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The Epistemological Significance of Reducing the Relativity Theories to First-order Zermelo-Fraenkel Set Theory Michèle Friend, Philosophy, George Washington University First International Conference on Logic and Relativity: Honouring István Németis 70 th Birthday Budapest, September, 2012 0 First International Conference on Logic and Relativity, Budapest Sept. 2012

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Three elements I shall bring together in this paper: 1.the project of Andréka, Madarász, Németi, Székely and others, represented in (Andréka et. al. 2002). 2.Molininis philosophical work on the nature of mathematical explanations in science (Molinini 2011). 3.my pluralist approach to mathematics (Friend forthcoming). 1 First International Conference on Logic and Relativity, Budapest Sept. 2012

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1. The Relativity Theory Project The researchers involved in the logical relativity theory project give genuine mathematical explanations for physical phenomena and for two very important theories in physics: the theories of special and general relativity. First International Conference on Logic and Relativity, Budapest Sept. 2012 2

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Philosophical way of thinking about the project: Andréka et. al. reduce the relativity theories to first-order ZF. The conceptual reverse of the reduction is that they give an explanation of the relativity theories in terms of ZF. One question is: what is explained by what? Or, what is the nature of the reduction/ explanation? To examine the nature we are after the philosophical characteristics of the explicans (the explanation) and the explicandum (the thing explained). The other question is: What is the significance of the explanation? To answer this question we want to discuss the philosophical context of the explicans. In section 2, I answer the first question. In section 3, I answer the second. First International Conference on Logic and Relativity, Budapest Sept. 2012 3

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2. The Nature of the Logical Explanation of the Relativity Theories Molinini takes a pluralist view of mathematical explanations of physical phenomena (henceforth: MEPP). This runs against Van Fraassen, Friedman (1974), Kitcher (1981) and Steiner (1978) who take the more common monist view. The monist view is that there has to be a single theory, or single model, of MEPPs. First International Conference on Logic and Relativity, Budapest Sept. 2012 4

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Pluralist MEPPs A pluralist in mathematical explanation thinks that: What counts as a good explanation can vary from case to case, and we cannot design a single model able to capture all of these instances. (Molinini 2010, 16). For Molinini, what makes cases different are: (a)the intellectual tools (b)the conceptual resources provided by different mathematical theories. An intellectual tool is: an ability to reason while used in the practice of explaining. (Molinini 2010, 352). A conceptual resource is a concept which permits the use of our intellectual tools in a particular situation. (Molinini 2010, 352). Conceptual resources in mathematics give us mathematical concepts which allow us to analyse or see a physical situation in a certain way. We then use mathematics, as a tool to reason over that situation. First International Conference on Logic and Relativity, Budapest Sept. 2012 5

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Explanations in science are acts of communication. The more basic the concepts, the more immediate they are. A logical axiom is an immediate judgment, in the sense that it cannot be further justified by recourse to the scientific or mathematical theory. Instead, if we require further justification, then it will take another form – in this case in terms of other mathematical theories or by reference to philosophical considerations and arguments. Thus, we can conclude that, under Molininis conceptions of explanation in science, the logical relativity theory project is a perfect example of a mathematical explanation for physical phenomena. First International Conference on Logic and Relativity, Budapest Sept. 2012 6

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3. The Significance of Explanation in ZF for the Mathematical Pluralist ZF is central to mathematics for a community of logicians and mathematicians, but not in a foundational way. By central I mean that it is a lingua franca, or a reference point. For this community, they find it reassuring when they can reduce their theory or their area of mathematics to ZF, or they compare it to ZF by means of an equi-consistency proof, or they indicate what is added to or subtracted from ZF to develop their theory. First International Conference on Logic and Relativity, Budapest Sept. 2012 7

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ZF receives a lot of confirmation in the form of crosschecking against other theories. The confirmation gives the theory a sense of objectivity and robustness. This is because the theory exercises cognitive command, in Wrights (1992) sense of the term. That is, it is not a subjective, or individual, matter whether 3 is a subset of 8. We prove this, and proofs are public displays of reasoning. This makes ZF internally robust and objective, that is, each theorem follows from the axioms. So, part of the significance is the enjoyed objectivity, brought by proofs subjected to scrutiny. These checks are also types of explanation. They are re- conceptualisations. When we can perceive the same object or theory from different angles, through the lens of different theories, we have another type of objectivity, what Wright calls wide cosmological role. First International Conference on Logic and Relativity, Budapest Sept. 2012 8

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Conclusion 1. The philosophical literature on the topic of mathematical explanations for physical phenomena has not (yet) engaged the logical relativity theory project. Instead the contributors rest content with isolated scientific phenomena in which the mathematics in the explanation seems to be indispensable. But said literature should engage this project because Andréka, Madarász, Németi, (2002) Székely (2012) give a mathematical explanation for whole physical theories, not just isolated phenomena. First International Conference on Logic and Relativity, Budapest Sept. 2012 9

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2. The epistemological significance is that the conceptual resource provided by ZF concerns the sorts of questions being asked. The questions are logicians questions. Moreover, the explanations are mathematical proofs. Thus, the explanations reach beyond the scientific community out to the mathematical community. First International Conference on Logic and Relativity, Budapest Sept. 2012 10

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3. For a pluralist, ZF is central to mathematics, but not foundational. Why does this pluralist approach to foundations in mathematics make the case for the significance of the logical relativity theory project more interesting than if we were foundationalists about mathematics? One reason is that because ZF is a lingua franca, and because it is tested against other theories, the spread of the results of the logical relativity theory project includes all of mathematics, as it is practiced today (beyond the bounds of ZF). So the conceptual resource includes all of mathematics, but through the lens of the language of ZF. An excursus outside ZF is reasonable under a pluralist conception of mathematics. The second reason is that ZF is a perfect tool for reasoning. This makes the logical relativity theory project objective in the sense that the project is subject to logical correction. This is related to three ideas in the literature, (i) one is Wrights ideas of cognitive command (we use and follow reasoning), (ii) width of cosmological role (that the ideas reach beyond the particular theory) and (iii) is the idea of error and logical justification. First International Conference on Logic and Relativity, Budapest Sept. 2012 11

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For a pluralist, like me, even logical justification in the form of a deductive proof is not an end. It is an invitation to investigate further and more deeply. This further investigation is already being carried out in the logical relativity theory project when they argue for their choice of mathematical theory (Andréka et. al. 2002, 1245ff). Logical justification is an invitation from the highest and purest ranks of thinking, where only our reasoning can guide us. For this reason, the tool for reasoning, ZF, reaches further than the mathematical theory. It invites an interplay between logic and philosophy, and we only develop our understanding of the relativity theories by extending our explanations and justification, not only in the direction of mathematics and logic, but beyond these to philosophy. First International Conference on Logic and Relativity, Budapest Sept. 2012 12

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