Abstract
Informally, structural properties of mathematical objects are usually characterized in one of two ways: either as properties expressible purely in terms of the primitive relations of mathematical theories, or as the properties that hold of all structurally similar mathematical objects. We present two formal explications corresponding to these two informal characterizations of structural properties. Based on this, we discuss the relation between the two explications. As will be shown, the two characterizations do not determine the same class of mathematical properties. From this observation we draw some philosophical conclusions about the possibility of a ‘correct’ analysis of structural properties.
Highlights
Structural properties play a central role in the contemporary philosophy of mathematics, in the debate about mathematical structuralism
This is the view that mathematics is not concerned with the ‘internal nature’ of its objects, but rather with how these objects ‘relate to each other’ [Shapiro, 1997; Resnik, 1997; Parsons, 1990]
The standard mathematical theory of the natural numbers is second-order Peano arithmetic. It is well-known that many different set-theoretic systems satisfy the axioms of Peano arithmetic, such as the von Neumann ordinals ∅, {∅}, {∅, {∅}}, . . . and the Zermelo ordinals ∅, {∅}, {{∅}}, . . . for example
Summary
Structural properties play a central role in the contemporary philosophy of mathematics, in the debate about mathematical structuralism. Present work on mathematical structuralism focuses mainly on structural properties in the latter sense, that is, on the structural properties of elements in mathematical systems This holds in particular for recent contributions to non-eliminative structuralism, in the debate on the identity of structurally indiscernible places in pure structures. The first main goal here will be to show that both the invariance account and the definability account can be made to work in a precise sense for both types of mathematical properties, that is, for properties of systems as well as properties of elements in such systems. We will show that the two accounts do not characterize the same concept Based on this observation, we propose a tolerant, Carnapian stance with respect to the choice of explication: we argue that neither of the two explications gives us the ‘correct’ notion of structural properties; instead both accounts have their philosophical and mathematical merits.
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