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](http://en.wikipedia.org/wiki/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?



Supervenience in mathematics
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.


