Solomon Feferman, Dept. of Mathematics, Stanford
INTRODUCTION: Feferman has a common cause with Penrose in opposing the dominant computational model of the mind, and considering that human thought, and in particular mathematical thought, is not achieved by the mechanical application of algorithms, but rather by trial-and-error, insight and inspiration, in a process that machines will never share with humans. His criticism of Penrose applies mainly to him extending his argument too far in areas such as mathematical soundness and consistency, and thus providing ammunition for the computational-mind camp.
This paper suggests that while Feferman makes some criticisms of Penrose, his position is nevertheless, much closer to Penrose than the view points found in mainstream consciousness studies. Feferman says, on only the second page of this paper, that he is convinced ‘of the extreme implausibility of a computational model of the mind.’ In a single sentence, he totally excludes himself from the Dennett/Churchland/Blackmore/Wegner etc. orthodoxy that dominates scientifically and philosophically respectable thinking about consciousness.
Feferman’s criticism of Penrose is that the latter’s argument relative to Godel’s theorem does not strengthen the argument against the computational model, but may actually give it support by it being possible to dismiss Penrose’s arguments at one point or another. There is a detailed and highly technical analysis of Penrose’s arguments, as presented in both the ‘Emperor’s New Mind’ and ‘Shadows of the Mind‘, that extends from page three to page ten of this paper. Feferman criticises Penrose’s version of mathematical soundness as ambiguous, and there are also considered to be problems with Penrose’s notion of consistency. Feferman says that even the tour-de-force of mathematical reasoning on pp. 3-10 of his paper does not cover all the technical errors made by Penrose. Feferman criticises Penrose for what he describes as ‘slapdash scholarship’ on the subject of the Godel theorem, which should have required particular care, given that it is central to his argument on consciousness. Having argued all this in pages of technical detail, Feferman then informs his exhausted reader that Penrose’s case would not be altered by putting right the logical flaws that Feferman has spent all this time discovering.
Feferman emphasises that Penrose’s argument relative to mathematical understanding rests primarily on the first part of Godel’s first incompleteness theorem, to the effect that in a formal system of axioms, a sentence G(F) is not provable within the system of axioms (F). Penrose is indicated to have pointed out in his books that a formal system of axioms can be reformulated as a Turing machine (a computer). Feferman stresses that ‘every theorem-generating machine can be recast as a formal system and vice versa’, meaning that a computer can be reformulated as a system of axioms.
Feferman’s own position is that the computational-mind argument is misleading in terms of the weight that it places on the equivalence between Turing machines and formal systems. The model of mathematical thought in terms of formal systems is considered to be closer to the nature of human thought, and particularly mathematical thought, than to the functioning of Turing machines. The Turing machine model would assume that given a problem, human reason would plug away, applying the same algorithm indefinitely, in the hope of finding an answer. Feferman says that it is ridiculous to think that mathematics is performed in this way. Trial-and-error reasoning, insight and inspiration, based on prior experience, but not on general rules, are seen as the basis of mathematical success. A more mechanical approach is only appropriate, after an initial proof has been arrived at. Then this approach can be used for mechanical checking of something initially arrived at by trial-and-error and insight.
Feferman views mathematical thought as being non-mechanical. He says that he agrees with Penrose that understanding is essential to mathematical thought, and that ‘it is just this area of mathematical thought that machines cannot share with us. ‘However, Feferman criticises Penrose for over stating his argument, and thus exposing it to criticism from the computational-mind majority. Much of this criticism relates to Penrose’s arguments about mathematical soundness. He also rejects Penrose’s platonism.