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**1 - 6**of**6**### THE NATURE OF CONTEMPORARY CORE MATHEMATICS

, 2010

"... Abstract. The goal of this essay is a description of modern mathematical practice, with emphasis on differences between this and practices in the nineteenth century. I explain how and why these differences greatly increased the effectiveness of mathematical methods and enabled sweeping developments ..."

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Abstract. The goal of this essay is a description of modern mathematical practice, with emphasis on differences between this and practices in the nineteenth century. I explain how and why these differences greatly increased the effectiveness of mathematical methods and enabled sweeping developments in the twentieth century. A particular concern is the significance for mathematics education: elementary education remains modeled on the mathematics of the nineteenth century and before, and use of modern methodologies might give advantages similar to those seen in mathematics. This draft is about 90 % complete, and comments are welcome. 1.

### Philosophy and Mathematics

, 2000

"... Philosophy is here conceived as an investigation into alternative courses of action and of the reasons for and against each. Philosophy can be relevant to the practice of mathematics|even though much prominent work in the philosophy of mathematics does not seem to be. 1 Introduction Paul Benace ..."

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Philosophy is here conceived as an investigation into alternative courses of action and of the reasons for and against each. Philosophy can be relevant to the practice of mathematics|even though much prominent work in the philosophy of mathematics does not seem to be. 1 Introduction Paul Benacerraf, a well-known philosopher of mathematics, recently wrote: \The philosophy of mathematics is . . . philosophy without the sugar coating of a pretense to Relevance to Life." [2, p. 10] There is good reason to believe that the philosophy of mathematics has not been relevant to the practice of mathematics. My own view is that the philosophy of mathematics can be relevant to the practice of mathematics. I discuss these claims in the following essay. These claims cannot be defended without clarifying how the phrase \philosophy of mathematics" is used. A single word or phrase can, of course, have 1 more than one use. This fact is important enough that I will illustrate it with a pair of exam...

### Journal of Pragmatics Creating Mathematical Infinities: Metaphor, Blending, and the Beauty of Transfinite Cardinals

"... The Infinite is one of the most intriguing ideas in which the human mind has ever engaged. Full of paradoxes and controversies, it has raised fundamental issues in domains as diverse and profound as theology, physics, and philosophy. The infinite, an elusive and counterintuitive idea, has even playe ..."

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The Infinite is one of the most intriguing ideas in which the human mind has ever engaged. Full of paradoxes and controversies, it has raised fundamental issues in domains as diverse and profound as theology, physics, and philosophy. The infinite, an elusive and counterintuitive idea, has even played a central role in defining mathematics, a fundamental field of human intellectual inquiry characterized by precision, certainty, objectivity, and effectiveness in modeling our real finite world. Particularly rich is the notion of actual infinity, that is, infinity seen as a “completed, ” “realized ” entity. This powerful notion has become so pervasive and fruitful in mathematics that if we decide to abolish it, most of mathematics as we know it would simply disappear, from infinitesimal calculus, to projective geometry, to set theory, to mention only a few. From the point of view of cognitive science, conceptual analysis, and cognitive semantics the study of mathematics, and of infinity in particular, raises several intriguing questions: How do we grasp the infinite if, after all, our bodies are finite, and so are our experiences and everything we encounter with our bodies? Where does then the infinite come from? What cognitive mechanisms make it possible? How an elusive and

### unknown title

, 2000

"... The words democracy, socialism, freedom, patriotic, realistic, justice, have each of them several different meanings which cannot be reconciled with one another. In the case of a word like democracy, not only is there no agreed definition, but the attempt to make one is resisted from all sides. It i ..."

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The words democracy, socialism, freedom, patriotic, realistic, justice, have each of them several different meanings which cannot be reconciled with one another. In the case of a word like democracy, not only is there no agreed definition, but the attempt to make one is resisted from all sides. It is almost universally felt that when we call a country democratic we are praising it: consequently the defenders of every kind of regime claim that it is a democracy, and fear that they might have to stop using the word if it were tied down to any one meaning. [19, p. 162]--George Orwell

### Aesthetic Analysis of Proofs of the Binomial Theorem

, 1999

"... This paper explores aesthetics of mathematical proof. Certain im-portant aspects of proofs are not relevant to aesthetics (validity, utility, exposition) but others are (immediacy, enlightenment, economy of means, establishment of connections, opening of mathematical vistas). Three dif-ferent proofs ..."

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This paper explores aesthetics of mathematical proof. Certain im-portant aspects of proofs are not relevant to aesthetics (validity, utility, exposition) but others are (immediacy, enlightenment, economy of means, establishment of connections, opening of mathematical vistas). Three dif-ferent proofs of the binomial theorem are used as illustrations. 1

### Prof. Daniel Goroff: Contextual Area History of the Mathematical Understanding of Dynamics

, 2002

"... 1) Describe a few of the most important developments in the history of dynamical systems theory. Understanding dynamical systems has required the development of many tools that radically altered our corresponding understanding of the physical world and mathematics itself. The tools of infinitesimal ..."

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1) Describe a few of the most important developments in the history of dynamical systems theory. Understanding dynamical systems has required the development of many tools that radically altered our corresponding understanding of the physical world and mathematics itself. The tools of infinitesimal calculus, the principle of least action, and topological descriptions of the qualitative behavior of dynamical systems are three of the most important contributions that dynamical systems theory has provided. These have greatly increased our insight into the causes and forces that give rise to motion. Infinitesimal Calculus Probably the most widely applied notion developed by those trying to understand mechanics is that of infinitesimal calculus. While many problems in statics could be understood without the aid of calculus, the dynamic problems of celestial mechanics and the computation of accurate ephemerides demanded more complex tools. The roots of infinitesimal calculus can be traced to the work of the ancient Greek geometers. The notion of quadrature of a shape, that is finding a square whose area equaled the shape, was of