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The Constructed Objectivity of Mathematics and the Cognitive Subject
, 2001
"... Introduction This essay concerns the nature and the foundation of mathematical knowledge, broadly construed. The main idea is that mathematics is a human construction, but a very peculiar one, as it is grounded on forms of "invariance" and "conceptual stability" that single out ..."
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Introduction This essay concerns the nature and the foundation of mathematical knowledge, broadly construed. The main idea is that mathematics is a human construction, but a very peculiar one, as it is grounded on forms of "invariance" and "conceptual stability" that single out the mathematical conceptualization from any other form of knowledge, and give unity to it. Yet, this very conceptualization is deeply rooted in our "acts of experience", as Weyl says, beginning with our presence in the world, first in space and time as living beings, up to the most complex attempts we make by language to give an account of it. I will try to sketch the origin of some key steps in organizing perception and knowledge by "mathematical tools", as mathematics is one of the many practical and conceptual instruments by which we categorize, organise and "give a structure" to the world. It is conceived on the "interface" between us and the world, or, to put it in husserlian terminology, it is "de
A short survey of automated reasoning
"... Abstract. This paper surveys the field of automated reasoning, giving some historical background and outlining a few of the main current research themes. We particularly emphasize the points of contact and the contrasts with computer algebra. We finish with a discussion of the main applications so f ..."
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Abstract. This paper surveys the field of automated reasoning, giving some historical background and outlining a few of the main current research themes. We particularly emphasize the points of contact and the contrasts with computer algebra. We finish with a discussion of the main applications so far. 1 Historical introduction The idea of reducing reasoning to mechanical calculation is an old dream [75]. Hobbes [55] made explicit the analogy in the slogan ‘Reason [...] is nothing but Reckoning’. This parallel was developed by Leibniz, who envisaged a ‘characteristica universalis’ (universal language) and a ‘calculus ratiocinator ’ (calculus of reasoning). His idea was that disputes of all kinds, not merely mathematical ones, could be settled if the parties translated their dispute into the characteristica and then simply calculated. Leibniz even made some steps towards realizing this lofty goal, but his work was largely forgotten. The characteristica universalis The dream of a truly universal language in Leibniz’s sense remains unrealized and probably unrealizable. But over the last few centuries a language that is at least adequate for
On the imaginative constructivist nature of design: a theoretical approach
"... Abstract: Most empirical accounts of design suggest that designing is an activity where objects and representations are progressively constructed. Despite this fact, whether design is a constructive process or not is not a question directly addressed in current design research. By contrast, in other ..."
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Abstract: Most empirical accounts of design suggest that designing is an activity where objects and representations are progressively constructed. Despite this fact, whether design is a constructive process or not is not a question directly addressed in current design research. By contrast, in other fields such as Mathematics or Psychology, the notion of constructivism is seen as a foundational issue. The present paper defends the point of view that forms of constructivism in design need to be identified and integrated as a foundational element in design research as well. In fact, a look at the literature reveals at least two types of constructive processes that are well embedded in design research. First, an interactive constructivism, where a designer engages a conversation with media, that allows changing the course of the activity as a result of this interaction. Second, a social constructivism, where designers need to handle communication and negotiation aspects, that allows integrating individuals ’ expertise into the global design process. A key feature lacking to these wellestablished paradigms is the explicit consideration of creativity as a central issue of design. To explore how creative and constructivist aspects of design can be taken into account conjointly, the present paper pursues a theoretical approach. We consider the roots of
Promoting Social Justice In and Through the Mathematics Curriculum: Exploring the Connections with Epistemologies of Mathematics
"... This article rehearses the argument that being a critical mathematics educator is associated with a particular epistemological stance, one which views the truths of mathematics as historically located, influenced by the knower and mutable. Case study data, collected in England, is offered which exem ..."
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This article rehearses the argument that being a critical mathematics educator is associated with a particular epistemological stance, one which views the truths of mathematics as historically located, influenced by the knower and mutable. Case study data, collected in England, is offered which exemplifies this connection between epistemology and openness to equity issues in the thinking of some beginning secondary mathematics teachers. Teachers ' responses are analysed around four themes: their beliefs about the nature of mathematics, how those beliefs affect their pedagogy, how they explain student failure, and their views on initial teacher education. These are linked to their commitment to social justice in and through mathematics. The links between subject studies in teacher education and equity issues in the classroom are discussed. The Status of Mathematical Knowledge The conventional view of the nature of mathematics, which has an entrenched position in mainstream contemporary Western culture, rests on a takenforgranted understanding of the nature of mathematical knowledge which accords it the
Grush & Churchland GAPS IN PENROSE'S TOILINGS
"... that the mechanism for consciousness involves quantum gravitational phenomena, acting through microtubules in neurons. We show that this hypothesis is implausible. First, the Gödel Result does not imply that human thought is in fact non algorithmic. Second, whether or not non algorithmic quantum gra ..."
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that the mechanism for consciousness involves quantum gravitational phenomena, acting through microtubules in neurons. We show that this hypothesis is implausible. First, the Gödel Result does not imply that human thought is in fact non algorithmic. Second, whether or not non algorithmic quantum gravitational phenomena actually exist, and if they did how that could conceivably implicate microtubules, and if microtubules were involved, how that could conceivably implicate consciousness, is entirely speculative. Third, cytoplasmic ions such as calcium and sodium are almost certainly present in the microtubule pore, barring the quantum mechanical effects Penrose envisages. Finally, physiological evidence indicates that consciousness does not directly depend on microtubule properties in any case, rendering doubtful any theory according to which consciousness is generated in the microtubules. I.
INTEGRATING MODERNIST AND POSTMODERNIST PERSPECTIVES ON ORGANIZATIONS: A COMPLEXITY
"... Competition between modernism and postmodernism has not been fruitful, and management researchers are divided in their preference, thereby undermining the legitimacy of truth claims in the field as a whole. Drawing on Ashby’s Law of Requisite Variety, on complexity science, and in particular on po ..."
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Competition between modernism and postmodernism has not been fruitful, and management researchers are divided in their preference, thereby undermining the legitimacy of truth claims in the field as a whole. Drawing on Ashby’s Law of Requisite Variety, on complexity science, and in particular on powerlawdistributed phenomena, we show how the orderseeking regime of the modernists and the richnessseeking regime of the postmodernists draw on different ontological assumptions that can be integrated within a single overarching framework. The study of social systems such as organizations has long been caught between two conflicting bases of legitimacy. On the one hand, we have positivism—a set of procedures for creating valid knowledge expressing a modernist outlook that originated in the eighteenth century Enlightenment project. Positivism presumes a real, relatively stable, and objectively given world, populated by phenomena that can be rationally known and rationally analyzed by independent observers. Such phenomena can be decomposed into observation protocols resting on sense data and predictively related to each other through stable laws integrated via a mathematical syntax (Benacerraf & Putnam, 1964; Lakatos, 1976). Positivism promotes the modernist agenda: the understanding, manipulation, and control of predominantly physical phenomena for beneficial social ends. In contemporary social sciences, neoclassical economics remains positivism’s foremost exemplar (Colan
Intuition in Mathematics
"... Abstract: The literature on mathematics suggests that intuition plays a role in it as a ground of belief. This article explores the nature of intuition as it occurs in mathematical thinking. Section 1 suggests that intuitions should be understood by analogy with perceptions. Section 2 explains what ..."
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Abstract: The literature on mathematics suggests that intuition plays a role in it as a ground of belief. This article explores the nature of intuition as it occurs in mathematical thinking. Section 1 suggests that intuitions should be understood by analogy with perceptions. Section 2 explains what fleshing out such an analogy requires. Section 3 discusses Kantian ways of fleshing it out. Section 4 discusses Platonist ways of fleshing it out. Section 5 sketches a proposal for resolving the main problem facing Platonists—the problem of explaining how our experiences make contact with mathematical reality. If you look at the literature on mathematics—the prefaces to math textbooks, discussion pieces by mathematicians, mathematical popularizations and biographies, philosophical works about the nature of mathematics, psychological studies of mathematical cognition, educational material on the teaching of mathematics—you will regularly find talk about intuition. This suggests that there is some role intuition plays in mathematics, specifically as a ground of belief about mathematical matters. The aim of the present chapter is to stake out some ideas
1 The Simile of the Line in Plato’s Republic VI1
, 1997
"... In the Republic Book V (476a9480a13), Plato distinguished the philosophers and the sightlovers in terms of the difference of the objects of their cognition and love; that is, the philosophers are those who recognize and love what is,2 while the sightlovers are those who recognize and love what is ..."
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In the Republic Book V (476a9480a13), Plato distinguished the philosophers and the sightlovers in terms of the difference of the objects of their cognition and love; that is, the philosophers are those who recognize and love what is,2 while the sightlovers are those who recognize and love what is and is not. What is is the knowable