What Structural Properties Are All About Johannes Korbmacher jkorbmacher@googlemail.com Georg Schiemer georg.schiemer@univie.ac.at June 2016 Abstract The paper develops a philosophical analysis of the concept of structural proper- ties in mathematics. Structural properties are generally characterized as properties that are invariant under isomorphic transformations of the mathematical objects considered. By drawing to a Lewis’ metaphysical theory of subject-matters and aboutness, it is shown here that structural properties so conceived are in a precise sense about the structure of a given class of mathematical objects. 1 Introduction The concept of structural properties is central to mathematical structuralism. Indeed, we can sum up mathematical structuralism as the view that mathematics is concerned precisely with the structural properties of its objects ([14], [12]). Moreover, structural properties figure prominently in several structuralist debates. Consider the following three theoretical contexts where the concept is explicitly discussed: 1. Non-Eliminative Structuralism: the debate on the identity of structurally indis- cernible places in abstract mathematical structures focuses on the question whether places, e.g. certain numbers in a number structure, that share the same structural properties should be identified ([3], [15], [4], [5]). 2. Homotopy Type Theory : It has been argued in [1] that homotopy type theory including the axiom of univalence presents a new approach to the foundations of mathematics that also captures a ‘principle of structuralism ’ present in mathemat- ics. This is the fact that, in modern mathematical practice, isomorphic objects are treated as identical. [1] further characterizes this view by stating that two objects (such as graphs or groups) are identical if and only if the share the same structural properties. 3. Abstraction Structuralism: [9] have recently outlined a new theory of non-eliminative structuralism based on Fregean abstraction principles for pure structures and po- sitions in such structures. It is argued that their account of structural abstraction (from concrete mathematical systems to abstract structures) validates what the 1