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The nonEuclidean style of Minkowskian relativity
 The Symbolic Universe: Geometry and Physics, 1890–1930
, 1999
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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
Mathematics and virtual culture: An evolutionary perspective on technology and mathematics education
 Educational Studies in Mathematics
, 1999
"... ABSTRACT. This paper suggests that from a cognitiveevolutionary perspective, computational media are qualitatively different from many of the technologies that have promised educational change in the past and failed to deliver. Recent theories of human cognitive evolution suggest that human cogniti ..."
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ABSTRACT. This paper suggests that from a cognitiveevolutionary perspective, computational media are qualitatively different from many of the technologies that have promised educational change in the past and failed to deliver. Recent theories of human cognitive evolution suggest that human cognition has evolved through four distinct stages: episodic, mimetic, mythic, and theoretical. This progression was driven by three cognitive advances: the ability to “represent ” events, the development of symbolic reference, and the creation of external symbolic representations. In this paper, we suggest that we are developing a new cognitive culture: a “virtual ” culture dependent on the externalization of symbolic processing. We suggest here that the ability to externalize the manipulation of formal systems changes the very nature of cognitive activity. These changes will have important consequences for mathematics education in coming decades. In particular, we argue that mathematics education in a virtual culture should strive to give students generative fluency to learn varieties of representational systems, provide opportunities to create and modify representational forms, develop skill in making and exploring virtual environments, and emphasize mathematics as a fundamental way of making sense of the world, reserving most exact computation and formal proof for those who will need those specialized skills.
Is probability the only coherent approach to uncertainty?, Risk Analysis, forthcoming
"... In this paper I discuss an argument that purports to prove that probability theory is the only sensible means of dealing with uncertainty. I show that this argument can succeed only if some rather controversial assumptions about the nature of uncertainty are accepted. I discuss these assumptions and ..."
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In this paper I discuss an argument that purports to prove that probability theory is the only sensible means of dealing with uncertainty. I show that this argument can succeed only if some rather controversial assumptions about the nature of uncertainty are accepted. I discuss these assumptions and provide reasons for rejecting them. I also present examples of what I take to be nonprobabilistic uncertainty. Key Words: Cox’s Theorem, NonClassical Logic, Probability, Uncertainty, Vagueness Uncertainties are ubiquitous in risk analysis and on the face of it, we must contend with a number of quite distinct sorts of uncertainty. There are, of course, many methods on hand to deal with uncertainty, so it is important to select the method best suited to the uncertainty in question. There is, however, a growing push towards dealing with all uncertainty in one fell swoop. That is, it is thought to be desirable to employ a single method capable of quantifying all sources of uncertainty. One candidate for this task is probability theory. For such a program to succeed, a demonstration that all uncertainty is,
Leibniz’s infinitesimals: Their fictionality, their modern implementations, and their foes from Berkeley to Russell and beyond
, 2012
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Geometric Algebra and its Application to Mathematical Physics
, 1994
"... This dissertation is the result of work carried out in the Department of Applied Mathem ..."
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This dissertation is the result of work carried out in the Department of Applied Mathem
An Infinite Set Of Heron Triangles With Two Rational Medians
 American Mathematical Monthly
, 1997
"... Introduction If we denote the sides of a triangle by (a; b; c) then the area is given by (1) = p s(s a)(s b)(s c) where s = (a + b + c)=2 is the semiperimeter. This formula is usually attributed to Heron of Alexandria circa 100 BC  100 AD. However, it was already known to Archimedes prior to 2 ..."
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Introduction If we denote the sides of a triangle by (a; b; c) then the area is given by (1) = p s(s a)(s b)(s c) where s = (a + b + c)=2 is the semiperimeter. This formula is usually attributed to Heron of Alexandria circa 100 BC  100 AD. However, it was already known to Archimedes prior to 212 BC [5, p. 105]. Our investigation is limited to triangles with rational sides. Even with sides of rational length, \Heron's" formula shows that the area need not be rational; any triangle with three rational sides and rational area is called a Heron triangle. The smallest such triangle with integer sides is the familiar (5; 4; 3) right triangle (with area 6) shown in Figure 1. B C a = 5<F
Euler and infinite series
 ISSN 0025570X. D O I : 10.2307/2690371. URL http: //dx.doi.org/10.2307/2690371
, 1983
"... Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at ..."
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Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at
Reformulation and Convex Relaxation Techniques for Global Optimization
 4OR
, 2004
"... Many engineering optimization problems can be formulated as nonconvex nonlinear programming problems (NLPs) involving a nonlinear objective function subject to nonlinear constraints. Such problems may exhibit more than one locally optimal point. However, one is often solely or primarily interested i ..."
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Many engineering optimization problems can be formulated as nonconvex nonlinear programming problems (NLPs) involving a nonlinear objective function subject to nonlinear constraints. Such problems may exhibit more than one locally optimal point. However, one is often solely or primarily interested in determining the globally optimal point. This thesis is concerned with techniques for establishing such global optima using spatial BranchandBound (sBB) algorithms.
Advanced MathematicalThinking at Any Age: Its Nature and Its Development
"... This article argues that advanced mathematical thinking, usually conceived as thinking in advanced mathematics, might profitably be viewed as advanced thinking in mathematics (advanced mathematicalthinking). Hence, advanced mathematicalthinking can properly be viewed as potentially starting in ele ..."
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This article argues that advanced mathematical thinking, usually conceived as thinking in advanced mathematics, might profitably be viewed as advanced thinking in mathematics (advanced mathematicalthinking). Hence, advanced mathematicalthinking can properly be viewed as potentially starting in elementary school. The definition of mathematical thinking entails considering the epistemological and didactical obstacles to a particular way of thinking. The interplay between ways of thinking and ways of understanding gives a contrast between the two, to make clearer the broader view of mathematical thinking and to suggest implications for instructional practices. The latter are summarized with a description of the DNR system (Duality, Necessity, and Repeated Reasoning). Certain common assumptions about instruction are criticized (in an effort to be provocative) by suggesting that they can interfere with growth in mathematical thinking. The reader may have noticed the unusual location of the hyphen in the title of this article. We relocated the hyphen in “advancedmathematical thinking ” (i.e., thinking in advanced mathematics) so that the phrase reads, “advanced mathematicalthinking” (i.e., mathematical thinking of an advanced nature). This change in emphasis is to argue that a student’s growth in mathematical thinking is an evolving process, and that the nature of mathematical thinking should be studied so as to