“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

"Proposed Automatic Calculating Machine" (1937)

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

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Howard H. Aiken 4

pioneer in computing, original conceptual designer behind I… 1900–1973

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“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.

“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

“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

“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.

“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.

“I regard the whole of arithmetic as a necessary, or at least natural, consequence of the simplest arithmetic act, that of counting, and counting itself as nothing else than the successive creation of the infinite series of positive integers in which each individual is defined by the one immediately preceding; the simplest act is the passing from an already-formed individual to the consecutive new one to be formed. The chain of these numbers forms in itself an exceedingly useful instrument for the human mind; it presents an inexhaustible wealth of remarkable laws obtained by the introduction of the four fundamental operations of arithmetic.”

Richard Dedekind (1831–1916) German mathematician

p, 125
Stetigkeit und irrationale Zahlen (1872)

“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

“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)

“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

“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

“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.”

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