Supervenience in mathematics I'm not quite sure if this is the right place to ask, and if this is the right way to ask, but I dare.
In philosophy (of mind, e.g.) the concept of supervenience is used:

"Supervenience [is] used to describe
  relationships between sets of
  properties in a manner which does not
  imply a strong reductive
  relationship."

That means an object might possess higher properties that depend on some base properties, but cannot be reduced to (defined by) them.
My question is: Can this situation occur in mathematics?
As I see it, for every mathematical object - be it a set with a structure or a vertex in an abstract graph or an object in a category - all its (relevant) properties are determined by its inner structure or its relations/morphisms to other objects. To me it seems unconceivable that a mathematical object can have any extra properties, let alone in a supervenient manner, that is, two isomorphic objects would have to share them.
But maybe I'm wrong. Can anyone point me to an example?
 A: It seems to me that even if the exact philosophical notion doesn't quite apply to mathematics, there are other notions, similar but a bit more precise, that do. For example, mathematical structures can have high-level properties that are definable in terms of low-level properties but are not easily computable. In that case, it may be that the reduction, even though it exists, is not useful. This seems to me to be fairly like a physical example such as the difficulty of defining what a liquid is in microscopic terms.
What I'm getting at (but also slightly struggling with) is that reductionism has its defects in mathematics just as it does in philosophy. To give an example, to prove the prime number theorem one doesn't break it up into lots of small statements, prove those, and put them together again. That would be quite impossible, given that the largest known prime is very finite. Rather, one somehow looks at all the primes at once. It's tempting to say that the true meaning of the prime number theorem is not that all those numbers out there are prime, but rather a more global statement about density that supervenes on the properties of the numbers themselves.  
A: When I read 

To me it seems unconceivable that a mathematical object can have any extra properties, let alone in a supervenient manner, that is, two isomorphic objects would have to share them.

I first of all agreed that mathematics should be like that, but the potential counterexample that sprang to mind is set theory, specifically membership-based set theory with urelements. 
By "membership-based set theory", I mean the traditional sort of set theory like ZF which is based on a membership relation $\in$; I do not mean a structural or categorical set theory like Lawvere's Elementary Theory of the Category of Sets. This comes to mind because two sets that are "isomorphic", which in the first instance might mean there is a bijection between them, may have very different set-theoretic properties; for example they may have different ranks. 
Upon further reflection, I felt this notion of isomorphism as bijection could be a loaded way to interpret "relation to other objects" and that one should look further to the structure of membership trees. Are two sets with the same inner structure (i.e., that have isomorphic membership trees) distinguishable as sets? In ordinary ZF, I believe one can prove by recursion that two sets with isomorphic membership trees are in fact equal. But this is not the case in a set theory with urelements. If we permit ordinary objects to be urelements of sets, then I can't think of any mathematical properties based on $\in$ alone which might distinguish two sets with three urelements each (thinking of say a box with three cats and another with three dogs), although we'd certainly want to distinguish them. My admittedly cursory reading leads me to believe that set theorists who work with New Foundations take this possibility seriously; I was glancing in particular at this Wikipedia article. I'd be happy to hear from set theorists who derive a different conclusion. 
A: Renowned popularizer of computability, S. Barry Cooper, has written a paper on this topic: From Descartes to Turing: the Computational Content of Supervenience
A: 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.
A: I think that an interesting phenomenon analogous to the relation of
informal mathematics to its set theoretical foundations described by
Joel David Hamkins is the relation between those meta-arithmetical
notions, theorems, and proofs that we formulate by the help of Gödel
numbering.  Actually, it is a perplexing fact that we
can establish the truth of arithmetical theorems without having the
faintest idea of their arithmetical content. For example, we know that
the Gödel sentence of a consistent theory is true. The statement
carrying this metamathematical content is actually a sentence of the
language of arithmetic. But, obviously, its arithmetical content 
is incomprehensible by any human being.
A: I want to try another answer, not because I think you will necessarily accept it, but because if you don't then I think your reasons for not doing so will clarify the question.
One definition of supervenience is that A supervenes on B if you can't have a change to A without a change to B. For instance, some people hold that the mind supervenes on the physical properties of the brain because you could not have two distinct mental states arising from brains that were physically identical. (Others dispute this, but that does not matter here.)
Now consider the halting property. That clearly supervenes on the specification of the given Turing machine (encoded as a sequence of 0s and 1s, say), since if one Turing machine halts and another doesn't, then they can't have identical specifications. But it's not clear that there's any sense in which the property of halting or otherwise is reducible to the specification of the Turing machine.
A: Properties which depend on a sequence but not on any finite number of terms are pervasive in mathematics (e.g. having a limit, having limit equal to $0$, or being summable).
The existence of such tail properties of sequences might be considered an example of some sort of (weak) supervenience.
As noted in the Addendum to Joel David Hamkins answer proofs involving these types of properties require types of reasoning that are of "higher order" than most people (including beginning students) use in their daily lives.   But even so there are many examples of the "high order" reasoning giving non-trivial insight into the finite order setting and vice verse (Example:  The whole interaction between number theory and ergodic theory starting with Szemeredi's theorem).
