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Metamathematics is the study of mathematics itself using mathematical methods. This study produces metatheories, which are mathematical theories about other mathematical theories. Metamathematical metatheorems about mathematics itself were originally differentiated from ordinary mathematical theorems in the 19th century, to focus on what was then called the foundational crisis of mathematics. Richard's paradox (Richard 1905) concerning certain 'definitions' of real numbers in the English language is an example of the sort of contradictions which can easily occur if one fails to distinguish between mathematics and metamathematics. Something similar can be said around the well-known Russell's paradox (Does the set of all those sets that do not contain themselves contain itself?).

The term "metamathematics" is sometimes used as a synonym for certain elementary parts of formal logic, including propositional logic and predicate logic.

History[link]

Metamathematics was intimately connected to mathematical logic, so that the early histories of the two fields, during the late 19th and early 20th centuries, largely overlap. More recently, mathematical logic has often included the study of new pure mathematics, such as set theory, recursion theory and pure model theory, which is not directly related to metamathematics.

Serious metamathematical reflection began with the work of Gottlob Frege, especially his Begriffsschrift.

David Hilbert was the first to invoke the term "metamathematics" with regularity (see Hilbert's program). In his hands, it meant something akin to contemporary proof theory, in which finitary methods are used to study various axiomatized mathematical theorems.

Other prominent figures in the field include Bertrand Russell, Thoralf Skolem, Emil Post, Alonzo Church, Stephen Kleene, Willard Quine, Paul Benacerraf, Hilary Putnam, Gregory Chaitin, Alfred Tarski and Kurt Gödel. In particular, arguably the greatest achievement of metamathematics and the philosophy of mathematics to date is Gödel's incompleteness theorem: proof that given any finite number of axioms for Peano arithmetic, there will be true statements about that arithmetic that cannot be proved from those axioms.

Milestones[link]

• Principia Mathematica (Whitehead and Russell 1925)
• Gödel's completeness theorem, 1930
• Gödel's incompleteness theorem, 1931
• Tarski's definition of model-theoretic satisfaction, now called the T-schema
• The proof of the impossibility of the Entscheidungsproblem, obtained independently in 1936–1937 by Church and Turing.

See also[link]

• Meta-
• Model theory
• Philosophy of mathematics
• Proof theory

References[link]

• W. J. Blok and Don Pigozzi, "Alfred Tarski's Work on General Metamathematics", The Journal of Symbolic Logic, v. 53, No. 1 (Mar., 1988), pp. 36–50.
• I. J. Good. "A Note on Richard's Paradox". Mind, New Series, Vol. 75, No. 299 (Jul., 1966), p. 431. JStor
• Douglas Hofstadter, 1980. Gödel, Escher, Bach. Vintage Books. Aimed at laypeople.
• Stephen Cole Kleene, 1952. Introduction to Metamathematics. North Holland. Aimed at mathematicians.
• Jules Richard, Les Principes des Mathématiques et le Problème des Ensembles, Revue Générale des Sciences Pures et Appliquées (1905); translated in Heijenoort J. van (ed.), Source Book in Mathematical Logic 1879-1931 (Cambridge, Mass., 1964).
• Alfred North Whitehead, and Bertrand Russell. Principia Mathematica, 3 vols, Cambridge University Press, 1910, 1912, and 1913. Second edition, 1925 (Vol. 1), 1927 (Vols 2, 3). Abridged as Principia Mathematica to *56, Cambridge University Press, 1962.

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Friedrich A. Kittler (June 12, 1943 – October 18, 2011) was a literary scholar and a media theorist. His works relate to media, technology, and the military.

Biography[link]

Friedrich Adolf Kittler was born in 1943 in Rochlitz in Saxony. His family fled with him to West Germany in 1958, where from 1958 to 1963 he went to a natural sciences and modern languages Gymnasium in Lahr in the Black Forest, and thereafter, until 1972, he studied German studies, Romance philology and philosophy at the Albert Ludwigs University of Freiburg in Freiburg im Breisgau. During his studies, he was influenced by Jacques Lacan's and Michel Foucault's writings.

In 1976, Kittler received his doctorate in philosophy after a thesis on the poet Conrad Ferdinand Meyer. Between 1976 and 1986 he worked as academic assistant at the university's Deutsches Seminar. In 1984, he earned his Habilitation in the field of Modern German Literary History.

He had several stints as a Visiting Assistant Professor or Visiting Professor at universities in the United States, such as the University of California, Berkeley, the University of California, Santa Barbara and Stanford University.

From 1986 to 1990, he headed the DFG's Literature and Media Analysis project in Kassel and in 1987 he was appointed Professor of Modern German Studies at the Ruhr University. In 1993 he was appointed to the chair for Media Aesthetics and History at the Humboldt University of Berlin. Between 2005 and 2011 he was a professor at the European Graduate School in Saas-Fee, Switzerland.^[1]

In 1993, Kittler was awarded the "Siemens Media Arts Prize" (Siemens-Medienkunstpreis) by ZKM Karlsruhe (Zentrum für Kunst und Medientechnologie, or "Centre for Art and Media Technology") for his research in the field of media theory.^[2]

He was recognized in 1996 as a Distinguished Scholar at Yale University and in 1997 as a Distinguished Visiting Professor at Columbia University in New York. Kittler was a member of the Hermann von Helmholtz Centre for Culture and the research group Bild Schrift Zahl ("Picture Writing Number") (DFG). He died in Berlin. His last words were reportedly "Alle Apparate auschalten", which translates, referring to the machines which were keeping him alive, as "switch off all apparatuses".^[3]

Work and its importance[link]

Friedrich Kittler is seminal in the new approach to media theory that grew popular starting in the 1980s, new media (German: Technische Medien, which translates roughly to "technical media"). Kittler's central project is to "prove to the human sciences [...] their technological-media a priori" (Hartmut Winkler), or in his own words: "Driving the spirit out of the humanities",^[4] a title that he gave a work that he published in 1980.

Kittler sees an autonomy in technology and therefore disagrees with Marshall McLuhan's reading of the media as "extensions of man": "Media are not pseudopods for extending the human body. They follow the logic of escalation that leaves a written history behind it. (Kittler in Geschichte der Kommunikationsmedien. In: Jörg Huber, Alois Martin Müller (publishers): Raum und Verfahren).

Consequently, he sees in writing literature, in writing programmes and in burning structures into silicon chips a complete continuum: "As we know and simply do not say, no human being writes anymore. [...] Today, human writing runs through inscriptions burnt into silicon by electronic lithography [...]. The last historic act of writing may thus have been in the late seventies when a team of Intel engineers [plotted] the hardware architecture of their first integrated microprocessor." (Kittler, Es gibt keine Software. In: ders.: Draculas Vermächtnis. Technische Schriften).

See also[link]

• Media influence
• Paul Virilio

Notes[link]

Publications[link]

• 2009: Musik und Mathematik. Band 1: Hellas, Teil 2: Eros. Wilhelm Fink Verlag, Paderborn.
• 2006: Musik und Mathematik. Band 1: Hellas, Teil 1: Aphrodite. Wilhelm Fink Verlag, Paderborn.
• 2004: Unsterbliche. Nachrufe, Erinnerungen, Geistergespräche. Wilhelm Fink Verlag, Paderborn.
• 2002: Zwischen Rauschen und Offenbarung. Zur Kultur- und Mediengeschichte der Stimme (as publisher). Akademie Verlag, Berlin
• 2002: Optische Medien. Merve: Berlin. ISBN 3-88396-183-3 (English edition: Optical Media, with an introduction by John Durham Peters. Polity Press 2010)
• 2001: Vom Griechenland (with Cornelia Vismann; Internationaler Merve Diskurs Bd.240). Merve: Berlin. ISBN 3-88396-173-6
• 2000: Nietzsche – Politik des Eigennamens: wie man abschafft, wovon man spricht (with Jacques Derrida). Berlin.
• 2000: Eine Kulturgeschichte der Kulturwissenschaft. München
• 1999: Hebbels Einbildungskraft – die dunkle Natur. Frankfurt, New York, Vienna
• 1998: Zur Theoriegeschichte von Information Warfare
• 1998: Hardware das unbekannte Wesen
• 1997: Literature, Media, Information Systems: Essays (published by John Johnston). Amsterdam
• 1993: Draculas Vermächtnis: Technische Schriften. Leipzig: Reclam. ISBN 3-379-01476-1 Essays zu den "Effekten der Sprengung des Schriftmonopols", zu den Analogmedien Schallplatte, Film und Radio sowie "technische Schriften, die numerisch oder algebraisch verfasst sind".
• 1991: Dichter – Mutter – Kind. Munich
• 1990: Die Nacht der Substanz. Bern
• 1986: Grammophon Film Typewriter. Berlin: Brinkmann & Bose. ISBN 3-922660-17-7 (English edition: Gramophone, Film, Typewriter, Stanford 1999)
• 1985: Aufschreibesysteme 1800/1900. Fink: Munich. ISBN 3-7705-2881-6 (English edition: Discourse Networks 1800 / 1900, with a foreword by David E. Wellbery. Stanford 1990)
• 1979: Dichtung als Sozialisationsspiel. Studien zu Goethe und Gottfried Keller (with Gerhard Kaiser). Göttingen
• 1977: Der Traum und die Rede. Eine Analyse der Kommunikationssituation Conrad Ferdinand Meyers. Bern-Munich

Literature[link]

Secondary literature on Friedrich Kittler:

• Frank Hartmann: Friedrich Kittler. In: Information Philosophie 25 (1997) 4, S. 40-44.
• Josef Wallmannsverger: Friedrich Kittler. In: Helmut Schanze (publisher): Metzler Lexikon Medientheorie/ Medienwissenschaft, S. 162 f. Stuttgart 2002.
• Geoffrey Winthrop-Young: Friedrich Kittler zur Einführung, Hamburg: Junius Verlag 2005.

Lectures[link]

• The Relation of Art and Techne, European Graduate School
• Friedrich Kittler. Ein Medienphilosoph, European Graduate School
• Die Endlichkeit der Algorithmen
• Aphrodite mit Stefan Münker
• Farben und/oder Maschinen denken
• Vom Gott der Diebe zum gerechneten Raum
• Kittler liest Gedichte von Sappho und spricht über die Liebe
• Blitz und Serie, Ereignis und Donner
• Ontologie der Medien

External links[link]

• Friedrich Kittler Faculty page at European Graduate School. Biography, bibliography, articles and video lectures
• Friedrich Kittler in the German National Library catalogue (German)
• Auf dem Weg in den Maschinenpark - Rezension Friedrich Kittler: Eine Kulturgeschichte der Kulturwissenschaft - von Frank Hartmann, 28 November 2000
• Telepolis: Friedrich Kittler meets Stephan Schambach von Stefan Krempl, 26. Mai 2000
• Telepolis: Vom Sündenfall der Software - Medientheorie mit Entlarvungsgestus: Friedrich Kittler - von Frank Hartmann, 22 December 1998
• Telepolis: Nicht Cyborg, sondern Affe 17 April 2005
• "Rock me, Aphrodite!" - Interview mit Kittler auf Telepolis, 24 May 2006.
• "Hegel is dead. Miscellanea on Friedrich A. Kittler (1943-2011)" - Obituary on Kittler, Telepolis, 17 November 2011.
• The Sigmund H. Danziger, Jr. Memorial Lecture in the Humanities (Sigmund H. Danziger, Jr Lecture 1993-1994
• "We only have ourselves to draw upon"" - An interview by Andreas Rosenfelder

Notes[link]