„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, buch The Foundations of Arithmetic

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

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Gottlob Frege5
deutscher Mathematiker, Logiker und Philosoph 1848 - 1925

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Gottlob Frege Foto

„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, buch 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.

Howard H. Aiken Foto

„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 pioneer in computing, original conceptual designer behind IBM's Harvard Mark I computer 1900 - 1973

"Proposed Automatic Calculating Machine" (1937)

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George Holmes Howison Foto

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

—  George Holmes Howison American philosopher 1834 - 1916

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

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

—  Francis Wayland Parker Union Army officer 1837 - 1902

Quelle: 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

George Peacock Foto
Stanisław Lem Foto
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Jean-Baptiste Say Foto
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Henri Poincaré Foto

„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, buch Wissenschaft und Hypothese

Quelle: Science and Hypothesis (1901), Ch. I. (1905) Tr. George Bruce Halstead
Kontext: 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

„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 British educationalist 1824 - 1903

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

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