I've been thinking about the Lucas/Penrose view of the purported limitations of computation as the basis for human thought. I know that Bruno has given a technical refutation of this position, but I'm insufficiently competent in the relevant areas for this to be intuitively convincing for me. So I've been musing on a more personally intuitive explication, perhaps along the following lines.
The mis-step on the part of L/P, ISTM, is that they fail to distinguish between categorically distinct 3p and 1p logics which, properly understood, should in fact be seen as the stock-in-trade of computationalism. The limitation they point to is inherent in incompleteness - i.e. the fact that there are more (implied) truths than proofs within the scope of any consistent (1p) formal system of sufficient power. L/P point out that despite this we humans can 'see' the missing truths, despite the lack of a formal proof, and hence it must follow that we have access to some non-algorithmic method inaccessible to computation. What I think they're missing here - because they're considering the *extrinsic or external* (3p) logic to be exclusively definitive of what they mean by computation - is the significance in this regard of the *intrinsic or internal* (1p) logic. This is what Bruno summarises as Bp and p, or true, justified belief, in terms of which perceptual objects are indeed directly 'seen' or apprehended. Hence a computational subject will have access not only to formal proof (3p) but also to direct perceptual apprehension (1p). It is this latter which then constitutes the 'seeing' of the truth that (literally) transcends the capabilities of the 3p system considered in isolation. If the foregoing makes sense, it may also give a useful clue in the debate over intuitionism versus Platonism in mathematics. Indeed, perceptual mathematics (as we might term it) - i.e. the mathematics we derive from the study of the relations obtaining between objects in our perceptual reality - may well be "considered to be purely the result of the constructive mental activity of humans" (Wikipedia). However, under computationalism, this very 'perceptual mathematics' can itself be shown to be the consequence of a deeper, underlying Platonist mathematics (if we may so term the bare assumption of the sufficiency of arithmetic for computation and its implications). Is this intelligible? David -- You received this message because you are subscribed to the Google Groups "Everything List" group. To unsubscribe from this group and stop receiving emails from it, send an email to everything-list+unsubscr...@googlegroups.com. To post to this group, send email to everything-list@googlegroups.com. Visit this group at https://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/d/optout.
