Representing structures, universal isomorphisms
I’d like to post some thoughts about one particular problem I face very often. This time I’d like to take it from a more abstract point of view than usually, because I think the generality of this problem is impressive.
Given any sort of structure, it is possible to present it in many different ways. Consider for example a two-dimensional structure like a 2D-graph, drawn on paper vs. represented in a computer’s memory.
Of particular interest to me is representing multi-dimensional structures with fewer (hopefully one) dimensions. Take the case of mathematical notation vs. computer math notation, for instance.
Quite often it is the case that we know multiple ways of representing our structure, but that we lose some information in the process. In our graph example, we wouldn’t know as easily by reading the computer’s memory that the graph represented an octagon, for instance.
Are there transformations that can map all the information from a structure (even information belonging to the original structure) to another, different structure? Probably not. In some sense, the answer is yes though, because we can construct the original graph out of a linear description in a computer’s memory, so the information of the original structure must exist in the linear description as well. However, there is clearly something missing from our linear description. What is going on here?
A bit more mathematically, we can ask whether there exists certain kinds of isomorphisms (in the sense of preserving essential information) between structures?
Wikipedia has some examples of “everyday” isomorphisms, but they do not work for my problem. The first example (with the decks of cards) makes me think “what if the colour is an essential piece of information?” Then these structures are not isomorphic in the sense I use here. Now let us look at the second example, in which we compare a wristwatch to a clock tower. In a way they are isomorphic, but not in my sense of the word. Looking at a wristwatch does not help you understand how a clocktower looks at all, so it is a different kind of isomorphism. This information is stored in the structure, and is not preserved in this case.
But of course… these fail because I used a very bad word in my definition of isomorphism; “essential” information.
We can not exclude any information, for it might be essential for someone or for some purposes.
What is useless to someone might as well be essential for someone else, so that leads us back to my first question, of whether there exists isomorphisms between different structures that preserve ALL information.
Please prove me wrong on this one!
