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Mathematics, Time, and Confirmation Mathematics, Time, and Confirmation Ulrich Meyer Doctor of Philosophy in Mathematics University of Cambridge, 1995 Submitted to the Department of Linguistics and Philosophy in partial fulfil- ment of the requirements for the degree of DOCTOR OF PHILOSOPHY at the Massachusetts Institute of Technology. September 2001 Author: ................................. .. .... Department of Linguistics aKPhilosophy August 10, 2001 Certified by: .......................... Robert C. Stalnaker Professor of Philosophy Thesis Supervisor Accepted by: .............. Vann McGee Professor of Philosophy Chair, Committee on Graduate Students @Ulrich Meyer 2001. All rights reserved. The author hereby grants permission to the Massachusetts Institute of Technology to reproduce and distribute publicly paper and electronic copies of this thesis in whole or in part. MASSACHUSETTS INSTITUTE OF TECHNOLOGY OCT 11 2001 ARCMly LIBRARIES Mathematics, Time, and Confirmation by Ulrich Meyer Submitted to The Department of Linguistics and Philosophy on 10 August 2001 in partial fulfilment of the requirements for the degree of Doctor of Philosophy ABSTRACT This dissertation discusses two issues about abstract objects: their role in scientific theories, and their relation to time. Chapter 1, "Why Apply Mathematics?" argues that scientific theories are not about the mathematics that is applied in them, and defends this thesis against the Quine-Putnam Indispensability Argument. Chapter 2, "Scientific Ontology," is a critical study of W. V. Quine's claim that metaphysics and mathematics are epistemologically on a par with nat- ural science. It is argued that Quine's view relies on a unacceptable account of empirical confirmation. Chapter 3, "Prior and the Platonist," demonstrates the incompatibility of two popular views about time: the "Platonist" thesis that some objects exist "outside" time, and A. N. Prior's proposal for treating tense on the model of modality. Chapter 4, "What has Eternity Ever Done for You?" argues against the widely held view that abstract objects exist eternally ("outside" time), and presents a defense of the rival view that they exist sempiternally (at all times). Thesis Supervisor: Robert C. Stalnaker Title: Professor of Philosophy Acknowledgements I would like to thank all the people who.have made MIT such a great place for doing philosophy: Sylvain Bromberger, Alex Byrne, Richard Cartwright, Joshua Cohen, Tyler Doggett, An- drew Egan, Iris Einheuser, Matti Eklund, Adam Elga, David Etlin, Roxanne Fay, Joshua Flaherty, Alex Friedman, Michael Glanzberg, Ned Hall, Elizabeth Harman, Sally Haslanger, James John, Ishani Maitra, Vann McGee, Sarah McGrath, Anthony Newman, Dean Pettit, Adina Roskies, Agustin Rayo, Carolina Sartorio, Irving Singer, Neema Sofaer, Robert Stalnaker, Asta Sveinsdottir, Judith Jarvis Thomson, Brian Trimmer, Gabriel Uzquiano, Ralph Wedgwood, and Stephen Yablo. 3 Contents Abstract 2 1 Why Apply Mathematics? 6 1 Radical Monism is Anti-Scientific .......... 9 2 The Indispensability Argument ........... 12 3 A Cardinality Argument ..... ........ 16 4 Indexing Properties .............. .. 18 5 A Strategy for the Dualist ............. 27 6 Parasitic Dualism .................. 29 7 Inferential Structure ................ 36 2 Scientific Ontology 42 1 Curing Empiricism . ................ 42 2 Duhem's Observation ................ 47 3 Holism .... ... .... ... .... ... ... 50 4 Ayer's Disease .................... 53 5 The Loop Problem ................. 60 6 Retreat to Safety? .................. 63 7 Living with Triviality .. .... ..... .... 68 8 Prediction and Confirmation ............. 73 4 Contents 5 3 Prior and the Platonist 78 1 Eternity and Sempiternity .. ...... .. 78 2 Varieties of Tense Logic . ...... .. 80 3 The Metaphysics of Eternity . .. ..... 85 4 Axioms for Eternity . .. ... ... ... .. 87 5 The Problem of Change . ... ... ... 91 4 What has Eternity Ever Done for You? 94 1 God and Time. ..................... 95 2 Can Chairs be Eternal? ............... 97 3 The Tenseless Present ................ 99 4 Arguing for Eternity ................. 104 5 The Argument from Necessity ............ 106 6 Simplicity and Temporal Parts ............ 109 7 Changelessness and the Flow of Time ....... 113 8 Eternity and the Atheist .............. 115 Bibliography 129 1 Why Apply Mathematics? In modern physical theories, the use of mathematics is both per- vasive and seemingly unavoidable. Even the most basic princi- ples of quantum mechanics and general relativity are incompre- hensible to anyone without detailed knowledge of higher math- ematics. Though a familiar feature of science, this is puzzling. Mathematics is usually thought to be about abstract mathemat- ical objects (numbers, functions, sets, etc.), and these objects do not participate in the causal processes physics is concerned with. The physical realm is causally closed and mathematical objects form no part of it. Yet if their subject matters are both disjoint and non-interacting, how can mathematics be relevant to physics? What business do numbers and functions have ap- pearing in physical theories? We used to have a very nice answer to these questions: logi- cism. If mathematics were part of logic, as the logicist contends, 6 Why Apply Mathematics? 7 then there would be no more of a mystery about what math- ematical objects are doing in scientific theories than there is about why they employ conjunction, or quantification. The ap- peal to mathematics would just be part of the logical form of these theories. Alas, logicism never fully recovered from Frege's debacle with Russell's paradox, and most philosophers nowadays think that logicism is based on a mistaken view of logicality. Logic is topic-neutral and does not have ontological commit- ments, as mathematics clearly does. The prevailing response to the demise of logicism has been to opt for W. V. Quine's radical monism.1 While the logicist regards mathematics as part of the logical form of a scientific theory, the radical monist takes them to be part of their content. The monist denies that there is a principled difference between the theoretical roles of mathematical and physical objects. On his view, numbers and functions are indispensable in science for exactly the same reason electrons and quarks are: they are part of what scientific theories are about. But that reply seems to be (almost) as bad as the logicist account. Physical objects causally interact with other physical objects, but they just don't interact with numbers or functions. It is obvious that there is a distinction to be made here, and we should be suspicious of any philosophical view that denies this. I think we can do better than monism. The present paper IQuine 1953c: 45; Quine 1969b: 97; Quine 1981d: 150. Why Apply Mathematics? 8 is a defence of what I call commonsensical dualism. Unlike the monist, the dualist wants to keep the subject matters of mathe- matics and physics separate. He distinguishes a theory's physi- cal content from the mathematical form in which it is conveyed. But he is neither like the logicist, who claims that mathemati- cal form is logical form, nor is he like the eliminative nominalist, who claims that scientific theories could be formulated without any appeal to mathematical objects. 2 His view is that math- ematics functions as a "representational aid" that boosts the ideology of scientific theories without inflating their ontological commitments. 3 The main challenge for the dualist is to show that the dis- tinction between mathematical form and physical content that he wants to make is tenable. I claim that it is, but the dualist account of applied mathematics that I will present here does not come for free. It requires more generous philosophical resources than radical monism does. But we will see that what we get in return is an account of applied mathematics that is superior to the monist's view. 4 21 will say more about eliminative nominalism in Section 2. 3 Here I am using 'ideology' in the idiosyncratic (but very useful) sense of Quine 1951a. 4There are other views about applied mathematics that I am ignoring here: neo-logicist, nominalist, and structuralist accounts are the most notable amongst them. I think dualism is better than either of these rivals, but I don't have the space to argue for this claim here. Why Apply Mathematics? 9 1 RADICAL MONISM IS ANTI-SCIENTIFIC Before I sketch what I regard as the correct, dualist, account of applied mathematics, let me begin by explaining in more detail why I think that radical monism is unsatisfactory. My main objection to monism is that it is an anti-scientific doctrine: sci- entists are dualists.P This commitment to dualism is shown by the fact that scientists take their theories to be invariant under certain replacements of the mathematics employed in them. But if it is possible for two formulations of the same theory to appeal to different parts of the ontology of mathematics, then monism cannot be right. A prominent example from the history of physics is Joseph Louis Lagrange's 1788 treatise on mechanics, M6canique: ana- lytique. Curiously, the theory of bodily motion presented in this book found immediate acceptance amongst scientists even though not a single experiment was performed to test its empir- ical claims. This looks like a lapse of scientific standards, but scientists have a ready explanation. On their view, Lagrange did not in fact present a new theory, but merely gave a more con- venient formulation of the same theory that Isaac Newton had proposed in
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