Irrational Philosophy? Kronecker's Constructive Philosophy and Finding the Real Roots of a Polynomial

Authors

Richard B. Schneider, University of Missouri - Kansas CityFollow

Abstract

The prominent mathematician Leopold Kronecker (1823 – 1891) is often relegated to footnotes and mainly remembered for his strict philosophical position on the foundation of mathematics. He held that only the natural numbers are intuitive, thus the only basis for all mathematical objects. In fact, Kronecker developed a complete school of thought on mathematical foundations and wrote many significant algebraic works, but his enigmatic writing style led to his historical marginalization. In 1887, Kronecker published an extended version of his paper, “On the Concept of Number,” translated into English in 2010 for the first time by Edward T. Dean, who confirms that Kronecker is “notoriously difficult to read.” In his paper, Kronecker proves that a so-called “algebraic number”, meaning any root of a polynomial with integer coefficients, can be isolated from the other roots of that polynomial, as Dean says, “using solely talk of natural numbers.” To ease the reader’s comprehension of Kronecker’s prose, here we explicate in detail the argument contained in that paper.

Author Bio

Richard Schneider is a junior at the University of Missouri — Kansas City studying Mathematics and Computer Science. He is involved on campus in Sigma Tau Delta and Lucerna, UMKC’s Undergraduate Research Journal, and enjoys reading, biking, and being outdoors. He composed his work in his History of Mathematics course taught by his sponsor, Dr. Richard Delaware.

Faculty Sponsor

Richard Delaware

Recommended Citation

Schneider, Richard B. (2021) "Irrational Philosophy? Kronecker's Constructive Philosophy and Finding the Real Roots of a Polynomial," Rose-Hulman Undergraduate Mathematics Journal: Vol. 22: Iss. 1, Article 1.
Available at: https://scholar.rose-hulman.edu/rhumj/vol22/iss1/1