This article analyzes the epistemic value of understanding and offers an account of the role of understanding in science. First, I discuss the objectivist view of the relation between explanation and understanding, defended by Carl Hempel and J. D. Trout. I challenge this view by arguing that pragmatic aspects of explanation are crucial for achieving the epistemic aims of science. Subsequently, I present an analysis of these pragmatic aspects in terms of ‘intelligibility’ and a contextual account of scientific understanding based (...) on this notion. †To contact the author, please write to: Faculty of Philosophy, VU University Amsterdam, De Boelelaan 1105, 1081 HV Amsterdam, The Netherlands; e‐mail: [email protected]. (shrink)

Since the beginning of the 20th century, philosophers of science have asked, "what kind of thing is a scientific theory?" The logical positivists answered: a scientific theory is a mathematical theory, plus an empirical interpretation of that theory. Moreover, they assumed that a mathematical theory is specified by a set of axioms in a formal language. Later 20th century philosophers questioned this account, arguing instead that a scientific theory need not include a mathematical component; or that the mathematical component need (...) not be specified by a set of axioms in a formal language. We survey various accounts of scientific theories entertained in the 20th century -- removing some misconceptions, and clearing a path for future research. (shrink)

One can (for the most part) formulate a model of a classical system in either the Lagrangian or the Hamiltonian framework. Though it is often thought that those two formulations are equivalent in all important ways, this is not true: the underlying geometrical structures one uses to formulate each theory are not isomorphic. This raises the question of whether one of the two is a more natural framework for the representation of classical systems. In the event, the answer is yes: (...) I state and sketch proofs of two technical results—inspired by simple physical arguments about the generic properties of classical systems—to the effect that, in a precise sense, classical systems evince exactly the geometric structure Lagrangian mechanics provides for the representation of systems, and none provided by Hamiltonian. The argument not only clarifies the conceptual structure of the two systems of mechanics, but also their relations to each other and their respective mechanisms for representing physical systems. It also shows why naïvely structural approaches to the representational content of physical theories cannot work. [Lagrange] grasped that he had gained a method of stating dynamical truths in a way, which is perfectly indifferent to the particular methods of measurement employed in fixing the positions of the various parts of the system. Accordingly, he went on to deduce equations of motion, which are equally applicable whatever quantitative measurements have been made, provided that they are adequate to fix positions. The beauty and almost divine simplicity of these equations is such that these formulae are worthy to rank with those mysterious symbols which in ancient times were held directly to indicate the Supreme Reason at the base of all things. (Whitehead [1948], p. 63)1. Introduction2.Classical Systems3. The Possible Interactions of a Classical System and the Structure of Its Space of States4. Classical Systems Are Lagrangian5. Classical Systems Are Not Hamiltonian6. How Lagrangian and Hamiltonian Mechanics Represent Classical Systems7. The Conceptual Structure of Classical Mechanics. (shrink)

In this article, I examine whether or not the Hamiltonian and Lagrangian formulations of classical mechanics are equivalent theories. I do so by applying a standard for equivalence that was recently introduced into philosophy of science by Halvorson and Weatherall. This case study yields three general philosophical payoffs. The first concerns what a theory is, while the second and third concern how we should interpret what our physical theories say about the world. 1Introduction 2When Are Two Theories Equivalent? 3Preliminaries on (...) Classical Mechanics 3.1Hamiltonian mechanics 3.2Lagrangian mechanics 4Are Hamiltonian and Lagrangian Mechanics Equivalent Theories? 4.1Tangent bundle versus cotangent bundle 4.2Tangent bundle versus symplectic manifold 4.3Lagrangian vector field versus Hamiltonian vector field 5Conclusion Appendix. (shrink)

We contend that diagrams are tools not only for communication but also for supporting the reasoning of biologists. In the mechanistic research that is characteristic of biology, diagrams delineate the phenomenon to be explained, display explanatory relations, and show the organized parts and operations of the mechanism proposed as responsible for the phenomenon. Both phenomenon diagrams and explanatory relations diagrams, employing graphs or other formats, facilitate applying visual processing to the detection of relevant patterns. Mechanism diagrams guide reasoning about how (...) the parts and operations work together to produce the phenomenon and what experiments need to be done to improve on the existing account. We examine how these functions are served by diagrams in circadian rhythm research. (shrink)

Woodward's long awaited book is an attempt to construct a comprehensive account of causation explanation that applies to a wide variety of causal and explanatory claims in different areas of science and everyday life. The book engages some of the relevant literature from other disciplines, as Woodward weaves together examples, counterexamples, criticisms, defenses, objections, and replies into a convincing defense of the core of his theory, which is that we can analyze causation by appeal to the notion of manipulation.

I argue that a criterion of theoretical equivalence due to Glymour :227–251, 1977) does not capture an important sense in which two theories may be equivalent. I then motivate and state an alternative criterion that does capture the sense of equivalence I have in mind. The principal claim of the paper is that relative to this second criterion, the answer to the question posed in the title is “yes”, at least on one natural understanding of Newtonian gravitation.

In this paper, we examine the relationship between general relativity and the theory of Einstein algebras. We show that according to a formal criterion for theoretical equivalence recently proposed by Halvorson and Weatherall, the two are equivalent theories.

Achieving understanding of nature is one of the aims of science. In this paper we offer an analysis of the nature of scientific understanding that accords with actual scientific practice and accommodates the historical diversity of conceptions of understanding. Its core idea is a general criterion for the intelligibility of scientific theories that is essentially contextual: which theories conform to this criterion depends on contextual factors, and can change in the course of time. Our analysis provides a general account of (...) how understanding is provided by scientific explanations of diverse types. In this way, it reconciles conflicting views of explanatory understanding, such as the causal-mechanical and the unificationist conceptions. (shrink)

In this article, I use a mental models computational account of representation to illustrate some details of my previously presented inferential model of scientific understanding. The hope is to shed some light on possible mechanisms behind the notion of scientific understanding. I argue that if mental models are a plausible approach to modelling cognition, then understanding can best be seen as the coupling of specific rules. I present our beliefs as ?ordinary? conditional rules, and the coupling process as one where (...) the consequent of one ordinary rule (OR) matches and activates the antecedent of the rule to which it is coupled in virtue of the activation of an intermediate ?inference? rule. I argue that on this approach knowledge of an explanation is the activation of ORs in a cognitive hierarchy, while understanding is achieved when those activated ORs are also coupled via correct inference rules. I do not directly address issues regarding the plausibility of mental models themselves. This article should therefore be seen as an exercise in refining the inferential model within an already presupposed computational setting, not one of arguing for the psychological adequacy of computational approaches. (shrink)

Peter Lipton has argued that understanding can exist in the absence of explanation. We argue that this does not denigrate explanation's importance to understanding. Specifically, we show that all of Lipton's examples are consistent with the idea that explanation is the ideal of understanding, i.e. other modes of understanding ought to be assessed by how well they replicate the understanding provided by a good and correct explanation. We defend this idea by showing that for all of Lipton's examples of non-explanatory (...) understanding of why p , there exists a correct and reasonably good explanation that would provide greater understanding of p. (shrink)

Recently, several authors have argued that scientific understanding should be a new topic of philosophical research. In this article, I argue that the three most developed accounts of understanding--Grimm's, de Regt's, and de Regt and Dieks's--can be replaced by earlier accounts of scientific explanation without loss. Indeed, in some cases, such replacements have clear benefits.

In his “EMU and Inference,” Mark Newman European Journal for Philosophy of Science, 4:55–74, 2014 provides several interesting challenges to my explanatory model of understanding :15–37, 2012). I offer three replies to Newman’s paper. First, Newman incorrectly attributes to EMU an overly restrictive view about the role of abilities in understanding. Second, his main argument against EMU rests on this incorrect attribution, and would still face difficulties even if this attribution were correct. Third, contrary to his stated ambitions, his own, (...) inferential model of understanding does not have any distinctive advantages over EMU. These three points defend EMU against Newman’s objections. (shrink)

Science frequently gives us multiple, compatible ways of solving the same problem or formulating the same theory. These compatible formulations change our understanding of the world, despite providing the same explanations. According to what I call "conceptualism," reformulations change our understanding by clarifying the epistemic structure of theories. I illustrate conceptualism by analyzing a typical example of symmetry-based reformulation in chemical physics. This case study poses a problem for "explanationism," the rival thesis that differences in understanding require ontic explanatory differences. (...) To defend conceptualism, I consider how prominent accounts of explanation might accommodate this case study. I argue that either they do not succeed, or they generate a skeptical challenge. (shrink)

I argue that understanding why p involves a kind of intellectual know how and differsfrom both knowledge that p and knowledge why p (as they are standardly understood).I argue that understanding, in this sense, is valuable.

I defend the claim that understanding is the goal of explanation against various persistent criticisms, especially the criticism that understanding is not truth-connected in the appropriate way, and hence is a merely psychological state. Part of the reason why understanding has been dismissed as the goal of explanation, I suggest, is because the psychological dimension of the goal of explanation has itself been almost entirely neglected. In turn, the psychological dimension of understanding—the Aha! experience, the sense that a certain explanation (...) “feels right”, and so on—has been conspicuously overemphasized. I try to correct for both of these exaggerations. Just as the goal of explanation includes a richer psychological—including phenomenological—dimension than is generally acknowledged, so too understanding has a stronger truth connection than is generally acknowledged.Keywords: Understanding; Explanation; Philosophy of science; Epistemology. (shrink)

Not all scientific explanations work by describing causal connections between events or the world's overall causal structure. In addition, mathematicians regard some proofs as explaining why the theorems being proved do in fact hold. This book proposes new philosophical accounts of many kinds of non-causal explanations in science and mathematics.

This book first argues that what philosophers are really after, or at least should be after, when they seek a theory of explanation, is a theory of answers to why-questions. The book's main thesis, then, is a theory of reasons why. Every reason why some event happened is either a cause, or a ground, of that event. Challenging this thesis are many examples philosophers have thought they have found of "non-causal explanations." Reasons Why uses two ideas to show that these (...) examples are not counterexamples to the theory it defends. First is the idea that not every part of a good response to a why-question is part of an answer to that why-question. Second is the idea that not every reason why something is a reason why an event happened is itself a reason why that event happened. (shrink)

In order to perfectly describe the world, it is not enough to speak truly. One must also use the right concepts - including the right logical concepts. One must use concepts that "carve at the joints", that give the world's "structure". There is an objectively correct way to "write the book of the world". Much of metaphysics, as traditionally conceived, is about the fundamental nature of reality; in the present terms, this is about the world's structure. Metametaphysics - inquiry into (...) the status of metaphysical questions - turns on structure. The question of whether ontological, causal, or modal questions are "substantive" is in large part a question of whether the world has ontological, causal, and modal structure - whether quantifiers, causal relations, and modal operators carve at the joints. (shrink)

Approaches to explanation -- Causal and explanatory relevance -- The kairetic account of /D making -- The kairetic account of explanation -- Extending the kairetic account -- Event explanation and causal claims -- Regularity explanation -- Abstraction in regularity explanation -- Approaches to probabilistic explanation -- Kairetic explanation of frequencies -- Kairetic explanation of single outcomes -- Looking outward -- Looking inward.

We are used to talking about the “structure” posited by a given theory of physics, such as the spacetime structure of relativity. What is “structure”? What does the mathematical structure used to formulate a theory tell us about the physical world according to the theory? What if there are different mathematical formulations of a given theory? Do different formulations posit different structures, or are they merely notational variants? I consider the case of Lagrangian and Hamiltonian classical mechanics. I argue that, (...) contrary to standard wisdom, these are not genuinely equivalent theories: they differ in statespace structure. I suggest that we should be realists about statespace structure. (shrink)

I review the philosophical literature on the question of when two physical theories are equivalent. This includes a discussion of empirical equivalence, which is often taken to be necessary, and sometimes taken to be sufficient, for theoretical equivalence; and "interpretational" equivalence, which is the idea that two theories are equivalent just in case they have the same interpretation. It also includes a discussion of several formal notions of equivalence that have been considered in the recent philosophical literature, including definitional equivalence (...) and categorical equivalence. The article concludes with a brief discussion of the relationship between equivalence and duality. (shrink)

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According to one large family of views, scientific explanations explain a phenomenon (such as an event or a regularity) by subsuming it under a general representation, model, prototype, or schema (see Bechtel, W., & Abrahamsen, A. (2005). Explanation: A mechanist alternative. Studies in History and Philosophy of Biological and Biomedical Sciences, 36(2), 421–441; Churchland, P. M. (1989). A neurocomputational perspective: The nature of mind and the structure of science. Cambridge: MIT Press; Darden (2006); Hempel, C. G. (1965). Aspects of scientific (...) explanation. In C. G. Hempel (Ed.), Aspects of scientific explanation (pp. 331–496). New York: Free Press; Kitcher (1989); Machamer, P., Darden, L., & Craver, C. F. (2000). Thinking about mechanisms. Philosophy of Science, 67(1), 1–25). My concern is with the minimal suggestion that an adequate philosophical theory of scientific explanation can limit its attention to the format or structure with which theories are represented. The representational subsumption view is a plausible hypothesis about the psychology of understanding. It is also a plausible claim about how scientists present their knowledge to the world. However, one cannot address the central questions for a philosophical theory of scientific explanation without turning one’s attention from the structure of representations to the basic commitments about the worldly structures that plausibly count as explanatory. A philosophical theory of scientific explanation should achieve two goals. The first is explanatory demarcation. It should show how explanation relates with other scientific achievements, such as control, description, measurement, prediction, and taxonomy. The second is explanatory normativity. It should say when putative explanations succeed and fail. One cannot achieve these goals without undertaking commitments about the kinds of ontic structures that plausibly count as explanatory. Representations convey explanatory information about a phenomenon when and only when they describe the ontic explanations for those phenomena. (shrink)

A classic result in the foundations of Yang-Mills theory, due to J. W. Barrett ["Holonomy and Path Structures in General Relativity and Yang-Mills Theory." Int. J. Th. Phys. 30, ], establishes that given a "generalized" holonomy map from the space of piece-wise smooth, closed curves based at some point of a manifold to a Lie group, there exists a principal bundle with that group as structure group and a principal connection on that bundle such that the holonomy map corresponds to (...) the holonomies of that connection. Barrett also provided one sense in which this "recovery theorem" yields a unique bundle, up to isomorphism. Here we show that something stronger is true: with an appropriate definition of isomorphism between generalized holonomy maps, there is an equivalence of categories between the category whose objects are generalized holonomy maps on a smooth, connected manifold and whose arrows are holonomy isomorphisms, and the category whose objects are principal connections on principal bundles over a smooth, connected manifold. This result clarifies, and somewhat improves upon, the sense of "unique recovery" in Barrett's theorems; it also makes precise a sense in which there is no loss of structure involved in moving from a principal bundle formulation of Yang-Mills theory to a holonomy, or "loop", formulation. (shrink)

This paper expounds the relations between continuous symmetries and conserved quantities, i.e. Noether's ``first theorem'', in both the Lagrangian and Hamiltonian frameworks for classical mechanics. This illustrates one of mechanics' grand themes: exploiting a symmetry so as to reduce the number of variables needed to treat a problem. I emphasise that, for both frameworks, the theorem is underpinned by the idea of cyclic coordinates; and that the Hamiltonian theorem is more powerful. The Lagrangian theorem's main ``ingredient'', apart from cyclic coordinates, (...) is the rectification of vector fields afforded by the local existence and uniqueness of solutions to ordinary differential equations. For the Hamiltonian theorem, the main extra ingredients are the asymmetry of the Poisson bracket, and the fact that a vector field generates canonical transformations iff it is Hamiltonian. (shrink)