Extending, Expanding, and Laying Bare: A Unified Account of Generalization in Mathematics
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2015
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Haverford College. Department of Philosophy
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Thesis
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The Charles Schwartz Memorial Prize in Philosophy
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eng
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Open Access
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Abstract
It is quite common for mathematicians to refer to theorems or definitions as generalizations of others. Although one gets a very good sense of what the term means by doing enough mathematics, it is not a term that mathematicians typically formally define. Indeed, with a little consideration, it can be seen that the task of giving a proper and comprehensive definition is highly non-trivial, because there are various different applications of the term in widely disparate contexts. Nonetheless, the use of the term is rarely, if ever controversial within the mathematical community. This suggests that there is something, albeit difficult to articulate, that mathematicians intuitively recognize the disparate cases to have in common. The primary goal of my thesis is to explain precisely what this commonality is, by giving a definition of generalization that is applicable to each of the various cases of the term's use. In service of this goal, I lay out an ontological picture of mathematics that borrows both from the long-standing structuralist ontological view of mathematics, and from the work of Danielle Macbeth, who claims that mathematics is a study of objective concepts, the referents of mathematical definitions, which definitions give Fregean senses. Having given my account of generalization, I then elaborate on generalization's roles in mathematical practice, using my account to shed some light on its utility in these roles.