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15
Mathematical proofs at a crossroad
 Theory Is Forever, Lectures Notes in Comput. Sci. 3113
, 2004
"... Abstract. For more than 2000 years, from Pythagoras and Euclid to Hilbert and Bourbaki, mathematical proofs were essentially based on axiomaticdeductive reasoning. In the last decades, the increasing length and complexity of many mathematical proofs led to the expansion of some empirical, experimen ..."
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Abstract. For more than 2000 years, from Pythagoras and Euclid to Hilbert and Bourbaki, mathematical proofs were essentially based on axiomaticdeductive reasoning. In the last decades, the increasing length and complexity of many mathematical proofs led to the expansion of some empirical, experimental, psychological and social aspects, yesterday only marginal, but now changing radically the very essence of proof. In this paper, we try to organize this evolution, to distinguish its different steps and aspects, and to evaluate its advantages and shortcomings. Axiomaticdeductive proofs are not a posteriori work, a luxury we can marginalize nor are computerassisted proofs bad mathematics. There is hope for integration! 1
Computing the Perfect Model: Why Do Economists Shun Simulation?*
, 2006
"... Like other mathematically intensive sciences, economics is becoming increasingly computerized. Despite the extent of the computation, however, there is very little true simulation. Simple computation is a form of theory articulation, whereas true simulation is analogous to an experimental procedure. ..."
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Like other mathematically intensive sciences, economics is becoming increasingly computerized. Despite the extent of the computation, however, there is very little true simulation. Simple computation is a form of theory articulation, whereas true simulation is analogous to an experimental procedure. Successful computation is faithful to an underlying mathematical model, whereas successful simulation directly mimics a process or a system. The computer is seen as a legitimate tool in economics only when traditional analytical solutions cannot be derived, i.e., only as a purely computational aid. We argue that true simulation is seldom practiced because it does not fit the conception of understanding inherent in mainstream economics. According to this conception, understanding is constituted by analytical derivation from a set of fundamental economic axioms. We articulate this conception using the concept of economists ’ perfect model. Since the deductive links between the assumptions and the consequences are not transparent in ‘bottomup ’ generative microsimulations, microsimulations cannot correspond to the perfect model and economists do not therefore consider them viable candidates for generating theories that enhance economic understanding. 1. Introduction. Economics
AN EPISTEMIC STRUCTURALIST ACCOUNT OF MATHEMATICAL KNOWLEDGE
, 2003
"... This thesis aims to explain the nature and justification of mathematical knowledge using an epistemic version of mathematical structuralism, that is a hybrid of Aristotelian structuralism and Hellman’s modal structuralism. Structuralism, the theory that mathematical entities are recurring structures ..."
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This thesis aims to explain the nature and justification of mathematical knowledge using an epistemic version of mathematical structuralism, that is a hybrid of Aristotelian structuralism and Hellman’s modal structuralism. Structuralism, the theory that mathematical entities are recurring structures or patterns, has become an increasingly prominent theory of mathematical ontology in the later decades of the twentieth century. The epistemically driven version of structuralism that is advocated in this thesis takes structures to be primarily physical, rather than Platonically abstract entities. A fundamental benefit of epistemic structuralism is that this account, unlike other accounts, can be integrated into a naturalistic epistemology, as well as being congruent with mathematical practice. In justifying mathematical knowledge, two levels of abstraction are introduced. Abstraction by simplification is how we extract mathematical structures from our experience of the physical world. Then, abstraction by extension, simplification or recombination are used to acquire concepts of derivative mathematical structures.
52 LIVING WITH A NEW MATHEMATICAL SPECIES
"... Computers are both the creature and the creator of mathematics. They are, in the apt phrase of Seymour Papert, "mathematicsspeaking beings". More recently J. David Bolter in his stimulating book Turing's Man [4] calls computers "embodied mathematics". Computers shape and en ..."
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Computers are both the creature and the creator of mathematics. They are, in the apt phrase of Seymour Papert, "mathematicsspeaking beings". More recently J. David Bolter in his stimulating book Turing's Man [4] calls computers "embodied mathematics". Computers shape and enhance the power of mathematics, while mathematics shapes and enhances the power of computers. Each forces the other to grow and change, creating, in Thomas Kuhn's language, a new mathematical paradigm. Until recently, mathematics was a strictly human endeavor. But suddenly, in a brief instant on the time scale of mathematics, a new species has entered the mathematical ecosystem. Computers speak mathematics, but in a dialect that is difficult for some humans to understand. Their number systems are finite rather than infinite; their addition is not commutative; and they don't really understand "zero", not to speak of "infinity". Nonetheless, they do embody
URI WILENSKY STATISTICAL MECHANICS FOR SECONDARY SCHOOL:
"... ABSTRACT. In the Connected Learning projects, we are studying students ’ learning of content through exploring and constructing computerbased models of that content. This paper presents a case study of a high school physics teacher’s design and exploration of a computerbased model of gas molecules ..."
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ABSTRACT. In the Connected Learning projects, we are studying students ’ learning of content through exploring and constructing computerbased models of that content. This paper presents a case study of a high school physics teacher’s design and exploration of a computerbased model of gas molecules in a box. We follow up the case study with shorter vignettes of students ’ exploration and elaboration of the GasinaBox model. The cases lead us to analyze and discuss the role of modelbased inquiry in science and mathematics education as well as to draw some general conclusions with respect to the design of modeling languages and the design of pedagogies and activities appropriate for modelbased inquiry in classroom settings. 1.
Connected Mathematics  Building . . . with Mathematical Knowledge
, 1993
"... The context for this thesis is the conflict between two prevalent ways of viewing mathematics. The first way is to see mathematics as primarily a formal enterprise, concerned with working out the syntactic/formal consequences of its definitions. In the second view, mathematics is a creative enterpri ..."
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The context for this thesis is the conflict between two prevalent ways of viewing mathematics. The first way is to see mathematics as primarily a formal enterprise, concerned with working out the syntactic/formal consequences of its definitions. In the second view, mathematics is a creative enterprise concerned primarily with the construction of new entities and the negotiation of their meaning and value. Among teachers of mathematics the formal view dominates. The consequence for learners is a shallow brittle understanding of the mathematics they learn. Even for mathematics that they can do, in the sense of calculating an answer, they often can't explain why they're doing what they're doing, relate it to other mathematical ideas or operations, or connect the mathematics to any idea or problem they may encounter in their lives. The aim of this thesis is to develop alternative ways of teaching mathematics which strengthen the informal, intuitive and creative in mathematics. This research develops an approach to learning mathematics called "connected mathematics" which emphasizes learners’ negotiation of mathematical meaning. I have
What’s experimental about experimental mathematics?
, 2008
"... From a philosophical viewpoint, mathematics has often and traditionally been distinguished from the natural sciences by its formal nature and emphasis on deductive reasoning. Experiments — one of the corner stones of most modern natural science — have had no role to play in mathematics. However, dur ..."
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From a philosophical viewpoint, mathematics has often and traditionally been distinguished from the natural sciences by its formal nature and emphasis on deductive reasoning. Experiments — one of the corner stones of most modern natural science — have had no role to play in mathematics. However, during the last three decades, high speed computers and sophisticated software packages such as Maple and Mathematica have entered into the domain of pure mathematics, bringing with them a new experimental flavor. They have opened up a new approach in which computerbased tools are used to experiment with the mathematical objects in a dialogue with more traditional methods of formal rigorous proof. At present, a subdiscipline of experimental mathematics is forming with its own research problems, methodology, conferences, and journals. In this paper, I first outline the role of the computer in the mathematical experiment and briefly describe the impact of high speed computing on mathematical research within the emerging subdiscipline of experimental mathematics. I then consider in more detail the epistemological claims put forward within experimental mathematics and comment on some of the discussions that experimental mathematics has provoked within the mathematical community in recent years. In the second part of the paper, I suggest the notion of exploratory experimentation as a possible framework for understanding experimental mathematics. This is illustrated by discussing the socalled PSLQ algorithm.
Mathematical Explanation: Problems and Prospects
"... Since this issue is devoted to the interaction between philosophy of mathematics and mathematical practice, I would like to begin with an introductory reflection on this topic, before I enter the specifics of my contribution. ..."
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Since this issue is devoted to the interaction between philosophy of mathematics and mathematical practice, I would like to begin with an introductory reflection on this topic, before I enter the specifics of my contribution.