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. November 25 is the 329th (in leap years the 330th) day of the year in the Gregorian calendar. ... 1841 is a common year starting on Friday (link will take you to calendar). ... June 16 is the 167th day of the year in the Gregorian calendar (168th in leap years), with 198 days remaining. ... Events January-April January 28 - The Carnegie Institution is founded in Washington, DC with a $10 million gift from Andrew Carnegie. ... Alternative meaning: Nineteenth Century (periodical) (18th century — 19th century — 20th century — more centuries) As a means of recording the passage of time, the 19th century was that century which lasted from 1801-1900 in the sense of the Gregorian calendar. ... Mathematical logic is a discipline within mathematics, studying formal systems in relation to the way they encode intuitive concepts of proof and computation as part of the foundations of mathematics. ... This article is not about George Boolos, another mathematical logician. ... Augustus De Morgan (June 27, 1806 - March 18, 1871) was an Indian-born British mathematician and logician. ... Charles Sanders Peirce Charles Sanders Peirce (September 10, 1839 – April 19, 1914) was an American logician, philosopher, scientist, and mathematician. ...

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. Logic (from ancient Greek λόγος (logos), meaning reason) is the study of arguments. ... (19th century - 20th century - 21st century - more centuries) Decades: 1900s 1910s 1920s 1930s 1940s 1950s 1960s 1970s 1980s 1990s The 20th century lasted from 1901 to 2000 in the Gregorian calendar (often from (1900 to 1999 in common usage). ...

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. This article is about the German city. ... Mathematics, often abbreviated maths in Commonwealth English and math in American English, is the study of abstraction. ... Events January - April January 1 - New York City annexes The Bronx January 23 - Marriage of the Duke of Edinburgh, second son of Queen Victoria, to Grand Duchess Marie Alexandrovna of Russia, only daughter of Emperor Alexander III of Russia. ... Map of Germany showing Karlsruhe Coat of Arms of Karlsruhe Karlsruhe castle at night Karlsruhe (population 282,595 in December 2003) is a city of Germany, in the Baden-Württemberg Bundesland, located near the French-German border. ...

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. The word Britain is used to refer to the United Kingdom (UK) the island of Great Britain, which consists of the countries of England, Scotland, and Wales sometimes the Roman province called Britain or Britannia The word British generally means belonging to or associated with Britain in one of the... In language and logic, quantification is a construct that specifies the extent of validity of a predicate, that is the extent to which a predicate holds over a range of things. ...

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. Algebra is a branch of mathematics which may be roughly characterized as a generalization and extension of arithmetic, in which symbols are employed to denote operations, and letters to represent number and quantity; it also refers to a particular kind of abstract algebra structure, the algebra over a field. ... Set theory is the mathematical theory of sets, which represent collections of abstract objects. ... Mathematical logic is a discipline within mathematics, studying formal systems in relation to the way they encode intuitive concepts of proof and computation as part of the foundations of mathematics. ... Ordinal numbers, or ordinals for short, are numbers used to denote the position in an ordered sequence: first, second, third, fourth, etc. ... In set theory, the Cantor-Bernstein-Schroeder theorem, named after Georg Cantor, Felix Bernstein, and Ernst Schröder, states that, if there exist injective functions f : A → B and g : B → A between the sets A and B, then there exists a bijective function h : A → B. In terms of the...

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). Spoken language is a language that people utter words of the language. ... Philosophy (from the Greek words philos and sophia meaning love of wisdom) is understood in different ways historically and by different philosophers. ... The idea of a universal language is at least as old as the Biblical story of Babel and its fall — the mythical point of which is that there was once a time of a universal Adamic language (and then something happened, analogous to the Fall of Man). ... First-order predicate calculus or first-order logic (FOL) is a theory in symbolic logic that permits the formulation of quantified statements such as there is at least one X such that. ... Friedrich Ludwig Gottlob Frege Friedrich Ludwig Gottlob Frege (November 8, 1848 - July 26, 1925) was a German mathematician, logician, and philosopher who is regarded as a founder of both modern mathematical logic and analytic philosophy. ... Giuseppe Peano (August 27, 1858 – April 20, 1932) was an Italian mathematician and philosopher. ...

On Frege versus Schröder, Hilary Putnam (1982) writes: Hilary Whitehall Putnam (born July 31, 1926) is a key figure in the philosophy of mind during the 20th century. ...

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?

The history of logic began in three cultures. ... Albert Thoralf Skolem (May 23, 1887 - March 23, 1963) was a Norwegian mathematician. ... Ernst Friedrich Ferdinand Zermelo (July 27, 1871 – May 21, 1953) was a German mathematician and philosopher. ... In epistemology, an axiom is a self-evident truth upon which other knowledge must rest, from which other knowledge is built up. ... A statue of Leif Ericson in front of the Hallgrímskirkja in Reykjavik Leif Ericson (old Icelandic: Leifr Eiríksson) was an explorer, the son of Eric the Red (Eiríkr rauði), a Norwegian outlaw, who was the son of another Norwegian outlaw, Þorvaldr Ásvaldsson. ... The Americas (sometimes referred to as America) is the area including the land mass located between the Pacific Ocean and the Atlantic Ocean, generally divided into North America and South America. ... -1... No authentic contemporary portrait of Columbus has been found; this late 19th-century engraving is one of many conjectural images For information about the director, see the article on Chris Columbus. ...

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 (online at http://www.meta-religion.com/Mathematics/Philosophy_of_mathematics/19_century_logic.htm, original link at http://www.phil.uni-erlangen.de/~p1phil/personen/peckhaus/texte/logic_phil_math.html seems broken)
• Peckhaus, V. 1997 ------
• Schröder, Ernst Vorlesungen über die Algebra der Logik. 1890-1905 Edition, Thoemmes Press
• 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.

Peter Geach is one of the foremost contemporary British philosophers. ... Max Black (1909 - 1988) was a distinguished Anglo-American philosopher, who has been a leading influence in analytic philosophy in the first half of the twentieth century. ... Infinity is a word carrying a number of different meanings in mathematics, philosophy, theology and everyday life. ... Georg Ferdinand Ludwig Philipp Cantor (March 3, 1845 – January 6, 1918) was a mathematician who was born in Russia and lived in Germany for most of his life. ...

External links

• http://intranet.woodvillehs.sa.edu.au/pages/resources/maths/History/Schrdr.htm