Mancosu - Philosophy of Mathematical Practice (Oxford, 2008).pdf

9Mathematical Conceptsand DefinitionsJAMIE TAPPENDENThese are some of the rules of classification and definition. But although nothingis more important in science than classifying and defining well, we need say nomore about it here, because it depends much more on our knowledge of thesubject matter being discussed than on the rules of logic.(Arnauld and Nicole, 1683, p.128)9.1 Definition and mathematical practiceThe basic observation structuring this survey is that mathematicians often setfinding the ‘right’/‘proper’/‘correct’/‘natural’ definition as a research objective,and success—finding ‘the proper’ definition—can be counted as a significantadvance in knowledge. Remarks like these from a retrospective on 20thcentury algebraic geometry are common:... the thesis presented here [is that] the progress of algebraic geometry is reflectedas much in its definitions as in its theorems (Harris, 1992, p.99)I am indebted to many people. Thanks to Paolo Mancosu both for comments on an unwieldy first draftand for bringing together the volume. Colin McClarty and Ian Proops gave detailed and illuminatingcomments. Colin also alerted me to a classic treatment by Emmy Noether (1921, pp. 25–29) ofthemultiple meanings of ‘prime’. An exciting conversation with Steven Menn about quadratic reciprocityset me on a path that led to some of the core examples in this paper. Early versions of this materialwere presented at Wayne State University and Berkeley. I’m grateful to both audiences, especiallyEric Hiddleston, Susan Vineberg, and Robert Bruner. Thanks to the members of my philosophyof mathematics seminar for discussing this material, especially Lina Jansson and Michael Dochertyfor conversation about the ‘bad company’ objection. As usual, I would have been lost withoutfriendly and patient guidance of the U. of M. mathematicians, especially (on this topic) Jim Milne(and his class notes on class field theory, algebraic number theory, and Galois theory, posted athttp://www.jmilne.org/math/) and Brian Conrad.