This is just something I thought of and I am not sure if this makes sense but mathematicians explore objects much in the same way particle physicists do. So the

This is just something I thought of and I am not sure if this makes sense but mathematicians explore objects much in the same way particle physicists do. So the physicists bounces particles off each other and see what happens to understand how these particles work. Similarly, mathematicians study objects by looking how they act on other objects. So if it turns out that two differently defined objects have the same actions on other objects, we call them isomorphic and don't really distinguish between them. So for instance, we say that the set of rigid motions that preserve the triangle and it's orientation is the same as the set of permutations of the roots of say: x^3-3x+1 even if the two sets are absolutely not defined in the same way. Hope that makes some sense.