Isomorphisms and Essences

I do want to follow up on the previous post; I think that there are some interesting ways in which the notion of chance in the third case can be applied to other situations, such as Advaita Vedanta and Neoplatonic emanation to help make sense of them (I know you're all terribly excited by the prospect). But, I was thinking about Avicenna's notion of essence today, especially as worked out by certain Scholastic thinkers, and I think that I made sense of something and I wanted to jot it down while I remembered it.

For Avicenna, essences exist either in reality or in the mind. I can talk about real, individual horses, or I can talk about the concept "horse" in my mind. Now, these can't be the same. If the essence of horse as it exists in the world were what it really is to be a horse, then horse would have to be an individual; but the idea of horse applies to many individuals. Similar considerations prevent us from taking horse as it exists as an idea in the mind to be what horseness really is. So, there is some way in which we can consider "horse" in itself apart from either real individuals or general concepts. But "horse" only exists in one way or the other. So what could we mean by horse in itself?

I thought, as usual, about mathematical systems. Let us take the natural numbers (N), that is, all whole numbers from 1 on up. Let us also take just the even natural numbers (2N). We'll just be adding numbers; other operations would make this more difficult. We can take any natural number x and transform it into a 2N number y using the formula x*2, and we can take a 2N number y and turn it into a regular natural number x using the formula y/2. It doesn't matter whether we add first and then switch systems, or switch systems then add. For an example, take 1 + 2 = 3:
1 + 2 = 3; 3*2 = 6
1*2 + 2*2 = 2 + 4 = 6
And similarly if we wanted to go in the opposite direction. This is a mathematical isomorphism between N and 2N under the operation of addition.

What the isomorphism means is this: there is the same structure between N and 2N under addition. On the one hand, it doesn't make any sense to say that this structure exists independently of some system; the structure simply is the way the different symbols interrelate and it makes no sense without such symbols. But there is still a sense in saying that there is a structure which is in both N and 2N. This seems to me to be the same logical move as Avicenna is making with essences, and as there is nothing wrong with it in the mathematical case (it at least makes perfect sense to me), the are grounds for thinking that it is intelligible in the metaphysical case as well.