J W Richard Dedekind - Collected Quotes

"[...] as the work of man, science is subject to his arbitrariness and to all the imperfections of his mental powers. There would essentially be no more science for a man gifted with an unbounded understanding - a man for whom the final conclusions, which we attain through a long chain of inferences, would be immediately evident truths." (Richard Dedekind, 1854)

"I find the essence of continuity [...] in the following principle: If all points of the straight line fall into two classes such that every point of the first class lies to the left of every point of the second class, then there exists one and only one point which produces this division of all points into these two classes, this severing of the straight line into two portions." (Richard Dedekind, "Stetigkeit und Irrationale Zahlen" ["Continuity and Irrational Numbers", 1872)

"I regard the whole of arithmetic as a necessary, or at least natural, consequence of the simplest arithmetic 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; the simplest act is the passing from an already-formed individual to the consecutive new one to be formed. The chain of these numbers forms in itself an exceedingly useful instrument for the human mind; it presents an inexhaustible wealth of remarkable laws obtained by the introduction of the four fundamental operations of arithmetic. Addition is the combination of any arbitrary repetitions of the above-mentioned simplest act into a single act; from it in a similar way arises multiplication. While the performance of these two operations is always possible, that of the inverse operations, subtraction and division, proves to be limited. Whatever the immediate occasion may have been, whatever comparisons or analogies with experience, or intuition, may have led thereto; it is certainly true that just this limitation in performing the indirect operations has in each case been the real motive for a new creative act; thus negative and fractional numbers have been created by the human mind; and in the system of all rational numbers there has been gained an instrument of infinitely greater perfection." (Richard Dedekind, "On Continuity and Irrational Numbers", 1872)

"Just as negative and fractional rational numbers are formed by a new creation, and as the laws of operating with these numbers must and can be reduced to the laws of operating with positive integers, so we must endeavor completely to define irrational numbers by means of the rational numbers alone. The question only remains how to do this." (Richard Dedekind, "On Continuity and Irrational Numbers", 1872)

"The above comparison of the domain R of rational numbers with a straight line has led to the recognition of the existence of gaps, of a certain incompleteness or discontinuity of the former, while we ascribe to the straight line completeness, absence of gaps, or continuity. In what then does this continuity consist? Everything must depend on the answer to this question, and only through it shall we obtain a scientific basis for the investigation of all continuous domains." (Richard Dedekind,"Stetigkeit und irrationale Zahle", 1872) 

"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)

"[…] numbers are free creations of the human mind; they serve as a means of apprehending more easily and more sharply the difference of things." (Richard Dedekind, "Essays on the Theory of Numbers", 1901)

"The comparison of the rational numbers with a straight line has led to the recognition of the existence of gaps, of a certain incompleteness or discontinuity of the rationals, while we ascribe to the straight line completeness, absence of gaps, or continuity." (Richard Dedekind, "Continuity and Irrational Numbers", 1901) (Richard Dedekind, "Continuity and Irrational Numbers", 1901)

"Of all the aids which the human mind has for simplifying its life, i.e., the work in which thinking consists, none is so rich in consequences and so inseparably bound up with its innermost nature as the concept of number. Arithmetic, whose sole object is this concept, is already a discipline of insurmountable breadth, and there is no doubt that there are absolutely no limits to its further development. Equally insurmountable is its field of application, for every thinking man, even if he does not clearly realize it, is a man of numbers, an arithmetician." (Dedekind)

"The splendid creations of this theory have excited the admiration of mathematicians mainly because they have enriched our science in an almost unparalleled way with an abundance of new ideas and opened up heretofore wholly unknown fields to research. The Cauchy integral formula, the Riemann mapping theorem and the Weierstrass power series calculus not only laid the groundwork for a new branch of mathematics but at the same time they furnished the first and till now the most fruitful example of the intimate connections between analysis and algebra. But it isn't just the wealth of novel ideas and discoveries which the new theory furnishes; of equal importance on the other hand are the boldness and profundity of the methods by which the greatest of difficulties are overcome and the most recondite of truths, the mysteria functiorum, are exposed tothe brightest." (Richard Dedekind)