NACIMIENTO DE LA SECCIÓN DE CENTROS DE ENSEÑANZA

We made reference above to the fact that Button & Smith(2012) are themselves sceptical regarding the viability of model-theoretic scepticism. They ‘do not suppose that Thoralf’s problem is a genuine problem’. (Button & Smith, 2012 : 119). This because in order for the sceptic to argue for the indeterminacy of mathematical notions she has to suppose their own determinacy. To clarify what is meant here, first, it is worth recalling that first-order logic lacks expressive power to capture the notion of finiteness or recursive procedure. Still, technically we do need to assume a notion of finiteness in order to define the syntax of first-order logic: we say that a first-order formula has an arbitrary but finite length, that it is a finite sequence of symbols arranged with certain rules. Similarly, well-formed formulas or a satisfaction relation are defined first by a base-clause for atomic formulas and then by recursive conditions for the others. Hence, it is clear that the use of first-order logic and first-order model theory presuppose a firm understanding of the relevant notions – finiteness and recursive procedure. As a consequence, in order for the skolemite sceptic to put forward his model-theoretic scepticism, he must make obvious use of first-order model theory and assume a grasp of the notions for which he argues there can be no such grasp. More generally, in using some model theory to argue for the indeterminacy of

18_{Of course this relies on the assumption that we do have a thorough understanding of the induction}

the concept of natural number, the sceptic must make use of conceptual tools that allow her (and us) to determine the natural numbers.19

To illustrate the latter point, consider again the argument from the Initial Segment Theorem. The argument pins down the standard model, up to isomorphism, as intended of our practice. But the sceptic is keen on pointing out that when using the Initial Segment Theorem we make implicit use of the (determinacy of the) notion offiniteness, interdefinable with natural number. Therefore she will argue that even when stating the argument we already assume what we wanted to prove: thatnatural number is already a determinate concept. Yet, in order for the sceptic to argue for the indeterminacy of the natural number sequence, she herself must implicitly assume the (determinacy of the) notion of finiteness when employing her model- theoretic paraphernalia. And this leads to the following dilemma: either (a)finiteness is not determinate, in which case the sceptical problem cannot even be posed; or (b) finiteness is determinate, in which case the sceptical problem can be easily solved20.

The point of all this is the following. In order to understand why model theory is supposed to push us towards sceptical concerns, we must possess certain model-theoretic concepts. However, possession of those model- theoretic concepts enables us to brush aside the sceptical concerns. Ac- cordingly: insofar as we can understand the sceptical challenge, we can dismiss it. (Button & Walsh, 2018 : 208)

By itself, this does not mean that everything is fine with the moderate realist view. As we saw, Skolem-Putnam’s challenge is a very natural problem to pose to this position: if we have only modest epistemic access to numbers, how can we know how they are like? What the above considerations show is rather that moderate realism is led to a deeply incoherent sceptical position, highlighting that something must be wrong not only with skolemite scepticism but with moderate realism itself too.21 Further, this considerations still do not suffice to show how we come to acquire determinate mathematical beliefs or that we can acquire them; only that the sceptical challenge does not stand.

Hence, the dissolution claims that the model-theoretic challenges cannot even be posed in a non-self-refuting way; in contrast, the solution assumes that the model- theoretic challenges can be meaningfully posed, but also that they can be meaningfully answered given that we already possess a primitive notion offiniteness.

3.7

Summary

In this chapter we have presented the argument from Tennenbaum’s Theorem and how it can be put to work against the sceptical challenge. After surveying its different uses

19_{See Bays, 2001 : § IV. ‘This, then, is what I like to call the stability objection to Putnam’s}

argument. The argument rests on the assumption that we cannot use semantically indeterminate language to describe “intended interpretation”. But, by Putnam’s own standards, the notions needed to formulate first-order model theory turn out to be semantically indeterminate. So, since Putnam’s own techniques for obtaining intended interpretations involve first-order model theory, his position is logically unstable.’ (Bays, 2001 : 346)

20_{For example, again, by the argument from the Initial Segment Theorem.}

21_{The proper assessment of this issue is heavily dependent on the exegetics of Putnam’s own work.}

Therefore we do not have much to add here except directing the reader to Button(2013) for an elaborate exposition.

and foibles, we concluded that perhaps the sceptical challenge is by itself incoherent there being no need for an argument from Tennenbaum’s Theorem at all.

In what follows we wish to give new life to the sceptical challenge, showing that neither the solution nor the dissolution are enough to resist it. Further, we want to present a new argument (what we will call the LP-argument) against the argument from Tennenbaum’s Theorem. We will see the unexpected consequences of our argument; if they can motivate skolemite scepticism depends on minute philosophical details that we postpone until the next chapter.

Chapter 4

Non-Classical Skolemite

Scepticism

4.1

Introduction

We have just covered two main results: first, the argument from Tennenbaum’s The- orem and, second, the sceptical solution/disolution. Let us quickly recap the former: the argument starts by assuming that in intended models the operation denoted by + is a recursive function. In fact, putting it in this way is already assuming too much. When properly seen, the basic claim here is that in intended models the operation denoted by + is an ‘informally-computable’ function. (Below we elaborate on this point, but for now it suffices to keep the above in mind.) Given this assumption, coupled with the Church-Turing Thesis and with Tennenbaum’s Theorem, it follows that the intended model of arithmetic is restricted to a single isomorphism type, the class of models isomorphic to N. Yet things are not so simple: the argument pre- supposes more than what makes explicit, being the purpose of this chapter to show exactly what the hidden assumptions amount to. As we will argue, for the argument to work as expected a ‘classicality-constraint’ must be assumed. We will show this by considering the non-classical (paraconsistent) semanticsLP (see Priest, 1979) and the models of pa obtained with this logic. A by-product of our presentation will be the reintroduction of the sceptical challenge against the solution/dissolution propos- als. In arguing against the argument from Tennenbaum’s Theorem we will see how the sceptic is happy to embrace a determinate notion of finiteness without leading to a full-blown determinacy of arithmetic’s interpretation nor contradicting his own sceptical qualms. It is the goal of this chapter to see how neither the strength of the argument from Tennenbaum’s Theorem nor of the sceptical (dis)solution remain unaffected by considering non-classical models of arithmetic.