# Ernst Schröder

**Ernst Schröder** (25 November, 1841 - 16 June, 1902) was the most significant representative of the algebraic logic school in Germany in the second half of the nineteenth century. He was important figure in the development of mathematical logic (a term he is thought to have invented), by drawing attention to the work of George Boole, Augustus De Morgan, Hugh MacColl and particularly Charles Peirce and others. His monumental work was the *Vorlesungen über die Algebra der Logik* (1890, 1891, 1895, 1905) some of which appeared posthumously.

His achievement was to assimilate and organize the disparate systems and notations of algebraic logic that were current in his day, and to present a systematic treatment of formal logic. This prepared the way for the development of mathematical logic as a separate discipline in the twentieth century.

## Contents

## Life

Schröder was born in Mannheim, Germany. He got his first chair of mathematics at Darmstadt University in 1874. He studied under Hesse and Kirchhoff then under Franz Neumann. He died in Karlsruhe, Germany.

## Work

Schröder's early work on formal algebra and logic did not benefit from work in the British school of algebraic logic. His sources were the textbooks of Ohm, Hermann Grassmann, Hankel and Robert Grassmann, which were written in the tradition of German combinatorial algebra and algebraic analysis (see Peckhaus 1997, ch. 6). However, from 1873 onwards, he learned of Boole's and De Morgan's work on logic, which he improved by adding Peirce's system of quantification.

Schröder also made original contributions in the fields of algebra, set theory and logic, and ordered sets and ordinal numbers. He was one of the two creators of the Cantor–Bernstein–Schroeder theorem, though there was an error in his original paper (Schröder 1898). Felix Bernstein (1878-1956) completed it as part of his Ph.D. dissertation.

His life's work *Vorlesungen über die Algebra der Logik*, was published between 1890 and 1905, completed by Müller after his death. This was a comprehensive and scholarly survey of "algebraic" (i.e. symbolic) logic up to the end of the nineteenth century, that had a great impact on the development of mathematical logic in the twentieth century. He 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.

His claim to have influenced the early development of the predicate calculus (via his popularisation of Peirce's work) is at least as great as Frege or Peano. Frege, however, was highly contemptuous of his work (Frege 1895).

On *Frege versus Schröder*, 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*. This book, which appeared in three volumes, has a third volume on the logic of relations (*Algebra und Logik der Relative*, 1895). The three volumes were the best-known logic text in the world among advanced students, and they can safely be taken to represent what any mathematician interested in the study of logic would have had to know, or at least become 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-Schroeder notation, and famous results and systems were published in it. Loewenheim stated and proved the Loewenheim theorem (later reproved and strengthened by Thoralf Skolem, whose name became attached to it together with Loewenheim's) in Peircian notation. In fact, there is no reference in Loewenheim's paper to any logic other than Peirce's. To cite another example, Zermelo presented his axioms for set theory in Peirce-Schroeder 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?

## References

- Dipert, R R. The life and work of Ernst Schröder, Modern Logic 1 (2-3) (1990/91), 117-139.
- Frege, G. "A critical elucidation of some points in E. Schroeder"s Vorlesungen Ueber Die Algebra der Logik",
*Archiv fur systematische Philosophie*1895, pp 433-456, transl. Geach, in Geach & Black 86-106 - Frege, Grundgesetze
- Peckhaus, V. "19th Century Logic between Philosophy and Mathematics", Bulletin of Symbolic Logic 5 (1999), 433-450; reprinted in Glen van Brummelen/Michael Kinyon (eds.), Mathmatics and the Historian's Craft. The Kenneth O. May Lectures, Springer: New York 2005, 203-220. (Download of the printed version: http://www.math.ucla.edu/%7Easl/bsl/0504/0504-001.ps; http://www-fakkw.upb.de/institute/philosophie/Personal/Peckhaus/Schriftenverzeichnis/Text__19th_century_logic.html).
- Peckhaus, V. Logik, Mathesis universalis und allgemeine Wissenschaft. Leibniz und die Wiederentdeckung der formalen Logik im 19. Jahrhundert, Akademie-Verlag: Berlin 1997, ch. 6, 233-296.
- Schröder, Ernst Vorlesungen über die Algebra der Logik, 3 vols., B.G. Teubner: Leipzig 1890-1905. Reprints Chelsea: Bronx, N.Y. 1966; Edition, Thoemmes Press.
- Peckhaus, V. "Schröder's Logic", in: Dov M. Gabbay/John Woods (Hgg.), Handbook of the History of Logic, vol. 3: The Rise of Modern Logic: From Leibniz to Frege, Elsevier North Holland: Amsterdam etc. 2004, 557-609.
- Putnam, H. "Peirce the Logician", Historia Mathematica, vol. 9, 1982, pp. 290-301, reprinted in H. Putnam,
*Realism with a Human Face*, Harvard University Press, 1990, pp. 252-260. - Schröder, Ernst "Uber zwei Definitionen der Endlichkeit und G. Cantor'sche Sätze "
*Abh. Kaiserl. Leop.-Car. Akad. Naturf*71, 301-62 - Thiel, C. A portrait, or, how to tell Frege from Schröder, Hist. Philos. Logic 2 (1981), 21-23.