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Friedrich Wilhelm Karl Ernst Schröder (; 25 November 1841 – 16 June 1902) was a German
mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...
mainly known for his work on
algebraic logic In mathematical logic, algebraic logic is the reasoning obtained by manipulating equations with Free variables and bound variables, free variables. What is now usually called classical algebraic logic focuses on the identification and algebraic de ...
. He is a major figure in the history of
mathematical logic Mathematical logic is the study of Logic#Formal logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory (also known as computability theory). Research in mathematical logic com ...
, by virtue of summarizing and extending the work of
George Boole George Boole ( ; 2 November 1815 – 8 December 1864) was a largely self-taught English mathematician, philosopher and logician, most of whose short career was spent as the first professor of mathematics at Queen's College, Cork in Ireland. H ...
, 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'', 1890–1905), 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 Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the study of deductively valid inferences or logical truths. It examines how conclusions follow from premises based on the structure o ...
of the day.


Life

Schröder learned mathematics at
Heidelberg Heidelberg (; ; ) is the List of cities in Baden-Württemberg by population, fifth-largest city in the States of Germany, German state of Baden-Württemberg, and with a population of about 163,000, of which roughly a quarter consists of studen ...
,
Königsberg Königsberg (; ; ; ; ; ; , ) is the historic Germany, German and Prussian name of the city now called Kaliningrad, Russia. The city was founded in 1255 on the site of the small Old Prussians, Old Prussian settlement ''Twangste'' by the Teuton ...
, and
Zürich Zurich (; ) is the list of cities in Switzerland, largest city in Switzerland and the capital of the canton of Zurich. It is in north-central Switzerland, at the northwestern tip of Lake Zurich. , the municipality had 448,664 inhabitants. The ...
, 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 Karlsruhe Polytechnische Schule, 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 George Boole ( ; 2 November 1815 – 8 December 1864) was a largely self-taught English mathematician, philosopher and logician, most of whose short career was spent as the first professor of mathematics at Queen's College, Cork in Ireland. H ...
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 Charles Sanders Peirce ( ; September 10, 1839 – April 19, 1914) was an American scientist, mathematician, logician, and philosopher who is sometimes known as "the father of pragmatism". According to philosopher Paul Weiss (philosopher), Paul ...
, including subsumption and quantification. Schröder also made original contributions to
algebra Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...
,
set theory Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory – as a branch of mathema ...
,
lattice theory A lattice is an abstract structure studied in the mathematical subdisciplines of order theory and abstract algebra. It consists of a partially ordered set in which every pair of elements has a unique supremum (also called a least upper bou ...
,"The Algebra of Logic Tradition"
''
Stanford Encyclopedia of Philosophy The ''Stanford Encyclopedia of Philosophy'' (''SEP'') is a freely available online philosophy resource published and maintained by Stanford University, encompassing both an online encyclopedia of philosophy and peer-reviewed original publication ...
''. ordered sets and
ordinal number In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, th, etc.) aimed to extend enumeration to infinite sets. A finite set can be enumerated by successively labeling each element with the leas ...
s. Along with
Georg Cantor Georg Ferdinand Ludwig Philipp Cantor ( ; ;  – 6 January 1918) was a mathematician who played a pivotal role in the creation of set theory, which has become a foundations of mathematics, fundamental theory in mathematics. Cantor establi ...
, 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 John Venn, Fellow of the Royal Society, FRS, Fellow of the Society of Antiquaries of London, FSA (4 August 1834 – 4 April 1923) was an English mathematician, logician and philosopher noted for introducing Venn diagrams, which are used in l ...
and Christine Ladd-Franklin both warmly cited this short book of Schröder's, and
Charles Sanders Peirce Charles Sanders Peirce ( ; September 10, 1839 – April 19, 1914) was an American scientist, mathematician, logician, and philosopher who is sometimes known as "the father of pragmatism". According to philosopher Paul Weiss (philosopher), Paul ...
used it as a text while teaching at
Johns Hopkins University The Johns Hopkins University (often abbreviated as Johns Hopkins, Hopkins, or JHU) is a private university, private research university in Baltimore, Maryland, United States. Founded in 1876 based on the European research institution model, J ...
. 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 logic In mathematical logic, algebraic logic is the reasoning obtained by manipulating equations with Free variables and bound variables, free variables. What is now usually called classical algebraic logic focuses on the identification and algebraic de ...
up to the end of the 19th century, one that had a considerable influence on the emergence of mathematical logic in the 20th century. He developed Boole's algebra into a calculus of relations, based on composition of relations as a multiplication. The Schröder rules relate alternative interpretations of a product of relations. 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 The ''Principia Mathematica'' (often abbreviated ''PM'') is a three-volume work on the foundations of mathematics written by the mathematician–philosophers Alfred North Whitehead and Bertrand Russell and published in 1910, 1912, and 1 ...
'' are very much owed to the ''Vorlesungen'', cited in ''Principias Preface and in
Bertrand Russell Bertrand Arthur William Russell, 3rd Earl Russell, (18 May 1872 – 2 February 1970) was a British philosopher, logician, mathematician, and public intellectual. He had influence on mathematics, logic, set theory, and various areas of analytic ...
'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 Hilary Whitehall Putnam (; July 31, 1926 – March 13, 2016) was an American philosopher, mathematician, computer scientist, and figure in analytic philosophy in the second half of the 20th century. He contributed to the studies of philosophy of ...
(1982) writes:


Works

* 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. **''Vorlesungen über die Algebra der Logik (Exakte Logik)''
Volume 1
**''Vorlesungen über die Algebra der Logik (Exakte Logik)''
Volume 2, Abt. 1
**''Vorlesungen über die Algebra der Logik (Exakte Logik)''
Volume 2, Abt. 2
**''Algebra und Logik der Relative, der Vorlesungen über die Algebra der Logik 3''
Volume 3, Abt. 1
* Schröder, E., 1898. "Über zwei Definitionen der Endlichkeit und G. Cantor'sche Sätze", ''Abh. Kaiserl. Leop.-Car. Akad. Naturf 71'': 301–362. ;Anthologies * Brady, Geraldine, 2000. ''From Peirce to Skolem''. North Holland. Includes an English translation of parts of the ''Vorlesungen''.


See also

* Schröder's equation * Schröder number * Schröder–Bernstein property * Schröder–Bernstein theorem for measurable spaces * Schröder–Hipparchus number


References


Further reading

* Irving Anellis, 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 Black is a color that results from the absence or complete absorption of visible light. It is an achromatic color, without chroma, like white and grey. It is often used symbolically or figuratively to represent darkness.Eva Heller, ''P ...
, ''Translations from the philosophical writings of Gottlob Frege''. Blackwell: 86–106. Original: 1895, ''Archiv für 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. Onlin
here
or

* 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 Hilary Whitehall Putnam (; July 31, 1926 – March 13, 2016) was an American philosopher, mathematician, computer scientist, and figure in analytic philosophy in the second half of the 20th century. He contributed to the studies of philosophy of ...
, 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.


External links

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requires login, unprovided) * * {{DEFAULTSORT:Schroder, Ernst 1841 births 1902 deaths 19th-century German mathematicians German logicians Scientists from Mannheim People from the Grand Duchy of Baden Academic staff of Technische Universität Darmstadt