Are isomorphisms crummy?

Raymond Smullyan, in his delightful book The Tao Is Silent, rejects the picture theory of ideas, which states that an idea resembles its object. What could be more ridiculous? The idea of Tao being vague because Tao is vague? The idea of red being red because red is red? The idea of stickiness being sticky because stickiness is sticky? Smullyan adds that some say there can be isomorphisms between thoughts and objects. But this strikes him as a crummy kind of resemblance!

I wasn't so quick to dismiss isomorphism between thoughts and objects. Why that is is worth a quick digression. Primarily it is the ways in which I see isomorphisms between thought and objects as nontrivial and meaningful. But I can also easily see that this is in large part due to my readings of mindologist extraordinaire Douglas Hofstadter, who has taught me the incredible value of isomorphism and analogy in human thought (and possibly porpoises too).

In a simple case, a thought may just be a label for an object, with no greater resemblance (or isomorphism) to it than the fact that we relate the two. This is a bit like language, in which some argue the labels we place on objects are completely arbitrary be could interchanged and, so long as we all agreed beforehand on what's attached to what.

But then I thought of Daniel Dennett's description of a computer simulation of a hurricane. How detailed does the simulation need to get - how many variables does it need to include - before we argue that we're actually getting kind of close to a real hurricane? It doesn't have to rain inside the hard drive for us to appreciate that the model may actually capture some aspects of this big, wet object.

So what happens when our thoughts are complicated, vague, multimeaninged and in other ways related to the objects they are about? I would argue that in precisely these cases, there are isomorphisms between the thoughts and the objects and this resemblance can be rather uncanny at times!