On Arithmetic (1900 - 1924)

"All knowledge must be recognition, on pain of being mere delusion; Arithmetic must be discovered in just the same sense in which Columbus discovered the West Indies, and we no more create numbers than he created the Indians…. Whatever can be thought of has being and its [arithmetic] being is a precondition, not a result, of its being thought of." (Bertrand A W Russell, "Is Position in Space and Time Absolute or Relative", Mind Vol. X, 1901)

"Great numbers are not counted correctly to a unit, they are estimated; and we might perhaps point to this as a division between arithmetic and statistics, that whereas arithmetic attains exactness, statistics deals with estimates, sometimes very accurate, and very often sufficiently so for their purpose, but never mathematically exact." (Arthur L Bowley, "Elements of Statistics", 1901)

"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, "Essays on the Theory of Numbers", 1901)

"Symbolism is useful because it makes things difficult. Now in the beginning everything is self-evident, and it is hard to see whether one self-evident proposition follows from another or not. Obviousness is always the enemy to correctness. Hence we must invent a new and difficult symbolism in which nothing is obvious. [...] Thus the whole of Arithmetic and Algebra has been shown to require three indefinable notions and five indemonstrable propositions." Bertrand Russell, International Monthly, 1901)

"Arithmetical symbols are written diagrams and geometrical figures are graphic formulas." (David Hilbert, Bulletin of the American Mathematical Society Mathematical Problems Vol. 8, 1902)

"Arithmetic must be discovered in just the same sense in which Columbus discovered the West Indies, and we no more create numbers than he created the Indians." (Bertrand Russell, "The Principles of Mathematics", 1903)

"For we do not listen with the best regard to the verses of a man who is only a poet, nor to his problems if he is only an algebraist; but if a man is at once acquainted with the geometric foundations of things and with their festal splendor, his poetry is exact and his arithmetic musical." (Ralph Waldo Emerson, "Works and Days" [in "Society and solitude: Twelve chapters", 1903])

"We believe that in our reasonings we no longer appeal to intuition; the philosophers will tell us this is an illusion. Pure logic could never lead us to anything but tautologies; it could create nothing new; not from it alone can any science issue. In one sense these philosophers are right; to make arithmetic, as to make geometry, or to make any science, something else than pure logic is necessary. To designate this something else we have no word other than intuition. But how many different ideas are hidden under thi4s same word?" (Henri Poincaré , "Intuition and Logic in Mathematics", 1905)

"Arithmetic does not present to us that feeling of continuity which is such a precious guide; each whole number is separate from the next of its kind and has in a sense individuality; each in a manner is an exception and that is why general theorems are rare in the theory of numbers; and that is why those theorems which may exist are more hidden and longer escape those who are searching for them." (Henri Poincaré, "Annual Report of the Board of Regents of the Smithsonian Institution", 1909)

"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: An outline of the theory of concentration", 1909)

"The student of arithmetic who has mastered the first four rules of his art, and successfully striven with money sums and fractions, finds himself confronted by an unbroken expanse of questions known as problems." (Stephen Leacock, "Literary Lapses", 1911)

"Geometry formerly was the chief borrower from arithmetic and algebra, but it has since repaid its obligation with abundant usury; and if I were asked to name, in one word, the pole-star round which the mathematical firmament revolves, the central idea which pervades as a hidden spirit the whole corpus of mathematical doctrine, I should point to Continuity as contained in our notions of space, and say, it is this, it is this!" (James J. Sylvester, Presidential Address to the British Association, [The Collected Mathematical Papers of James Joseph Sylvester Vol. 2, cca. 1904–1912])

"Time was when all the parts of the subject were dissevered, when algebra, geometry, and arithmetic either lived apart or kept up cold relations of acquaintance confined to occasional calls upon one another; but that is now at an end; they are drawn together and are constantly becoming more and more intimately related and connected by a thousand fresh ties, and we may confidently look forward to a time when they shall form but one body with one soul." (James J. Sylvester, Presidential Address to the British Association, [The Collected Mathematical Papers of James Joseph Sylvester Vol. 2, cca. 1904–1912])

"Arithmetic is the science of the Evaluation of Functions, Algebra is the science of the Transformation of Functions." (George H Howison, Journal of Speculative Philosophy Vol. 5, 1914)

"The way to enable a student to apprehend the instrumental value of arithmetic is not to lecture him on the benefit it will be to him in some remote and uncertain future, but to let him discover that success in something he is interested in doing depends on ability to use numbers." (John Dewey, "Democracy and Education: An Introduction to the Philosophy of Education", 1916)