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 always true, and what is true is not always provable.
I see no reason to doubt that the infinite is a necessary concept for the human mind.
The definition of a real number as a cut in the system of rational numbers is, in my opinion, the only one that is logically satisfactory.
Mathematics is the queen of the sciences and arithmetic the queen of mathematics.
Every infinite system S can be mapped onto a proper part of itself.
The true value of a mathematical concept lies in its applicability to the real world.
We must always be prepared to question our most fundamental assumptions.
A system S is infinite if it is similar to a proper part of itself.
The concept of number is entirely independent of the concept of space.
Logic and arithmetic are not two different sciences, but one and the same.
The continuity of the real numbers is a fundamental property that must be rigorously defined.
To understand is to perceive the connections between things.
Every theorem in mathematics should be proven with the utmost rigor.
The concept of a 'cut' provides a precise way to define irrational numbers.
Mathematics is not about numbers, but about relations between numbers.
The progress of science depends on the clarity of our definitions.
The natural numbers are the foundation upon which all of mathematics is built.
The infinite is not a mere potentiality, but an actual existence.
A good definition is half the battle in mathematics.
Mathematics is an art as much as it is a science.
Contemporaries of Richard Dedekind
Other Mathematicss born within 50 years of Richard Dedekind (1831–1916).