Introduction (Full Text)
- In these comments I want to leave aside entirely whether human mathematical understanding is achieved solely through the manipulation of linguistic symbols by syntactically specifiable rules, i.e. whether it is solely a matter of humans performing a computation. I also want to leave aside the problems that arise in interpreting quantum theory1, in particular the measurement problem. Those problems stand on their own quite independent of Gödel's theorem. Rather, I want to focus explicitly on how Gödel's theorem, together with facts about human mathematical understanding, could conceivably have any bearing on physics, that is, on how the first part of Shadows of the Mind is related to the second. I want chiefly to argue the reflections arising from Gödel's theorem and human cognitive capacities do not, and could not, have any bearing on physics.
- That there might be any connection at all would be surprising for the following reason. Ultimately, the empirical data of physics resolve themselves into claims about the positions of material bodies. Any physical theory that correctly predicts or accounts for the positions of bodies -- including the positions of needles on complicated scientific instruments, the positions of ink particles on computer printouts, and the positions of dots on photographic plates -- cannot be objected to on empirical grounds. One might object on aesthetic or other grounds (e.g. one might object in principle to a theory that postulates unmediated action at a distance) but this would not be an empirical failure of the theory. So if Professor Penrose's argument somehow shows that classical physics or quantum physics cannot be complete and correct accounts of physical reality, then Gödel's theorem must somehow have implications about how material bodies can move.
- The overall strategy for connecting Gödel's result to physics would have to be to show that some actual motion of bodies cannot in principle be accommodated within a physical theory of a certain kind. Just as analysis can show that the physical behavior of planets whose orbits precess cannot be accounted for by Newtonian gravitational theory, so Penrose seems to claim that all of classical and quantum physics (as well as a large class of possible extensions or emendations of those theories) cannot account for the physical motions of some known physical bodies: those of human mathematicians. How, in detail, could this connection between a mathematical theorem and physical action possibly be made?
Review of "Penrose (Roger) - Shadows of the Mind"; Link (Defunct).
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