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)
Mathematics is the science of what is clear and simple.
The continuity of the line is not given by intuition, but is a logical construction.
The concept of number is not an empirical concept, but an a priori concept.
The definition of number must be independent of any particular representation.
The concept of a mapping or function is one of the most fundamental in mathematics.
The theory of ideals is nothing but a generalization of the theory of numbers.
The introduction of ideals allows us to restore unique factorization in algebraic number fields.
The concept of a field is fundamental for understanding algebraic structures.
The whole of mathematics rests upon the concept of number.
The continuity of the real numbers is a postulate, not a theorem.
The concept of a system (set) is the most general concept in mathematics.
The definition of number is not a matter of convention, but of logical necessity.
The concept of infinity is not a vague notion, but a precise mathematical concept.
The properties of numbers are not discovered, but are consequences of their definition.
The introduction of cuts provides a rigorous foundation for the theory of real numbers.
The concept of a field is a natural generalization of the concept of rational numbers.
The theory of numbers is the most fundamental branch of mathematics.
The concept of a group is essential for understanding symmetry in mathematics.
Mathematics is not about symbols, but about the concepts they represent.
The existence of a one-to-one correspondence is the essence of similarity between systems.
Contemporaries of Richard Dedekind
Other Mathematicss born within 50 years of Richard Dedekind (1831–1916).