Importance and Explanatory Relevance: The Case of Mathematical Explanations

A way to argue that something (e.g. mathematics, idealizations, moral properties, etc.) plays an explanatory role in science is by linking explanatory relevance with importance in the context of an explanation. The idea is deceptively simple: a part of an explanation is an explanatorily relevant part of that explanation if removing it affects the explanation either by destroying it or by diminishing its explanatory power, i.e. an important part (one that if removed affects the explanation) is an explanatorily relevant part. This can be very useful in many ontological debates. My aim in this paper is twofold. First of all, I will try to assess how this view on explanatory relevance can affect the recent ontological debate in the philosophy of mathematics—as I will argue, contrary to how it may appear at first glance, it does not help very much the mathematical realists. Second of all, I will show that there are big problems with it.     

Viciousness and the structure of reality

Given the centrality of arguments from vicious infinite regress to our philosophical reasoning, it is little wonder that they should also appear on the catalogue of arguments offered in defense of theses that pertain to the fundamental structure of reality. In particular, the metaphysical foundationalist will argue that, on pain of vicious infinite regress, there must be something fundamental. But why think that infinite regresses of grounds are vicious? I explore existing proposed accounts of viciousness cast in terms of contradictions, dependence, failed reductive theories and parsimony. I argue that no one of these accounts adequately captures the conditions under which an infinite regress—any infinite regress—is vicious as opposed to benign. In their place, I suggest an account of viciousness in terms of explanatory failure. If this account is correct, infinite grounding regresses are not necessarily vicious; and we must be much more careful employing such arguments to the conclusion that there has to be something fundamental.     

A Role for Mathematics in the Physical Sciences

Conflicting accounts of the role of mathematics in our physical theories can be traced to two principles. Mathematics appears to be both (1) theoretically indispensable, as we have no acceptable non‐mathematical versions of our theories, and (2) metaphysically dispensable, as mathematical entities, if they existed, would lack a relevant causal role in the physical world. I offer a new account of a role for mathematics in the physical sciences that emphasizes the epistemic benefits of having mathematics around when we do science. This account successfully reconciles theoretical indispensability and metaphysical dispensability and has important consequences for both advocates and critics of indispensability arguments for platonism about mathematics.     

LOT 2: The Language of Thought Revisited

A main thread of the debate over mathematical realism has come down to whether mathematics does explanatory work of its own in some of our best scientific explanations of empirical facts. Realists argue that it does; anti-realists argue that it doesn't. Part of this debate depends on how mathematics might be able to do explanatory work in an explanation. Everyone agrees that it's not enough that there merely be some mathematics in the explanation. Anti-realists claim there is nothing mathematics can do to make an explanation mathematical; realists think something can be done, but they are not clear about what that something is.

I argue that many of the examples of mathematical explanations of empirical facts in the literature can be accounted for in terms of Jackson and Pettit's [1990] notion of program explanation, and that mathematical realists can use the notion of program explanation to support their realism. This is exactly what has happened in a recent thread of the debate over moral realism (in this journal). I explain how the two debates are analogous and how moves that have been made in the moral realism debate can be made in the mathematical realism debate. However, I conclude that one can be a mathematical realist without having to be a moral realist.     

Laws and Models in a Theory of Idealization

I first give a brief summary of a critique of the traditional theories of approximation and idealization; and after identifying one of the major roles of idealization as detaching component processes or systems from their joints, a detailed analysis is given of idealized laws – which are discoverable and/or applicable – in such processes and systems (i.e., idealized model systems). Then, I argue that dispositional properties should be regarded as admissible properties for laws and that such an inclusion supplies the much needed connection between idealized models and the laws they `produce' or `accommodate'. And I then argue that idealized law-statements so produced or accommodated in the models may be either true simpliciter or true approximately, but the latter is not because of the idealizations involved. I argue that the kind of limiting-case idealizations that produce approximate truth is best regarded as approximation; and finally I compare my theory with some existing theories of laws of nature.We seem to trace [in KingLear] ... the tendency of imagination toanalyse and abstract, to decomposehuman nature into its constituentfactors, and then to construct beings in whomone or more of these factors isabsent or atrophied or only incipient     

Access Problems and explanatory overkill

I argue that recent attempts to deflect Access Problems for realism about a priori domains such as mathematics, logic, morality, and modality using arguments from evolution result in two kinds of explanatory overkill: (1) the Access Problem is eliminated for contentious domains, and (2) realist belief becomes viciously immune to arguments from dispensability, and to non-rebutting counter-arguments more generally.     

Idealizations,essential self-adjointness,and minimal model explanation in the Aharonov–Bohm effect

Two approaches to understanding the idealizations that arise in the Aharonov–Bohm (AB) effect are presented. It is argued that a common topological approach, which takes the non-simply connected electron configuration space to be an essential element in the explanation and understanding of the effect, is flawed. An alternative approach is outlined. Consequently, it is shown that the existence and uniqueness of self-adjoint extensions of symmetric operators in quantum mechanics have important implications for philosophical issues. Also, the alleged indispensable explanatory role of said idealizations is examined via a minimal model explanatory scheme. Last, the idealizations involved in the AB effect are placed in a wider philosophical context via a short survey of part of the literature on infinite and essential idealizations.     

General theories of explanation: buyer beware

We argue that there is no general theory of explanation that spans the sciences, mathematics, and ethics, etc. More specifically, there is no good reason to believe that substantive and domain-invariant constraints on explanatory information exist. Using Nickel (Noûs 44(2):305–328, 2010) as an exemplar of the contrary, generalist position, we first show that Nickel’s arguments rest on several ambiguities, and then show that even when these ambiguities are charitably corrected, Nickel’s defense of general theories of explanation is inadequate along several different dimensions. Specifically, we argue that Nickel’s argument has three fatal flaws. First, he has not provided any compelling illustrations of domain-invariant constraints on explanation. Second, in order to fend off the most vehement skeptics of domain-invariant theories of explanation, Nickel must beg all of the important questions. Third, Nickel’s examples of explanations from different domains with common explanatory structure rely on incorrect formulations of the explanations under consideration, circular justifications, and/or a mischaracterization of the position Nickel intends to critique. Given that the best and most elaborate defense of the generalist position fails in so many ways, we conclude that the standard practice in philosophy (and in philosophy of science in particular), which is to develop theories of explanation that are tailored to specific domains, still is justified. For those who want to buy into a more ambitious project: beware of the costs!     

Properties,laws, and worlds

Jonathan Schaffer argues against a necessary connection between properties and laws. He takes this to be a question of what possible worlds we ought to countenance in our best theories of modality, counterfactuals, etc. In doing so, he unfairly rigs the game in favor of contingentism. I argue that the necessitarian can resist Schaffer’s conclusion while accepting his key premise that our best theories of modality, counterfactuals, etc. require a very wide range of things called ‘possible worlds’. However, the necessitarian can and should insist that, in many cases, these worlds are not metaphysically possible. I will further argue that, having taken such a stance, the necessitarian has additional resources to respond to Schaffer’s other arguments against the view.     

Explaining with Models: The Role of Idealizations

Because they contain idealizations, scientific models are often considered to be misrepresentations of their target systems. An important question is therefore how models can explain the behaviours of these systems. Most of the answers to this question are representationalist in nature. Proponents of this view are generally committed to the claim that models are explanatory if they represent their target systems to some degree of accuracy; in other words, they try to determine the conditions under which idealizations can be made without jeopardizing the representational function of models. In this article, we first outline several forms of this representationalist view. We then argue that this view, in each of these forms, omits an important role of idealizations: that of facilitating the identification of the explanatory components within a model. Via examination of a case study from contemporary astrophysics, we show that one way in which idealizations can do this is by creating a comparison case that serves to highlight the relevant features of the target system.     

Infinite Reasoning

Our relationship to the infinite is controversial. But it is widely agreed that our powers of reasoning are finite. I disagree with this consensus; I think that we can, and perhaps do, engage in infinite reasoning. Many think it is just obvious that we can't reason infinitely. This is mistaken. Infinite reasoning does not require constructing infinitely long proofs, nor would it gift us with non-recursive mental powers. To reason infinitely we only need an ability to perform infinite inferences. I argue that we have this ability. My argument looks to our best current theories of inference and considers examples of apparent infinite reasoning. My position is controversial, but if I'm right, our theories of truth, mathematics, and beyond could be transformed. And even if I'm wrong, a more careful consideration of infinite reasoning can only deepen our understanding of thinking and reasoning.     

In Defence of Metametasemantics

In the paper I defend the idea of metametasemantics against the arguments recently presented by Ori Simchen (2017). Simchen attacks the view, according to which metametasemantics incorporating all possible metasemantic accounts is necessary to protect the metasemantic theories from the notorious problem of inscrutability of reference (see Sider 2011). Simchen claims that if metametasemantics is allowed it ‘absorbs’ metasemantic theories to the extent that it diminishes their explanatory value. Furthermore, in this way Simchen sets up two main metasemantic paradigms i.e. productivism (roughly speaking: speaker’s metasemantics) and interpretationism (audience’s metasemantics) as the rival theories inevitably excluding each other. I endeavour to undermine Simchen’s point by demonstrating that his argumentation mixes up deflationary reading of the predicate ‘is true’ with its substantial reading. Consequently, I demonstrate that accepting metametasemantics does not diminish explanatory value of various metasemantic theories and thus that there is no good reason to forbid metametasemantics. I also argue that even if we ignore the above-mentioned confusion in Simchen’s reasoning, his arguments still fail when considering various problems with the notion of diminishment of explanatory value and because the analogy that his arguments are based on is fairly weak. Eventually, I conclude that metametasemantics does not pose any danger to metasemantics and that it provides a solid ground for developing a theory that benefits from both productivism and interpretationism.

     

Irrational desires

Conclusion I believe that the attempts discussed above fail to show that logically satisfiable basic desires can be rationally impotent. Obviously, this does not entail that they cannot be. Nevertheless, I think it is reasonable to accept a Neo-Humean view. Such acceptence need not be based on burden of proof arguments, about which there is well-grounded skepticism. I prefer instead to base it on a burden of introduction argument; because of the initial plausibility of the Neo-Humean view, critics carry the burden of introducing theories that entail that basic desires can be non-instrumentally irrational (in the relevant sense). Once such theories are introduced, the philosophical court can rule without imposing a burden of proof. I have tried to establish that three recently introduced theories in fact give us no grounds for rejecting Neo-Humeanism.     

Infinite lies and explanatory ties: idealization in phase transitions

Synthese - Infinite idealizations appear in our best scientific explanations of phase transitions. This is thought by some to be paradoxical. In this paper I connect the existing literature on the...     

From Standard Scientific Realism and Structural Realism to Best Current Theory Realism

I defend a realist commitment to the truth of our most empirically successful current scientific theories—on the ground that it provides the best explanation of their success and the success of their falsified predecessors. I argue that this Best Current Theory Realism (BCTR) is superior to preservative realism (PR) and the structural realism (SR). I show that PR and SR rest on the implausible assumption that the success of outdated theories requires the realist to hold that these theories possessed truthful components. PR is undone by the fact that past theories succeeded even though their ontological claims about unobservables are false. SR backpeddles to argue that the realist is only committed to the truth about the structure of relations implied by the outdated theory, in order to explain its success. I argue that the structural component of theories is too bare-bones thin to explain the predictive/explanatory success of outdated theories. I conclude that BCTR can meet these objections to PR and SR, and also overcome the pessimistic meta-induction.     

Consciousness-Dependence and the Explanatory Gap

Contrary to certain rumours, the mind-body problem is alive and well. So argues Joseph Levine in Purple Haze: The Puzzle of Consciousness . The main argument is simple enough. Considerations of causal efficacy require us to accept that subjective experiential, or 'phenomenal', properties are realized in basic non-mental, probably physical properties. But no amount of knowledge of those physical properties will allow us conclusively to deduce facts about the existence and nature of phenomenal properties. This failure of deducibility constitutes an explanatory problem - an explanatory gap - but does not imply the existence of immaterial mental properties. Levine introduced this notion of the explanatory gap almost two decades ago. Purple Haze allows Levine to situate the explanatory gap in a broader philosophical context. He engages with those who hold that the explanatory gap is best understood as implying anti-materialist metaphysical conclusions. But he also seeks to distance himself from contemporary naturalistic philosophical theorizing about consciousness by arguing that reductive and eliminative theories of consciousness all fail. Levine's work is best seen as an attempt to firmly establish a definite status for the mind-body problem, i.e. that the mind-body problem is a real, substantive epistemological problem but emphatically not a metaphysical one. Because Levine's work is tightly focused upon contemporary Anglophone analytic philosophy of mind, there is little discussion of the broader conceptual background to the mind-body problem. My aim here is to place Levine's work in a broader conceptual context. In particular, I focus on the relationship between consciousness and intentionality in the belief that doing so will allow us better to understand and evaluate Levine's arguments and their place in contemporary theorizing about mentality and consciousness.     

Strict Finitism and the Happy Sorites

Call an argument a ‘happy sorites’ if it is a sorites argument with true premises and a false conclusion. It is a striking fact that although most philosophers working on the sorites paradox find it at prima facie highly compelling that the premises of the sorites paradox are true and its conclusion false, few (if any) of the standard theories on the issue ultimately allow for happy sorites arguments. There is one philosophical view, however, that appears to allow for at least some happy sorites arguments: strict finitism in the philosophy of mathematics. My aim in this paper is to explore to what extent this appearance is accurate. As we shall see, this question is far from trivial. In particular, I will discuss two arguments that threaten to show that strict finitism cannot consistently accept happy sorites arguments, but I will argue that (given reasonable assumptions on strict finitistic logic) these arguments can ultimately be avoided, and the view can indeed allow for happy sorites arguments.     

Arithmetic,set theory,reduction and explanation

Philosophers of science since Nagel have been interested in the links between intertheoretic reduction and explanation, understanding and other forms of epistemic progress. Although intertheoretic reduction is widely agreed to occur in pure mathematics as well as empirical science, the relationship between reduction and explanation in the mathematical setting has rarely been investigated in a similarly serious way. This paper examines an important and well-known case: the reduction of arithmetic to set theory. I claim that the reduction is unexplanatory. In defense of this claim, I offer some evidence from mathematical practice, and I respond to contrary suggestions due to Steinhart, Maddy, Kitcher and Quine. I then show how, even if set-theoretic reductions are generally not explanatory, set theory can nevertheless serve as a legitimate and successful foundation for mathematics. Finally, some implications of my thesis for philosophy of mathematics and philosophy of science are discussed. In particular, I suggest that some reductions in mathematics are probably explanatory, and I propose that differing standards of theory acceptance might account for the apparent lack of unexplanatory reductions in the empirical sciences.     

Approximations, Idealizations, and Models in Statistical Mechanics

In this paper, a criticism of the traditional theories of approximation and idealization is given as a summary of previous works. After identifying the real purpose and measure of idealization in the practice of science, it is argued that the best way to characterize idealization is not to formulate a logical model – something analogous to Hempel's D-N model for explanation – but to study its different guises in the praxis of science. A case study of it is then made in thermostatistical physics. After a brief sketch of the theories for phase transitions and critical phenomena, I examine the various idealizations that go into the making of models at three difference levels. The intended result is to induce a deeper appreciation of the complexity and fruitfulness of idealization in the praxis of model-building, not to give an abstract theory of it.