“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

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

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George Ballard Mathews 6

British mathematician 1861–1922

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“The statement is so frequently made that the differential calculus deals with continuous magnitude, and yet an explanation of this continuity is nowhere given; even the most rigorous expositions of the differential calculus do not base their proofs upon continuity but, with more or less consciousness of the fact, they either appeal to geometric notions or those suggested by geometry, or depend upon theorems which are never established in a purely arithmetic manner. Among these, for example, belongs the above mentioned theorem, and a more careful investigation convinced me that this theorem, or any one equivalent to it, can be regarded in some way as a sufficient basis for infinitesimal analysis. It then only remained to discover its true origin in the elements of arithmetic and thus at the same time to secure a real definition of the essence of continuity. I succeeded Nov. 24, 1858.”

Richard Dedekind (1831–1916) German mathematician

p, 125
Stetigkeit und irrationale Zahlen (1872)

“The analysis of variance is not a mathematical theorem, but rather a convenient method of arranging the arithmetic.”

Ronald Fisher (1890–1962) English statistician, evolutionary biologist, geneticist, and eugenicist

Discussion to ‘Statistics in agricultural research’ by J.Wishart, Journal of the Royal Statistical Society, Supplement, 1, 26-61, 1934.
1930s

“It is plain, of course, that any proof of this depends upon the validity of the doctrine of a priori cognition; only by our proved possession of such cognition can there be any evidence that we are self-active realities. It is in this reference noteworthy, therefore, that Lange, as defender of agnosticism, sees he cannot afford to admit the theory upon which alone cognition strictly a priori can be established.”

George Holmes Howison (1834–1916) American philosopher

Source: The Limits of Evolution, and Other Essays, Illustrating the Metaphysical Theory of Personal Ideaalism (1905), Later German Philosophy, p.175

“As Gauss first pointed out, the problem of cyclotomy, or division of the circle into a number of equal parts, depends in a very remarkable way upon arithmetical considerations. We have here the earliest and simplest example of those relations of the theory of numbers to transcendental analysis, and even to pure geometry, which so often unexpectedly present themselves, and which, at first sight, are so mysterious.”

George Ballard Mathews (1861–1922) British mathematician

Part 1, sect. 167.
Theory of Numbers, 1892

“It is an arithmetic, moreover, which cannot be denied even though we nearly all try to deny it. The arithmetic is simply this: Any positive rate of growth whatever eventually carries a human population to an unacceptable magnitude, no matter how small the rate of growth may be unless the rate of population growth can be reduced to zero before the population reaches an unacceptable magnitude. There is a famous theorem in economics, one which I call the dismal theorem, which states that if the only thing which can check the growth of population is starvation and misery, then the population will grow until it is sufficiently miserable and starving to check its growth. There is a second, even worse theorem which I call the utterly dismal theorem. It says that if the only thing which can check the growth of population is starvation and misery, then the ultimate result of any technological improvement is to enable a larger number of people to live in misery than before and hence to increase the total sum of human misery.”

Kenneth E. Boulding (1910–1993) British-American economist

Source: 1960s, The meaning of the twentieth century: the great transition, 1964, p. 126

“Legendre's Law of Quadratic Reciprocity' … is …the most important general truth in the science of integral numbers which has been discovered since the time of Fermat. It has been called by Gauss 'the gem of the higher arithmetic,' and is equally remarkable whether we consider the simplicity of its enunciation, the difficulties which for a long time attended its demonstration, or the number and variety of the results which have been obtained by its means. …[W]e find in the 'Opuscula Analytica' of Euler… a memoir… which contains a general and very elegant theorem from which the Law of Reciprocity is immediately deducible, and which is, vice versâ, deducible from that law. But Euler… expressly observes that the theorem is undemonstrated; and this would seem to be the only place in which he mentions it in connexion with the theory of the Residues of Powers; though in other researches he has frequently developed results which are consequences of the theorem, and which relate to the linear forms of the divisors of quadratic formulae. But here also his conclusions repose on induction only; though in one memoir he seems to have imagined… that he had obtained a satisfactory demonstration.”

Henry John Stephen Smith (1826–1883) mathematician

Report on the Theory of Numbers (1859) Part I, pp. 56-57.
The Collected Mathematical Papers of Henry John Stephen Smith (1894) Vol. 1

“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

“In arithmetical algebra we consider symbols as representing numbers, and the operations to which they are submitted as included in the same definitions as in common arithmetic; the signs + and - denote the operations of addition and subtraction in their ordinary meaning only, and those operations are considered as impossible in all cases where the symbols subjected to them possess values which would render them so in case they were replaced by digital numbers; thus in expressions such as a + b we must suppose a and b to be quantities of the same kind; in others, like a - b, we must suppose a greater than b and therefore homogeneous with it; in products and quotients, like ab and \frac{a}{b} we must suppose the multiplier and divisor to be abstract numbers; all results whatsoever, including negative quantities, which are not strictly deducible as legitimate conclusions from the definitions of the several operations must be rejected as impossible, or as foreign to the science.”

George Peacock (1791–1858) Scottish mathematician

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

“Most, if not all, of the great ideas of modern mathematics have had their origin in observation. Take, for instance, the arithmetical theory of forms, of which the foundation was laid in the diophantine theorems of Fermat, left without proof by their author, which resisted all efforts of the myriad-minded Euler to reduce to demonstration, and only yielded up their cause of being when turned over in the blow-pipe flame of Gauss’s transcendent genius; or the doctrine of double periodicity, which resulted from the observation of Jacobi of a purely analytical fact of transformation; or Legendre’s law of reciprocity; or Sturm’s theorem about the roots of equations, which, as he informed me with his own lips, stared him in the face in the midst of some mechanical investigations connected (if my memory serves me right) with the motion of compound pendulums; or Huyghen’s method of continued fractions, characterized by Lagrange as one of the principal discoveries of that great mathematician, and to which he appears to have been led by the construction of his Planetary Automaton; or the new algebra, speaking of which one of my predecessors (Mr. Spottiswoode) has said, not without just reason and authority, from this chair, “that it reaches out and indissolubly connects itself each year with fresh branches of mathematics, that the theory of equations has become almost new through it, algebraic 31 geometry transfigured in its light, that the calculus of variations, molecular physics, and mechanics” (he might, if speaking at the present moment, go on to add the theory of elasticity and the development of the integral calculus) “have all felt its influence.”

James Joseph Sylvester (1814–1897) English mathematician

James Joseph Sylvester. "A Plea for the Mathematician, Nature," Vol. 1, p. 238; Collected Mathematical Papers, Vol. 2 (1908), pp. 655, 656.

“If I compare arithmetic with a tree that unfolds upward into a multitude of techniques and theorems while its root drives into the depths, then it seems to me that the impetus of the root.”

Gottlob Frege (1848–1925) mathematician, logician, philosopher

Gottlob Frege, Montgomery Furth (1964). The Basic Laws of Arithmetic: Exposition of the System. p. 10

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George Ballard Mathews 6

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