Abstract
In this paper I shall consider two related avenues of argument that have been used to make the case for the inconsistency of mathematics: firstly, Gödel’s paradox which leads to a contradiction within mathematics and, secondly, the incompatibility of completeness and consistency established by Gödel’s incompleteness theorems. By bringing in considerations from the philosophy of mathematical practice on informal proofs, I suggest that we should add to the two axes of completeness and consistency a third axis of formality and informality. I use this perspective to respond to the arguments for the inconsistency of mathematics made by Beall and Priest, presenting problems with the assumptions needed concerning formalisation, the unity of informal mathematics and the relation between the formal and informal.
This work was supported by the Caroline Elder Scholarship and a St Andrews/Stirling Philosophy Scholarship. Many thanks to two anonymous referees, Aaron Cotnoir, Benedikt Löwe, Noah Friedman-Biglin, Jc Beall, Graham Priest, Alex Yates, Ryo Ito, Morgan Thomas, Brian King and audiences in St Andrews, Cambridge and Munich.
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Notes
- 1.
A terminological note: while I speak of ‘informal proofs’ and ‘formal proofs’, some of the literature on this subject instead speaks of ‘proofs’ and ‘derivations’ to get at the same distinction. In [15], Priest also uses the term ‘naïve proof’ to refer to the informal proofs.
- 2.
- 3.
At this point we are only concerned with the informal version of the paradox. Later I take on the formal results of Gödel’s theorems.
- 4.
A final note on Beall: although the argument I am criticising is from an older paper, the response offered here would fit well with Beall’s more recent work in Beal [4]. The suggestion I have made may be appropriated to make the case that informal proof should join truth in the category of useful devices, which when introduced bring ‘merely’ semantic paradoxes as by-products or ‘spandrels’ without thereby rendering the base language (in this case, that of mathematics) inconsistent.
- 5.
As the target of his argument, Priest needs to explain what he takes naïve or informal mathematics to be exactly. He says:
Proof, as understood by mathematicians (not logicians), is that process of deductive argumentation by which we establish certain mathematical claims to be true. [15, p. 40]
His distinction is, in effect, the same as the distinction between formal and informal mathematics as found in Sect. 2.
- 6.
The matter is somewhat more complicated than this suggests, of course. Milne discusses in [12] the many ways that Gödel sentences can be constructed and what exactly they ‘say’.
- 7.
Not just this, though, since Priest takes it that the theory given by the formalisation of informal mathematics can prove its own soundness and hence must be able to give its own semantics. From here he takes it to follow that it must be able to prove the T-scheme for this theory inside the theory, giving him all of the paradoxes he describes as semantic (as opposed to set-theoretic paradoxes). For example, he lists the liar, Grelling’s paradox, Berry’s paradox, Richard’s paradox and Koenig’s paradox as falling under the umbrella of semantic paradoxes. In fact, then, Priest argues that “Our naive theory is semantically closed and inconsistent. By contrast, any consistent theory cannot be semantically closed.” [15, p. 47].
- 8.
The ‘many’ here is due to the fact that it might end up being case that multiple informal proofs are mapped to the same formal proof.
- 9.
I use the terms ‘super-theory’ and ‘super-system’ throughout this paper. I do not intend anything of the ‘super-’ prefix besides that it is all-encompassing of mathematics in the way described.
- 10.
An anonymous referee suggests that we may be able to distinguish between a plurality of results which are equivalent under translation and those which genuinely disagree. I believe, however, that this will not save the argument. In a critical discussion of Azzouni’s formalist account of proofs [20], I have previously argued that such a move is not going to deliver the substantial kind of formalisation required for the argument to proceed.
- 11.
An anonymous referee proposes an additional argument against Priest based on this section: that the translation on the many-many case is not effective means that informal proof can therefore not meet the minimum requirements for falling under Gödel’s theorems. Grist to the mill!
- 12.
And we are well used to theories being incomplete for more reasons than Gödel theorem. For instance, Peano arithmetic also has examples like Goodstein’s theorem and the Paris-Harrington theorem.
- 13.
Note that this cannot be avoided by insisting that the Gödel sentence must be part of naïve arithmetic without running afoul of the distinction of Sect. 3.
- 14.
I take it that, as mathematicians, we don’t need to commit ourselves to the truth, in some philosophical sense, of the mathematics that is being carried out.
- 15.
Or, at least, to see incomplete and inconsistent systems as both serving purposes which may be justified by key pragmatic principles.
References
Azzouni, J. (2004). The derivation-indicator view of mathematical practice. Philosophia Mathematica (III), 12, 81–105.
Azzouni, J. (2005). How to nominalize formalism. Philosophia Mathematica (III), 13, 135–159.
Beall, J. (1999). From full blooded platonism to really full blooded platonism. Philosophia Mathematica (III), 7, 322–325.
Beall, J. (2009). Spandrels of Truth. Oxford: Clarendon Press.
Benacerraf, P. (1965). What numbers could not be. The Philosophical Review, 74, 47–73.
Carnap, R. (1945). The two concepts of probability. Philosophy and Phenomenological Research, 5, 513–532.
Dutilh Novaes, C. (2011). The different ways in which logic is (said to be) formal. History and Philosophy of Logic, 32, 303–332.
Hersh, R. (1997). Prove–once more and again. Philosophia Mathematica (III), 5, 153–165.
Larvor, B. (2012). How to think about informal proofs. Synthese, 187, 715–730.
Leitgeb, H. (2009). “On Formal and Informal Provability”. In O. Bueno, Ø. Linnebo(Eds.), (2009): New Waves in Philosophy of Mathematics, Palgrave Macmillan, pp. 263–299.
Marfori, M. A. (2010). Informal proofs and mathematical rigour. Studia Logica, 96, 261–272.
Milne, P. (2007). On Gödel sentences and what they say. Philosophia Mathematica (III), 15, 193–226.
Nelsen, R. B. (1993). Proofs Without Words: Exercises in Visual Thinking. Washington: The Mathematical Association of America.
Nelsen, R. B. (2000). Proofs Without Words II: More Exercises in Visual Thinking. Washington: The Mathematical Association of America.
Priest, G. (1987). In Contradiction: A Study of the Transconsistent. Oxford: Oxford University Press.
Priest, G. (2012). “Mathematical Pluralism”, Logic Journal of the IGPL, pp. 4–13.
Rav, Y. (1999). Why do we prove theorems? Philosophia Mathematica (III), 7, 5–41.
Rav, Y. (2007). “A critique of a formalist-mechanist version of the justification of arguments in mathematicians’ proof practices”, Philosophia Mathematica (III) 15, pp. 291-320.
Robinson, J.A. (1991). “Formal and Informal Proofs” In R.S. Boyer (Ed.),: Automated Reasoning, pp. 267–282. Berlin: Kluwer Academic Publishers.
Tanswell, F.S. (2015). “A Problem with the Dependence of Informal Proofs on Formal Proofs”, Philosophia Mathematica (III), Vol. 23(3), currently available through advance access.
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Tanswell, F.S. (2016). Saving Proof from Paradox: Gödel’s Paradox and the Inconsistency of Informal Mathematics. In: Andreas, H., Verdée, P. (eds) Logical Studies of Paraconsistent Reasoning in Science and Mathematics. Trends in Logic, vol 45. Springer, Cham. https://doi.org/10.1007/978-3-319-40220-8_11
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