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The Ignorance of Bourbaki
, 1990
"... this article writes: "Which half of his brains did Bourbaki use ? My impression is, the left half. Perhaps I am projecting. The Bourbachistes were uncomfortable with the rightbrain mathematics of the Italian geometers, and for good reason: significant portions were suspect and might, if one takes ` ..."
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this article writes: "Which half of his brains did Bourbaki use ? My impression is, the left half. Perhaps I am projecting. The Bourbachistes were uncomfortable with the rightbrain mathematics of the Italian geometers, and for good reason: significant portions were suspect and might, if one takes `true' and `false' to be leftbrain notions and `right' and `wrong' to be rightbrain ones, be justifiably described as right, but false.
Representational formalisms: What they are and why we haven’t had any, submitted to a special issue of Pattern Recognition (2007) http://www.cs.unb.ca/~goldfarb/ETS special issue/Repr formalisms.pdf
, 2006
"... Abstract. Currently, the only discipline that has dealt with scientific representations— albeit nonstructural ones—is mathematics (as distinct from logic). I suggest that it is this discipline, only vastly expanded based on a new, structural, foundation, that will also deal with structural represen ..."
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Abstract. Currently, the only discipline that has dealt with scientific representations— albeit nonstructural ones—is mathematics (as distinct from logic). I suggest that it is this discipline, only vastly expanded based on a new, structural, foundation, that will also deal with structural representations. Logic (including computability theory) is not concerned with the issues of various representations useful in natural sciences. Artificial intelligence was supposed to address these issues but has, in fact, hardly advanced them at all. How do we, then, approach the development of representational formalisms? It appears that the only reasonable starting point is the primordial point at which all of mathematics began, i.e. we should start with the generalization of the process of construction of natural numbers, replacing the identical structureless units, out of which numbers are built, by structural ones, each signifying an atomic “transforming ” event. This paper is conceived as a companion to [1], and is a revised version of [2]. Mathematics is the science of the infinite, its goal is the symbolic comprehension of the infinite with human, that is finite, means.
Loss of vision: How mathematics turned blind while it learned to see more clearly
 In B. Löwe and T. Müller (Eds.), Philosophy of Mathematics: Sociological Aspects and Mathematical Practice
, 2010
"... The aim of this paper is to provide a framework for the discussion of mathematical ontology that is rooted in actual mathematical practice, i.e., the way in which mathematicians have introduced and dealt with mathematical ..."
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The aim of this paper is to provide a framework for the discussion of mathematical ontology that is rooted in actual mathematical practice, i.e., the way in which mathematicians have introduced and dealt with mathematical
ARISTOTELIAN REALISM
"... Aristotelian, or nonPlatonist, realism holds that mathematics is a science of the real world, just as much as biology or sociology are. Where biology studies living things and sociology studies human social relations, mathematics studies the quantitative or structural aspects of things, such as rat ..."
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Aristotelian, or nonPlatonist, realism holds that mathematics is a science of the real world, just as much as biology or sociology are. Where biology studies living things and sociology studies human social relations, mathematics studies the quantitative or structural aspects of things, such as ratios, or patterns, or complexity,
Category Theory and Structuralism
, 2009
"... The term structuralism occurred in several branches of the humanities and the sciences in the period 1929 – 1970: in Linguistics (Ferdinand de Saussure, Roman Jakobson), Anthropology (Claude LéviStrauss), Developmental psychology (Jean Piaget), Literature (Workshop for potential literature, Raymond ..."
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The term structuralism occurred in several branches of the humanities and the sciences in the period 1929 – 1970: in Linguistics (Ferdinand de Saussure, Roman Jakobson), Anthropology (Claude LéviStrauss), Developmental psychology (Jean Piaget), Literature (Workshop for potential literature, Raymond Queneau) and in Mathematics (Nicolas Bourbaki). To the layman the structuralist movement in mathematics was perhaps most visible the form of New Math, which was strongly influenced by the Bourbaki school. It has been argued in (Aubin 1997) that there were cultural connections between these movements. (See also A. Aczel 2007.) Some of these interactions may be regarded as rather superficial. The epistemologist Piaget however was very much influenced by Bourbaki, and seems to have suggested that those patterns of thought used to explain cognitive development were closely related to the mathematical “mother structures ” found by Bourbaki. On a very general level, structuralism refers to a mode of thinking involving abstraction from specifics and systematic identification and naming of common patterns. It is the relation of objects under study to each other that is of importance rather than their specific appearance, or “nature”. In mathematics, Richard Dedekind may be said to be the first structuralist. He described the positive integers (1, 2, 3,...) as positions in an infinite progression of elements (a socalled simply infinite system) 1 � � 2
LEARNING AND UNDERSTANDING IN ABSTRACT ALGEBRA BY
, 2001
"... who exceed all hopes and make the world grow with possibilities iv ACKNOWLEDGMENTS Perhaps it takes several villages to raise a scholar, for if I do indeed become a scholar, it will be largely because of the support, encouragement, and prodding I have received from individuals of many villages in th ..."
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who exceed all hopes and make the world grow with possibilities iv ACKNOWLEDGMENTS Perhaps it takes several villages to raise a scholar, for if I do indeed become a scholar, it will be largely because of the support, encouragement, and prodding I have received from individuals of many villages in the world of mathematics education and beyond. I thank my advisor, Joan FerriniMundy, for helping me craft this study and its account into a coherent whole, for pushing me to articulate and refine many of the premises that were implicitly guiding my thinking, and for providing support and encouragement throughout the process. I am indebted to Joan not only for directing my graduate program and this dissertation but also for providing an abundance of professional opportunities over the past eight years. In particular, I am grateful for the opportunity to join her at the National Research Council, where she again served as my supervisor and mentor. Thanks go to Karen Graham, for stepping in as codirector of this study, for helping me