DOCSLIB.ORG

Carnap, Tarski, and Quine's Year Together: Conversations on Logic

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Carnap, Tarski, and Quine's Year Together: Conversations on Logic Carnap, Tarski, and Quine's Year Together: Conversations on Logic, Mathematics, and Science Greg Frost-Arnold September 26, 2008 ii Contents 1 Background and Overview 1 1.1 Introduction: Setting the Historical Stage . 1 1.2 Outline of the Primary Project: A `Finitist-Nominalist' Total Language of Science . 3 1.3 Mathematics in a Finitist-Nominalist Language . 11 1.3.1 Number . 11 1.3.2 Interpreting numerals that are `too large' . 13 1.4 Pre-history of the 1941 Finitist-Nominalist Project . 15 1.4.1 The Poles: Chwistek, Kotarbinski, Lesniewski . 15 1.4.2 Russellian influences . 21 1.4.3 Logical Empiricists skeptical of infinity . 23 2 Justifications 27 2.1 First Justification for the FN Conditions: Verst¨andlichkeit . 28 2.1.1 Assertions meeting the FN criteria are verst¨andlich . 28 2.1.2 What does `understandable' [verst¨andlich] mean in the discussion notes? . 28 2.1.3 What does `understandable' or `intelligible' mean in Car- nap's publications? . 30 2.1.4 What does `understandable' or `intelligible' mean for Quine? 34 2.2 Second Rationale for the FN Conditions: Overcoming Metaphysics 37 2.3 Third Rationale: Inferential Safety, or Taking the Paradoxes Se- riously . 43 2.4 Fourth Rationale for the FN Conditions: Natural Science . 45 2.5 Current Justifications for Nominalist Projects . 47 2.5.1 The positive argument: from a causal theory of knowledge and/or reference . 47 2.5.2 The negative argument: rebutting the indispensibility ar- gument . 48 3 Responses 51 3.1 Why Does Carnap Participate in the FN Project, Given his Reser- vations? . 51 iii iv CONTENTS 3.2 Answering the Finitist-Nominalist's Arguments: Higher Mathe- matics is Meaningful . 53 3.2.1 An analogy between higher mathematics and theoretical physics . 54 3.2.2 Potential infinity to the rescue? . 58 3.3 Are There Any Infinities Compatible with Nominalism? (and a Detour through Finitist Syntax) . 60 3.4 Attacking the FN Conditions . 65 3.4.1 Arithmetic is distorted . 66 3.4.2 Problems with proofs . 70 3.5 An Objection to the FN Project not in the Notes . 71 4 Nominalism and Analyticity 73 4.1 Under a Finitist-Nominalist Regime, Arithmetic becomes Synthetic 74 4.2 Radicalization of Quine's Critique of Analyticity: A Historical Conjecture . 81 5 Analyticity in 1941 89 5.1 What is Analyticity, circa 1940? . 90 5.1.1 Carnap and Tarski on Analyticity . 91 5.1.2 Quine: analysis of language{including analyticity{should be syntactic and extensional . 94 5.2 Tarski's Objections to Analyticity . 98 5.2.1 Tarski's first objection: any sign can be converted into a logical sign . 100 5.2.2 Tarski's second objection: G¨odelsentences . 102 5.3 Quine's Disagreements with Carnap circa 1940 . 107 5.3.1 Analyticity is an empirical concept, not a logical one . 107 5.3.2 Modal and intensional languages are unacceptable . 111 5.3.3 A further step in Quine's radicalization . 113 6 Metaphysics and the Unity of Science 117 6.1 Unity of Science: Unity of Language, not of Laws . 118 6.2 Overcoming Metaphysics . 122 6.2.1 Aufbau .............................123 6.2.2 \Overcoming Metaphysics through Logical Analysis of Lan- guage" . 125 6.2.3 Logical Syntax of Language . 127 6.2.4 \Empiricism, Semantics, and Ontology" . 129 6.2.5 Neurath . 130 6.3 A Difficulty: What Cannot be Incorporated into a Language of Science? . 132 6.4 Conclusion: The Origin of the Term ‘Unified Science' . 135 CONTENTS v Preface Several years ago, I began combing through the Rudolf Carnap Collection at the University of Pittsburgh's Archives of Scientific Philosophy, searching for material about Alfred Tarski's theory of truth. I fortuitously came across a folder full of dictation notes Carnap had taken during conversations with Tarski and others, mostly from 1941. Although I did not fully recognize then what I had stumbled upon, those notes are the originating cause of this book. The first portion of this book consists of my attempts to come to grips with those documents, both in terms of their place in the history of analytic philosophy, and their often surprising philosophical content. As the subtitle of this book suggests, I have grouped this content into three interdependent sections: math- ematical nominalism (mathematics), analytic truth (logic), and the unity of science. This choice of what to focus on in these wide-ranging notes certainly reflects my own, sometimes idiosyncratic interests in and perspective on the his- tory of analytic philosophy; another, almost entirely different book could most likely be written about the same archival material. Because the documents would amply reward such study, an edited version of the archival material itself can be found in the Appendix, along with an English translation. I do not feel that I wrote this book. The final product is very much the result of many people|I just happen to be the person who put the most time into the group's project. My inability to see obvious errors that others spotted easily, and others' insightful and creative suggestions for new directions, truly made this a collaborative effort. John Earman and Nuel Belnap gave me very helpful guidance as this material coalesced slowly into a dissertation. An audience at HOPOS 2004 helped improve the core of what is now the first chapter; Alan Richardson and Chris Pincock, in particular, both provided fruitful suggestions. A more intimate audience at the 2004 PSA helped iron out some of the deficien- cies of the final chapter; Rick Creath provided especially useful feedback both then and later, as the larger project progressed. His Dear Carnap, Dear Van was not only an invaluable research tool for present purposes, but also served as an exemplar after which this book is patterned. Michael Friedman, Don Howard, and Thomas Ricketts each brought their substantial erudition to bear on various ideas presented here; their ability to see the historical `big picture' was a very helpful corrective to my myopia and ignorance. Andre Carus gave me very helpful suggestions about the historical big picture as well, and Chap- ter 1 in particular was thereby much improved. Marion Ledwig, Greg Lavers, and Jon Tsou each undertook the task of reading the entire manuscript when it was close to completion; their fresh eyes provided wonderful new perspective on ideas that had been bouncing around in my head for years. Jon, in partic- ular, engaged my arguments seriously, and (thankfully!) would not let me get away with anything that was too quick. I am certain I am leaving out several people who helped me recognize and rectify deficiencies in the ideas presented here, only because I've been overeager to discuss this material with anyone who would listen for the last five years or so. I apologize to you all. I have saved this book's three greatest intellectual debts for last. First vi CONTENTS is Paolo Mancosu, who found the records of these conversations at roughly the same time I did, and began working on them shortly thereafter. Writing about documents that no one has ever seen before is not something for which the usual historical training on Plato or Descartes really prepares you. I am extremely fortunate that a philosopher as able|and as intellectually generous| as Paolo began studying these works as well: his perspective kept me from being completely lost in the dark as I attempted to wade through Carnap's notes. More than he perhaps realizes, his published work and private comments helped iron out serious distortions and outright misunderstandings in my conception of what is going on in this material, and he helped me to recognize better what is central and what is peripheral. Steve Awodey gave me copious and insightful feedback every step of the way, especially when the project was just beginning: he read very, very rough drafts with more care and respect than they deserved at the time. His ability to comment constructively on ideas that were highly underdeveloped|instead of simply saying `think about this more,' the common and easy response|was a great boon. And Steve has shown excellent editorial touch as well, giving me all the freedom I could want. Plus, he prevented me from saying some very wrong-headed things about Russell's theory of types. Laura Ruetsche's help has been essential from the very beginning of this work. She possesses the enviable knack of seeing to the core of an issue whose surface I'll only scratch for 20 pages or so. Laura has been a true mentor, not merely a very dedicated and astute reader: she struck the perfect balance of encouragement for work I had already done, while showing me how much improvement could yet be done. I am extremely fortunate that she supervised my dissertation. My wife Karen has not only given me a wonderful life, but has been a fantastic philosophical rudder as well. Whenever I get lost in the philosophical trees, she shows me the forest again. More importantly, when I'm confronted with an apparently insuperable obstacle, she shows me that it can be overcome, and when I stumble, she helps me up. Chapter 1 Historical Background and Overview of Harvard Activities 1.1 Introduction: Setting the Historical Stage During the academic year 1940{1941, several giants of analytic philosophy| both established and budding|congregated at Harvard University. The list of philosophers is impressive. Bertrand Russell, who was only at Harvard during the Fall semester of 1940, originally emigrated from Britain in 1938. During 1940, he was embroiled in his infamous legal battles with the City College of New York.
Load more

Recommended publications
  • Some Strands of Wittgenstein's Normative Pragmatism
    Some Strands of Wittgenstein’s Normative Pragmatism, and Some Strains of his Semantic Nihilism ROBERT B. BRANDOM ABSTRACT WORK TYPE In this reflection I address one of the critical questions this monograph is Article about: How to justify proposing yet another semantic theory in the light of Wittgenstein’s strong warnings against it. I see two clear motives for ARTICLE HISTORY Wittgenstein’s semantic nihilism. The first one is the view that philosophical Received: problems arise from postulating hypothetical entities such as ‘meanings’. To 27–January–2018 dissolve the philosophical problems rather than create new ones, Wittgenstein Accepted: suggests substituting ‘meaning’ with ‘use’ and avoiding scientism in philosophy 3–March–2018 together with the urge to penetrate in one's investigation to unobservable depths. I believe this first motive constitutes only a weak motive for ARTICLE LANGUAGE Wittgenstein’s quietism, because there are substantial differences between English empirical theories in natural sciences and semantic theories in philosophy that KEYWORDS leave Wittgenstein’s assimilation of both open to criticism. But Wittgenstein is Meaning and Use right, on the second motive, that given the dynamic character of linguistic Hypothetical Entities practice, the classical project of semantic theory is a disease that can be Antiscientism removed or ameliorated only by heeding the advice to replace concern with Semantic Nihilism meaning by concern with use. On my view, this does not preclude, however, a Linguistic Dynamism different kind of theoretical approach to meaning that avoids the pitfalls of the Procrustean enterprise Wittgenstein complained about. © Studia Humanitatis – Universidad de Salamanca 2019 R. Brandom (✉) Disputatio. Philosophical Research Bulletin University of Pittsburgh, USA Vol.
    [Show full text]
  • Semantic Analysis of the First Cities from a Deconstruction Perspective Doi: 10.23968/2500-0055-2020-5-3-43-48
    Mojtaba Valibeigi, Faezeh Ashuri— Pages 43–48 SEMANTIC ANALYSIS OF THE FIRST CITIES FROM A DECONSTRUCTION PERSPECTIVE DOI: 10.23968/2500-0055-2020-5-3-43-48 SEMANTIC ANALYSIS OF THE FIRST CITIES FROM A DECONSTRUCTION PERSPECTIVE Mojtaba Valibeigi*, Faezeh Ashuri Buein Zahra Technical University Imam Khomeini Blvd, Buein Zahra, Qazvin, Iran *Corresponding author: [email protected] Abstract Introduction: Deconstruction is looking for any meaning, semantics and concepts and then shows how all of them seem to lead to chaos, and are always on the border of meaning duality. Purpose of the study: The study is aimed to investigate urban identity from a deconstruction perspective. Since two important bases of deconstruction are text and meaning and their relationships, we chose the first cities on a symbolic level as a text and tried to analyze their meanings. Methods: The study used a deductive content analysis in three steps including preparation, organization and final report or conclusion. In the first step, we argued deconstruction philosophy based on Derrida’s views accordingly. Then we determined some common semantic features of the first cities. Finally, we presented some conclusions based on a semantic interpretation of the first cities’ identity.Results: It seems that all cities as texts tend to provoke a special imaginary meaning, while simultaneously promoting and emphasizing the opposite meaning of what they want to show. Keywords Deconstruction, text, meaning, urban semantics, the first cities. Introduction to have a specific meaning (Gualberto and Kress, 2019; Expressions are an act of recognizing or displaying Leone, 2019; Stojiljković and Ristić Trajković, 2018).
    [Show full text]
  • Matrix Decompositions and Latent Semantic Indexing
    Online edition (c)2009 Cambridge UP DRAFT! © April 1, 2009 Cambridge University Press. Feedback welcome. 403 Matrix decompositions and latent 18 semantic indexing On page 123 we introduced the notion of a term-document matrix: an M N matrix C, each of whose rows represents a term and each of whose column× s represents a document in the collection. Even for a collection of modest size, the term-document matrix C is likely to have several tens of thousands of rows and columns. In Section 18.1.1 we first develop a class of operations from linear algebra, known as matrix decomposition. In Section 18.2 we use a special form of matrix decomposition to construct a low-rank approximation to the term-document matrix. In Section 18.3 we examine the application of such low-rank approximations to indexing and retrieving documents, a technique referred to as latent semantic indexing. While latent semantic in- dexing has not been established as a significant force in scoring and ranking for information retrieval, it remains an intriguing approach to clustering in a number of domains including for collections of text documents (Section 16.6, page 372). Understanding its full potential remains an area of active research. Readers who do not require a refresher on linear algebra may skip Sec- tion 18.1, although Example 18.1 is especially recommended as it highlights a property of eigenvalues that we exploit later in the chapter. 18.1 Linear algebra review We briefly review some necessary background in linear algebra. Let C be an M N matrix with real-valued entries; for a term-document matrix, all × RANK entries are in fact non-negative.
    [Show full text]
  • L. Maligranda REVIEW of the BOOK by MARIUSZ URBANEK
    Математичнi Студiї. Т.50, №1 Matematychni Studii. V.50, No.1 УДК 51 L. Maligranda REVIEW OF THE BOOK BY MARIUSZ URBANEK, “GENIALNI – LWOWSKA SZKOL A MATEMATYCZNA” (POLISH) [GENIUSES – THE LVOV SCHOOL OF MATHEMATICS] L. Maligranda. Review of the book by Mariusz Urbanek, “Genialni – Lwowska Szko la Matema- tyczna” (Polish) [Geniuses – the Lvov school of mathematics], Wydawnictwo Iskry, Warsaw 2014, 283 pp. ISBN: 978-83-244-0381-3 , Mat. Stud. 50 (2018), 105–112. This review is an extended version of my short review of Urbanek's book that was published in MathSciNet. Here it is written about his book in greater detail, which was not possible in the short review. I will present facts described in the book as well as some false information there. My short review of Urbanek’s book was published in MathSciNet [24]. Here I write about his book in greater detail. Mariusz Urbanek, writer and journalist, author of many books devoted to poets, politicians and other figures of public life, decided to delve also in the world of mathematicians. He has written a book on the phenomenon in the history of Polish science called the Lvov School of Mathematics. Let us add that at the same time there were also the Warsaw School of Mathematics and the Krakow School of Mathematics, and the three formed together the Polish School of Mathematics. The Lvov School of Mathematics was a group of mathematicians in the Polish city of Lvov (Lw´ow,in Polish; now the city is in Ukraine) in the period 1920–1945 under the leadership of Stefan Banach and Hugo Steinhaus, who worked together and often came to the Scottish Caf´e (Kawiarnia Szkocka) to discuss mathematical problems.
    [Show full text]
  • Conceptual Metaphors and Concepts of Multiple Realities: Points in Common
    KWARTALNIK NEOFILOLOGICZNY, LVIII, 3/2011 KRZYSZTOF KOSECKI (ŁÓDŹ) CONCEPTUAL METAPHORS AND CONCEPTS OF MULTIPLE REALITIES: POINTS IN COMMON The paper discusses the concept of reality and four attempts to account for it: William James’s conception of “sub-universes” and Lawrence LeShan’s idea of “alternate realities”, both developed in psychology; Leon Chwistek’s conception of multiple realities in philosophy and art; Alfred Schütz’s idea of “limited areas of sense (meaning)”, advanced in sociology. It sets them against the second generation cognitive linguistic view of conceptual metaphor, developed by George Lakoff and Mark Johnson. The cognitive linguists claim that metaphors shape people’s ontological orientations and so largely defi ne their everyday realities. Various points in common between the respective conceptions of realities and the cognitive linguistic view of metaphor are considered. The theory of conceptual metaphor is presented as another attempt to cope with the concept of reality. Like the other conceptions, it claims that reality is not homogenous; instead, metaphorical pluralism is the only way that allows one to deal effectively with the complexity of the surrounding world. INTRODUCTION: THE CONCEPT OF REALITY The concept of reality is a vague one, which is refl ected in its diverse de- scriptions. A popular view of it was expressed by the Spanish philosopher José Ortega y Gasset (1986: 167-168): A man tends to regard as real those things which he knows best and perceives with least effort. In each of us, the attention seems to be inclined to turn of its own accord to one defi - nite class of objects or another.
    [Show full text]
  • Linguistic Relativity Hyp
    THE LINGUISTIC RELATIVITY HYPOTHESIS by Michele Nathan A Thesis Submitted to the Faculty of the College of Social Science in Partial Fulfillment of the Requirements for the Degree of Master of Arts Florida Atlantic University Boca Raton, Florida December 1973 THE LINGUISTIC RELATIVITY HYPOTHESIS by Michele Nathan This thesis was prepared under the direction of the candidate's thesis advisor, Dr. John D. Early, Department of Anthropology, and has been approved by the members of his supervisory committee. It was submitted to the faculty of the College of Social Science and was accepted in partial fulfillment of the requirements for the degree of Master of Arts. SUPERVISORY COMMITTEE: &~ rl7 IC?13 (date) 1 ii ABSTRACT Author: Michele Nathan Title: The Linguistic Relativity Hypothesis Institution: Florida Atlantic University Degree: Master of Arts Year: 1973 Although interest in the linguistic relativity hypothesis seems to have waned in recent years, this thesis attempts to assess the available evidence supporting it in order to show that further investigation of the hypothesis might be most profitable. Special attention is paid to the fact that anthropology has largely failed to substantiate any claims that correlations between culture and the semantics of language do exist. This has been due to the impressionistic nature of the studies in this area. The use of statistics and hypothesis testing to provide mor.e rigorous methodology is discussed in the hope that employing such paradigms would enable anthropology to contribute some sound evidence regarding t~~ hypothesis. iii TABLE OF CONTENTS Page Introduction • 1 CHAPTER I THE.HISTORY OF THE FORMULATION OF THE HYPOTHESIS.
    [Show full text]
  • L. Maligranda REVIEW of the BOOK BY
    Математичнi Студiї. Т.50, №1 Matematychni Studii. V.50, No.1 УДК 51 L. Maligranda REVIEW OF THE BOOK BY MARIUSZ URBANEK, “GENIALNI – LWOWSKA SZKOL A MATEMATYCZNA” (POLISH) [GENIUSES – THE LVOV SCHOOL OF MATHEMATICS] L. Maligranda. Review of the book by Mariusz Urbanek, “Genialni – Lwowska Szko la Matema- tyczna” (Polish) [Geniuses – the Lvov school of mathematics], Wydawnictwo Iskry, Warsaw 2014, 283 pp. ISBN: 978-83-244-0381-3 , Mat. Stud. 50 (2018), 105–112. This review is an extended version of my short review of Urbanek's book that was published in MathSciNet. Here it is written about his book in greater detail, which was not possible in the short review. I will present facts described in the book as well as some false information there. My short review of Urbanek’s book was published in MathSciNet [24]. Here I write about his book in greater detail. Mariusz Urbanek, writer and journalist, author of many books devoted to poets, politicians and other figures of public life, decided to delve also in the world of mathematicians. He has written a book on the phenomenon in the history of Polish science called the Lvov School of Mathematics. Let us add that at the same time there were also the Warsaw School of Mathematics and the Krakow School of Mathematics, and the three formed together the Polish School of Mathematics. The Lvov School of Mathematics was a group of mathematicians in the Polish city of Lvov (Lw´ow,in Polish; now the city is in Ukraine) in the period 1920–1945 under the leadership of Stefan Banach and Hugo Steinhaus, who worked together and often came to the Scottish Caf´e (Kawiarnia Szkocka) to discuss mathematical problems.
    [Show full text]
  • Polish Mathematicians and Mathematics in World War I. Part I: Galicia (Austro-Hungarian Empire)
    Science in Poland Stanisław Domoradzki ORCID 0000-0002-6511-0812 Faculty of Mathematics and Natural Sciences, University of Rzeszów (Rzeszów, Poland) [email protected] Małgorzata Stawiska ORCID 0000-0001-5704-7270 Mathematical Reviews (Ann Arbor, USA) [email protected] Polish mathematicians and mathematics in World War I. Part I: Galicia (Austro-Hungarian Empire) Abstract In this article we present diverse experiences of Polish math- ematicians (in a broad sense) who during World War I fought for freedom of their homeland or conducted their research and teaching in difficult wartime circumstances. We discuss not only individual fates, but also organizational efforts of many kinds (teaching at the academic level outside traditional institutions, Polish scientific societies, publishing activities) in order to illus- trate the formation of modern Polish mathematical community. PUBLICATION e-ISSN 2543-702X INFO ISSN 2451-3202 DIAMOND OPEN ACCESS CITATION Domoradzki, Stanisław; Stawiska, Małgorzata 2018: Polish mathematicians and mathematics in World War I. Part I: Galicia (Austro-Hungarian Empire. Studia Historiae Scientiarum 17, pp. 23–49. Available online: https://doi.org/10.4467/2543702XSHS.18.003.9323. ARCHIVE RECEIVED: 2.02.2018 LICENSE POLICY ACCEPTED: 22.10.2018 Green SHERPA / PUBLISHED ONLINE: 12.12.2018 RoMEO Colour WWW http://www.ejournals.eu/sj/index.php/SHS/; http://pau.krakow.pl/Studia-Historiae-Scientiarum/ Stanisław Domoradzki, Małgorzata Stawiska Polish mathematicians and mathematics in World War I ... In Part I we focus on mathematicians affiliated with the ex- isting Polish institutions of higher education: Universities in Lwów in Kraków and the Polytechnical School in Lwów, within the Austro-Hungarian empire.
    [Show full text]
  • Quine and Whitehead: Ontology and Methodology 3
    McHENRY I QUINE AND WHITEHEAD: ONTOLOGY AND METHODOLOGY 3 philosophical scene as of late (e.g., deconstructionism, postmodem relativism), the and Whitehead: Ontology and gulf that separates Whitehead and analytical philosophy is not as great as it used to Quine 'be.3 Before I begin to explore the affinities, as well as some contrasts, between Methodology . Quine and Whitehead, a word of caution is in order. It is well known that Quine wrote his doctoral dissertation, "The Logic of Sequences: A Generalization of LeemonMcHenry Principia Mathematica," under Whitehead's direction.4 Quine also took two of Whitehead's seminars at Harvard,"Science and the Modem World" and "Cosmol­ I LEEMON McHENRY reaches philosophy at Loyola Marymount University, Los ogies Ancient and Modem," but he says that he "responded little to these courses" Angeles, CA 90045, E-mail: [email protected] and "took refuge in his relatively mathematical material on 'extensive abstrac­ tion. "'s Despite Quine's statement that he "retained a vivid sense of being in the presence of the great," he does not acknowledge any philosophical influence from · Introduction . the ph!losoph!es o!'W.V . Qu�e an d Whitehead. Rather, Camap and Russell are cited as the inspirations of Quine's The very idea· of a basis for comparing . , mcluding Qume himself. early development. A.N. Whitehead may be surprising to most philosophe� . philosophy two completely Both produced systems of thought that have taken m When Quine overlapped with Whitehead at Harvard, Whitehead was deeply at the forefron: of c�ntemp�­ involved in working out the details of his metaphysics of process--a philosophic different directions.
    [Show full text]
  • Philosophy of Language in the Twentieth Century Jason Stanley Rutgers University
    Philosophy of Language in the Twentieth Century Jason Stanley Rutgers University In the Twentieth Century, Logic and Philosophy of Language are two of the few areas of philosophy in which philosophers made indisputable progress. For example, even now many of the foremost living ethicists present their theories as somewhat more explicit versions of the ideas of Kant, Mill, or Aristotle. In contrast, it would be patently absurd for a contemporary philosopher of language or logician to think of herself as working in the shadow of any figure who died before the Twentieth Century began. Advances in these disciplines make even the most unaccomplished of its practitioners vastly more sophisticated than Kant. There were previous periods in which the problems of language and logic were studied extensively (e.g. the medieval period). But from the perspective of the progress made in the last 120 years, previous work is at most a source of interesting data or occasional insight. All systematic theorizing about content that meets contemporary standards of rigor has been done subsequently. The advances Philosophy of Language has made in the Twentieth Century are of course the result of the remarkable progress made in logic. Few other philosophical disciplines gained as much from the developments in logic as the Philosophy of Language. In the course of presenting the first formal system in the Begriffsscrift , Gottlob Frege developed a formal language. Subsequently, logicians provided rigorous semantics for formal languages, in order to define truth in a model, and thereby characterize logical consequence. Such rigor was required in order to enable logicians to carry out semantic proofs about formal systems in a formal system, thereby providing semantics with the same benefits as increased formalization had provided for other branches of mathematics.
    [Show full text]
  • Mathematics, Time, and Confirmation
    Mathematics, Time, and Confirmation Ulrich Meyer Doctor of Philosophy in Mathematics University of Cambridge, 1995 Submitted to the Department of Linguistics and Philosophy in partial fulfil- ment of the requirements for the degree of DOCTOR OF PHILOSOPHY at the Massachusetts Institute of Technology. September 2001 Author: ................................. .. .... Department of Linguistics aKPhilosophy August 10, 2001 Certified by: .......................... Robert C. Stalnaker Professor of Philosophy Thesis Supervisor Accepted by: .............. Vann McGee Professor of Philosophy Chair, Committee on Graduate Students @Ulrich Meyer 2001. All rights reserved. The author hereby grants permission to the Massachusetts Institute of Technology to reproduce and distribute publicly paper and electronic copies of this thesis in whole or in part. MASSACHUSETTS INSTITUTE OF TECHNOLOGY OCT 11 2001 ARCMly LIBRARIES Mathematics, Time, and Confirmation by Ulrich Meyer Submitted to The Department of Linguistics and Philosophy on 10 August 2001 in partial fulfilment of the requirements for the degree of Doctor of Philosophy ABSTRACT This dissertation discusses two issues about abstract objects: their role in scientific theories, and their relation to time. Chapter 1, "Why Apply Mathematics?" argues that scientific theories are not about the mathematics that is applied in them, and defends this thesis against the Quine-Putnam Indispensability Argument. Chapter 2, "Scientific Ontology," is a critical study of W. V. Quine's claim that metaphysics and mathematics are epistemologically on a par with nat- ural science. It is argued that Quine's view relies on a unacceptable account of empirical confirmation. Chapter 3, "Prior and the Platonist," demonstrates the incompatibility of two popular views about time: the "Platonist" thesis that some objects exist "outside" time, and A.
    [Show full text]
  • A History of Semantics Naming
    1/27 A history of semantics [in Nick Riemer (ed.) Routledge Handbook of Semantics] Naming [M]an “makes” his territory by naming the “things” in it. (Chatwin 1988: 301) Human beings name things in their environment. The name helps to distinguish and identify the denotatum (thing named) and is essential to communication with fellow humans about 1 such denotata. In Plato’s Cratylus (Plato 1997) c. 385 BCE, Socrates advances the hypothesis that the earliest name-giver (onomatourgos) selected a name that captures the essence of its denotatum, that is in some way iconic as with onomatopoeic bird names like cuckoo or whippoorwill. On this hypothesis the meaning of a word would be ‘natural’ because directly recognizable from the form of the word. Many of the Ancients sought to demonstrate that names are far more descriptive than the facts allow. For example Socrates in Cratylus 406c derives the name Dionusos (god of Bacchanalia) from didous ton oinon ‘giving wine’. In De lingua latina V: 101 (c. 45 BCE) Varro suggests that because the fox is fleet-footed, volpes ‘fox’ is a blend of volare ‘fly’ and pes ‘foot’ (Varro 1938). Isidore of Seville suggested c. 625 CE that oratio ‘utterance’ derives from oris ratio ‘the mouth’s reason’. None of these is correct and many such ‘etymologies’ are utterly absurd (see Allan 2010). Indeed, the implausibility of such accounts was recognized by Socrates in Cratylus (426b-427b, 434e- 435c), but a clear statement that names are symbols which denote by convention is first found some 25 years after Cratylus in Aristotle’s On Interpretation 16a3, 16a20 (Aristotle 1984).
    [Show full text]