1. Heyting Arend Heyting. Born 9 May 1898 Arend Heyting's father was Johannes Heytingand his mother was Clarissa Kok. Both Arend's parents were http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Heyting.html

Arend Heyting Born: 9 May 1898 in Amsterdam, Netherlands Died: 9 July 1980 in Lugano, Switzerland Click the picture above to see a larger version Show birthplace location Previous (Chronologically) Next Biographies Index Previous (Alphabetically) Next Main index Arend Heyting 's father was Johannes Heyting and his mother was Clarissa Kok. Both Arend's parents were school teachers and Johannes Heyting was particularly successful in his profession being appointed as head of a secondary school. Arend spent his school years with the intention that he would make a career in engineering. Only near the end of his schooling did his love and ability in mathematics mean that the course of his career changed and he went to university to study mathematics. Although Heyting's father was a successful school teacher, the family were still in financial problems when Heyting began his studies in 1916 at the University of Amsterdam. Both Heyting and his father earned the extra money necessary to pay for his studies by taking on private tutoring work. At the University of Amsterdam Heyting was taught by Brouwer who had a large influence on his future work. In 1922 Heyting graduated with a degree of master's standard.

2. Haken Hermann SYNERGETICS (Springer-Verlag, 1977) The distributed system can only exhibit global coherence . heyting arendINTUITIONISM (North Holland, 1956). A classic textbook for intuitionism. http://www.thymos.com/mind/h.html

Haken Hermann: SYNERGETICS (Springer-Verlag, 1977) Synergetics is a theory of pattern formation in complex systems. It tries to explain structures that develop spontaneously in nature. Since order emerges out of chaos, and chaos is not well defined, synergetics employs probabilities (to describe uncertainty) and information (to describe approximation). Entropy becomes a central concept, relating physics to information theory. Synergetics revolves around the concepts of: compression of the degrees of freedom of a complex system into dynamic patterns that can be expressed as a collective variable; behavioral attractors of changing stabilities; and the appearance of new forms as nonequilibrium phase transitions. Systems at instability points are driven by a slaving principle: long-lasting quantities can enslave short-lasting quantities (i.e., they can act as order parameters). Close to instability, stable motions (or "modes") are enslaved by unstable modes and can be ignored, thereby reducing the degrees of freedom of the system. The macroscopic behavior of the system is determined by the unstable modes. The dynamic equations of the system reflect the interplay between stochastic forces ("chance") and deterministic forces ("necessity"). Synergetics deals with self-organization, how collections of parts can produce structures. Synergetics therefore applies to systems driven far from equilibrium, where the classic concepts of thermodynamics are no longer adequate. Order can arise from chaos and can be maintained by flows of energy/matter.

3. Virtual Encyclopedia Of Mathematics alexandria herschel caroline lucretia herschel john frederick william herstein israelnathan hesse ludwig otto heuraet hendrik van heyting arend hilbert david http://www.lacim.uqam.ca/~plouffe/Simon/supermath.html

Super-Index of Biographies of Mathematicians abel niels henrik abraham bar hiyya ha-nasi abraham max abu kamil shuja ibn aslam ibn muhammad ... zygmund antoni This index was automatically generated using a new tagging program written by Simon Plouffe at LaCIM

4. Sylaby Logika, SYLABY. heyting arend Heyting 18981980. LOG101. Základy matematickéanalýzy (Introduction to Calculus). Jednosemestrální kurs, rozsah 2/1, Zk. http://logika.ff.cuni.cz/newpages/czech/sylaby.htm

Logický semináø 2000/2001 Sylabus kursu zimní semestr 2000/2001, výbìrová pøednáška typu A/C Katedra logiky, Filosofická fakulta, Universita Karlova TomᚠHoleèek, Vojtìch Kolman Cíl kurzu tento kurz je tvoøen nìkolika bloky, volnì spojenými snahou o rozbor obecných pøedpokladù formálního myšlení Program kurzu Komentáø k vybraným paragrafùm Filosofických zkoumání jedná se o poznámky týkající se logiky; §§23, 48, 81, 89-108, 109, 240-2; komentáø veden s dùrazem na užití logiky Pøíklady aplikace logiky ve spoleèenských vìdách odmítnutí Russellovy koncepce základních výrokù (Inquiry); vícejazyèné pojetí vìdecké teorie; návrh logické rekonstrukce vztahù komplementárních teorií, založené na tetralektice Poznámky k Lorenzenovì konstruktivismu operativní pojetí logiky a matematiky, dialogická logika, kritika pokusù o 'rozumné' založení logiky (Grundprobleme) Atestace zkouška: pøimìøená úèast, prostudování literatury (po domluvì), orientace v probírané látce kolokvium: pøimìøená úèast Literatura Wittgenstein, Ludwig: Filosofická zkoumání, Filosofia, Praha 1998 Russell, Bertrand: An Inquiry into Meaning and Truth, George Allen, London 1961

5. Heyting Arend Heyting. Born 9 May 1898 Arend Heyting was taught by Brouwer whohad a large influence on his future work. He received his doctorate http://www.math.hcmuns.edu.vn/~algebra/history/history/Mathematicians/Heyting.ht

6. Arend Heyting Biography of arend heyting (18981980) arend heyting's father was Johannes heyting and his mother was Clarissa Kok. http://www.philosophenlexikon.de/heyting.htm

Begriffe Abaelard - Ayer Baader - Byron Cabanis - Czezowski Ebbinghaus - Ewald ... Geschichte der Philosophie Diskussion PhilTalk Philosophieforen Andere Lexika PhilLex -Lexikon der Philosophie Lexikon der griechischen Mythologie PhiloThek Bibliothek der Klassiker Zeitschriftenlesesaal Nachschlagewerke Allgemeine Information ... Dokumentenlieferdienste Spiele Philosophisches Galgenraten PhilSearch.de Shops PhiloShop PhiloShirt Service Kontakt Impressum eMail Arend Heyting (1898 - 1980) Logiker powered by Uwe Wiedemann

7. Thematic Afternoon On Constructivism Including the first arend heyting Lecture. Amsterdam, the Netherlands; 14 December 2001. http://turing.wins.uva.nl/~anne/heyting.html

Thematic Afternoon on CONSTRUCTIVISM including the first Arend Heyting Lecture Date: Friday, December 14, 2001 Place: De Rode Hoed, Keizersgracht 102, 1015 CV AMSTERDAM, tel. 020 638 5606 ABOUT THE AREND HEYTING LECTURE The Arend Heyting Foundation came into being as the result of the last will of ms J.F. Heyting-van Anrooij, who died in september 1998. The aims of the Foundation are to support mathematical logic, and more specifically to organize every three years the Arend Heyting Lecture. Ms. heyting stipulated in he last will that the first Arend Heyting lecture should be devoted to workconnected with Heytings own work on intuitionism. If you wish to attend, please notify Marjan Veldhuisen: marjanv@science.uva.nl PROVISIONAL PROGRAMME (version 6-11-2001) 13.30-13.45 Reception 13.45-14.00 Introduction by the chairman of the Arend Heyting Foundation 14.00-15.00 F. Richman, Constructive algebra (1st Arend Heyting Lecture) 15.10-15.50 J. van Oosten, Modified realizability 16.30-17.30 A. Simpson, After the Interval ABSTRACTS OF THE TALKS F.Richman, Constructive Algebra (1st AREND HEYTING Lecture)

8. Sobre El Axioma X De Heyting SOBRE EL AXIOMA X DE heyting En la formalización que presenta arend heyting. para la lógica intuicionista esta expresión corresponde al axioma X. http://galileo.fcien.edu.uy/sobre_el_axioma_x_de_heyting.htm

SOBRE EL AXIOMA X DE HEYTING Robert Calabria La ley de Scoto o ex falso quodlibet , esto es ØA®(A®B) fue considerada desde siempre una paradoja para la Lógica Clásica al menos en el débil sentido de “resultado chocante”; así que uno no puede menos de preguntarse cómo en un sistema intuicionista llega a tener la dignidad de un axioma. En la formalización que presenta Arend Heyting para la lógica intuicionista esta expresión corresponde al axioma X . Ciertamente, el mismo Heyting se siente obligado a dar una especie de justificación del mismo-y, como anota Susan Haack en , p.109, es significativo que considerara que sólo este axioma requiriese justificación-, ofreciéndola en , p.102, como sigue: Axiom X may not seem intuitively clear. As a matter of fact, it adds to the precision of the definition of implication. You remember that A®B can be asserted if and only if we possess a construction which , joined to the construction A , would prove B Now suppose that A that is, we have deduced a contradiction from the supposition that

9. UntitledA Division Algorithm Florida Atlantic University Boca Raton, FL 33431 Ab lived from 1898 to 1980 arend heyting is important in the development of intuitionistic logic and algebra. Find out more at http://www.math.fau.edu/Richman/docs/heyting2.html

A division algorithm Fred Richman Florida Atlantic University Boca Raton, FL 33431 richman@fau.edu Abstract: A divisibility test of Arend Heyting, for polynomials over a field in an intuitionistic setting, may be thought of as a kind of division algorithm. We show that such a division algorithm holds for divisibility by polynomials of content 1 over any commutative ring in which nilpotent elements are zero. In addition, for an arbitary commutative ring R , we characterize those polynomials g such that the R -module endomorphism of R X ] given by multiplication by g has a left inverse. 1 Introduction In [ ], Arend Heyting talks about polynomials over what has become known as a Heyting field ]. This is a commutative ring with an apartness relation a b , that satisfies the following properties Not a a . (consistent) If a b , then b a . (symmetric) If a b , then either a c or b c . (cotransitive) If not a b , then a b . (tight) If a b , then a c b c . (shift invariant) a if and only if a is invertible. (field) If g a a X a n X n is a polynomial with coefficients in a Heyting field, then

10. References For Heyting References for arend heyting. Biography Articles J Alcolea Banegas, arendheyting (Spanish), Mathesis (México) 4 (2) (1988), 189220. M http://www-gap.dcs.st-and.ac.uk/~history/References/Heyting.html

References for Arend Heyting Biography in Dictionary of Scientific Biography (New York 1970-1990). Articles: J Alcolea Banegas, Arend Heyting (Spanish), M Franchella, Heyting's contribution to the change in research into the foundations of mathematics, Hist. Philos. Logic Arend Heyting (Dutch), Nederl. Akad. Wetensch. Verslag Afd. Natuurk. B A Kushner, Arend Heyting : a short sketch of his life and work (Russian), Methodological analysis of the foundations of mathematics 'Nauka' (Moscow, 1988), 121-135. Letters from A N Kolmogorov to A Heyting (Russian), Uspekhi Mat. Nauk Letters from A N Kolmogorov to A Heyting, Russian Math. Surveys N H Niekus, H van Riemsdijk and A S Troelstra, Bibliography of A Heyting, Nieuw Arch. Wisk. A S Troelstra, Arend Heyting and his Contribution to Intuitionism, Nieuw archief voor wiskunde A S Troelstra, The scientific work of A Heyting, Compositio Math. D van Dalen, Heyting and intuitionistic geometry, in Mathematical logic (New York, 1990), 19-27. Main index Birthplace Maps Biographies Index History Topics ... Societies, honours, etc.

11. References For Heyting References for the biography of arend heyting J Alcolea Banegas, arend heyting (Spanish), Mathesis (México) 4 (2) (1988), 189220. http://www-history.mcs.st-and.ac.uk/References/Heyting.html

References for Arend Heyting Biography in Dictionary of Scientific Biography (New York 1970-1990). Articles: J Alcolea Banegas, Arend Heyting (Spanish), M Franchella, Heyting's contribution to the change in research into the foundations of mathematics, Hist. Philos. Logic Arend Heyting (Dutch), Nederl. Akad. Wetensch. Verslag Afd. Natuurk. B A Kushner, Arend Heyting : a short sketch of his life and work (Russian), Methodological analysis of the foundations of mathematics 'Nauka' (Moscow, 1988), 121-135. Letters from A N Kolmogorov to A Heyting (Russian), Uspekhi Mat. Nauk Letters from A N Kolmogorov to A Heyting, Russian Math. Surveys N H Niekus, H van Riemsdijk and A S Troelstra, Bibliography of A Heyting, Nieuw Arch. Wisk. A S Troelstra, Arend Heyting and his Contribution to Intuitionism, Nieuw archief voor wiskunde A S Troelstra, The scientific work of A Heyting, Compositio Math. D van Dalen, Heyting and intuitionistic geometry, in Mathematical logic (New York, 1990), 19-27. Main index Birthplace Maps Biographies Index History Topics ... Societies, honours, etc.

12. Philosophenlexikon.de heyting,arend; Hieronymos aus Rhodos; Hilbert, David; Hildebrand, Dietrich http://www.philosophenlexikon.de/index-hh.htm

Begriffe Abaelard - Ayer Baader - Byron Cabanis - Czezowski Ebbinghaus - Ewald ... Geschichte der Philosophie Diskussion PhilTalk Philosophieforen Andere Lexika PhilLex -Lexikon der Philosophie Lexikon der griechischen Mythologie PhiloThek Bibliothek der Klassiker Zeitschriftenlesesaal Nachschlagewerke Allgemeine Information ... Dokumentenlieferdienste Spiele Philosophisches Galgenraten PhilSearch.de Shops PhiloShop PhiloShirt Service Kontakt Impressum eMail Philosophen und Logiker [a] [b] [c] [d] ... [z] H Habrotelia von Tarent Haeckel, Ernst Haecker, Theodor Hahn, Hans ... Hyppolite, Jean powered by Uwe Wiedemann

13. Neil Tennant Reading (on Reserve in Department) 1930. heyting, arend. 1931. heyting, arend.`Die intuitionistische Grundlegung der Mathematik', Erkenntnis, vol. 2, pp. http://philosophy.ohio-state.edu/people/Faculty/tennant.9/652.html

NEIL W. TENNANT tennant.9@osu.edu If you email me, please use the header PHIL 652: YOURNAME. Professor Department of Philosophy Autumn Term 2001 PHIL 653: Non-Classical Logics Lecture/seminar Aims of this course. We aim to impart an appreciation of the great variety of non-classical logics, their philosophical motivations, and their proof-theoretic and semantic properties. We shall examine both extensions of classical logic, such as modal logic, and deviant subsystems of classical logic, such as intuitionistic, relevant, and paraconsistent logics. Topics . We shall be covering topics drawn from the following list: The logical systems C (classical), I (intuitionistic), M (minimal), IR (intuitionistic relevant), Anderson-Belnap R (relevant), E (entailment), T, B, K, S4, S5 (modal). Proof-theoretic systems for these logics. Semantical treatments. Relations among systems. Philosophical motivations. Applications. Textbook: Alan Ross Anderson and Nuel D. Belnap, eds., Entailment: The Logic of Relevance and Necessity , Vol. I, Princeton University Press, 1975. Chapter 1.

14. Heyting Portrait Portrait of arend heyting arend heyting. JOC/EFR August 2001 http://www-history.mcs.st-and.ac.uk/history/PictDisplay/Heyting.html

Arend Heyting JOC/EFR August 2001 The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/PictDisplay/Heyting.html

15. Godel 3 Galley. Project for Ergebnisse , A proposed joint book with arend heyting (seeCorrespondence Series I heyting, arend) 11, Drafts by arend heyting http://libweb.princeton.edu/libraries/firestone/rbsc/aids/godel/godel3.html

IV. Drafts and Offprints Box/Folder AMs Notebook (in Gabelsberger shorthand) labelled "Diss. unrein," written both directions [1929?] TMs [carbon] (in German) labelled "Dissertation," with autograph corrections, 34 pp. [1929?] TMs (in German), labelled "Vollstandigkeit d. Axiome" with autograph corrections, 20 pp. [1930?] Printed page proof with autograph corrections [1930?] Offprint 1930 Galley with autograph annotations [1930] TMs (in German) with autograph corrections, pp. 10 [1930] AC describing contents of original file n.d. Erkenntnis 2 TMs [carbon] of discussion (in German) with autograph corrections, p. 23 ca.1930 TMs of Nachtrag ("Supplement"), with autograph corrections, 3 pp., back labelled "Erkenntnis" [1931?] TMs of Nachtrag with autograph corrections, p. 3 Offprint with autograph annotations [1931] Copy of Erkenntnis 2 with autograph annotations 1931 Undecidability Results (early drafts of AMs (in Gabelsberger shorthand) in 2 Notebooks, one inserted in the other, labelled "Unentsch. unrein," written both directions [1930?] AMs Notebook (in Gabelsberger shorthand), labelled "Unentsch. unrein," written both directions [1930?]

16. Godel 1 64, Herbrand, Jacques, 1931. heyting, arend 65, 19311933. 66, 1957, 1969. 32,Zentralblatt für Mathematik (re Gödel/arend heyting Collaboration) 1931-1935. http://libweb.princeton.edu/libraries/firestone/rbsc/aids/godel/godel1.html

I. Personal and Scientific Correspondence, 1929-1978 Box/Folder Addison, John Miscellaneous "A" Behman, Heinrich Bernays, Paul: [See also Series XIII: Folders 6 and 7] Boone, William: January-May, 1958 June-August, 1958 n.d. Brutian, George A. Burks, Arthur Miscellaneous "B": T. R. Bachiller to Errett Bishop Max Black to Terrell Ward Bynum Carnap, Rudolf [See also Series XIV: Folder 1] Chang, C. C. Chomsky, Noam Chuaqui Kettlun, Rolando B. 1969-1972, n.d. Church, Alonzo Cohen, Paul J.: April 24-July 17, 1963 July 20-September 27, 1963 October 4-December 13, 1963 Miscellaneous "C": Ronald Calinger to D. V. Choodnovsky Jeffrey Cohen to Haskell B. Curry Davis, Martin: 1965, n.d. Dreben, Burton S. Miscellaneous "D" [Einstein, Albert: see Miscellaneous "E"] Ellentuck, Erik: Miscellaneous "E" Feferman, Solomon: Feigl, Herbert: Fisher, Edward R., Jr. Flexner, Abraham Ford, Lester R. (re: Friedburg, Robert Friedman, Harvey: Miscellaneous "F" Gandy, R. O. Grandjean, Burke 1957-1961, includes undated notes Miscellaneous "G" Halpern, James, includes discussion notes Hasenjaeger, G.

17. Logician's Year May. 5 May, Lowenheim, Leopold, (+ 1957). 9 May, heyting, arend, (* 1898). July.1 Jul, Leibniz, Gottfried Wilhelm von, (* 1646). 9 Jul, heyting, arend, (+ 1980). http://www.volny.cz/logici/vyroci/english.html

The Logician's Year January February March April ... December January 5 Jan Kleene, Stephen Cole 6 Jan Cantor, Georg 12 Jan Hintikka, Jaakko 14 Jan Carroll, Lewis Tarski, Alfred Gödel, Kurt 19 Jan Ramsey, Frank Plumpton 23 Jan Hilbert, David 26 Jan Kleene, Stephen Cole 27 Jan Carroll, Lewis February 2 Feb Russell, Bertrand Artur William 3 Feb Lewis, Clarence Irving 6 Feb Arnauld, Antoine 8 Feb von Neumann, Johannes 11 Feb Post, Emil Leon 12 Feb Herbrand, Jean Dedekind, Richard 13 Feb £ukasiewicz, Jan 14 Feb Hilbert, David 15 Feb Whitehead, Alfred North 16 Feb Nicod, Jean 17 Feb Fraenkel, Adolf Abraham 22 Feb Ramsey, Frank Plumpton 27 Feb Brouwer, Luitzgen Egbertus Jan March 3 Mar Cantor, Georg 5 Mar Barwise, Jon 6 Mar Davidson, Donald 7 Mar Montague, Richard 18 Mar de Morgan, Augustus Carnap Rudolf 23 Mar Skolem, Thoralf 24 Mar Lorenzen, Paul 25 Mar Ackermann, Wilhelm April 2 Apr Vaught, Robert Lawson 4 Apr Venn, John Vaught, Robert Lawson 12 Apr Lewis, Clarence Irving 19 Apr Peirce, Charles Sanders 20 Apr Peano, Giuseppe 21 Apr Post, Emil Leon 26 Apr Wittgenstein, Ludwig 28 Apr Gödel, Kurt

18. Logikùv Rok Translate this page Kveten. 5. 5. Lowenheim, Leopold, (+ 1957). 9. 5. heyting, arend, (* 1898). 13. Cervenec.1. 7. Leibniz, Gottfried Wilhelm von, (* 1646). 9. 7. heyting, arend, (+ 1980). http://www.volny.cz/logici/vyroci/

Logikùv rok Leden Únor Bøezen Duben ... Prosinec Leden Kleene, Stephen Cole Cantor, Georg Hintikka, Jaakko Carroll, Lewis Tarski, Alfred Gödel, Kurt Ramsey, Frank Plumpton Hilbert, David Kleene, Stephen Cole Carroll, Lewis Únor Russell, Bertrand Artur William Lewis, Clarence Irving Arnauld, Antoine von Neumann, Johannes Post, Emil Leon Herbrand, Jean Dedekind, Richard £ukasiewicz, Jan Hilbert, David Whitehead, Alfred North Nicod, Jean Fraenkel, Adolf Abraham Ramsey, Frank Plumpton Brouwer, Luitzgen Egbertus Jan Bøezen Cantor, Georg Barwise, Jon Davidson, Donald Montague, Richard de Morgan, Augustus Carnap Rudolf Skolem, Thoralf Lorenzen, Paul Ackermann, Wilhelm Duben Vaught, Robert Lawson Venn, John Vaught, Robert Lawson Lewis, Clarence Irving Peirce, Charles Sanders Peano, Giuseppe Post, Emil Leon Wittgenstein, Ludwig Gödel, Kurt Wittgenstein, Ludwig Kvìten Löwenheim, Leopold Heyting, Arend Wang, Hao Russell, Bertrand Arthur William Wang, Hao Zermelo, Ernst Skolem, Thoralf Èerven Turing, Alan Mathison Church, Alonzo von Wright, Georg Henrik Turing, Alan Mathison Quine, Willard Van Orman

19. The Mathematics Genealogy Project - Index Of HEY 1957. Heyneman, Robert, Heynig, Frank, Technische Universität Chemnitz,1978. heyting, arend, Universiteit van Amsterdam, 1925. Heywang http://genealogy.math.ndsu.nodak.edu/html/letter.phtml?letter=HEY

20. Stránka Studentù Logiky Kveten. 5. 5. Löwenheim, Leopold, (+ 1957). 9. 5. heyting, arend, (* 1898). Cervenec.1. 7. Leibniz, Gottfried Wilhelm von, (* 1646). 9. 7. heyting, arend, (+ 1980). http://www.cuni.cz/singleton/vyroci/vyroci.html

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