Richard Dedekind
A German mathematician who made important contributions to abstract algebra, particularly in algebraic number theory.
Most quoted
"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 two classes, this severing of the straight line into two portions."
— from Stetigkeit und irrationale Zahlen, 1872
"The continuity of the domain of real numbers is the property that if all its elements are divided into two classes, such that every element of the first class is less than every element of the second class, then there exists one and only one number which produces this division."
— from Stetigkeit und irrationale Zahlen, 1872
"The way in which the irrational numbers are usually introduced is based directly upon the conception of extensive magnitudes—which itself is nowhere carefully defined—and explains number as the result of measuring such a magnitude by another of the same kind."
— from Stetigkeit und irrationale Zahlen, 1872
All quotes by Richard Dedekind (399)
The simplest and most natural way of founding the theory of irrational numbers is the method of Dedekind cuts.
The finite is that which is not infinite.
The theory of functions is the true foundation of analysis.
The integers are the only numbers that are directly given to us by nature.
The concept of a cut in the rational numbers is a purely logical concept.
The whole of arithmetic and analysis can be derived from the properties of the natural numbers.
The infinite is not a quantity but a property of certain sets.
The definition of the number 1 is the first and most important step in the foundation of arithmetic.
The rational numbers are insufficient for the purposes of analysis.
The real numbers are the cuts in the set of rational numbers.
The concept of a one-to-one correspondence is fundamental to the theory of sets.
The natural numbers are the model for all mathematical thinking.
The theory of algebraic numbers is the most beautiful part of mathematics.
The principle of mathematical induction is not a mere trick but a fundamental property of the natural numbers.
The definition of addition and multiplication for natural numbers must be based on the successor function.
The concept of a function is more general than that of a formula.
The theory of ideals was created to restore the law of unique factorization in certain rings of algebraic integers.
The notion of a field is essential for the modern development of algebra.
The real numbers form a continuous domain; the rational numbers do not.
Contemporaries of Richard Dedekind
Other Mathematicss born within 50 years of Richard Dedekind (1831–1916).