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)
What is provable is not necessarily true.
Numbers are free creations of the human mind; they serve as a means of apprehending more easily and more sharply the difference of things.
I regard the whole of arithmetic as a necessary, or at least natural, consequence of the simplest act of thought, the counting of things.
The definition of the real number by means of a cut (Schnitt) is the true foundation for the analysis of real numbers.
In the infinite realm of numbers, there is no end to the creation of new numbers.
Every infinite system is similar to a part of itself.
The essence of mathematics lies in its freedom.
A system S is infinite if it is similar to a proper part of itself; otherwise it is finite.
The true value of a mathematical concept lies in its power to simplify and unify.
The continuity of the real numbers is the most important property of the number system.
My aim was to show that the whole of arithmetic can be developed from the concept of number alone, without recourse to intuition.
The concept of a 'cut' is a purely logical one, independent of any geometric intuition.
Mathematics is the science of the infinite.
The natural numbers are the foundation upon which all other numbers are built.
The essence of number is not in its magnitude, but in its relation to other numbers.
The definition of a number system must be purely abstract, free from any specific representation.
The concept of a mapping (Abbildung) is fundamental to all of mathematics.
The irrational numbers fill the gaps in the rational number line, making it continuous.
The power of abstraction is the greatest tool of the mathematician.
The concept of a field is essential for understanding algebraic structures.
Contemporaries of Richard Dedekind
Other Mathematicss born within 50 years of Richard Dedekind (1831–1916).