Categories_for_the_working_mathematician

Categories_for_the_working_mathematician - Synthese(2008...

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Synthese (2008) 162:1–13 DOI 10.1007/s11229-007-9166-9 ORIGINAL PAPER Categories for the working mathematician: making the impossible possible Jessica Carter Received: 1 June 2006 / Accepted: 12 February 2007 / Published online: 16 March 2007 © Springer Science+Business Media, Inc. 2007 Abstract This paper discusses the notion of necessity in the light of results from contemporary mathematical practice. Two descriptions of necessity are considered. According to the first, necessarily true statements are true because they describe ‘unchangeable properties of unchangeable objects’. The result that I present is argued to provide a counterexample to this description, as it concerns a case where objects are moved from one category to another in order to change the properties of these objects. The second description concerns necessary ‘structural properties’. Although I grant that mathematical statements could be considered as necessarily true in this sense, I question whether this justifies the claim that mathematics as a whole is necessary. Keywords Philosophy of mathematical practice · Necessity The history of mathematics abounds with results that could be described as ‘making the impossible possible’. For example, introducing the imaginary unit makes it pos- sible that the equation x 2 + 1 = 0 has a solution; working in Z 3 makes it possible that 2 + 2 is equal to 1; by rejecting Euclid’s fifth postulate, it becomes possible that the sum of the angles of a triangle is not equal to two right angles. On the other hand, it is a common view that mathematical statements are necessarily true. Neces- sary, true statements are often described as ‘statements that are true in all possible worlds’. But on considering examples like the ones above, one could question whether it makes sense to claim that mathematical statements are necessarily true. In order to A version of this paper was presented at the meeting, ‘Impact des Categories: Aspects Historiques et Philosophiques’, held in Paris, Autumn 2005. I thank the audience for valuable comments and discussion. I especially wish to thank Colin McLarty for his invaluable support. Also thanks to the two anonymous referees for their helpful comments. J. Carter ( B ) Department of Mathematics and Computer Science, University of Southern Denmark, Campusvej 55, Odense 5230, Denmark e-mail: [email protected]
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2 Synthese (2008) 162:1–13 address this question two descriptions of necessity will be discussed: one in terms of unchangeable properties of unchangeable objects leading to the notion of possible worlds, and another in terms of propositions following necessarily from certain assumptions. I will consider these descriptions of necessity in the light of an example from contemporary mathematics. The example concern mathematics using category theory. In the light of the fact that category theory has today become an integrated part of many branches of mathematics, note that the use of category theory in this example does not make it exceptional.
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