Is it beneficial to think about what natural numbers/object of sets are?

Solution 1:

All that matters about mathematical objects is their properties and how they relate to each other. Their “nature” is irrelevant. The idea of the “essence” is a bit nonsensical, even if applied to real objects. Anything is just the sum of its properties. When you use a hammer, all that matters is it’s solidity, weight, rigidity, etc. Is there anything beyond that? Mathematicians are just very honest people and do not pretend to know the final reality, as others do.

Solution 2:

I am not sure I understand what your question is, so let me know if this does not approach an answer to your question. Here's how I see it:

Objects are whatever you define them to be, and to prove that an object has a property, you don't really have to assume the object itself exists or imagine how it can be visualised. Instead, you can show that anything satisfying the definition of the object also satisfies the property.

In a way this is even necessary, because we cannot prove that a model of $\mathsf{PA}$ or $\mathsf{ZFC}$ even exists, without assuming stronger axioms. We can prove what kind of properties things like natural numbers have if they were to exist, by reasoning with the description of the object instead of with the object itself. This way you don't have to contemplate the actual nature of natural numbers, nor their actual existence, but you can still agree on how they behave.