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Computing in Cognitive Science
, 1989
"... Introduction Nobody doubts that computers have had a profound influence on the study of human cognition. The very existence of a discipline called Cognitive Science is a tribute to this influence. One of the principal characteristics that distinguishes Cognitive Science from more traditional studies ..."
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Introduction Nobody doubts that computers have had a profound influence on the study of human cognition. The very existence of a discipline called Cognitive Science is a tribute to this influence. One of the principal characteristics that distinguishes Cognitive Science from more traditional studies of cognition within Psychology, is the extent to which it has been influenced by both the ideas and the techniques of computing. It may come as a surprise to the outsider, then, to discover that there is no unanimity within the discipline on either (a) the nature (and in some cases the desireabilty) of the influence and (b) what computing is  or at least on its  essential character, as this pertains to Cognitive Science. In this essay I will attempt to comment on both these questions. The first question will bring us to a discussion of the role that computing plays in our understanding of human (and perhaps animal) cognition. I wi
"Clarifying the Nature of the Infinite": the development of metamathematics and proof theory
, 2001
"... We discuss the development of metamathematics in the Hilbert school, and Hilbert's prooftheoretic program in particular. We place this program in a broader historical and philosophical context, especially with respect to nineteenth century developments in mathematics and logic. Finally, we sho ..."
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Cited by 5 (2 self)
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We discuss the development of metamathematics in the Hilbert school, and Hilbert's prooftheoretic program in particular. We place this program in a broader historical and philosophical context, especially with respect to nineteenth century developments in mathematics and logic. Finally, we show how these considerations help frame our understanding of metamathematics and proof theory today.
In Defense of the Ideal 2nd DRAFT
"... This paper lies at the edge of the topic of the workshop. We can write down a Π1 1 axiom whose models are precisely the ∈structures 〈Rα, ∈ ∩R2 α〉 where α> 0 and Rα is the collection of all (pure) sets of rank < α. From this, one can consider the introduction of new axioms concerning the size ..."
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This paper lies at the edge of the topic of the workshop. We can write down a Π1 1 axiom whose models are precisely the ∈structures 〈Rα, ∈ ∩R2 α〉 where α> 0 and Rα is the collection of all (pure) sets of rank < α. From this, one can consider the introduction of new axioms concerning the size of α. The question of the grounds for doing so is perhaps the central question of the workshop. But I want to discuss another question which, as I said, arises at the periphery: How do we know that there are structures 〈Rα, ∈ ∩R2 α〉? How do we know that there exist such things as sets and how do we know that, given such things, the axioms we write down are true of them? These seem very primitive questions, but the skepticism implicit in them has deep (and ancient) roots. In particular, they are questions about ideal objects in general, and not just about the actual infinite. I want to explain why I think the questions (as intended) are empty and the skepticism unfounded. 1 I will be expanding the argument of the first part of my paper “Proof and truth: the Platonism of mathematics”[1986a]. 2 The argument in question
39 DIAGRAMS AS MEANS AND OBJECTS OF MATHEMATICAL REASONING
"... According to Peirce a great part of mathematical thinking consists in observing or imagining the outcomes and regularities of manipulations of all sorts of diagrams. This tenet is explained and substantiated by expounding the notion of diagram and by analyzing several specific cases from different p ..."
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According to Peirce a great part of mathematical thinking consists in observing or imagining the outcomes and regularities of manipulations of all sorts of diagrams. This tenet is explained and substantiated by expounding the notion of diagram and by analyzing several specific cases from different parts of mathematics.
EDUCATION
"... This research reviewed literature on proof in mathematics education. Several views of proof classified and identified such as psychological approach, (Platonism, empiricism), structural approach, (logicism, formalism, intuitionism), social approach, (ontology, axiomatic systems). All these views ..."
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This research reviewed literature on proof in mathematics education. Several views of proof classified and identified such as psychological approach, (Platonism, empiricism), structural approach, (logicism, formalism, intuitionism), social approach, (ontology, axiomatic systems). All these views of proof are valuable in mathematics education society. The concept of proof can be found in the form of analytic knowledge not of constructive knowledge. Human beings developed their knowledge in the sequence of constructive knowledge to analytic knowledge. Therefore, in mathematics education, the curriculum of mathematics should involve the process of cognitive knowledge development.
Gödel and the Metamathematical Tradition
, 2007
"... The metamathematical tradition that developed from Hilbert’s program is based on syntactic characterizations of mathematics and the use of explicit, finitary methods in the metatheory. Although Gödel’s work in logic fits squarely in that tradition, one often finds him curiously at odds with the asso ..."
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The metamathematical tradition that developed from Hilbert’s program is based on syntactic characterizations of mathematics and the use of explicit, finitary methods in the metatheory. Although Gödel’s work in logic fits squarely in that tradition, one often finds him curiously at odds with the associated methodological orientation. This essay explores that tension and what lies behind it. 1
Introducing Formalism in Economics: The Growth Model
, 2010
"... Summary: The objective is to interpret John von Neumann's growth model as a decisive step of the forthcoming formalist revolution of the 1950s in economics. This model gave rise to an impressive variety of comments about its classical or neoclassical underpinnings. We go beyond this traditional ..."
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Summary: The objective is to interpret John von Neumann's growth model as a decisive step of the forthcoming formalist revolution of the 1950s in economics. This model gave rise to an impressive variety of comments about its classical or neoclassical underpinnings. We go beyond this traditional criterion and interpret rather this model as the manifestation of von Neumann's involvement in the formalist programme of mathematician David Hilbert. We discuss the impact of Kurt Gödel’s discoveries on this programme. We show that the growth model reflects the pragmatic turn of the formalist programme after Gödel and proposes the extension of modern axiomatisation to economics.