To my way of thinking, the most natural example of
supervenience in mathematics---and the most similar to how
this term is used in the philosophy of mind, where one uses
it to describe the relation of the higher order properties,
such as features of the mind, to the lower order
properties, such as molecular structures in the brain---is
provided by the sense in which set theory is viewed as
forming a foundation for mathematics.
On that view of the foundations of mathematics (and there
are many other views), the set-theoretic universe is seen
to provide an ontological foundation for mathematics, in
the sense that every mathematical object is regarded
fundamentally as a set. One builds the natural numbers from
sets as ordinals and then the integers and the rationals
and the reals in any of the usual set-theoretic
constructions; a group is a set with a binary operation (a
set) having certain properties; a topological space is a
set together with a set of subsets having a certain nature;
and so on. On this view, every mathematical object is
regarded as a set and the context of set theory is taken to
provide a common forum in which to treat mathematical
objects and constructions from what would otherwise be
diverse forums. The existence of such a common forum allows
us sensibly to apply knowledge from one area of mathematics
to arguments in a distantly related area, and this is
important.
So the view is that the basic features of the reals or of
any mathematical object ultimately reduce to set theory in
the sense that that object is fundamentally a set. But
meanwhile, although this reduction of mathematics to set
theory is important foundationally (and there are resulting
a number of intriguing or even startling conclusions about
ZFC-independence and paradox in non-set-theoretic
contexts), the main view is also that the set-theoretic
reduction is largely irrelevant for ordinary mathematics.
We don't want to undertake most arguments in number theory
or algebraic geometry or whatever with constant reference
to the complete set-theoretic reduction of the subject, for
example, by speaking of the "elements" of $\pi$. Thus,
mathematics can be seen to reduce to set theory, but for
most higher level mathematics, this reduction is either
very complicated or not seen as illuminating of the
interesting mathematical phenomenon at hand.
This relation seems very similar to the relation between our current understanding of mental properties and molecular structures in the brain. In principle, we believe that there is a reduction, but that reduction is either very complicated or not particularly illuminating of the mental phenomenon. We seem to fulfill the following analogy:
Higher-order Higher-order
mental features mathematical objects
and properties and relations
----------------- ------------------
molecular structure sets and the
of the brain membership relation
So this situation seems to accord accurately with your
description of supervenience.
Addendum. Let me also mention another sense of supervenience, related to the point made by Gowers, in his second paragraph. The truth of a universal statement $\forall n\ \varphi(n)$ in arithmetic, say, reduces to the instances $\varphi(0), \varphi(1),\varphi(2)$, and so on. But by the Compactness theorem, one cannot prove the universal statement merely from those assertions in first order logic. Thus, the truth of $\forall n\ \varphi(n)$ would seem to supervene on those instances in the sense of the question. We don't prove a universal statement by proving each instance separately.
