This chapter discusses the complex conditions for the emergence of 19th-century symbolic logic. The main scope will be on the mathematical motives leading to the interest in logic; the philosophical context will be dealt with only in passing. The main object of study will be the algebra of logic in its British and German versions. Special emphasis will be laid on the systems of George Boole and above all of his German follower Ernst Schröder.
It is shown that David Hilbert's formalistic approach to axiomaticis accompanied by a certain pragmatism that is compatible with aphilosophical, or, so to say, external foundation of mathematics.Hilbert's foundational programme can thus be seen as areconciliation of Pragmatism and Apriorism. This interpretation iselaborated by discussing two recent positions in the philosophy ofmathematics which are or can be related to Hilbert's axiomaticalprogramme and his formalism. In a first step it is argued that thepragmatism of Hilbert's axiomatic contradicts the opinion thatHilbert style (...) axiomatical systems are closed systems, a reproachposed by Carlo Cellucci. In the second section the question isdiscussed whether Hilbert's pragmatism in foundational issuescomes close to an a-philosophical ``naturalism in mathematics'' assuggested by Penelope Maddy. The answer is ``no'', because forHilbert philosophy had its specific tasks in the general projectto found mathematics. This is illuminated in the concludingsection giving further evidence for Hilbert's foundationalapriorism by discussing his ``axiom of the existence of mind'' andrelating it to the ``one and only axiom'' of the German algebraistof logic, Ernst Schröder, postulating the inherence of signs onthe paper. (shrink)
Zermelos Zeit in Göttingen (1897?1910) kann als wissenschaftlich fruchtbarste Periode in seiner Karriere angesehen werden. Gleichwohl stehen bisher Untersuchungen aus. die eine Einbettung von Zermelos Werk in den biographischen und sozialen Kontext ermöglichen Die vorliegende Studie will diese Lücke unter Konzentration auf zwei Gegenstandsbereiche teileweise ausfüllen: (1) den historischen Entstehungskontext von Zermelos ersten Arbeiten über die Grundlagen der Mengenlehre; (2) die Vorgeschichte und näheren Umstände des 1907 an Zermelo verliehenen Lehrauftrages für mathematische Logik und verwandte Gegenstände. mit dem ein erster (...) Schritt zur Institutionalisierung dieses Faches als mathematischer Teildisziplin gemacht wurde. Beides wird in enger Verbindung zur ersten Phase des Hilbertschen Programms zur Grundlegung der Mathematik gesehen. Es wird aber auch gezeigt, daß für die Erteilung des Lehrauftrages neben diesen systematischen und forschungspolitischen Motiven auch persönliche und institutspolitische Rücksichten ausschlaggebend waren. Im Anhang werden Teile eines Briefwechsels zwischen Ernst Zermelo und dem Göttinger Philosophen Leonard Nelson ediert sowie die Anträge der Direktoren des Göttinger mathematisch-physikalischen Seminars auf Erteilung des Lehrauftrags und auf Ernennung Zermelos zum außerordentlichen Professor. Zermelo's Göttingen period (1897?1910) can be regarded as the most fruitful period in his scientific career. Nevertheless, up till now there have been no investigations which enable us to embed his work in its biographical and social contexts. This study tries to partially fill the gap, concentrating on two major themes: (I) the historical context of the development of Zermelo's early work on the foundations of set theory; (2) the prehistory and particulars of his lectureship for mathematical logic and related topics, this lectureship, which he was awarded in 1907, being the first step taken in Germany towards the establishment of this subject as a separate mathematical discipline. Both will be seen in close connection with the first phase of Hilbert's programme of founding mathematics. It is shown, however, that this lectureship was not only created to further the aims of Hilbert's research programme but that personal and institutional motives also played a role. In the appendices, parts of a correspondence between Ernst Zermelo and the Göttingen Philosopher Leonard Nelson are edited, as well as the applications of the directors of the Göttingen Mathematisch-physikalisches Seminar for awarding the lectureship to Zermelo in 1907, and for appointing Zermelo to an extraordinary professorship in 1910. (shrink)
This paper gives a survey of David Hilbert's (1862â1943) changing attitudes towards logic. The logical theory of the GÃ¶ttingen mathematician is presented as intimately linked to his studies on the foundation of mathematics. Hilbert developed his logical theory in three stages: (1) in his early axiomatic programme until 1903 Hilbert proposed to use the traditional theory of logical inferences to prove the consistency of his set of axioms for arithmetic. (2) After the publication of the logical and set-theoretical paradoxes by (...) Gottlob Frege and Bertrand Russell it was due to Hilbert and his closest collaborator Ernst Zermelo that mathematical logic became one of the topics taught in courses for GÃ¶ttingen mathematics students. The axiomatization of logic and set-theory became part of the axiomatic programme, and they tried to create their own consistent logical calculi as tools for proving consistency of axiomatic systems. (3) In his struggle with intuitionism, represented by L. E. J. Brouwer and his advocate Hermann Weyl, Hilbert, assisted by Paul Bemays, created the distinction between proper mathematics and meta-mathematics, the latter using only finite means. He considerably revised the logical calculus of thePrincipia Mathematica of Alfred North Whitehead and Bertrand Russell by introducing the Îµ-axiom which should serve for avoiding infinite operations in logic. (shrink)
The history of modern logic is usually written as the history of mathematical or, more general, symbolic logic. As such it was created by mathematicians. Not regarding its anticipations in Scholastic logic and in the rationalistic era, its continuous development began with George Boole's The Mathematical Analysis of Logic of 1847, and it became a mathematical subdiscipline in the early 20th century. This style of presentation cuts off one eminent line of development, the philosophical development of logic, although logic is (...) evidently one of the basic disciplines of philosophy. One needs only to recall some of the standard 19th century definitions of logic as, e.g., the art and science of reasoning (Whateley) or as giving the normative rules of correct reasoning (Herbart). In the paper the relationship between the philosophical and the mathematical development of logic will be discussed. Answers to the following questions will be provided: 1. What were the reasons for the philosophers' lack of interest in formal logic? 2. What were the reasons for the mathematicians' interest in logic? 3. What did "logic reform" mean in the 19th century? Were the systems of mathematical logic initially regarded as contributions to a reform of logic? 4. Was mathematical logic regarded as art, as science or as both? (shrink)
This paper presents the history of the first German lectureship for mathematical logic based on a ministerial commission, to which the Göttingen mathematician Ernst Zermelo was appointed in 1907. The lectureship is shown as imbedded in the intellectual history of mathematical logic which was at that time determined by the discussion of the set theoretical and logical paradoxes. Although Zermelo's early set theoretic papers can be regarded, and were in fact regarded in the Göttingen mathematicians' application for the lectureship, as (...) contributions to mathematical logic, the close connection between set theory and logic was at that time not evident. It is shown, however, that such “internal” motives were not the only reasons for proposing the lectureship, since the application was also highly influenced by personal and institutional factors and the attempt to find a paid position for Zermelo at the University of Göttingen. In addition, Zermelo's lectureship is presented in the context of the history of academic teaching in mathematical and symbolic logic with an overview of German lecture courses in this topic in the period between 1830 and 1915. The results are correlated with the reception of George Boole's algebra of logic and other symbolic logical systems in Germany at that time. Zermelo's lectureship is regarded as the first step towards an institutionalization of mathematical logic as a subdiscipline of mathematics. This step is interpreted by critically applying the results of Hubert Laitko's and Martin Guntau's considerations on the genesis of scientific disciplines. (shrink)
The Deutsche Forschungsgemeinschaft (DFG)is supporting a research project entitled ?Case studies towards the establishment of a social history of logic? with a grant, initially for two years. The project is being carried out by a team of five members under the direction of Professor Christian Thiel in the Institut für Philosophie and the Interdisziplinäres Institut für Wissenschaftstheorie und Wissenschaftsgeschichte (IIWW) of the University of Erlangen-Nürnberg.
When Benno Kerry (1858?89) died at the age of 30 he was already well?known for his competent and thoroughgoing philosophical criticism of Cantor?s set theory and Frege?s early philosophy of mathematics.Before his death he was working on a theory of limits (Grenzbegriffe) which was an elaboration of his Habilitationsschrift of 1884 and of which only a first part was published posthumously.This paper gives a survey of Kerry?s basic biographical data, and a first description of his Habilitationsschrift which had been missing (...) for a long time but was found by chance in the Nachlass of the German philosopher Leonard Nelson. (shrink)
Using a contextual method the specific development of logic between c. 1830 and 1930 is explained. A characteristic mark of this period is the decomposition of the complex traditional philosophical omnibus discipline logic into new philosophical subdisciplines and separate disciplines such as psychology, epistemology, philosophy of science, and formal logic. In the 19th century a growing foundational need in mathematics provoked the emergence of a structural view on mathematics and the reformulation of logic for mathematical means. As a result formallogic (...) was taken over by mathematics in the beginning of the 20th century as is shown by sketching the German example. (shrink)
The distinction between the contexts of discovery and justification has left a turbulent wake in the philosophy of science. This book recognizes the need to re-open the debate about the nature, development, and significance of the context distinction, about its merits and flaws. The discussion clears the ground for the productive and fruitful integration of these new developments into philosophy of science.
P. M. S. Hacker, Wittgenstein?s place in twentieth-century analytic philosophy:Oxford Blackwell, 1996. xviii + 346 pp. £50.00 $54.00 (cloth); £14.99 $21.95 (paper) Jarmo Pulkkinen, The threat of logical mathematism. A study on the critique of mathematical logic in Germany at the turn of the 20th century. Frankfurt a.M:Peter Lang, 1994. Scandinavian University Studies in the Humanities and Social Sciences; 7). 186 pp. 24 DM. ISBN 3-631-47409-1.
The German debates concerning the need for a reform of logic in post-Hegelian times took place under the label “The logical question”, a label introduced by Friedrich Adolf Trendelenburg. The main objective of these debates was to overcome the Hegelian identification of logic and metaphysics without re-establishing the old Aristotelian-scholastic formal logic. This paper presents the positions developed by Friedrich Adolf Trendelenburg, Otto Friedrich Gruppe, and Carl v. Prantl, each of whom advocated the importance of language in logic in order (...) to introduce a more dynamical element into the alleged static character of formal logic. (shrink)