# Ernst Schröder

# Ernst Schröder

**Friedrich Wilhelm Karl Ernst Schröder** (25 November 1841 in Mannheim, Baden, Germany – 16 June 1902 in Karlsruhe, Germany) was a German mathematician mainly known for his work on algebraic logic. He is a major figure in the history of mathematical logic (a term he may have invented), by virtue of summarizing and extending the work of George Boole, Augustus De Morgan, Hugh MacColl, and especially Charles Peirce. He is best known for his monumental *Vorlesungen über die Algebra der Logik* (Lectures on the algebra of logic), in three volumes, which prepared the way for the emergence of mathematical logic as a separate discipline in the twentieth century by systematizing the various systems of formal logic of the day.

Life

Schröder learned mathematics at Heidelberg, Königsberg, and Zürich, under Otto Hesse, Gustav Kirchhoff, and Franz Neumann. After teaching school for a few years, he moved to the Technische Hochschule Darmstadt in 1874. Two years later, he took up a chair in mathematics at the Polytechnische Schule in Karlsruhe, where he spent the remainder of his life. He never married.

Work

Schröder's early work on formal algebra and logic was written in ignorance of the British logicians George Boole and Augustus De Morgan. Instead, his sources were texts by Ohm, Hankel, Hermann Grassmann, and Robert Grassmann (Peckhaus 1997: 233–296). In 1873, Schröder learned of Boole's and De Morgan's work on logic. To their work he subsequently added several important concepts due to Charles Sanders Peirce, including subsumption and quantification.

Schröder also made original contributions to algebra, set theory, lattice theory, ordered sets and ordinal numbers. Along with Georg Cantor, he codiscovered the Cantor–Bernstein–Schröder theorem, although Schröder's proof (1898) is flawed. Felix Bernstein (1878–1956) subsequently corrected the proof as part of his Ph.D. dissertation.

Schröder (1877) was a concise exposition of Boole's ideas on algebra and logic, which did much to introduce Boole's work to continental readers. The influence of the Grassmanns, especially Robert's little-known *Formenlehre*, is clear. Unlike Boole, Schröder fully appreciated duality. John Venn and Christine Ladd-Franklin both warmly cited this short book of Schröder's, and Charles Sanders Peirce used it as a text while teaching at Johns Hopkins University.

Schröder's masterwork, his *Vorlesungen über die Algebra der Logik*, was published in three volumes between 1890 and 1905, at the author's expense. Vol. 2 is in two parts, the second published posthumously, edited by Eugen Müller. The *Vorlesungen* was a comprehensive and scholarly survey of "algebraic" (today we would say "symbolic") logic up to the end of the 19th century, one that had a considerable influence on the emergence of mathematical logic in the 20th century. The *Vorlesungen* is a prolix affair, only a small part of which has been translated into English. That part, along with an extended discussion of the entire *Vorlesungen*, is in Brady (2000). Also see Grattan-Guinness (2000: 159–76).

Schröder said his aim was:

Influence

Schröder's influence on the early development of the predicate calculus, mainly by popularising C. S. Peirce's work on quantification, is at least as great as that of Frege or Peano. For an example of the influence of Schröder's work on English-speaking logicians of the early 20th century, see Clarence Irving Lewis (1918). The relational concepts that pervade *Principia Mathematica* are very much owed to the *Vorlesungen*, cited in *Principia'*s Preface and in Bertrand Russell's Principles of Mathematics.

Frege (1960) dismissed Schröder's work, and admiration for Frege's pioneering role has dominated subsequent historical discussion. Contrasting Frege with Schröder and C. S. Peirce, however, Hilary Putnam (1982) writes:

See also

Schröder's equation

Schröder number

Schröder rules