In this paper we discuss three interrelated questions. First: is explanation in mathematics a topic that philosophers of mathematics can legitimately investigate? Second: are the specific aims that philosophers of mathematical explanation set themselves legitimate? Finally: are the models of explanation developed by philosophers of science useful tools for philosophers of mathematical explanation? We argue that the answer to all these questions is positive. Our views are completely opposite to the views that Mark Zelcer has put forward recently. Throughout this (...) paper, we show why Zelcer’s arguments fail. (shrink)

By a macro-level feature, I understand any feature that supervenes on, and is thus realized in, lower-level features. Recent discussions by Kim have suggested that such features cannot be causally relevant insofar as they are not classically reducible to lower-level features. This seems to render macro-level features causally irrelevant. I defend the causal relevance of some such features. Such features have been thought causally relevant in many examples that have underpinned philosophical work on causality. Additionally, in certain typical biological cases, (...) we conceive of causally relevant features at various compatible levels of analysis. When elaborated, these points make a strong prima facie case for macro-level causal relevance. However, we might abandon both the philosophical guideposts and the corresponding explanatory practice in the special sciences were we convinced that no reflective philosophical account could provide for the causal relevance there supposed. I show that such drastic measures are not necessary, for we can make sense of macro-level causal relevance by drawing on Paul Humphreys' recent work in ways suggested by the concrete examples considered here. (shrink)

This paper characterizes the increase in ‘scientific understanding’ that resulted from the Ingold Revolution in organic chemistry. By describing both the sorts of explanations facilitated by Ingold’s Revolution and the sense in which organic chemistry was ‘unified’ by adopting these approaches to explanation, one can appreciate how this revolution led to a dramatic qualitative improvement in organic chemists’ understanding of the phenomena that they study. The explanatory unification responsible for this transformation in organic chemistry is contrasted with contemporary philosophical accounts (...) of unification and its relationship to both scientific understanding and explanation. (shrink)

The goal of this paper is to present and defend an empiricist, neo-Hempelian account of scientific explanation as ampliative, specialized embedding. The proposal aims to preserve what I take to be the core of Hempel’s empiricist account, by weakening it in some respects and strengthening it in others, introducing two new conditions that solve most of Hempel’s problems without abandoning his empiricist strictures. According to this proposal, to explain a phenomenon is to make it expectable by introducing new conceptual/ontological machinery (...) and using special, and non-ad hoc, non-accidental regularities. The new conditions are elaborated making essential use of two central structuralist ideas, namely T-theoreticity and specialization. I first introduce and qualify the project, then present the new account in detail and assess it vis-à-vis its rivals, and finally discuss some possible objections, concluding that the account fares better than its monistic rivals and well enough to qualify as a promising neo-Hempelian account. Even for those unpersuaded by its monistic goals, it has the merit of calling attention to two new necessary conditions not explicitly emphasized thus far and showing how they serve to answer many of the criticisms addressed against Hempel’s account. (shrink)

The aim of this dissertation is to clarify the debate over the explanation of quantum speedup and to submit, for the reader's consideration, a tentative resolution to it. In particular, I argue, in this dissertation, that the physical explanation for quantum speedup is precisely the fact that the phenomenon of quantum entanglement enables a quantum computer to fully exploit the representational capacity of Hilbert space. This is impossible for classical systems, joint states of which must always be representable as product (...) states. I begin the dissertation by considering, in Chapter 2, the most popular of the candidate physical explanations for quantum speedup: the many worlds explanation of quantum computation. I argue that, although it is inspired by the neo-Everettian interpretation of quantum mechanics, unlike the latter it does not have the conceptual resources required to overcome objections such as the so-called `preferred basis objection'. I further argue that the many worlds explanation, at best, can serve as a good description of the physical process which takes place in so-called network-based computation, but that it is incompatible with other models of computation such as cluster state quantum computing. I next consider, in Chapter 3, a common component of most other candidate explanations of quantum speedup: quantum entanglement. I investigate whether entanglement can be said to be a necessary component of any explanation for quantum speedup, and I consider two major purported counter-examples to this claim. I argue that neither of these, in fact, show that entanglement is unnecessary for speedup, and that, on the contrary, we should conclude that it is. In Chapters 4 and 5 I then ask whether entanglement can be said to be sufficient as well. In Chapter 4 I argue that despite a result that seems to indicate the contrary, entanglement, considered as a resource, can be seen as sufficient to enable quantum speedup. Finally, in Chapter 5 I argue that entanglement is sufficient to explain quantum speedup as well. (shrink)

This research offers a new perspective by reframing the positivist-interpretive debate as a distinction between the functional outcomes of research: explanation and understanding. Based on an older and well-established literature in philosophy, this distinction can reinvigorate important differences in research outcomes that have been lost. Understanding or “subjective meaning” is connected to the intentionality, thoughts, and motivations of the human subjects under study. From this perspective, understanding is the type of knowledge gained from determining the meanings, categories, and symbols humans (...) attach to actions, knowledge, and systems. In contrast, explanation is achieved by subsuming individual instantiations of the phenomenon under broad general laws, or identifying causal mechanisms that support antecedent-consequent pairs. Researchers can proactively use the understanding-explanation distinction as a heuristic to create new lines of research questions based on what has not been explained or understood rather than on which ontology or methodology has not been used. (shrink)

Many philosophers of science believe that empirical psychology can contribute little to the philosophical investigation of explanations. They take this to be shown by the fact that certain explanations fail to elicit any relevant psychological events (e.g., familiarity, insight, intelligibility, etc.). We report results from a study suggesting that, at least among those with extensive science training, a capacity to render an event intelligible is considered a requirement for explanation. We also investigate for whom explanations must be capable of rendering (...) events intelligible and whether or not accuracy is also viewed as a requirement. (shrink)

Of the many and varied applications of quantum information theory, perhaps the most fascinating is the sub-field of quantum computation. In this sub-field, computational algorithms are designed which utilise the resources available in quantum systems in order to compute solutions to computational problems with, in some cases, exponentially fewer resources than any known classical algorithm. While the fact of quantum computational speedup is almost beyond doubt, the source of quantum speedup is still a matter of debate. In this paper I (...) argue that entanglement is a necessary component for any explanation of quantum speedup and I address some purported counter-examples that some claim show that the contrary is true. In particular, I address Cleve et al.'s solution to Deutsch's problem, Biham et al.'s mixed-state version of the Deutsch-Jozsa algorithm, and Knill & Laflamme's deterministic quantum computation with one qubit model of quantum computation. I argue that these examples do not demonstrate that entanglement is unnecessary for the explanation of quantum speedup, but that they rather illuminate and clarify the role that entanglement does play. (shrink)