Papers by Volker Peckhaus
Modern Logic, Jun 1, 1993
The book also contains four papers of a less technical nature. I refer to William Aspray's articl... more The book also contains four papers of a less technical nature. I refer to William Aspray's article, "Oswald Veblen and the Origins of Mathematical Logic at Princeton" (where the role of extra scientific factors in the process of forming a scientific centre are explained); Stephen C. Kleene's story about writing his famous Introduction to Metamathematics; the paper by Jonathan P. Seldin, "In Memóriám: Haskell Brooks Curry" (where the importance of Curry's work both for logic and computer science is assessed); and the paper by Dirk Siefkes, "The Work of J. Richard Buchi" (which is supplemented with a list of Buchi's publications and a list of his doctoral students). Though the aim of the book and the intention of the editor is not to give a full history of mathematical logic in the last century, it is a pity that some important parts of this history were not presented. There is for example, no paper devoted to the meaning and the role of the Polish school of logic between the Wars. The editor acknowledges that omission in the Introduction but the reasons given for it are not convincing. In conclusion, this is an important book. It exposes the richness of ideas and viewpoints, the difficult and not always direct pathways taken in the development of mathematical logic in the last century, and the various factors which did and continue to affect that development.
Modern Logic, Apr 1, 1995
In 1910 the B.G. Teubner Verlag in Leipzig published a book entitled Der Lügner. Theorie, Geschic... more In 1910 the B.G. Teubner Verlag in Leipzig published a book entitled Der Lügner. Theorie, Geschichte und Auflösung by Alexander Rüstow , who was at that time member of the scientific staff of that publishing house and later became famous as one of the fathers of German social market economy. It was the printed version of his dissertation presented already in 1908 to the Philosophical Faculty of the University of Erlangen. The printed version has three parts. After a discussion of the logical and set-theoretical paradoxes (I "Theorie") Rüstow treats the history of the paradoxes (II "Geschichte"), presenting above all several versions of the Liar Paradox in ancient philosophy. In the last part (III "Auflösung") he attempts a solution of the paradoxes, among them Russell's paradox. As Kenneth Blackwell pointed out to me, Rüstow's doctoral thesis might be one of the oldest theses on Russell. Therefore it seems to be worthwhile to sketch its historical context. Rüstow was already in his school times in Berlin a friend of Leonard Nelson (1882-1927) who became later the founder of the Neue Fries'sche Schule in Göttingen. Rüstow's thesis is an outgrowth of the early philosophical discussion of Russell's paradox in Göttingen. This discussion was evoked by Hubert's early reaction on the publication of the paradoxes in his Heidelberg lecture "Über die Grundlagen der Logik und der Arithmetik" (1904, published 1905). It became relevant for the Nelson circle because Gerhard Hessenberg , another friend of Nelson's, and co-editor of the circle's journal Abhandlungen der Fries'sehen Schule. N.F., intended to write a report on set-theory for this * Reprinted, with additions and corrections, from History of Logic Newsletter, no. 13 (September -October 1994), pp. 2-3, by permission of the publisher and author.
The Baltic International Yearbook of Cognition, Logic and Communication, 2008
The German debates concerning the need for a reform of logic in post-Hegelian times took place un... more 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 reestablishing 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.
Modern Logic, 1991
Es wird gezeigt, daß Schröders Auseinandersetzung mit der Relativlogik Peircescher Prägung zu ein... more Es wird gezeigt, daß Schröders Auseinandersetzung mit der Relativlogik Peircescher Prägung zu einem tiefgreifenden Wandel in seiner Auffassung von der Rolle der Logik bei der Begründung der Mathematik führte. Die Algebra und Logik der Relative wurde zum pasigraphischen Schlüssel zur Schaffung einer schon in den frühen zeichentheoretischen Schriften programmatisch geforderten wissenschaftlichen Universalsprache und zu einem Instrument für den Aufbau der "absoluten Algebra", einer allgemeinen Theorie der Verknüpfung. Daher steht in Schröders logischen Schriften der Jahre 1895 bis 1901 nicht mehr die Anwendung mathematischer Methoden auf die Analyse der Logik im Vordergrund, sondern entsprechend der logizistischen Grundthese eine Darstellung und Analyse der Mathematik mit den Mitteln der Logik. Die Wendung Schröders zum Logizismus wird anhand veröffentlichter und bisher unveröffentlichter Zeugnisse dokumentiert. Im Anhang werden einschlägige Stücke seines Briefwechsels mit Paul Carus und Felix Klein ediert.
In this paper the Indian Buddhist logic of the Middle Ages is connected to methodological aspects... more In this paper the Indian Buddhist logic of the Middle Ages is connected to methodological aspects of logic, especially to the idea that logic can serve as an organon, e. g., as a tool for discovering novelties. Research on this idea can be characterized by questions like “Is there a logic of invention?”, “What is the role, if any, of deductive logic in creative processes?”, “What is the role of logic in philosophy of science, especially in a philosophy of science which takes the context of discovery into account?”
Historisches Wörterbuch der Philosophie
Handbook of the History of Logic, 2004
The German mathematician, Ernst Schroder (1841-1902), was one of the most important representativ... more The German mathematician, Ernst Schroder (1841-1902), was one of the most important representatives of the algebra of logic. Schroder considered himself an algebraist. It is only by chance that his life's work is usually connected to logic. No doubt, most of his life he was concerned with logic, always regarding it as means to an end, the vision of a scientific universal language. Both, logic and a universal language, are based on algebra as general theory of connecting operations. His contributions to set theory are the results of his research on models of algebraic logical structures. Besides this, his name is recognized in eponyms—such as Schroder domain, Schroder equation, Schroder functional equation—that is, in the context of functional theory and complex dynamics.
Boston Studies in the Philosophy of Science, 1996
Today the achievements of Hermann Gunther and at least part of Robert Grassmann’s results have fi... more Today the achievements of Hermann Gunther and at least part of Robert Grassmann’s results have finally found the recognition they deserve. This is true for Hermann Gunther Grassmann’s Ausdehnungslehre, also for Robert Grassmann’s algebraico-logical Formenlehre (1872), works which were partly written in collaboration. Robert Grassmann (1815–1901)1 is now placed undoubtedly among the representatives of the algebra of logic, and his logic system is positioned close to that of George Boole (1815–1864), although published 25 years after the first logical book of the British founder of this symbolic approach to logic (Boole 1847). It is stated with benevolence that Grassmann obviously created his logic without any knowledge of his British predecessors. In order to appraise this last judgement it has to be regarded, however, that in 1872, when Grassmann published his Formenlehre oder Mathematik, the British algebra of logic had not yet made the leap across the channel anyway.
The use of general descriptive names, registered names, trademarks, service marks, etc. in this p... more The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein.
NTM International Journal of History and Ethics of Natural Sciences, Technology and Medicine, 1995
This paper gives a survey of David Hilbert's (1862-1943) changing attitudes towards logic. The lo... more This paper gives a survey of David Hilbert's (1862-1943) changing attitudes towards logic. The logical theory of the G6ttingen 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 G6ttingen 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 met,a-mathematics, the latter using only finite means. He considerably revised the logical calculus of the Principia Mathematica of Alfred North Whitehead and Bertrand Russell by introducing the e-axiom which should serve for avoiding infinite operations in logic. Vortrag, gehalten am 6. Mai 1993 im Rahmen einer Vortragsreihe zum 10ilbert-Jubiliiumsjahr des Fachbereichs Mathematik der Georg-August-Universittit G&tingen und am 16. Dezember 1993 im Rahmen des Symposiums ,,David Hilbert und die Herausbildung der modernen Mathematik" der S~ichsischen Akademie der Wissenschaften zu Leipzig und des Karl-Sudhoff-Instituts der Universit~it Leipzig.
Modern Logic, Oct 1, 1994
At the end of his life Ernst Schröder (1841-1902) regarded the formulation of a general theory of... more At the end of his life Ernst Schröder (1841-1902) regarded the formulation of a general theory of operations as his "very own field of research." Already in his Lehrbuch der Arithmetik und Algebra (1873) and especially in Über die formalen Elemente der absoluten Algebra (1874) he had made the first steps towards establishing a "formal" or, in its final stage, an "absolute algebra" proceeding from the assumption that there are operations which allow to connect two objects of a given domain (not restricted to mathematical objects) to a third, which belongs also to the domain. By this means Schroder tried to go beyond the narrow boundaries of traditional arithmetic, and to embrace also non-commutative numbers like quaternions. Through Robert Grassmann's Formenlehre (1872), Schröder discovered the analogy between arithmetical and logical connectives, but already late in 1874 he went further: he then treated formal logic and arithmetic as two different models of formal algebra. His subsequent research was devoted to the analysis of logic as such a model. Schröder considered his proof that the "second subsumption of the distributive law" was not provable in the identical calculus without negation as one of the main results of his Vorlesungen über die Algebra der Logik (vol. I, 1890). As a conclusion from this proof, he distinguished between a "really logical calculus" (of groups, algorithms etc.) without complete distributivity, and the identical calculus which had to contain a special postulate to provide the problematic second subsumption. When Schröder studied Peirce's algebra of relatives in the beginning 1890s, the focal point of his research returned to his early program of an absolute algebra. The logic of relatives with its relative operations following the laws of the absolute algebra seemed to provide the language for applying the intended general theory of operations to all fields of mathematics and, beyond this, to all fields of knowledge containing formal structures. In this modified conception Schröder regarded arithmetic as part of a "general logic".
History and Philosophy of Logic, 2009
Mathematische Semesterberichte
Wolfgang Tschirk hat zu seinen 2021 erschienenen Geistesgeschichten der Mathematik ("Vom Spiegel ... more Wolfgang Tschirk hat zu seinen 2021 erschienenen Geistesgeschichten der Mathematik ("Vom Spiegel des Universums") und der Physik ("Vom Universum") eine weitere gesellt, eine Geistesgeschichte der Logik, die uns durch das "Universum des Denkens" leiten will. Seit Anbeginn der Philosophie gehört die Logik in deren Kernbestand, und zwar als Denklehre neben der Ethik als Lehre vom guten Handeln und der (Meta-)Physik, die sich mit der nicht-menschlichen Welt befasst. Im Verbund mit den sieben freien Künsten konstituierte sich daraus die allumfassende Weltweisheit als Summe aller Wissenschaften. Die Auffassung der Logik als Denklehre hielt sich bis ins ausgehende 19. Jahrhundert, allerdings spezialisiert als normative Theorie des Denkens, die lehrt, wie gedacht werden soll, um etwa aus gegebenen Prämissen einen korrekten Schluss ziehen zu können. Sie wurde dadurch von der Psychologie unterschieden, die das Denken, wie es sich in der Denkpraxis manifestiert, empirisch untersucht. Heute wird man allerdings mit logischen Problemen noch am ehesten in den Grundlagendisziplinen der Mathematik und in der theoretischen Informatik konfrontiert. Obwohl sich Wolfgang Tschirks Büchlein als logikhistorische Skizze präsentiert, ist es auch systematisch orientiert und nach für die Logik zentralen Grundbegriffen gegliedert. Wer nun eine Einteilung nach Art traditioneller Logiklehrbücher erwar
The Paideia Archive: Twentieth World Congress of Philosophy, 1998
This lecture will deal with the heuristic power of the deductive method and its contributions to ... more This lecture will deal with the heuristic power of the deductive method and its contributions to the scientific task of finding new knowledge. I will argue for a new reading of the term 'deductive method.' It will be presented as an architectural scheme for the reconstruction of the processes of gaining reliable scientific knowledge. This scheme combines the activities of doing science ('context of discovery') with the activities of presenting scientific results ('context of justification'). It combines the heuristic and the deductive side of science. The heuristic side is represented, e.g., by the creative methods to find the 'best' hypotheses (abduction), to design experimental systems for empirical research in order to formulate general laws (induction), or to create axiomatic systems. The other side consists of the production of deductive knowledge. This combination leads to a clear hierarchy: the heuristic side provides the basic presuppositions ...
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Papers by Volker Peckhaus