The assumption that we can effectively recognize a proof of a given statement of some mathematical theory, say elementary number theory, lies at the basis of all intuitionistic mathematics; but to hold that there is any recursive procedure for recognizing proofs of arithmetical statements would be to run foul of Gödel’s Incompleteness Theorem.—M. Dummett Elements of Intuitionism
Michael Dummett pointed out when I read the original paper to the Oxford Philosophical Society, I could be helped, indeed helped by all the mathematicians in the world, who might be keen to see a mind, even mine, defeat mechanism.—J. R. Lucas “The Gödelian Argument: Turn Over the Page”
Abstract
We first discuss Michael Dummett’s philosophy of mathematics and Robert Brandom’s philosophy of language to demonstrate that inferentialism entails the falsity of Church’s Thesis and, as a consequence, the Computational Theory of Mind. This amounts to an entirely novel critique of mechanism in the philosophy of mind, one we show to have tremendous advantages over the traditional Lucas-Penrose argument.
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Notes
Exactly how this must go is much more complicated, and philosophically interesting, than would first appear (Cogburn 2004).
It must be noted that this argument is independent of issues concerning intuitionism. While the Dummettian intuitionist must accept the conclusion, so must one who (albeit perhaps mistakenly) believes that verificationism, the manifestation requirement, and molecularism can be consistently held by a defender of the universal validity of strictly classical (non-intuitionist) inferences such as the law of excluded middle or double negation elimination.
If it weren’t for the role molecularism plays, our argument might be considered to be merely an interpretation of what Dummett seemed to have in mind in the epigram at the beginning of this paper. Dummett seems to be asserting that the manifestation requirement is inconsistent with Church’s Thesis. However, we’ve shown this not to be true. Rather, the manifestation requirement, verificationism, and molecularism are jointly inconsistent with Church’s Thesis. Also note that it is not at all clear to many contemporary intuitionist anti-realists that Church’s Thesis is false. In fact, two noted arguments for intuitionism (Dragalin 1988; Tenant 1997) use Church’s Thesis as a premise! It is a consequence of the current paper that it is no accident that Dummett’s own arguments (Cogburn 2000; Cogburn 2005d) never do.
Of course, Dummett’s own views apply outside of mathematics, though their original motivation concerned worries about infinity. By focusing on one each of Dummett and Sellars’ intellectual heirs, we hope that our discussion will further rapprochement between these two schools of thought.
In the case of cashing out distinctively empirical concepts this is a non-trivial and important addition, as Dummettian harmony does fail in important ways once we leave the realm of logic and mathematics. Explicating this issue would bring us too far afield of the current argument though.
This comes from Sellar’s classic paper, “Meaning as Functional Classification,” which describes the three types of pattern governed linguistic behavior.
(1) Language Entry Transitions: The speaker responds to objects in perceptual situations, and in certain states of himself, with appropriate linguistic activity.
(2) Intra-linguistic Moves: The speaker’s linguistic conceptual episodes tend to occur in patters of valid inferences (theoretical and practical), and tend not to occur in patterns which violate logical principles.
(3) Language Departure Transitions: The speaker responds to such linguistic conceptual episodes as ‘I will now raise my hand’ with an upward motion of the hand, etc. (Sellars 1980, 423–424)
Elsewhere (Cogburn 2005a) one of the authors ties this to both Dummett and Christopher Peacocke’s views about content.
Incidentally, this shows why Brandom should not find intuitionistically unacceptable principles to be logical principles. Classical negation does not conservatively extend intuitionistic logic. When one adds classical negation rules to the introduction and elimination rules for the other logical operators, one can derive Putnam’s Theorem (P → Q) ∨ (Q → P), which does not contain negation. Intuitionist negation, on the other hand, does conservatively extend the negation free fragment of the logic.
As is pointed out by Wright, this ‘disjunctive view’ seems to be the view endorsed by Gödel himself in his famous Gibbs lecture.
The source material for these quotes (Lucas 2002) is a web page, so page numbers are not given.
One must note that our proof does rely on the performance-competence distinction. The ability to recognize proofs at the heart of mathematical understanding is an idealized procedure instantiated by understanders. However, as has been shown (Cogburn 2005b) the idealization must be pretty tightly tied to inferential behavior of actual people. Unlike the case of proof recognition, it is not clear how the idealized ability to recognize one’s own Gödel sentence is tied to actual inferential behavior. So there is an important distinction between our appeal and Wright’s. However, conceding the point to Wright is not a large concession. It will be clear that Detlefsen’s critique of Wright does not affect our argument.
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Cogburn, J., Megil, J. Are Turing Machines Platonists? Inferentialism and the Computational Theory of Mind. Minds & Machines 20, 423–439 (2010). https://doi.org/10.1007/s11023-010-9203-1
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DOI: https://doi.org/10.1007/s11023-010-9203-1