"I regard the whole of arithmetic as a necessary, or at least natural, consequence of the simplest arithmetical act, that of counting, and counting itself as nothing else than the successive creation of the infinite series of positive integers in which each individual is defined by the one immediately preceding […]" (Richard Dedekind, "On Continuity and Irrational Numbers", 1872)
"The most distinct and beautiful statement of any truth [in science] must take at last the mathematical form. We might so simplify the rules of moral philosophy, as well as of arithmetic, that one formula would express them both." (Henry Thoreau, "A Week on the Concord and Merrimack Rivers", 1873)
"Thought is symbolical of Sensation as Algebra is of Arithmetic, and because it is symbolical, is very unlike what it symbolises. For one thing, sensations are always positive; in this resembling arithmetical quantities. A negative sensation is no more possible than a negative number. But ideas, like algebraic quantities, may be either positive or negative. However paradoxical the square of a negative quantity, the square root of an unknown quantity, nay, even in imaginary quantity, the student of Algebra finds these paradoxes to be valid operations. And the student of Philosophy finds analogous paradoxes in operations impossible in the sphere of Sense. Thus although it is impossible to feel non-existence, it is possible to think it; although it is impossible to frame an image of Infinity, we can, and do, form the idea, and reason on it with precision." (George H Lewes "Problems of Life and Mind", 1873)
"Thus numbers may be said to rule the whole world of quantity, and the four rules of arithmetic may be regarded as the complete equipment of the mathematician." James C Maxwell, "Electricity and Magnetism", 1873)
"The rules of Arithmetic operate in Algebra; the logical operations supposed to be peculiar to Ideation operate in Sensation, There is but one Calculus, but one Logic; though for convenience we divide the one into Arithmetic the calculus of values, and Algebra the calculus of relations; the other into the Logic of Feeling and the Logic of Signs." (George H Lewes "Problems of Life and Mind", 1873)
"Algebra is but written geometry and geometry is but figured algebra." (Sophie Germain, "Mémoire sur la surfaces élastiques", 1880)
"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, "The Foundations of Arithmetic", 1884)
"In science nothing capable of proof ought to be accepted without proof. Though this demand seems so reasonable yet I cannot regard it as having been met even in […] that part of logic which deals with the theory of numbers. In speaking of arithmetic (algebra, analysis) as a part of logic I mean to imply that I consider the number concept entirely independent of the notions of intuition of space and time, that I consider it an immediate result from the laws of thought." (Richard Dedekind, "Was sind und was sollen die Zahlen?", 1888)
"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 B Mathews, "Theory of Numbers", 1892)
"I compare arithmetic with a tree that unfolds upwards in a multitude of techniques and theorems while the root drives into the depths." (Gottlob Frege, "Grundgesetze der Arithmetik", 1893)
"Considering the remarkable elegance, generality, and simplicity of the method [Homer’s Method of finding the numerical values of the roots of an equation], it is not a little surprising that it has not taken a more prominent place in current mathematical textbooks. [...] As a matter of fact, its spirit is purely arithmetical; and its beauty, which can only be appreciated after one has used it in particular cases, is of that indescribably simple kind, which distinguishes the use of position in the decimal notation and the arrangement of the simple rules of arithmetic. It is, in short, one of those things whose invention was the creation of a commonplace." (George Chrystal, "Algebra", 1893)
"The object of all arithmetical operations is to save direct enumeration, by utilizing the results of our old operations of counting. Our endeavor is, having done a sum once, to preserve the answer for future use [...]. Such, too, is the purpose of algebra, which, substituting relations for values, symbolizes and definitely fixes all numerical operations which follow the same rule." (Ernst Mach, "The Science of Mechanics", 1893)
"The science of arithmetic may be called the science of exact limitation of matter and things in space, force, and time." (Francis W Parker, "Talks on Pedagogics", 1894),
"The best review of arithmetic consists in the study of algebra." (Florian Cajori, "Teaching and History of Mathematics in U. S.", 1896)
"Anyone who understands algebraic notation, reads at a glance in an equation results reached arithmetically only with great labour and pains." (A Augustin Cournot, "Researches Into the Mathematical Principles of the Theory of Wealth", 1897)
"In order to comprehend and fully control arithmetical concepts and methods of proof, a high degree of abstraction is necessary, and this condition has at times been charged against arithmetic as a fault. I am of the opinion that all other fields of knowledge require at least an equally high degree of abstraction as mathematics, - provided, that in these fields the foundations are also everywhere examined with the rigour and completeness which is actually necessary." (David Hilbert, "Die Theorie der algebraischen Zahlkorper", 1897)
"Mathematics in its pure form, as arithmetic, algebra, geometry, and the applications of the analytic method, as well as mathematics applied to matter and force, or statics and dynamics, furnishes the peculiar study that gives to us, whether as children or as men, the command of nature in this its quantitative aspect; mathematics furnishes the instrument, the tool of thought, which we wield in this realm." (William T Harris, "Psychologic Foundations of Education", 1898)
"The laws of algebra, though suggested by arithmetic, do not depend on it. They depend entirely on the conventions by which it is stated that certain modes of grouping the symbols are to be considered as identical. This assigns certain properties to the marks which form the symbols of algebra. The laws regulating the manipulation of algebraic symbols are identical with those of arithmetic. It follows that no algebraic theorem can ever contradict any result which could be arrived at by arithmetic; for the reasoning in both cases merely applies the same general laws to different classes of things. If an algebraic theorem can be interpreted in arithmetic, the corresponding arithmetical theorem is therefore true." (Alfred N Whitehead, "Universal Algebra", 1898)
"The method of arithmetical teaching is perhaps the best understood of any of the methods concerned with elementary studies." (Alexander Bain, "Education as a Science", 1898)
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