November 25, 2010 <Back to Index>
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Ernst Schröder (25 November 1841 Mannheim, Germany – 16 June 1902 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 3 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. Schröder learned mathematics at Heidelberg, Königsberg, and Zürich, under Hesse, 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. 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 the proof in Schröder (1898) is flawed. Felix Bernstein (1878 – 1956) subsequently corrected the proof as part of his Ph.D. dissertation. Schröder wrote in 1877 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. Schröder said his aim was: 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: While,
to my knowledge, no one except Frege ever published a single paper in
Frege's notation, many famous logicians adopted Peirce-Schröder
notation, and famous results and systems were published in it. Löwenheim stated and proved the Löwenheim theorem (later reproved and strengthened by Thoralf Skolem,
whose name became attached to it together with Löwenheim's) in
Peircian notation. In fact, there is no reference in Löwenheim's
paper to any logic other than Peirce's. To cite another example, Zermelo presented his axioms for set theory in Peirce-Schröder notation, and not, as one might have expected, in Russell-Whitehead notation. One
can sum up these simple facts (which anyone can quickly verify) as
follows: Frege certainly discovered the quantifier first (four years
before Oscar Howard Mitchell, going by publication dates, which are all we have as far as I know). But Leif Ericson probably discovered America "first" (forgive me for not counting the native Americans, who of course really discovered it "first"). If the effective discoverer, from a European point of view, is Christopher Columbus,
that is because he discovered it so that it stayed discovered (by
Europeans, that is), so that the discovery became known (by Europeans).
Frege did "discover" the quantifier in the sense of having the rightful
claim to priority; but Peirce and his students discovered it in the
effective sense. The fact is that until Russell appreciated what he had
done, Frege was relatively obscure, and it was Peirce who seems to have
been known to the entire world logical community. How many of the
people who think that "Frege invented logic" are aware of these facts? |