„An ancient writer said that arithmetic and geometry are the wings of mathematics; I believe one can say without speaking metaphorically that these two sciences are the foundation and essence of all the sciences which deal with quantity. Not only are they the foundation, they are also, as it were, the capstones; for, whenever a result has been arrived at, in order to use that result, it is necessary to translate it into numbers or into lines; to translate it into numbers requires the aid of arithmetic, to translate it into lines necessitates the use of geometry.“

Dans Les Leçons Élémentaires sur les Mathématiques (1795) Leçon cinquiéme, Tr. McCormack, cited in Moritz, Memorabilia mathematica or, The philomath's quotation-book (1914) Ch. 15 Arithmetic, p. 261. https://archive.org/stream/memorabiliamathe00moriiala#page/260/mode/2up

Joseph-Louis de Lagrange Foto
Joseph-Louis de Lagrange3
matemático, físico y astrónomo italiano 1736 - 1813

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„If in other sciences we should arrive at certainty without doubt and truth without error, it behooves us to place the foundations of knowledge in mathematics…“

—  Roger Bacon, libro Opus Majus

Bk. 1, ch. 4. Translated by Robert B. Burke, in: Edward Grant (1974) Source Book in Medieval Science. Harvard University Press. p. 93
Opus Majus, c. 1267

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David Eugene Smith Foto
David Eugene Smith Foto
George Boole Foto

„It is not of the essence of mathematics to be conversant with the ideas of number and quantity.“

—  George Boole English mathematician, philosopher and logician 1815 - 1864

Fuente: 1850s, An Investigation of the Laws of Thought (1854), p. 12; Cited in: Alexander Bain (1870) Logic, p. 191

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, libro Science and Hypothesis

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

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„Newton… (after having remarked that geometry only requires two of the mechanical actions which it postulates, namely, to describe a straight line and a circle) says: geometry is proud of being able to achieve so much while taking so little from extraneous sources. One might say of metaphysics, on the other hand: it stands astonished, that with so much offered it by pure mathematics it can effect so little.“

—  Immanuel Kant, libro Metaphysical Foundations of Natural Science

In the meantime, this little is something which mathematics indispensably requires in its application to natural science, which, inasmuch as it must here necessarily borrow from metaphysics, need not be ashamed to allow itself to be seen in company with the latter.
Preface, Tr. Bax (1883) citing Isaac Newton's Principia
Metaphysical Foundations of Natural Science (1786)

George Peacock Foto

„Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Etiam egestas wisi a erat. Morbi imperdiet, mauris ac auctor dictum.“