Preface. PART I BACKGROUND MATERIAL. 1 Sets and Functions. 1.1 Sets in General. 1.2 Sets of Numbe... more Preface. PART I BACKGROUND MATERIAL. 1 Sets and Functions. 1.1 Sets in General. 1.2 Sets of Numbers. 1.3 Functions. 2 Real Numbers. 2.1 Review of the Order Relations. 2.2 Completeness of Real Numbers. 2.3 Sequences of Real Numbers. 2.4 Subsequences. 2.5 Series of Real Numbers. 2.6 Intervals and Connected Sets. 3 Vector Functions. 3.1 Vector Spaces: The Basics. 3.2 Bilinear Functions. 3.3 Multilinear Functions. 3.4 Inner Products. 3.5 Orthogonal Projections. 3.6 Spectral Theorem. PART II DIFFERENTIATION. 4 Normed Vector Spaces. 4.1 Preliminaries. 4.2 Convergence in Normed Spaces. 4.3 Norms of Linear and Multilinear Transformations. 4.4 Continuity in Normed Spaces. 4.5 Topology of Normed Spaces. 5 Derivatives. 5.1 Functions of a Real Variable. 5.2 Differentiable Functions. 5.3 Existence of Derivatives. 5.4 Partial Derivatives. 5.5 Rules of Differentiation. 5.6 Differentiation of Products. 6 Diffeomorphisms and Manifolds. 6.1 The Inverse Function Theorem. 6.2 Graphs. 6.3 Manifolds in Parametric Representations. 6.4 Manifolds in Implicit Representations. 6.5 Differentiation on Manifolds. 7 Higher-Order Derivatives. 7.1 Definitions. 7.2 Change of Order in Differentiation. 7.3 Sequences of Polynomials. 7.4 Local Extremal Values. PART III INTEGRATION. 8 Multiple Integrals. 8.1 Jordan Sets and Volume. 8.2 Integrals. 8.3 Images of Jordan Sets. 8.4 Change of Variables. 9 Integration on Manifolds. 9.1 Euclidean Volumes. 9.2 Integration on Manifolds. 9.3 Oriented Manifolds. 9.4 Integrals of Vector Fields. 9.5 Integrals of Tensor Fields. 9.6 Integration on Graphs. 10 Stokes' Theorem. 10.1 Basic Stokes' Theorem. 10.2 Flows. 10.3 Flux and Change of Volume in a Flow. 10.4 Exterior Derivatives. 10.5 Regular and Almost Regular Sets. 10.6 Stokes' Theorem on Manifolds. PART IV APPENDICES. Appendix A: Construction of the Real Numbers. A.1 Field and Order Axioms in Q. A.2 Equivalence Classes of Cauchy Sequences in Q. A.3 Completeness of R. Appendix B: Dimension of a Vector Space. B.1 Bases and Linearly Independent Subsets. Appendix C: Determinants. C.1 Permutations. C.2 Determinants of Square Matrices. C.3 Determinant Functions. C.4 Determinant of a Linear Transformation. C.5 Determinants on Cartesian Products. C.6 Determinants in Euclidean Spaces. C.7 Trace of an Operator. Appendix D: Partitions of Unity. D.1 Partitions of Unity. Index.
A rigorous introduction to calculus in vector spaces The concepts and theorems of advanced calcul... more A rigorous introduction to calculus in vector spaces The concepts and theorems of advanced calculus combined with related computational methods are essential to understanding nearly all areas of quantitative science. Analysis in Vector Spaces presents the central results of this classic subject through rigorous arguments, discussions, and examples. The book aims to cultivate not only knowledge of the major theoretical results, but also the geometric intuition needed for both mathematical problem-solving and modeling in the formal sciences. The authors begin with an outline of key concepts, terminology, and notation and also provide a basic introduction to set theory, the properties of real numbers, and a review of linear algebra. An elegant approach to eigenvector problems and the spectral theorem sets the stage for later results on volume and integration. Subsequent chapters present the major results of differential and integral calculus of several variables as well as the theory o...
In analogical reasoning, observations about one or more source domains provide varying degrees of... more In analogical reasoning, observations about one or more source domains provide varying degrees of support for a conjecture about a target domain. Norton (2021) challenges the usefulness of formal models of analogical inference. Other philosophers (Dardashti et al 2019) develop just such formal models in order to show how analogue experiments can confirm a hypothesis, even when the target domain is inaccessible. This paper defends the value of quasi-formal models of analogical reasoning. Such models are broadly compatible with Norton’s position, but help to clarify the structure of analogical reasoning and to identify basic requirements for a good analogical inference.
The precautionary principle is often argued to be irrational because it cannot adequately explain... more The precautionary principle is often argued to be irrational because it cannot adequately explain how resources should be distributed across multiple possible catastrophes or between catastrophic and noncatastrophic risks. We address this problem of trade-offs by extending a recently proposed formal interpretation of the precautionary principle (PP) within a lexical utility framework and using it to prove results about which distribution of resources maximizes lexical utility when several catastrophic risks exist, given different assumptions. We also explain how our lexical utility interpretation of PP can recommend balanced distributions of resources between disaster prevention and other concerns.
This paper argues that analogical arguments, especially in science, are often employed to show th... more This paper argues that analogical arguments, especially in science, are often employed to show that their conclusions are prima facie plausible, or serious possibilities. Prima facie plausibility is not a matter of degree; rather, it implies the existence of a threshold below which analogical arguments provide no justification for their conclusions. This position is supported by means of analogical arguments in mathematics. The paper then argues that structure-mapping theories cannot easily accommodate the notion of prima facie plausibility.
Abstract: The Principle of Indifference, which dictates that we ought to assign two outcomes equa... more Abstract: The Principle of Indifference, which dictates that we ought to assign two outcomes equal probability in the absence of known reasons to do otherwise, is vulnerable to well-known objections. Nevertheless, the appeal of the principle, and of symmetry-based assignments of equal probability, persists. We show that, relative to a given class of symmetries satisfying certain properties, we are justified in calling certain outcomes equally probable, and more generally, in defining what we call relative probabilities. Relative probabilities are useful in providing a generalized approach to conditionalization. The technique is illustrated by application to simple examples. Word Count: 4,948 1. Introduction. A recent statement of the classical Principle of Indifference (PI) runs as follows: If there are n mutually exclusive possibilities h1, …, hn, and e gives no more reason to believe any one of these more likely to be true than any other, then P(hi / e) is the same for all i. 1
How can self-locating propositions be integrated into normal patterns of belief revision? Puzzles... more How can self-locating propositions be integrated into normal patterns of belief revision? Puzzles such as Sleeping Beauty seem to show that such propositions lead to violation of ordinary principles for reasoning with subjective probability, such as Conditionalization and Reflection. I show that sophisticated forms of Conditionalization and Reflection are not only compatible with self-locating propositions, but also indispensable in understanding how they can function as evidence in Sleeping Beauty and similar cases.
We examine a distinctive kind of problem for decision theory, involving what we call discontinuit... more We examine a distinctive kind of problem for decision theory, involving what we call discontinuity at infinity. Roughly, it arises when an infinite sequence of choices, each apparently sanctioned by plausible principles, converges to a 'limit choice' whose utility is much lower than the limit approached by the utilities of the choices in the sequence. We give examples of this phenomenon, focusing on Arntzenius et al.'s Satan's apple, and give a general characterization of it. In these examples, repeated dominance reasoning (a paradigm of rationality) apparently gives rise to a situation closely analogous to having intransitive preferences (a paradigm of irrationality). Indeed, the agents in these examples are vulnerable to a money pump set-up despite having preferences that exhibit no obvious defect of rationality. We explore several putative solutions to such problems, particularly those that appeal to binding and to deliberative dynamics. We consider the prospects ...
Confronted with the possibility of severe environmental harms, such as catastrophic climate chang... more Confronted with the possibility of severe environmental harms, such as catastrophic climate change, some researchers have suggested that we should abandon the principle at the heart of standard decision theory—the injunction to maximize expected utility—and embrace a different one: the Precautionary Principle. Arguably, the most sophisticated philosophical treatment of the Precautionary Principle (PP) is due to Steel (2015). Steel interprets PP as a qualitative decision rule and appears to conclude that a quantitative decision-theoretic statement of PP is both impossible and unnecessary. In this article, we propose a decision-theoretic formulation of PP in terms of lexical (or lexicographic) utilities. We show that this lexical model is largely faithful to Steel’s approach, but also that it corrects three problems with Steel’s account and clarifies the relationship between PP and standard decision theory. Using a range of examples, we illustrate how the lexical model can be used to ...
There is a long history of fruitful connections between work in probability theory and the philos... more There is a long history of fruitful connections between work in probability theory and the philosophy of religion. This chapter explores these connections through discussion of two classic arguments: the fine-tuning argument and Pascal’s Wager. The formulation and assessment of both arguments relies upon increasingly sophisticated applications of the probability calculus and other formal tools. Two themes emerge from a survey of recent work. First, diverse forms of ‘philosophical technology’ are invaluable in constructing precise models, clarifying objections and identifying new approaches to venerable arguments concerning the existence of God and the rationality of religious belief. Second, benefits flow in the reverse direction as well: the philosophy of religion is fertile ground for testing ideas in formal epistemology and decision theory.
Preface. PART I BACKGROUND MATERIAL. 1 Sets and Functions. 1.1 Sets in General. 1.2 Sets of Numbe... more Preface. PART I BACKGROUND MATERIAL. 1 Sets and Functions. 1.1 Sets in General. 1.2 Sets of Numbers. 1.3 Functions. 2 Real Numbers. 2.1 Review of the Order Relations. 2.2 Completeness of Real Numbers. 2.3 Sequences of Real Numbers. 2.4 Subsequences. 2.5 Series of Real Numbers. 2.6 Intervals and Connected Sets. 3 Vector Functions. 3.1 Vector Spaces: The Basics. 3.2 Bilinear Functions. 3.3 Multilinear Functions. 3.4 Inner Products. 3.5 Orthogonal Projections. 3.6 Spectral Theorem. PART II DIFFERENTIATION. 4 Normed Vector Spaces. 4.1 Preliminaries. 4.2 Convergence in Normed Spaces. 4.3 Norms of Linear and Multilinear Transformations. 4.4 Continuity in Normed Spaces. 4.5 Topology of Normed Spaces. 5 Derivatives. 5.1 Functions of a Real Variable. 5.2 Differentiable Functions. 5.3 Existence of Derivatives. 5.4 Partial Derivatives. 5.5 Rules of Differentiation. 5.6 Differentiation of Products. 6 Diffeomorphisms and Manifolds. 6.1 The Inverse Function Theorem. 6.2 Graphs. 6.3 Manifolds in Parametric Representations. 6.4 Manifolds in Implicit Representations. 6.5 Differentiation on Manifolds. 7 Higher-Order Derivatives. 7.1 Definitions. 7.2 Change of Order in Differentiation. 7.3 Sequences of Polynomials. 7.4 Local Extremal Values. PART III INTEGRATION. 8 Multiple Integrals. 8.1 Jordan Sets and Volume. 8.2 Integrals. 8.3 Images of Jordan Sets. 8.4 Change of Variables. 9 Integration on Manifolds. 9.1 Euclidean Volumes. 9.2 Integration on Manifolds. 9.3 Oriented Manifolds. 9.4 Integrals of Vector Fields. 9.5 Integrals of Tensor Fields. 9.6 Integration on Graphs. 10 Stokes' Theorem. 10.1 Basic Stokes' Theorem. 10.2 Flows. 10.3 Flux and Change of Volume in a Flow. 10.4 Exterior Derivatives. 10.5 Regular and Almost Regular Sets. 10.6 Stokes' Theorem on Manifolds. PART IV APPENDICES. Appendix A: Construction of the Real Numbers. A.1 Field and Order Axioms in Q. A.2 Equivalence Classes of Cauchy Sequences in Q. A.3 Completeness of R. Appendix B: Dimension of a Vector Space. B.1 Bases and Linearly Independent Subsets. Appendix C: Determinants. C.1 Permutations. C.2 Determinants of Square Matrices. C.3 Determinant Functions. C.4 Determinant of a Linear Transformation. C.5 Determinants on Cartesian Products. C.6 Determinants in Euclidean Spaces. C.7 Trace of an Operator. Appendix D: Partitions of Unity. D.1 Partitions of Unity. Index.
A rigorous introduction to calculus in vector spaces The concepts and theorems of advanced calcul... more A rigorous introduction to calculus in vector spaces The concepts and theorems of advanced calculus combined with related computational methods are essential to understanding nearly all areas of quantitative science. Analysis in Vector Spaces presents the central results of this classic subject through rigorous arguments, discussions, and examples. The book aims to cultivate not only knowledge of the major theoretical results, but also the geometric intuition needed for both mathematical problem-solving and modeling in the formal sciences. The authors begin with an outline of key concepts, terminology, and notation and also provide a basic introduction to set theory, the properties of real numbers, and a review of linear algebra. An elegant approach to eigenvector problems and the spectral theorem sets the stage for later results on volume and integration. Subsequent chapters present the major results of differential and integral calculus of several variables as well as the theory o...
In analogical reasoning, observations about one or more source domains provide varying degrees of... more In analogical reasoning, observations about one or more source domains provide varying degrees of support for a conjecture about a target domain. Norton (2021) challenges the usefulness of formal models of analogical inference. Other philosophers (Dardashti et al 2019) develop just such formal models in order to show how analogue experiments can confirm a hypothesis, even when the target domain is inaccessible. This paper defends the value of quasi-formal models of analogical reasoning. Such models are broadly compatible with Norton’s position, but help to clarify the structure of analogical reasoning and to identify basic requirements for a good analogical inference.
The precautionary principle is often argued to be irrational because it cannot adequately explain... more The precautionary principle is often argued to be irrational because it cannot adequately explain how resources should be distributed across multiple possible catastrophes or between catastrophic and noncatastrophic risks. We address this problem of trade-offs by extending a recently proposed formal interpretation of the precautionary principle (PP) within a lexical utility framework and using it to prove results about which distribution of resources maximizes lexical utility when several catastrophic risks exist, given different assumptions. We also explain how our lexical utility interpretation of PP can recommend balanced distributions of resources between disaster prevention and other concerns.
This paper argues that analogical arguments, especially in science, are often employed to show th... more This paper argues that analogical arguments, especially in science, are often employed to show that their conclusions are prima facie plausible, or serious possibilities. Prima facie plausibility is not a matter of degree; rather, it implies the existence of a threshold below which analogical arguments provide no justification for their conclusions. This position is supported by means of analogical arguments in mathematics. The paper then argues that structure-mapping theories cannot easily accommodate the notion of prima facie plausibility.
Abstract: The Principle of Indifference, which dictates that we ought to assign two outcomes equa... more Abstract: The Principle of Indifference, which dictates that we ought to assign two outcomes equal probability in the absence of known reasons to do otherwise, is vulnerable to well-known objections. Nevertheless, the appeal of the principle, and of symmetry-based assignments of equal probability, persists. We show that, relative to a given class of symmetries satisfying certain properties, we are justified in calling certain outcomes equally probable, and more generally, in defining what we call relative probabilities. Relative probabilities are useful in providing a generalized approach to conditionalization. The technique is illustrated by application to simple examples. Word Count: 4,948 1. Introduction. A recent statement of the classical Principle of Indifference (PI) runs as follows: If there are n mutually exclusive possibilities h1, …, hn, and e gives no more reason to believe any one of these more likely to be true than any other, then P(hi / e) is the same for all i. 1
How can self-locating propositions be integrated into normal patterns of belief revision? Puzzles... more How can self-locating propositions be integrated into normal patterns of belief revision? Puzzles such as Sleeping Beauty seem to show that such propositions lead to violation of ordinary principles for reasoning with subjective probability, such as Conditionalization and Reflection. I show that sophisticated forms of Conditionalization and Reflection are not only compatible with self-locating propositions, but also indispensable in understanding how they can function as evidence in Sleeping Beauty and similar cases.
We examine a distinctive kind of problem for decision theory, involving what we call discontinuit... more We examine a distinctive kind of problem for decision theory, involving what we call discontinuity at infinity. Roughly, it arises when an infinite sequence of choices, each apparently sanctioned by plausible principles, converges to a 'limit choice' whose utility is much lower than the limit approached by the utilities of the choices in the sequence. We give examples of this phenomenon, focusing on Arntzenius et al.'s Satan's apple, and give a general characterization of it. In these examples, repeated dominance reasoning (a paradigm of rationality) apparently gives rise to a situation closely analogous to having intransitive preferences (a paradigm of irrationality). Indeed, the agents in these examples are vulnerable to a money pump set-up despite having preferences that exhibit no obvious defect of rationality. We explore several putative solutions to such problems, particularly those that appeal to binding and to deliberative dynamics. We consider the prospects ...
Confronted with the possibility of severe environmental harms, such as catastrophic climate chang... more Confronted with the possibility of severe environmental harms, such as catastrophic climate change, some researchers have suggested that we should abandon the principle at the heart of standard decision theory—the injunction to maximize expected utility—and embrace a different one: the Precautionary Principle. Arguably, the most sophisticated philosophical treatment of the Precautionary Principle (PP) is due to Steel (2015). Steel interprets PP as a qualitative decision rule and appears to conclude that a quantitative decision-theoretic statement of PP is both impossible and unnecessary. In this article, we propose a decision-theoretic formulation of PP in terms of lexical (or lexicographic) utilities. We show that this lexical model is largely faithful to Steel’s approach, but also that it corrects three problems with Steel’s account and clarifies the relationship between PP and standard decision theory. Using a range of examples, we illustrate how the lexical model can be used to ...
There is a long history of fruitful connections between work in probability theory and the philos... more There is a long history of fruitful connections between work in probability theory and the philosophy of religion. This chapter explores these connections through discussion of two classic arguments: the fine-tuning argument and Pascal’s Wager. The formulation and assessment of both arguments relies upon increasingly sophisticated applications of the probability calculus and other formal tools. Two themes emerge from a survey of recent work. First, diverse forms of ‘philosophical technology’ are invaluable in constructing precise models, clarifying objections and identifying new approaches to venerable arguments concerning the existence of God and the rationality of religious belief. Second, benefits flow in the reverse direction as well: the philosophy of religion is fertile ground for testing ideas in formal epistemology and decision theory.
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