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A COGNITIVE APPROACH to BENACERRAF's DILEMMA Thesis Format: Monograph Luke V. Jerzykiewicz Graduate Program in Philosophy a Th

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A COGNITIVE APPROACH to BENACERRAF's DILEMMA Thesis Format: Monograph Luke V. Jerzykiewicz Graduate Program in Philosophy a Th A COGNITIVE APPROACH TO BENACERRAF'S DILEMMA Thesis format: Monograph by Luke V. Jerzykiewicz Graduate Program in Philosophy A thesis submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy The School of Graduate and Postdoctoral Studies The University of Western Ontario London, Ontario, Canada c Luke V. Jerzykiewicz 2009 Abstract One of the important challenges in the philosophy of mathematics is to account for the se- mantics of sentences that express mathematical propositions while simultaneously explaining our access to their contents. This is Benacerraf's Dilemma. In this dissertation, I argue that cognitive science furnishes new tools by means of which we can make progress on this problem. The foundation of the solution, I argue, must be an ontologically realist, albeit non-platonist, conception of mathematical reality. The semantic portion of the problem can be addressed by accepting a Chomskyan conception of natural languages and a matching internalist, mentalist and nativist view of semantics. A helpful perspective on the epistemic aspect of the puzzle can be gained by translating Kurt G¨odel'sneo-Kantian conception of the nature of mathematics and its objects into modern, cognitive terms. Keywords: mathematical cognition, philosophy of mathematics, realism, psycholo- gism, functional architecture, conceptualist semantics, cognition, mathematics, Benacerraf, Chomksy, G¨odel,Kant. iii Acknowledgements I have accrued a great many personal and professional debts over the years. Let me restrict myself here to acknowledging explicitly only those that bear directly on the writing of this monograph. Above all, let me express my heartfelt thanks to my thesis supervisor, Robert Stainton, whose firm guidance, patient instruction, unfailing support, and kind help over many years have seen me though this project. It's largely by observing Rob work that I have learned what I know of the philosopher's craft. For this I am very grateful. As well, let me extend my thanks to Andy Brook who took a chance on me many years ago and who also taught me how to read Kant slightly askance. My thanks also to the Carleton Institute of Cognitive Science. At the Institute I learned to view cognitive science as an unified endeavour spanning many fields and methodologies, an endeavour with deep historical roots and an open future. Furthermore, I am very grateful to John L. Bell for his insight and wisdom. I regret that I was not able to benefit more from his instruc- tion. His work remains for me a signpost on the horizon toward which I hope slowly to make my way. I have had the good fortune to study under some remarkable teachers. I would especially like to thank Paul Pietroski, Susan Dwyer, James McGilvray, Stephen Houlgate, Chris Herdman, Marie Odile Junker, Storrs McCall, Philip Buckley, and Angela Cozea. Their ideas and their methods have, in various ways, shaped my approach to research and to writing|sometimes with results they may not have expected or perhaps even approved of. I'd like to thank Penelope Maddy, Susan Carey, and Ray Jackendoff for brief but extraordinarily enlightening conversations. Each of them has been an inspiration. I have also benefited from the thoughtful criticisms of Georges Rey, Edouard Machery, Mary Kate McGowan, Ileana Paul, Chris Viger, Pete Mandik, Eric Thomson, and Philip Grier. Let me also acknowledge the many friends and acquaintances off of whom I have bounced ideas over the years. Here I should mention Dan Robins and Sam Scott in particular. Their help and patience with my little obsessions is greatly appreciated. Of course, I owe a very special debt of gratitude to my parents Janina and Tom Jerzykiewicz. They have been wonderfully supportive of me along the way. It was from my Mom and Dad that I first learned the meaning and the importance of intellectual labour as well as the habits that it calls for. It's impossible to imagine more loving or understanding parents. Finally, let me express my warm gratitude to my best friend and partner for many years, Catherine Wearing. Her advice and her criticisms have been invaluable. Most of the ideas in this dissertation were first tested in discussion with her. Above all though, I'd like to thank her for sharing with me some very happy years. I gratefully acknowledge the Ontario Graduate Scholarship Program and the Social Sciences and Humanities Research Council of Canada who helped fund this research. iv Contents 0 Introduction 1 1 Minimal Realism 3 Definition . 3 Evidence . 4 Indispensability and truth . 5 Semantics and entities . 20 Objectivity . 25 Conclusion . 32 2 Troubles for Platonism 33 Abstract Realism . 33 Platonism . 34 Structuralism . 36 Knowledge . 38 Ante rem account . 39 Counter-evidence . 43 Large structures . 43 Small structures . 54 Conclusion . 58 3 The Semantic Problem 59 Benacerraf's assumptions . 60 Limitations of formal semantics . 62 Linguistic internalism . 69 Realism redefined . 84 Conclusion . 87 4 The Epistemic Problem 88 Psychologism . 88 Limitations . 97 Mathematical `perception' . 103 Background . 104 Kant's Transcendental Psychology . 106 G¨odel'sTranscendental Realism . 117 Knowledge of architecture . 128 Conclusion . 133 v 5 Final Remarks 134 Curriculum Vitae 150 List of Tables 1 Semantic dementia impacts arithmetic cognition . 46 2 Agrammatic aphasia impacts arithmetic cognition . 48 3 Savant performance on arithmetic tasks . 52 List of Figures 1 Tokens of a single type. 40 2 Cortical atrophy in semantic dementia. 45 3 Parallel cognitive architecture . 76 4 Cerebral localization of mathematical cognition . 90 5 Cognitive processing at language-math interface . 91 vi 1 0 Introduction The appearance of an antinomy is for me a symptom of disease. Whenever this happens, we have to submit our ways of thinking to a thorough revision, to reject some premisses in which we believed, or to improve some forms of argument which we used. | Alfred Tarski One of the important unsolved problems in the philosophy of mathematics was first artic- ulated in Paul Benacerraf's [1973] paper, Mathematical Truth. There, Benacerraf argued that it is difficult (surprisingly difficult, in fact) to offer an adequate semantic account of sentences that, to all appearances, express mathematical knowledge, while simultaneously also providing an explanation of our epistemic access to their contents. On a standard real- ist conception, mathematics studies a body of facts|ones not unlike those investigated by the geologist or the chemist, albeit significantly more `abstract' in nature. Many working mathematicians and philosophers find such a conception attractive. However, it is widely recognized that mathematical realism faces serious difficulties in epistemology: how, short of magic, can our cognitive access to abstract states of affairs be explained? An alternative conception proposes that we view the practice of mathematics as the playing out of a formal game with correct and incorrect moves. Some who consider themselves `hard-nosed' natu- ralists opt for this view. To be sure, abandoning the supposition that mathematics studies full-blooded facts does some violence to our commonsense interpretation of the meaning of mathematical expressions. It does however have the advantage of explaining (or, more ac- curately, explaining away) mathematical `knowledge,' thereby making the epistemologist's task tractable. This approach encounters its own set of problems. Chief among them is the difficulty of adequately explaining maths' uncanny utility in natural science. Thesis, antithesis, stalemate. The past two decades have witnessed a steady increase in the interest paid to mathe- matics (especially arithmetic) by cognitive scientists, including cognitive psychologists and cognitive neuroscientists.1 As their projects gather momentum researchers are faced with a choice whether or not to interpret the study of mathematics|including algebra, analysis, set theory, and so on|as the investigation of an objectively existing domain. The choice has direct implications for which empirical research programs are taken seriously and which are shelved as prima facie implausible. At the moment, the balance is swinging in the anti- realists' favour. This, I will argue, is a mistake. My central thesis in what follows is that 1See, for example, Butterworth [1999], Carey [2001], Dehaene [1997], Gallistel and Gelman [2000], Gelman and Gallistel [1978], Hauser and Spelke [2004], Spelke and Tsivkin [2001], Wynn [1998], Xu and Spelke [2000]. There have also been a few attempts by sympathetic, naturalist philosophers to apply empirical results and techniques to problems in the philosophy of mathematics. They include Kitcher [1983] and Maddy [1990], and more recently Giaquinto [2001], and Laurence and Margolis [2005]. 2 cognitive science should opt for an unflinchingly realist conception of mathematicalia. But in order for this to be possible, ontological realism about mathematics must itself undergo a transformation both with respect to the semantics it endorses and the epistemic picture it relies on. The realist must be able to offer a coherent, naturalist hypothesis regarding the correct solution to Benacerraf's dilemma. What follows is intended as a contribution to that project. I argue for my thesis in four Chapters. In Chapters 1 and 2, I articulate the kernel of the ontological realist view and show that it is well-motivated. I try to demonstrate however that the standard, platonist elaboration of minimal ontological realism is untenable and so should be abandoned. The positive
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