Mark Colyvan (2010) raises two problems for ‘easy road’ nominalism about mathematical objects. The first is that a theory’s mathematical commitments may run too deep to permit the extraction of nominalistic content. Taking the math out is, or could be, like taking the hobbits out of Lord of the Rings. I agree with the ‘could be’, but not (or not yet) the ‘is’. A notion of logical subtraction is developed that supports the possibility, questioned by Colyvan, of bracketing a theory’s (...) mathematical aspects to obtain, as remainder, what it says ‘mathematics aside’. The other problem concerns explanation. Several grades of mathematical involvement in physical explanation are distinguished, by analogy with Quine’s three grades of modal involvement. The first two grades plausibly obtain, but they do not require mathematical objects. The third grade is likelier to require mathematical objects. But it is not clear from Colyvan’s example that the third grade really obtains. (shrink)

I respond to the frequent objection that structural realism fails to sharply state an alternative to the standard predicate-logic, object / property / relation, way of doing metaphysics. The approach I propose is based on what I call a ‘math-first’ approach to physical theories (close to the so-called ‘semantic view of theories') where the content of a physical theory is to be understood primarily in terms of its mathematical structure and the representational relations it bears to physical systems, rather than (...) as a collection of sentences that attempt to make true claims about those systems (a ‘language-first’ approach). I argue that adopting the math-first approach already amounts to a form of structural realism, and that the choice between epistemic and ontic versions of structural realism is then a choice between a language-first and math-first view of metaphysics; I then explore the status of objects (and properties and relations) in fundamental and non-fundamental physics for both versions of math-first structural realism. (shrink)

Philosophers often aim to demonstrate that the things we ordinarily think and say can be reconciled with our considered beliefs about the world. To this end, many philosophers try to paraphrase ordinary language claims by finding equivalent sentences that are less misleading. For instance, though we know that there is no British family that is the average one, we want to say that the average British family has 1.8 children, and we might do that by paraphrasing this claim as: there (...) are nearly twice as many British children as British families. This article aims to introduce and distinguish different paraphrase strategies, sketch their role in metaphysical debates and highlight some crucial questions that proponents of such strategies face. (shrink)

Humean Supervenience is the view that (a) there are a plurality of fundamental beings, (b) there are no inexplicable constraints on modal space, and hence the fundamental nature of each such being is independent of those of all the rest and of the fundamental relations in which it stands to the rest, (c) the fundamental beings stand in no fundamental causal or nomic relations, and hence (d) the distribution of any causal or nomic relations in which they do stand globally (...) supervenes on their fundamental natures and the non‐nomic, non‐causal fundamental relations in which they stand. If Humean Supervenience is true, then as A.J. Ayer put it, it's just one damn thing after another. Radical Pluralism is the view that Humean Supervenience is true, and, moreover, that none of the fundamental beings stands in any fundamental relations at all. If Radical Pluralism is true, then, as William James puts it, the world's pieces are held together by nothing more than conjunction: it's just one damn thing and another. I argue that Radical Pluralism is very likely true, conditional upon Humean Supervenience. Any philosopher pluralist enough to be Humean ought to be radically pluralistic. (shrink)

I will contrast two conceptions of the nature of mathematical objects: the conception of mathematical objects as preconceived objects, and heavy duty platonism. I will argue that friends of the indispensability argument are committed to some metaphysical theses and that one promising way to motivate such theses is to adopt heavy duty platonism. On the other hand, combining the indispensability argument with the conception of mathematical objects as preconceived objects yields an unstable position. The conclusion is that the metaphysical commitments (...) of the indispensability argument should be carefully scrutinized. (shrink)

Mathematical platonism is the view that abstract mathematical objects exist. Ontological pluralism is the view that there are many modes of existence. This paper examines the prospects for...

Mathematical platonism is the view that abstract mathematical objects exist. Ontological pluralism is the view that there are many modes of existence. This paper examines the prospects for plural platonism, the view that results from combining mathematical platonism and ontological pluralism. I will argue that some forms of platonism are in harmony with ontological pluralism, while other forms of platonism are in tension with it. This shows that there are some interesting connections between the platonism–antiplatonism dispute and recent debates over (...) ontological pluralism. (shrink)

According to the doctrine of Super-Humeanism, the world’s mosaic consists only of permanent matter points and changing spatial relations, while all the other entities and features figuring in scientific theories are nomological parameters, whose role is merely to build the best law system. In this paper, I develop an argument against Super-Humeanism by pointing out that it is vulnerable to and does not have the resources to solve the well-known problem of immanent comparisons. Firstly, I show that it cannot endorse (...) a fundamentalist solution à la Lewis, since its two pillars—a minimalist ontology and a best system account of lawhood—would generate, together, a tedious problem of internal coherence. Secondly, I consider anti-fundamentalist strategies, proposed within Humeanism, and find them inapplicable to the Super-Humean doctrine. The concern is that, since it is impossible to choose the best law system within Super-Humeanism, this doctrine may be charged with incoherence. (shrink)

Conversational exculpature is a pragmatic process whereby information is subtracted from, rather than added to, what the speaker literally says. This pragmatic content subtraction explains why we can say “Rob is six feet tall” without implying that Rob is between 5'0.99" and 6'0.01" tall, and why we can say “Ellen has a hat like the one Sherlock Holmes always wears” without implying Holmes exists or has a hat. This article presents a simple formalism for understanding this pragmatic mechanism, specifying how, (...) in context, the result of such subtractions is determined. And it shows how the resulting theory of conversational exculpature accounts for a varied range of linguistic phenomena. A distinctive feature of the approach is the crucial role played by the question under discussion in determining the result of a given exculpature. (shrink)

Standard definitions of causal closure focus on where the causes in question are. In this paper, the focus is changed to where they are not. Causal closure is linked to the principle that no cause of another universe causes an event in a particular universe. This view permits the one universe to be affected by the other via an interface. An interface between universes can be seen as a domain that violates the suggested account of causal closure, suggesting a view (...) in which universes are causally closed whereas interfaces are not. On this basis, universes are not affected by other universes directly but rather indirectly. (shrink)

I explore some ways in which one might base an account of the fundamental metaphysics of geometry on the mathematical theory of Linear Structures recently developed by Tim Maudlin (2010). Having considered some of the challenges facing this approach, Idevelop an alternative approach, according to which the fundamental ontology includes concrete entities structurally isomorphic to functions from space-time points to real numbers.

Do numbers exist? Do properties? Do possible worlds? Do fictional characters? Many metaphysicians spend time and effort trying to answer these and other questions about the existence of various entities. These inquiries have recently encountered opposition: a group of philosophers, drawing inspiration from Aristotle, have argued that many or all of the existence questions debated by metaphysicians can be answered trivially, and so are not worth debating. Our task is to defend existence questions from the neo-Aristotelians' attacks.

Inspired by Cantor's Theorem (CT), orthodoxy takes infinities to come in different sizes. The orthodox view has had enormous influence in mathematics, philosophy, and science. We will defend the contrary view---Countablism---according to which, necessarily, every infinite collection (set or plurality) is countable. We first argue that the potentialist or modal strategy for treating Russell's Paradox, first proposed by Parsons (2000) and developed by Linnebo (2010, 2013) and Linnebo and Shapiro (2019), should also be applied to CT, in a way that (...) vindicates Countabilism. Our discussion dovetails with recent independently developed treatments of CT in Meadows (2015), Pruss (2020), and Scambler (2021), aimed at establishing the mathematical viability, and therefore epistemic possibility, of Countabilism. Unlike these authors, our goal isn't to vindicate the mathematical underpinnings of Countabilism. Rather, we aim to argue that, given that Countabilism is mathematically viable, Countabilism should moreover be regarded as true. After clarifying the modal content of Countabilism, we canvas some of Countabilism's many positive implications, including that Countabilism provides the best account of the pervasive independence phenomena in set theory, and that Countabilism has the power to defuse several persistent puzzles and paradoxes found in physics and metaphysics. We conclude that in light of its theoretical and explanatory advantages, Countabilism is more likely true than not. (shrink)

Easy-road mathematical fictionalists grant for the sake of argument that quantification over mathematical entities is indispensable to some of our best scientific theories and explanations. Even so they maintain we can accept those theories and explanations, without believing their mathematical components, provided we believe the concrete world is intrinsically as it needs to be for those components to be true. Those I refer to as “mathematical surrealists” by contrast appeal to facts about the intrinsic character of the concrete world, not (...) to explain why our best mathematically imbued scientific theories and explanations are acceptable in spite of having false components, but in order to replace those theories and explanations with parasitic, nominalistically acceptable alternatives. I argue that easy-road fictionalism is viable only if mathematical surrealism is and that the latter constitutes a superior nominalist strategy. Two advantages of mathematical surrealism are that it neither begs the question concerning the explanatory role of mathematics in science nor requires rejecting the cogency of inference to the best explanation. (shrink)

An increasing amount of contemporary philosophy of mathematics posits, and theorizes in terms of special kinds of mathematical modality. The goal of this paper is to bring recent work on higher-order metaphysics to bear on the investigation of these modalities. The main focus of the paper will be views that posit mathematical contingency or indeterminacy about statements that concern the ‘width’ of the set theoretic universe, such as Cantor’s continuum hypothesis. Within a higher-order framework I show that contingency about the (...) width of the set-theoretic universe refutes two orthodoxies concerning the structure of modal reality: the view that the broadest necessity has a logic of S5, and the ‘Leibniz biconditionals’ stating that what is possible, in the broadest sense of _possible_, is what is true in some possible world. Nonetheless, I suggest that the underlying picture of modal set-theory is coherent and has attractions. (shrink)

We should see the debate over the existence of spacetime as a debate about the fundamentality of spatiotemporal structure to the physical world. This is a non-traditional conception of the debate, which captures the spirit of the traditional one. At the same time, it clarifies the point of contention between opposing views and offsets worries that the dispute is stagnant or non-substantive. It also unearths a novel argument for substantivalism, given current physics. Even so, that conclusion can be overridden by (...) future physics. I conclude that this debate is a substantive one, which the substantivalist is currently winning. (shrink)

In a quantum universe with a strong arrow of time, it is standard to postulate that the initial wave function started in a particular macrostate---the special low-entropy macrostate selected by the Past Hypothesis. Moreover, there is an additional postulate about statistical mechanical probabilities according to which the initial wave function is a ''typical'' choice in the macrostate. Together, they support a probabilistic version of the Second Law of Thermodynamics: typical initial wave functions will increase in entropy. Hence, there are two (...) sources of randomness in such a universe: the quantum-mechanical probabilities of the Born rule and the statistical mechanical probabilities of the Statistical Postulate. I propose a new way to understand time's arrow in a quantum universe. It is based on what I call the Thermodynamic Theories of Quantum Mechanics. According to this perspective, there is a natural choice for the initial quantum state of the universe, which is given by not a wave function but by a density matrix. The density matrix plays a microscopic role: it appears in the fundamental dynamical equations of those theories. The density matrix also plays a macroscopic / thermodynamic role: it is exactly the projection operator onto the Past Hypothesis subspace. Thus, given an initial subspace, we obtain a unique choice of the initial density matrix. I call this property "the conditional uniqueness" of the initial quantum state. The conditional uniqueness provides a new and general strategy to eliminate statistical mechanical probabilities in the fundamental physical theories, by which we can reduce the two sources of randomness to only the quantum mechanical one. I also explore the idea of an absolutely unique initial quantum state, in a way that might realize Penrose's idea of a strongly deterministic universe. (shrink)