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Switch to: References
Citations of:
Vorlesungen über die algebra der logik
Leipzig: B. G. Teubner. Edited by Jakob Lüroth & Karl Eugen Müller (1890)
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David Hilbert’s early foundational views, especially those corresponding to the 1890s, are analysed here. I consider strong evidence for the fact that Hilbert was a logicist at that time, following upon Dedekind’s footsteps in his understanding of pure mathematics. This insight makes it possible to throw new light on the evolution of Hilbert’s foundational ideas, including his early contributions to the foundations of geometry and the real number system. The context of Dedekind-style logicism makes it possible to offer a new (...) |
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We study algebras whose elements are relations, and the operations are natural “manipulations” of relations. This area goes back to 140 years ago to works of De Morgan, Peirce, Schröder . Well known examples of algebras of relations are the varieties RCAn of cylindric algebras of n-ary relations, RPEAn of polyadic equality algebras of n-ary relations, and RRA of binary relations with composition. We prove that any axiomatization, say E, of RCAn has to be very complex in the following sense: (...) |
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We offer an interpretation of the words and works of Richard Dedekind and the David Hilbert of around 1900 on which they are held to entertain diverging views on the structure of a deductive science. Firstly, it is argued that Dedekind sees the beginnings of a science in concepts, whereas Hilbert sees such beginnings in axioms. Secondly, it is argued that for Dedekind, the primitive terms of a science are substantive terms whose sense is to be conveyed by elucidation, whereas (...) |
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Set theory is an autonomous and sophisticated field of mathematics, enormously successful not only at its continuing development of its historical heritage but also at analyzing mathematical propositions cast in set-theoretic terms and gauging their consistency strength. But set theory is also distinguished by having begun intertwined with pronounced metaphysical attitudes, and these have even been regarded as crucial by some of its great developers. This has encouraged the exaggeration of crises in foundations and of metaphysical doctrines in general. However, (...) |
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In my dissertation, I present Hermann Cohen's foundation for the history and philosophy of science. My investigation begins with Cohen's formulation of a neo-Kantian epistemology. I analyze Cohen's early work, especially his contributions to 19th century debates about the theory of knowledge. I conclude by examining Cohen's mature theory of science in two works, The Principle of the Infinitesimal Method and its History of 1883, and Cohen's extensive 1914 Introduction to Friedrich Lange's History of Materialism. In the former, Cohen gives (...) |
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In this paper I suggest an answer to the question of what Frege means when he says that his logical system, the Begriffsschrift, is like the language Leibniz sketched, a lingua characteristica, and not merely a logical calculus. According to the nineteenth century studies, Leibniz’s lingua characteristica was supposed to be a language with which the truths of science and the constitution of its concepts could be accurately expressed. I argue that this is exactly what the Begriffsschrift is: it is (...) |
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Three different ways in which systems of axioms can contribute to the discovery of new notions are presented and they are illustrated by the various ways in which lattices have been introduced in mathematics by Schröder et al. These historical episodes reveal that the axiomatic method is not only a way of systematizing our knowledge, but that it can also be used as a fruitful tool for discovering and introducing new mathematical notions. Looked at it from this perspective, the creative (...) |
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In the nineteenth century, the separation of naturalist or psychological accounts of validity from normative validity came into question. In his 1877 Logical Studies (Logische Studien), Friedrich Albert Lange argues that the basis for necessary inference is demonstration, which takes place by spatially delimiting the extension of concepts using imagined or physical diagrams. These diagrams are signs or indications of concepts' extension, but do not represent their content. Only the inference as a whole captures the objective content of the proof. (...) |
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Computational complexity is a discipline of computer science and mathematics which classifies computational problems depending on their inherent difficulty, i.e. categorizes algorithms according to their performance, and relates these classes to each other. P problems are a class of computational problems that can be solved in polynomial time using a deterministic Turing machine while solutions to NP problems can be verified in polynomial time, but we still do not know whether they can be solved in polynomial time as well. A (...) |
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Contact relations have been studied in the context of qualitative geometry and physics since the early 1920s, and have recently received attention in qualitative spatial reasoning. In this paper, we present a sound and complete proof system in the style of Rasiowa and Sikorski (1963) for relation algebras generated by a contact relation. |
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Henry M. Sheffer is well known to logicians for the discovery (or rather, the rediscovery) of the ?Sheffer stroke? of propositional logic. But what else did Sheffer contribute to logic? He published very little, though he is known to have been carrying on a rather mysterious research program in logic; the only substantial result of this research was the unpublished monograph The General Theory of Notational Relativity. The main aim of this paper is to explain, as far as possible (given (...) |
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In this essay, I discuss some observations by Peirce which suggest he had some idea of the substantive metalogical differences between logics which permit both quantifiers and relations, and those which do not. Peirce thus seems to have had arguments?which even De Morgan and Frege lacked?that show the superior expressiveness of relational logics. |
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This study seeks to trace the boundaries of the sign in the phenomenological tradition of Edmund Husserl. The approach adopted here is largely historical and has no other ambition that to identify those questions that pertain to the sign and have been of interest for phenomenology. The article is divided in four parts : the first examines an essay from 1890 entitled Semiotik and situates it in the context of the young Husserl's work in the philosophy of mathematics ; the (...) |
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In the Logical Investigations Husserl announced a critique of Lotze’s epistemology, but it was never included in the printed text. The aim of my paper is to investigate the remnant of Husserl’s planned text with special emphasis on the question of whether it goes beyond the obvious aspects of Husserl’s indebtedness to Lotze. Using Husserl’s student notes, excerpts, and book annotations, I refine the dating of Husserl’s encounter with Lotze and separate the various layers of influence. I argue that Husserl’s (...) |
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In this paper, I aim to present the main components of my non-standard interpretation of Frege’s views on existence to the English-speaking public. First, I will outline the standard interpretation and show how to a great but not full extent the standard interpretation can be justified on the basis of Frege’s writings. Second, I show that the main error of the standard interpretation consists in the assimilation of the contents of the ordinary language expressions “exist” and “there is” according to (...) |
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In what follows, the difference between Frege’s and Schröder’s understanding of logical connectives will be investigated. It will be argued that Frege thought of logical connectives as concepts, whereas Schröder thought of them as operations. For Frege, logical connectives can themselves be connected. There is no substantial difference between the connectives and the concepts they connect. Frege’s distinction between concepts and objects is central to this conception, because it allows a method of concept formation which enables us to form concepts (...) |
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Recent historical studies have investigated the first proponents of methodological structuralism in late nineteenth-century mathematics. In this paper, I shall attempt to answer the question of whether Peano can be counted amongst the early structuralists. I shall focus on Peano’s understanding of the primitive notions and axioms of geometry and arithmetic. First, I shall argue that the undefinability of the primitive notions of geometry and arithmetic led Peano to the study of the relational features of the systems of objects that (...) |
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On 5 May 1957, Leopold Löwenheim passed away in a Berlin hospital following a short but severe illness, unnoticed by the community of mathematical logicians who believed that he had perished in a Nazi concentration camp in or shortly after 1940 (the year of publication in the Journal of Symbolic Logic of his last paper before the end of World War II). The 50th anniversary of his death seems an appropriate date for the posthumous publication of a paper that was (...) |
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A common aim of elimination problems for languages of logic is to express the entire content of a set of formulas of the language, or a certain part of it, in a way that is more elementary or more informative. We want to bring out that as the languages for logic grew in expressive power and, at the same time, our knowledge of their expressive limitations also grew, elimination problems in logic underwent some change. For languages other than that for (...) |
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Niklas Luhmann verwendet in seiner soziologischen Systemtheorie offenbar etwas, das er den Büchern des englischen Mathematikers George Spencer Brown entnimmt. Dessen Formenkalkül ist für Luhmann, wie Günther Schulte treffend bemerkt, “Mädchen für alles, mit dem er nicht nur in der Lage ist Teezukochen, sondern auch Auto oder Straßenbahn zu fahren”. Der erste Blick in Spencer Browns Laws of Form vermittelt einen anderen Eindruck: nichts scheinen sie mit soziologischer Systemtheorie zu tun zu haben. Der vorliegende Text bearbeitet hieran anknüpfend eine recht (...) |
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In this paper I discuss Ernst Zermelo’s ideas concerning the possibility of developing a system of infinitary logic that, in his opinion, should be suitable for mathematical inferences. The presentation of Zermelo’s ideas is accompanied with some remarks concerning the development of infinitary logic. I also stress the fact that the second axiomatization of set theory provided by Zermelo in 1930 involved the use of extremal axioms of a very specific sort.1. |
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This paper's goal is very simple, that of trying to look at the broad and multifaceted concept of uncertainty from a typically scientific point of view, namely, that consisting in representing concepts through numerical quantities. For the goal, it is supposed that the universe of discourse is amorphe, and that the basic treats of the meaning shown by the mother-predicate ‘uncertain’, from which it comes the concept of uncertainty, can be empirically captured in, at least, for what concerns its variability (...) |
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Para Frege las relaciones lógicas se dan entre contenidos judicables, entre pensamientos. Aquellas relaciones son inferenciales. Los pensamientos se definen a través de sus relaciones inferenciales con otros. De acuerdo con esto es discutible afirmar, como lo hizo Schröder, que el proyecto lógico de Frege es como el proyecto lógico de Boole. También es cuestionable afirmar, como lo hizo Dummett, que la relación inferencial no es siempre central en el proyecto fregeano. En este texto defenderé una lectura del proyecto lógico (...) |
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'Reason, Causation and Compatibility with the Phenomena' strives to give answers to the philosophical problem of the interplay between realism, explanation and experience. This book is a compilation of essays that recollect significant conceptions of rival terms such as determinism and freedom, reason and appearance, power and knowledge. This title discusses the progress made in epistemology and natural philosophy, especially the steps that led from the ancient theory of atomism to the modern quantum theory, and from mathematization to analytic philosophy. (...) |
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This paper contains some reflections going from pinpointing one of the main goals of fuzzy logic, that of representing linguistic terms whose use is gradable in a given context, to show a mathematical model able to capture what is a fuzzy set labeled P in a universe of discourse, throughout a way in which some comments on what a mathematical model is, and to what it serves, are done. It is shown that such a model for a fuzzy set is (...) |
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