On Philosophy

June 6, 2007

Common Confusions: Uninstantiated Abstractions And Arbitrary Isomorphisms

Filed under: Metaphysics — Peter @ 12:00 am

Confusion 1: Uninstantiated Abstractions

Let us consider the abstraction of color. Being a specific color is an abstract property in the sense that while the color of an object is completely determined by its basic properties two objects with different basic properties may have the same color. Let’s consider a particular shade of color, #991724, which itself has the property of being a kind of red. Shade #991724 may be instantiated any number of times, but the number of times it is instantiated seems to have no effect on the fact that shade #991724 is a kind of red. Thus, we might reason, even if shade #991724 isn’t instantiated it is still a kind of red.

This, I think is a confusion; it doesn’t make sense to say things of uninstantiated properties. However as things stand the question may seem completely irrelevant. And except for considerations of realism it is irrelevant in the case of color. But a doctrine about abstract properties can equally apply to functional properties as well. And so we might consider the functional properties Q, which support a particular set of conscious experiences. Reasoning in the same way we did with color we might be led to think that the nature of the conscious experiences is unaffected by the number of times Q is instantiated (and indeed in a sense this is true). And thus even when Q isn’t instantiated that the conscious experiences, and thus an experiencer of those conscious experiences, still exists. This seems to imply that consciousness always exists regardless of the state of the physical world and thus implies a kind of immortality. And there are many who are willing to accept a position such as this and ignore any flaws it might have merely because it promises them a kind of immortality.

But no matter how desirable this position may seem to some it must be rejected because it is based on a confusion about what properties apply to. Let’s go back to the color example. It was wrong (or at least implied a faulty understanding) in this case to say that shade #991724 was a kind of red. What is true is that an objects being shade #991724 implies that it also can be described as red; both being shade #991724 and being red are properties of the object. Now, to go back to the functional properties Q, the best way of talking about this case is to say that having a system functional properties Q implies that a certain consciousness is present in it. Thus if Q is not instantiated then no systems have that consciousness, and hence that consciousness doesn’t exist. Moral of this story: if you absolutely must believe in some kind of immortality just buy into some kind of epiphenomenal dualism; at least immortality is perfectly compatible with that doctrine and so won’t lead to adoption of additional unsound claims just to work in some form of it.

Confusion 2: Arbitrary Isomorphisms

How do we know when two systems have the same functional properties? Well, if the first system changes from state A to state B to state C and the second system changes from state X to state Y to state Z, and we can devise some systematic way of drawing a correspondence between A and X, B and Y, and C and Z (an isomorphism) then we say that there is some functional property that is shared by both these systems (note: states may describe the systems at a level of abstraction).

Consider then three arbitrary sets of particles, G H and I. It may be possible to construct an isomorphism (although an extremely complicated one) that maps G to A, H to B, and I to C. Thus by the above definition wouldn’t the system that is the sum of G H and I have the same functional properties as the A-B-C system? This seems to imply that every arbitrarily defined system has every possible set of functional properties. And if consciousness is just some functional property this may again seem to guarantee some kind of immortality.

Of course the immortality tie-in should make us suspicious. In this case rightfully so. The problem here is not that those who hold that arbitrary isomorphisms can be established have added something ridiculous, but rather that the original definition was a bit too vague, either in what the states consist in or what the correspondence needs to be drawn between. I’ll go with the second route here, and improve the definition by elaborating more on what correspondences have to hold. Let us say that Î is our isomorphism, and let A and B be a sequence of states that some functional properties Q hold for and let Y and Z be another sequence of states that we are interested in determining whether the same functional properties hold for. If we say that a∈A that means that a is some part of state A, and let the operator T evolve the states some amount of time, such that T(A) = B and T(Y) = Z. Finally, let A[a/b] denote the state A changed by changing the part of a designated by a into the part designated by b. Now if the isomorphism holds between A and Y then Î(Y) = A, and ∀a∈Y Î(a)∈A, and the same must hold for B and Z. Let us simply grant that such an isomorphism can be constructed. But if the isomorphism is to imply that they have the same functional properties an additional identity must hold, namely that ∀a∀bT(A[Î(a)/Î(b)]) = Î(T(Y[a/b])). (Note: in a complete treatment we would also want to restrict b to reasonable replacements, but that is an unnecessary detail here.) This asserts that the parts identified by the isomorphism in A and Y, B and Z, have the same causal pattern of interactions, and thus that information is passed in the same way through Y and Z as it is through A and B, etc. Now although we might be able to establish an isomorphism between arbitrary states it is not the case that these arbitrary isomorphisms will preserve the causal relations as well, except when they do in fact share the same functional properties, as is suggested by intuition. And thus arbitrary isomorphisms need not bother us, since the chance of being able to construct an isomorphism that implies that a random cloud or bucket of water has the same functional properties as a human brain is infinitesimally small, at least using the fully detailed definition I advocate here.

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