This paper studies the epistemic failures to reach understanding in relation to scientific explanations. We make a distinction between genuine understanding and its negative phenomena—lack of understanding and misunderstanding. We define explanatory understanding as inclusive as possible, as the epistemic success that depends on abilities, skills, and correct explanations. This success, we add, is often supplemented by specific positive phenomenology which plays a part in forming epistemic inclinations—tendencies to receive (...) an insight from familiar types of explanations. We define lack of understanding as the epistemic failure that results from a lack of an explanation or from an incorrect one. This can occur due to insufficient abilities and skills, or to fallacious explanatory information. Finally, we characterize misunderstanding by cases where one’s epistemic inclinations do not align with an otherwise correct explanation. We suggest that it leads to potential debates about the explanatory power of different explanatory strategies. We further illustrate this idea with a short meta-philosophical study on the current debates about distinctively mathematical explanations. (shrink)

Lange argues that some natural phenomena can be explained by appeal to mathematical, rather than natural, facts. In these “distinctively mathematical” explanations, the core explanatory facts are either modally stronger than facts about ordinary causal law or understood to be constitutive of the physical task or arrangement at issue. Craver and Povich argue that Lange’s account of DME fails to exclude certain “reversals”. Lange has replied that his account can avoid these directionality charges. Specifically, Lange (...) argues that in legitimate DMEs, but not in their “reversals,” the empirical fact appealed to in the explanation is “understood to be constitutive of the physical task or arrangement at issue” in the explanandum. I argue that Lange’s reply is unsatisfactory because it leaves the crucial notion of being “understood to be constitutive of the physical task or arrangement” obscure in ways that fail to block “reversals” except by an apparent ad hoc stipulation or by abandoning the reliance on understanding and instead accepting a strong realism about essence. (shrink)

I first quickly outline what I think grasping is, and suggest that it is both among our basic aims of inquiry and not essentially tied to belief, justification, or knowledge. Then, I briefly look at some places in the metaphysics of science in which it looks like our aim of grasping and our aim in knowing—or perhaps more specifically in knowing the explanations for things—might seem to conflict. I will use this conflict to support a broader view: sometimes, (...) we might develop philosophical views or theories—and even endorse them—in order to better grasp them, regardless of whether we genuinely believe them, or are justified in so doing. At other times, we may be aiming at propositional knowledge. These aims can come apart, and perhaps even systematically come apart. A pluralism about the value of those aims doesn’t entail an “anything goes” attitude with respect to how we think about what philosophical views to put forward, defend, or endorse, however—far from it. Instead, it suggests that what counts as a virtue of a philosophical theory depends on our aims in espousing it. If philosophers have distinct aims at the meta-level, it will be hard for them to engage in joint theory evaluation at the first-order level. At least all three of theoretical virtues, evaluative judgments of philosophical views, and what attitudes we ought to have towards the views we ourselves espouse will vary according to our guiding metaphilosophical aims. (shrink)

Mathematicians suggest that some proofs are valued for their explanatory value. This has led to a philosophical debate about the distinction between explanatory and non-explanatory proofs. In this paper, we explore whether contrasting views about the explanatory value of proof are possible and how to understand these diverging assessments. By considering an epistemic and contextual conception of explanation, we can make sense of disagreements about explanatoriness in mathematics by identifying differences in the background knowledge, skill (...) corpus, or epistemic aims of mathematicians or mathematical communities. We focus on the relation between explanation, epistemic aims and diverging explanatory assessments by looking at cases from mathematical practice. (shrink)

Philosophers have paid little attention to mathematical explanations . I present a variety of examples of mathematical explanation and examine two cases in detail. I argue that mathematical explanations have important implications for the philosophy of mathematics and of science. ;The first case study compares many proofs of Pick's theorem, a simple geometrical result. Though a simple proof surfaces to establish the result, some of the proofs explain the result better than others. The (...) second case study comes from George Polya's Mathematics and Plausible Reasoning. He gives a proof that, while entirely satisfactory in establishing its conclusion, is insufficiently explanatory. To provide a better explanation, he supplements the proof with additional exposition. ;These case studies illustrate at least two distinct explanatory virtues, and suggest there may be more. First, an explanatory improvement occurs when a sense of "arbitrariness" is reduced in the proofs. Proofs more explanatory in this way place greater restrictions on the steps that can be used to reach the conclusion. Second, explanatoriness is judged by directness of representation. More explanatory proofs allow one to ascribe geometric meaning to the terms of Pick's formula as they arise. ;I trace the lack of attention to mathematical explanations to an implicit assumption, justificationism, that only justificational aspects of mathematical reasoning are epistemically important. I propose an anti-justificationist epistemic position, the epistemic virtues view, which holds that justificational virtues, while important, are not the only ones of philosophical interest in mathematics. Indeed, explanatory benefits are rarely justificational. I show how the epistemic virtues view and the recognition of mathematical explanation can shed new light on philosophical debates. ;Mathematical explanations have consequences for philosophy of science as well. I show that mathematical explanations provide serious challenges to any theory, such as Bas van Fraassen's, that considers explanations to be fundamentally answers to why-questions. I urge a closer interaction between philosophy of mathematics and philosophy of science; both will be needed for a fuller understanding of mathematical explanation. (shrink)

The increasing preponderance of opinion that some natural phenomena can be explained mathematically has inspired a search for a viable account of distinctively mathematical explanation. Among the desiderata for an adequate account is that it should solve the problem of directionality and the reversals of distinctively mathematical explanations should not count as members among the explanatory fold but any solution must also avoid the exclusion of genuine explanations. In what follows, I introduce and (...) defend what I refer to as a quasi-erotetic solution which provides a remedy to the problem in the form of an additional necessary condition on explanation. (shrink)

Philosophy of Science in Practice (PoSiP) has the “practice of science” as its object of research. Notwithstanding, it does not possess yet any general or specific methodology in order to achieve its goal. Instead of sticking to one protocol, PoSiP takes advantage of a set of approaches from different fields. This thesis takes as a starting point a collaborative and interdisciplinary research between two Ph.D. students from distinct areas: ecology and philosophy. This collaboration showed how a (...) scientist could benefit from philosophy of science (in this case study the philosophical approach of. mechanistic explanation) to construct a model of his explanandum, by means of heuristics approach (heuristics as an instrument but also a methodological approach) and, also allowed philosophy of science take a closer look into the scientific practice to investigate how explanations are constructed and how scientific understanding is achieved (in this thesis, with a dialogue with the contextual theory of scientific understanding). As a result, it is asserted that (i) mechanistic explanation possess limitations but may work as epistemic instruments that mediate between theories, data, scientists, and models; (ii) explanation construction and scientific understanding deeply relies on intuition; (iii) scientific understanding is an instant, a moment, a temporary achievement, and its process may happen in degrees; (iv) philosophy of science, by means of heuristics process, may enhance scientists’ epistemic virtues, improving his academic skills, by means of self-evaluation. This research shows that interdisciplinarity and collaborative work can act, through heuristics, as a toolbox for PoSiP to achieve its goal of understanding how science is made. Despite its success, an analysis of this collaborative practice leads to some fundamental issues. First, philosophy of science in practice is a philosophy of past practice, in that the majority of examples used by mainstream PoSiP come from the final products of science. Second, is it philosophy of [science in practice] or philosophy of science [in practice]? How to practice philosophy of scientific practice and, how to practice interdisciplinarity in the philosophy of scientific practices simultaneously to its scientific activity? This research exposes the epistemic role heuristics and interdisciplinarity possess as methodological toolboxes for philosophy of science in practice. It is defended that other ways of constructing sciences would be through different dynamics such as collaborative networks and interdisciplinarity research contributing to the vision of Trading Zones from Peter Galison, in which bridges between specialized disciplines are created in order to exchange knowledge and information. (shrink)

In “What Makes a Scientific Explanation Distinctively Mathematical?” (2013b), Lange uses several compelling examples to argue that certain explanations for natural phenomena appeal primarily to mathematical, rather than natural, facts. In such explanations, the core explanatory facts are modally stronger than facts about causation, regularity, and other natural relations. We show that Lange's account of distinctively mathematical explanation is flawed in that it fails to account for the implicit directionality in each of (...) his examples. This inadequacy is remediable in each case by appeal to ontic facts that account for why the explanation is acceptable in one direction and unacceptable in the other direction. The mathematics involved in these examples cannot play this crucial normative role. While Lange's examples fail to demonstrate the existence of distinctively mathematical explanations, they help to emphasize that many superficially natural scientific explanations rely for their explanatory force on relations of stronger-than-natural necessity. These are not opposing kinds of scientific explanations; they are different aspects of scientific explanation. (shrink)

In Cognitive science and philosophy of consciousness, the explanatory gap, following Joseph Levine, refers to the unintelligible link between our conscious mental life and its corresponding objective physical explanation; the gap in our understanding of how consciousness is related to a physical or a physiological substrate :354–361, 1983). David Chalmers holds the explanatory gap as the evidence for a form of metaphysical dualism between consciousness and physical reality. On the other hand, McGinn takes it as (...) an epistemic rather than an ontological gap. Considering the recent advances in neuroscience, however, the reductionist approaches have become replete, which attempt to reduce consciousness and subjectivity to neurobiological accounts. On this view, consciousness mirrors the brain mechanisms, and the self is formulated as an illusory product or construct of neuronal processes. Thomas Fuchs, who is committed to embodied and enactive approaches toward cognition, in Ecology of the Brain, espouses an enactive-ecological perspective concerning the problem of the explanatory gap. He develops an account of human’s life in its dual aspects of the living body and lived body which, on the one hand, defies the ontological dualism and, on the other hand, avoids drifting towards any form of reductionism. Often, the ontologically monistic approaches to the explanatory gap have inclined to a form of reductionism because they conceive consciousness as either identical with its physiological substrate or caused by it, where in both cases, consciousness is claimed to be explainable within the framework of physicalism. Fuchs, however, defends an ontological monism which remains irreconcilable with reductionism. In his account of dual aspects, there is no interaction or impact between the sphere of subjectivity and nature; however, these two aspects imply one another. In this essay, I will develop the philosophical justification of the above reformulation of the mind–body problem and employing the analogy of light will canvass the paradoxical relationship between dual aspects in a phenomenological framework. (shrink)

Science and mathematics continually change in their tools, methods, and concepts. Many of these changes are not just modifications but progress---steps to be admired. But what constitutes progress? This dissertation addresses one central source of intellectual advancement in both disciplines: reformulating a problem-solving plan into a new, logically compatible one. For short, I call these cases of compatible problem-solving plans "reformulations." Two aspects of reformulations are puzzling. First, reformulating is often unnecessary. Given that we could already solve a problem (...) using an older formulation, what do we gain by reformulating? Second, some reformulations are genuinely trivial or insignificant. Merely replacing one symbol with another does not lead to intellectual progress. What distinguishes significant reformulations from trivial ones? According to what I call "conceptualism" (or "conceptual empiricism"), reformulations are intellectually significant when they provide a different plan for solving problems. Significant reformulations provide inferentially different routes to the same solution. In contrast, trivial reformulations provide exactly the same problem-solving plans, and hence they do not change our understanding. This answers the second question about what distinguishes trivial from significant reformulations. However, the first question remains: what makes a new way of solving an old problem valuable? Here, a bevy of practical considerations come to mind: one formulation might be faster, less complicated, or use more familiar concepts. According to "instrumentalism," these practical benefits are all there is to reformulating. Some reformulations are simply more instrumentally valuable for meeting the aims of science than others. At another extreme, "fundamentalism" contends that a reformulation is valuable when it provides a more fundamental description of reality. According to this view, some reformulations directly contribute to the metaphysical aim of carving reality at its joints. Conceptualism develops a middle ground between instrumentalism and fundamentalism, preserving their benefits without their costs. I argue that the epistemic value of significant reformulations does not reduce to either practical or metaphysical value. Reformulations are valuable because they are a constitutive part of problem-solving. Both science and mathematics aim at solving all possible problems within their respective domains. Meeting this aim requires being able to plan for any possible problem-solving context, and this requires reformulating. By reformulating, we clarify what we need to know to solve problems. Still, one might wonder whether the value of reformulations requires underlying differences in explanatory power. According to "explanationism," a reformulation is valuable only when it provides a better explanation. Explanationism stands as a rival middle ground position to my own. However, it faces numerous counterexamples. In many cases, two reformulations provide the same explanation while nonetheless providing different ways of understanding a phenomenon. Hence, reformulating can be valuable even when neither formulation is more explanatory. Methodologically, I draw on a variety of case studies to support my account of reformulation. These range from classical mechanics to quantum chemistry, along with examples from mathematics. Symmetry arguments provide a paradigmatic example: the mathematics of symmetry groups radically recasts quantum mechanics and quantum chemistry. Nevertheless, elementary approaches exist that eschew this additional mathematical apparatus, solving problems in a more tedious but less mathematically-demanding manner. Further examples include reformulations of quantum field theory, Arabic vs. Roman numerals, and Fermat's little theorem in number theory. In each case, my account identifies how reformulations change and improve our understanding of science and mathematics. (shrink)

An account of distinctively mathematical explanation (DME) should satisfy three desiderata: it should account for the modal import of some DMEs; it should distinguish uses of mathematics in explanation that are distinctively mathematical from those that are not (Baron [2016]); and it should also account for the directionality of DMEs (Craver and Povich [2017]). Baron’s (forthcoming) deductive-mathematical account, because it is modelled on the deductive-nomological account, is unlikely to satisfy these desiderata. I provide a counterfactual (...) account of DME, the Narrow Ontic Counterfactual Account (NOCA), that can satisfy all three desiderata. NOCA appeals to ontic considerations to account for explanatory asymmetry and ground the relevant counterfactuals. NOCA provides a unification of the causal and the non-causal, the ontic and the modal, by identifying a common core that all explanations share and in virtue of which they are explanatory. (shrink)

In this contribution, I comment on Raffaella Campaner’s defense of explanatory pluralism in psychiatry (in this volume). In her paper, Campaner focuses primarily on explanatory pluralism in contrast to explanatory reductionism. Furthermore, she distinguishes between pluralists who consider pluralism to be a temporary state on the one hand and pluralists who consider it to be a persisting state on the other hand. I suggest that it would be helpful to distinguish more than those two (...) versions of pluralism – different understandings of explanatory pluralism both within philosophy of science and psychiatry – namely moderate/temporary pluralism, anything goes pluralism, isolationist pluralism, integrative pluralism and interactive pluralism. Next, I discuss the pros and cons of these different understandings of explanatory pluralism. Finally, I raise the question of how to implement or operationalize explanatory pluralism in scientific practice; how to structure the “genuine dialogue” or shape “the pluralistic attitude” Campaner is referring to. As tentative answers, I explore a question-based framework for explanatory pluralism as well as social-epistemological procedures for interaction among competing approaches and explanations. (shrink)

In the last couple of years, a few seemingly independent debates on scientific explanation have emerged, with several key questions that take different forms in different areas. For example, the questions what makes an explanation distinctly mathematical and are there any non-causal explanations in sciences (i.e., explanations that don’t cite causes in the explanans) sometimes take a form of the question of what makes mathematical models explanatory, especially whether highly idealized models in science can (...) be explanatory and in virtue of what they are explanatory. These questions raise further issues about counterfactuals, modality, and explanatory asymmetries: i.e., do mathematical and non-causal explanations support counterfactuals, and how ought we to understand explanatory asymmetries in non-causal explanations? Even though these are very common issues in the philosophy of physics and mathematics, they can be found in different guises in the philosophy of biology where there is the statistical interpretation of the Modern Synthesis theory of evolution, according to which the post-Darwinian theory of natural selection explains evolutionary change by citing statistical properties of populations and not the causes of changes. These questions also arise in philosophy of ecology or neuroscience in regard to the nature of topological explanations. The question here is can the mathematical (or more precisely topological) properties in network models in biology, ecology, neuroscience, and computer science be explanatory of physical phenomena, or are they just different ways to represent causal structures. The aim of this special issue is to unify all these debates around several overlapping questions. These questions are: are there genuinely or distinctively mathematical and non-causal explanations?; are all distinctively mathematical explanations also non-causal; in virtue of what they are explanatory; does the instantiation, implementation, or in general, applicability of mathematical structures to a variety of phenomena and systems play any explanatory role? This special issue provides a platform for unifying the debates around several key issues and thus opens up avenues for better understanding of mathematical and non-causal explanations in general, but also, it will enable even better understanding of key issues within each of the debates. (shrink)

Explanatory pluralism has been defended by several philosophers of history and social science, recently, for example, by Tor Egil Førland in this journal. In this article, we provide a better argument for explanatory pluralism, based on the pragmatist idea of epistemic interests. Second, we show that there are three quite different senses in which one can be an explanatory pluralist: one can be a pluralist about questions, a pluralist about answers to questions, and (...) a pluralist about both. We defend the last position. Finally, our third aim is to argue that pluralism should not be equated with “anything goes”: we will argue for non-relativistic explanatory pluralism. This pluralism will be illustrated by examples from history and social science in which different forms of explanation (for example, structural, functional, and intentional explanations) are discussed, and the fruitfulness of our framework for understanding explanatory pluralism is shown. (shrink)

In the last couple of years a few seemingly independent debates on scientific explanation have emerged, with several key questions that take different forms in different areas. For example, the question what makes an explanation distinctly mathematical and are there any non-causal explanations in sciences (i.e. explanations that don’t cite causes in the explanans) sometimes take a form of the question what makes mathematical models explanatory, especially whether highly idealized models in science can be (...) explanatory and in virtue of what they are explanatory. These questions raise further issues about counterfactuals, modality and explanatory asymmetries, i.e. do mathematical and non-causal explanations support counterfactuals, and how to understand explanatory asymmetries in non-causal explanations. Even though these are very common issues in the philosophy of physics and mathematics, they can be found in different guises in the philosophy of biology, where there is the statistical interpretation of the Modern Synthesis theory of evolution, according to which the post-Darwinian theory of natural selection explains evolutionary change by citing statistical properties of populations and not the causes of changes. These questions also arise in philosophy of ecology or neuroscience in regard to the nature of topological explanations. The question here is whether in network models in biology, ecology, neuroscience and computer science the mathematical or more precisely topological properties can be explanatory of physical phenomena, or they are just different ways to represent causal structures. The aim of the special issue is to unify all these debates around several overlapping questions. (shrink)

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 particular case: the reduction of arithmetic to set theory. I claim that the (...) reduction is unexplanatory. In defense of this claim, I offer 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 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. (shrink)

Philosophical discussions of mechanisms and mechanistic explanation have often been framed by contrast to laws and deductive-nomological explanation. A more adequate conception of lawfulness and nomological necessity, emphasizing the role of modal considerations in scientific reasoning, circumvents such contrasts and enhances understanding of mechanisms and their scientific significance. The first part of the paper sketches this conception of lawfulness, drawing upon Haugeland, Lange, and Rouse. This conception emphasizes the role of lawful stability under relevant counterfactual suppositions in scientific reasoning (...) across the sciences, in place of traditional conceptions of law that are primarily confined to the physical sciences. It also extends lawful stability beyond verbally or mathematically expressed law-statements, to encompass other ways of conjoining patterns in the world with scientific pattern recognition. The remainder of this paper shows how and why mechanisms constructively exemplify this conception of lawfulness in scientific practice: • Mechanisms are robust, counterfactually stable and inductively projectible patterns, even though they are not exceptionless “laws of nature”. • Mechanistic explanations often take non-verbal forms, which consequently resist philosophical inclinations to semantic ascent, but understanding lawfulness in terms of counterfactually stable pattern recognition accounts for these ways in which scientific understanding outruns the expressive capacities of natural languages; • Mechanisms are sometimes characterized as real patterns in the world, and sometimes as epistemic representations; understanding mechanisms as modal patterns shows why both conceptions are needed, as mutually supportive. • Mechanisms are typically open-ended, and only partially specified, in ways open to and directive toward further articulation and revision. Understanding mechanisms as modal patterns incorporates this aspect of mechanistic understanding within a broader conception of scientific understanding as embedded in research practice, rather than in bodies of knowledge extracted from it. • Mechanistic explanation has often been placed on the causal side of an opposition between causal and nomological explanation, but understanding mechanisms as modal patterns helps overcome that opposition, and contributes to a pluralist conception of causal relations and their characteristic forms of counterfactual invariance. • The recognition of mechanisms as modal patterns allows for a new way to think about the relations among distinct levels of a mechanistic hierarchy, and the broader scientific significance of mechanistic understanding. (shrink)

Pluralism is widely appealed to in many areas of philosophy of science, though what is meant by ‘pluralism’ may profoundly vary. Because explanations of behaviour have been a favoured target for pluralistic theses, the sciences of behaviour offer a rich context in which to further investigate pluralism. This is what the topical collection The Biology of Behaviour: Explanatory pluralism across the life sciences is about. In the present introduction, we briefly review major (...) strands of pluralist theses and their motivations. We highlight three distinct types of pluralisms—type pluralism, fragmentation pluralism and insular pluralism—and introduce the articles of the topical collection. (shrink)

Nowadays there is a growing tendency in the philosophy of science to think that some phenomena cannot be exhaustively explained, or even described, by a single theory or a particular approach. Thus, we are occasionally required to use various approaches in order to give account of the phenomenon we are analyzing. And sometimes, we can appreciate this as an invitation to be pluralist in certain respects about our understanding of a particular aspect in science. -/- During (...) the last decade applications of pluralism have increased and led to several other relevant debates in the philosophy of science. By the strengthening of pluralism, new questions have emerged, and more importantly, new alternatives have been offered in dealing with problems about how to interpret the ontology and the ontological commitments of particular theories, or how to apply strategies to analyze and understand specific episodes from the history of science, or how to work within different, and sometimes incompatible, explanations about a particular phenomenon, among others. In general, pluralism seems to be a very rich and yet not enough explored path for the philosophy of science in general. -/- Today, it is recognized that, at least for methodological purposes, the application of pluralism to the study of science can offer a great number of benefits. One of them would be the opportunity of analyzing the role that some epistemic virtues -such as scope, fruitfulness, consistency, and simplicity, to name just a few- play in the scientific activity. From the different pluralist positions, a lot has been said about empirical adequacy, refutability and explanatory power yet consistency power, yet, consistency has not been equally dealt with. -/- As a matter of fact, the lack of consistency and its philosophical implications have been studied from an angle that does not necessarily involve a pluralism of any kind. At the moment, it is commonly accepted that inconsistencies are more frequent in scientific development that the traditional philosophy of science could have expected and the idea that inconsistency is not always a synonym of logical anarchy, as it was suggested in the classical literature of logic and the philosophy of science, has been gaining support. All this has been possible, mostly, thanks to the emergence of paraconsistent logics and the availability of case studies that show how inconsistency is not an uncommon phenomenon in science. -/- But pluralism does not necessarily entail inconsistency toleration nor vice versa. Accordingly the main motivation for this volume is to explore the links between pluralism and inconsistency toleration in science, in order to connect the reflections on inconsistency toleration with broader and major issues in philosophy of science. In order to do so, we will suggest two different lines of investigation: first, to focus on the implication of some pluralistic accounts in the philosophy of science, regarding inconsistency; and second to analyze the implications of some paraconsistent approaches regarding pluralism in science. (shrink)

Scientific explanation is a perennial topic in philosophy of science, but the literature has fragmented into specialized discussions in different scientific disciplines. An increasing attention to scientific practice by philosophers is (in part) responsible for this fragmentation and has put pressure on criteria of adequacy for philosophical accounts of explanation, usually demanding some form of pluralism. This commentary examines the arguments offered by Fagan and Woody with respect to explanation and understanding in scientific practice. I begin (...) by scrutinizing Fagan's concept of collaborative explanation, highlighting its distinctive advantages and expressing concern about several of its assumptions. Then I analyze Woody's attempt to reorient discussions of scientific explanation around functional considerations, elaborating on the wider implications of this methodological recommendation. I conclude with reflections on synergies and tensions that emerge when the two papers are juxtaposed and how these draw attention to critical issues that confront ongoing philosophical analyses of scientific explanation. (shrink)

This thesis looks at explanation in the natural sciences, the social sciences, and in religious reflection. Although these fields differ radically in the objects studied and the methods employed, they do evidence certain formal commonalities when one inquires into the nature of the explanatory endeavor as it is manifested in each. By exploring the links between explanations and the various contexts or disciplines in which they occur, I attempt to provide a general framework for speaking of rational (...) class='Hi'>explanations in these diverse areas. ;In an opening chapter I consider several alternatives regarding the epistemic status of religious explanations, focusing finally on the model of "intersubjective explanation." Contemporary defenses of intersubjective religious explanation are placed within the context of the broader faith/reason debate, and a quick survey is made of recent advocates of this position. I then turn to the methodology dispute in the philosophy of the natural sciences . My aim is to mediate between purely formal analyses of explanatory structure and the more recent emphasis on contextual and pragmatic factors. ;In the social sciences, many have argued, explanation is subordinate to intuitive understanding. After tracing the explanation versus understanding debate from Dilthey to the present, I present the recent work of Jurgen Habermas as a case study in the problems of social scientific rationality. Explanations in the social sciences are rational reconstructions of human meaning contexts, and "Verstehen" is required as a precondition for material adequacy. Such explanations are linked to the individual or communal effort to "make sense" of, or bring coherence into, subjective and intersubjective "worlds." ;After an excursus on philosophical explanations and the problem of philosophical rationality, I attempt a brief phenomenology of religious beliefs as explanations. As in the social sphere, religious explanations represent the believer's attempt to "make sense" of her experience in light of a given religious tradition. Although the comprehensive nature of religious explanations makes comparisons difficult--in the limit case they verge on ineffability--the concepts of meaning and coherence allow us to speak of the rationality of religious belief without total equivocation. ;The final chapter turns to the study and evaluation of religious explanations as they occur in the discipline of theology. Under the rubric "theology as a science," I defend theology's dual status as an academic discipline and as believing reflection in the service of the religious community. Through vitally concerned with the truth of its assertions, theology also shares the goals of coherence and intersubjective criticizability with its scientific counterparts. (shrink)

Certain scientific explanations of physical facts have recently been characterized as distinctively mathematical –that is, as mathematical in a different way from ordinary explanations that employ mathematics. This article identifies what it is that makes some scientific explanations distinctively mathematical and how such explanations work. These explanations are non-causal, but this does not mean that they fail to cite the explanandum’s causes, that they abstract away from detailed causal histories, or (...) that they cite no natural laws. Rather, in these explanations, the facts doing the explaining are modally stronger than ordinary causal laws or are understood in the why question’s context to be constitutive of the physical arrangement at issue. A distinctively mathematical explanation works by showing the explanandum to be more necessary than ordinary causal laws could render it. Distinctively mathematical explanations thus supply a kind of understanding that causal explanations cannot. 1 Introduction2 Some Distinctively Mathematical Scientific Explanations3 Are Distinctively Mathematical Explanations Set Apart by their Failure to Cite Causes? 4 Distinctively Mathematical Explanations do not Exploit Causal Powers5 How these Distinctively Mathematical Explanations Work6 Conclusion. (shrink)

One of the central aims of the philosophical analysis of mathematical explanation is to determine how one can distinguish explanatory proofs from non-explanatory proofs. In this paper, I take a closer look at the current status of the debate, and what the challenges for the philosophical analysis of explanatory proofs are. In order to provide an answer to these challenges, I suggest we start from analysing the concept understanding. More precisely, I will defend four claims: (...) understanding is a condition for explanation, unificatory understanding is a type of explanatory understanding, unificatory understanding is valuable in mathematics, and mathematical proofs can contribute to unificatory understanding. As a result, in a context where the epistemic aim is to unify mathematical results, I argue it is fruitful to make a distinction between proofs based on their explanatory value. (shrink)

A finer-grained delineation of a given explanandum reveals a nexus of closely related causal and non- causal explanations, complementing one another in ways that yield further explanatory traction on the phenomenon in question. By taking a narrower construal of what counts as a causal explanation, a new class of distinctively mathematical explanations pops into focus; Lange’s characterization of distinctively mathematical explanations can be extended to cover these. This new class of distinctively (...) mathematical explanations is illustrated with the Lotka-Volterra equations. There are at least two distinct ways those equations might hold of a system, one of which yields straightforwardly causal explanations, but the other of which yields explanations that are distinctively mathematical in terms of nomological strength. In the first, one first picks out a system or class of systems, finds that the equations hold in a causal -explanatory way; in the second, one starts with the equations and explanations that must apply to any system of which the equations hold, and only then turns to the world to see of what, if any, systems it does in fact hold. Using this new way in which a model might hold of a system, I highlight four specific avenues by which causal and non- causal explanations can complement one another. (shrink)

In this paper I present a case study of mathematical explanation in science that is new to the philosophical literature, and that arises in the context of estimating the energetic costs of running in bipedal animals. I refer to this as the Bipedal Gait Costs explanation. I argue that it is important for examples of applied mathematics to be driven not just by philosophical and mathematical concerns but also by scientific concerns. After a detailed presentation of the (...) BGC case study, I discuss ways in which it is different from standard examples of MES. One distinctive aspect of BGC is that substantive mathematics occurs at two different levels. I argue for a distinction to be drawn between the mathematical modeling that occurs at one level, which is non-explanatory, and the mathematical theorem which is applied to results gleaned from these models, which is explanatory. I conclude by drawing some connections with broader indispensability-based arguments for platonism in the philosophy of mathematics. (shrink)

This article outlines a philosophy of science in practice that focuses on the engineering sciences. A methodological issue is that these practices seem to be divided by two different styles of scientific reasoning, namely, causal-mechanistic and mathematical reasoning. These styles are philosophically characterized by what Kuhn called ?disciplinary matrices?. Due to distinct metaphysical background pictures and/or distinct ideas of what counts as intelligible, they entail distinct ideas of the character of phenomena and what counts as a scientific (...) explanation. It is argued that the two styles cannot be reduced to each other. At the same time, although they are incompatible, they must not be regarded as competing. Instead, they produce different kinds of epistemic results, which serve different kinds of epistemic functions. Moreover, some scientific breakthroughs essentially result from relating them. This view of complementary styles of scientific reasoning is supported by pluralism about metaphysical background pictures. (shrink)

Nietzsche offers us an account of how different drives interact with one another; it is rich but also appears to risk the homunculus fallacy. Competing attempts to deflect this charge on his behalf share an implicit consensus about the ‘epistemic ends’ of the account: they assume Nietzsche is trying to provide true explanations of psychological phenomena. I argue against this consensus. I claim that Nietzsche's characterisations of drive interaction are to be taken as fictive and are not intended (...) to have explanatory value. They nevertheless facilitate genuine epistemic achievement. Drawing on Catherine Elgin's account of the epistemic role of idealised models in science, I argue that Nietzsche's account of drive interaction is a ‘model of the mind’ that, despite relying on falsehoods, can exemplify features of our psychology that aid us in making novel predictions. We then see that Nietzsche neatly sidesteps the homunculus fallacy; we can further understand more fully what Nietzsche hopes his drive psychology will teach us. We can now resolve, for example, outstanding interpretative puzzles about the relationship between psychic integration and Nietzsche’s distinctive notion of spiritual health. (shrink)

Both philosophers and psychologists have argued for the existence of distinct kinds of explanations, including teleological explanations that cite functions or goals, and mechanistic explanations that cite causal mechanisms. Theories of causation, in contrast, have generally been unitary, with dominant theories focusing either on counterfactual dependence or on physical connections. This paper argues that both approaches to causation are psychologically real, with different modes of explanation promoting judgments more or less consistent with each approach. Two sets of (...) experiments isolate the contributions of counterfactual dependence and physical connections in causal ascriptions involving events with people, artifacts, or biological traits, and manipulate whether the events are construed teleologically or mechanistically. The findings suggest that when events are construed teleologically, causal ascriptions are sensitive to counterfactual dependence and relatively insensitive to the presence of physical connections, but when events are construed mechanistically, causal ascriptions are sensitive to both counterfactual dependence and physical connections. The conclusion introduces an account of causation, an "exportable dependence theory," that provides a way to understand the contributions of physical connections and teleology in terms of the functions of causal ascriptions. (shrink)

A finer-grained delineation of a given explanandum reveals a nexus of closely related causal and non-causal explanations, complementing one another in ways that yield further explanatory traction on the phenomenon in question. By taking a narrower construal of what counts as a causal explanation, a new class of distinctively mathematical explanations pops into focus; Lange’s characterization of distinctively mathematical explanations can be extended to cover these. This new class of distinctively (...) class='Hi'>mathematical explanations is illustrated with the Lotka–Volterra equations. There are at least two distinct ways those equations might hold of a system, one of which yields straightforwardly causal explanations, and another that yields explanations that are distinctively mathematical in terms of nomological strength. In the first case, one first picks out a system or class of systems, and finds that the equations hold in a causal–explanatory way. In the second case, one starts with the equations and explanations that must apply to any system of which the equations hold, and only then turns to the world to see of what, if any, systems it does in fact hold. Using this new way in which a model might hold of a system, I highlight four specific avenues by which causal and non-causal explanations can complement one another. _1_. Introduction _2._ Delineating the Boundaries of Causal Explanation _2.1._ Why construe causal explanation narrowly? The land of explanation versus grain-focusing _2.2._ Reasons to narrow the scope of causal explanation _3._ Broadening the Scope of Mathematical Explanation _4._ Lotka–Volterra: Same Model, Different Explanation Types _4.1._ General biocide in the Lotka–Volterra model _4.2._ Two ways a model can hold, yielding causal versus mathematical explanations _5._ Four Complementary Relationships between Mathematical and Causal Explanation _5.1._ Slight reformulations of explananda _5.2._ Causal distortion of idealized mathematical models _5.3._ Partial explanations requiring supplementation _5.4._ Explanatory dimensionality _6._ Conclusion. (shrink)

1. Introduction: Naturalism and Psychological Explanations To a large extent, contemporary philosophical debate takes place within a framework of naturalistic assumptions. From the perspective of the history of philosophy, naturalism is the legacy of positivism without its empiricist epistemology and empiricist conception of meaning and cognitive significance. Systematically, it is best to characterize naturalism as the philosophical articulation of the underlying presuppositions of a reductive scientific research program that was rather successful in the last few centuries and, equally (...) important, promises to be so in the future particularly in the biological sciences and the neurosciences. It seems as if the secrets of human life and behavior and the mysteries of the mind will be cracked on the molecular level of the genes or the brain, or at least so we are told. Viewed in this manner it is understandable why philosophical naturalism tends to be committed to monism, both as a metaphysical or ontological claim and as a methodological position in the philosophy of social science. Naturalists are inclined to adopt a physicalist ontology that rejects free floating Cartesian substances and they view higher order macroscopic facts and properties as being dependent or supervenient on basic micro-physical facts. Naturalists, furthermore, expect that any scientific explanation of higher order properties has to provide an account of why and how these lower order facts give rise to higher order ones. These ontological and epistemic commitments also underpin a position of methodological monism in regard to the social sciences and the explanation of human agency. If the above ontological picture is correct then there is no reason to expect that the structure of the sciences dealing with higher order properties on the social level should fundamentally differ in their methodology from the natural sciences. In both domains of investigation, scientists will develop and make explanatory use of comprehensive and empirically well supported theories with adequate predictive powers that describe.... (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Response to Elvira Panaiotidi, “The Nature of Paradigms and Paradigm Shifts in Music Education”Carlos Xavier RodriguezElvira Panaiotidi has delivered a very useful and appealing paper on the topic of how the music education community decides it is time to change the way it thinks and acts. Her primary focus is whether the concept of "paradigms" proposed by Thomas Kuhn in The Structure of Scientific Revolutions reasonably explains how change (...) occurs in music education theory and practice and, if not, whether it can be modified so that it does. Following Peter Abbs' account of paradigm shifts in arts education, she proposes that paradigm shifts occur when methods of inquiry change even if what is being analyzed does not change. More specifically, paradigm shifts involve gradual transformations such that certain principles and terminology are carried over from the previous paradigm to the new one, and change at a slower rate than the methodologies adopted to make sense of them. Such discrepancy creates the kind of communicative discord she alludes to in her introduction: the aesthetic/praxial debate. Her further analysis of Abbs' argument leads her to conclude that the paradigm shifts he describes in arts education are different in kind from those intended by Kuhn. They apparently lack conceptual modification, or "conversion experience," and seem motivated by the need for increasingly comprehensive explanations of artistic behavior and thinking.Panaiotidi's synopsis of Kuhn's original criteria for paradigms reminded me [End Page 108] that his Structure of Scientific Revolutions as my first college textbook, exerted a powerful influence on me by its power to explain how humans formulate and react to new and better ideas. Panaiotidi seems to share this feeling, and scrutinizes Kuhn's original intention to use paradigms as a way of delineating the natural sciences from the humanities. Kuhn himself cautioned that his characterization of change and progress in science is not readily applicable to the arts and, more importantly, that interpretive frameworks for understanding art are not paradigms because paradigms are not theories, a distinction he echoed throughout his career. I argue that Kuhn would not consider the aesthetic and praxial approaches to music education paradigms at all; rather they are more a search for "a more inclusive theory" as Panaiotidi states. However, the ready applicability of Kuhn's criteria for paradigms to the work of music education philosophers is apparent. Their writings have been "sufficiently unprecedented to attract an enduring group of adherents away from competing modes of scientific activity." They have become associated with new methodological strategies because their writings are "sufficiently open-ended to leave all sorts of problems for the redefined group of practitioners to solve."There is, however, quite a difference in the way that science and art communities function. As Ted Gracyk writes in his book, I Wanna Be Me: Rock Music and the Politics of Identity:Where a scientific community singles out one paradigm as the foundation for further research, artistic communities do not. Once Newtonian mechanics was accepted as a scientific paradigm, no other approach to mechanics counted as a rational basis for mechanics. "Unlike art," Kuhn observes, "science destroys its past." The artworld can admire multiple paradigms, representing different periods in art history, without risking incoherence: Picasso's success did not lead museums to put their Rembrandts into storage."1Perhaps, then, the discord noted by Panaiotidi between "aestheticians" and "praxialists" arises not from a shortcoming of music education theory to provide what she calls "tenacious theoretical footing which could help settle the dispute," but from a too-literal interpretation of Kuhn's original criterion that applies to the natural sciences: new paradigms must necessarily refute those that have preceded them. Indeed, the music education research community has grappled with this very issue in its polarized views of acceptable research practice, which is perhaps more fundamentally a misunderstanding regarding the nature of the sciences versus the nature of the humanities. We in arts education have long admired scientific inquiry for its relative stability, what Kuhn calls a "normal" versus "revolutionary" mode of activity. However, as Panaiotidi explains, "Normal science [End Page 109] is characterized by a broad consensus of the specialists and is not by its nature... (shrink)

Explanatory inquiry characteristically begins with a certain puzzlement about the world. But why do certain situations elicit our puzzlement while others leave us, in some epistemically relevant sense, cold? Moreover, what exactly is involved in the move from a state of puzzlement to a state where one's puzzlement is satisfied? In this paper I try to answer both of these questions. I also suggest ways in which our account of scientific rationality might benefit from having a better sense of (...) the kind of epistemic goal we are trying to realize, when we engage in our explanatory inquiries. Two Senses The Need for Explanation An Example Proto-understanding Conclusion CiteULike Connotea Del.icio.us What's this? (shrink)

Mathematicians distinguish between proofs that explain their results and those that merely prove. This paper explores the nature of explanatory proofs, their role in mathematical practice, and some of the reasons why philosophers should care about them. Among the questions addressed are the following: what kinds of proofs are generally explanatory (or not)? What makes a proof explanatory? Do all mathematical explanations involve proof in an essential way? Are there really such things as (...) class='Hi'>explanatory proofs, and if so, how do they relate to the sorts of explanation encountered in philosophy of science and metaphysics? (shrink)

The paper provides a systematic overview of recent debates in epistemology and philosophy of science on the nature of understanding. We explain why philosophers have turned their attention to understanding and discuss conditions for “explanatory” understanding of why something is the case and for “objectual” understanding of a whole subject matter. The most debated conditions for these types of understanding roughly resemble the three traditional conditions for knowledge: truth, justification and belief. We (...) discuss prominent views about how to construe these conditions for understanding, whether understanding indeed requires conditions of all three types and whether additional conditions are needed. (shrink)

Call an explanation in which a non-mathematical fact is explained—in part or in whole—by mathematical facts: an extra-mathematical explanation. Such explanations have attracted a great deal of interest recently in arguments over mathematical realism. In this article, a theory of extra-mathematical explanation is developed. The theory is modelled on a deductive-nomological theory of scientific explanation. A basic DN account of extra-mathematical explanation is proposed and then redeveloped in the light of two difficulties that (...) the basic theory faces. The final view appeals to relevance logic and uses resources in information theory to understand the explanatory relationship between mathematical and physical facts. 1Introduction2Anchoring3The Basic Deductive-Mathematical Account4The Genuineness Problem5Irrelevance6Relevance and Information7Objections and Replies 7.1Against relevance logic7.2Too epistemic7.3Informational containment8Conclusion. (shrink)

We argue that two well-known examples (strawberry distribution and Konigsberg bridges) generally considered genuine cases of distinctively _mathematical_ explanation can also be understood as cases of distinctively _generic_ explanation. The latter answer resemblance questions (e.g., why did neither person A nor B manage to cross all bridges) by appealing to ‘generic task laws’ instead of mathematical necessity (as is done in distinctively mathematical explanations). We submit that distinctively generic explanations derive their (...) class='Hi'>explanatory force from their role in ontological unification. Additionally, we argue that distinctively generic explanations are better seen as standardly mathematical instead of distinctively mathematical. Finally, we compare and contrast our proposal with the work of Christopher Pincock on abstract explanations in science and the views of Michael Strevens on abstract causal event explanations. (shrink)

Both Carnap and Quine made significant contributions to the philosophy of mathematics despite their diversedviews. Carnap endorsed the dichotomy between analytic and synthetic knowledge and classified certainmathematical questions as internal questions appealing to logic and convention. On the contrary, Quine wasopposed to the analytic-synthetic distinction and promoted a holistic view of scientific inquiry. The purpose of thispaper is to argue that in light of the recent advancement of experimental mathematics such as Monte Carlosimulations, limiting mathematical inquiry to the (...) domain of logic is unjustified. Robustness studies implemented inMonte Carlo Studies demonstrate that mathematics is on par with other experimental-based sciences. (shrink)

Climatology is a paradigmatic complex systems science. Understanding the global climate involves tackling problems in physics, chemistry, economics, and many other disciplines. I argue that complex systems like the global climate are characterized by certain dynamical features that explain how those systems change over time. A complex system's dynamics are shaped by the interaction of many different components operating at many different temporal and spatial scales. Examining the multidisciplinary and holistic methods of climatology can help us better understand (...) the nature of complex systems in general. -/- Questions surrounding climate science can be divided into three rough categories: foundational, methodological, and evaluative questions. "How do we know that we can trust science?" is a paradigmatic foundational question (and a surprisingly difficult one to answer). Because the global climate is so complex, questions like "what makes a system complex?" also fall into this category. There are a number of existing definitions of `complexity,' and while all of them capture some aspects of what makes intuitively complex systems distinctive, none is entirely satisfactory. Most existing accounts of complexity have been developed to work with information-theoretic objects (signals, for instance) rather than the physical and social systems studied by scientists. -/- Dynamical complexity, a concept articulated in detail in the first third of the dissertation, is designed to bridge the gap between the mathematics of contemporary complexity theory (in particular the formalism of "effective complexity" developed by Gell-Mann and Lloyd [2003]) and a more general account of the structure of science generally. Dynamical complexity provides a physical interpretation of the formal tools of mathematical complexity theory, and thus can be used as a framework for thinking about general problems in the philosophy of science, including theories, explanation, and lawhood. -/- Methodological questions include questions about how climate science constructs its models, on what basis we trust those models, and how we might improve those models. In order to answer questions about climate modeling, it's important to understand what climate models look like and how they are constructed. Climate model families are significantly more diverse than are the model families of most other sciences (even sciences that study other complex systems). Existing climate models range from basic models that can be solved on paper to staggeringly complicated models that can only be analyzed using the most advanced supercomputers in the world. I introduce some of the central concepts in climatology by demonstrating how one of the most basic climate models might be constructed. I begin with the assumption that the Earth is a simple featureless blackbody which receives energy from the sun and releases it into space, and show how to model that assumption formally. I then gradually add other factors (e.g. albedo and the greenhouse effect) to the model, and show how each addition brings the model's prediction closer to agreement with observation. After constructing this basic model, I describe the so-called "complexity hierarchy" of the rest of climate models, and argue that the sense of "complexity" used in the climate modeling community is related to dynamical complexity. -/- With a clear understanding of the basics of climate modeling in hand, I then argue that foundational issues discussed early in the dissertation suggest that computation plays an irrevocably central role in climate modeling. "Science by simulation" is essential given the complexity of the global climate, but features of the climate system--the presence of non-linearities, feedback loops, and chaotic dynamics--put principled limits on the effectiveness of computational models. This tension is at the root of the staggering pluralism of the climate model hierarchy, and suggests that such pluralism is here to stay, rather than an artifact of our ignorance. Rather than attempting to converge on a single "best fit" climate model, we ought to embrace the diversity of climate models, and view each as a specialized tool designed to predict and explain a rather narrow range of phenomena. Understanding the climate system as a whole requires examining a number of different models, and correlating their outputs. This is the most significant methodological challenge of climatology. -/- Climatology's role contemporary political discourse raises an unusually high number of evaluative questions for a physical science. The two leading approaches to crafting policy surrounding climate change center on mitigation (i.e. stopping the changes from occurring) and adaptation (making post hoc changes to ameliorate the harm caused by those changes). Crafting an effective socio-political response to the threat of anthropogenic climate change, however, requires us to integrate multiple perspectives and values: the proper response will be just as diverse and pluralistic as the climate models themselves, and will incorporate aspects of both approaches. I conclude by offering some concrete recommendations about how to integrate this value pluralism into our socio-political decision making framework. (shrink)

This paper addresses the theoretical notion of a game as it arisesacross scientific inquiries, exploring its uses as a technical andformal asset in logic and science versus an explanatory mechanism. Whilegames comprise a widely used method in a broad intellectual realm(including, but not limited to, philosophy, logic, mathematics,cognitive science, artificial intelligence, computation, linguistics,physics, economics), each discipline advocates its own methodology and aunified understanding is lacking. In the first part of this paper, anumber of game theories (...) in formal studies are critically surveyed. Inthe second part, the doctrine of games as explanations for logic isassessed, and the relevance of a conceptual analysis of games tocognition discussed. It is suggested that the notion of evolution playsa part in the game-theoretic concept of meaning. (shrink)

We explore the prospects of a monist account of explanation for both non-causal explanations in science and pure mathematics. Our starting point is the counterfactual theory of explanation for explanations in science, as advocated in the recent literature on explanation. We argue that, despite the obvious differences between mathematical and scientific explanation, the CTE can be extended to cover both non-causal explanations in science and mathematical explanations. In particular, a successful application (...) of the CTE to mathematical explanations requires us to rely on counterpossibles. We conclude that the CTE is a promising candidate for a monist account of explanation in both science and mathematics. (shrink)

Instances of explanatory reduction are often advocated on metaphysical grounds; given that the only real things in the world are subatomic particles and their interaction, we have to try to explain everything in terms of the laws of physics. In this paper, we show that explanatory reduction cannot be defended on metaphysical grounds. Nevertheless, indispensability arguments for reductive explanations can be developed, taking into account actual scientific practice and the role of epistemic interests. Reductive explanations (...) might be indispensable to address some epistemic interest answering a specific explanation-seeking question in the most accurate, adequate and efficient way. Just like explanatory pluralists often advocate the indispensability of higher levels of explanation pointing at the pragmatic value of the explanatory information obtained on these higher levels, we argue that explanatory reduction—traditionally understood as the contender of pluralism—can be defended in a similar way. The pragmatic value reductionist, lower level explanations might have in the biomedical sciences and the social sciences is illustrated by some case studies. (shrink)

In this paper, I argue that the newly developed network approach in neuroscience and biology provides a basis for formulating a unique type of realization, which I call topological realization. Some of its features and its relation to one of the dominant paradigms of realization and explanation in sciences, i.e. the mechanistic one, are already being discussed in the literature. But the detailed features of topological realization, its explanatory power and its relation to another prominent view of realization, namely (...) the semantic one, have not yet been discussed. I argue that topological realization is distinct from mechanistic and semantic ones because the realization base in this framework is not based on local realisers, regardless of the scale but on global realizers. In mechanistic approach, the realization base is always at the local level, in both ontic and epistemic accounts. The explanatory power of realization relation in mechanistic approach comes directly from the realization relation-either by showing how a model is mapped onto a mechanism, or by describing some ontic relations that are explanatory in themselves. Similarly, the semantic approach requires that concepts at different scales logically satisfy microphysical descriptions, which are at the local level. In topological framework the realization base can be found at different scales, but whatever the scale the realization base is global, within that scale, and not local. Furthermore, topological realization enables us to answer the “why” questions, which according to Polger 2010 make it explanatory. The explanatoriness of topological realization stems from understanding mathematical consequences of different topologies, not from the mere fact that a system realizes them. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:Thales of Miletus: The Beginnings of Western Science and PhilosophyKevin RobbPatricia F. O’Grady. Thales of Miletus: The Beginnings of Western Science and Philosophy. Aldershot, England: Ashgate, 2002. Pp xxii + 310. Paper, $84.95.This book has a consistent thesis: Thales of Miletus was the first Western scientist and philosopher not just for what he began, but for what he himself said (or, as O'Grady believes, wrote). (...) On this view, Thales made significant contributions to metaphysics ("He asked the question: 'What is real?' no less than Plato and Aristotle did," 249), cosmology (on the earth and earthquakes, 87 ff.), astronomy (notably the eclipse, 126 ff.), mathematics (Proclus attributes Euclidean theorems to Thales, 191 ff.), psychology (on psyche) and theology (on "full of the gods," 108 ff.). O'Grady's thesis is that we know enough about Thales' thoughts in these areas (each a treacherous minefield of scholarly dispute, as O'Grady admits) that, after a clarified understanding of "Scientificity and Rationality" (her awkward title for Chapter 11), we can confidently conclude that philosophy and science were well begun by Thales. "I believe I have drawn a clear and detailed picture of a brilliant, veracious, and courageously speculative Greek. I see him as a questioner of almost everything, an investigator of nature...Thales is not the mere shadowy figure of Guthrie's perception. He is a man of substance, a man of distinction and of lasting importance" (251).O'Grady had early noted that W. K. C. Guthrie considered Thales a shadowy figure historically, made all the more so, Guthrie believed, because the legend kept expanding. For example, a sixth-century Thales as the seeker after the material archê, and so the first important physikos, is unknown before fourth-century Aristotle's Metaphysics Alpha, and by the time Euclidean theorems are attributed to Thales the tendency to foist imposing discoveries on famous persons was prevalent.This raises the major issue concerning O'Grady's exhaustively researched book. To what extent can the doxographic sources (the "A" sections of Diels's 1903Die Fragmente der Vorsokratiker), be used to interpret the Presocratics, especially if the original wording from a thinker perished (any "B" section quotations). It is clear that in the fourth century nothing in writing from Thales survived for the Lyceum to read and catalogue although its scholars clearly sought it; the repeated legetai ("it is said") of Aristotle in Metaphysics Alpha and elsewhere is decisive. It is his standard way of indicating that no written work on the matter survives to be consulted, but stories or oral accounts do.O'Grady's thesis requires that the "A" section material be exploited fully, especially to present Thales as a philosopher (29-70). Many historians will be unwilling to take the entire journey with her. Some (Harold Cherniss) would balk at the first step. The demand for confirming ipsissima verba (or "B" quotations) to reconstruct the thought of any Presocratic with real confidence (as once proposed by G. S. Kirk) threatens to exclude Thales from histories of philosophy, an unsatisfactory result. Even so, Harold Cherniss proposed that drastic solution in 1935 in his magisterial Aristotle's Criticism of Presocratric Philosophy. Presocratic philosophy, he argued, should begin with Anaximander (375).The heart of Cherniss's numbingly exhaustive argument (400 + pages) was that Aristotle's purpose in surveying his predecessors's views was never historical, but dialectical. His intent was to argue for a fourfold causality (on the grounds only four archai had been turned up by all his predecessors, starting with Thales' material cause), and to fit those predecessors's views into a theory of natural causality as he had finally formulated it in his sophisticated, [End Page 107] largely technical vocabulary. Aristotle's "historical" notices on which all later accounts must depend were, thus, never history; the problem and the solution, along with their language, were Aristotle's alone. If Theophrastus paraphrased what he found in Aristotle (McDiarmid), and if the post-Aristotelian doxographic notices were largely dependent on Theophrastus (so already Diels in 1879, in Doxographi Graeci), then any attempt to extract a philosophical Thales from the later doxographic sources (or Aristotle) is methodologically doomed from... (shrink)

Abstract This essay will argue systematically and from a historical perspective that there is something to be said for the traditional claim that the human and natural sciences are distinct epistemic practices. Yet, in light of recent developments in contemporary philosophy of science, one has to be rather careful in utilizing the distinction between understanding and explanation for this purpose. One can only recognize the epistemic distinctiveness of the human sciences by recognizing the epistemic (...) centrality of reenactive empathy for our understanding of rational agency, that is, by emphasizing the psychological component in the concept of understanding that nineteenth-century philosophers like Droysen, in contrast to twentieth-century hermeneutic philosophers, still acknowledged. In addition, the essay will show in detail that merely pointing to the fact that narratives have a cognitive function in the domain of the human sciences, as is common among philosophers of history, does not provide us with a sufficient demarcation criterion for distinguishing between the human and natural sciences. (shrink)

Both Carnap and Quine made significant contributions to the philosophy of mathematics despite their diversed views. Carnap endorsed the dichotomy between analytic and synthetic knowledge and classified certain mathematical questions as internal questions appealing to logic and convention. On the contrary, Quine was opposed to the analytic-synthetic distinction and promoted a holistic view of scientific inquiry. The purpose of this paper is to argue that in light of the recent advancement of experimental mathematics such as Monte Carlo simulations, (...) limiting mathematical inquiry to the domain of logic is unjustified. Robustness studies implemented in Monte Carlo Studies demonstrate that mathematics is on par with other experimental-based sciences. (shrink)

Distinctively mathematical explanations (DMEs) explain natural phenomena primarily by appeal to mathematical facts. One important question is whether there can be an ontic account of DME. An ontic account of DME would treat the explananda and explanantia of DMEs as ontic structures and the explanatory relation between them as an ontic relation (e.g., Pincock 2015, Povich 2021). Here I present a conventionalist account of DME, defend it against objections, and argue that it should be considered (...) ontic. Notably, if indeed it is ontic, the conventionalist account seems to avoid a convincing objection to other ontic accounts (Kuorikoski 2021). (shrink)

A way to argue that something 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 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. (shrink)

Can mathematics contribute to our understanding of physical phenomena? One way to try to answer this question is by getting involved in the recent philosophical dispute about the existence of mathematical explanations of physical phenomena. If there is such a thing, given the relation between explanation and understanding, we can say that there is an affirmative answer to our question. But what if we do not agree that mathematics can play an explanatory role in (...) class='Hi'>science? Can we still consider that the above question can have an affirmative answer? My main aim here is to give an account that takes mathematics, in some of the cases discussed in the literature, as contributing to our understanding of physical phenomena despite not being explanatory. (shrink)

Does mathematics ever play an explanatory role in science? If so then this opens the way for scientific realists to argue for the existence of mathematical entities using inference to the best explanation. Elsewhere I have argued, using a case study involving the prime-numbered life cycles of periodical cicadas, that there are examples of indispensable mathematical explanations of purely physical phenomena. In this paper I respond to objections to this claim that have been made by (...) various philosophers, and I discuss potential future directions of research for each side in the debate over the existence of abstract mathematical objects. (shrink)

The philosophical analysis of mathematical explanations concerns itself with two different, although connected, areas of investigation. The first area addresses the problem of whether mathematics can play an explanatory role in the natural and social sciences. The second deals with the problem of whether mathematical explanations occur within mathematics itself. Accordingly, this entry surveys the contributions to both areas, it shows their relevance to the history of philosophy and science, it articulates their connection, (...) and points to the philosophical pay-offs to be expected by deepening our understanding of the topic. (shrink)