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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 ta ..."
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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
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
NICOLAS BOURBAKI AND THE CONCEPT OF
"... JSTOR is a notforprofit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JS ..."
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JSTOR is a notforprofit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org. Springer is collaborating with JSTOR to digitize, preserve and extend access to Synthese.
Universal Logic as a Science of Patterns
"... Abstract. This article addresses Béziau’s [11] vision that universal logic should be capable of helping other fields of knowledge to build the right logic for the right situation, and that for some disciplines mathematical abstract conceptualization is more appropriate than symbolic formalization. ..."
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Abstract. This article addresses Béziau’s [11] vision that universal logic should be capable of helping other fields of knowledge to build the right logic for the right situation, and that for some disciplines mathematical abstract conceptualization is more appropriate than symbolic formalization. Hertz’s [67] diagrams of logical inference patterns are formalized and extended to present the universal logic conceptual framework as a comprehensible science of patterns. This facilitates those in other disciplines to develop, visualize and apply logical representation and inference structures that emerge from their problématique. A family of protologics is developed by resemantifying the sign for deduction,→, with inference patterns common to many logics, and specifying possible constraints on its use to represent the structural connectives and defeasible reasoning. Prooftheoretic, truththeoretic, intensional and extensional protosemantics are derived that supervene on the inference patterns. Examples are given of applications problem areas in a range of other disciplines, including the representation of states of affairs, individuals and relations.
Categories in Context: Historical, Foundational, and Philosophical †
"... The aim of this paper is to put into context the historical, foundational and philosophical significance of category theory. We use our historical investigation to inform the various categorytheoretic foundational debates and to point to some common elements found among those who advocate adopting ..."
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The aim of this paper is to put into context the historical, foundational and philosophical significance of category theory. We use our historical investigation to inform the various categorytheoretic foundational debates and to point to some common elements found among those who advocate adopting a foundational stance. We then use these elements to argue for the philosophical position that category theory provides a framework for an algebraic in re interpretation of mathematical structuralism. In each context, what we aim to show is that, whatever the significance of category theory, it need not rely upon any settheoretic underpinning. 1. History Any (rational) reconstruction of a history, if it is not merely to consist in a list of dates and ‘facts’, requires a perspective. Noting this, the perspective taken in our detailing the history of category theory will be bounded by our investigation of category theorists ’ topdown approach towards analyzing mathematical concepts in a categorytheoretic context. Any perspective too has an agenda: ours is that, contrary to popular belief, whatever the
Chapter Three Modal Semantics for Logics of Formal Inconsistency
"... This chapter collects three papers: 3.1 brings ‘Logics of essence and accident’, ..."
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This chapter collects three papers: 3.1 brings ‘Logics of essence and accident’,
here with the kind permission of Marcel Dekker.
"... The paper that constitutes this chapter was published as [12]. It is reproduced ..."
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The paper that constitutes this chapter was published as [12]. It is reproduced