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MATHEMATICAL THOUGHT AND ITS OBJECTS
In Mathematical Thought and Its Objects,Charles Parsons examines the
notion of object, with the aim of navigating between nominalism, which
denies that distinctively mathematical objects exist, and forms of Platonism
that postulate a transcendent realm of such objects. He introduces the central
mathematical notion of structure and defends a version of the structuralist
view of mathematical objects, according to which their existence is relative
to a structure and they have no more of a “nature” than that confers on them.
Parsons also analyzes the concept of intuition and presents a conception
of it distantly inspired by that of Kant, which describes a basic kind of access
to abstract objects and an element of a first conception of the infinite. An
intuitive model witnesses the possibility of the structure of natural numbers.
However, the full concept of number and knowledge of numbers involve
more that is conceptual and rational. Parsons considers how one can talk
about numbers, even though they are not objects of intuition. He explores
the conceptual role of the principle of mathematical induction and the sense
in which it determines the natural numbers uniquely.
Parsons ends with a discussion of reason and its role in mathematical
knowledge, attempting to do justice to plementary roles in mathe-
matical knowledge of rational insight, intuition, and the integration of our
theory as a whole.
Charles Parsons is Edgar Pierce Professor of Philosophy, Emeritus, at Harvard
University. He is a former editor of the Journal of author
of Mathematics in Philosophy and co-editor of the posthumous works of Kurt
Godel.¨
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