Philosophy of Mathematics and Its Logic: Introduction
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Russell's 1903 – 1905 Anticipation of the Lambda Calculus
HISTORY AND PHILOSOPHY OF LOGIC, 24 (2003), 15–37 Russell’s 1903 – 1905 Anticipation of the Lambda Calculus KEVIN C. KLEMENT Philosophy Dept, Univ. of Massachusetts, 352 Bartlett Hall, 130 Hicks Way, Amherst, MA 01003, USA Received 22 July 2002 It is well known that the circumflex notation used by Russell and Whitehead to form complex function names in Principia Mathematica played a role in inspiring Alonzo Church’s ‘Lambda Calculus’ for functional logic developed in the 1920s and 1930s. Interestingly, earlier unpublished manuscripts written by Russell between 1903 and 1905—surely unknown to Church—contain a more extensive anticipation of the essential details of the Lambda Calculus. Russell also anticipated Scho¨ nfinkel’s Combinatory Logic approach of treating multi- argument functions as functions having other functions as value. Russell’s work in this regard seems to have been largely inspired by Frege’s theory of functions and ‘value-ranges’. This system was discarded by Russell due to his abandonment of propositional functions as genuine entities as part of a new tack for solving Rus- sell’s paradox. In this article, I explore the genesis and demise of Russell’s early anticipation of the Lambda Calculus. 1. Introduction The Lambda Calculus, as we know it today, was initially developed by Alonzo Church in the late 1920s and 1930s (see, e.g. Church 1932, 1940, 1941, Church and Rosser 1936), although it built heavily on work done by others, perhaps most notably, the early works on Combinatory Logic by Moses Scho¨ nfinkel (see, e.g. his 1924) and Haskell Curry (see, e.g. -
Pluralisms About Truth and Logic Nathan Kellen University of Connecticut - Storrs, [email protected]
University of Connecticut OpenCommons@UConn Doctoral Dissertations University of Connecticut Graduate School 8-9-2019 Pluralisms about Truth and Logic Nathan Kellen University of Connecticut - Storrs, [email protected] Follow this and additional works at: https://opencommons.uconn.edu/dissertations Recommended Citation Kellen, Nathan, "Pluralisms about Truth and Logic" (2019). Doctoral Dissertations. 2263. https://opencommons.uconn.edu/dissertations/2263 Pluralisms about Truth and Logic Nathan Kellen, PhD University of Connecticut, 2019 Abstract: In this dissertation I analyze two theories, truth pluralism and logical pluralism, as well as the theoretical connections between them, including whether they can be combined into a single, coherent framework. I begin by arguing that truth pluralism is a combination of realist and anti-realist intuitions, and that we should recognize these motivations when categorizing and formulating truth pluralist views. I then introduce logical functionalism, which analyzes logical consequence as a functional concept. I show how one can both build theories from the ground up and analyze existing views within the functionalist framework. One upshot of logical functionalism is a unified account of logical monism, pluralism and nihilism. I conclude with two negative arguments. First, I argue that the most prominent form of logical pluralism faces a serious dilemma: it either must give up on one of the core principles of logical consequence, and thus fail to be a theory of logic at all, or it must give up on pluralism itself. I call this \The Normative Problem for Logical Pluralism", and argue that it is unsolvable for the most prominent form of logical pluralism. Second, I examine an argument given by multiple truth pluralists that purports to show that truth pluralists must also be logical pluralists. -
The Analytic-Synthetic Distinction and the Classical Model of Science: Kant, Bolzano and Frege
Synthese (2010) 174:237–261 DOI 10.1007/s11229-008-9420-9 The analytic-synthetic distinction and the classical model of science: Kant, Bolzano and Frege Willem R. de Jong Received: 10 April 2007 / Revised: 24 July 2007 / Accepted: 1 April 2008 / Published online: 8 November 2008 © The Author(s) 2008. This article is published with open access at Springerlink.com Abstract This paper concentrates on some aspects of the history of the analytic- synthetic distinction from Kant to Bolzano and Frege. This history evinces con- siderable continuity but also some important discontinuities. The analytic-synthetic distinction has to be seen in the first place in relation to a science, i.e. an ordered system of cognition. Looking especially to the place and role of logic it will be argued that Kant, Bolzano and Frege each developed the analytic-synthetic distinction within the same conception of scientific rationality, that is, within the Classical Model of Science: scientific knowledge as cognitio ex principiis. But as we will see, the way the distinction between analytic and synthetic judgments or propositions functions within this model turns out to differ considerably between them. Keywords Analytic-synthetic · Science · Logic · Kant · Bolzano · Frege 1 Introduction As is well known, the critical Kant is the first to apply the analytic-synthetic distinction to such things as judgments, sentences or propositions. For Kant this distinction is not only important in his repudiation of traditional, so-called dogmatic, metaphysics, but it is also crucial in his inquiry into (the possibility of) metaphysics as a rational science. Namely, this distinction should be “indispensable with regard to the critique of human understanding, and therefore deserves to be classical in it” (Kant 1783, p. -
Haskell Before Haskell. an Alternative Lesson in Practical Logics of the ENIAC.∗
Haskell before Haskell. An alternative lesson in practical logics of the ENIAC.∗ Liesbeth De Mol† Martin Carl´e‡ Maarten Bullynck§ January 21, 2011 Abstract This article expands on Curry’s work on how to implement the prob- lem of inverse interpolation on the ENIAC (1946) and his subsequent work on developing a theory of program composition (1948-1950). It is shown that Curry’s hands-on experience with the ENIAC on the one side and his acquaintance with systems of formal logic on the other, were conduc- tive to conceive a compact “notation for program construction” which in turn would be instrumental to a mechanical synthesis of programs. Since Curry’s systematic programming technique pronounces a critique of the Goldstine-von Neumann style of coding, his “calculus of program com- position”not only anticipates automatic programming but also proposes explicit hardware optimisations largely unperceived by computer history until Backus famous ACM Turing Award lecture (1977). This article frames Haskell B. Curry’s development of a general approach to programming. Curry’s work grew out of his experience with the ENIAC in 1946, took shape by his background as a logician and finally materialised into two lengthy reports (1948 and 1949) that describe what Curry called the ‘composition of programs’. Following up to the concise exposition of Curry’s approach that we gave in [28], we now elaborate on technical details, such as diagrams, tables and compiler-like procedures described in Curry’s reports. We also give a more in-depth discussion of the transition from Curry’s concrete, ‘hands-on’ experience with the ENIAC to a general theory of combining pro- grams in contrast to the Goldstine-von Neumann method of tackling the “coding and planning of problems” [18] for computers with the help of ‘flow-charts’. -
Bernard Bolzano. Theory of Science
428 • Philosophia Mathematica Menzel, Christopher [1991] ‘The true modal logic’, Journal of Philosophical Logic 20, 331–374. ———[1993]: ‘Singular propositions and modal logic’, Philosophical Topics 21, 113–148. ———[2008]: ‘Actualism’, in Edward N. Zalta, ed., Stanford Encyclopedia of Philosophy (Winter 2008 Edition). Stanford University. http://plato.stanford.edu/archives/win2008/entries/actualism, last accessed June 2015. Nelson, Michael [2009]: ‘The contingency of existence’, in L.M. Jorgensen and S. Newlands, eds, Metaphysics and the Good: Themes from the Philosophy of Robert Adams, Chapter 3, pp. 95–155. Oxford University Press. Downloaded from https://academic.oup.com/philmat/article/23/3/428/1449457 by guest on 30 September 2021 Parsons, Charles [1983]: Mathematics in Philosophy: Selected Essays. New York: Cornell University Press. Plantinga, Alvin [1979]: ‘Actualism and possible worlds’, in Michael Loux, ed., The Possible and the Actual, pp. 253–273. Ithaca: Cornell University Press. ———[1983]: ‘On existentialism’, Philosophical Studies 44, 1–20. Prior, Arthur N. [1956]: ‘Modality and quantification in S5’, The Journal of Symbolic Logic 21, 60–62. ———[1957]: Time and Modality. Oxford: Clarendon Press. ———[1968]: Papers on Time and Tense. Oxford University Press. Quine, W.V. [1951]: ‘Two dogmas of empiricism’, Philosophical Review 60, 20–43. ———[1969]: Ontological Relativity and Other Essays. New York: Columbia University Press. ———[1986]: Philosophy of Logic. 2nd ed. Cambridge, Mass.: Harvard University Press. Shapiro, Stewart [1991]: Foundations without Foundationalism: A Case for Second-order Logic.Oxford University Press. Stalnaker, Robert [2012]: Mere Possibilities: Metaphysical Foundations of Modal Semantics. Princeton, N.J.: Princeton University Press. Turner, Jason [2005]: ‘Strong and weak possibility’, Philosophical Studies 125, 191–217. -
Bolzano and the Traditions of Analysis
Bolzano and the Traditions of Analysis Paul Rusnock (Appeared in Grazer Phil. Studien 53 (1997) 61-86.) §1 Russell’s discussion of analytic philosophy in his popular History begins on a sur- prising note: the first analytic philosopher he mentions is . Weierstrass. His fur- ther remarks—in which he discusses Cantor and Frege, singling out their work in the foundations of mathematics—indicate that he thought that the origin of mod- ern philosophical analysis lay in the elaboration of modern mathematical analysis in the nineteenth century [13, 829-30]. Given the markedly different meanings attached to the word “analysis” in these two contexts, this juxtaposition might be dismissed as merely an odd coincidence. As it turns out, however, modern philo- sophical and mathematical analysis are rather closely linked. They have, for one thing, a common root, albeit one long since buried and forgotten. More important still, and apparently unknown to Russell, is the circumstance that one individual was instrumental in the creation of both: Bolzano. Russell’s account could easily leave one with the impression that analytic phi- losophy had no deep roots in philosophical tradition; that, instead, it emerged when methods and principles used more or less tacitly in mathematics were, af- ter long use, finally articulated and brought to the attention of the philosophical public. A most misleading impression this would be. For right at the begin- ning of the reconstruction of the calculus which Russell attributed to Weierstrass we find Bolzano setting out with great clarity the methodology guiding these de- velopments in mathematics—a methodology which, far from being rootless, was developed in close conjunction with Bolzano’s usual critical survey of the relevant philosophical literature. -
Strategic Maneuvering in Mathematical Proofs
CORE Metadata, citation and similar papers at core.ac.uk Provided by Springer - Publisher Connector Argumentation (2008) 22:453–468 DOI 10.1007/s10503-008-9098-7 Strategic Maneuvering in Mathematical Proofs Erik C. W. Krabbe Published online: 6 May 2008 Ó The Author(s) 2008 Abstract This paper explores applications of concepts from argumentation theory to mathematical proofs. Note is taken of the various contexts in which proofs occur and of the various objectives they may serve. Examples of strategic maneuvering are discussed when surveying, in proofs, the four stages of argumentation distin- guished by pragma-dialectics. Derailments of strategies (fallacies) are seen to encompass more than logical fallacies and to occur both in alleged proofs that are completely out of bounds and in alleged proofs that are at least mathematical arguments. These considerations lead to a dialectical and rhetorical view of proofs. Keywords Argumentation Á Blunder Á Fallacy Á False proof Á Mathematics Á Meno Á Pragma-dialectics Á Proof Á Saccheri Á Strategic maneuvering Á Tarski 1 Introduction The purport of this paper1 is to show that concepts from argumentation theory can be fruitfully applied to contexts of mathematical proof. As a source for concepts to be tested I turn to pragma-dialectics: both to the Standard Theory and to the Integrated Theory. The Standard Theory has been around for a long time and achieved its final formulation in 2004 (Van Eemeren and Grootendorst 1984, 1992, 2004). According to the Standard Theory argumentation is a communication process aiming at 1 The paper was first presented at the NWO-conference on ‘‘Strategic Manoeuvring in Institutionalised Contexts’’, University of Amsterdam, 26 October 2007. -
Reason, Causation and Compatibility with the Phenomena
Reason, causation and compatibility with the phenomena Basil Evangelidis Series in Philosophy Copyright © 2020 Vernon Press, an imprint of Vernon Art and Science Inc, on behalf of the author. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior permission of Vernon Art and Science Inc. www.vernonpress.com In the Americas: In the rest of the world: Vernon Press Vernon Press 1000 N West Street, C/Sancti Espiritu 17, Suite 1200, Wilmington, Malaga, 29006 Delaware 19801 Spain United States Series in Philosophy Library of Congress Control Number: 2019942259 ISBN: 978-1-62273-755-0 Cover design by Vernon Press. Cover image by Garik Barseghyan from Pixabay. Product and company names mentioned in this work are the trademarks of their respective owners. While every care has been taken in preparing this work, neither the authors nor Vernon Art and Science Inc. may be held responsible for any loss or damage caused or alleged to be caused directly or indirectly by the information contained in it. Every effort has been made to trace all copyright holders, but if any have been inadvertently overlooked the publisher will be pleased to include any necessary credits in any subsequent reprint or edition. Table of contents Abbreviations vii Preface ix Introduction xi Chapter 1 Causation, determinism and the universe 1 1. Natural principles and the rise of free-will 1 1.1. “The most exact of the sciences” 2 1.2. -
CURRICULUM VITAE MICHAEL DAVID RESNIK Born: March 20, 1938, New Haven, Connecticut Education
CURRICULUM VITAE MICHAEL DAVID RESNIK Born: March 20, 1938, New Haven, Connecticut Education: Yale University, New Haven, Connecticut, B.A. as a Scholar of the House in Mathematics and Philosophy, 1960 Harvard University, Cambridge, Massachusetts, M.A. in Philosophy, 1962; Ph.D. in Philosophy, 1964 Academic Experience: University Distinguished Professor, The University of North Carolina at Chapel Hill, 1988- Professor, The University of North Carolina at Chapel Hill, 1975-1988 Member, Social Science Research Institute, The University of North Carolina at Chapel Hill, 1985-present; Fellow of the UNC Institute of Arts and Humanities, 1988- Visiting Fellow, Center for The Study of Science in Society, Virginia Polytechnic Institute and State University, Blacksburg, Virginia, Spring, 1984 Chairman, The University of North Carolina at Chapel Hill, 1975-1983 Associate Professor, The University of North Carolina at Chapel Hill, 1967- 1975 Visiting Assistant Professor, Washington University, St. Louis, Missouri, 1966-1967 Assistant Professor, The University of Hawaii, Honolulu, 1964-1967; Member, Social Science Research Institute, 1966-1967 Memberships: American Philosophical Association Association for Symbolic Logic Philosophy of Science Association Grants: Co-principal investigator with C. Y. Cheng on N.S.F. Grant G.S. 835,"Classical Chinese Logic and Scientific Methodology", 1965-1966 Director, National Endowment for the Humanities Summer Seminar for College Teachers, "Frege and the Philosophy of Mathematics", 1980 Participant in a Sloan Foundation -
Hunting the Story of Moses Schönfinkel
Where Did Combinators Come From? Hunting the Story of Moses Schönfinkel Stephen Wolfram* Combinators were a key idea in the development of mathematical logic and the emergence of the concept of universal computation. They were introduced on December 7, 1920, by Moses Schönfinkel. This is an exploration of the personal story and intellectual context of Moses Schönfinkel, including extensive new research based on primary sources. December 7, 1920 On Tuesday, December 7, 1920, the Göttingen Mathematics Society held its regular weekly meeting—at which a 32-year-old local mathematician named Moses Schönfinkel with no known previous mathematical publications gave a talk entitled “Elemente der Logik” (“Elements of Logic”). This piece is included in S. Wolfram (2021), Combinators: A Centennial View, Wolfram Media. (wolframmedia.com/products/combinators-a-centennial-view.html) and accompanies arXiv:2103.12811 and arXiv:2102.09658. Originally published December 7, 2020 *Email: [email protected] 2 | Stephen Wolfram A hundred years later what was presented in that talk still seems in many ways alien and futuristic—and for most people almost irreducibly abstract. But we now realize that that talk gave the first complete formalism for what is probably the single most important idea of this past century: the idea of universal computation. Sixteen years later would come Turing machines (and lambda calculus). But in 1920 Moses Schönfinkel presented what he called “building blocks of logic”—or what we now call “combinators”—and then proceeded to show that by appropriately combining them one could effectively define any function, or, in modern terms, that they could be used to do universal computation. -
CURRY, Haskell Brooks (1900–1982) Haskell Brooks Curry Was Born On
CURRY, Haskell Brooks (1900–1982) Haskell Brooks Curry was born on 12 September 1900 at Millis, Massachusetts and died on 1 September 1982 at State College, Pennsylvania. His parents were Samuel Silas Curry and Anna Baright Curry, founders of the School for Expression in Boston, now known as Curry College. He graduated from high school in 1916 and entered Harvard College with the intention of going into medicine. When the USA entered World War I in 1917, he changed his major to mathematics because he thought it might be useful for military purposes in the artillery. He also joined the Student Army Training Corps on 18 October 1918, but the war ended before he saw action. He received his B.A. in mathematics from Harvard University in 1920. From 1920 until 1922, he studied electrical engineering at MIT in a program that involved working half-time at the General Electric Company. From 1922 until 1924, he studied physics at Harvard; during the first of those two years he was a half-time research assistant to P. W. Bridgman, and in 1924 he received an A. M. in physics. He then studied mathematics, still at Harvard, until 1927; during the first semester of 1926–27 he was a half-time instructor. During 1927–28 he was an instructor at Princeton University. During 1928–29, he studied at the University of Göttingen, where he wrote his doctoral dissertation; his oral defense under David Hilbert was on 24 July 1929, although the published version carries the date of 1930. In the fall of 1929, Haskell Curry joined the faculty of The Pennsylvania State College (now The Pennsylvania State University), where he spent most of the rest of his career. -
An Introduction to Bolzano's Essay on Beauty Aesthetics
Zlom2_2015_Sestava 1 27.11.15 11:45 Stránka 203 AESTHETICS IN CENTRAL EUROPE ARTICLES AND ARCHIVE DOCUMENTS AN INTRODUCTION TO BOLZANO’S ESSAY ON BEAUTY PAISLEY LIVINGSTON A neglected gem in the history of aesthetics, Bolzano’s essay on beauty is best understood when read alongside his other writings and philosophical sources. 1 This introduction is designed to contribute to such a reading. In Part I, I identify and discuss three salient ways in which Bolzano’s account can be misunderstood. As a lack of familiarity with Bolzano’s background assumptions is one source of these misunderstandings, in Part II, I elucidate some of his ideas about the psychological processes involved in the contemplation and enjoyment of beauty. In Part III, I situate Bolzano’s discussion of beauty within the more general framework of his ideas about the nature of philosophy, the relation between philosophy and aesthetics, and the place of the concept of beauty within the latter. Part IV is devoted to Bolzano’s discussion of some of his philosophical antecedents, including Kant. In Part V, I raise some objections to Bolzano’s account and indicate how his advocates might respond to them. Many thanks to the editors of Estetika for inviting me to submit this introduction and for taking the initiative to commission and publish Adam Bresnahan’s translation of the first half of Bolzano’s essay. Special thanks are due to Tereza Hadravová and Jakub Stejskal for helpful comments on a draft of the introduction. The work for this essay was supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No.