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The chapter introduces the iterative conception, according to which every set appears at one level or another of the mathematical structure known as the cumulative hierarchy, as well as theories based on the conception. The chapter presents various accounts of the iterative conception: the constructivist account, the dependency account and my own minimalist account. It is argued that the minimalist account is to be preferred to the others. A method – which I call inference to the best conception – is then described to defend the correctness of the iterative conception so understood. This method requires one to show that the iterative conception fares better than other conceptions with respect to a number of desiderata on conceptions of set. This provides additional motivation for exploring alternative conceptions of set in the remainder of the book.
Eliminative and non-eliminative forms of mathematical structuralism contain insights that every decent philosophical account of mathematicalstructures has to accommodate. So the challenge is to incorporate the insights in the nature of structure that have been acquired by eliminative and non-eliminative mathematical structuralists and to do better where existing accounts have fallen short. It is in this spirit that I try to connect my account of mathematical structure with elements of leading versions of mathematical structuralism.
This chapter is of a transitional nature. I turn to the connection between arbitrary object theory and the notion of mathematical structuralism. Arbitrary object theory is put asidefor a while, and I concentrate on forms of eliminative and non-eliminative structuralism in the philosophy of mathematics.
Building on the seminal work of Kit Fine in the 1980s, Leon Horsten here develops a new theory of arbitrary entities. He connects this theory to issues and debates in metaphysics, logic, and contemporary philosophy of mathematics, investigating the relation between specific and arbitrary objects and between specific and arbitrary systems of objects. His book shows how this innovative theory is highly applicable to problems in the philosophy of arithmetic, and explores in particular how arbitrary objects can engage with the nineteenth-century concept of variable mathematical quantities, how they are relevant for debates around mathematical structuralism, and how they can help our understanding of the concept of random variables in statistics. This fully worked through theory will open up new avenues within philosophy of mathematics, bringing in the work of other philosophers such as Saul Kripke, and providing new insights into the development of the foundations of mathematics from the eighteenth century to the present day.
The present work is a systematic study of five frameworks or perspectives articulating mathematical structuralism, whose core idea is that mathematics is concerned primarily with interrelations in abstraction from the nature of objects. The first two, set-theoretic and category-theoretic, arose within mathematics itself. After exposing a number of problems, the Element considers three further perspectives formulated by logicians and philosophers of mathematics: sui generis, treating structures as abstract universals, modal, eliminating structures as objects in favor of freely entertained logical possibilities, and finally, modal-set-theoretic, a sort of synthesis of the set-theoretic and modal perspectives.
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