Results 1  10
of
12
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 ..."
Abstract

Cited by 7 (0 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
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
SET THEORY FROM CANTOR TO COHEN
"... Set theory is an autonomous and sophisticated field of mathematics, enormously successful not only at its continuing development of its historical heritage but also at analyzing mathematical propositions and gauging their consistency strength. But set theory is also distinguished by having begun int ..."
Abstract
 Add to MetaCart
Set theory is an autonomous and sophisticated field of mathematics, enormously successful not only at its continuing development of its historical heritage but also at analyzing mathematical propositions and gauging their consistency strength. But set theory is also distinguished by having begun intertwined with pronounced metaphysical attitudes, and these have even been regarded as crucial by some of its great developers. This has encouraged the exaggeration of crises in foundations and of metaphysical doctrines in general. However, set theory has proceeded in the opposite direction, from a web of intensions to a theory of extension par excellence, and like other fields of mathematics its vitality and progress have depended on a steadily growing core of mathematical proofs and methods, problems and results. There is also the stronger contention that from the beginning set theory actually developed through a progression of mathematical moves, whatever and sometimes in spite of what has been claimed on its behalf. What follows is an account of the development of set theory from its beginnings through the creation of forcing based on these contentions, with an avowedly Whiggish emphasis on the heritage that has been retained and developed by the current theory. The
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 ..."
Abstract
 Add to MetaCart
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 ..."
Abstract
 Add to MetaCart
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
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, stu ..."
Abstract
 Add to MetaCart
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
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 ..."
Abstract
 Add to MetaCart
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 ..."
Abstract
 Add to MetaCart
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