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29
Categories and groupoids
, 1971
"... In 1968, when this book was written, categories had been around for 20 years and groupoids for twice as long. Category theory had by then become widely accepted as an essential tool in many parts of mathematics and a number of books on the subject had appeared, or were about to appear (e.g. [13, 22, ..."
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In 1968, when this book was written, categories had been around for 20 years and groupoids for twice as long. Category theory had by then become widely accepted as an essential tool in many parts of mathematics and a number of books on the subject had appeared, or were about to appear (e.g. [13, 22, 37, 58, 65] 1). By contrast, the use of groupoids was confined to a small number of pioneering articles, notably by Ehresmann [12] and Mackey [57], which were largely ignored by the mathematical community. Indeed groupoids were generally considered at that time not to be a subject for serious study. It was argued by several wellknown mathematicians that group theory sufficed for all situations where groupoids might be used, since a connected groupoid could be reduced to a group and a set. Curiously, this argument, which makes no appeal to elegance, was not applied to vector spaces: it was well known that the analogous reduction in this case is not canonical, and so is not available, when there is extra structure, even such simple structure as an endomorphism. Recently, Corfield in [41] has discussed methodological issues in mathematics with this topic, the resistance to the notion of groupoids, as a prime example. My book was intended chiefly as an attempt to reverse this general assessment of the time by presenting applications of groupoids to group theory
Philosophies of probability: objective Bayesianism and its challenges
 Handbook of the philosophy of mathematics. Elsevier, Amsterdam. Handbook of the Philosophy of Science
, 2004
"... This chapter presents an overview of the major interpretations of probability followed by an outline of the objective Bayesian interpretation and a discussion of the key challenges it faces. ..."
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This chapter presents an overview of the major interpretations of probability followed by an outline of the objective Bayesian interpretation and a discussion of the key challenges it faces.
Mathematical method and proof
"... Abstract. On a traditional view, the primary role of a mathematical proof is to warrant the truth of the resulting theorem. This view fails to explain why it is very often the case that a new proof of a theorem is deemed important. Three case studies from elementary arithmetic show, informally, that ..."
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Abstract. On a traditional view, the primary role of a mathematical proof is to warrant the truth of the resulting theorem. This view fails to explain why it is very often the case that a new proof of a theorem is deemed important. Three case studies from elementary arithmetic show, informally, that there are many criteria by which ordinary proofs are valued. I argue that at least some of these criteria depend on the methods of inference the proofs employ, and that standard models of formal deduction are not wellequipped to support such evaluations. I discuss a model of proof that is used in the automated deduction community, and show that this model does better in that respect.
Computers in mathematical inquiry
 in The Philosophy of Mathematical Practice
, 2008
"... Computers are playing an increasingly central role in mathematical practice. What are we to make of the new methods of inquiry? In Section 2, I survey some of the ways that computers are used in mathematics. These raise questions that seem to have a generally epistemological character, ..."
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Computers are playing an increasingly central role in mathematical practice. What are we to make of the new methods of inquiry? In Section 2, I survey some of the ways that computers are used in mathematics. These raise questions that seem to have a generally epistemological character,
Philosophies of probability
 Handbook of the Philosophy of Mathematics, Volume 4 of the Handbook of the Philosophy of Science
"... This chapter presents an overview of the major interpretations of probability followed by an outline of the objective Bayesian interpretation and a discussion of the key challenges it faces. I discuss the ramifications of interpretations of probability and objective Bayesianism for the philosophy of ..."
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This chapter presents an overview of the major interpretations of probability followed by an outline of the objective Bayesian interpretation and a discussion of the key challenges it faces. I discuss the ramifications of interpretations of probability and objective Bayesianism for the philosophy of mathematics in general.
To appear in Structural Foundations of Quantum Gravity,
, 2004
"... General relativity may seem very different from quantum theory, but work on quantum gravity has revealed a deep analogy between the two. General relativity makes heavy use of the category nCob, whose objects are (n − 1)dimensional manifolds representing ‘space ’ and whose morphisms are ndimensiona ..."
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General relativity may seem very different from quantum theory, but work on quantum gravity has revealed a deep analogy between the two. General relativity makes heavy use of the category nCob, whose objects are (n − 1)dimensional manifolds representing ‘space ’ and whose morphisms are ndimensional cobordisms representing ‘spacetime’. Quantum theory makes heavy use of the category Hilb, whose objects are Hilbert spaces used to describe ‘states’, and whose morphisms are bounded linear operators used to describe ‘processes’. Moreover, the categories nCob and Hilb resemble each other far more than either resembles Set, the category whose objects are sets and whose morphisms are functions. In particular, both Hilb and nCob but not Set are ∗categories with a noncartesian monoidal structure. We show how this accounts for many of the famously puzzling features of quantum theory: the failure of local realism, the impossibility of duplicating quantum information, and so on. We argue that these features only seem puzzling when we try to treat Hilb as analogous to Set rather than nCob, so that quantum theory will make more sense when regarded as part of a theory of spacetime. 1
Probabilistic Proofs and Transferability
 Philosophia Mathematica
"... One of the central questions in the philosophy of mathematics concerns the nature of mathematical knowledge. The version of this question familiar from [Benacerraf, 1973] asks how knowledge of any mathematical proposition could be consistent with any picture of the semantics of mathematical language ..."
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One of the central questions in the philosophy of mathematics concerns the nature of mathematical knowledge. The version of this question familiar from [Benacerraf, 1973] asks how knowledge of any mathematical proposition could be consistent with any picture of the semantics of mathematical language (and in particular with the apparently abstract and acausal nature of mathematical objects). However, there is a further question even granting existing knowledge of mathematical propositions, one may wonder what exactly it takes for a mathematician to come to know yet more propositions. To begin to address this question, I note that there is some extremely close connection in mathematics between knowledge and proof. Mathematicians often say that a claim is not known until a proof has been given, and an account somewhat like this is presupposed in some naturalistic discussions of mathematical knowledge (see [Horsten, 2001, pp. 1869], where he concedes that other means may provide knowledge of mathematical propositions, but suggests that proof must underlie a notion of “mathematical knowledge”).
On the persuasiveness of visual arguments in mathematics
"... Abstract. Two experiments are reported which investigate the factors that influence how persuaded mathematicians are by visual arguments. We demonstrate that if a visual argument is accompanied by a passage of text which describes the image, both researchactive mathematicians and successful underg ..."
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Abstract. Two experiments are reported which investigate the factors that influence how persuaded mathematicians are by visual arguments. We demonstrate that if a visual argument is accompanied by a passage of text which describes the image, both researchactive mathematicians and successful undergraduate mathematics students perceive it to be significantly more persuasive than if no text is given. We suggest that mathematicians ’ epistemological concerns about supporting a claim using visual images are less prominent when the image is described in words. Finally we suggest that empirical studies can make a useful contribution to our understanding of mathematical practice.
Mathematics and Argumentation
"... Abstract. Some authors have begun to appeal directly to studies of argumentation in their analyses of mathematical practice. These include researchers from an impressively diverse range of disciplines: not only philosophy of mathematics and argumentation theory, but also psychology, education, and c ..."
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Abstract. Some authors have begun to appeal directly to studies of argumentation in their analyses of mathematical practice. These include researchers from an impressively diverse range of disciplines: not only philosophy of mathematics and argumentation theory, but also psychology, education, and computer science. This introduction provides some background to their work.
Justifying definitions in mathematics—going beyond Lakatos
 Philosophia Mathematica
"... This paper addresses the actual practice of justifying definitions in mathematics. First, I introduce the main account of this issue, namely Lakatos’s proofgenerated definitions. Based on a case study of definitions of randomness in ergodic theory, I identify three other common ways of justifying ..."
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This paper addresses the actual practice of justifying definitions in mathematics. First, I introduce the main account of this issue, namely Lakatos’s proofgenerated definitions. Based on a case study of definitions of randomness in ergodic theory, I identify three other common ways of justifying definitions: naturalworldjustification, conditionjustification and redundancyjustification. Also, I clarify the interrelationships between the different kinds of justification. Finally, I point out how Lakatos’s ideas are limited: they fail to show that various kinds of justification can be found and can be reasonable, and they fail to acknowledge the interplay between the different kinds