Leopold Kronecker

Leopold Kronecker (December 7, 1823 - December 29, 1891) was a German mathematician and logician who argued that arithmetic and analysis must be founded on "whole numbers", saying, "God made the integers; all else is the work of man" (Bell 1986, p. 477). This put Kronecker in opposition to some of the mathematical extensions of Georg Cantor. Kronecker was a student and lifelong friend of Ernst Kummer. public domain image File links The following pages link to this file: Leopold Kronecker ... public domain image File links The following pages link to this file: Leopold Kronecker ... December 7 is the 341st day (342nd on leap years) of the year in the Gregorian calendar. ... 1823 was a common year starting on Wednesday (see link for calendar). ... December 29 is the 363rd day of the year (364th in leap years) in the Gregorian Calendar, with 2 days remaining. ... 1891 was a common year starting on Thursday (see link for calendar). ... A mathematician is a person whose area of study and research is mathematics. ... The article titled Logicians treats the ancient Chinese philosophers known by that name (with a capital L). List of logicians (with a lower-case l) treats philosophers, mathematicians, and others whose topic of scholarly study is logic. ... Arithmetic or arithmetics (from the Greek word αριθμός = number) in common usage is a branch of (or the forerunner of) mathematics which records elementary properties of certain operations on numerals, though in usage by professional mathematicians, it often is treated as a synonym for number theory. ... An analysis is a critical evaluation, usually made by breaking a subject (either material or intellectual) down into its constituent parts, then describing the parts and their relationship to the whole. ... The integers consist of the positive natural numbers (1, 2, 3, …) the negative natural numbers (−1, −2, −3, ...) and the number zero. ... Georg Cantor Georg Ferdinand Ludwig Philipp Cantor (March 3, 1845 – January 6, 1918) was a mathematician who was born in Russia and lived in Germany for most of his life. ... Ernst Eduard Kummer (29 January 1810 in Sorau, Brandenburg, Prussia - 14 May 1893 in Berlin, Germany) was a German mathematician. ...

Kronecker wrote 1845 his dissertation, at the University of Berlin, on number theory, giving special formulation to units in certain algebraic number fields. Peter Gustav Dirichlet was his teacher. 1845 was a common year starting on Wednesday (see link for calendar). ... There is no institution called the University of Berlin, but there are four universities in Berlin, Germany: Humboldt University of Berlin (Humboldt-Universität zu Berlin) Technical University of Berlin (Technische Universität Berlin) Free University of Berlin (Freie Universität Berlin) Berlin University of the Arts (Universität der Künste Berlin) This is... Traditionally, number theory is that branch of pure mathematics concerned with the properties of integers. ... The word unit means any of several things: Unit of measurement or physical unit, a fundamental quantity of measurement in science or engineering. ... In mathematics, an algebraic number field (or simply number field) is a finite field extension of the rational numbers Q. That is, it is a field which contains Q and has finite dimension when considered as a vector space over Q. The study of algebraic number fields, and these days... Johann Peter Gustav Lejeune Dirichlet (February 13, 1805 - May 5, 1859) was a German mathematician credited with the modern formal definition of a function. ...

After obtaining his degree, however, Kronecker managed the estate and business of his uncle, producing nothing mathematical for eight years. In his 1853 memoir on the algebraic solvability of equations, Kronecker extended the work of Évariste Galois on the theory of equations. He accepted a professorship at the University of Berlin in 1883. 1853 was a common year starting on Saturday (see link for calendar). ... Evariste Galois - Wikipedia /**/ @import /skins-1. ... In mathematics, the theory of equations comprises a major part of traditional algebra. ... 1883 was a common year starting on Monday (see link for calendar). ...

Kronecker also contributed to the concept of continuity, reconstructing the form of irrational numbers in real numbers. In analysis, Kronecker rejected the formulation of a continuous, nowhere differentiable function by his colleague, Karl Weierstrass. In his 1850 paper, On the Solution of the General Equation of the Fifth Degree, Kronecker solved the quintic equation by applying group theory. In mathematics, a continuous function is one in which arbitrarily small changes in the input produce arbitrarily small changes in the output. ... In mathematics, an irrational number is any real number that is not a rational number, i. ... In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ... An analysis is a critical evaluation, usually made by breaking a subject (either material or intellectual) down into its constituent parts, then describing the parts and their relationship to the whole. ... In mathematics, the derivative of a function is one of the two central concepts of calculus. ... Look up Function on Wiktionary, the free dictionary In general (not in the mathematical but in the engineering sense), a function is a goal-oriented property of an entity (according to the Adam Maria Gadomskis TOGA meta-theory, 1993). ... Karl Theodor Wilhelm Weierstrass (Weierstraß) (October 31, 1815 – February 19, 1897) was a German mathematician who is often cited as the father of modern analysis. He was born in Ostenfelde, Westphalia (today Germany) and died in Berlin, Germany. ... 1850 was a common year starting on Tuesday (see link for calendar). ... In mathematics, a quintic equation is a polynomial equation in which the greatest exponent on the independent variable is five. ... Group theory is that branch of mathematics concerned with the study of groups. ...

Kronecker's finitism made him a forerunner of intuitionism in foundations of mathematics. In the philosophy of mathematics, finitism is an extreme form of constructivism, according to which a mathematical object does not exist unless it can be constructed from natural numbers in a finite number of steps. ... In the philosophy of mathematics, intuitionism, or neointuitionism (opposed to preintuitionism), is an approach to mathematics as the constructive mental activity of humans. ... The term foundations of mathematics is sometimes used for certain fields of mathematics itself, namely for mathematical logic, axiomatic set theory, proof theory, model theory, and recursion theory. ...

The Kronecker delta and Kronecker product are named after Kronecker, as are the Kronecker-Weber theorem and Kronecker's theorem in number theory. In mathematics, the Kronecker delta or Kroneckers delta, named after Leopold Kronecker (1823-1891), is a function of two variables, usually integers, which is 1 if they are equal, and 0 otherwise. ... In mathematics, the Kronecker product, denoted by ⊗, is an operation on two matrices of arbitrary size resulting in a block matrix. ... In algebraic number theory, the Kronecker-Weber theorem states that every finite abelian extension of the field of rational numbers , or in other words every algebraic number field whose Galois group over is abelian, is a subfield of a cyclotomic field, i. ... In mathematics, Kroneckers theorem is a result in diophantine approximation applying to several real numbers xi, for 1 ≤ i ≤ N, which generalises the fact that an infinite cyclic subgroup of the unit circle group is a dense subset. ...

References

• Eric Temple Bell, Men of Mathematics, Simon and Schuster, New York, 1986.

Eric Temple Bell (1883 - 1960) was a mathematician born in Scotland who lived in the USA from 1903 until his death. ...

External link

• Biography at the MacTutor archive