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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 the mathematical con ..."
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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
Statement
"... legal responsibility for the information which this document contains or the use to which this information is subsequently put. Although every step is taken to ensure that the information is as accurate as possible, it is understood that this material is supplied on the basis that there is no legal ..."
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legal responsibility for the information which this document contains or the use to which this information is subsequently put. Although every step is taken to ensure that the information is as accurate as possible, it is understood that this material is supplied on the basis that there is no legal responsibility for these materials or resulting from the use to which these can or may be put. Note: the telephone and fax numbers given in this guide for addresses outside the United Kingdom are those to be used if you are in that country. If you are telephoning or faxing from another country, we suggest you contact your local telecommunications provider for details of the country code and area code that you should use. Main contents
Student’s Conceptions of Mathematical Truth 1 Exploring the Student’s Conception of Mathematical Truth in Mathematical Reasoning
"... Since the recent inclusion of mathematical reasoning as an aspect of student’s mathematical competence, the development of students ’ mathematical reasoning has become one of the main issues in mathematics education (NCTM, 1989, 1991, 2000; Stiff & Curcio, 1999). From a cognitive standpoint, student ..."
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Since the recent inclusion of mathematical reasoning as an aspect of student’s mathematical competence, the development of students ’ mathematical reasoning has become one of the main issues in mathematics education (NCTM, 1989, 1991, 2000; Stiff & Curcio, 1999). From a cognitive standpoint, students ’ processes of reasoning can be influenced by their embodied
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
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