„Symbolical algebra adopts the rules of arithmetical algebra, but removes altogether their restrictions: thus symbolical subtraction differs from the same operation in arithmetical algebra in being possible for all relations of value of the symbols or expressions employed… all the results of arithmetical algebra which are deduced by the application of its rules, and which are general in form, though particular in value, are results likewise of symbolical algebra, where they are general in value as well as in form: thus the product of a^{m} and a^{n}, which is a^{m+n} when m and n are whole numbers, and therefore general in form though particular in value, will be their product likewise when m and n are general in value as well as in form: the series for (a+b)^{n}, determined by the principles of arithmetical algebra, when n is any whole number, if it be exhibited in a general form, without reference to a final term, may be shewn, upon the same principle, to the equivalent series for (a+b)^n, when n is general both in form and value.“

Vol. I: Arithmetical Algebra Preface, p. vi-vii
A Treatise on Algebra (1842)

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George Peacock
1791 - 1858

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Duncan Gregory photo

„There are a number of theorems in ordinary algebra, which, though apparently proved to be true only for symbols representing numbers, admit of a much more extended application. Such theorems depend only on the laws of combination to which the symbols are subject, and are therefore true for all symbols, whatever their nature may be, which are subject to the same laws of combination. The laws with which we have here concern are few in number, and may be stated in the following manner. Let a, b represent two operations, u, v two subjects on which they operate, then the laws are
(1) ab(u) = ba (u),
(2) a(u + v) = a (u) + a (v),
(3) am. an. u = am + n. u.
The first of these laws is called the commutative law, and symbols which are subject to it are called commutative symbols. The second law is called distributive, and the symbols subject to it distributive symbols. The third law is not so much a law of combination of the operation denoted by a, but rather of the operation performed on a, which is indicated by the index affixed to a. It may be conveniently called the law of repetition, since the most obvious and important case of it is that in which m and n are integers, and am therefore indicates the repetition m times of the operation a.“

—  Duncan Gregory British mathematician 1813 - 1844

That these are the laws employed in the demonstration of the principal theorems in Algebra, a slight examination of the processes will easily shew ; but they are not confined to symbols of numbers ; they apply also to the symbol used to denote differentiation.
p. 237 http://books.google.com/books?id=8lQ7AQAAIAAJ&pg=PA237; Highlighted section cited in: George Boole " Mr Boole on a General Method in Analysis http://books.google.com/books?pg=PA225-IA15&id=aGwOAAAAIAAJ&hl," Philosophical Transactions, Vol. 134 (1844), p. 225; Other section (partly) cited in: James Gasser (2000) A Boole Anthology: Recent and Classical Studies in the Logic of George Boole,, p. 52
Examples of the processes of the differential and integral calculus, (1841)

Hermann Weyl photo

„Cartan developed a general scheme of infinitesimal geometry in which Klein's notions were applied to the tangent plane and not to the n-dimensional manifold M itself.“

—  Hermann Weyl German mathematician 1885 - 1955

On the foundations of general infinitesimal geometry. Bull. Amer. Math. Soc. 35 (1929) 716–725 [10.1090/S0002-9904-1929-04812-2] (quote on p. 716)

E. W. Hobson photo

„Perhaps the least inadequate description of the general scope of modern Pure Mathematics—I will not call it a definition—would be to say that it deals with form, in a very general sense of the term; this would include algebraic form, functional relationship, the relations of order in any ordered set of entities such as numbers, and the analysis of the peculiarities of form of groups of operations.“

—  E. W. Hobson British mathematician 1856 - 1933

Fonte: Presidential Address British Association for the Advancement of Science, Section A (1910), p. 287; Cited in: Robert Edouard Moritz. Memorabilia mathematica; or, The philomath's quotation-book https://archive.org/stream/memorabiliamathe00moriiala#page/4/mode/2up, (1914), p. 5: Definitions and objects of mathematics.

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„Patience, n. A minor form of despair, disguised as a virtue.“

—  Ambrose Bierce American editorialist, journalist, short story writer, fabulist, and satirist 1842 - 1914

The Devil's Dictionary (1911)
Fonte: The Unabridged Devil's Dictionary

Antoine Augustin Cournot photo

„Anyone who understands algebraic notation, reads at a glance in an equation results reached arithmetically only with great labour and pains.“

—  Antoine Augustin Cournot, Researches into the Mathematical Principles of the Theory of Wealth

Fonte: Researches into the Mathematical Principles of the Theory of Wealth, 1897, p. 4; Cited in: Moritz (1914, 197): About mathematics as language

George Holmes Howison photo

„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
Journals

John Nash photo

„The writer has, by such a treatment, obtained values for all finite two-person cooperative games, and some special n-person games.“

—  John Nash American mathematician and Nobel Prize laureate 1928 - 2015

"Non-cooperative Games" in Annals of Mathematics, Vol. 54, No. 2 (September 1951)<!-- ; as cited in Can and should the Nash program be looked at as a part of mechanism theory? (2003) by Walter Trockel -->
1950s
Contexto: The writer has developed a “dynamical” approach to the study of cooperative games based upon reduction to non-cooperative form. One proceeds by constructing a model of the preplay negotiation so that the steps of negotiation become moves in a larger non-cooperative game [which will have an infinity of pure strategies] describing the total situation. This larger game is then treated in terms of the theory of this paper [extended to infinite games] and if values are obtained they are taken as the values of the cooperative game. Thus the problem of analyzing a cooperative game becomes the problem of obtaining a suitable, and convincing, non-cooperative model for the negotiation.
The writer has, by such a treatment, obtained values for all finite two-person cooperative games, and some special n-person games.

William John Macquorn Rankine photo

„In treating of the practical application of scientific principles, an algebraical formula should only be employed when its shortness and simplicity are such as to render it a clearer expression of a proposition or rule than common language would be, and when there is no difficulty in keeping the thing represented by each symbol constantly before the mind.“

—  William John Macquorn Rankine civil engineer 1820 - 1872

"On the Harmony of Theory and Practice in Mechanics" (Jan. 3, 1856)
Contexto: In treating of the practical application of scientific principles, an algebraical formula should only be employed when its shortness and simplicity are such as to render it a clearer expression of a proposition or rule than common language would be, and when there is no difficulty in keeping the thing represented by each symbol constantly before the mind.<!--p. 177

George Boole photo

„There is not only a close analogy between the operations of the mind in general reasoning and its operations in the particular science of Algebra, but there is to a considerable extent an exact agreement in the laws by which the two classes of operations are conducted.“

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

Fonte: 1850s, An Investigation of the Laws of Thought (1854), p. 6; As cited in: Leandro N. De Castro, Fernando J. Von Zuben, Recent Developments in Biologically Inspired Computing, Idea Group Inc (IGI), 2005 p. 236

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