Abstract

Finally, chapter five presents and discusses my proposed structuralist view of the foundations of mathematics. This view can be summarized as follows: mathematics studies integrated wholes, not atomistic aggregates; the relevant properties of an individual in a mathematical system are extrinsic rather than intrinsic; mathematics studies structures , and not only systems; and the structures mathematics studies are objective, independent, non-artifactual abstracts. We are acquainted with these structures by pattern cognition. It is argued that under this view, mathematical epistemology presents no problems which are not also encountered in "real world" epistemology. I develop some of the consequences of this view, discuss the issues raised in chapters one and four from a structuralist viewpoint, and conclude by raising some open questions. ;The dissertation consists of five chapters. Chapter one discusses prephilosophical but reflective views mathematicians have held concerning mathematics which suggest a structuralist understanding. Issues raised here include the view that consistence implies existence, the view of axiomatizations as implicit definitions of their primitives, some confusions surrounding isomorphism, how to understand natural orderings, unproblematic identification of different theory discourses as expressing the same theory, and reinterpretations of different kinds of numbers. ;Chapter two discusses six ways 'structure' is understood, identifies one as the meaning central to the dissertation, arguing briefly that this is also the meaning most central to the structuralist movement, and finally outlines several major structuralist concerns. ;Chapter three characterizes structuralist views in general and briefly discusses the structuralist movement in the social sciences in light of that characterization. Figures who receive particular attention are Ferdinand de Saussure, Claude Levi-Strauss, and Jacques Lacan. ;Chapter four begins with a brief discussion of what I take mathematics and its philosophy and foundations to be. There follows a discussion of ten issues which adequate philosophical views of the foundations of mathematics should shed light on. In particular, such a view should account for the objectivity of mathematical knowledge; account for the degree of agreement and disagreement among mathematicians on issues like what axioms are true; allow for the growth of mathematical knowledge; account for the connection between knowledge and truth in mathematics; allow for mathematical discovery; provide a framework for understanding the history of mathematics; account for the applications of mathematics to the empirical sciences; not inhibit mathematical practice or alter the field beyond recognitions; and enhance mathematical pedagogy. ;Philosophies of mathematics tend to fall into two categories: those which account for mathematics only in passing and those which account for mathematics only. This dissertation aims at developing a structuralist account of mathematical foundations which sheds light on important features of mathematical knowledge and practice and which also forms part of a coherent general view of experience as a whole