A Route to Agnosticism in Mathematics Stefano Boscolo Anno Accademico 2014/15 Contents Introduction 5 1 Arguments for and against Platonism 12 1.1 Between a rock and a hard place: Benacerraf's argument . 15 1.2 Benacerraf revisited: the reliability claim . 22 1.2.1 Field's argument against Platonism . 26 1.3 The indispensability argument(s) . 29 1.3.1 Three forms of naturalism . 32 1.3.2 Dodging the indispensability argument: an outline . 35 1.4 Burgess and Rosen's challenge to nominalism . 36 2 Removing Plato's Beard: Hard Roads to anti-Platonism 40 2.1 Physics without numbers . 40 2.1.1 Truth and conservativeness . 42 2.1.2 Dispensing with mathematical objects in physics . 45 2.1.3 Shapiro's objections . 48 2.1.4 Field's modal deflationism . 49 2.1.5 Extending Field's program . 51 2.2 Mathematics and constructibility quantifiers . 53 2.2.1 The constructibility theory . 53 2.2.2 Modality without possible worlds . 60 2.2.3 Towards modal fictionalism . 61 2.2.4 Burgess' objection . 64 2.3 Mathematics without numbers . 66 1 Contents 2 2.3.1 Hellman's program . 67 2.3.2 Beyond the modal interpretation of arithmetic . 69 2.3.3 The wolf and the lamb . 70 2.3.4 Ontological innocence . 71 3 Trimming Plato's Beard: Easy Roads to anti-Platonism 74 3.1 Deflating existential consequences . 74 3.1.1 A criterion for what exists . 77 3.1.2 Existence and ontological commitment . 78 3.1.3 Grades of ontological commitment . 86 3.1.4 Ontological nihilism and objectivity . 92 3.1.5 Defeasibility condition and fuzzy posits . 97 3.1.6 Bueno and Zalta's objection . 101 3.2 How logical subtraction works . 104 3.2.1 Aboutness and possible worlds . 106 3.2.2 Logical subtraction . 113 3.2.3 Mathematics is extricable . 117 3.2.4 Some remarks on Yablo's approach . 120 3.3 On confirmation holism . 127 3.3.1 Maddy's non-factualistic view . 128 3.3.2 Sober's objection to the indispensability argument . 130 3.3.3 Confirmation holism and Platonism . 134 4 Agnosticism in the Philosophy of Mathematics 139 4.1 Balaguer's anti-metaphysical conclusion . 139 4.1.1 Full-blooded Platonism and fictionalism . 140 4.1.2 There is no fact of the matter . 148 4.1.3 Non-factualism and tertium non datur . 150 4.2 Agnostic nominalism . 152 4.2.1 Azzouni's objection to agnostic nominalism . 157 Contents 3 5 Conclusion 163 5.1 Four claims . 163 5.2 My own agnostic view . 165 Bibliography 176 Acknowledgements I have received helpful comments from several people who have played a major role in developing my doctoral dissertation, and I would like to express my appreciation to all of them. Among these people, two deserve to be singled out: Gianluigi Oliveri and Jody Azzouni. From Gianluigi Oliveri, I have learned much about realism in the philosophy of mathematics. I owe him a debt of gratitude for all the time he has spent monitoring my work and progress during the last three years. From Jody Azzouni, I have learned much about nominalism and this has helped me to understand the so-called `easy road' in the metaphysical debate. His comments have made a significant contribution to this work. I would also like to thank Ot´avio Bueno for all the support he has given to my own agnostic view, Stephen Yablo for assisting me with several issues related to logical subtraction, and Mark Balaguer for having made useful comments on section 4.1. I am also grateful to Vincenzo Teresi for all our conversations and for being my workmate during these years. I wish to thank the students, staff and faculty members of the PhD program in Filosofia del Linguaggio at the Universit`a di Palermo. I am extremely grateful to the Department of Philosophy at Tufts University for having provided an ideal working environment during my stay in Boston. Introduction The ontological problem is characterized by Quine as the question on what there is. This question can be split into smaller issues according to the way we describe the world: common sense, chemistry, computer science, physics, and so on. Mathematics plays a crucial role in our web of beliefs, because it seems indispensable to the scientific picture of the world. Because this is so, part of the ontological problem takes the form of what mathematics is about: that is, whether mathematical objects exist or not. If there are such objects, like numbers, it seems we ought to regard them as abstracta: causally inert objects that are outside of space-time and mind-language in- dependent. This is because no one can build a telescope to detect where mathematical objects are, no one can break a Turing machine, and no one can causally interact with numbers. The view that mathematical abstracta exist is called Platonism, whereas the view according to which there is no commitment to such objects is called anti-Platonism. Anti-Platonism is a broad name for a large number of different positions that are divided into two classes: anti-Platonist realism and anti-Platonist anti-realism. According to the former, mathematical objects exist but are not abstracta, whereas, according to the latter, mathematical objects do not exist at all. Both views are generally driven by the weak epistemology of abstract objects, that is, by the fact that it is hard to account for how the correlation between belief states and abstract objects occurs. Episte- mological arguments against Platonism are based on manifold theories of knowledge (e.g. the causal theory of knowledge, reliabilism, etc.) but they 5 Contents 6 all point to the same problem: no epistemic story can bridge the gap be- tween belief states and abstract objects. And if we lack such an epistemic story, it might be questionable whether or not to include abstracta in our ontology. The Platonist has a straightforward reply to such epistemological wor- ries: the indispensability arguments. According to one of the most impor- tant versions of the indispensability argument, if mathematical objects are indispensable to our best scientific theories, we ought to have ontological commitment to such objects independently of any epistemological concern. Because the indispensability argument aims to overcome any epistemologi- cal objection, it is perhaps the best argument for Platonism. In this regard, there have been many attempts at disarming the indispensability argument without committing to any sort of revisionism of actual scientific practice. I should like to present the best examples of such attempts by advocating the distinction between the `hard road' and the `easy road' to anti-Platonism: hard-road strategies paraphrase physical, or mathematical, statements in order to dispense with mathematical objects; easy-road strategies also re- ject abstract objects but do not require that physics and mathematics be paraphrase-bound. I will examine three `hard roads': Field's nominalization of classical me- chanics and Newtonian gravitational theory, Chihara's constructibility the- ory, and Hellman's modal reconstruction of arithmetic. I intend to consider the best-developed of all nominalist reconstructions: the best anti-realist hard roads available to us. According to Field, it is possible to show that the mathematical objects we refer to in classical mechanics and Newtonian gravitational theory are dispensable. Roughly speaking, Field's argument proceeds as follows: if the physical consequences we can derive by using mathematics are the same as those we can derive from a body of pure physical assertions, then mathemat- ical objects are dispensable. As a result, the statements that existentially quantify over mathematical objects, such as `there are infinitely many prime Contents 7 numbers', turn out to be false. Under the assumption that mathematical objects do not exist, a large number of mathematical assertions turn out to be false. But according to Field, mathematics does not need to be true to be useful: it merely needs to be a device for making calculations easier. And if it is possible to extend Field's approach to all our best scientific theories, the indispensability argument will be disarmed. Chihara provides an extensive reconstruction of mathematics without ontological commitment to mathematical abstracta. In this regard, Chi- hara settles on a nominalistic interpretation of mathematics through the constructibility theory, namely a first-order language equipped with con- structibility quantifiers. Constructibility quantifiers are concrete tokens (marks on paper, or on screen, etc.) that explain the rules for building se- quences of concrete tokens. Such sequences are in turn employed to replace mathematical objects, formal relations, mathematical properties and so on. To put it another way, Chihara's goal is to find a surrogate of mathematical objects that is nominalistically acceptable. Moreover, Chihara makes use of the modal operator `it is possible to construct X', where X is conceived as a token located in space and time. Hellman develops a modal strategy that offers some analogies to Chi- hara's. According to Hellman, mathematics is about possible structures. For instance, Peano Arithmetic is the study of the possible structures (i.e. progressions or !-sequences) that satisfy Peano axioms. Hellman's modal structuralism assumes that mathematical structures are possible, but does not force any commitment to actual structures. In addition, there is no ontological commitment to the objects that occur in structures. Hellman also presents a truth-translation scheme according to which all ordinary mathematical statements can be translated into statements about possible structures. The translation scheme allows Hellman to recover arithmetic, real analysis, and set theory partially, given the hypothetical existence of suitable mathematical structures. For a long time, `hard-road' strategies attracted the attention of anti- Contents 8 Platonists.