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What Structural Objects Could Be – Mathematical Structuralism and Its Prospects – What Structural Objects Could Be { Mathematical Structuralism and its Prospects { MSc Thesis (Afstudeerscriptie) written by Teodor Tiberiu C˘alinoiu (born March 5, 1993 in Bucharest, Romania) under the supervision of Lect Dr Bahram Assadian, and submitted to the Examinations Board in partial fulfillment of the requirements for the degree of MSc in Logic at the Universiteit van Amsterdam. Date of the public defense: Members of the Thesis Committee: July 6, 2020 Asist Prof Luca Incurvati Dr Julian Schl¨oder Asist Prof Katrin Schulz Prof Dr Yde Venema (chair) Abstract This thesis covers structuralism in the philosophy of mathematics, focusing on non-eliminative versions thereof and zooming in on three fresh and promising contemporary articulations. After introducing the topic and essential piece of terminology, we follow a quasi-historical route to modern mathematical structuralism: starting with Paul Benacerraf's seminal articles and after drawing a taxonomy of views playing out in the contemporary field, we discuss eliminative structuralism alongside introducing useful ideology, and we formulate eliminativist discontents which feed a line of reasoning which is crucially invoked by non-eliminativists to motivate their view. Moving thus on to non-eliminativism, we introduce Stewart Shapiro's early articulation thereof: Sui Generis Structuralism, followed by an extensive discussion of many of the the problems and ensuing objections leveraged against it. Gathered together, all these concerns constitute the canon we use to assess the three newly emerging articulations of positionalist non-eliminativist structuralism. After taking a motivated detour through non-positionalist non-eliminativism, we introduce in some detail Øystein Linnebo and Richard Pettigrew's Fregean Abstractionist Structuralism, Edward Zalta and Uri Nodelman's Object Theoretic Structuralism and Hannes Leitgeb's Graph Theoretic Structuralism. Assessing each of these views against our canon, we find that, for the most part, each of these is successfully replied. Our thesis is that in spite of sustained criticism, there is still fuel in the realist's tank, meaning that each of the three views is left standing following their assessment against the canon, albeit this claim will be qualified in Conclusion. Acknowledgements There are many people who deserve to be acknowledged for their direct or indirect contribution to this thesis; feeling as if committing an injustice, I'll only mention a few. This essay would not have been possible without the supervision of Bahram Assadian, whose professional expertise, careful guiding and own work in the field of mathematical structuralism, as well as his sustained moral support, kind encouragement and everlasting patience made possible the eventual virtues and depth of the present work. Katrin Schulz deserves special mention for her active mentoring, joyful planning and thorough understanding; our conversations always turned complicated situations into highways to the desired result. I owe to Tanja Kasenaar the taming of the bureaucratic beast through swift practical support and solutions to all things administrative, even in the worst of times. Hrafn Valtyr Oddson, my fellow master logician, passionate mathematician, flatmate and friend proved an invaluable source of meaning for complex proofs and mathematical results; our long discussions about anything mathematical { always turned set theoretical { have always been a source of light in my early pilgrimage through the formal jungle. The Master of Logic has been a roller coaster. The ILLC is ultimately an organism and the people mentioned above may also stand as representatives of the organs making it up and running it. The ILLC collective { researchers and teachers, TA's, students and administration { all deserve a heartful round of applause. I am grateful to Daniele Garancini for the many short hours { those past as well as those to come { spent discussing all things philosophical; moreover, for his unexpectedly fruitful intervention, for the DnD game taking us through the eventful last months, and so on and so forth. I owe too much to tell to C˘at˘alinaFr^ancu,for being everything, from maecenas to the ultimate conversation partner to brilliant designer of worlds and much more. Needless to say, any shortcomings of the present essay are of my own making; all its virtues are a collective fruit. Contents 1 Introduction 4 1.1 Historical note......................................7 1.2 A Taxonomy of MS.................................... 12 2 Eliminativist MS 14 2.1 Relativist MS....................................... 15 2.2 Universalist MS...................................... 20 2.3 Modal MS......................................... 24 2.4 Concluding to non-eliminativism............................ 25 3 Non-eliminativist MS 27 3.1 Non-eliminativism in 1997................................ 28 3.2 Non-eliminativist troubles................................ 31 3.2.1 Problems of Identity............................... 32 3.2.1.1 Cross-structural Identities Problem.................. 32 3.2.1.2 The Automorphism Problem...................... 33 3.2.1.3 The Individuation Objection...................... 34 3.2.2 Problems of Objects............................... 38 3.2.2.1 The Permutation Problem....................... 38 3.2.2.2 The Circularity Problem........................ 40 3.2.2.3 The Problem of Structural Properties................. 41 3.2.2.4 MacBride's Objection.......................... 42 3.2.3 Problems of Reference.............................. 45 3.2.3.1 The Problem of Singular Reference.................. 45 3.2.3.2 The Semantic Objections........................ 47 3.2.4 Summing Up................................... 50 3.3 A Detour Through Non-Positionalism......................... 51 3.4 Positionalism....................................... 54 3.4.1 Fregean Abstractionist Structuralism...................... 54 3.4.1.1 Theory.................................. 55 3.4.1.2 Against the Canon........................... 63 3.4.1.3 Summing up............................... 71 3.4.2 Object Theoretic Structuralism......................... 71 3.4.2.1 Theory.................................. 71 3.4.2.2 Against the Canon........................... 77 3.4.2.3 Summing up............................... 79 3.4.3 Unlabeled Graph-theoretic Structuralism................... 80 3.4.3.1 Theory.................................. 80 3.4.3.2 Against the Canon........................... 88 3.4.3.3 Summing up............................... 91 4 Conclusion 92 Bibliography 95 Introduction 1 Introduction The last five decades brought structuralism into the spotlight of philosophy, especially mathematical structuralism (MS) in the philosophy of mathematics. Ever since Paul Benacerraf's seminal article,1 the philosophical community engaged closer and closer with structuralist themes, leading to the emergence of several different structuralist views to the point that no currently available taxonomy may comfortably accommodate them all. John Burgess sketches a history of the evolution of structuralist views in the last century.2 The following essay is concerned with central topics in the contemporary debates around MS. Coming of age against the background of a mathematical practice which emerged radically transformed at the end of more than a century of foundational disputes, modern structuralism in the philosophy of mathematics is a paradigm whose core claim is that structures constitute the subject matter of mathematical theories: such theories, the structuralist holds, are about structures.3 The main motivation behind mathematical structuralism rests on peculiar phaenomena of indifference in the modern mathematical practice: the number theorist, for instance, doesn't care whether natural numbers are set theoretic systems, a sequence of Roman emperors or another of watermelons, at least as long as there are enough of them. As much is a common to all sorts of currently trending structuralist views, but the agreement stops here. On the metaphysical side, divisions appear with respect to the status and nature of structures. On the ontological side, similar divisions emerge about the existence, nature and identity of mathematical objects understood as positions in structures; we are going to indiscriminately use `metaphysical' and `ontological' when referring to either cluster of issues in what follows. Such issues are paralleled by deep disagreements concerning the proper construal of ordinary mathematical discourse; such concerns are labeled `semantic'. Further, yet mostly neglected topics are epistemological,4 but these will be almost entirely bracketed in what follows. In what follows we are chiefly concerned with metaphysical and semantic aspects of MS. Some of these disagreements are resolved along two main separation lines well known in philosophical circles: eliminativism and non-eliminativism, a.k.a. 1Benacerraf[1965]. 2Burgess[2015, x3]. See also Hellman and Shapiro[2019, x2]. 3 Alongside mathematical structuralism, scientific structuralism is a boiling hot topic in the philosophy of science. See Ladyman et al.[2007] introducing ontic structural realism and Ladyman [2020] for a review of the contemporary field. We will not discuss scientific structuralism per se. Many problems and potential solutions discussed below have correspondents in scientific structuralism and a joint assessment would certainly prove most interesting. However, this is a topic for further work. 4See Shapiro[1997, x4] for an epistemology
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