Curriculum Vitæ JOHN PATTON BURGESS
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Naturalism and the a Priori Penelope Maddy
5 Naturalism and the A Priori PenelopeMaddy The naturalism I aim to practise is a descendantof Quine's. I My goal here is to place this naturalism in what I hope to be an illuminating historical context, to trace the status of the a priori through its various twists and turns, and eventu- ally to draw some tentative conclusions about the naturalistic status of the a priori. To do this, I first return to Kant. While it's surely no surprise that an examination of the a priori should start from Kant, perhapshis relevanceto natu- ralism is less obvious. Let me begin, then, with an introductory word on that connection. Though my naturalism differs from Quine's in a couple of significant ways, these disagreements won't matter until the very end. so we can begin with Quine's leading idea: the 'abandonmentof the goal of a first philosophy' (Quine 1975: 72). The interconnectionsbetween Quine and Carnap will take centre stage later, but for now we needonly note that the bare rejection of first philosophy can be seen as evolving out of Carnap's classification of many traditional metaphys- ical claims as 'pseudo-statement[s] without cognitive content' (Carnap 1950: 250). Carnap's idea is that legitimate scientific questions, 'theoretical questions', are asked within the linguistic framework of scientific language,with its associ- ated principles of evidence; in contrast, metaphysical pseudo-questionsare posed outside of all linguistic frameworks; perhaps as preamble to the adoption of a linguistic framework, as such, they are asked without the backing of associated evidential rules that would make them answerable,and indeed, that would give them sense. -
Forcopy-Editing Only
i i “Abstractionism_OUP_for_copyediting” — 2016/4/20 — 11:40 — page 1 — #10 i i 1 Introduction to Abstractionism Philip A. Ebert and Marcus Rossberg 1.1 WHAT IS ABSTRACTIONISM? Abstractionism in philosophy of mathematics has its origins in Gottlob Frege’s logicism—a position Frege developed in the late nineteenth and early twenti- eth century. Frege’s main aim wasfor to reduce copy-editing arithmetic and analysis to only logic in order to provide a secure foundation for mathematical knowledge. As is well known, Frege’s development of logicism failed. The infamous Basic Law V— one of the six basic laws of logic— Frege proposed in his magnum opus Grund- gesetze der Arithmetik—is subject to Russell’s Paradox. The striking feature of Frege’s Basic Law V is that it takes the form of an abstraction principle. The general form of an abstraction principle can by symbolised like this:1 ↵ = β ↵ β §( ) §( ) $ ⇠ where ‘ ’ is a term-forming operator applicable to expression of the type of ↵ § and β, and is an equivalence relation on entities denoted by expressions of ⇠ that type. Accordingly, abstraction principles are biconditionals that feature an equivalence relation on the right-hand side and an identity statement on the left-hand side. The abstracta denoted by the terms featuring in the identity statement on the left are taken to be introduced, in some sense, by the ab- straction principle, giving the equivalence on the right-hand side conceptual priority over them. More on this below. Frege’s ill-fated Basic Law V, involves co-extentionality (of functions) as the relevant equivalence relation on the right-hand side, introducing, what Frege – termed value-ranges, "' " , on the left:2 ( ) 1Here and below, we will omit prefixed universal quantifiers in abstraction principles. -
The Logic of Provability
The Logic of Provability Notes by R.J. Buehler Based on The Logic of Provability by George Boolos September 16, 2014 ii Contents Preface v 1 GL and Modal Logic1 1.1 Introduction..........................................1 1.2 Natural Deduction......................................2 1.2.1 ...........................................2 1.2.2 3 ...........................................2 1.3 Definitions and Terms....................................3 1.4 Normality...........................................4 1.5 Refining The System.....................................6 2 Peano Arithmetic 9 2.1 Introduction..........................................9 2.2 Basic Model Theory..................................... 10 2.3 The Theorems of PA..................................... 10 2.3.1 P-Terms........................................ 10 2.3.2Σ,Π, and∆ Formulas................................ 11 2.3.3 G¨odelNumbering................................... 12 2.3.4 Bew(x)........................................ 12 2.4 On the Choice of PA..................................... 13 3 The Box as Bew(x) 15 3.1 Realizations and Translations................................ 15 3.2 The Generalized Diagonal Lemma............................. 16 3.3 L¨ob'sTheorem........................................ 17 3.4 K4LR............................................. 19 3.5 The Box as Provability.................................... 20 3.6 GLS.............................................. 21 4 Model Theory for GL 23 5 Completeness & Decidability of GL 25 6 Canonical Models 27 7 On -
Yesterday's Algorithm: Penrose on the Gödel Argument
Yesterday’s Algorithm: Penrose and the Gödel Argument §1. The Gödel Argument. Roger Penrose is justly famous for his work in physics and mathematics but he is notorious for his endorsement of the Gödel argument (see his 1989, 1994, 1997). This argument, first advanced by J. R. Lucas (in 1961), attempts to show that Gödel’s (first) incompleteness theorem can be seen to reveal that the human mind transcends all algorithmic models of it1. Penrose's version of the argument has been seen to fall victim to the original objections raised against Lucas (see Boolos (1990) and for a particularly intemperate review, Putnam (1994)). Yet I believe that more can and should be said about the argument. Only a brief review is necessary here although I wish to present the argument in a somewhat peculiar form. Let us suppose that the human cognitive abilities that underlie our propensities to acquire mathematical beliefs on the basis of what we call proofs can be given a classical cognitive psychological or computational explanation (henceforth I will often use ‘belief’ as short for ‘belief on the basis of proof’). These propensities are of at least two types: the ability to appreciate chains of logical reasoning but also the ability to recognise ‘elementary proofs’ or, it would perhaps be better to say, elementary truths, such as if a = b and b = c then a = c, without any need or, often, any possibility of proof in the first sense. On this supposition, these propensities are the realization of a certain definite algorithm or program somehow implemented by the neural hardware of the brain. -
Believing the Axioms. I Penelope Maddy the Journal of Symbolic
Believing the Axioms. I Penelope Maddy The Journal of Symbolic Logic, Vol. 53, No. 2. (Jun., 1988), pp. 481-511. Stable URL: http://links.jstor.org/sici?sici=0022-4812%28198806%2953%3A2%3C481%3ABTAI%3E2.0.CO%3B2-3 The Journal of Symbolic Logic is currently published by Association for Symbolic Logic. Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/about/terms.html. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/journals/asl.html. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is an independent not-for-profit organization dedicated to and preserving a digital archive of scholarly journals. For more information regarding JSTOR, please contact [email protected]. http://www.jstor.org Tue Apr 3 15:14:09 2007 THEJOURNALOF SYMBOLICLOGIC Volume 53, Number 2, June 1988 BELIEVING THE AXIOMS. I PENELOPE MADDY Nothing venture, nothing win, Blood is thick, but water's thin. -Gilbert & Sullivan $0. Introduction. Ask a beginning philosophy of mathematics student why we believe the theorems of mathematics and you are likely to hear, "because we have proofs!" The more sophisticated might add that those proofs are based on true axioms, and that our rules of inference preserve truth. -
Logic Colloquium 2007 Edited by Francoise¸ Delon, Ulrich Kohlenbach, Penelope Maddy and Frank Stephan Frontmatter More Information
Cambridge University Press 978-0-521-76065-2 - Logic Colloquium 2007 Edited by Francoise¸ Delon, Ulrich Kohlenbach, Penelope Maddy and Frank Stephan Frontmatter More information Logic Colloquium 2007 The Annual European Meeting of the Association for Symbolic Logic, also known as the Logic Colloquium, is among the most prestigious annual meetings in the field. The current volume, Logic Colloquium 2007, with contributions from ple- nary speakers and selected special session speakers, contains both expository and research papers by some of the best logicians in the world. This volume covers many areas of contemporary logic: model theory, proof theory, set theory, and computer science, as well as philosophical logic, including tutorials on cardinal arithmetic, on Pillay’s conjecture, and on automatic structures. This volume will be invaluable for experts as well as those interested in an overview of central contemporary themes in mathematical logic. Fran¸coiseDelon was Directrice d’´etudesat the Centre de Formation des PEGC of Reims and Humboldt Stipendiatin at Freiburg and is presently a Directrice de Recherche at Centre National de la Recherche Scientifique. Ulrich Kohlenbach is a Professor of Mathematics at TU Darmstadt (Germany). He is the coordinating editor of Annals of Pure and Applied Logic and the president of the Deutsche Vereinigung f¨urMathematische Logik und f¨urGrundlagen der Exakten Wissenschaften. Penelope Maddy is a Distinguished Professor of Logic and Philosophy of Science at the University of California, Irvine. She is a Fellow of the American Academy of Arts and Sciences and is currently the president of the Association for Symbolic Logic. Frank Stephan is an Associate Professor in the departments of mathematics and computer science at the National University of Singapore. -
CURRICULUM VITAE Penelope Maddy
CURRICULUM VITAE Penelope Maddy Department of Logic and Philosophy of Science 949-824-1520 (dep’t) University of California at Irvine 949-824-4133 (office) Irvine, CA 92697-5100 [email protected] Education Princeton University, 1974-1978 PhD in Philosophy, January, 1979 University of California at Berkeley, 1968-1972 B. A. in Mathematics, June, 1972 Professional Positions UCI Distinguished Professor, 2007-present Professor of Logic and Philosophy of Science, 1998-present Professor of Mathematics, 1989 - present Professor of Philosophy, 1989 - 1998, 2014-present Associate Professor of Philosophy and of Mathematics, 1987-1989 University of California at Irvine Associate Professor, 1983-1987 Department of Philosophy University of Illinois at Chicago Assistant Professor, 1979-1983 Lecturer, 1978-1979 Department of Philosophy Department of Mathematics University of Notre Dame Honors Phi Beta Kappa-Romanell Professor, 2014-2015 UCI Distinguished Professor, November 2007- present Ambrose/Tymoczko Lecture, Smith College, February 2007 Gauss Lecture, Dresden, Germany, October 2006 Lakatos Prize for Naturalism in Mathematics, November 2002 Chancellor’s Professor, April 2002-November 2007 UCI Distinguished Faculty Lecturership for Research, March 2002 Patrick Romanell Lecture on Philosophical Naturalism, APA, December 2001 Elected to American Academy of Arts and Sciences, April 1998 NSF Research Grant, 1994-95 NSF Research Grant, 1990-91 UCI Academic Senate Research Fellowship, 1990-91 NSF Research Grant, 1988-89 UC President's Research Fellowship, 1988-89 Postdoctoral Scholar, UCLA Department of Mathematics, 1986 NSF Research Grant, Calendar Year 1986 NEH Fellowship, Summer 1984 UIC Short Research Leave, Spring 1984 AAUW Fellowship, 1982-1983 Notre Dame Faculty Research Grant, Summer 1981 Marshall Scholarship, 1972-1973 UC Berkeley, Mathematics Department Citation, 1972 B. -
Set-Theoretic Foundations
Contemporary Mathematics Volume 690, 2017 http://dx.doi.org/10.1090/conm/690/13872 Set-theoretic foundations Penelope Maddy Contents 1. Foundational uses of set theory 2. Foundational uses of category theory 3. The multiverse 4. Inconclusive conclusion References It’s more or less standard orthodoxy these days that set theory – ZFC, extended by large cardinals – provides a foundation for classical mathematics. Oddly enough, it’s less clear what ‘providing a foundation’ comes to. Still, there are those who argue strenuously that category theory would do this job better than set theory does, or even that set theory can’t do it at all, and that category theory can. There are also those who insist that set theory should be understood, not as the study of a single universe, V, purportedly described by ZFC + LCs, but as the study of a so-called ‘multiverse’ of set-theoretic universes – while retaining its foundational role. I won’t pretend to sort out all these complex and contentious matters, but I do hope to compile a few relevant observations that might help bring illumination somewhat closer to hand. 1. Foundational uses of set theory The most common characterization of set theory’s foundational role, the char- acterization found in textbooks, is illustrated in the opening sentences of Kunen’s classic book on forcing: Set theory is the foundation of mathematics. All mathematical concepts are defined in terms of the primitive notions of set and membership. In axiomatic set theory we formulate . axioms 2010 Mathematics Subject Classification. Primary 03A05; Secondary 00A30, 03Exx, 03B30, 18A15. -
Modal Realism and Metaphysical Nihilism Gonzalo Rodriguez-Pereyra
Mind, 2004, 113 (452), pp. 683-704. Modal Realism and Metaphysical Nihilism Gonzalo Rodriguez-Pereyra 1. Modal Realism is an ontological doctrine whose most characteristic thesis is that there exist non-actual possible individuals which are of a kind with actual individuals. That is, there are non-actual chairs, tables, donkeys, people and stars. As developed by David Lewis, Modal Realism is accompanied by a cluster of theses, for instance that all possible worlds (i.e. actual and non-actual possible worlds) exist, that all possible worlds are of a kind, that possible worlds are maximal sums of spatiotemporally related objects, and that a sentence like ‘it is possible that p’ is true just in case there is a possible world where p. Modal Realism has, among its theoretical benefits, a reductive account, within limits, of modality. Among its costs, it counts clashing with several intuitive views. One of these is the view that it is possible that nothing exists, that is, that there could have been nothing. Lewis saw that his Modal Realism is incompatible with this view (Lewis 1986, p. 73 and Lewis 1991, p. 13, footnote 6). Another closely related intuitive view with which Lewis’s Modal Realism is incompatible is what has recently been called Metaphysical Nihilism, namely that it is possible that nothing concrete exists, that is, that there could have been nothing concrete. Metaphysical Nihilism is not only intuitive, there are persuasive arguments in its favour. So, other things being equal, to be compatible with Metaphysical Nihilism is a theoretical virtue. In this paper I shall argue that Modal Realism can be modified so as to be compatible with Metaphysical Nihilism. -
Sometime a Paradox, Now Proof: Non-First-Order-Izability of Yablo's
Sometime a Paradox, Now Proof: Non-First-Order-izability of Yablo’s Paradox Saeed Salehi, Research Institute for Fundamental Sciences (RIFS), University of Tabriz, P.O.Box 51666-16471, Tabriz, Iran. School of Mathematics, Institute for Research in Fundamental Sciences, P.O.Box 19395–5746, Tehran, Iran. E-mail: [email protected] Abstract Paradoxes are interesting puzzles in philosophy and mathematics, and they could be even more fascinating when turned into proofs and theorems. For example, Liar’s paradox can be translated into a propositional tautology, and Barber’s paradox turns into a first-order tautology. Russell’s paradox, which collapsed Frege’s foundational framework, is now a classical theorem in set theory, implying that no set of all sets can exist. Paradoxes can be used in proofs of some other theorems; Liar’s paradox has been used in the classical proof of Tarski’s theorem on the undefinability of truth in sufficiently rich languages. This paradox (and also Richard’s paradox) appears implicitly in G¨odel’s proof of his celebrated first incompleteness theorem. In this paper, we study Yablo’s paradox from the viewpoint of first and second order logics. We prove that a formalization of Yablo’s paradox (which is second-order in nature) is non-first-order-izable in the sense of George Boolos (1984). 2010 AMS Subject Classification: 03B05 · 03B10 · 03C07. Keywords: Yablo’s Paradox · Non-first-orderizability. This was sometime a paradox, but now the time gives it proof. — William Shakespeare (Hamlet, Act 3, Scene 1). 1 Introduction If mathematicians and philosophers have come to the conclusion that some (if not almost all) of the paradoxes cannot be (re)solved, or as Priest [10, p. -
A Logical Framework†
Revista del Instituto de Filosofía, Universidad de Valparaíso, Año 1, N° 1. Junio 2013. Pags. 35 – 40 A logical framework† Göran Sundholm Resumen El artículo presenta un marco de distinciones para la filosofía de la lógica en la que las interrelaciones entre algunas nociones lógicas centrales, como la de declaración, juicio (el acto), juicio, (el resultado de juzgar), proposición (contenido), consecuencia e inferencia, se detallan. PALABRAS CLAVE: Declaración, juicio, proposición, consecuencia, inferencia. Abstract The paper presents a framework of distinctions for the philosophy of logic in which the interrelations between some central logical notions, such as statement, judgement (-act), judgement (made), proposition (al content), consequence, and inference are spelled out. KEY WORDS: Statement, judgement, proposition, consequence, inference. 1. Hilary Putnam, and, following him, George Boolos, have, on different occasions, taken exception to Quine's dictum that "Logic is an old subject, and since 1879 it has been a great one", with which he opened the first editions of his Methods of Logic.1 In their opinion, Quine's implicit preference for Frege's Begriffsschrift does an injustice to Boole (Boolos and Putnam) and the Booleans, of whom Peirce in particular (Putnam). Ten years ago, in an inaugural lecture at Leyden, also I argued that Quine presented too narrow a view of logic, and, that as far as the nineteenth century was concerned, the crucial date in the development of logical doctrine is not 1879 (nor 1847, I would add today, disagreeing -
SPREAD WORLDS, PLENITUDE and MODAL REALISM: a PROBLEM for DAVID LEWIS by Charles R
1 SPREAD WORLDS, PLENITUDE AND MODAL REALISM: A PROBLEM FOR DAVID LEWIS By Charles R. Pigden and Rebecca E. B. Entwisle 1. Introduction: Modal Realism and Meta-Philosophy David Lewis was famous (among other things) for the meta-philosophical claim that knock-down refutations are rare to non-existent in philosophy. You can argue that the costs of a theory exceed the intellectual benefits but it is difficult, if not impossible, to prove that it is false. When it comes to choosing between philosophical theories, it is generally a matter of ‘the price is right’, and when it comes to refuting a theory the best you can generally do is to drive up the costs whilst diminishing the benefits. There may come a time when the costs are so high and the benefits are so low that the theory can’t attract any takers, but this is usually due to a cumulative process of philosophical debate not to a sudden drop in net value brought about by a single decisive counterargument. When philosophers win, it’s nearly always a victory on points and almost never a knock-out. At the same time, Lewis subscribed to a philosophical thesis so outrageous as to cry out for knock-down refutation, namely that in order to make sense of modal truths, we must postulate an infinity of possible universes, all just as real as the one that we inhabit, but cut off from us in space and time. (This has the corollary that in order to make sense of causality we must postulate an infinity of items that have no causal connection to anything that happens in this universe.) Thus the temptation is to go for the double-whammy, the knock-down refutation of modal realism which is also a counterexample to the meta- philosophical claim.