyrus, included in the Rhind collection of the British Museum, was deciphered by Eisenlohr in 1877, and found to be a mathematical manual containing problems in arithmetic and geometry. It was written by Ahmes some time before 1700 B.C., and was founded on an older work believed by Birch to date back as far as 3400 B.C.! This curious papyrus -- the most ancient mathematical handbook known to us -- puts us at once in contact with the mathematical thought in Egypt of three or five thousand years ago. It is entitled "Directions for obtaining the Knowledge of all Dark Things." We see from it that the Egyptians cared but little for theoretical results. Theorems are not found in it at all. It contains "hardly any general rules of procedure, but chiefly mere statements of results intended possibly to be explained by a teacher to his pupils."
  • "I couldn't afford to learn it," said the Mock Turtle with a sigh. "I only took the regular course."

    "What was that?" inquired Alice.

    "Reeling and Writhing, of course, to begin with," the Mock Turtle replied; "and then the different branches of Arithmetic - Ambition, Distraction, Uglification, and Derision."

  • No man acquires property without acquiring with it a little arithmetic, also.
  • 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.
  • 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, Montgomery Furth (1964) The Basic Laws of Arithmetic: Exposition Ofthe System. p.10
  • But in our opinion truths of this kind should be drawn from notions rather than from notations.
  • 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.
God made the integers, all the rest is the work of man.
- L. Kroneckker (1823-1891)
  • A great part of its theories derives an additional charm from the peculiarity that important propositions, with the impress of simplicity on them, are often easily discovered by induction, and yet are of so profound a character that we cannot find the demonstrations till after many vain attempts; and even then, when we do succeed, it is often by some tedious and artificial process, while the simple methods may long remain concealed.
  • Statistics began as the systematic study of quantitative facts about the state.
    • Ian Hacking (1975) The Emergence Of Probability Chapter 12, Political Arithmetic, p. 102
  • If the potential of every number is in the monad, then the monad would be intelligible number in the strict sense, since it is not yet manifesting anything actual, but everything conceptually together in it.
    • Iamblichus (c. 245 - c. 325) The Theology of Arithmetic On the Monad
  • Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk.
    • God made the integers, all the rest is the work of man.
    • Leopold Kronecker (1823-1891) quoted in A. George, D.J. Velleman (2002) "Philosophies of Mathematics". p.13
  • I do hate sums, There is no greater mistake than to call arithmetic an exact science. There are Permutations and Aberrations discernible to minds entirely noble like mine; subtle variations which ordinary accountants fail to discover; hidden laws of Numbers which it requires a mind like mine to perceive. For instance, if you add a sum from the bottom up, and then again from the top down, the result is always different.
    • Maria Price La Touche, The Letters of a Noble Woman (1908), ed. Margaret Ferrier Young, letter dated July 1878, p. 49 (quoted in Mathematical Gazette, Vol. 12, 1924)
  • 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.
  • Population, when unchecked, increases in a geometrical ratio, Subsistence, increases only in an arithmetical ratio.
    • Thomas Malthus (1798) An Essay on The Principle of Population. Chapter I, paragraph 18, lines 1-2
  • He who refuses to do arithmetic is doomed to talk nonsense.
  • 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.
  • 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.
    • John von Neumann, "Various techniques used in connection with random digits" by John von Neumann in Monte Carlo Method (1951), ed. A.S. Householder, G.E. Forsythe, and H.H. Germond
  • Questions that pertain to the foundations of mathematics, although treated by many in recent times, still lack a satisfactory solution. Ambiguity of language is philosophy's main source of problems. That is why it is of the utmost importance to examine attentively the very words we use.
    • Giuseppe Peano (1889) Arithmetices Principia, nova methodo exposita [The Principles of Arithmetic, presented by a new method]
  • 1. 0 is a number.
    2. The immediate successor of a number is also a number.
    3. 0 is not the immediate successor of any number.
    4. No two numbers have the same immediate successor.
    5. Any property belonging to 0 and to the immediate successor of any number that also has that property belongs to all numbers.
    • Giuseppe Peano As expressed in Galileo's Finger: The Ten Great Ideas of Science (2003) by Peter Atkins, Ch. 10 "Arithmetic : The Limits of Reason", p. 333
  • All relations are either qualitative or quantitative. Qualitative relations can be considered by themselves without regard to quantity. The algebra of such enquiries may be called logical algebra, of which a fine example is given by Boole.
    Quantitative relations may also be considered by themselves without regard to quality. They belong to arithmetic, and the corresponding algebra is the common or arithmetical algebra.
    In all other algebras both relations must be combined, and the algebra must conform to the character of the relations.
  • The most savage controversies are those about matters as to which there is no good evidence either way. Persecution is used in theology, not in arithmetic, because in arithmetic there is knowledge, but in theology there is only opinion.
  • 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 (1832) A Treatise On Political Economy Book I, On Production, Chapter XVII, Section III, p. 188

A History of Western Philosophy, 1945[edit]

Source: A History of Western Philosophy (1945) by the philosopher Bertrand Russell

Pythagoras, as everyone knows, said that "all things are numbers."
  • Pythagoras, as everyone knows, said that "all things are numbers." This statement, interpreted in a modern way, is logical nonsense, but what he meant was not exactly nonsense. He discovered the importance of numbers in music and the connection which he established between music and arithmetic survives in the mathematical terms "harmonic mean" and "harmonic progression." He thought of numbers as shapes, as they appear on dice or playing cards. We still speak of squares or cubes of numbers, which are terms that we owe to him. He also spoke of oblong numbers, triangular numbers, pyramidal numbers, and so on. These were the numbers of pebbles (or as we would more naturally say, shot) required to make the shapes in question. ...He presumably thought of the world as atomic, and of bodies as built up of molecules composed of atoms arranged in various shapes. In this way he hoped to make arithmetic the fundamental study in physics as in aesthetics.
    • Book One, Part I, Chapter III, Pythagoras, p. 35
  • Unfortunately for Pythagoras, his theorem led at once to the discovery of incommensurables, which appeared to disprove his whole philosophy. So long as no adequate arithmetical theory on incommensurables existed, the method of Euclid was the best that was possible in geometry. When Descartes introduced co-ordinate geometry, thereby again making arithmetic supreme, he [Descartes] assumed the possibility of a solution of the problem of incommensurables, though in his day no such solution had been found.
    • Book One, Part I, Chapter III, Pythagoras, p. 35-6
  • Mystical doctrines as to the relation of time to eternity are also reinforced by pure mathematics, for the mathematical objects, such as numbers, if real at all, are eternal and not in time. Such eternal objects can be conceived as God's thoughts. Hence Plato's doctrine that God is a geometer, and Sir James Jeans' belief that He is addicted to arithmetic.
    • Book One, Part I, Chapter III, Pythagoras, p. 37
  • Not only philosophers were influenced by Plato. Why did the Puritans object to the music and painting and gorgeous ritual of the Catholic Church? You will find the answer in the tenth book of the Republic. Why are children compelled to learn arithmetic? The reasons are given in the seventh book.
    • Book One, Part II, Chapter XV, The Theory of Ideas, p. 120
  • Plato proceeds to an interesting sketch of the education proper to a young man who is to be a guardian. ...The young man chosen for these merits will spend the years from twenty to thirty on the four Pythagorean studies: arithmetic, geometry (plane and solid), astronomy, and harmony. These studies are not to be pursued in any utilitarian spirit, but in order to prepare his mind for the vision of eternal things. In astronomy, for example, he is not to trouble himself too much about the actual heavenly bodies, but rather with the mathematics of motion of ideal heavenly bodies. This may seem absurd to modern ears, but, strange to say, it proved to be a fruitful point of view in connection with empirical astronomy.
    • Book One, Part II, Chapter XV, The Theory of Ideas, p. 130
  • It is noteworthy that modern Platonists, almost without exception, are ignorant of mathematics, in spite of the immense importance that Plato attached to arithmetic and geometry, and the immense influence that they [these studies] had on his philosophy. This is an example of the evils of specialization: a man must not write on Plato unless he has spent so much of his youth on Greek as to have no time for the things that Plato thought important.
    • Book One, Part II, Chapter XV, The Theory of Ideas, p. 132
  • I should agree with Plato that arithmetic, and pure mathematics generally, is not derived from perception. Pure mathematics consists of tautologies, analogous to "men are men," but usually more complicated. To know that a mathematical proposition is correct, we do not have to study the world, but only the meanings of symbols; and the symbols, when we dispense with definitions (of which the purpose is merely abbreviation), are found to be such words as "or" and "not," and "all" and "some," which do not, like "Socrates," denote anything in the actual world. A mathematical equation asserts that two groups of symbols have the same meaning; and so long as we confine ourselves to pure mathematics, this meaning must he one that can be understood without knowing anything about what can be perceived. Mathematical truth, therefore, is, as Plato contends, independent of perception; but it is truth of a very peculiar sort, and is concerned only with symbols.
    • Book One, Part II, Chapter XVIII, Knowledge and Perception in Plato, p. 155

About the arithmetical machine[edit]

Analytical Machine Babbage in London.
Main article: computers
  • 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.
  • 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.
  • 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.
  • 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.
  • The arithmetical machine produces effects which approach nearer to thought than all the actions of animals. But it does nothing which would enable us to attribute will to it, as to the animals.
  • 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." Need the arithmetic be so bad!

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