While 'analytic' truths might seem to be independent of any facts about the world, the possibility of an analytic engine is an empirical matter. That some machine can be made which does analytic computations is a fact about the world - that it contains this possibility.
Arguably we, either individually or communally (I'm thinking of a language community here), are just such analytic engines. I say 'arguably' since anyone who has followed the general trend of my arguments so far will see that I think a claim to the contrary would turn out to be unintelligible - any argument for such a contrary claim would rely upon the very analytic 'machinations' which it would want to render questionable. If I can't produce reliable arguments, then I can't reliably argue that position either.
Does this tell us something about analyticity itself? I'm not sure. I have a moderately (though not completely) Quinean view on this - what we count as conceptual seems to be at least partly contextual. There is this, though:
To say that 'we' are an analytical engine is, I think, to say that we can produce reliable arguments for something being true on the grounds that some other things are true. We can deduce one truth from another, and we can show that this deduction at least makes sense, and, in some cases, is unavoidable. Scepticism about this must entail a scepticism about sceptical arguments as well, and so about its own intelligibility.
Taken this way, taking the possibility of argument (entailed by the possibility of language use), as the context, the possibility of analytical argument depends on the possibility of the engine - an empirical, not a logical (or analytic) possibility.
Can we imagine the world being otherwise? Not if we imagine an orderly world which we can talk about - i.e. one which we can both imagine and describe. A world with definite characteristics is a world whose parts are related in a grammatical way, so to speak. And if they are related in such a way, they comprise, to a minimal degree, an analytical engine.
That fact that we can build computers as well as produce arguments tells us this about the world - that it's character can be rendered grammatically, and so that it can be reorganised into a grammatical computation and prediction engine of sorts. We are such an engine.
Can we imagine a world without us in it? Of course, to an extent. But not one that we aren't imagining - that we haven't brought into our linguistic context. (How does the sentence 'I am presently imagining a world that cannot be described' work? And is 'a world that cannot be described' a description? - OQ paradoxes again ...)
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