“Addition is the combination of any arbitrary repetitions of the above-mentioned simplest act into a single act; from it in a similar way arises multiplication. While the performance of these two operations is always possible, that of the inverse operations, subtraction and division, proves to be limited. Whatever the immediate occasion may have been, whatever comparisons or analogies with experience, or intuition, may have led thereto; it is certainly true that just this limitation in performing the indirect operations has in each case been the real motive for a new creative act; thus negative and fractional numbers have been created by the human mind; and in the system of all rational numbers there has been gained an instrument of infinitely greater perfection. This system, which I shall denote by R, possesses first of all a completeness and self-containedness which I have designated… as characteristic of a body of numbers [Zahlkőrper] and which consists in this, that the four fundamental operations are always performable with any two individuals in R, i. e., the result is always an individual of R, the single case of division by the number zero being excepted.”

p, 125
Stetigkeit und irrationale Zahlen (1872)

Adopted from Wikiquote. Last update June 3, 2021. History

Help us to complete the source, original and additional information

Do you have more details about the quote "Addition is the combination of any arbitrary repetitions of the above-mentioned simplest act into a single act; from it…" by Richard Dedekind?
Richard Dedekind photo
Richard Dedekind 13
German mathematician 1831–1916

Related quotes

Ebenezer Howard photo

“Whatever may have been the causes which have operated in the past, and are operating now, to draw the people into the cities, those causes may all be summed up as "attractions "”

Ebenezer Howard (1850–1928) British writer, founder of the garden city movement

Introduction.
Garden Cities of To-morrow (1898)
Context: Whatever may have been the causes which have operated in the past, and are operating now, to draw the people into the cities, those causes may all be summed up as "attractions "; and it is obvious, therefore, that no remedy can possibly be effective which will not present to the people, or at least to considerable portions of them, greater "attractions " than our cities now possess, so that the force of the old "attractions" shall be overcome by the force of new "attractions" which are to be created. Each city may be regarded as a magnet, each person as a needle; and, so viewed, it is at once seen that nothing short of the discovery of a method for constructing magnets of yet greater power than our cities possess can be effective for redistributing the population in a spontaneous and healthy manner.

Duncan Gregory photo

“There are a number of theorems in ordinary algebra, which, though apparently proved to be true only for symbols representing numbers, admit of a much more extended application. Such theorems depend only on the laws of combination to which the symbols are subject, and are therefore true for all symbols, whatever their nature may be, which are subject to the same laws of combination. The laws with which we have here concern are few in number, and may be stated in the following manner. Let a, b represent two operations, u, v two subjects on which they operate, then the laws are
(1) ab(u) = ba (u),
(2) a(u + v) = a (u) + a (v),
(3) am. an. u = am + n. u.
The first of these laws is called the commutative law, and symbols which are subject to it are called commutative symbols. The second law is called distributive, and the symbols subject to it distributive symbols. The third law is not so much a law of combination of the operation denoted by a, but rather of the operation performed on a, which is indicated by the index affixed to a. It may be conveniently called the law of repetition, since the most obvious and important case of it is that in which m and n are integers, and am therefore indicates the repetition m times of the operation a.”

Duncan Gregory (1813–1844) British mathematician

That these are the laws employed in the demonstration of the principal theorems in Algebra, a slight examination of the processes will easily shew ; but they are not confined to symbols of numbers ; they apply also to the symbol used to denote differentiation.
p. 237 http://books.google.com/books?id=8lQ7AQAAIAAJ&pg=PA237; Highlighted section cited in: George Boole " Mr Boole on a General Method in Analysis http://books.google.com/books?pg=PA225-IA15&id=aGwOAAAAIAAJ&hl," Philosophical Transactions, Vol. 134 (1844), p. 225; Other section (partly) cited in: James Gasser (2000) A Boole Anthology: Recent and Classical Studies in the Logic of George Boole,, p. 52
Examples of the processes of the differential and integral calculus, (1841)

Jane Roberts photo
Richard von Mises photo

“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)

James Mill photo

“A certain immense aggregate of operations, is subservient to the production of the commodities useful and agreeable to man. It is of the highest importance that this aggregate should be divided into portions, consisting, each, of as small a number of operations as possible, in order that every operation may be the more quickly and perfectly, performed. If each man could, by the more frequent repetition thus occasioned, perform two of these operations, instead of one, and also perform each of them better, the powers of the community, in producing articles useful and agreeable to them, would, upon this supposition, be more than doubled. Not only would they be doubled in quantity, but a great advantage would be gained in point of quality.
This subject has been fully illustrated by Dr. Smith, in the first chapter of the first book of the "Inquiry into the Nature and Causes of the Wealth of Nations," where the extraordinary effect of the division of labour in increasing its productive powers, in the more complicated cases, is displayed in some very remarkable instances. He states that a boy, who has been accustomed to make nothing but nails, can make-upwards of two thousand three hundred in a day; while a common blacksmith, whose operations are nevertheless so much akin to those of the nailer, cannot make above three hundred, and those very bad ones.”

James Mill (1773–1836) Scottish historian, economist, political theorist and philosopher

Ch 1 : Production
Elements of Political Economy (1821)

John Marshall photo
Mwanandeke Kindembo photo
Bahá'u'lláh photo

“Wherefore, if those who have come to the sea of His presence are found to possess none of the limited things of this perishable world, whether it be outer wealth or personal opinions, it mattereth not. For whatever the creatures have is limited by their own limits, and whatever the True One hath is sanctified therefrom; this utterance must be deeply pondered that its purport may be clear”

Bahá'u'lláh (1817–1892) founder of the Bahá'í Faith

The Valley of True Poverty and Absolute Nothingness
The Seven Valleys Of Bahá’u’lláh
Context: He who hath attained this station is sanctified from all that pertaineth to the world. Wherefore, if those who have come to the sea of His presence are found to possess none of the limited things of this perishable world, whether it be outer wealth or personal opinions, it mattereth not. For whatever the creatures have is limited by their own limits, and whatever the True One hath is sanctified therefrom; this utterance must be deeply pondered that its purport may be clear. “Verily the righteous shall drink of a winecup tempered at the camphor fountain.” If the interpretation of “camphor” become known, the true intention will be evident. This state is that poverty of which it is said, “Poverty is My glory.” And of inward and outward poverty there is many a stage and many a meaning which I have not thought pertinent to mention here; hence I have reserved these for another time, dependent on what God may desire and fate may seal.

Adam Smith photo

“Each person, therefore, making a tenth part of forty-eight thousand pins, might be considered as making four thousand eight hundred pins in a day. But if they had all wrought separately and independently, and without any of them having been educated to this peculiar business, they certainly could not each of them have made twenty, perhaps not one pin in a day; that is, certainly, not the two hundred and fortieth, perhaps not the four thousand eight hundredth, part of what they are at present capable of performing, in consequence of a proper division and combination of their different operations.”

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

Source: The Wealth of Nations (1776), Book I, Chapter I, p. 8-9.
Context: To take an example, therefore, from a very trifling manufacture, but one in which the division of labour has been very often taken notice of, the trade of a pin-maker: a workman not educated to this business (which the division of labour has rendered a distinct trade, nor acquainted with the use of the machinery employed in it (to the invention of which the same division of labour has probably given occasion), could scarce, perhaps, with his utmost industry, make one pin in a day, and certainly could not make twenty. But in the way in which this business is now carried on, not only the whole work is a peculiar trade, but it is divided into a number of branches, of which the greater part are likewise peculiar trades. One man draws out the wire; another straights it; a third cuts it; a fourth points it; a fifth grinds it at the top for receiving the head; to make the head requires two or three distinct operations; to put it on is a peculiar business; to whiten the pins is another; it is even a trade by itself to put them into the paper; and the important business of making a pin is, in this manner, divided into about eighteen distinct operations, which, in some manufactories, are all performed by distinct hands, though in others the same man will sometimes perform two or three of them. I have seen a small manufactory of this kind, where ten men only were employed, and where some of them consequently performed two or three distinct operations. But though they were very poor, and therefore but indifferently accommodated with the necessary machinery, they could, when they exerted themselves, make among them about twelve pounds of pins in a day. There are in a pound upwards of four thousand pins of a middling size. Those ten persons, therefore, could make among them upwards of forty-eight thousand pins in a day. Each person, therefore, making a tenth part of forty-eight thousand pins, might be considered as making four thousand eight hundred pins in a day. But if they had all wrought separately and independently, and without any of them having been educated to this peculiar business, they certainly could not each of them have made twenty, perhaps not one pin in a day; that is, certainly, not the two hundred and fortieth, perhaps not the four thousand eight hundredth, part of what they are at present capable of performing, in consequence of a proper division and combination of their different operations.

Richard Arden, 1st Baron Alvanley photo

“Inconvenience arising from the operation of an Act of Parliament can be no ground of argument in a Court of law.”

Richard Arden, 1st Baron Alvanley (1744–1804) British judge and politician

Grigby v. Oakes (1801), 1 Bos. & Pull. 528.

Related topics