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Mathematical Treasure: H. C. Schwarz's Philosophy of Mathematics

Frank J. Swetz (The Pennsylvania State University)

Georg Wilhelm Hegel (1770-1831) was a German philosopher who, in seeking the relationship between “mind” and “nature,” developed a philosophy of absolute idealism. Among the realities of his time that Hegel questioned was that of mathematics. He felt that many mathematical topics were externally defined and lacked expression of the true essence or dialectic from which the concept or topic arose: continuity and infinity were two such questionable topics. Hegel also rejected “the law of excluded middle” as a method of logical or mathematical proof. Another biography of Hegel is available from the Stanford Encyclopedia of Philosophy.

Hermann Cuno Schwarz was a mathematics instructor based at Halle who, in his treatise Versuch einer Philosophie der Mathematik (Toward a Philosophy of Mathematics, 1853), refuted Hegel’s theories concerning higher mathematics. While he was not the best known Hermann Schwarz in mathematics, the same author published Elemente der Zahlen-Theorie (Elements of Number Theory) in 1855.                                              

Schwarz’s “Table of Contents” provides an outline of the main areas of disagreement.                                               

On page 68 of the Philosophy is a discussion and definition of the process of differentiation that a contemporary calculus student can easily follow.                                               

The images above were obtained from the Rare Book Collection at Franklin and Marshall College, Lancaster, Pennsylvania.

Index to Mathematical Treasures

Frank J. Swetz (The Pennsylvania State University), "Mathematical Treasure: H. C. Schwarz's Philosophy of Mathematics," Convergence (February 2016)