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- Michael A. Arbib (1990). A Piagetian Perspective on Mathematical Construction. Synthese 84 (1):43 - 58.In this paper, we offer a Piagetian perspective on the construction of the logico-mathematical schemas which embody our knowledge of logic and mathematics. Logico-mathematical entities are tied to the subject's activities, yet are so constructed by reflective abstraction that they result from sensorimotor experience only via the construction of intermediate schemas of increasing abstraction. The axiom set does not exhaust the cognitive structure (schema network) which the mathematician thus acquires. We thus view truth not as something to be defined within the closed world of a formal system but rather in terms of the schema network within which the formal system is embedded. We differ from Piaget in that we see mathematical knowledge as based on social processes of mutual verification which provide an external drive to any necessary dynamic of reflective abstraction within the individual. From this perspective, we argue that axiom schemas tied to a preferred interpretation may provide a necessary intermediate stage of reflective abstraction en route to acquisition of the ability to use formal systems in abstracto.Reading list | Discuss | Edit | Categorize |My bibliography
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In this paper it is argued that the fundamental difference of the formal and the informal position in the philosophy of mathematics results from the collision of an object and a process centric perspective towards mathematics. This collision can be overcome by means of dialectical analysis, which shows that both perspectives essentially depend on each other. This is illustrated by the example of mathematical proof and its formal and informal nature. A short overview of the employed materialist dialectical approach is given that rationalises mathematical development as a process of model production. It aims at placing more emphasis on the application aspects of mathematical results. Moreover, it is shown how such production realises subjective capacities as well as objective conditions, where the latter are mediated by mathematical formalism. The approach is further sustained by Polanyi’s theory of problem solving and Stegmaier’s philosophy of orientation. In particular, the tool and application perspective illuminates which role computer-based proofs can play in mathematics.
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| | Other links: dx.doi.org springerlink.com | Scholar | At my library | More options ... While Gödel's (first) incompleteness theorem has been used to refute the main contentions of Hilbert's program, it does not seem to have been generally used to stress that a basic ingredient of that program, the concept of formal system as a closed system - as well as the underlying view, embodied in the axiomatic method, that mathematical theories are deductions from first principles must be abandoned. Indeed the logical community has generally failed to learn Gödel's lesson that Hilbert's concept of formal system as a closed system is inadequate and continues to use it as if there were no incompleteness theorem.
In this paper I will stress the role of Gödel's incompleteness theorem in showing the inadequacy of such a concept of formal system and the need for a more articulated view of mathematical theories. More generally I will argue that Gödel's result entails that, as an alternative to mathematical logic, a new concept of logic is required: logic as the theory of communicating inference processes.
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| | Scholar | At my library | More options ... In this book, Michael Arbib, a researcher in artificial intelligence and brain theory, joins forces with Mary Hesse, a philosopher of science, to present an integrated account of how humans "construct" reality through interaction with the social and physical world around them. The book is a major expansion of the Gifford Lectures delivered by the authors at the University of Edinburgh in the autumn of 1983. The authors reconcile a theory of the individual's construction of reality as a network of schemas "in the head" with an account of the social construction of language, science, ideology, and religion to provide an integrated schema-theoretic view of human knowledge. The authors still find scope for lively debate, particularly in their discussion of free will and of the reality of God. The book integrates an accessible exposition of background information with a cumulative marshalling of evidence to address fundamental questions concerning human action in the world and the nature of ultimate reality.
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| | Scholar | At my library | More options ... Cases where analogy has played a significant role in the formation of a new scientific concept are well-documented. Yet, how is it that genuinely new representations can be constructed from existing representations? It is argued that the process of âgeneric modelingâ enables abstraction of features common to both the domain of the source of the analogy and of the target phenomena. The analysis focuses on James Clerk Maxwell's construction of the electromagnetic field concept. The mathematical representation Maxwell constructed turned out to be a system of abstract laws that when applied to electromagnetic systems yield laws of a dynamical system that will not map back onto the mechanicals domains used in their construction.
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| | Other links: springerlink.com dx.doi.org | Scholar | At my library | More options ... That Piagetian epistemology has the dynamics of knowledge growth as its core consideration predetermines a need to consider it as potentially applicable to teaching. This paper addresses that need by first outlining the Piagetian theory of equilibration and then applying it to the construction of methods of teaching science.
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| | Other links: jstor.org | Scholar | At my library | More options ... In this paper, I introduce and examine the notion of “mathematical engineering” and its impact on mathematical change. Mathematical engineering is an important part of contemporary mathematics and it roughly consists of the “construction” and development of various machines, probes and instruments used in numerous mathematical fields. As an example of such constructions, I briefly present the basic steps and properties of homology theory. I then try to show that this aspect of contemporary mathematics has important consequences on our conception of mathematical knowledge, in particular mathematical growth.
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| Discuss | Edit | Categorize | My bibliography | Export citation | Other links: informaworld.com tandfonline.com dx.doi.org | Scholar | At my library | More options ...
| | Other links: informaworld.com tandfonline.com dx.doi.org | Scholar | At my library | More options ... In this expository article one of the contributions of Jean Cavailles to the philosophy of mathematics is presented: the analysis of ‘mathematical experience’. The place of Cavailles on the logico-philosophical scene of the 30s and 40s is sketched. I propose a partial interpretation of Cavailles's epistemological program of so-called ‘conceptual dialectics’: mathematical holism, duality principles, the notion of formal contents, and the specific temporal structure of conceptual dynamics. The structure of mathematical abstraction is analysed in terms of its complementary dimensions: paradigmatic generalization (domain extension, descriptive definitions, creative role of the symbolism...) and thematic reflexivity of concepts (promotion of operations to objects of a higher type).
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| | Scholar | At my library | More options ... Kit Fine develops a Fregean theory of abstraction, and suggests that it may yield a new philosophical foundation for mathematics, one that can account for both our reference to various mathematical objects and our knowledge of various mathematical truths. The Limits of Abstraction breaks new ground both technically and philosophically.
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| | Scholar | At my library | More options ... I discuss a puzzle that shows there is a need to develop a new metaphysical interpretation of mathematical theories, because all well-known interpretations conflict with important aspects of mathematical activities. The new interpretation, I argue, must authenticate the ontological commitments of mathematical theories without curtailing mathematicians’ freedom and authority to creatively introduce mathematical ontology during mathematical problem-solving. Further, I argue that these two constraints are best met by a metaphysical interpretation of mathematics that takes mathematical entities to be constitutively constructed by human activity in a manner similar to the constitutive construction of the US Supreme Court by certain legal and political activities. Finally, I outline some of the philosophical merits of metaphysical interpretations of mathematical theories of this type.
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| | Other links: dx.doi.org | Scholar | At my library | More options ... To explore the relation between mathematical models and reality, four different domains of reality are distinguished: observer-independent reality (to which there is no direct access), personal reality, social reality and mathematical/formal reality. The concepts of personal and social reality are strongly inspired by constructivist ideas. Mathematical reality is social as well, but constructed as an autonomous system in order to make absolute agreement possible. The essential problem of mathematical modelling is that within mathematics there is agreement about ‘truth’, but the assignment of mathematics to informal reality is not itself formally analysable, and it is dependent on social and personal construction processes. On these levels, absolute agreement cannot be expected. Starting from this point of view, repercussion of mathematical on social and personal reality, the historical development of mathematical modelling, and the role, use and interpretation of mathematical models in scientific practice are discussed.
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| | Scholar | At my library | More options ... Discussion of Michael A. Arbib, A Piagetian perspective on mathematical construction
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