Source: Science and Hypothesis (1901), Ch. I. (1905) Tr. George Bruce Halstead
Context: 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
“Stereotypes are, in fact, merely one aspect of the mind’s ability to make generalizations, without which science and mathematics, not to mention much of everyday life, would be impossible.”
Source: Derb Quotes https://www.nationalreview.com/corner/derb-quotes-john-derbyshire/, National Review, November 20, 2003.
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Richard Courant in: The Australian Mathematics Teacher, Volumes 39-40 http://books.google.co.in/books?id=CofxAAAAMAAJ, Australian Association of Mathematics Teachers, 1983, p. 3
Source: The mechanization of the world picture, 1961, p. 499
The Notebooks of Leonardo da Vinci (1883), XIX Philosophical Maxims. Morals. Polemics and Speculations.
Quotes 1990s, 1990-1994, Noam Chomsky: A Life of Dissent, 1992
Context: There is a noticeable general difference between the sciences and mathematics on the one hand, and the humanities and social sciences on the other. It's a first approximation, but one that is real. In the former, the factors of integrity tend to dominate more over the factors of ideology. It's not that scientists are more honest people. It's just that nature is a harsh taskmaster. You can lie or distort the story of the French Revolution as long as you like, and nothing will happen. Propose a false theory in chemistry, and it'll be refuted tomorrow.
Reported in Josiah Hotchkiss Gilbert, Dictionary of Burning Words of Brilliant Writers (1895), p. 36.
July 28, 1788, p. 150.
North Carolina's Debates, in Convention, on the adoption of the Federal Constitution (1787)