In Defense of the Implicit Commitment Thesis

2.1. A Description of Axioms and Rules

I will begin with the less controversial aspect of my formulation of ICT, namely that $\text{S}$ must be given under some fixed description of its axioms and rules which the agent in question is able to grasp and assent to. Here is why we need that condition. As a theory, in the technical sense, $\text{S}$ is just a set of sentences closed under deduction. But my topic here is epistemological, so I am interested specifically in theories that can be known, or at least believed; and thus we must have some way of grasping or apprehending the set of sentences $\text{S}$ . Since we cannot directly apprehend infinite sets of sentences off in Plato’s heaven (whatever that would even mean), we must have some more immediate, concrete way of apprehending a theory. Typically, we grasp a set of sentences by means of an axiomatization. But there may be multiple distinct axiomatizations of a given theory, and we may be in a position to justifiably assent to some of those axiomatizations and not others. Thus, we want to hold fixed the particular axiomatization that we are in a position to grasp and accept.

In order to apprehend a finitely axiomatizable theory, it suffices to write down its finitely many axioms. Many theories of interest are not finitely axiomatizable, however. In such cases, we presumably need a finite description of the infinite axiomatization, and the standard way of providing such a description is by means of axiom schemas, as with the schema of induction in arithmetic or the schema of separation in set theory. In this case, in order to be said to grasp and accept the axiomatization, the agent should be able to grasp and recognize the schema qua schema; that is, she should recognize its intended range of generality and be disposed to assent to each instance of the schema as such.

This takes us back to the restriction that ${\mathrm{Prov}}_{\text{S}}$ be a canonical provability predicate. Let me detail what I mean by that. I will assume we have some straightforward presentation of the axioms of $\text{S}$ as a finite list of axioms and axiom schemas (and rules, if any). For brevity, let ${\text{S}}_a$ denote a particular description of the axioms (and rules, if any) of $\text{S}$ . I also assume we have some standard deductive system, and that we have developed a coding apparatus in $\text{S}$ capable of representing the usual properties such as is a formula, is a sentence, etc.

Now a canonical provability predicate is a formula defined in two steps: first, we transcribe directly into the coding apparatus our definition of the axioms of $\text{S}$ as being a member of ${\text{S}}_a$ . This is contrasted with ‘non-standard’ ways of defining the axioms of the theory $\text{S}$ , such as by defining $\text{S}$ as the maximal consistent subtheory of some set of axioms given by a list. This way, the agent in question is able to grasp and assent to the formalized definition of ${\text{S}}_a$ just as they were able to grasp and assent to the unformalized definition.

The second step is to take this formalized definition of ${\text{S}}_a$ and transcribe directly into the coding apparatus the definition of proof whose premises are among  ${\text{S}}_a$ in our chosen deductive system. Then a canonical provability predicate is the formula that ‘says’ there exists a proof from  ${\text{S}}_a$  whose conclusion is  $\varphi$ . (Thus, it would be more perspicuous to abbreviate the canonical provability predicate as ${\mathrm{Prov}}_{{\text{S}}_a}$ rather than ${\mathrm{Prov}}_{\text{S}}$ , and I will do so.)

The significance of all this is that, because the agent is able to grasp the theory $\text{S}$ under the description ${\text{S}}_a$ , when ${\mathrm{Prov}}_{{\text{S}}_a}$ is a canonical provability predicate, the agent is also able to recognize that it expresses the property of being provable in $\text{S}$ .

2.2. Mathematical Certainty

Let us turn now to mathematical certainty. I will adopt a notion of certainty as the absence of reasons for doubt.⁵ On my view, what is distinctive of mathematical certainty, as opposed to practical certainty, empirical certainty, or absolute certainty, concerns the nature of the grounds for doubt. To have mathematical certainty in $\varphi$ is to have no mathematical grounds for doubting $\varphi$ . Thus, for instance, I can have mathematical certainty in the claim that $1+1=2$ , even if philosophical worries about the existence of abstract objects give me reason to doubt the literal truth of this claim. These grounds of doubt are not mathematical in character. By contrast, an experienced mathematician might have very reliable intuitions about, say, whether a given series converges. On this basis they might justifiably have very high confidence that a series converges. Nevertheless, this intuition, though reliable, is not conclusive, and there may remain some mathematical grounds for doubt that the series converges.

There will of course be examples of doubts that are more difficult to classify as mathematical or non-mathematical in character. For instance, consider the debate between intuitionists and classical mathematicians about whether the Law of Excluded Middle holds for arithmetical sentences. Does the viability of intuitionism provide mathematical reason for doubting LEM? Or is it a philosophical reason, being based in the intuitionist’s metaphysical views about mathematical objects? In my opinion, these doubts are better classified as philosophical rather than mathematical, but the matter is not entirely clear.⁶ As another example, many set-theorists doubt $V=L$ on the grounds that $L$ is much too small and restrictive to comprise the entire universe of set theory. On the one hand, this might seem mathematical, because it is grounded in certain expectations about which mathematical claims should be true.⁷ On the other hand, it might seem to be a philosophical doubt because it is based on certain philosophical ideas about how the universe of sets should look.

Other cases are easier to classify, though. For instance, the $P$ vs. $NP$ problem is a clear example of an open mathematical problem. There are plausibility arguments and some inductive evidence for the claim that $P$ ≠ $NP$ .⁸ But these fall short of conclusive mathematical evidences, and there is clear room for mathematical doubt about $P$ vs. $NP$ .

In general, wherever we have mathematical arguments that fall short of being dispositive we can find mathematical room for doubt. Plausibility arguments, limiting-case considerations, trained intuition, enumerative induction, and probabilistic methods can all provide mathematical reasons to believe some proposition. But the reasons that these methods provide will not be conclusive and will leave open the possibility of doubt. As a concrete example, a probabilistic primality test can make it very likely that a given number is prime. But the test is merely probabilistic; indeed, it is provable that the test has a certain error rate. Hence, you still have mathematical grounds for doubting that the number is in fact prime.

What matters for present purposes is whether there are some clear, non-trivial examples of theories that we can have mathematical certainty in and which exhibit Gödelian incompleteness. If so, then there are non-trivial examples of theories to which the ICT applies. To be clear, I am not assuming that all of mathematics is certain, and I do not need to take a stand on where or how the line might be drawn between the certain and the less-than-certain (though I will return to this question in §2.4). I am also not committed to any particular view on the nature of mathematical justification. A traditional view held that mathematical axioms must be self-evident; and while this plausibly applies to some axioms such as $0+1=1$ , it has more recently become widely acknowledged that holistic, abductive, and pragmatic factors can also play a role in justifying a choice of axioms.⁹ Despite invoking a notion of mathematical certainty, I am not reverting to the traditional Euclidean view that all of mathematics consists in deduction from self-evident axioms. I am not even committed to the claim that self-evidence is the source of whatever certainty there is in mathematics. (For all I have said here, it may be possible to obtain mathematical certainty from holistic considerations!) All that I require is the claim that there is some theory which does enjoy mathematical certainty and which exhibits Gödelian incompleteness.

I offer Robinson arithmetic $\text{Q}$ as such an example. It is a classic result that $\text{Q}$ suffices for proving Gödel’s incompleteness theorem.¹⁰ It also seems to me that the axioms of $\text{Q}$ are beyond any mathematical doubt. After all, they are only the simplest axioms characterizing $0,\hspace{0.17em}\hspace{0.17em}1,\hspace{0.17em}\hspace{0.17em}+,\hspace{0.17em}\hspace{0.17em}\times$ , and . To the skeptic, I ask: what are your mathematical grounds for doubting the axioms of $\text{Q}$ ?

2.3. The Argument for ICT

The argument for ICT is very simple. At the heart of the argument is the idea that justification flows through deduction, so that if some evidence provides justification for belief in a set of statements, then that same evidence provides justification for belief in anything those axioms entail. Note that I am not assuming that knowledge is closed under deduction, only that evidential support flows through deduction. If this were not so, then it would be difficult to explain the role of deduction in our reasoning practices or how it is even possible to gain knowledge by inference. This idea is the fundamental motivation for the argument, but the precise claim the argument relies on is this: if $\varphi$ is provable from $\Delta$ , and you have some grounds for doubting $\varphi$ , then that same ground provides reason for doubting $\Delta$ .¹¹ It is helpful heuristically to suppose that there exists reason to doubt $\varphi$ just in case there is a plausible scenario where $\varphi$ does not hold. Under this paraphrase, the main premise of the argument can be stated as: if $\varphi$ is provable from $\Delta$ , then if there is a plausible scenario where $\varphi$ fails, then there is a plausible scenario where some member of $\Delta$ fails.

Now suppose ICT is false, so you have mathematical certainty in a list of axioms ${\text{S}}_a$ , but for some sentence $\varphi$ , you do not have mathematical certainty in the conditional ${\mathrm{Prov}}_{{\text{S}}_a}(⌜\varphi ⌝)\to \varphi$ . Then there are some mathematical reasons for doubting ${\mathrm{Prov}}_{{\text{S}}_a}(⌜\varphi ⌝)\to \varphi$ . This would require a mathematically plausible scenario where $\varphi$ is indeed provable from the axioms ${\text{S}}_a$ , but nevertheless $\varphi$ fails to hold. But this, I claim, is ruled out by the fact that you have mathematical certainty in ${\text{S}}_a$ . I take it as obvious that there is no mathematically plausible scenario in which the axioms ${\text{S}}_a$ all hold but some provable consequence $\varphi$ does not hold. And since ${\text{S}}_a$ are all mathematically certain, there is no mathematically plausible scenario where ${\text{S}}_a$ do not all hold. And thus there is no mathematically plausible scenario where $\varphi$ is a provable consequence of ${\text{S}}_a$ but does not hold.

There is a clear connection between this argument and the truth-theoretic argument for ICT sketched earlier, and one might object that my argument actually presupposes the truth-theoretic argument. While I have avoided use of the word ‘true’, I referred to things that hold or do not hold. Saying $\varphi$ holds is plausibly equivalent to saying that $\varphi$ is true, and I assumed implicitly that when some sentences $\Delta$ hold, so do any of their consequences. In other words, the objection goes, my argument requires all of the assumptions of the truth-theoretic argument and hence does not provide any new defense of ICT.

In response to this objection, I would point out that the talk of sentences holding in some scenarios is a vivid way of grasping the argument, but is ultimately dispensible. The essential argument can be expressed entirely in terms of doubt and certainty. (This point is analogous to the common observation that it is often helpful to reason about modal matters using possible worlds, but this can ultimately be dispensed with in favor of a primitive modal operator.) Write $C\psi$ to mean that $\psi$ is mathematically certain. Then the two key premises of the argument are:

• $\mathcal{C}\neg ({\text{S}}_a\wedge {\mathrm{Prov}}_{{\text{S}}_a}(⌜\varphi ⌝)\wedge \neg \varphi )$ .

• $\mathcal{C}(\psi \to \theta )\to (\mathcal{C}\psi \to \mathcal{C\theta })$ .

From these two premises it is easy to derive ICT.¹²

Thus, my argument does not clearly rely on the same assumptions as the truth-theoretic argument. It is true, however, that there is some similarity between the two arguments, in that both arguments rely on some property being preserved under deduction. For the truth-theoretic argument, this property is truth, while in my argument the property is mathematical certainty. At the same time, there is an important difference between the arguments in their dialectical role and in who is supposed to grasp them. Recall the structure of the truth-theoretic argument: if the axioms of some theory are true, then by induction on the length of proof we can argue that anything provable from them must also be true. So we can conclude that the local reflection principles are true. On its face this says nothing about the epistemic standing of the reflection principles. To infer that some agent is committed to the local reflection principles we need further assumptions. The most natural way to fill in this gap is with the two assumptions that, first, the agent is explicitly or implicitly justified in believing the relevant truth theory, and, second, one is justified in believing the consequences of whatever one is justified in believing.¹³

The important difference between this line of reasoning and my own argument is that this line of reasoning requires the agent in question to have some justification for believing the truth theory. Furthermore, in order for the agent to believe the reflection principles with doxastic justification, they presumably have to be able to perform the relevant deduction of the reflection principle from the truth theory. If that is correct, then in order for the agent to be in a position to justifiably believe the reflection principles, they have to be able to grasp and use the truth theory. By contrast, my argument solely concerns the structure of epistemic reasons—reasons for believing and doubting propositions. The agent in question does not have to be able to articulate those reasons or the structure between them, provided only that they are appropriately responsive to those reasons in their epistemic behavior. This is why the commitment in question is implicit: the agent implicitly has the justification to believe the reflection principles just in virtue of the nature of their epistemic reasons.

I have defended the following version of the implicit commitment thesis: Whenever one is mathematically certain of some theory, under a fixed description ${\text{S}}_a$ of its axioms and rules, one is also in a position to be mathematically certain of the local reflection principles ${\mathrm{Prov}}_{{\text{S}}_a}(⌜\varphi ⌝)\to \varphi$ . In the following subsections, I will address a few objections and discuss possible extensions of my argument.

2.4. Is the ICT Trivial?

One possible response to this argument is to grant its cogency, but object that it is not interesting because it only applies to comparatively weak theories for which we already have independent means of establishing the reflection principles. For instance, say that we can be mathematically certain of $\mathrm{PA}$ . Then my argument would show that we can be mathematically certain of the theory $\mathrm{PA}+\{{\mathrm{Prov}}_{PA}(⌜\varphi ⌝)\to \varphi :\varphi \in {ℒ}_{\mathrm{PA}}\}$ . But this theory is not that much stronger than $\mathrm{PA}$ itself, and is properly subsumed by other theories, such as $\mathrm{ZFC}$ , that we already have good reason to believe. So, one may object, we have not made any epistemic progress.

Let us grant for the sake of argument that whenever my version of the ICT guarantees us justification for some reflection principles, we also have independent reason to believe those reflection principles. Even so, we can reply to this objection by noting that it is generally of epistemological interest to note what grounds there are for a certain belief even when some of those grounds might be redundant. For instance, my visual perception of an event does not make my auditory perception of the same event epistemologically irrelevant or uninteresting.

Building on this observation, we can note two things that are distinctive of the justification guaranteed by the ICT. The first is the strength of the justification guaranteed by the ICT. In the version I have defended, the ICT says not merely that we have some justification for believing reflection principles, but that we can believe them with certainty. While we might be able to prove the reflection principles for a theory $\text{S}$ in some stronger theory ${\text{S}}^{\prime }$ , there is no guarantee that ${\text{S}}^{\prime }$ is mathematically certain. For instance, one might think that Zermelo set theory enjoys mathematical certainty, but that the Axiom of Replacement is not certain. One can prove the consistency of Zermelo set theory in $\mathrm{ZFC}$ , but if the relevant instances of Replacement are not mathematically certain, then the proof will not warrant mathematical certainty in its conclusion.

The second distinctive feature of the ICT is the structure of the justification it guarantees. The ICT, as I have defended it, shows that mathematical certainty has a self-reproducing character. When a theory $\text{S}$ enjoys mathematical certainty, then the ICT tells us that so does a stronger theory ${\text{S}}^{\prime }$ obtained by adding to $\text{S}$ its local reflection principles. Then ${\text{S}}^{\prime }$ can be extended to another mathematically certain theory by adding its reflection principles, and so on.

This second observation has an interesting corollary, namely that, under some general assumptions, we cannot characterize with mathematical certainty the portion of mathematics that enjoys mathematical certainty. Specifically, it is impossible to be mathematically certain that a recursive axiom system includes all and only the mathematically certain propositions. For suppose that we had some recursive axiomatization $A$ , and we were mathematically certain that $A$ entailed all and only the sentences that enjoyed mathematical certainty. Then the ICT would say that we can also be certain of $\mathrm{Con}(A)$ , along with the other local reflection principles. But since $A$ is supposed to be recursive, and given the definition of a canonical provability predicate above, ${\mathrm{Prov}}_A(x)$ will be a ${\Sigma }_1$ formula and hence the incompleteness theorem entails that $A⊬\mathrm{Con}(A)$ . So, although we can be certain of $\mathrm{Con}(A)$ , it is not entailed by $A$ itself, contradicting the assumption that $A$ entails all mathematically certain sentences.

There is no guarantee that the only way for us to apprehend theories is by recursive axiomatizations. So this observation does not guarantee that there is no way to characterize the mathematically certain propositions with mathematical certainty. But any theory that we apprehend by a finite list of axioms and schemas will fit this description.¹⁴ Since this includes all familiar mathematical theories, we can reasonably summarize the conclusion here by saying that if any theory $T$ captured, with mathematical certainty, exactly the mathematically certain sentences, then $T$ must be in some sense unfamiliar.

2.5. Commitments of Uncertain Theories

Although my version of the ICT is not trivial, one might still think that it does not go far enough. For simplicity, I will mostly focus here on the single instance of local reflection $\mathrm{Con}({\text{S}}_a)$ , though my comments will generalize. One might think that accepting any theory commits one to the consistency of that theory, regardless of whether that theory is certain or not. So even if your belief in ${\text{S}}_a$ falls short of mathematical certainty, you still have a commitment to $\mathrm{Con}({\text{S}}_a)$ . After all, if ${\text{S}}_a$ were inconsistent, then anything at all could be derived from ${\text{S}}_a$ , which would undermine the acceptance of ${\text{S}}_a$ as a mathematical theory.

There are two ways of fleshing out this essential idea. The first way is to observe that accepting a theory has a pragmatic aspect as well as a doxastic aspect. When you accept a theory $\text{S}$ , you are not only assenting to its axioms ${\text{S}}_a$ , you are adopting a commitment to rely on $\text{S}$ in your reasoning. (You are possibly also adopting other commitments such as regarding open questions about $\text{S}$ as being legitimate issues of mathematical inquiry, though this is not important for what follows.)¹⁵ In order for this practical commitment to be reasonable, though, reliance on $\text{S}$ should not render all reasoning trivial. And if $\text{S}$ were inconsistent, all reasoning would be trivial because everything would follow from $\text{S}$ . So accepting $\text{S}$ rationally presupposes the consistency of $\text{S}$ .

This line of reasoning is fine as far as it goes. It has two shortcomings, however. The first shortcoming is that, even by the lights of this pragmatic argument, it is not clear that if $\text{S}$ were inconsistent then all reasoning would be trivial. For instance, ‘almost-consistent’ theories are theories that may be inconsistent, but where the only proofs of inconsistency are unfeasibly long.¹⁶ In an almost consistent theory, any reasoning that one could practically undertake would be non-trivial and, depending on the example, may even be perfectly reliable.¹⁷ This is a comparatively minor point, however. The more significant shortcoming with this line of reasoning is that it is entirely pragmatic. It shows that as a matter of practical reason, the acceptance of a theory presupposes its consistency. In this paper, however, I am concerned with the epistemic standing of reflection principles. Showing that our practical commitments presuppose $\mathrm{Con}({\text{S}}_a)$ does not show that we have any epistemic reason to believe $\mathrm{Con}({\text{S}}_a)$ .

As I said, there are two ways of fleshing out the original objection we started with. If the first way fails to bridge the gap between the practical standing and the epistemic standing of $\mathrm{Con}({\text{S}}_a)$ , the second way tries to overcome this gap by taking some ideas of the first approach but looking at them in an epistemic light. Similar to the first approach, we begin by noting that believing a theory S is not an isolated cognitive act or state, but it is part of a larger cognitive project of mathematical inquiry. And it is not just a pragmatic requirement, but indeed an (epistemically) rationally required presupposition of that cognitive practice that our beliefs be consistent. (Of course, we do not always live up to this requirement, but that only shows that we are not perfectly rational, not that the requirement is not genuine.) And when $\varphi$ is some rational presupposition of a cognitive project, the argument goes, an agent engaged in that project has a default rational entitlement to believe $\varphi$ .¹⁸ Thus, being engaged in a project of mathematical inquiry, if one believes $\text{S}$ , then one is rationally entitled to believe $\mathrm{Con}({\text{S}}_a)$ .¹⁹

This argument is again fine as far as it goes. Entitlement, however, is generally taken to be a weaker notion than justification; so while this argument applies to a wider range of theories than my argument does, it yields a strictly weaker conclusion about them. The notion of entitlement is also more controversial than that of justification, so there is interest in determining the extent to which one has full-fledged justification to believe reflection principles rather than mere default entitlement.

This raises the question of whether my argument that we have justification to believe reflection principles can be extended beyond those theories that are mathematically certain. I cannot rule out that there is some argument for this stronger claim,²⁰ but I am doubtful that it would proceed similarly to the argument I have given here. Here is why.²¹ The motivation for the argument I gave above was that justification flows through deduction. Without such an assumption, it is not clear my argument even gets off the ground. But that assumption also motivates the following claim, where $J\psi$ is to be read as “you are in a position to justifiably believe $\psi$ ”:

$J({\text{S}}_a)\to ({\mathrm{Prov}}_{{\text{S}}_a}(⌜\varphi ⌝)\to \mathrm{J\varphi })$

After all, if you are in a position to justifiably believe ${\text{S}}_a$ , then by competently performing the deduction of $\varphi$ from ${\text{S}}_a$ you would put yourself in a position to justifiably believe $\varphi$ . Taking the instance of $\varphi :=0=1$ and using propositional logic, this gives:

$\neg J(0=1)\to (J({\text{S}}_a)\to \neg {\mathrm{Prov}}_{{\text{S}}_a}(⌜0=1⌝))$

Assuming, quite reasonably, that one is never in a position to justifiably believe $0=1$ , by modus ponens we have:

$J({\text{S}}_a)\to \neg {\mathrm{Prov}}_{{\text{S}}_a}(⌜0=1⌝)$

That is, the relevant notion of justification requires that it be impossible to justifiably believe an inconsistent theory. This requires a very strong notion of justification. So even if this includes justification that is somewhat short of mathematical certainty, it is unlikely that the argument I have given, or one akin to it, would apply to theories that enjoy a significantly weaker form of justification.

2.6. The Need to Learn Gödel Coding?

I have so far argued at an informal level, ignoring the fact the local reflection principles are formal sentences in the language of $\text{S}$ . Might that make a difference to their epistemic standing? For my part, I cannot see why it should make any more difference to the epistemic standing of a belief that it be expressed in the language of arithmetic (say) than if it were expressed in the language of French (say). If one does not know French, then one may need to acquire new conceptual resources to appreciate that they have reason to believe le chat est noir. But if one already knows French, then seeing the black cat will suffice to give them reason to believe le chat est noir. Moreover, in learning French, one has not fundamentally altered their epistemic state, at least as far as the black cat is concerned—one simply put oneself in a position to take advantage of their epistemic reasons in favor of assenting to ‘le chat est noir’. Similarly, if one were not familiar with how to use the coding apparatus to represent the syntax of $\text{S}$ within $\text{S}$ , then they may require some new conceptual resources to appreciate that they have reason to believe ${\mathrm{Prov}}_{{\text{S}}_a}(⌜\varphi ⌝)\to \varphi$ .²² But since we are assuming that ${\mathrm{Prov}}_{{\text{S}}_a}$ is a canonical provability predicate, once one learned how to transcribe an ordinary finite list of axioms or axiom schemas and the definition of a proof into the coding apparatus, one would then be in a position to appreciate that the formal sentence ${\mathrm{Prov}}_{{\text{S}}_a}(⌜\varphi ⌝)\to \varphi$ expresses the same thing as if there is an  $\text{S}$ -proof of  $\varphi$ , then $\varphi$ . Of course, learning how to use a coding apparatus can be a non-trivial task. But then so can learning French. And just as learning French does not affect the epistemic standing of one’s belief that the cat is black, but only gives one a new way of expressing that belief, so learning the formal coding apparatus for $\text{S}$ does not affect the epistemic standing of one’s belief that if there is an $\text{S}$ -proof of $\varphi$ then $\varphi$ , but merely gives one a new way of expressing that belief.

A related objection one often hears is that there is an important difference between an informal claim about the consistency of a theory $\text{S}$ and the formal consistency sentence $\mathrm{Con}(\text{S})$ , because the formal arithmetized sentence is about numbers rather than being about the consistency a formal theory. This ostensible difference in subject matter is then alleged to be of some epistemic import. If so, however, the point would seem to generalize beyond the formalization of syntax to other uses of formalization. For instance, graph theory can be formalized in arithmetic. Is there some important difference between the epistemic standing of sentences in graph-theoretic language and sentences in the language of arithmetic that formalize those same graph-theoretic claims? I can see no basis for thinking so.

2.7. Other Reflection Principles?

I have cast my argument in terms of local reflection principles. Does it extend also to the uniform reflection principle:

$\forall x{\mathrm{Prov}}_{{\text{S}}_a}(⌜\varphi (\dot{x})⌝)\to \forall x\varphi (x) ?$

Or to the global reflection principle, where $\mathrm{Sent}(x)$ means that $x$ is a (code of a) sentence and $\mathrm{Tr}$ is a truth predicate:

$\forall x[\mathrm{Sent}(x)\wedge {\mathrm{Prov}}_{{\text{S}}_a}(x)\to \mathrm{Tr}(x)] ?$

Nothing I have said requires me to either accept or deny ICT applied to uniform or global reflection principles, and I am happy to remain officially neutral. I will, however, note a few complications that would arise in extending the above argument to apply to global and uniform reflection.

Obviously, the global reflection principle only carries any substance when there is a truth theory in the background.²³ Thus, extending the implicit commitment argument to apply to the global reflection principle involves taking a stand in the fraught debate surrounding theories of truth.

A second point about the global reflection principle is that it seems to require more conceptual resources on the part of the believing agent. For an agent to be in a position to know the global reflection principle for $\text{S}$ , they must have the concept of truth. This may or may not be a substantial assumption, depending on the nature of truth. But I have tried to show how it can be avoided by explaining implicit commitment to local reflection principles without appealing to the concept of truth.

On the question of uniform reflection principles: their formulation assumes that every object in the domain has a name (this is a background assumption of the dot-notation). This is of course problematic for most theories other than arithmetic. So while my argument for implicit commitment to local reflection principles is fully general, the potential scope of a similar argument for implicit commitment to uniform reflection principles would be severely restricted. Even in the restricted case of arithmetic, adapting the argument above to show that belief in, say, $\mathrm{PA}$ incurs a commitment to the uniform reflection principles would seem to require the assumption that the believing agent knows that every object in the domain has a name.²⁴ If the agent’s names for numbers are just the usual numerals, this commitment would apparently allow the agent to rule out non-standard models. As with a theory of truth, this may or may not be a substantial assumption. There may or may not be a plausible account of how the agent is able to distinguish standard and non-standard models of arithmetic. I am simply observing that extra complications arise in trying to extend my argument to global or uniform reflection principles.²⁵

3.1. Finitism

With finitism, Dean has in mind the influential analysis of Tait (1981). Tait aims to delimit the functions and number-theoretic statements which are meaningful from the finitist point of view. He argues that the general conception of a function as an arbitrary mapping between objects of a domain is not finitistically meaningful. Rather, we have to ask about the specific mappings that can be finitistically constructed on a specific finitistically acceptable domain. Thus Tait aims to give an account of the functions that can be finitistically constructed on the basis of the finitist view of the natural numbers.

Tait’s account begins from the idea that, according to the finitist, numbers are apprehended as (finite) sequences; the concept Number, then, is the “generic form of a finite sequence” (Tait 1981: 530). Accordingly, the basic operation implicit in the finitist picture of the natural numbers is iteration, as we obtain one finite sequence from another by the mapping $n\mapsto n+1$ . On this basis, Tait argues that the finitistically acceptable functions are exactly the primitive recursive functions. From this claim, Tait then further argues that the finitistically acceptable theorems of arithmetic are precisely those of primitive recursive arithmetic ( $\mathrm{PRA}$ ).

However, the coincidence of primitive recursive arithmetic and finitist arithmetic is not something the finitist can recognize. Having no general conception of a function, the finitist cannot understand primitive recursion as a higher-level operation of functions; they can merely apply the schema of recursion to particular functions that are already given to them. As Tait explains, “For the finitist to recognize the validity of primitive recursive arithmetic, he must recognize the general validity of definition of functions by primitive recursion. But he cannot even formulate this since it involves the notion of function” (1981: 545). Since the finitist is not able to recognize the validity of $\mathrm{PRA}$ , they are also not in a position to justifiably accept the reflection principle ${\mathrm{Prov}}_{\mathrm{PRA}}(⌜\varphi ⌝)\to \varphi$ .²⁸

The point here is that, although the finitist can justifiably accept each axiom of $\mathrm{PRA}$ , there is no means by which the finitist can apprehend the whole of $\mathrm{PRA}$ . Accordingly, there is no fixed description of the axioms and rules of $\mathrm{PRA}$ such that the finitist is in a position to justifiably accept the whole of $\mathrm{PRA}$ under that description. My claim about the implicit commitment to reflection principles, however, applied only when one is in a position to justifiably accept a mathematical theory under a fixed description of its axioms and rules. Thus, Dean’s argument from the epistemic stability of finitism does not undermine the implicit commitment claim as I have formulated it here. Moreover, my formulation of the ICT is independently motivated by the considerations about how we apprehend and accept theories. If I had simply added an extra condition to the ICT to avoid Dean’s counterexample, that would be ad hoc. But I have shown that there is an attractive, well-motivated version of the ICT to which this first counterexample of Dean’s does not apply.

3.2. Isaacson’s Thesis

Let us turn now to Dean’s second case study, Isaacson’s thesis. Isaacson argues for the view that $\mathrm{PA}$ “occupies an intrinsic, conceptually well-defined region of arithmetical truth” (1987: 147), or as he phrases his view later in that paper, “[Peano Arithmetic] is complete with respect to purely arithmetical truth” (1987: 166). In a subsequent paper, the epistemological character of Isaacson’s thesis comes out more clearly, where he says that for a statement to belong to this distinguished region of arithmetical truths and falsehoods—in short, to be ‘arithmetical’—that statement must be expressed in the language of arithmetic and further “that its truth or falsity be perceivable directly on the basis of an articulation of our grasp of the fundamental nature and structure of the natural numbers, or directly from statements which themselves are arithmetical” (Isaacson 1992: 95). This is clearly an epistemological criterion for being arithmetical, in Isaacson’s sense of the word. Furthermore, on this characterization of arithmeticity, Isaacson’s thesis would entail that $\mathrm{PA}$ is epistemically stable in Dean’s sense, because our grasp of the fundamental structure of the natural numbers would justify accepting $\mathrm{PA}$ but nothing further.

Isaacson offers several considerations in favor of the view that $\mathrm{PA}$ occupies a distinguished region of arithmetical truth, drawing on results such as the categoricity of second-order $\mathrm{PA}$ and the bi-interpretability of $\mathrm{PA}$ with the theory of hereditarily finite sets. Of more direct relevance to present concern, however, is why Isaacson thinks that Gödel sentences and reflection principles are excluded from this conceptually distinguished region of arithmetical truth.

Isaacson does not mention local reflection principles in general, but he does discuss the case of the Gödel sentence that expresses its own unprovability. So why does Isaacson think that our grasp of the natural number structure does not by itself justify accepting a Gödel sentence? In considering the possibility of adding a Gödel sentence G as an axiom to $\mathrm{PA}$ , Isaacson (1987: 159) writes:

Such a move would be unnatural. An axiom in this context should be an evident truth, in the terms in which it is expressed. But the truth of this statement, as a statement of arithmetic, is not directly perceivable. $\mathrm{PA}+G$ would not constitute, in this way, a purely arithmetical extension of $\mathrm{PA}$ . The Gödel sentence thus offers an instance of the general thesis of this paper that any axiomatic extension of Peano Arithmetic must be motivated by considerations for establishing its truth which rely essentially on non-arithmetical notions.

What are the ‘non-arithmetical notions’ that are involved in establishing the truth of the Gödel sentence? The answer is clearer in the later paper, where he writes that accepting such sentences would require hidden ‘higher-order’ concepts:

In the case of the Gödel sentence for Peano arithmetic, the hidden concepts are [1] provability in the formal system of Peano arithmetic and, most crucially, [2] consistency of Peano arithmetic. That is, to perceive the truth of the Gödel sentence (presented purely in the first-order language of arithmetic) we must [1] understand that it expresses the condition that this sentence is not provable in this given formal system and [2] see that this formal system is consistent. (Isaacson 1992: 96; numbering added for clarity)

Isaacson’s point here is that we do not perceive the truth of the Gödel sentence $G$ “directly on the basis of an articulation of our grasp of the fundamental nature and structure of the natural numbers, or directly from statements which themselves are arithmetical”, as we would have to for $G$ to count as arithmetical in Isaacson’s sense. Rather, our belief in $G$ relies on our grasp of the concept of provability in $\mathrm{PA}$ and some beliefs about what is provable. We need something beyond a mere grasp of the natural numbers to perceive the truth of $G$ .

Following Isaacson’s analysis of the basis for belief in $G$ , we can say something similar about what is involved in accepting a local reflection principle: to perceive the truth of the reflection principle ${\mathrm{Prov}}_{PA}(⌜\varphi ⌝)\to \varphi$ we need three things. We need to have the abstract concept of proof, we need to be able to grasp the axioms of $\mathrm{PA}$ as a collection of statements we accept, and we need to be able to recognize that ${\mathrm{Prov}}_{PA}$ expresses the abstract concept of proof from the axioms of $\mathrm{PA}$ .

Now, I was at pains to emphasize above that for you to be justified in accepting a reflection principle ${\mathrm{Prov}}_{\text{S}}(⌜\varphi ⌝)\to \varphi$ , the description of the axioms of $\text{S}$ —what I referred to as ${\text{S}}_a$ —must be one that you can recognize as a body of statements you accept. I also argued that if ${\mathrm{Prov}}_{{\text{S}}_a}$ is a canonical provability predicate, then one will be able to recognize that ${\mathrm{Prov}}_{{\text{S}}_a}$ expresses the abstract concept of proof from the axioms of $\text{S}$ . On these points, then, I agree with the Isaacsonian analysis, but this does not undermine ICT as I have defended it.

However, with the claim that we also need the abstract concept of proof, I must admit that the Isaacsonian has a point. This is a relatively sophisticated, high-level concept, and it is possible that one might have a sufficiently developed conception of the natural numbers to justify accepting the axioms of $\mathrm{PA}$ , but not yet have the general, abstract concept of proof. For instance, it is conceivable that a very cognitively advanced animal would grasp the axioms of $\mathrm{PA}$ and be able to perform reasoning with them without having the concept of a proof in general. To this extent, then, Isaacson is right that the local reflection principles for $\mathrm{PA}$ are not justified directly and exclusively on the basis of our grasp of the natural number structure, and hence that local reflection principles are not purely arithmetical in his sense.

Thus I grant that Isaacson’s thesis does undermine the following interpretation of ICT:

Any creature whatsoever who warrantedly accepts a theory $\text{S}$ under a recognizable description of its axioms is thereby in a position to warrantedly accept the local reflection principles for $\text{S}$ .

On the other hand, Isaacson’s thesis does not undermine this interpretation of ICT:

Any creature who has the general concept of proof and who warrantedly accepts a theory $\text{S}$ under a recognizable description of its axioms is thereby in a position to warrantedly accept the local reflection principles for $\text{S}$ .

This is a slightly weaker, but still quite interesting thesis. And humans generally do have the abstract concept of proof. Or, to hedge somewhat, humans are generally capable of acquiring the abstract concept of proof. Acquiring this concept can be a difficult task, as witnessed by the training it takes to impart the concept of a proof to undergraduate math students. On the other hand, the concept is not outside the ken of human cognitive capacities, as witnessed by the fact that it is acquired by thousands of undergraduate math students every year. So we can paraphrase this final version of ICT as saying that any creature like us who recognizably accepts a theory $\text{S}$ is thereby implicitly committed to further sentences not deductively entailed by $\text{S}$ .

Since this weaker thesis has a slightly more restricted range of applicability, it tells us less about the epistemology of mathematics in full generality. But this, I want to suggest, is not much loss. There is certainly interest in studying abstract epistemological questions in full generality. For instance, one might study rationality as such, or one might try to give a fully general conceptual analysis of knowledge. And likewise there is interest in asking about the epistemology of mathematics in full generality, as it applies to all creatures capable of mathematical knowledge. But most epistemological questions are more parochial. For instance, the epistemology of perception is going to depend on what a given creature’s perceptual apparatus is like. What counts as a reasonable failure of logical omniscience for non-ideal agents is going to depend on what the particular non-ideal agents in question are like. Similarly, if we are interested in better understanding our mathematical knowledge, we will focus on the epistemology of mathematics for agents like us. And for that purpose, ICT as restricted to agents like us is all that we really need.

On reflection, it should not be surprising that Isaacson’s thesis says little about the epistemology of actual agents like us. After all, Isaacson’s thesis concerns only what is implicit in a grasp of the fundamental nature and structure of the natural numbers. But any actual agent will bring numerous other concepts and capacities to bear on their mathematical knowledge and there is no reason to expect that those other concepts and capacities cannot be combined with a grasp of the natural numbers in a way that yields new mathematical fruit.