“Now as to what pertains to these Surd numbers (which, as it were by way of reproach and calumny, having no merit of their own are also styled Irrational, Irregular, and Inexplicable) they are by many denied to be numbers properly speaking, and are wont to be banished from arithmetic to another Science, (which yet is no science) viz. algebra.”

— Isaac Barrow

Source: Mathematical Lectures (1734), p. 44

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Isaac Barrow 20

English Christian theologian, and mathematician 1630–1677

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“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.”

Joseph Louis Lagrange (1736–1813) Italian mathematician and mathematical physicist

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

“The way in which the irrational numbers are usually introduced is based directly upon the conception of extensive magnitudes—which itself is nowhere carefully defined—and explains number as the result of measuring such a magnitude by another of the same kind. Instead of this I demand that arithmetic shall be developed out of itself.”

Richard Dedekind (1831–1916) German mathematician

Footnote: The apparent advantage of the generality of this definition of number disappears as soon as we consider complex numbers. According to my view, on the other hand, the notion of the ratio between two numbers of the same kind can be clearly developed only after the introduction of irrational numbers.
Stetigkeit und irrationale Zahlen (1872)

“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)

“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
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“I have separated arithmetical from symbolical algebra, and I have devoted the present volume entirely to the exposition of the principles of the former science and their application to the theory of numbers and of arithmetical processes: the second volume, which is now in the press, will embrace the principles of symbolical algebra: it will be followed, if other and higher duties should allow me the leisure to complete them, by other works, embracing all the more important departments of analysis, with the view of presenting their principles in such a form, as may make them component parts of one uniform and connected system.”

George Peacock (1791–1858) Scottish mathematician

Vol. I: Arithmetical Algebra Preface, p. iii
A Treatise on Algebra (1842)

“The seventeenth century witnessed the birth of modern science as we know it today. The science was something new, based on a direct confrontation of nature by experiment and observation. But there was another feature of the new science—a dependence on numbers, on real numbers of actual experience.
…The ancients knew a few numerical laws… But prior to the Scientific Revolution, the goal of science (or the study of nature) was not to seek laws of nature expressed in terms of numbers or number relations. …the new science …not only found laws based on numbers but they were also willing to express these laws in terms of higher powers of numbers—squares and cubes.”

I. Bernard Cohen (1914–2003) American historian of science

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“It wasn’t businessmen who developed new technology and science but amateur inventors or salaried professors instead; from the foundation of this country to the invention of the Internet, continuing with Einstein and finally, the arrival of the first man on the moon. Not to mention, the basis of the sciences—which were shaped by those horrible and uncivilized Arabs centuries before—from the numbers that we use to Algebra to algorithms and many other sciences and philosophies that are part of Western civilization today, continuing with the Europeans in the 17th century. None of these men were businessmen, of course.”

Jorge Majfud (1969) Uruguayan-American writer

Carta abierta a Donald Trump http://www.huffingtonpost.es/jorge-majfud/carta-abierta-a-donald-tr_b_10218246.html Translation at The Huffington Post http://www.huffingtonpost.com/entry/57dc39fee4b0d5920b5b2aac?timestamp=1474051083758.

“The Hindus saw clearly that if the arithmetic operations… were properly defined for negative numbers, these numbers could be employed to as good advantage as people had previously derived from positive numbers. …To people to whom the word number had always meant positive whole numbers and positive fractions, the very idea that there could be other numbers came hard. For many centuries negative numbers were either rejected or treated as second-class citizens.
What was especially difficult for mathematicians to swallow was that negative numbers could be acceptable roots of equations.”

Morris Kline (1908–1992) American mathematician

Source: Mathematics and the Physical World (1959), p. 51.

“While the mathematicians were still looking askance at the Greek gift of the irrational number, the Hindus of India were preparing another brain-teaser, the negative number, which they introduced about A. D. 700. The Hindus saw that when the usual, positive numbers were used to represent assets, it was helpful to have other number represent debts.”

Morris Kline (1908–1992) American mathematician

Source: Mathematics and the Physical World (1959), pp. 49-50.

“Strictly speaking, the theory of numbers has nothing to do with negative, or fractional, or irrational quantities, as such. No theorem which cannot be expressed without reference to these notions is purely arithmetical: and no proof of an arithmetical theorem, can be considered finally satisfactory if it intrinsically depends upon extraneous analytical theories.”

George Ballard Mathews (1861–1922) British mathematician

Part 1, sect. 1.
Theory of Numbers, 1892

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