Truth and Provability— A Comment on Redhead

Michael Redhead’s recent argument aiming to show that humanly certifiable truth outruns provability is critically evaluated. It is argued that the argument is at odds with logical facts and fails. Michael Redhead puts forward, in his ambitious paper ‘Mathematics and the Mind’ (Redhead [2004]), a simple argument which aims to show that humanly certifiable truth outruns provability. Redhead’s arguments require a comment. Redhead first discusses two possible answers to the question of how we know that the Gödel sentence G (for Peano arithmetic, PA) is true. He dismisses them both by stating that they presuppose that the axioms of PA are true. Strictly speaking, this is wrong—they require only that PA is consistent, which is a much weaker assumption. But Redhead is certainly on the right track in rebutting these strategies. Redhead’s own argument focuses on the weaker Robinson arithmetic Q (Redhead calls it, following Lucas, ‘sorites arithmetic’, but I prefer to use the standard name), which, unlike PA, does not have the induction scheme. His reason for this is that its axioms are ‘arguably analytic’: ‘If any of these axioms were false we would not be talking about numbers.’ Redhead contrasts them with the induction axiom (or scheme), which he calls ‘notorious’ and ‘more mysterious’. With a reference to Poincaré, he concludes that the induction scheme is not analytically true. 1 At one point (see p. 735), Redhead’s wording seems to suggest that Gödel’s theorem shows that there are true sentences of arithmetic which cannot be proved in any consistent, axiomatizable extension of Robinson arithmetic. But of course it shows no such thing. It shows only that no such extension can be complete. Different extensions have different undecidable Gödel sentences. In all likelihood, Redhead did not really intend to claim the contrary, but given his misleading formulation, this point is perhaps worth making. The Author (2005). Published by Oxford University Press on behalf of British Society for the Philosophy of Science. All rights reserved. doi:10.1093/bjps/axi134 For Permissions, please email: [email protected] Advance Access published on August 10, 2005. Now, the distinction between the analytic and the synthetic is famously elusive and problematic, but it is far from clear that the induction scheme is in any way less analytic than the other axioms. It is equivalent (assuming classical logic) to the least number principle, that is, to the claim that if there exists a number with a property P, there exists the smallest number with the property P. But it is quite plausible to say that if this principle fails, one is not talking about natural numbers, in other words, that the principle, and hence, induction, is analytic in Redhead’s sense. But be that as it may, let us now consider Redhead’s main argument. It begins with the well-known fact that while For all pairs m, n ð Þ, it is provable in Q that m n 1⁄4 n m ð1Þ holds, the following is not true: It is provable in Q that for all pairs m, n ð Þ, m n 1⁄4 n m: ð2Þ The universal generalization For all pairs m, n ð Þ, m n 1⁄4 n m ð3Þ can be proved (e.g. in PA) with the help of the induction scheme, which Q does not have. Redhead next submits that we can argue—presumably without using the induction scheme—that (3) is nevertheless true. Redhead introduces the notion of truth (or, ‘is true’) and argues that since the axioms of Q are analytically true, we can replace (1) by For all pairs m, n ð Þ, it is true that m n 1⁄4 n m, ð4Þ which, according to Redhead, is strictly equivalent to It is true that for all pairs m, n ð Þ, m n 1⁄4 n m: ð5Þ By eliminating the truth predicate, one gets (3). Redhead concludes that we have here a case in which certifiable truth outruns provability. One problem with Redhead’s discussion is that he does not make explicit which kind of notion of truth he is assuming in the above reasoning. His remarks at the end of the paper suggest that he has a Tarskian definition of truth in mind. However, such a definition can be given only in a sufficiently strong metatheory, a theory which must certainly contain the induction scheme. Hence there is a risk here that one smuggles in the very principle one is trying to avoid. However, the most serious problem in Readhead’s reasoning concerns (1) and the grounds of our knowledge of it. It is a statement about provability in Robinson arithmetic Q, but this does not guarantee that it is itself provable 612 Panu Raatikainen