Ernst Schröder

Ernst Schröder

:"For the actor, see Ernst Schröder (actor)."Ernst Schröder (25 November, 1841 Mannheim, Germany16 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", 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, all written in the tradition of German combinatorial algebra and algebraic analysis (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 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 (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 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:


Schröder's influence on the early development of the predicate calculus, mainly by popularising 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 Charles Peirce, however, Hilary Putnam (1982) writes:

cquote|When I started to trace the later development of logic, the first thing I did was to look at Schröder's "Vorlesungen über die Algebra der Logik", ... [whose] third volume is on the logic of relations ("Algebra und Logik der Relative", 1895). The three volumes immediately became the best-known advanced logic text, and embody what any mathematician interested in the study of logic should have known, or at least have been acquainted with, in the 1890s.

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?


** Schröder, E., 1877. "Der Operationskreis des Logikkalküls". Leipzig: B.G. Teubner.
** Schröder, E., 1890-1905. "Vorlesungen über die Algebra der Logik", 3 vols. Leipzig: B.G. Teubner. Reprints: 1966, Chelsea; 2000, Thoemmes Press.
** Schröder, E., 1898. "Über zwei Definitionen der Endlichkeit und G. Cantor'sche Sätze ", "Abh. Kaiserl. Leop.-Car. Akad. Naturf 71": 301-362.
*Both Primary and Secondary
** Brady, Geraldine, 2000. "From Pierce to Skolem". North Holland. Includes an English translation of parts of the "Vorlesungen".
** Anellis, I. H., 1990-91, "Schröder Materials at the Russell Archives," "Modern Logic 1": 237-247.
** Dipert, R. R., 1990/91. "The life and work of Ernst Schröder," "Modern Logic 1": 117-139.
** Frege, G., 1960, "A critical elucidation of some points in E. Schröder's "Vorlesungen über die Algebra der Logik", translated by Geach, in Geach & Black, "Translations from the philosophical writings of Gottlob Frege". Blackwell: 86-106. Original: 1895, "Archiv fur systematische Philosophie 1": 433-456.
**Ivor Grattan-Guinness, 2000. "The Search for Mathematical Roots 1870-1940". Princeton University Press.
**Clarence Irving Lewis, 1960 (1918). " A Survey of Symbolic Logic". Dover.
** Peckhaus, V., 1997. "Logik, Mathesis universalis und allgemeine Wissenschaft. Leibniz und die Wiederentdeckung der formalen Logik im 19. Jahrhundert". Akademie-Verlag.
** Peckhaus, V., 1999, "19th Century Logic between Philosophy and Mathematics," "Bulletin of Symbolic Logic 5": 433-450. Reprinted in Glen van Brummelen and Michael Kinyon, eds., 2005. "Mathematics and the Historian's Craft. The Kenneth O. May Lectures". Springer: 203-220. Online [ here] or [ here] .
** Peckhaus, V., 2004. "Schröder's Logic" in Gabbay, Dov M., and John Woods, eds., "Handbook of the History of Logic. Vol. 3: The Rise of Modern Logic: From Leibniz to Frege". North Holland: 557-609.
** Hilary Putnam, 1982, "Peirce the Logician," "Historia Mathematica 9": 290-301. Reprinted in his 1990 "Realism with a Human Face". Harvard University Press: 252-260. [ Online fragment.]
** Thiel, C., 1981. "A portrait, or, how to tell Frege from Schröder," "History and Philosophy of Logic 2": 21-23.

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