Richard Dedekind: an Algebraic Foundation for Calculus

In 1858, while giving lectures on differential calculus, mathematician Richard Dedekind noted the lack of a truly scientific foundation of the arithmetic with which he taught his class. Realizing this flaw, Dedekind was motivated to improve matters in the foundations of the calculus. Dedekind made a number of highly significant contributions to mathematics and his work would change the style of mathematics into what is familiar to us today. One remarkable piece of work was his redefinition of irrational numbers in terms of “Dedekind cuts”. His work on mathematical induction, including the definition of finite and infinite sets, and his work in number theory, particularly in algebraic number fields, is also of major importance.

Julius Wilhelm Richard Dedekind was born October 6, 1831 in Brunswick, in what is today the country of Germany. The youngest of four children, his father was a professor at the Collegium Carolinum, and his mother the daughter of a professor at the same school. Dedekind attended school in Brunswick at the Gymnasium Martino-Catharineum from the age of seven. Early on, Dedekind was very interested in physics, however, he began to lose interest an

And yet despite these advances in technique, calculus remained without logical foundations. In 1821, French mathematician Cauchy began to clarify the basis of calculus with his theory of limits, a purely arithmetic theory that did not depend on geometric intuition or infinitesimals. Cauchy then showed how this could be used to give a logical account of the ideas of continuity, derivatives, integrals, and infinite series. In the next decade, the Russian mathematician Lobachevsky and Dedekind’s mentor Dirichlet both gave the definition of a function as a correspondence between two sets of real numbers.

During this time period, there were two schools of thought regarding the nature of math, the classical mathematicians and the constructive mathematicians, and this difference sometimes resulted in completely different mathematical theories. Dedekind could be described as an idealistic classical mathematician, in that he thought of mathematical concepts as things thought up rather than discovered, but that they formed “a system of simultaneously existing inter-related entities” (Ferreiros 92). Constructive mathematicians tended to think of mathematical concepts as ideas that are individually created and come into being one after another, and traditional classicisists tended to think of mathematical concepts as abstract entities that existed independently of thought.

Thus many mathematicians that held different philosophical views concerning math didn’t want to follow Dedekind’s reasoning that defined real numbers as creations of the mind corresponding to cuts in the system of rational numbers. They accepted his cuts, but just felt the necessity to describe them differently. For example, Weber addressed a letter to Dedekind suggesting that an irrational number should be taken to be the cut, instead of something newly created by the mind. It seems Dedekind was not insulted or in the least bit annoyed with other mathematicians defining his cuts in terms with which they were more comfortable. In fact, he wrote to Lipschitz “that if one does not wish to introduce numbers in his sense, “I have nothing against it; the theorem I prove (on completeness) then reads: the system of all cuts in the discontinuous domain of rational numbers forms a continuous manifold” (Bunn 224).

In part two of Dedekind’s book, The Nature and Meaning of Numbers, Dedekind explains his view of just that, the nature and meaning of number, as it was a topic of much debate at the time (mentioned above). In a later publication, Was sind und as Sollen die Zahlen? (1887), Dedekind plunged further into number theory. Instead of defining a particular set of objects as natural numbers, he defined a class of structures, which he called “simply infinite systems.” Dedekind defines:

Although Dedekind was now an instructor, he was still very much a student as well. In addition to the Dirichlet lectures he attended, Dedekind also attended Riemann’s lectures on Abelian and elliptic functions, as well as giving his own lectures on Galois theory. In 1858, Dedekind was offered and accepted a position in the Polytechnikum in Zurich; Riemann accepted a position there as well. In 1862, he was appointed to a position at the Polytechnikum in Brunswick, which had been created from the Collegium Carolinum, and he remained there until his death. From 1872 to 1875, Dedekind assumed directorship of the Polytechnikum, to a certain extent the successor to his father.

The primary motivation behind part one of his book, Continuity and Irrational Numbers, was the desire to replace loose geometrical concepts and intuitive justifications with exact

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