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Who gave you the Cauchy–Weierstrass tale? The dual history of rigorous calculus
 FOUNDATIONS OF SCIENCE
, 2012
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Leibniz’s infinitesimals: Their fictionality, their modern implementations, and their foes from Berkeley to Russell and beyond
, 2012
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A Revolution in Mathematics? What Really Happened a Century Ago and Why It Matters Today
"... The physical sciences all went through “revolutions”: wrenching transitions in which methods changed radically and became much more powerful. It is not widely realized, but there was a similar transition in mathematics between about 1890 and 1930. The first section briefly describes the changes that ..."
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The physical sciences all went through “revolutions”: wrenching transitions in which methods changed radically and became much more powerful. It is not widely realized, but there was a similar transition in mathematics between about 1890 and 1930. The first section briefly describes the changes that took place and why they qualify as a “revolution”, and the second describes turmoil and resistance to the changes at the time. The mathematical event was different from those in science, however. In science, most of the older material was wrong and discarded, while old mathematics needed precision upgrades but was mostly correct. The sciences were completely transformed while mathematics split, with the core changing profoundly but many applied areas, and mathematical science outside the core, relatively unchanged. The strangest difference is that the scientific revolutions were highly visible, while the significance of the mathematical event is essentially unrecognized. The section “Obscurity” explores factors contributing to this situation and suggests historical turning points that might have changed it. The main point of this article is not that a revolution occurred, but that there are penalties for not being aware of it. First, precollege mathematics education is still based on nineteenthcentury methodology, and it seems to me that we will not get satisfactory outcomes until this changes [9]. Second, the mathematical community is adapted to the social and intellectual environment of the mid and late twentieth century, and this environment is changing in ways likely to marginalize core mathematics. But core mathematics provides the skeleton that supports the muscles and sinews of
Contributions to a science of contemporary mathematics, preprint; current draft at http:// www.math.vt.edu/people/quinn
"... Abstract. This essay provides a description of modern mathematical practice, with emphasis on differences between this and practices in the nineteenth century, and in other sciences. Roughly, modern practice is well adapted to the structure of the subject and, within this constraint, much better ad ..."
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Abstract. This essay provides a description of modern mathematical practice, with emphasis on differences between this and practices in the nineteenth century, and in other sciences. Roughly, modern practice is well adapted to the structure of the subject and, within this constraint, much better adapted to the strengths and weaknesses of human cognition. These adaptations greatly increased the effectiveness of mathematical methods and enabled sweeping developments in the twentieth century. The subject is approached in a bottomup ‘scientific ’ way, finding patterns in concrete microlevel observations and being eventually lead by these to understanding at macro levels. The complex and intenselydisciplined technical details of modern practice are fully represented. Finding accurate commonalities that transcend technical detail is certainly a challenge, but any account that shies away from this cannot be complete. As in all sciences, the final result is complex, highly nuanced, and has many surprises. A particular objective is to provide a resource for mathematics education. Elementary education remains modeled on the mathematics of the nineteenth century and before, and outcomes have not changed much either. Modern methodologies might lead to educational gains similar to those seen in professional practice. This draft is about 90 % complete, and comments are welcome. 1.
Loss of vision: How mathematics turned blind while it learned to see more clearly
 In B. Löwe and T. Müller (Eds.), Philosophy of Mathematics: Sociological Aspects and Mathematical Practice
, 2010
"... The aim of this paper is to provide a framework for the discussion of mathematical ontology that is rooted in actual mathematical practice, i.e., the way in which mathematicians have introduced and dealt with mathematical ..."
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The aim of this paper is to provide a framework for the discussion of mathematical ontology that is rooted in actual mathematical practice, i.e., the way in which mathematicians have introduced and dealt with mathematical
The Total War of Paris Mathematicians
"... Abstract. From 1914 to 1918, Paris mathematicians were highly mobilized for war. In this paper, we argue that the wartime experience of influent Parisian mathematicians was total, in three areas especially: military expertise, scientific and innovation policies, and intellectual debates about the r ..."
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Abstract. From 1914 to 1918, Paris mathematicians were highly mobilized for war. In this paper, we argue that the wartime experience of influent Parisian mathematicians was total, in three areas especially: military expertise, scientific and innovation policies, and intellectual debates about the relation between science and society. We establish that Paris mathematicians’ early role as the unquestioned leaders of the scientific mobilization was anchored in a prewar context where mathematics held a high status in the sciences and society. In wartime, we focus on mathematicians ’ activities at the Academy of Sciences and in government and pay special attention to their contribution to sound–ranging and ballistics. We find that mathematicians’ roles underwent significant changes in the course of the conflict. We finally try to assess the effects of war on postwar images of mathematics focusing on three specific aspects: institutional reconstruction, internationalism, and modernism. In contrast to the view made popular by members of Bourbaki’s first generation, we argue that there was no sharp decline in postwar mathematics
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.
A Subjective Comparison Between a Historical and a Contemporary Textbook on Geometry1
"... ©2014 by the authors. This work is licensed under a Creative Commons License. JHM is an open access biannual journal sponsored by the Claremont Center for the Mathematical ..."
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©2014 by the authors. This work is licensed under a Creative Commons License. JHM is an open access biannual journal sponsored by the Claremont Center for the Mathematical