Last week, one of the campers asked me about Penrose’s arguments about the mind based on Gödel’s theorems. I wasn’t entirely sure about them, since it’s been years since I read Penrose, but I discussed it for about an hour with the camper and another staff member that was interested, and helped clarify (to myself as well) what seems to be going on here. In the process, I think I’ve found a connection between an issue of these arguments and John Perry’s “The Problem of the Essential Indexical”. (Of course, I should probably check out the book on Gödel’s theorem by the late Torkel Franzén for more about these arguments.)
Anyway, as far as I can remember (which may not be terribly accurate), Penrose’s arguments go something like the following. If humans were computational beings, then the set of mathematical truths knowable by humans would be recursively axiomatizable. The set of truths knowable by humans seems to include Peano arithmetic. Therefore, Gödel’s second incompleteness theorem tells us that the fact that this set of truths is consistent is independent of this set, and thus the consistency can’t be known. However, since we know that the truth is consistent, and we know this set of sentences to be true, this means that we know the set is consistent, which is a contradiction. Therefore, our initial assumption that humans are computational must be false.
Some versions of this argument might use some propositional attitude other than knowledge – I think those would generally be weaker arguments, because there’s no good reason to suppose that beliefs (or anything else) should be consistent. I will also bracket the contention by various non-classical logicians that the truth might not actually be consistent – I think all that we need is that 0=1 is not provable from the knowable truths, and that Gödel’s theorem tells us we can’t prove that. And besides, I can’t conceive of what it would mean for the truth to be inconsistent (except perhaps for some special cases like the liar paradox).
I think the important challenge to this argument comes when I say “we know this set of sentences to be true, [and thus] we know the set is consistent”. There seems to me to be an important ambiguity in knowing a set of sentences to be true. In particular, there is a difference between knowing, of each sentence in the set, that it is true, and knowing of the set as a whole, that all of its elements are true sentences. To know that the set is consistent, one needs the latter, rather than the former, because consistency is a property of the set as a whole, and not of its individual sentences, the way truth is.
In other words, although I might know that “the set of all mathematical truths that I know is a consistent set”, it seems plausible that if I was presented with this same set under a different mode of presentation, I might not recognize it as a consistent set. This seems to be an important step in this version of Penrose’s argument – I don’t think Gödel’s theorems would cause trouble for my knowledge of the statement “the set of all mathematical truths I know is consistent”, though it would cause trouble for any system S proving the statement “system S is consistent”, when system S is presented in some sort of transparent manner, say by listing its axioms. Presenting it as “my system” doesn’t let me prove much about its consequences, so I can know it to be consistent. Presenting it in a more extensional format lets me prove a lot about its consequences, perhaps at the cost of my knowing that it’s consistent. Penrose would need to bridge that gap in order for his argument to be valid. (At least as stated above – it seems quite plausible that he’s got a more subtle argument that gets around these points.)
Anyway, I found an interesting duality here to Perry’s point in “The Problem of the Essential Indexical”. In that paper, Perry points out the importance of learning some essentially indexical information in addition to purely descriptive information. In his example, one can be in a grocery store and see a trail of spilled sugar, and realize “someone has a leaky bag of sugar – I should go let him or her know”. After following the trail for a little bit, one realizes one is walking in a circle, and evetually gains the new information “I have a leaky bag of sugar”, at which point one can remedy the situation. This information must be presented with the indexical “I”, rather than with any description or other third-person presentation (unless one already has the connection between that description and the first person).
Conversely, in this case with the Penrose argument, it seems to me that one might have the indexical information, without the third-person description! That is, one can know “the set of mathematical truths I know is consistent” (because it is a set of truths), without knowing, “system S is consistent”, even if system S characterizes the set of mathematical truths I know. Perhaps all that’s important here is the difference between two different descriptions of the same set, but maybe there’s a role for Perry’s “essential indexical” as well, though the role is the opposite of the one he is concerned with.