10 - The Nature of Mathematical Objects

Summary

From the Editors

As we work on a mathematical problem, the mathematical objects we are working with seem very real and concrete. For example, when they do not behave as we had expected, they certainly seem quite objective and separate from our internal thoughts. As a result, many mathematicians tend toward a platonist/realist view of mathematical objects, at least until the problems with platonism are voiced. In this chapter, Øystein Linnebo attempts to resolve some of those problems by looking carefully at the language we use when referring to mathematical objects.

In ordinary speech, when we say “that's merely a semantic distinction,” we mean that there is no real difference—they are just two different ways of saying the same thing. But for philosophers, questions of semantics—what words mean and how they are used—are essential to unraveling many apparent disagreements or contradictions. Mathematicians, for example, generally are indifferent to the distinction between numbers and numerals (or tend to resolve them, as Errett Bishop did, by saying “I identify a number with its numeral”). But paradoxes appear when one is not careful about the distinction between an object and its name. In this chapter, Linnebo uses this distinction in his attempt to resolve some of the problems plaguing those who would like to view mathematical objects as objective.