simonofb@rckvm1.vnet.ibm.com (06/27/91)
How many possible thoughts are there? Given any of the assumptions below, it seems plausible that the number of possible thoughts is at most countably infinite. 1) If a thought corresponds to a finite statement in some natural language, then the set of possible thoughts is a subset of the set of all finite strings from a finite alphabet. This set is countably infinite. 2) If a thought corresponds to a finite computation of some Turing machine, then the set of possible thoughts (including the TM program as well as the computation) is a subset of the set of all finite sequences of finite strings from a finite alphabet. This set, like that in #1 above, is countably infinite. 3) If a thought corresponds to a finite sequence <A> of neural activations A in some brain, each A can be represented as a finite string of zeros and ones, A(i) = 1 if neuron "i" is firing, 0 otherwise (include the brain architecture as a prefix to <A>). As in #2, this again gives a subset of the set of finite sequences of finite strings from a finite alphabet. Notes: 1) Hardware/software details such as parallelism, recursion, etc should not affect these conclusions. 2) Cosmological boundary conditions, such as the predicted heat death of the universe, may actually constrain the number of possible thoughts to be finite -- in this universe at least.