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Mathematical explanation and the theory of whyquestions
 British Journal for the Philosophy of Science
, 1998
"... Van Fraassen and others have urged that judgements of explanations are relative to whyquestions; explanations should be considered good in so far as they effectively answer whyquestions. In this paper, I evaluate van Fraassen's theory with respect to mathematical explanation. I show that his theor ..."
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Van Fraassen and others have urged that judgements of explanations are relative to whyquestions; explanations should be considered good in so far as they effectively answer whyquestions. In this paper, I evaluate van Fraassen's theory with respect to mathematical explanation. I show that his theory cannot recognize any proofs as explanatory. I also present an example that contradicts the main thesis of the whyquestion approach—an explanation that appears explanatory despite its inability to answer the whyquestion that motivated it. This example shows how explanatory judgements can be contextdependent without being whyquestionrelative. 1
Computers, Reasoning and Mathematical Practice
"... ion in itself is not the goal: for Whitehead [117]"it is the large generalisation, limited by a happy particularity, which is the fruitful conception." As an example consider the theorem in ring theory, which states that if R is a ring, f(x) is a polynomial over R and f(r) = 0 for every element of ..."
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ion in itself is not the goal: for Whitehead [117]"it is the large generalisation, limited by a happy particularity, which is the fruitful conception." As an example consider the theorem in ring theory, which states that if R is a ring, f(x) is a polynomial over R and f(r) = 0 for every element of r of R then R is commutative. Special cases of this, for example f(x) is x 2 \Gamma x or x 3 \Gamma x, can be given a first order proof in a few lines of symbol manipulation. The usual proof of the general result [20] (which takes a semester's postgraduate course to develop from scratch) is a corollary of other results: we prove that rings satisfying the condition are semisimple artinian, apply a theorem which shows that all such rings are matrix rings over division rings, and eventually obtain the result by showing that all finite division rings are fields, and hence commutative. This displays von Neumann's architectural qualities: it is "deep" in a way in which the symbol manipulati...
Evolution of the function concept: A brief survey
 The College Mathematics Journal
, 1989
"... received his Ph.D. in ring theory at McGill University, and has been at York University for over twenty years. He has been involved in teacher education at the undergraduate and graduate levels and has given numerous talks to high school students and teachers. One of his major interests is the histo ..."
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received his Ph.D. in ring theory at McGill University, and has been at York University for over twenty years. He has been involved in teacher education at the undergraduate and graduate levels and has given numerous talks to high school students and teachers. One of his major interests is the history of mathematics and its use in the teaching of mathematics. Introduction. The evolution of the concept of function goes back 4000 years; 3700 of these consist of anticipations. The idea evolved for close to 300 years in intimate connection with problems in calculus and analysis. (A onesentence definition of analysis as the study of properties of various classes of functions would not be far off the mark.) In fact, the concept of function is one of the distinguishing features of
On The Purpose of Mathematics Education Research: Making productive contributions to policy and practice
 In L. English (Ed.), Handbook of International Research on Mathematics Education
, 2002
"... Lester and Dylan Wiliam and published in 2002. The complete reference citation for the ..."
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Lester and Dylan Wiliam and published in 2002. The complete reference citation for the
TopDown and BottomUp Philosophy of Mathematics
"... Abstract. The philosophy of mathematics of the last few decades is commonly distinguished into mainstream and maverick, to which a ‘third way ’ has been recently added, the philosophy of mathematical practice. In this paper the limitations of these trends in the philosophy of mathematics are pointed ..."
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Abstract. The philosophy of mathematics of the last few decades is commonly distinguished into mainstream and maverick, to which a ‘third way ’ has been recently added, the philosophy of mathematical practice. In this paper the limitations of these trends in the philosophy of mathematics are pointed out, and it is argued that they are due to the fact that all of them are based on a topdown approach, that is, an approach which explains the nature of mathematics in terms of some general unproven assumption. As an alternative, a bottomup approach is proposed, which explains the nature of mathematics in terms of the activity of real individuals and interactions between them. This involves distinguishing between mathematics as a discipline and the mathematics embodied in organisms as a result of biological evolution, which however, while being distinguished, are not opposed. Moreover, it requires a view of mathematical proof, mathematical definition and mathematical objects which is alternative to the topdown approach.
forthcoming)). Applying lakatosstyle reasoning to ai problems
 Thinking Machines and the philosophy of computer science: Concepts and principles. IGI Global
, 2010
"... One current direction in AI research is to focus on combining different reasoning styles such as deduction, induction, abduction, analogical reasoning, nonmonotonic reasoning, vague and uncertain reasoning. The philosopher Imre Lakatos produced one such theory of how people with different reasoning ..."
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One current direction in AI research is to focus on combining different reasoning styles such as deduction, induction, abduction, analogical reasoning, nonmonotonic reasoning, vague and uncertain reasoning. The philosopher Imre Lakatos produced one such theory of how people with different reasoning styles collaborate to develop mathematical ideas. Lakatos argued that mathematics is a quasiempirical, flexible, fallible, human endeavour, involving negotiations, mistakes, vague concept definitions and disagreements, and he outlined a heuristic approach towards the subject. In this chapter we apply these heuristics to the AI domains of evolving requirement specifications, planning and constraint satisfaction problems. In drawing analogies between Lakatos’s theory and these three domains we identify areas of work which correspond to each heuristic, and suggest extensions and further ways in which Lakatos’s philosophy can inform AI problem solving. Thus, we show how we might begin to produce a philosophicallyinspired AI theory of combined reasoning. 1
EMPIRICISM, CONTINGENCY AND EVOLUTIONARY METAPHORS: GETTING BEYOND THE “MATH WARS”
"... As a middle school mathematics teacher, I was frequently frustrated by what went on in the classroom. Theorists and practitioners in other subject areas have worked to explicitly link the role of human agency to their respective disciplines and to find ways to apply school knowledge in reasonably re ..."
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As a middle school mathematics teacher, I was frequently frustrated by what went on in the classroom. Theorists and practitioners in other subject areas have worked to explicitly link the role of human agency to their respective disciplines and to find ways to apply school knowledge in reasonably realistic contexts. In language arts, science, and history, emphasis on
Blending and Other Conceptual Operations in the Interpretation of Mathematical Proofs
"... this paper I benefited from discussions with Gilles Fauconnier. ..."
Ontological Dependency
"... Successful ontological analysis depends upon having the right underlying theory. The work described here, exploring how to understand organisations as systems of social norms found that the familiar objectivist position did not work, eventually replacing it with a radically subjectivist ontology whi ..."
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Successful ontological analysis depends upon having the right underlying theory. The work described here, exploring how to understand organisations as systems of social norms found that the familiar objectivist position did not work, eventually replacing it with a radically subjectivist ontology which treats every thing, relationship and attribute as a repertoire of behaviour as understood by some responsible agent. Gibson's Theory of Affordances supports this view in relation to our physical reality and the concept of norms extends the theory naturally into the social domain. A formalism, Norma, which captures the need always to specify the responsible agent and some more or less complex repertoire of behaviour, introduces the concept of ontological dependency where one repertoire depends for its existence on another. This unusual logical relationship allows one to devise schemas which can generate systems as byproducts; the paper ends with an example dealing with health insurance. T...
Helmholtz’s Naturalised Conception of Geometry and his Spatial Theory of Signs
"... this paper, I will try to answer these questions by looking at his "The Facts in Perception," a Rektoratsrede held at the FriedrichWilhelmsUniversitt in Berlin on 3 August 1878, when Helmholtz took up the chair in Physics. As this lecture weaves together a number of themes, I will focus on only tw ..."
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this paper, I will try to answer these questions by looking at his "The Facts in Perception," a Rektoratsrede held at the FriedrichWilhelmsUniversitt in Berlin on 3 August 1878, when Helmholtz took up the chair in Physics. As this lecture weaves together a number of themes, I will focus on only two: his manifold theory of perception, which construes the human perceptual field as a dataspace; and (Einstein 1917) (Helmholtz 1977) Helmholtz's use of this theory, particularly in the second appendix to the lecture, to rebut neoKantian criticisms of his earlier papers on nonEuclidean geometry. The argument that (some species of) geometry is itself a physical science is probably the best known aspect of Helmholtz's philosophical work. What is not so well known, however, is the role of the perceptual theory that he advanced simultaneously, and which influenced both Wittgenstein's and the Vienna Circle's notion that our experience plays out in a phenomenological manifold, or "space." Helmholtz's use of the term manifold was not metaphorical: Riemann calls a system of differences in which the individual element (das Einzelne) can be determined by n measurements, an nfold manifold, or a manifold of n dimensions. Thus the space that we know and in which we live is a threefold extended manifold, a plane a twofold, and a line a onefold one, as is indeed time. The system of colours also constitutes a threefold manifold, in that each colour can be represented ... as the mixture of three elementary colours, of each of which a definite quantum is to be chosen. ... We could just as well describe the domain of simple tones as a manifold of two dimensions, if we take them to be differentiated only by pitch and volume ..