On Sep 17, 2008, at 9:40 PM, Christopher Menzel wrote: On Sep 16, 2008, at 1:37 PM, Pat Hayes wrote: ...We have argued about that point before, and logicians and philosophers have taken many different positions on it. Although I do not believe that the real world *is* a model, it is at least conceivable that some people might make that claim. It might be conceivable, but I don't know how anyon e could *plausibly* make that claim. By definition a model is a mathematical entity of a certain sort, usually an ordered n-tuple of some ilk. The real world, whatever it is, isn't an n-tuple.
Well now, lets engage on this question. Because it seems to me that both of you are falling into a trap here, one that treats mathematical language as making a kind of restricted ontological commitment because it speaks about 'mathematical stuff'
No, I'm certainly not falling into that trap, as I think formal models can be full of ordinary (and not to ordinary) objects, properties, and relations.
I know you do, but you seem to stop short of following through on the consequences of that. (?) If someone objects: but such a description will always be incomplete, always omit some aspect of reality; then the reply is: yes, of course: so what? An incomplete description of something can still be of that complete thing. Semantics is not about being exhaustive, it is an analysis of what descriptions are, or can be, about.
Of course, but that's just the point, isn't it, Pat? If the model is incomplete, then it can't be identical to the world.
The mathematical description of it as a tuple is incomplete. But that description is in the metatheory of the semantics. There is more to the world than its tuple-hood. Nevertheless, it can still BE a tuple.
You are an American. This does not say everything there is to be said about you: there is more to you than your American-ness. Nevertheless, you *are* an American. The other facts about you do not make this fact false, do not make you non-American by virtue of your being also a cyclist. So, similarly: the world IS a tuple, and doesn't stop being a tuple by virtue of its also being made of, say, quarks and leptons.
Now, I am pretty sure that Chris agrees with all the above. But let me ask: why, then, is it implausible to claim that the real world is an n-tuple? After all, what does this mean? It means only that the real world exhibits a structure which can be mathematically described in terms of n-tuples.
Seems to me pretty clear that it does not mean that!
But that is exactly what it means. What else COULD it mean? Just apply model-theoretic thinking to the mathematical language itself. Or otherwise, seems to me, you must be assuming that some other semantics is being used when it comes to mathematical language. I don't buy this: but even if I did, I'd want you to tell me what kind of semantics you had in mind. Indeed, your last sentence is exactly what I'd say about the relation between a model -- the mathematical "description" -- and the world.
But that relation is between the description in the model theory itself, i.e. couched in the (mathematical, set-theoretic) language of model theory. Yes, of course that description, being a description, is a model and not the reality. But what it describes can be actual reality. And to say that this reality "is" what the description says it is, is simply to say that the description is true. But really now. Let's define a model (for a given first-order language L, say) to be a triple <D,R,V>, where D is a set of objects, R a set of extensional relations over D, and V is a mapping from L to appropriate semantic values in D and R. Surely neither *the world*, nor any physical part of it, is literally such a triple.
Yes, it is. Why not? I agree this isn't saying very much about the world, to call it a triple (and not saying very much about it is part of where model theory gets it power, in fact), but surely you don't want to say that this description of the world-as-a-triple is wrong. do you?
Pat
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