Thursday, October 18, 2007

Structure is as structure does

Philosophers like structural comparisons for various reasons. Establishing these are often important. For example, in the Tractatus, Wittgenstein seems to be trying to set up an isomorphism between language and the world. In other contexts, philosophers try to establish that structures are isomorphic. It seems like outside of model theory the only notion of structural comparison that gets any play is isomorphism. The model theory class I'm taking has started making me wonder why this is. There are lots of other interesting and useful sorts of comparisons: back-and-forth games, bisimulation, homomorphisms of various sorts, etc. We could even through out a lot of the homomorphisms, the ones that are one-way. I'm not sure why the others don't see more philosophical action. Bisimulation is the key notion of modal logics, at least in the Amsterdam school. Yet, philosophical discussions of modality don't mention it as far as I know (which, admittedly, isn't very far). Back-and-forth games (meant here in a more general sense of games of any finite length) seem like they would be fruitful and usefully combined with ideas from computability theory. This is because they emphasize the idea of being able to differentiate two structures using only a certain number of checks. The two structures could be non-isomorphic, but discovering this fact could take an infinite number of checks, something that finite little agents could not do. What lead to the popularity of isomorphism? It often seems a little heavy handed. Is it just that it gets covered in the standard logic classes, like induction, and so is the tool of choice amongst philosophers?

1 comment:

Greg said...

Very interesting point. I like your answer to the question you posed (philosophers use the tools we're given), but I would guess that part of the answer also has to do with philosophers trying to establish (or refute) identity claims ('Is the mind identical to the brain?' is an obvious example), and isomorphism being the right tool for that job.

I also wanted to mention/ ask whether you think the example you give at the end of the post (some back-and-forth games can only differentiate structures after an infinite number of checks) could serve as a nice, tidy little exemplar in the realism debate. These two structures are really different (i.e. non-isomorphic), but we would never know that they were.

Hmm. As I think about it for a second more, the more this seems like it would be used differently in different realism debates (scientific and mathematical). Thoughts?