According to realist structuralism, mathematics is the study of
structures. Structures are understood to be special kinds of complex
properties that can be instantiated by particulars together with
relations between these particulars. For example, the field of complex
numbers is assumed to be instantiated by any suitably large collection
of particulars in combination with four operations that satisfy certain
logical constraints. (The four operations correspond to addition,
subtraction, multiplication, and division.)
The fundamental properties provide a minimal basis for all intrinsic qualities of things. That is, whenever two things are not perfect qualitative duplicates, they differ in the distribution of fundamental properties over their parts; whenever two things do not differ by that distribution, they are perfect qualitative duplicates. It follows that all fundamental properties are intrinsic. But not all intrinsic properties are fundamental: the fundamental properties provide a minimal basis for all qualitaties. Hence there is no fundamental property of having a mass of either 1g or 2g, because instantiation of that property is already determined by the distribution of mass 1g and mass 2g. For the same reason, there is no fundamental property of being the fusion of a round thing and a distinct rectangular thing. By and large, fundamental properties are never logically complex (like A or B) and never structural (determined by the distribution of properties over the parts of their instances).
I've been invited to this year's German-Italian Colloquium in Analytic Philosophy, for which I've put together some remarks on the philosophy of mathematics: "Emperors, dragons and other
mathematicalia" (PDF). I mainly argue that mathematical sentences should be interpreted as quantifications over possibilia. Technically, this isn't really new. Daniel Nolan in particular has made a very similar suggestion (PDF). What hasn't been emphasized enough, I believe, is that this interpretation not only works from a technical point of view, but is quite attractive for various philosophical reasons. (Unlike Nolan, I argue that it isn't a reform, but a faithful interpretation of mathematics.)
Happy new year everybody. I'm still alive, and I still have questions and
comments on the metaphysics of David Lewis. This one is about Lewis'
philosophy of mathematics.
In "Mathematics is Megethology", Lewis argues
for structuralism in set theory: There is no particular relation of
membership, connecting particular things with particular classes. Instead,
there are just two sides of Reality, ordinary individuals on the one side,
proper-class many mereological atoms (called 'singletons') on the other.
Set theory is about all relations on this Reality that satisfy certain
constraints, like 'every individual stands in that relation to a singleton'.
Supervaluationsism and structuralism ('eliminative structuralism', not
the kind of structuralism that postulates structures) almost coincide.
Structuralism about something t says that any sentence 'F(t)' is to be
interpreted as 'for all x (if x is a candidate for t then F(x))'. For
example, arithmetical structuralism says that '2+2=4' is to be interpreted
as 'for all [N,0,',+,.] (if [N,0,',+,.] satisfies the peano
axioms then 0''+0''=0''''). If we translate 'x is a candidate for t' as 'x
is the referent of a precisification of 't'', we get: F(t) is (super-)true
iff it is true on all precisifications of 't'.