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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 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 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