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8 Aristotle and Modern Mathematical Theories of the Continuum *
"... The mathematical structure of the continuum, in the guise of the domain of continuous, differentiable functions, has proved immensely useful in the study of nature. However, we have learned to be sceptical of any claim to the effect that our current favourite ..."
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The mathematical structure of the continuum, in the guise of the domain of continuous, differentiable functions, has proved immensely useful in the study of nature. However, we have learned to be sceptical of any claim to the effect that our current favourite
M.: A Cognitive Analysis of Cauchy’s Conceptions of Continuity, Limit, and Infinitesimal, with implications for teaching the calculus (submitted for publication, 2011). Available on the internet at: http://www.warwick.ac.uk/staff/David.Tall/downloads.html
"... Opinions concerning Cauchy’s ideas of continuity, limit, and infinitesimal, and his role in the development of modern analysis are many and varied. Here we complement the range of views with a cognitive analysis of his work based on Merlin Donald’s notion of ‘three levels of consciousness ’ and Davi ..."
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Opinions concerning Cauchy’s ideas of continuity, limit, and infinitesimal, and his role in the development of modern analysis are many and varied. Here we complement the range of views with a cognitive analysis of his work based on Merlin Donald’s notion of ‘three levels of consciousness ’ and David Tall’s framework describing the development of mathematical thinking in terms of embodiment, symbolism and formalism. Cauchy lived in an era when modern formal proof from settheoretic axioms did not yet exist. His theoretical framework is based on a blend of geometric embodiment and manipulable symbolism where symbolic processes with sequences of numbers are conceptualized in such a manner that they can be verbalised as infinitesimal concepts. His insights provide the foundations for later developments as axiomatic formulations of both epsilondelta analysis and nonstandard analysis, though in his era there were no formal conceptions of either. When our students are introduced to calculus and analysis, they too build on geometric embodiment and manipulable symbolism. Research has revealed their difficulties with the concepts of continuity, limit and the intuitive notion of infinitesimal. We exploit the framework of superimposed levels of consciousness and the theory of development through embodiment, symbolism, and formalism. We explore the implications of such a framework for our current views on teaching calculus and analysis. 1.
Alternative Set Theories
, 2006
"... By “alternative set theories ” we mean systems of set theory differing significantly from the dominant ZF (ZermeloFrankel set theory) and its close relatives (though we will review these systems in the article). Among the systems we will review are typed theories of sets, Zermelo set theory and its ..."
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By “alternative set theories ” we mean systems of set theory differing significantly from the dominant ZF (ZermeloFrankel set theory) and its close relatives (though we will review these systems in the article). Among the systems we will review are typed theories of sets, Zermelo set theory and its variations, New Foundations and related systems, positive set theories, and constructive set theories. An interest in the range of alternative set theories does not presuppose an interest in replacing the dominant set theory with one of the alternatives; acquainting ourselves with foundations of mathematics formulated in terms of an alternative system can be instructive as showing us what any set theory (including the usual one) is supposed to do for us. The study of alternative set theories can dispel a facile identification of “set theory ” with “ZermeloFraenkel set theory”; they are not the same thing. Contents 1 Why set theory? 2 1.1 The Dedekind construction of the reals............... 3 1.2 The FregeRussell definition of the natural numbers....... 4
Conceptions of the Continuum
"... Abstract: A number of conceptions of the continuum are examined from the perspective of conceptual structuralism, a view of the nature of mathematics according to which mathematics emerges from humanly constructed, intersubjectively established, basic structural conceptions. This puts into question ..."
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Abstract: A number of conceptions of the continuum are examined from the perspective of conceptual structuralism, a view of the nature of mathematics according to which mathematics emerges from humanly constructed, intersubjectively established, basic structural conceptions. This puts into question the idea from current set theory that the continuum is somehow a uniquely determined concept. Key words: the continuum, structuralism, conceptual structuralism, basic structural conceptions, Euclidean geometry, Hilbertian geometry, the real number system, settheoretical conceptions, phenomenological conceptions, foundational conceptions, physical conceptions. 1. What is the continuum? On the face of it, there are several distinct forms of the continuum as a mathematical concept: in geometry, as a straight line, in analysis as the real number system (characterized in one of several ways), and in set theory as the power set of the natural numbers and, alternatively, as the set of all infinite sequences of zeros and ones. Since it is common to refer to the continuum, in what sense are these all instances of the same concept? When one speaks of the continuum in current settheoretical
The Mathematical Infinite as a Matter of Method
, 2010
"... Abstract. I address the historical emergence of the mathematical infinite, and how we are to take the infinite in and out of mathematics. The thesis is that the mathematical infinite in mathematics is a matter of method. The infinite, of course, is a large topic. At the outset, one can historically ..."
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Abstract. I address the historical emergence of the mathematical infinite, and how we are to take the infinite in and out of mathematics. The thesis is that the mathematical infinite in mathematics is a matter of method. The infinite, of course, is a large topic. At the outset, one can historically discern two overlapping clusters of concepts: (1) wholeness, completeness, universality, absoluteness. (2) endlessness, boundlessness, indivisibility, continuousness. The first, the metaphysical infinite, I shall set aside. It is the second, the mathematical infinite, that I will address. Furthermore, I will address mathematical infinite by considering its historical emergence in set theory and how we are to take it in and out of mathematics. Insofar as physics and, more broadly, science deals with the mathematical infinite through mathematical language and techniques, my remarks should be subsuming and consequent. The main underlying point is that how the mathematical infinite is approached, assimilated, and applied in mathematics is not a matter of “ontological commitment”, of coming to terms with whatever that might mean, but rather of epistemological articulation, of coming to terms through knowledge. The mathematical infinite in mathematics is a matter of method. How we deal with the specific individual issues involving the infinite turns on the narrative we present about how it fits into methodological mathematical frameworks established and being established. The first section discusses the mathematical infinite in historical context, and the second, set theory and the emergence of the mathematical infinite. The third section discusses the infinite in and out of mathematics, and how it is to be taken. §1. The Infinite in Mathematics What role does the infinite play in modern mathematics? In modern mathematics, infinite sets abound both in the workings of proofs and as subject matter in statements, and so do universal statements, often of ∀ ∃ “for all there exists” form, which are indicative of direct engagement with the infinite. In many ways the role of the infinite is importantly “secondorder ” in the sense that Frege regarded number generally, in that the concepts of modern mathematics are understood as having infinite instances over a broad range. 1 But
A COGNITIVE ANALYSIS OF CAUCHY’S CONCEPTIONS OF FUNCTION, CONTINUITY, LIMIT, AND INFINITESIMAL, WITH IMPLICATIONS FOR TEACHING THE CALCULUS
"... In this paper we use theoretical frameworks from mathematics education and cognitive psychology to analyse Cauchy’s ideas of function, continuity, limit and infinitesimal expressed in his Cours D’Analyse. Our analysis focuses on the development of mathematical thinking from human perception and acti ..."
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In this paper we use theoretical frameworks from mathematics education and cognitive psychology to analyse Cauchy’s ideas of function, continuity, limit and infinitesimal expressed in his Cours D’Analyse. Our analysis focuses on the development of mathematical thinking from human perception and action into more sophisticated forms of reasoning and proof, offering different insights from those afforded by historical or mathematical analyses. It reveals the conceptual power of Cauchy’s vision and the fundamental change involved in passing from the dynamic variability of the calculus to the modern settheoretic formulation of mathematical analysis. This offers a reevaluation of the relationship between the natural geometry and algebra of elementary calculus that survives in applied mathematics, and the formal set theory of mathematical analysis that develops in pure mathematics and evolves into the logical development of nonstandard analysis using infinitesimal concepts. It counsels us that educational theories developed to evaluate student learning are themselves based on the conceptions of the experts who formulate them. It encourages us to reflect on the principles that we use to analyse the developing mathematical thinking of students, and to make an effort to understand the rationale of differing theoretical viewpoints. 1.