Philosophers who regard some mathematical proofs as explaining why theorems hold, and others as m... more Philosophers who regard some mathematical proofs as explaining why theorems hold, and others as merely proving that they do hold, disagree sharply about the explanatory value of proofs by mathematical induction. I offer an argument that aims to resolve this conflict of intuitions without making any controversial presuppositions about what mathematical explanations would be. 1. The problem Proposed accounts of scientific explanation have long been tested against certain canonical examples. Various arguments are paradigmatically explanatory, such as Newton's explanations of Kepler's laws and the tides, Darwin's explanations of various biogeographical and anatomical facts, Wegener's explanation of the correspondence between the South American and African coastlines, and Einstein's explanation of the equality of inertial and gravitational mass. Any promising comprehensive theory of scientific explanation must deem these arguments to be genuinely explanatory. By the same token, there is widespread agreement that various other arguments are not explanatory-examples so standard that I need only give their familiar monikers: the flagpole, the eclipse, the barometer, the hexed salt and so forth. Although there are some controversial cases, of course (such as 'explanations' of the dormitive-virtue variety), philosophers who defend rival accounts of scientific explanation nevertheless agree to a large extent on the phenomena that they are trying to save. Alas, the same cannot be said when it comes to mathematical explanation. Philosophers disagree sharply about which proofs of a given theorem explain why that theorem holds and which merely prove that it holds. 1 This kind of disagreement makes it difficult to test proposed accounts of mathematical explanation without making controversial presuppositions about the phenomena to be saved.
espanolRoche y Sober (2013) han ofrecido un nuevo argumento de neutralizacion [“screening off” ar... more espanolRoche y Sober (2013) han ofrecido un nuevo argumento de neutralizacion [“screening off” argument] en contra de la inferencia a la mejor explicacion [“Inference to the Best Explanation”, “IBE” por sus siglas en ingles]. La forma en que Roche y Sober deben concebir a la IBE para que su argumento sea aplicable queda revelada en su reciente respuesta (2019) a las objeciones de Lange (2017) a su argumento de neutralizacion. El argumento de Roche y Sober en contra de la IBE requiere que bajo la IBE las “virtudes explicativas” que posee una hipotesis H modifiquen la credibilidad que tiene un agente racional en H independientemente de sus opiniones de fondo. Pero es impropio atribuir a la IBE tal conexion magica entre las consideraciones explicativas y la confirmacion. EnglishRoche and Sober (2013) have offered a new, “screening-off” argument against Inference to the Best Explanation (IBE). The way that Roche and Sober must conceive of IBE for their argument to apply is revealed by t...
We can form a six-digit number by taking the three digits on any row, column, or main diagonal on... more We can form a six-digit number by taking the three digits on any row, column, or main diagonal on the keyboard in forward and then in reverse order. For instance, the bottom row taken from left to right, and then right to left,
Kant saw science as presupposing that the natural laws bring maximal diversity under maximal unit... more Kant saw science as presupposing that the natural laws bring maximal diversity under maximal unity. Many philosophers, such as David Lewis, have regarded objective chances as upshots of science's aim at systematic unity-as ideal credences projected onto the world. This Kantian projectivism has seemed the only possible way to account for the rational constraint (codified by the 'Principal Principle') that our credences about chances impose on our credences regarding what they are chances of. This paper examines three ways of elaborating Lewis's Kantian strategy for explaining this rational constraint. After arguing that none of these three approaches is unproblematic, the paper proposes a non-Kantian alternative account according to which a chance measures the strength of a causal tendency.
In “Tense and Reality”, Kit Fine () proposed a novel way to think about realism about tense in th... more In “Tense and Reality”, Kit Fine () proposed a novel way to think about realism about tense in the metaphysics of time. In particular, he explored two non‐standard forms of realism about tense (“external relativism” and “fragmentalism”), arguing that they are to be preferred over standard forms of realism. In the process of defending his own preferred view, fragmentalism, he proposed a fragmentalist interpretation of the special theory of relativity (STR), which will be our focus in this paper. After presenting Fine's position, we will raise a problem for his fragmentalist interpretation of STR. We will argue that Fine's view is in tension with the proper explanation of why various facts (such as the Lorentz transformations) obtain. We will then consider whether similar considerations also speak against fragmentalism in domains other than STR, notably fragmentalism about tense.
THE JOURNAL OF PHILOSOPHY VOLUME XCVIII, NO. 5, MAY 2001-4-0-+ THE MOST FAMOUS EQUATION* typical ... more THE JOURNAL OF PHILOSOPHY VOLUME XCVIII, NO. 5, MAY 2001-4-0-+ THE MOST FAMOUS EQUATION* typical undergraduate physics textbook, written byJoseph W. A Kane and Morton M. Sternheim,' says: The equation E = Mc2 is perhaps the most famous equation of twentiethcentury physics. It is a statement that mass and energy are two forms of the same thing, and that one can be converted into the other (ibid., p. 493). With the first of these sentences, concerning the fame of Albert Einstein's equation, it is difficult to disagree. But the second sentence, which characterizes mass and energy as interconvertible and "two forms of the same thing," suggests that some care will need to be exercised in interpreting the famous equation. Indeed, it is difficult to find a scientific equation whose ontological implications have been misunderstood so widely and in so many ways. I shall begin by looking at some of the issues that arise when interpreting the famous equation. Along the way, I shall examine briefly some of the ways in which one could become confused by the equation's common interpretations-not only in ordinary physics texts, but also by some very distinguished physicists and philosophers. I shall then try to set the record straight.2 I. SOME ISSUES Let us begin to lay out some of these interpretive issues by taking seriously the textbook's remark that mass "can be converted into" * My thanks to Arthur Fine for comments on the penultimate draft of this essay, and to Alexander Rosenberg for his encouragement. 1 Physics (New York: Wiley, 1978). 2 To keep things simple and focused, I shall restrict myself throughout to interpretation of the special theory of relativity, leaving aside considerations arising from the general theory, quantum mechanics, and so forth.
JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, a... more JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact
Explanation in mathematics has recently attracted increased attention from philosophers. The cent... more Explanation in mathematics has recently attracted increased attention from philosophers. The central issue is taken to be how to distinguish between two types of mathematical proofs: those that explain why what they prove is true and those that merely prove theorems without explaining why they are true. This way of framing the issue neglects the possibility of mathematical explanations that are not proofs at all. This paper addresses what it would take for a non-proof to explain. The paper focuses on a particular example of an explanatory non-proof: an argument that mathematicians regard as explaining why a given theorem holds regarding the derivative of an infinite sum of differentiable functions. The paper contrasts this explanatory non-proof with various non-explanatory proofs (and non-explanatory nonproofs) of the same theorem. The paper offers an account of what makes the given non-proof explanatory. This account is motivated by investigating the difficulties that arise when we try to extend Mark Steiner's influential account of explanatory proofs to cover this explanatory non-proof.
Does smoke cause fire or does fire cause smoke? James Woodward’s “Flagpoles anyone? Causal and ex... more Does smoke cause fire or does fire cause smoke? James Woodward’s “Flagpoles anyone? Causal and explanatory asymmetries” argues that various statistical independence relations not only help us to uncover the directions of causal and explanatory relations in our world, but also are the worldly basis of causal and explanatory directions. We raise questions about Woodward’s envisioned epistemology, but our primary focus is on his metaphysics. We argue that any alleged connection between statistical (in)dependence and causal/explanatory direction is contingent, at best. The directions of causal/explanatory relations in our world seem not to depend on the statistical (in)dependence relations in our world (conceived of either as frequency patterns or as relations among chances). Thus, we doubt that statistical (in)dependence relations are the worldly basis of causal and explanatory directions
Why (according to classical physics) do forces compose according to the parallelogram of forces? ... more Why (according to classical physics) do forces compose according to the parallelogram of forces? This question has been controversial; it is one episode in a longstanding, fundamental dispute regarding which facts are not to be explained dynamically. If the parallelogram law is explained statically, then the laws of statics are separate from and (in an important sense) “transcend” the laws of dynamics. Alternatively, if the parallelogram law is explained dynamically, then statical laws become mere corollaries to the dynamical laws. I shall attempt to trace the history of this controversy in order to identify what it would be for one or the other of these rival views to be correct. I shall argue that various familiar accounts of natural law (Lewis’s Best System Account, laws as contingent relations among universals, and scientific essentialism) not only make it difficult to see what the point of this dispute could have been, but also improperly foreclose some serious scientific optio...
THEORIA. An International Journal for Theory, History and Foundations of Science, 2021
Does smoke cause fire or does fire cause smoke? James Woodward’s “Flagpoles anyone? Causal and e... more Does smoke cause fire or does fire cause smoke? James Woodward’s “Flagpoles anyone? Causal and explanatory asymmetries” argues that various statistical independence relations not only help us to uncover the directions of causal and explanatory relations in our world, but also are the worldly basis of causal and explanatory directions. We raise questions about Woodward’s envisioned epistemology, but our primary focus is on his metaphysics. We argue that any alleged connection between statistical (in)dependence and causal/explanatory direction is contingent, at best. The directions of causal/explanatory relations in our world seem not to depend on the statistical (in)dependence relations in our world (conceived of either as frequency patterns or as relations among chances). Thus, we doubt that statistical (in)dependence relations are the worldly basis of causal and explanatory directions.
This paper examines some recent attempts that use counterfactuals to understand the asymmetry of ... more This paper examines some recent attempts that use counterfactuals to understand the asymmetry of non-causal scientific explanations. These attempts recognize that even when there is explanatory asymmetry, there may be symmetry in counterfactual dependence. Therefore, something more than mere counterfactual dependence is needed to account for explanatory asymmetry. Whether that further ingredient, even if applicable to causal explanation, can fit non-causal explanation is the challenge that explanatory asymmetry poses for counterfactual accounts of non-causal explanation. This paper argues that several recent accounts (Woodward, in: Reutlinger and Saatsi (eds) Explanation beyond causation: philosophical perspectives on non-causal explanations,
Wilson (2017) derives various broad philosophical morals from the scientific role played by the P... more Wilson (2017) derives various broad philosophical morals from the scientific role played by the Principle of Virtual Work (PVW). He argues roughly (i) that PVW conditionals cannot be understood in terms of things as large as possible worlds; (ii) that PVW conditionals are peculiar and so cannot be accommodated by general accounts of counterfactuals, thereby reflecting the piecemeal character of scientific practice and standing at odds with the one-size-fits-all approach of Banalytic metaphysicians^; and (iii) that PVW counterfactuals are not made true partly by natural laws. I distinguish, elaborate and critically examine various arguments for these morals suggested by the PVW and Wilson's text, looking especially at what makes a displacement Bvirtual^and the operation of the conditionals that the PVW takes to express necessary and sufficient conditions for equilibrium. Ultimately, I do not find the PVW to be especially well suited to support Wilson's morals; some of these arguments fail, whereas others arise from general considerations rather than having to appeal to anything like the PVW.
Unlike explanation in science, explanation in mathematics has received relatively scant attention... more Unlike explanation in science, explanation in mathematics has received relatively scant attention from philosophers. Whereas there are canonical examples of scientific explanations (as well as canonical examples of nonexplanations, such as “the flagpole,” “the eclipse,” and “the barometer”), there are few (if any) examples that have become widely accepted as exhibiting the distinction between mathematical proofs that explain why some mathematical theorem holds and proofs that merely prove that the theorem holds without revealing the reason why it holds. This essay offers some examples of proofs that mathematicians have considered explanatory (or not), and it argues that these examples suggest a particular account of explanation in mathematics (at least, of those explanations consisting of proofs). The essay compares its account to Steiner's and Kitcher's. Among the topics that arise are proofs that exploit symmetries, mathematical coincidences, brute-force proofs, simplicity...
This paper argues that in at least some cases, one proof of a given theorem is deeper than anothe... more This paper argues that in at least some cases, one proof of a given theorem is deeper than another by virtue of supplying a deeper explanation of the theorem-that is, a deeper account of why the theorem holds. There are cases of scientific depth that also involve a common abstract structure explaining a similarity between two otherwise unrelated phenomena, making their similarity no coincidence and purchasing depth by answering why questions that separate, dissimilar explanations of the two phenomena cannot correctly answer. The connections between explanation, depth, unification, power, and coincidence in mathematics and science are compared. † My thanks to all of the participants at the UCI Workshop on Mathematical Depth for their helpful comments, and especially to Pen Maddy for organizing the workshop. 1 I include the qualification 'in at least some cases' because I have no general argument that the depth of all deep proofs arises from their explanatory power, and it may be that 'depth' is used in
The editorial staff of JHM works hard to make sure the scholarship disseminated in JHM is accurat... more The editorial staff of JHM works hard to make sure the scholarship disseminated in JHM is accurate and upholds professional ethical guidelines. However the views and opinions expressed in each published manuscript belong exclusively to the individual contributor(s). The publisher and the editors do not endorse or accept responsibility for them. See https://scholarship.claremont.edu/jhm/policies.html for more information.
My title is taken from a passage in Richard Feynman's classic book, The Character of Physical Law... more My title is taken from a passage in Richard Feynman's classic book, The Character of Physical Law : When learning about the laws of physics you fi nd that there are a large number of complicated and detailed laws, laws of gravitation, of electricity and magnetism, nuclear interactions, and so on, but across the variety of these detailed laws there sweep great general principles which all the laws seem to follow. Examples of these are the principles of conservation. . .
The British Journal for the Philosophy of Science, 2012
Certain scientific explanations of physical facts have recently been characterized as distinctive... more 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 (since they concern the framework of any possible causal relation) 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 (given the physical arrangement in question) than ordinary causal laws could render it. Distinctively mathematical explanations thus supply a kind of understanding that causal explanations cannot.
Philosophers who regard some mathematical proofs as explaining why theorems hold, and others as m... more Philosophers who regard some mathematical proofs as explaining why theorems hold, and others as merely proving that they do hold, disagree sharply about the explanatory value of proofs by mathematical induction. I offer an argument that aims to resolve this conflict of intuitions without making any controversial presuppositions about what mathematical explanations would be. 1. The problem Proposed accounts of scientific explanation have long been tested against certain canonical examples. Various arguments are paradigmatically explanatory, such as Newton's explanations of Kepler's laws and the tides, Darwin's explanations of various biogeographical and anatomical facts, Wegener's explanation of the correspondence between the South American and African coastlines, and Einstein's explanation of the equality of inertial and gravitational mass. Any promising comprehensive theory of scientific explanation must deem these arguments to be genuinely explanatory. By the same token, there is widespread agreement that various other arguments are not explanatory-examples so standard that I need only give their familiar monikers: the flagpole, the eclipse, the barometer, the hexed salt and so forth. Although there are some controversial cases, of course (such as 'explanations' of the dormitive-virtue variety), philosophers who defend rival accounts of scientific explanation nevertheless agree to a large extent on the phenomena that they are trying to save. Alas, the same cannot be said when it comes to mathematical explanation. Philosophers disagree sharply about which proofs of a given theorem explain why that theorem holds and which merely prove that it holds. 1 This kind of disagreement makes it difficult to test proposed accounts of mathematical explanation without making controversial presuppositions about the phenomena to be saved.
espanolRoche y Sober (2013) han ofrecido un nuevo argumento de neutralizacion [“screening off” ar... more espanolRoche y Sober (2013) han ofrecido un nuevo argumento de neutralizacion [“screening off” argument] en contra de la inferencia a la mejor explicacion [“Inference to the Best Explanation”, “IBE” por sus siglas en ingles]. La forma en que Roche y Sober deben concebir a la IBE para que su argumento sea aplicable queda revelada en su reciente respuesta (2019) a las objeciones de Lange (2017) a su argumento de neutralizacion. El argumento de Roche y Sober en contra de la IBE requiere que bajo la IBE las “virtudes explicativas” que posee una hipotesis H modifiquen la credibilidad que tiene un agente racional en H independientemente de sus opiniones de fondo. Pero es impropio atribuir a la IBE tal conexion magica entre las consideraciones explicativas y la confirmacion. EnglishRoche and Sober (2013) have offered a new, “screening-off” argument against Inference to the Best Explanation (IBE). The way that Roche and Sober must conceive of IBE for their argument to apply is revealed by t...
We can form a six-digit number by taking the three digits on any row, column, or main diagonal on... more We can form a six-digit number by taking the three digits on any row, column, or main diagonal on the keyboard in forward and then in reverse order. For instance, the bottom row taken from left to right, and then right to left,
Kant saw science as presupposing that the natural laws bring maximal diversity under maximal unit... more Kant saw science as presupposing that the natural laws bring maximal diversity under maximal unity. Many philosophers, such as David Lewis, have regarded objective chances as upshots of science's aim at systematic unity-as ideal credences projected onto the world. This Kantian projectivism has seemed the only possible way to account for the rational constraint (codified by the 'Principal Principle') that our credences about chances impose on our credences regarding what they are chances of. This paper examines three ways of elaborating Lewis's Kantian strategy for explaining this rational constraint. After arguing that none of these three approaches is unproblematic, the paper proposes a non-Kantian alternative account according to which a chance measures the strength of a causal tendency.
In “Tense and Reality”, Kit Fine () proposed a novel way to think about realism about tense in th... more In “Tense and Reality”, Kit Fine () proposed a novel way to think about realism about tense in the metaphysics of time. In particular, he explored two non‐standard forms of realism about tense (“external relativism” and “fragmentalism”), arguing that they are to be preferred over standard forms of realism. In the process of defending his own preferred view, fragmentalism, he proposed a fragmentalist interpretation of the special theory of relativity (STR), which will be our focus in this paper. After presenting Fine's position, we will raise a problem for his fragmentalist interpretation of STR. We will argue that Fine's view is in tension with the proper explanation of why various facts (such as the Lorentz transformations) obtain. We will then consider whether similar considerations also speak against fragmentalism in domains other than STR, notably fragmentalism about tense.
THE JOURNAL OF PHILOSOPHY VOLUME XCVIII, NO. 5, MAY 2001-4-0-+ THE MOST FAMOUS EQUATION* typical ... more THE JOURNAL OF PHILOSOPHY VOLUME XCVIII, NO. 5, MAY 2001-4-0-+ THE MOST FAMOUS EQUATION* typical undergraduate physics textbook, written byJoseph W. A Kane and Morton M. Sternheim,' says: The equation E = Mc2 is perhaps the most famous equation of twentiethcentury physics. It is a statement that mass and energy are two forms of the same thing, and that one can be converted into the other (ibid., p. 493). With the first of these sentences, concerning the fame of Albert Einstein's equation, it is difficult to disagree. But the second sentence, which characterizes mass and energy as interconvertible and "two forms of the same thing," suggests that some care will need to be exercised in interpreting the famous equation. Indeed, it is difficult to find a scientific equation whose ontological implications have been misunderstood so widely and in so many ways. I shall begin by looking at some of the issues that arise when interpreting the famous equation. Along the way, I shall examine briefly some of the ways in which one could become confused by the equation's common interpretations-not only in ordinary physics texts, but also by some very distinguished physicists and philosophers. I shall then try to set the record straight.2 I. SOME ISSUES Let us begin to lay out some of these interpretive issues by taking seriously the textbook's remark that mass "can be converted into" * My thanks to Arthur Fine for comments on the penultimate draft of this essay, and to Alexander Rosenberg for his encouragement. 1 Physics (New York: Wiley, 1978). 2 To keep things simple and focused, I shall restrict myself throughout to interpretation of the special theory of relativity, leaving aside considerations arising from the general theory, quantum mechanics, and so forth.
JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, a... more JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact
Explanation in mathematics has recently attracted increased attention from philosophers. The cent... more Explanation in mathematics has recently attracted increased attention from philosophers. The central issue is taken to be how to distinguish between two types of mathematical proofs: those that explain why what they prove is true and those that merely prove theorems without explaining why they are true. This way of framing the issue neglects the possibility of mathematical explanations that are not proofs at all. This paper addresses what it would take for a non-proof to explain. The paper focuses on a particular example of an explanatory non-proof: an argument that mathematicians regard as explaining why a given theorem holds regarding the derivative of an infinite sum of differentiable functions. The paper contrasts this explanatory non-proof with various non-explanatory proofs (and non-explanatory nonproofs) of the same theorem. The paper offers an account of what makes the given non-proof explanatory. This account is motivated by investigating the difficulties that arise when we try to extend Mark Steiner's influential account of explanatory proofs to cover this explanatory non-proof.
Does smoke cause fire or does fire cause smoke? James Woodward’s “Flagpoles anyone? Causal and ex... more Does smoke cause fire or does fire cause smoke? James Woodward’s “Flagpoles anyone? Causal and explanatory asymmetries” argues that various statistical independence relations not only help us to uncover the directions of causal and explanatory relations in our world, but also are the worldly basis of causal and explanatory directions. We raise questions about Woodward’s envisioned epistemology, but our primary focus is on his metaphysics. We argue that any alleged connection between statistical (in)dependence and causal/explanatory direction is contingent, at best. The directions of causal/explanatory relations in our world seem not to depend on the statistical (in)dependence relations in our world (conceived of either as frequency patterns or as relations among chances). Thus, we doubt that statistical (in)dependence relations are the worldly basis of causal and explanatory directions
Why (according to classical physics) do forces compose according to the parallelogram of forces? ... more Why (according to classical physics) do forces compose according to the parallelogram of forces? This question has been controversial; it is one episode in a longstanding, fundamental dispute regarding which facts are not to be explained dynamically. If the parallelogram law is explained statically, then the laws of statics are separate from and (in an important sense) “transcend” the laws of dynamics. Alternatively, if the parallelogram law is explained dynamically, then statical laws become mere corollaries to the dynamical laws. I shall attempt to trace the history of this controversy in order to identify what it would be for one or the other of these rival views to be correct. I shall argue that various familiar accounts of natural law (Lewis’s Best System Account, laws as contingent relations among universals, and scientific essentialism) not only make it difficult to see what the point of this dispute could have been, but also improperly foreclose some serious scientific optio...
THEORIA. An International Journal for Theory, History and Foundations of Science, 2021
Does smoke cause fire or does fire cause smoke? James Woodward’s “Flagpoles anyone? Causal and e... more Does smoke cause fire or does fire cause smoke? James Woodward’s “Flagpoles anyone? Causal and explanatory asymmetries” argues that various statistical independence relations not only help us to uncover the directions of causal and explanatory relations in our world, but also are the worldly basis of causal and explanatory directions. We raise questions about Woodward’s envisioned epistemology, but our primary focus is on his metaphysics. We argue that any alleged connection between statistical (in)dependence and causal/explanatory direction is contingent, at best. The directions of causal/explanatory relations in our world seem not to depend on the statistical (in)dependence relations in our world (conceived of either as frequency patterns or as relations among chances). Thus, we doubt that statistical (in)dependence relations are the worldly basis of causal and explanatory directions.
This paper examines some recent attempts that use counterfactuals to understand the asymmetry of ... more This paper examines some recent attempts that use counterfactuals to understand the asymmetry of non-causal scientific explanations. These attempts recognize that even when there is explanatory asymmetry, there may be symmetry in counterfactual dependence. Therefore, something more than mere counterfactual dependence is needed to account for explanatory asymmetry. Whether that further ingredient, even if applicable to causal explanation, can fit non-causal explanation is the challenge that explanatory asymmetry poses for counterfactual accounts of non-causal explanation. This paper argues that several recent accounts (Woodward, in: Reutlinger and Saatsi (eds) Explanation beyond causation: philosophical perspectives on non-causal explanations,
Wilson (2017) derives various broad philosophical morals from the scientific role played by the P... more Wilson (2017) derives various broad philosophical morals from the scientific role played by the Principle of Virtual Work (PVW). He argues roughly (i) that PVW conditionals cannot be understood in terms of things as large as possible worlds; (ii) that PVW conditionals are peculiar and so cannot be accommodated by general accounts of counterfactuals, thereby reflecting the piecemeal character of scientific practice and standing at odds with the one-size-fits-all approach of Banalytic metaphysicians^; and (iii) that PVW counterfactuals are not made true partly by natural laws. I distinguish, elaborate and critically examine various arguments for these morals suggested by the PVW and Wilson's text, looking especially at what makes a displacement Bvirtual^and the operation of the conditionals that the PVW takes to express necessary and sufficient conditions for equilibrium. Ultimately, I do not find the PVW to be especially well suited to support Wilson's morals; some of these arguments fail, whereas others arise from general considerations rather than having to appeal to anything like the PVW.
Unlike explanation in science, explanation in mathematics has received relatively scant attention... more Unlike explanation in science, explanation in mathematics has received relatively scant attention from philosophers. Whereas there are canonical examples of scientific explanations (as well as canonical examples of nonexplanations, such as “the flagpole,” “the eclipse,” and “the barometer”), there are few (if any) examples that have become widely accepted as exhibiting the distinction between mathematical proofs that explain why some mathematical theorem holds and proofs that merely prove that the theorem holds without revealing the reason why it holds. This essay offers some examples of proofs that mathematicians have considered explanatory (or not), and it argues that these examples suggest a particular account of explanation in mathematics (at least, of those explanations consisting of proofs). The essay compares its account to Steiner's and Kitcher's. Among the topics that arise are proofs that exploit symmetries, mathematical coincidences, brute-force proofs, simplicity...
This paper argues that in at least some cases, one proof of a given theorem is deeper than anothe... more This paper argues that in at least some cases, one proof of a given theorem is deeper than another by virtue of supplying a deeper explanation of the theorem-that is, a deeper account of why the theorem holds. There are cases of scientific depth that also involve a common abstract structure explaining a similarity between two otherwise unrelated phenomena, making their similarity no coincidence and purchasing depth by answering why questions that separate, dissimilar explanations of the two phenomena cannot correctly answer. The connections between explanation, depth, unification, power, and coincidence in mathematics and science are compared. † My thanks to all of the participants at the UCI Workshop on Mathematical Depth for their helpful comments, and especially to Pen Maddy for organizing the workshop. 1 I include the qualification 'in at least some cases' because I have no general argument that the depth of all deep proofs arises from their explanatory power, and it may be that 'depth' is used in
The editorial staff of JHM works hard to make sure the scholarship disseminated in JHM is accurat... more The editorial staff of JHM works hard to make sure the scholarship disseminated in JHM is accurate and upholds professional ethical guidelines. However the views and opinions expressed in each published manuscript belong exclusively to the individual contributor(s). The publisher and the editors do not endorse or accept responsibility for them. See https://scholarship.claremont.edu/jhm/policies.html for more information.
My title is taken from a passage in Richard Feynman's classic book, The Character of Physical Law... more My title is taken from a passage in Richard Feynman's classic book, The Character of Physical Law : When learning about the laws of physics you fi nd that there are a large number of complicated and detailed laws, laws of gravitation, of electricity and magnetism, nuclear interactions, and so on, but across the variety of these detailed laws there sweep great general principles which all the laws seem to follow. Examples of these are the principles of conservation. . .
The British Journal for the Philosophy of Science, 2012
Certain scientific explanations of physical facts have recently been characterized as distinctive... more 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 (since they concern the framework of any possible causal relation) 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 (given the physical arrangement in question) than ordinary causal laws could render it. Distinctively mathematical explanations thus supply a kind of understanding that causal explanations cannot.
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