DOCSLIB.ORG

NUMBERS WITHOUT SCIENCE by Russell Marcus a Dissertation

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

NUMBERS WITHOUT SCIENCE by Russell Marcus A dissertation submitted to the Graduate Faculty in Philosophy in partial fulfillment of the requirements for the degree of Doctor of Philosophy, The City University of New York 2007 Page ii © 2007 Russell Marcus All Rights Reserved Page iii This manuscript has been read and accepted for the Graduate Faculty in Philosophy in satisfaction of the dissertation requirement for the degree of Doctor of Philosophy. Elliott Mendelson October 26, 2006 _________________________________ Chair of Examining Committee Peter Simpson October 26, 2006 _________________________________ Executive Officer David Rosenthal (Adviser) Michael Devitt Jody Azzouni Alberto Cordero Supervisory Committee The City University of New York Page iv Abstract NUMBERS WITHOUT SCIENCE by Russell Marcus Adviser: Professor David Rosenthal Numbers without Science opposes the Quine-Putnam indispensability argument, seeking to undermine the argument and reduce its profound influence. Philosophers rely on indispensability to justify mathematical knowledge using only empiricist epistemology. I argue that we need an independent account of our knowledge of mathematics. The indispensability argument, in broad form, consists of two premises. The major premise alleges that we are committed to mathematical objects if science requires them. The minor premise alleges that science in fact requires mathematical objects. The most common rejection of the argument denies its minor premise by introducing scientific theories which do not refer to mathematical objects. Hartry Field has shown how we can reformulate some physical theories without mathematical commitments. I argue that Field’s preference for intrinsic explanation, which underlies his reformulation, is ill-motivated, and that his resultant fictionalism suffers unacceptable consequences. I attack the major premise instead. I argue that Quine provides a mistaken Page v criterion for ontic commitment. Our uses of mathematics in scientific theory are instrumental and do not commit us to mathematical objects. Furthermore, even if we accept Quine’s criterion for ontic commitment, the indispensability argument justifies only an anemic version of mathematics, and does not yield traditional mathematical objects. The first two chapters of the dissertation develop these results for Quine’s indispensability argument. In the third chapter, I apply my findings to other contemporary indispensabilists, specifically the structuralists Michael Resnik and Stewart Shapiro. In the fourth chapter, I show that indispensability arguments which do not rely on Quine’s holism, like that of Putnam, are even less successful. Also in Chapter 4, I show how Putnam’s work in the philosophy of mathematics is unified around the indispensability argument. In the last chapter of the dissertation, I conclude that we need an account of mathematical knowledge which does not appeal to empirical science and which does not succumb to mysticism and speculation. Briefly, my strategy is to argue that any defensible solution to the demarcation problem of separating good scientific theories from bad ones will find mathematics to be good, if not empirical, science. Page vi Preface The indispensability argument alleges that we have knowledge of the abstract objects of mathematics, and that this knowledge is justified by our uses of mathematics in empirical science. The evidence for mathematics is thus supposed to be empirical. In this dissertation, I defend mathematical knowledge while denying its empirical justification. Mathematics requires an epistemology independent of that for empirical science. There are at least three ways of arguing for empirical justification of mathematics. The first is to argue, as Mill did, that mathematical knowledge is knowledge of empirical objects. I will not pursue criticisms of this dead end; I refer the reader to Frege (1953). The second is to argue that while mathematical knowledge is knowledge of abstract objects, we have sensory access to such objects. I will not pursue this Aristotelian line, either. I take the fallout from Penelope Maddy’s attempt, including her own rejection of that position, to suffice.1 The currently most popular way to justify mathematics empirically is to argue: A) Mathematical knowledge is of abstract objects; B) We have experiences only with concrete objects; and yet C) Our experiences with concrete objects justify our mathematical knowledge. This is the Quine-Putnam Indispensability Argument, in its broadest form. In Chapter 1 of the dissertation I present and defend my interpretation of Quine’s version of 1 See Maddy (1980) for the position, and Balaguer (1994) for an excellent summary of problems with the position. Page vii the indispensability argument, which depends on his method for determining ontic commitment. I also show how the most popular and promising response to Quine’s argument, Field’s dispensabilist project, fails to defeat it. In Chapter 2, I provide three criticisms of Quine’s argument. In Part 1, I deny Quine’s method for determining ontic commitment. In Part 2, I defend instrumentalism as an alternative to Quine’s method. In Part 3, I show how even if we accept Quine’s method, the indispensability argument does not yield mathematical objects. In Chapters 3 and 4, I consider other versions of the indispensability argument. In Chapter 3, I present essential characteristics of indispensability arguments, and the unfortunate consequences of these arguments. I apply these general results to structuralists who rely on the indispensability argument. Part 1 of Chapter 4 is an aside on the central role that the indispensability argument plays in Hilary Putnam’s work. In Part 2, I show how non-holistic indispensability arguments, Michael Resnik’s pragmatic argument and Putnam’s success argument, fare no better than holistic versions. The concluding Chapter 5 has four parts. In Part 1, I argue for the legitimacy of mathematics as a science in its own right. In Part 2, I argue that we should pursue an epistemology for mathematics independent of that for empirical science. I characterize several elements of this approach, which I call autonomy realism, including its dependence on mathematical intuition. In Part 3, I revisit Field’s work, as he also seeks an alternative to the indispensability argument. I show that autonomy realism is a preferable alternative to both indispensabilism and Field’s fictionalism. I conclude by indicating a few areas for further research. Page viii Acknowledgments This dissertation was begun under the advisement of Jerrold Katz. Jerry and his work continue to inspire me. I wish he were still with us, if only to tell me where my errors are. Jared Blank, Jennifer Fisher, Eric Hetherington, and Mark McEvoy have been supportive friends and helpful critics over the years. I have spoken earnestly with Eric about both my concerns about the philosophical role of formal theories and about misuses of mathematics in philosophy. I fear that my attempts to flesh out the first concern have led me to commit errors of the second kind. I am grateful to Michael Devitt for reading several drafts and making helpful comments, and also for his friendly guidance. Elliott Mendelson has been supportive all the way. I deeply appreciate Jody Azzouni’s selfless assistance. Paul Horwich generously spent hours discussing with me my work on Quine, and helped me to formulate my position more clearly. Michael Levin read drafts and made many useful comments. Errors and infelicities which remain are of course solely my responsibility. Portions of the dissertation have been presented to audiences at the CUNY Graduate Philosophy Conference, Spring 2005; the Graduate Center Philosophy Student Colloquium series, Spring 2005; the UT Austin Graduate Philosophy Conference, April 2006; the University of British Columbia Student Philosophy Conference, April 2006; and the Rocky Mountain Student Philosophy Conference, March 2006. David Rosenthal has been invaluable in many ways, not least as a trusted adviser and confidant. I owe him immeasurable gratitude and a great debt. Page ix I owe my loving family both heartfelt thanks and sincere apologies for putting their empathetic hearts through prolonged strain. My parents supported me despite wondering why I would follow my mother’s lead and leave the beginnings of a successful career to pursue a Graduate Center Ph.D. My children, Marina and Isidor, have been profound inspirations and motivations. But most of all, Emily, whose confidence in me, puzzlingly, never wavered. Page x Table of Contents Abstract ....................................................................................................................... iv Preface ......................................................................................................................... vi Chapter 1: Quine’s Indispensability Argument and Field’s Response Part 1: Quine’s Indispensability Argument §1.1: Quine’s Argument ...................................................................... 1 §1.2: A Best Theory ............................................................................ 2 §1.3: Believing Our Best Theory ........................................................ 6 §1.4: Quine’s Procedure for Determining Ontic Commitments ......... 9 §1.4.1: First-Order Logic ........................................................ 10 §1.4.2: The Domain of Quantification .................................... 14 §1.5: Mathematization ........................................................................
Recommended publications
  • Jody Azzouni

    Jody Azzouni

    ON “ON WHAT THERE IS”* BY JODY AZZOUNI Abstract: All sides in the recent debates over the Quine-Putnam Indispensability thesis presuppose Quine’s criterion for determining what a discourse is ontologically committed to. I subject the criterion to scrutiny, especially in regard to the available competitor-criteria, asking what means of evaluation there are for comparing alternative criteria against each other. Finding none, the paper concludes that ontological questions, in a certain sense, are philosophically indeterminate. (What is C. trying to pull?) marginalia on Quine’s copy of a letter from Carnap 1. A lot of philosophy of mathematics is motivated by considerations arising from what has come to be called the Quine-Putnam indispensability thesis;1 the claim, roughly, that if one’s best scientific (physical) theory requires existential quantification over certain entities, then one is onto- logically committed to such entities.2 Many books in this area, such as Chihara (1990), Field (1980), Hellman (1989), and Maddy (1990), draw their philosophical raison d’être from the view that scientific theories commit us to the existence of mathematical objects this way. The indispensability thesis, it seems, drives philosophers to hard choices: rewrite one’s science, rewrite one’s mathematics, or regretfully embrace extravagant ontologies. It’s quite unsurprising, therefore, that such a seminal claim has once again come under intense scrutiny; and equally unsurprising, I guess, to find philosophers on both sides of the philosophical fence. Maddy strongly Pacific Philosophical Quarterly 79 (1998) 1–18 0031–5621/98/0100–0000 © 1998 University of Southern California and Blackwell Publishers Ltd.
  • False Dilemma Wikipedia Contents

    False Dilemma Wikipedia Contents

    False dilemma Wikipedia Contents 1 False dilemma 1 1.1 Examples ............................................... 1 1.1.1 Morton's fork ......................................... 1 1.1.2 False choice .......................................... 2 1.1.3 Black-and-white thinking ................................... 2 1.2 See also ................................................ 2 1.3 References ............................................... 3 1.4 External links ............................................. 3 2 Affirmative action 4 2.1 Origins ................................................. 4 2.2 Women ................................................ 4 2.3 Quotas ................................................. 5 2.4 National approaches .......................................... 5 2.4.1 Africa ............................................ 5 2.4.2 Asia .............................................. 7 2.4.3 Europe ............................................ 8 2.4.4 North America ........................................ 10 2.4.5 Oceania ............................................ 11 2.4.6 South America ........................................ 11 2.5 International organizations ...................................... 11 2.5.1 United Nations ........................................ 12 2.6 Support ................................................ 12 2.6.1 Polls .............................................. 12 2.7 Criticism ............................................... 12 2.7.1 Mismatching ......................................... 13 2.8 See also
  • Forensic Pattern Recognition Evidence an Educational Module

    Forensic Pattern Recognition Evidence an Educational Module

    Forensic Pattern Recognition Evidence An Educational Module Prepared by Simon A. Cole Professor, Department of Criminology, Law, and Society Director, Newkirk Center for Science and Society University of California, Irvine Alyse Berthental PhD Candidate, Department of Criminology, Law, and Society University of California, Irvine Jaclyn Seelagy Scholar, PULSE (Program on Understanding Law, Science, and Evidence) University of California, Los Angeles School of Law For Committee on Preparing the Next Generation of Policy Makers for Science-Based Decisions Committee on Science, Technology, and Law June 2016 The copyright in this module is owned by the authors of the module, and may be used subject to the terms of the Creative Commons Attribution-NonCommercial 4.0 International Public License. By using or further adapting the educational module, you agree to comply with the terms of the license. The educational module is solely the product of the authors and others that have added modifications and is not necessarily endorsed or adopted by the National Academy of Sciences, Engineering, and Medicine or the sponsors of this activity. Contents Introduction .......................................................................................................................... 1 Goals and Methods ..................................................................................................................... 1 Audience .....................................................................................................................................
  • Yesterday's Algorithm: Penrose on the Gödel Argument

    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.
  • Deflationary Nominalism and Puzzle Avoidance1 David Mark Kovacs

    Deflationary Nominalism and Puzzle Avoidance1 David Mark Kovacs

    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by PhilPapers Deflationary nominalism and puzzle avoidance1 David Mark Kovacs (Forthcoming in Philosophia Mathematica. Draft; please cite the final version!) Abstract: In a series of works, Jody Azzouni has defended deflationary nominalism, the view that certain sentences quantifying over mathematical objects are literally true, although such objects do not exist. One alleged attraction of this view is that it avoids various philosophical puzzles about mathematical objects. I argue that this thought is misguided. I first develop an ontologically neutral counterpart of Field’s reliability challenge and argue that deflationary nominalism offers no distinctive answer to it. I then show how this reasoning generalizes to other philosophically problematic entities. The moral is that puzzle avoidance fails to motivate deflationary nominalism. 1. Introduction In a series of works, Jody Azzouni has defended deflationary nominalism:2 the view that (1) reality contains no numbers and other mathematical objects, but (2) existentially quantified sentences (including those quantifying over mathematical objects), as well as their natural language counterparts, can be strictly and literally true without conferring ontological commitment to what they quantifier over (the “separation thesis”).3 The view is a version of nominalism because it dispenses with ontological commitment to mathematical objects, and it is deflationary because it maintains that 1 For helpful comments and discussion I thank Sandy Berkovski, Matti Eklund, Alexander Jones, Dan Korman, Ted Parent, Robert Schwartzkopff, the anonymous referees of Philosophia Mathematica, and audiences at a workshop titled “Existence” at Uppsala University, the 20th meeting of the Israeli Philosophical Association at Ben Gurion University, and the 9th European Congress of Analytic Philosophy at the Ludwig Maximilian University in Munich.
  • In Defence of a Princess Margaret Premise

    In Defence of a Princess Margaret Premise

    Fabrice Pataut: “In Defence of a Princess Margaret Premise,” Introductory chapter to Truth, Objects, Infinity – New Perspectives on the Philosophy of Paul Benacerraf, Fabrice Pataut, editor. Springer International Publishing, Logic, Epistemology, and the Unity of Science, vol. 28, Switzerland, 2016, pp. xvii-xxxviii. ISBN 978-3-319-45978-3 ISSN 2214-9775 ISBN 978-3-319-45980-6 (eBook) ISSN 2214-9783 (electronic) DOI 10.1007/978-3-319-45980-6 Library of Congress Control Number : 2016950889 © Springer International Publishing Swtizerland 2016 1 IN DEFENCE OF A PRINCESS MARGARET PREMISE Fabrice Pataut 1. Introductory remarks In their introduction to the volume Benacerraf and his Critics, Adam Morton and Stephen Stich remark that “[t]wo bits of methodology will stand out clearly in anyone who has talked philosophy with Paul Benacerraf”: (i) “[i]n philosophy you never prove anything; you just show its price,” and (ii) “[f]ormal arguments yield philosophical conclusions only with the help of hidden philosophical premises” (Morton and Stich 1996b: 5). The first bit will come to some as a disappointment and to others as a welcome display of suitable modesty. Frustration notwithstanding, the second bit suggests that, should one stick to modesty, one might after all prove something just in case one discloses the hidden premises of one’s chosen argument and pleads convincingly in their favor on independent grounds. I would like to argue that a philosophical premise may be uncovered — of the kind that Benacerraf has dubbed “Princess Margaret Premise” (PMP)1 — that helps us reach a philosophical conclusion to be drawn from an argument having a metamathematical result as one of its other premises, viz.
  • Volume 6, Number 6 (Optimised for Screen Readers)

    Volume 6, Number 6 (Optimised for Screen Readers)

    Volume 6, Number 6 June 2012 www.thereasoner.org ISSN 1757-0522 Contents Editorial Features News What’s Hot in . Events Courses and Programmes Jobs and Studentships Editorial Much ado about nothing My three year old doesn’t believe that Paris really exists. This isn’t the result of some sophisticated ontological position, borne out of an inherited and grossly exaggerated taste for desert landscapes (he is only three, after all). No, the reason is simpler than that: his only knowledge of Paris is through fiction. The Disney film, The Aristocats, to be precise. With his robust sense of reality, he knows full well that this is just a story. So when I tell him I’m off to Paris for a few days for a con- ference he looks at me sceptically and says, “Don’t be silly, mummy! There’s no such place. Cats don’t really talk, you know.” Well, I need to explain my movements somehow, so we sit down and have ‘the talk’. I explain that, while it’s true that Madame, Duchess, the kittens, and the villainous butler Edgar don’t really exist, some of the things we talk about in stories do, and Paris is one of them. And that’s where I’m off to. I think I’m getting through to him, but now he looks worried. “But mummy—what if Edgar gets you?” I remind him that Edgar doesn’t really exist, and that at any rate the last anyone saw of him was when the cats bundled him into a trunk and posted him off to Timbuktu.
  • Mcevoy on Benacerraf's Problem and the Epistemic Role Puzzle

    Mcevoy on Benacerraf's Problem and the Epistemic Role Puzzle

    McEvoy on Benacerraf’s problem and the epistemic role puzzle Jody Azzouni Tufts University [email protected] Forthcoming in New Perspectives on the Philosophy of Paul Benacerraf: Truth, Objects, Infinity (ed. Fabrice Pataut). Springer: Logic, Epistemology And the Unity of Science, 2015. 1. Benacerraf’s problem. Benacerraf’s problem is justly famous. It’s had a major influence on the philosophy of mathematics right from its initial appearance,1 an influence that continues up through the present moment. In its author’s supernaturally elegant prose, it lays out a tension between the possibility of an epistemic access to abstracta and the apparent semantics (truth conditions) of mathematical statements about those entities. Given a causal construal of epistemic access, on the one hand, it seems that we can’t have any epistemic access to the objects that our true mathematical statements must be about because those objects are causally inefficacious and causally insensitive; on the other hand, the mathematical truths in question are genuinely about those objects, and somehow we are adept at identifying some of the true mathematical statements and some of the false ones. Benacerraf’s problem long outlasted the faddish “causal theory of knowledge” that he originally couched it in terms of. Field (1989, 26), among others, generalized Benacerraf’s problem by writing: Benacerraf’s challenge … is to provide an account of the mechanisms that explains how our beliefs about these remote entities can so well reflect the facts about them. The idea is that if it appears in principle impossible to explain this, then that tends to undermine the 1 Benacerraf 1973.
  • A Searchable Bibliography of Fallacies – 2016

    A Searchable Bibliography of Fallacies – 2016

    A Searchable Bibliography of Fallacies – 2016 HANS V. HANSEN Department of Philosophy University of Windsor Windsor, ON CANADA N9B 3P4 [email protected] CAMERON FIORET Department of Philosophy University of Guelph Guelph, ON CANADA N1G 2W1 [email protected] This bibliography of literature on the fallacies is intended to be a resource for argumentation theorists. It incorporates and sup- plements the material in the bibliography in Hansen and Pinto’s Fallacies: Classical and Contemporary Readings (1995), and now includes over 550 entries. The bibliography is here present- ed in electronic form which gives the researcher the advantage of being able to do a search by any word or phrase of interest. Moreover, all the entries have been classified under at least one of 45 categories indicated below. Using the code, entered as e.g., ‘[AM-E]’, one can select all the entries that have been des- ignated as being about the ambiguity fallacy, equivocation. Literature about fallacies falls into two broad classes. It is either about fallacies in general (fallacy theory, or views about fallacies) or about particular fallacies (e.g., equivocation, appeal to pity, etc.). The former category includes, among others, con- siderations of the importance of fallacies, the basis of fallacies, the teaching of fallacies, etc. These general views about fallacies often come from a particular theoretical orientation about how fallacies are best understood; for example, some view fallacies as epistemological mistakes, some as mistakes in disagreement resolution, others as frustrations of rhetorical practice and com- munication. Accordingly, we have attempted to classify the en- © Hans V. Hansen & Cameron Fioret.
  • On Computer-Based Assessment of Mathematics

    On Computer-Based Assessment of Mathematics

    ON COMPUTER-BASED ASSESSMENT OF MATHEMATICS DANIEL ARTHUR PEAD, BA Thesis submitted to the University of Nottingham for the degree of Doctor of Philosophy December 2010 Abstract This work explores some issues arising from the widespread use of computer based assessment of Mathematics in primary and secondary education. In particular, it considers the potential of computer based assessment for testing “process skills” and “problem solving”. This is discussed through a case study of the World Class Tests project which set out to test problem solving skills. The study also considers how on-screen “eAssessment” differs from conventional paper tests and how transferring established assessment tasks to the new media might change their difficulty, or even alter what they assess. Once source of evidence is a detailed comparison of the paper and computer versions of a commercially published test – nferNelson's Progress in Maths - including a new analysis of the publisher's own equating study. The other major aspect of the work is a design research exercise which starts by analysing tasks from Mathematics GCSE papers and proceeds to design, implement and trial a computer-based system for delivering and marking similar styles of tasks. This produces a number of insights into the design challenges of computer-based assessment, and also raises some questions about the design assumptions behind the original paper tests. One unanticipated finding was that, unlike younger pupils, some GCSE candidates expressed doubts about the idea of a computer-based examination. The study concludes that implementing a Mathematics test on a computer involves detailed decisions requiring expertise in both assessment and software design, particularly in the case of richer tasks targeting process skills.
  • Fantasy & Science Fiction V023n04

    Fantasy & Science Fiction V023n04

    13th Anniversary ALL STAR THE MAGAZINE 0 E Fantasy FANTAiy and Science Fictio SCIENCE The Journey of Joenes fiCTION a new novel by ROBERT SHFCfCtfY A Kind of Artistry BRIAN W. ALDISS 6 There Was an Old Woman Who Lived in a Shoe ROBERT F. YOUNG 28 24 Hours in a Princess’s Life> With 'Fe6gs DON WHITE 34 Inquest in Kansas {verse) ;/ HYACINTHE HILL 38 Measure My Love ' MILDRED CLINGERMAN 39 Science', Slow Burn ISAAC ASIMOV 52 Books AVRAM DAVIDSON 63 The Unfortunate Mr. Morlcy VANCE AANDAHL 68 Ferdinand Feghoot: LV GRENDEL BRIARTON 70 The Journey of Joenes {1st 6j 2 parts ) ROBERT SHECKLEY 71 ^ Editorial > /* 4 / • In this issue . Coming si/on 5 F&SE Marketplace ' 129 Cover by Ed Emsh {illustrating ^'The Journey of Joenes”) The Magasine of Fantasy and Sciffiice Fiction, Vclume 23, No. 4, Whole No. 137, Oct. 1962. P-ttblisked monthly by Mercury Press, Inc., at 40^ a copy. Annual subscription S4 50 in U. S. and Possessions, $5.00 in Canada and the Pan American Union; $5.50 m all other countries. Publicaiion office, IQ Ferry Street, Concord, N. //. Editorial and general mail should be sent to 347 East 53rd St., New York 22, N. Y. Second Class postage paid at Concord, N. H. Printed in U. S. A. © 2^62 by Mercury Press, Inc. Ail rights, including translations into other languages, reserved. Submissions mtist be accom- panied by stamped, self-addressed envelopes; the Publisher assumes no responsibility for return of unsolicited manuscripts. Joseph W. Forman, publisher Avram Davidson, executive editor Robert P.
  • Observations on Sick Mathematics Andrew Aberdein

    Observations on Sick Mathematics Andrew Aberdein

    ESSAY 1 Observations on Sick Mathematics Andrew Aberdein And you claim to have discovered this . from observations on sick people? (Freud 1990: 13) This paper argues that new light may be shed on mathematical reason- ing in its non-pathological forms by careful observation of its patholo- gies. The first section explores the application to mathematics of recent work on fallacy theory, specifically the concept of an ‘argumentation scheme’: a characteristic pattern under which many similar inferential steps may be subsumed. Fallacies may then be understood as argumen- tation schemes used inappropriately. The next section demonstrates how some specific mathematical fallacies may be characterized in terms · Andrew Aberdein of argumentation schemes. The third section considers the phenomenon of correct answers which result from incorrect methods. This turns out to pose some deep questions concerning the nature of mathematical knowledge. In particular, it is argued that a satisfactory epistemology for mathematical practice must address the role of luck. Otherwise, we risk being unable to distinguish successful informal mathematical rea- soning from chance. We shall see that argumentation schemes may play a role in resolving this problem too.1 Mathematical errors occur in many different forms. In his classic short treatment of the topic, E.A. Maxwell distinguished the simple mistake, which may be caused by ‘a momentary aberration, a slip in writing, or the misreading of earlier work,’ from the howler, ‘an error which leads innocently to a correct result,’ and the fallacy, which ‘leads by guile to a wrong but plausible conclusion’ (Maxwell 1959: 9). We might gloss this preliminary typology of mathematical error as correlating the (in)correctness of the result to the (un)soundness of the method, as in Table 1.