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Mathematics as a Science of Patterns$
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Michael D. Resnik

Print publication date: 1999

Print ISBN-13: 9780198250142

Published to Oxford Scholarship Online: November 2003

DOI: 10.1093/0198250142.001.0001

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Doubts About Realism

Doubts About Realism

(p.82) 5 Doubts About Realism
Mathematics as a Science of Patterns

Michael D. Resnik (Contributor Webpage)

Oxford University Press

One of the strongest motivations for being an anti‐realist with regard to mathematics is the difficulty in formulating a plausible realist epistemology, given that there seems to be a lack of ties between the mathematical apparatus and observation. In this chapter, I discuss a few puzzles that the mathematical realist has to solve in order to formulate an acceptable epistemology, and I hint at the direction in which one might hope to find the solution to these puzzles. One of the puzzles, that was first clearly formulated by Paul Benacerraf, is that since mathematical objects are supposed to be causally inert entities existing outside of space and time, it is hard to see how we can ever get in contact with them, i.e. how we can ever gain knowledge of them or refer to them. A second puzzle concerns the incompleteness of mathematical entities that has it, that we have no conceivable evidence and so no answers to questions of whether the objects that one mathematical theory discusses are identical to those that another theory treats. Realists can reject the presuppositions of both these puzzles. The first presupposes that there must be some sort of direct or indirect interaction between us and the objects of our knowledge. The second assumes that there is always a fact of the matter as to whether the objects of one theory are the same or distinct, as those of another. In later chapters, I will try to undermine both of these presuppositions.

Keywords:   Benacerraf, identity, incompleteness, mathematical knowledge, mathematical objects, mathematical realism, observation, reference, reliabilism

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