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:
:"...to design logic as a calculating discipline, especially to give access to the exact handling of relative concepts, and, from then on, by emancipation from the routine claims of
natural language, to withdraw any fertile soil from "cliché" in the field of
philosophy as well. This should prepare the ground for a scientific
universal language that looks more like a sign language than like a sound language. "
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), however, 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:
:"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 O. H. 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?"
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