Abstract
Conventionalism about mathematics claims that mathematical truths are true by linguistic convention. This is often spelled out by appealing to facts concerning rules of inference and formal systems, but this leads to a problem: since the incompleteness theorems we’ve known that syntactic notions can be expressed using arithmetical sentences. There is serious prima facie tension here: how can mathematics be a matter of convention and syntax a matter of fact given the arithmetization of syntax? This challenge has been pressed in the literature by Hilary Putnam and Peter Koellner. In this paper I (i) sketch a conventionalist theory of mathematics, (ii) show that this conventionalist theory can meet the challenge just raised (this is done by considering how arithmetical coding works in non-standard models of arithmetic), and (iii) clarify the type of mathematical pluralism endorsed by the conventionalist by introducing the notion of a semantic counterpart. The paper’s aim is an improved understanding of conventionalism, pluralism, and the relationship between them.
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Notes
The focus on introduction and elimination rules in natural deduction systems derives from the work of Gentzen (1935). It should be noted that standard single conclusion natural deduction treatments of the logical constants fail to determine truth-functional meanings for most of the connectives; this was first noticed and patched in Carnap (1943).
See Prior (1960).
I defend unrestricted inferentialism from the challenge of tonk in my “Talking with Tonkers”, currently under review.
Unless otherwise indicated, when I say “Peano Arithmetic” I’ll always be talking about the first-order version of the Peano Axioms.
These axioms are, \((i)\, \forall x(x+0=x), (ii)\, \forall x\forall y(x+s(y)=s(a+b)), (iii)\, \forall y(y\times 0=0)\), and \((iv)\, \forall y\forall z(y\times s(z)=y\times z+y)\). They could each be added in rule form to be more in keeping with inferentialism about arithmetic.
Adding these rules to a standard system of first-order logic results in a non-conservative extension, i.e., we can prove sentences without any arithmetical vocabulary using the Peano rules when these sentences are not provable using only the non-arithmetical rules. To see this, simply note that every counting sentence (there are at least \(n\) things): (i) can be formulated using only standard logical vocabulary, (ii) is not a logical theorem, and (iii) can be proved using the Peano rules. This means that restricting the \(\mathbf {MRC}\) so as to apply only to rules that conservatively extend our base theory blocks the extension of inferentialism from logic to mathematics.
See Quine (1948).
There are other objections specifically aimed at mathematical conventionalism that I won’t directly address here, e.g., see Gödel (1953/1959) and Pollock (1967). Both logical and mathematical conventionalism are developed and defended at length in my forthcoming book, Syntactic Shadows: A Linguistic Theory of Logic and Mathematics.
Arithmetization was introduced in Gödel (1931).
Actually, care must be taken here: there are some seemingly natural ways of expressing the consistency of \(PA\) that are provable in \(PA\). Hilbert and Bernays (1939) gave three conditions on the provability predicate used in formulating the consistency sentence that, if satisfied, suffice for Gödel’s second theorem to hold for the associated consistency sentence. The conditions were later given in a more perspicuous form by Löb (1955): for theory \(T\), let \(\square \phi =\exists xPRF_{T}(x,y)\) (see below for an explanation of this relation) then the conditions are: \((i)\) if \(T\vdash \phi \) then \(T\vdash \square \phi \); \((ii)\, T\vdash \square (\phi \rightarrow \psi )\rightarrow (\square \phi \rightarrow \square \psi )\); \((iii)\, T\vdash \square \phi \rightarrow \square \square \phi \).
In Sect. 1 I didn’t offer a sufficient condition for mathematical truth, but however the conventionalist fleshes out the story of mathematical truth, truth will have to be owed to convention in some sense.
I am neglecting to use corners when employing quasi-quotation because I will later use corners for Gödel numbers.
Page 84 in Koellner (2009).
This might naturally be combined with the Putnam characterization since it seems like a stipulation or practice can be neither true nor false.
I am focusing on \(PA\) both for definiteness and because it is the case relevant to the theory of arithmetic given above, but of course we can apply arithmetization to theories other than \(PA\).
Throughout I’ll be assuming that “\(PRF_{PA}(x,y)\)” is the standard or canonical provability relation required for proofs of the second incompleteness theorem; this is necessary because there are other provability relations that won’t allow us to form consistency sentences for which the second incompleteness theorem holds see Feferman (1960). For helpful discussion of this, see Smith (2013), Sects. 36.1 and 36.2.
I have abbreviated “\(s(\hat{0})\)” as “\(\hat{1}\)”, I will continue to do this without comment below.
Recall once again the warning about provable “consistency” sentences.
I’m extrapolating a bit here, but I think Koellner would endorse this claim.
As noted above, it isn’t literally correct to claim that our arithmetical language’s rules include nothing beyond the Peano Rules. In addition to needing to add rules for addition and multiplication, our arithmetical practice includes various bridge laws that allow us to apply arithmetic to the world. I’ll be ignoring these complications here.
N.B., when a set of sentences appears on the left of the semantic turnstile (“\(\models \)”) the claim should be read as saying that any model in which the set of sentences on the left are all true is a model in which the sentence on the right is also true, i.e., the sentence on the right is a logical consequence of the set of sentences on the left.
In fact, because of the way these consistency sentences are constructed, if a theory \(S\) extends a theory \(T\) then \(\vdash CON_{S}\rightarrow CON_{T}\), so contrapose for our result.
This observation is related to an interpretation of some of Wittgenstein’s remarks on Gödel’s theorems offered in Floyd and Putnam (2000).
This shows that the provability predicate for \(PA\) so formed is not primitive recursive, despite its being expressed by a \(\Sigma _{1}^{0}\)-formula.
The type of logical pluralism entailed by conventionalism is different than the logical pluralism of Beall and Restall (2006) which concerns there being indeterminacy within our language about what counts as a legitimate means of making precise the “truth-preservation in all cases” core of our concept of consequence.
See Lewis (1968); in a full treatment, I would actually slightly generalize the notion of counterpart used in theories like Lewis’s so as to allow even greater flexibility in the intra-language case.
Of course, in some cases, we may resolve to split the relevant semantic role in two and switch back and forth between the “competing” theories whenever it suits us.
For a proof of this latter claim in addition to some discussion of equivalence claims in metaphysics, see the appendix to my Warren (2014b).
Here I’m just defining an “arithmetical” theory as one that can do enough arithmetic for Gödel’s results to hold. Although Gödel’s original proof was applied to the language of Principia Mathematica, subsequent work has shown that very weak arithmetical theories suffice for this.
In actual historical fact, Carnap’s base theory here is something like the weak theory Primitive Recursive Arithmetic.
In fact, this holds even for adding the \(\omega \)-rule to a weak arithmetical theory like Robinson Arithmetic.
Carnap also has another procedure which, from the modern point of view, can be seen as a convoluted variant of Tarski’s theory of truth.
I address this issue in work under review, especially my “The Determinacy of Arithmetic: The Puzzle and Its Solution”, where the problem is addressed in a way congenial to conventionalists but without assuming conventionalism.
Thanks to an audience at NYU and two referees.
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Warren, J. Conventionalism, consistency, and consistency sentences. Synthese 192, 1351–1371 (2015). https://doi.org/10.1007/s11229-014-0626-8
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DOI: https://doi.org/10.1007/s11229-014-0626-8