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.
I see no reason to doubt that the whole of arithmetic can be developed from the concept of the natural number.
A system S is said to be infinite if it is similar to a proper part of itself; in the contrary case S is called a finite system.
Every infinite system is similar to a proper part of itself.
A system S is said to be simply infinite if there exists a transformation φ of S into itself such that S is similar to the range of φ, and such that no proper part of S is similar to S.
The existence of such a simply infinite system is therefore a necessary condition for the existence of the natural numbers.
To every real number corresponds a cut in the system of rational numbers.
If the system R of all rational numbers is separated into two classes A1 and A2 such that every number of A1 is less than every number of A2, then there exists one and only one real number which produces this cut.
This property of continuity is the true and characteristic property of all real numbers.
The essence of arithmetic is the concept of number.
The definition of the real numbers by means of cuts is independent of any geometric intuition.
The natural numbers are the free creations of the human mind.
The concept of number is entirely independent of the concepts of space and time.
The theory of numbers is the most beautiful part of mathematics.
The concept of a set is fundamental to all mathematics.
The definition of a number system must be purely logical.
The introduction of irrational numbers is a necessity for the continuity of the number line.
The concept of a mapping or transformation is one of the most fruitful in mathematics.
Mathematics is the science of the infinitely many.
The notion of a 'cut' is a purely logical concept.
Contemporaries of Richard Dedekind
Other Mathematicss born within 50 years of Richard Dedekind (1831–1916).