Lorenzen's Proof of Consistency for Elementary Number Theory [With An
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1 Radical Besinnung in Formale Und Transzendentale Logik (1929)
1 Radical Besinnung in Formale und transzendentale Logik (1929)1 This is a pre-print of an article published in Husserl Studies. The final authenticated version is available online at: https://doi.org/DOI: 10.1007/s10743- 018-9228-5 ABSTRACT. This paper explicates Husserl’s usage of what he calls ‘radical Besinnung’ in Formale und transzendentale Logik (1929, henceforth FTL). Husserl introduces radical Besinnung as his method in the introduction to FTL. Radical Besinnung aims at criticizing the practice of formal sciences by means of transcendental phenomenological clarification of its aims and presuppositions. By showing how Husserl applies this method to the history of formal sciences down to mathematicians’ work in his time, the paper explains in detail the relationship between historical critical Besinnung and transcendental phenomenology. Ultimately the paper suggests that radical Besinnung should be viewed as a general methodological framework within which transcendental phenomenological descriptions are used to criticize historically given goal- directed practices. KEYWORDS: Husserl, Formal and Transcendental Logic, Besinnung, mathematics, apophantic logic, phenomenological method, criticism. Introduction This paper explicates Husserl’s usage of what he calls ‘radical Besinnung’ in Formale und transzendentale Logik (1929, henceforth FTL). Husserl started to emphasize the importance of Besinnung already in the early in the 1920s,2 but, for the sake of a clear argument, I limit myself 1 This paper has been greatly improved thanks to the detailed criticisms and useful and generous suggestions of George Heffernan. I am particularly thankful for him for drawing my attention to FTL §§102-105, where Husserl discusses transcendental Selbst-Besinnung. 2 The term is mentioned already in Logical Investigations (e.g., Hua1/1, 24-25; 304). -
'N Goue Draad Wat Deur Die Geskiedenis Van Die Wiskunde Loop
'n Goue Draad wat deur die Geskiedenis van die Wiskunde loop Prof D F M Strauss Dept Filosofie Fakulteit Geesteswetenskappe Universiteit van die Vrystaat Posbus 339, Bloemtontein 9300 [email protected] Opsomming Die aanvanklike sukses wat opgesluit gelê het in the Pythagoreïse oortuiging dat rasionale verhoudinge alles getalsmatig toeganklik maak het spoedig gegewens teëgekom wat nie met behulp van rasionale getalsverhoudinge hanteer kom word nie. Die onvermoë om hierdie probleem getalsmatig tot 'n oplossing tebring sou daartoe lei dat die Griekse wiskunde die weg van 'n geometrisering bewandel het. Daar is nie probeer om hierdie impasse met behulp van die idee van oneindige totaliteite te bowe te kom nie. Nie alleen het dit die aard van getal ondergeskik gemaak aan die ruimte nie, want dit het ook die middeleeuse spekulatiewe “metafisika van die syn” tot gevolg gehad. Eers via Descartes sou die moderne tyd 'n toenemende terugkeer na 'n aritmetiese perspektief vergestalt. Die ontdekking van die differensiaal- en integraalrekene deur Newton en Leibniz het egter opnuut probleme geskep, met name in die verantwoording van die aard van limiete. Die geykte benadering om limiete met behulp van konvergerende rye rasionale getalle te bepaal het 'n onverantwoorde sirkelredenasie bevat, want om as limiet te kan optree moes elke sodanige limiet rééds 'n getal gewees het. Skynbaar het Weierstrass daarin geslaag om 'n suksesvolle alternatiewe waardering van 'n veranderlike in sy statiese teorie te gee. Limiet- waardes kan willekeurig gekies word uit 'n domein van onderskeie getalle wat as 'n oneindige totaliteit opeens voorhande is. Toe hierdie benadering egter in die versamelingsteorie van Cantor tot Russel se kontradiksie lei, het twee teëgestelde reaksies die toneel oorheers. -
Gödel on Finitism, Constructivity and Hilbert's Program
Lieber Herr Bernays!, Lieber Herr Gödel! Gödel on finitism, constructivity and Hilbert’s program Solomon Feferman 1. Gödel, Bernays, and Hilbert. The correspondence between Paul Bernays and Kurt Gödel is one of the most extensive in the two volumes of Gödel’s collected works devoted to his letters of (primarily) scientific, philosophical and historical interest. It ranges from 1930 to 1975 and deals with a rich body of logical and philosophical issues, including the incompleteness theorems, finitism, constructivity, set theory, the philosophy of mathematics, and post- Kantian philosophy, and contains Gödel’s thoughts on many topics that are not expressed elsewhere. In addition, it testifies to their life-long warm personal relationship. I have given a detailed synopsis of the Bernays Gödel correspondence, with explanatory background, in my introductory note to it in Vol. IV of Gödel’s Collected Works, pp. 41- 79.1 My purpose here is to focus on only one group of interrelated topics from these exchanges, namely the light that ittogether with assorted published and unpublished articles and lectures by Gödelthrows on his perennial preoccupations with the limits of finitism, its relations to constructivity, and the significance of his incompleteness theorems for Hilbert’s program.2 In that connection, this piece has an important subtext, namely the shadow of Hilbert that loomed over Gödel from the beginning to the end of his career. 1 The five volumes of Gödel’s Collected Works (1986-2003) are referred to below, respectively, as CW I, II, III, IV and V. CW I consists of the publications 1929-1936, CW II of the publications 1938-1974, CW III of unpublished essays and letters, CW IV of correspondence A-G, and CW V of correspondence H-Z. -
Common Sense for Concurrency and Strong Paraconsistency Using Unstratified Inference and Reflection
Published in ArXiv http://arxiv.org/abs/0812.4852 http://commonsense.carlhewitt.info Common sense for concurrency and strong paraconsistency using unstratified inference and reflection Carl Hewitt http://carlhewitt.info This paper is dedicated to John McCarthy. Abstract Unstratified Reflection is the Norm....................................... 11 Abstraction and Reification .............................................. 11 This paper develops a strongly paraconsistent formalism (called Direct Logic™) that incorporates the mathematics of Diagonal Argument .......................................................... 12 Computer Science and allows unstratified inference and Logical Fixed Point Theorem ........................................... 12 reflection using mathematical induction for almost all of Disadvantages of stratified metatheories ........................... 12 classical logic to be used. Direct Logic allows mutual Reification Reflection ....................................................... 13 reflection among the mutually chock full of inconsistencies Incompleteness Theorem for Theories of Direct Logic ..... 14 code, documentation, and use cases of large software systems Inconsistency Theorem for Theories of Direct Logic ........ 15 thereby overcoming the limitations of the traditional Tarskian Consequences of Logically Necessary Inconsistency ........ 16 framework of stratified metatheories. Concurrency is the Norm ...................................................... 16 Gödel first formalized and proved that it is not possible -
On Proofs of the Consistency of Arithmetic
STUDIES IN LOGIC, GRAMMAR AND RHETORIC 5 (18) 2002 Roman Murawski Adam Mickiewicz University, Pozna´n ON PROOFS OF THE CONSISTENCY OF ARITHMETIC 1. The main aim and purpose of Hilbert’s programme was to defend the integrity of classical mathematics (refering to the actual infinity) by showing that it is safe and free of any inconsistencies. This problem was formulated by him for the first time in his lecture at the Second International Congress of Mathematicians held in Paris in August 1900 (cf. Hilbert, 1901). Among twenty three problems Hilbert mentioned under number 2 the problem of proving the consistency of axioms of arithmetic (under the name “arithmetic” Hilbert meant number theory and analysis). Hilbert returned to the problem of justification of mathematics in lectures and papers, especially in the twentieth 1, where he tried to describe and to explain the problem more precisely (in particular the methods allowed to be used) and simultaneously presented the partial solutions obtained by his students. Hilbert distinguished between the unproblematic, finitistic part of mathematics and the infinitistic part that needed justification. Finitistic mathematics deals with so called real sentences, which are completely meaningful because they refer only to given concrete objects. Infinitistic mathematics on the other hand deals with so called ideal sentences that contain reference to infinite totalities. It should be justified by finitistic methods – only they can give it security (Sicherheit). Hilbert proposed to base mathematics on finitistic mathematics via proof theory (Beweistheorie). It should be shown that proofs which use ideal elements in order to prove results in the real part of mathematics always yield correct results, more exactly, that (1) finitistic mathematics is conservative over finitistic mathematics with respect to real sentences and (2) the infinitistic 1 More information on this can be found for example in (Mancosu, 1998). -
What Is Mathematics: Gödel's Theorem and Around. by Karlis
1 Version released: January 25, 2015 What is Mathematics: Gödel's Theorem and Around Hyper-textbook for students by Karlis Podnieks, Professor University of Latvia Institute of Mathematics and Computer Science An extended translation of the 2nd edition of my book "Around Gödel's theorem" published in 1992 in Russian (online copy). Diploma, 2000 Diploma, 1999 This work is licensed under a Creative Commons License and is copyrighted © 1997-2015 by me, Karlis Podnieks. This hyper-textbook contains many links to: Wikipedia, the free encyclopedia; MacTutor History of Mathematics archive of the University of St Andrews; MathWorld of Wolfram Research. Are you a platonist? Test yourself. Tuesday, August 26, 1930: Chronology of a turning point in the human intellectua l history... Visiting Gödel in Vienna... An explanation of “The Incomprehensible Effectiveness of Mathematics in the Natural Sciences" (as put by Eugene Wigner). 2 Table of Contents References..........................................................................................................4 1. Platonism, intuition and the nature of mathematics.......................................6 1.1. Platonism – the Philosophy of Working Mathematicians.......................6 1.2. Investigation of Stable Self-contained Models – the True Nature of the Mathematical Method..................................................................................15 1.3. Intuition and Axioms............................................................................20 1.4. Formal Theories....................................................................................27 -
Biographies.Pdf
Chapter udf Biographies bio.1 Georg Cantor his:bio:can: An early biography of Georg Cantor sec (gay-org kahn-tor) claimed that he was born and found on a ship that was sailing for Saint Petersburg, Russia, and that his parents were unknown. This, however, is not true; although he was born in Saint Petersburg in 1845. Cantor received his doctorate in mathematics at the University of Berlin in 1867. He is known for his work in set theory, and is credited with founding set theory as a distinctive research discipline. He was the first to prove that there are infinite sets of different sizes. His theo- ries, and especially his theory of infini- ties, caused much debate among mathe- maticians at the time, and his work was controversial. Cantor's religious beliefs and his Figure bio.1: Georg Cantor mathematical work were inextricably tied; he even claimed that the theory of transfinite numbers had been communicated to him directly by God. Inlater life, Cantor suffered from mental illness. Beginning in 1894, and more fre- quently towards his later years, Cantor was hospitalized. The heavy criticism of his work, including a falling out with the mathematician Leopold Kronecker, led to depression and a lack of interest in mathematics. During depressive episodes, Cantor would turn to philosophy and literature, and even published a theory that Francis Bacon was the author of Shakespeare's plays. Cantor died on January 6, 1918, in a sanatorium in Halle. 1 Further Reading For full biographies of Cantor, see Dauben(1990) and Grattan-Guinness(1971). -
Gerhard Gentzen
bio.1 Gerhard Gentzen his:bio:gen: Gerhard Gentzen is known primarily sec as the creator of structural proof the- ory, and specifically the creation of the natural deduction and sequent calculus derivation systems. He was born on November 24, 1909 in Greifswald, Ger- many. Gerhard was homeschooled for three years before attending preparatory school, where he was behind most of his classmates in terms of education. De- spite this, he was a brilliant student and Figure 1: Gerhard Gentzen showed a strong aptitude for mathemat- ics. His interests were varied, and he, for instance, also write poems for his mother and plays for the school theatre. Gentzen began his university studies at the University of Greifswald, but moved around to G¨ottingen,Munich, and Berlin. He received his doctorate in 1933 from the University of G¨ottingenunder Hermann Weyl. (Paul Bernays supervised most of his work, but was dismissed from the university by the Nazis.) In 1934, Gentzen began work as an assistant to David Hilbert. That same year he developed the sequent calculus and natural deduction derivation systems, in his papers Untersuchungen ¨uber das logische Schließen I{II [Inves- tigations Into Logical Deduction I{II]. He proved the consistency of the Peano axioms in 1936. Gentzen's relationship with the Nazis is complicated. At the same time his mentor Bernays was forced to leave Germany, Gentzen joined the university branch of the SA, the Nazi paramilitary organization. Like many Germans, he was a member of the Nazi party. During the war, he served as a telecommu- nications officer for the air intelligence unit. -
Husserlian and Fichtean Leanings: Weyl on Logicism, Intuitionism, and Formalism
Philosophia Scientiæ Travaux d'histoire et de philosophie des sciences 13-2 | 2009 Varia Husserlian and Fichtean Leanings: Weyl on Logicism, Intuitionism, and Formalism Norman Sieroka Electronic version URL: http://journals.openedition.org/philosophiascientiae/295 DOI: 10.4000/philosophiascientiae.295 ISSN: 1775-4283 Publisher Éditions Kimé Printed version Date of publication: 1 October 2009 Number of pages: 85-96 ISBN: 978-2-84174-504-3 ISSN: 1281-2463 Electronic reference Norman Sieroka, « Husserlian and Fichtean Leanings: Weyl on Logicism, Intuitionism, and Formalism », Philosophia Scientiæ [Online], 13-2 | 2009, Online since 01 October 2012, connection on 02 November 2020. URL : http://journals.openedition.org/philosophiascientiae/295 ; DOI : https:// doi.org/10.4000/philosophiascientiae.295 Tous droits réservés Husserlian and Fichtean Leanings: Weyl on Logicism, Intuitionism, and Formalism Norman Sieroka ETH Zürich Résumé : Vers 1918 Hermann Weyl abandonnait le logicisme et donc la ten- tative de réduire les mathématiques à la logique et la théorie des ensembles. Au niveau philosophique, ses points de référence furent ensuite Husserl et Fichte. Dans les années 1920 il distingua leurs positions, entre une direc- tion intuitionniste-phénoménologique d’un côté, et formaliste-constructiviste de l’autre. Peu après Weyl, Oskar Becker adopta une distinction similaire. Mais à la différence du phénoménologue Becker, Weyl considérait l’approche active du constructivisme de Fichte comme supérieure à la vision d’essences passive de Husserl. Dans cet essai, je montre ce développement dans la pensée de Weyl. Bien que certains points dans sa réception de Husserl et Fichte ne soient pas soutenables, je vais argumenter en faveur de la distinction fonda- mentale mentionnée ci-dessus. -
Hilbert's Program Then And
Hilbert’s Program Then and Now Richard Zach Abstract Hilbert’s program was an ambitious and wide-ranging project in the philosophy and foundations of mathematics. In order to “dispose of the foundational questions in mathematics once and for all,” Hilbert proposed a two-pronged approach in 1921: first, classical mathematics should be formalized in axiomatic systems; second, using only restricted, “finitary” means, one should give proofs of the consistency of these axiomatic sys- tems. Although G¨odel’s incompleteness theorems show that the program as originally conceived cannot be carried out, it had many partial suc- cesses, and generated important advances in logical theory and meta- theory, both at the time and since. The article discusses the historical background and development of Hilbert’s program, its philosophical un- derpinnings and consequences, and its subsequent development and influ- ences since the 1930s. Keywords: Hilbert’s program, philosophy of mathematics, formal- ism, finitism, proof theory, meta-mathematics, foundations of mathemat- ics, consistency proofs, axiomatic method, instrumentalism. 1 Introduction Hilbert’s program is, in the first instance, a proposal and a research program in the philosophy and foundations of mathematics. It was formulated in the early 1920s by German mathematician David Hilbert (1862–1943), and was pursued by him and his collaborators at the University of G¨ottingen and elsewhere in the 1920s and 1930s. Briefly, Hilbert’s proposal called for a new foundation of mathematics based on two pillars: the axiomatic method, and finitary proof theory. Hilbert thought that by formalizing mathematics in axiomatic systems, and subsequently proving by finitary methods that these systems are consistent (i.e., do not prove contradictions), he could provide a philosophically satisfactory grounding of classical, infinitary mathematics (analysis and set theory). -
Hilbert's Program Then And
Hilbert’s Program Then and Now Richard Zach Abstract Hilbert’s program was an ambitious and wide-ranging project in the philosophy and foundations of mathematics. In order to “dispose of the foundational questions in mathematics once and for all,” Hilbert proposed a two-pronged approach in 1921: first, classical mathematics should be formalized in axiomatic systems; second, using only restricted, “finitary” means, one should give proofs of the consistency of these axiomatic sys- tems. Although G¨odel’s incompleteness theorems show that the program as originally conceived cannot be carried out, it had many partial suc- cesses, and generated important advances in logical theory and meta- theory, both at the time and since. The article discusses the historical background and development of Hilbert’s program, its philosophical un- derpinnings and consequences, and its subsequent development and influ- ences since the 1930s. Keywords: Hilbert’s program, philosophy of mathematics, formal- ism, finitism, proof theory, meta-mathematics, foundations of mathemat- ics, consistency proofs, axiomatic method, instrumentalism. 1 Introduction Hilbert’s program is, in the first instance, a proposal and a research program in the philosophy and foundations of mathematics. It was formulated in the early 1920s by German mathematician David Hilbert (1862–1943), and was pursued by him and his collaborators at the University of G¨ottingen and elsewhere in the 1920s and 1930s. Briefly, Hilbert’s proposal called for a new foundation of mathematics based on two pillars: the axiomatic method, and finitary proof theory. Hilbert thought that by formalizing mathematics in axiomatic systems, arXiv:math/0508572v1 [math.LO] 29 Aug 2005 and subsequently proving by finitary methods that these systems are consistent (i.e., do not prove contradictions), he could provide a philosophically satisfactory grounding of classical, infinitary mathematics (analysis and set theory). -
History and Philosophy of Logic
This article was downloaded by:[Universidad Granada] [Universidad Granada] On: 22 May 2007 Access Details: [subscription number 773444453] Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK History and Philosophy of Logic Publication details, including instructions for authors and subscription information: http://www.informaworld.com/smpp/title~content=t713812075 Book Review To cite this Article: , 'Book Review', History and Philosophy of Logic, 27:2, 200 - 203 To link to this article: DOI: 10.1080/01445340600563135 URL: http://dx.doi.org/10.1080/01445340600563135 PLEASE SCROLL DOWN FOR ARTICLE Full terms and conditions of use: http://www.informaworld.com/terms-and-conditions-of-access.pdf This article maybe used for research, teaching and private study purposes. Any substantial or systematic reproduction, re-distribution, re-selling, loan or sub-licensing, systematic supply or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material. © Taylor and Francis 2007 200 Book Reviews (IV) Topics and persons loosely related to Becker. The relation of the remaining three articles to Oskar Becker’s philosophy of mathematics is only slim.