Leibniz law's states that if A and B have the same properties then A and B are one and the same, however we can consider mathematical objects that are isomoprhic but not identitical, they have the same properties but they are not the same. Is this a contradiction of Lebiniz Law or do mathematical objects have a 'sense' which must be considered for 'identity'?
For physical objects they can be the same in all observable properties except their 'sense' the idea that they are themselves and not the other. They are in one physical location at one time, and this is one property that only they can have.
If I define a set of abstract 'objects' whose mathematical properties are identical to that of the real numbers, would my objects in fact be the real numbers, or is there some 'sense' or lurking properties that differentiate the 'objects' that I have defined and the real numbers themselves. Would it be possible to create a set of objects which are in every way similar to the real numbers but where each is distinct from the real number whose properties it shares?
If I define my set and define it to have the properties under addition, multiplication and to form the same structure as the real numbers. Is my set in fact the set of real numbers? What are the properties (if any) that would differentiate it from the real numbers?
Are mathematical/abstract objects defined by their properties, in which case what properties am I ommitting which are required in order for the object I define to simply be a real number?