Just how is it that I manage to see that red Lamborghini Countach I'm planning to buy as soon as my luck turns? Pretty much everyone agrees that when I see it, I must have some sort of representation of it, but how does that work, exactly? I don't have an exact copy; the Countach is not a big car, but it's still too big to park under my skull. Nor is it a miniature. I'm quite sure there are absolutely no tires in there, however small. No, the representation has to be done in neurons, synapses, neurotransmitters, action potentials and such -- all very much unlike a red Lamborghini Countach -- but that's okay, because the symbols involved in a representation don't have to resemble the things they represent; they just have to play their part in the overall scheme.
There's still a problem, though. In general, any sort of representation can be fleshed out by showing how the various symbols in the representation refer to the things being represented. This "fleshing out" is called an interpretation, and it means pretty much what it seems to mean -- an assignment of meanings ("referents") to symbols: the name "George" refers to George, the word "snow" refers to snow, etc. Very simple stuff. Once a collection of symbols is given an interpretation, we can see whether sentences composed of those symbols are true in that interpretation. If "snow" refers to snow, and "white" refers to that very pale color, then the sentence "Snow is white" is true.
This works very nicely when we're dealing with nice public stuff like snow, because we have no trouble assigning referents to the words we use: Here's some snow. When I say "snow," I'm referring to stuff like that. But now we're asking a trickier question: "How is my internal representation of the external world to be interpreted?" The problem here is that I'm in no position to give an interpretation. All I've got is the representation. I don't have any access to the external world except by way of the representation. So any interpretation might be as good as any other! My "internal" Lamborghini Countach might refer to a really fantastic car, or it might refer to a pile of bat guano, and there's no way I can tell the difference!
Maybe it's not as bad as all that. Interpretations are not all created equal. Some -- maybe most -- just don't fit with the representations they interpret; they don't fit in the sense that the sentences we form just can't be true in those interpretations. For example, if "7" means the number of days in a week, "5" means the number of corners on the Pentagon, "12" means the number of months in a year, "=" means identity, and "+" means the subtraction operation, then the sentence "7 + 5 = 12" is false. Clearly, that interpretation isn't very useful or interesting. So let's agree to consider only those interpretations in which our beliefs -- i.e., sentences in our representations of the world -- are actually true. (These interpretations are called models of our beliefs about the world.) That should cut the problem down to a manageable size.
Well, no, not really. It's a consequence of the Löwenheim–Skolem theorem -- trust me -- that every theory (set of sentences) that has a model, has a model whose domain (the set of things to which symbols refer) is the set of natural numbers (0, 1, 2, 3, ...). So if our internal representation of the world has an interpretation that makes it true -- a model, it also has a model where the "world" consist of absolutely nothing whatever except for natural numbers!
"Big deal," I hear you cry, "who cares if such a model exists? That's not the interpretation I'm using, so it has nothing to do with my representation of the world." Oh yeah? How do you know? Everything you know about the world comes from your representation of it. You can't escape the representation to point out what your symbols mean. So, I'm afraid, you have no reason whatsoever to imagine that anything other than natural numbers exists!
This is ridiculous, of course, but it follows from the assumption that my "access" to the external world is by way of representation. I don't know about you, but I'm not prepared to believe with Pythagoras that "all is number," so I'm giving up the representation idea. The external world and I have a much more intimate relationship than that. When I figure out what it is, you'll be the first to know.
There's still a problem, though. In general, any sort of representation can be fleshed out by showing how the various symbols in the representation refer to the things being represented. This "fleshing out" is called an interpretation, and it means pretty much what it seems to mean -- an assignment of meanings ("referents") to symbols: the name "George" refers to George, the word "snow" refers to snow, etc. Very simple stuff. Once a collection of symbols is given an interpretation, we can see whether sentences composed of those symbols are true in that interpretation. If "snow" refers to snow, and "white" refers to that very pale color, then the sentence "Snow is white" is true.
This works very nicely when we're dealing with nice public stuff like snow, because we have no trouble assigning referents to the words we use: Here's some snow. When I say "snow," I'm referring to stuff like that. But now we're asking a trickier question: "How is my internal representation of the external world to be interpreted?" The problem here is that I'm in no position to give an interpretation. All I've got is the representation. I don't have any access to the external world except by way of the representation. So any interpretation might be as good as any other! My "internal" Lamborghini Countach might refer to a really fantastic car, or it might refer to a pile of bat guano, and there's no way I can tell the difference!
Maybe it's not as bad as all that. Interpretations are not all created equal. Some -- maybe most -- just don't fit with the representations they interpret; they don't fit in the sense that the sentences we form just can't be true in those interpretations. For example, if "7" means the number of days in a week, "5" means the number of corners on the Pentagon, "12" means the number of months in a year, "=" means identity, and "+" means the subtraction operation, then the sentence "7 + 5 = 12" is false. Clearly, that interpretation isn't very useful or interesting. So let's agree to consider only those interpretations in which our beliefs -- i.e., sentences in our representations of the world -- are actually true. (These interpretations are called models of our beliefs about the world.) That should cut the problem down to a manageable size.
Well, no, not really. It's a consequence of the Löwenheim–Skolem theorem -- trust me -- that every theory (set of sentences) that has a model, has a model whose domain (the set of things to which symbols refer) is the set of natural numbers (0, 1, 2, 3, ...). So if our internal representation of the world has an interpretation that makes it true -- a model, it also has a model where the "world" consist of absolutely nothing whatever except for natural numbers!
"Big deal," I hear you cry, "who cares if such a model exists? That's not the interpretation I'm using, so it has nothing to do with my representation of the world." Oh yeah? How do you know? Everything you know about the world comes from your representation of it. You can't escape the representation to point out what your symbols mean. So, I'm afraid, you have no reason whatsoever to imagine that anything other than natural numbers exists!
This is ridiculous, of course, but it follows from the assumption that my "access" to the external world is by way of representation. I don't know about you, but I'm not prepared to believe with Pythagoras that "all is number," so I'm giving up the representation idea. The external world and I have a much more intimate relationship than that. When I figure out what it is, you'll be the first to know.
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