[FOM] The Lucas-Penrose Thesis

[Rupert McCallum]

A point worth making about Penrose's argument, which I think was made
by David Chalmers, is that it can be generalized. The same argument
works just as well to show that the set of sentences in a given
language which humans can perceive to be true, cannot be definable by a
description in that language such that humans can know that description
to define a sound set of sentences. For example, the set of sentences
in the first-order language of set theory which humans can perceive to
be true could not be definable by an expression in the first-order
language of set theory such that humans can perceive that that
expression defines a sound set of sentences. The obvious point to make
in reply to this example is that it is possible that such an expression
exists, but that we cannot perceive it to define a sound set of
sentences - there is no reason why we should always be able to perceive
such a thing. Similarly, it may be that a Turing machine generating the
set of sentences we can perceive to be true exists, but that we cannot
perceive that the Turing machine generates a sound set of sentences -
there is no reason why we should always be able to perceive this.
Penrose tries hard to argue that this is an unlikely possibility but I
think ultimately his arguments are not very persuasive.


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