# Identity of mathematical objects

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?

• Can you give an example of two mathematical objects that have all of the same properties but are two different objects? I can't think of one. Oct 30, 2022 at 21:49
• Mathematical objects that are isomorphic but not identical do not have all properties the same. The vector space of linear polynomials with real coefficients is isomorphic to the geometric plane, for example, but they have plenty of properties that distinguish them (e.g. polynomials can be multiplied while geometric points cannot be). Isomorphism is always isomorphism relative to a certain structure (of a vector space, in this case), properties beyond that structure remain distinct. Oct 31, 2022 at 2:43
• @DavidGudeman that is true, I guess I was thinking each element of the corresponding elements in the reals and naturals are isomorphic to each other but considered distinct by some. Oct 31, 2022 at 10:58
• It's going to be hard to give a definitive answer to this, but might it be useful to talk a bit about Hume's principle, abstraction and Frege's programme? Nov 6, 2022 at 22:59