(Probably) not companions in guilt

Abstract

In this paper, I will attempt to develop and defend a common form of intuitive resistance to the companions in guilt argument. I will argue that (contra the companions in guilt argument) one can reasonably believe there are promising solutions to the access problem for mathematical realism that don’t translate to moral realism. In particular, I will suggest that the structuralist project of accounting for mathematical knowledge in terms of some form of logical knowledge offers significant hope of success while no analogous approach offers such hope for moral realism.

1 Introduction

If moral realism is correct, what can explain human accuracy about moral facts? The supposed match between moral facts and our beliefs about them can seem quite mysterious. This worry is sometimes called the ‘access problem’ (for moral realism) and is a popular motivation for moral anti-realism.

One popular response to this challenge is the Companions in Guilt argument, which maintains that accepting mathematical realism generates an exactly parallel access problem. If this analogy holds up, then one cannot reject moral realism (purely) on the basis of the access problem above, while accepting mathematical realism. But, many philosophers who happily take anti-realist positions about morals are loath to do so about mathematics.

I will suggest that a justification for this differential treatment can be found in the following difference between mathematical and moral investigation. Mathematicians (and most philosophers of a realist persuasion) are willing to accept that investigations of alternate logically coherent mathematical structures would yield knowledge of mathematical truths of equal metaphysical status. For example, mathematicians could decide to investigate the natural numbers under plus and times except for 17 (though such a structure is unlikely to be interesting).¹ In contrast, moral realists hold that almost no alternate practice of deciding what behaviors count as “permissible” would yield true beliefs about some other concept of equal metaphysical status (and the same holds for other moral vocabulary). Baring the success of a very controversial Kantian program, this implies that very few logically coherent moral practices would yield truths of equal status. Thus, there seems to be a close relationship between mathematics and logic (specifically logical coherence) which makes reducing access worries about mathematics to access worries about logical coherence a promising project, in a way that the analogous reduction of moral facts to logical coherence facts it is not.

The strategy of trying to explain mathematics via ‘logic’ and thereby address access worries, goes back to Frege (1980) and Hilbert′s program Zach (2016), but aggressive claims in this area have left something of a philosophical stain on this approach. However, I will suggest that an appropriately modest (and modern) version of this strategy actually fares quite well and need not be in tension with metamathematical results that posed problems for previous proposals (e.g., Gödelian proof transcendence), and the philosophical insights derived from them. In particular, we will see that a range of popular contemporary philosophies of mathematics (within what I will call the structuralist consensus) reconcile realism about mathematics (including proof transcendent mathematical facts and even, in many cases, the literal existence of mathematical objects) with the idea that almost any coherent mathematical structure can be posited.

I will then delineate a strategy for solving the residual access problem for knowledge of logical coherence that I believe to be quite promising. In so doing, I hope to show that, plausibly, the mathematical realist can explain access to mathematical facts through access to facts about logical coherence, while the moral realist cannot successfully employ an analogous strategy.

Admittedly, this point alone does not suffice to show that the access problem for moral realism is (ultimately) worse than that for mathematical realism. For maybe there is some other, even better, strategy for addressing access worries about morals (unrelated to exploiting a connection to logic) which has no analog for mathematics. Or maybe current appearances regarding the prospects for Kantianism about morals and/or the ‘structuralist consensus’ regarding mathematics are deceptive. But it does suffice to explain and defend the current position of many philosophers, who accept mathematical realism while rejecting moral realism on the basis of access worries. Contrary to the Companions in Guilt argument no hypocrisy is needed to take this position, for there are credible lines of attack on the mathematical access problem which appear (at least in light of the current development of relevant philosophical research programs) to be much less promising when applied in the moral domain.²

2 Setting up the question

Let me begin by clarifying how I will understand the Companions in Guilt thesis and what it means for a theory to face an access problem.

Mackie introduced the idea of a ‘Companions in Guilt’ argument in (1977) pg 39, as a possible response to his argument from queerness, which contends that moral realist facts would have to be somehow deeply (and implausibly) metaphysically or epistemologically different from all other kinds of facts which we reasonably accept. In this paper, I will be considering the more recent formulation of the Companions in Guilt argument given in Clarke-Doane (2012), which appeals to a comparison with mathematics to diffuse access worries about moral realism. So let me now clarify what I mean by ‘an access problem’, and what it means for one theory’s access problem to be worse than another’s.

In Realism, Mathematics and Modality Field (1989) influentially proposes that we should think of the access problem for (mathematical) realists as arising from a challenge for the realist to “explain how our beliefs about [mathematical objects] can so well reflect the facts about them” in some internally coherent fashion. More specifically, Field demands that we explain the truth of ‘reliably, if mathematicians believe that $\phi$ then $\phi$’, for various mathematical statements $\phi$. And he notes that, “[I]f it appears in principle impossible to explain this, then that tends to undermine ... belief in mathematical entities, despite whatever reason we might have for believing in them.” It has been very popular to construe access worries about morals, metaphysical possibility, mathematical truth-value realism, aesthetics etc. analogously. On this view, access worries provide us ceterus paribus reason for rejecting a theory (in this case realism about mathematical objects), by appealing to a kind of cumulative gestalt impression that no adequate answer to this explanatory demand seems possible.

Accordingly, I will say that the access worry for realism about a given domain is worse, to the extent that there seems to be more reason to think a satisfying explanation of human accuracy about that domain is impossible. And we can think of Companions in Guilt arguments as saying that the Moral Realists’ access problem is no worse than the Mathematical Realists’ access problem because any roadblocks to explaining access to morals also apply to the mathematical case.

In this paper I will attack the above companions in guilt argument by providing (via a simplified model) an example of an attractive mechanism which could explain our accuracy about realist mathematics, and arguing that no comparably attractive analog to this mechanism can be used to explain our accuracy about moral realist facts.³

3 Apparent contrast between moral and mathematical realism

Now let us turn to the contrast between moral and mathematical realism. The term ‘realism’ has infamously been used in many different ways by many different philosophers (Wright 1993). With this in mind, I will lay out how I will use the terms moral and mathematical realism, attempting to evoke concepts typically at issue in Companions in Guilt arguments.⁴

3.1 Mathematical realism

By mathematical realism, I mean what is sometimes called truthvalue realism, i.e., the idea that there are definite right answers to certain mathematical questions, whether we can ever discover them or not, and which we are not free to stipulate.⁵

For instance, I take mathematical realism to entail that every first order sentence in the language of arithmetic (e.g., the claim there are infinitely many twin primes) is either true or false, even Gödelian statements whose truth isn’t provable from our current axioms. Even if we learned that for some such sentence P, neither P nor $\lnot P$ is provable from axioms we accept, we would not be free to toss a coin and extend our current number theoretic practice by adding P if we got heads and $\lnot P$ if we got tails. Something in our current conception of the numbers (e.g., our expectation that if the numbers exist they satisfy $PA_2$), already suffices to ensure that only one of these options expresses a truth.⁶

In contrast, this mathematical realism does not limit which coherent putative structures are proper objects of study. Indeed, contemporary mathematical practice seems to push in the opposite direction. For example, see Kitcher on the history of the complex numbers,⁷ Coles’s autobiographical remarks about mathematicians’ apparent freedom to introduce new structures⁸ and Lockhart’s comments about mathematical creativity.⁹

In this paper I will consider what people who accept mathematical realism (in the sense above) can say about access worries. But perhaps I should stress that I do not purport to argue for mathematical realism here. I simply evaluate whether the popular position of accepting mathematical realism while rejecting moral realism on the basis of access worries is coherent.

3.2 Moral realism

In contrast, moral realism of the kind which has raised intuitive access worries seems to require something more than mere truth-value realism. Many philosophers (like moral projectivists and sentimentalists Hume 2000) have been quite willing to judge moral claims as either true or false without (in a sense) taking them to be ‘about’ anything more than social norms, sentiments etc. The property that is both responsible for moral realism’s access problem and the intuitive sense that it refers to what’s ‘really’ moral is sometimes referred to as mind-independence¹⁰ (Street 2006). But spelling out just what it means to be mind-independent in this sense is notoriously fraught (e.g., facts about human psychology and beliefs would seem to be ‘mind-dependent’ in a way that doesn’t prevent them from being objective and separate from our recognition practices). Luckily however, we don’t need the full power of this notion to raise strong access worries for moral realism. The following disagreement thesis is sufficient to illustrate a substantial dissimilarity with mathematical realism.

The intuition behind the disagreement thesis is that our moral knowledge is more than a mere elaboration of definitional freedom and/or exploration of proof-transcendent facts about logical coherence and consequence. A powerful moral realist intuition holds that which laws relating moral to descriptive facts express truths should reflect something deeper than mere contingencies about what kinds of things human beings tend to feel positive emotions towards, or how we have chosen to use our words. It maintains that people with different morality-like practices (people who had analogous tendencies to pursue, repent, blame etc tied to descriptive facts in a different way) would be wrong about morality rather than right about something of equal metaphysical interest.¹¹ Accordingly, for the purposes of this paper, I will take moral realism to require accepting the following disagreement thesis (or an analogous principled formulated by replacing permissibility with the primitive of your choice):

Disagreement Thesis: Necessarily, if two thinkers (or communities) appear to be disagreeing about the permissibility of some action (where all parties agree on the usual action guiding role of permissibility)¹² there is a single proposition which they are disagreeing about.¹³, ¹⁴

Such speakers aren’t like the protagonists in Frege’s example of two people arguing about whether a coin is heavy where each is talking about the coin in his own pocket.

Projectivists about morality would reject this claim. For, on the projectivist picture the English language term “permissible”(rigidly) applies to the kind of things that bear a certain relationship to our actual moral sentiments, just like ‘edible’ refers to those actions which bear a certain relationship to what we can digest. So beings who had different moral sentiments and/or inclinations to admire praise and blame,¹⁵ and correspondingly different judgments about what is “permissible”, would also count as expressing truths about a different concept PERMISSIBILITY*, with equal metaphysical status to our own.

3.3 Contrasting relationships to logical coherence

Now let’s turn to the access worries faced by moral and mathematical realists, and the plausiblity of appealing to some kind of general logical accuracy to solve them.¹⁶

Moral realists face an access problem to the extent that they are unable to explain how our beliefs about morality (such as what actions are permissible) so well match the truth about morality. They must explain why we label just the right descriptive class of actions as permissible¹⁷ since, per the Disagreement Thesis, any (substantially different) alternate practice would be wrong about PERMISSIBILITY rather than right about some other concept.

At first glance, it might seem mathematical realists face the same challenge. After all, they must explain why our beliefs about arithmetic sentences are true rather than false, even though there are logically coherent structures in which these sentences turn out to have a different truth values. For example, there is (on the, presumably true, assumption that PA is consistent) a model of the integers satisfying the first order axioms of arithmetic (PA) in which Con(PA) turns out to be false. Thus, it might seem that the mathematical realist’s position regarding our beliefs about arithmetic is very similar to the moral realist’s position regarding our beliefs about permissibility.

However, unlike the moral realist, the mathematical realist is free to appeal to our choice of topic in this explanation and say that any that logically coherent¹⁸ variant on our arithmetical practice would have expressed a truth. Specifically, the mathematical realist can appeal to the fact that we take the numbers to satisfy $PA_2$ (a categorical second order axiomatization of the natural numbers)¹⁹ to specify the structure under investigation in a way that determines the truth or falsity of all arithmetical sentences. Importantly, the mathematical realist need not explain why we decided to investigate $PA_2$ rather than some other structure. They can say that any logically coherent description of structures for pure mathematical investigation would succeed (see appendix A for an important caveat about applied mathematics), but given that we accept $PA_2$ there are definite right answers to all questions which are logically necessitated by $PA_2$.

This is a critical difference. While the mathematical realist only needs to explain how we came to accept a some logically coherent characterization of ‘the numbers’ and derive our beliefs from that characterization, the moral realist must explain not only how we work out the consequences of some particular permissibility-adjacent concept but why, of all the logically coherent permissibility-like concepts,²⁰ we choose the right one.

This doesn’t make the access problem trivial for the mathematical realist. That is, one might worry that our ability to postulate coherent, rather than incoherent, structures (and in some cases our ability to recognize that stipulations are categorical)²¹ itself raises an access problem.

Also, as such a categorical description must necessarily be non-first-order, the mathematical realist will need to invoke a notion of logical coherence and consequence for non-first order descriptions of states of affairs. This requires that we embrace a notion of logical coherence which is distinct from the mere inability to derive a contradiction in some (computably specified) deductive system.²² But embracing such a powerful syntax transcendent notion of logic is fairly popular and independently motivated.²³

This brings us to the following (crude) example of the apparent contrast between the relationship of mathematics and morals’ to logic:

• The mathematical realist can (and usually does) say: nearly all (conservative)²⁴ logically coherent (in the sense above) variants on our number practices would express a truth of equal metaphysical status to the one we currently express—if perhaps with different pragmatic usefulness or aesthetic interest.

• The moral realist will say²⁵: almost no (conservative) logically coherent variants on our permissibility practices would express a truth of equal metaphysical status to the ones we currently express.

3.4 Support from the philosophy of mathematics literature

Further support for the idea that realist mathematical accuracy could be (somehow) attractively explained in terms of mere accuracy about logical coherence is provided by the existence of a number of contemporary philosophical views which converge on this point. These views (forming, what we might call, the structuralist consensus) are committed to a close relationship between logical coherence and mathematical fact. Such views endorse the above mathematical realist idea that there are definite proof transcendent right answers to mathematical questions, while taking mathematics to be ‘the science of structure’, and maintain that any (or nearly any) choice of new mathematical structures coherently extending one’s current mathematical practice would succeed.²⁶ I have in mind views like classic Set Theoretic Foundationalism and other truth-value realist forms of Plenetudinous Platonism, Neo-Fregeianism, Hellman (1994)′s modal structuralism, Shapiro (1997)′s ante rem structuralism, Quantifier Variance fueled neo-Carnapian realism about mathematical objects, and Fictionalism.²⁷, ²⁸

As these views allow any coherent mathematical structure²⁹ to be posited, they transform an explanation of our access to logical coherence facts into an explanation of our access to mathematical facts. For if we take our ability to postulate logically coherent, rather than incoherent, mathematical structures for granted, mathematical knowledge flows simply from application of logical inferences (which are themselves expressible as claims about coherence)³⁰.

To see how this plays out in more detail, remember that resolving Field’s access problem requires explaining why ‘if mathematicians believe that p then p’ holds reliably (where p ranges over mathematical claims). Now any explanation of our access to logical coherence facts, i.e., why reliably (descriptions of) structures we believe to be coherent (incoherent) are actually coherent (incoherent), gives us an explanation of why mathematicians only study coherent posits. By definition, views in the Modal Structuralist consensus imply that one can express truths by adopting any³¹ coherent mathematical posit and deriving logical consequences from it.³² So this suffices to explain the desired reliability claim.³³

For example, Plenitudinous Platonists and (many) Neo-Fregeans think that nearly all coherent mathematical descriptions D which mathematicians are likely to consider will truly describe some portion of the mathematical universe (and that all our further purely mathematical knowledge can be gained by recognizing the logical consequences of D). Ante rem Structualists like Shapiro believe that there are a wide range of special abstract objects, called structures, corresponding to different coherent descriptions, like the second order description of the natural numbers $PA_{2}$ mentioned above.³⁴

Modal Structuralists like Hellman take knowledge of pure mathematics to just be knowledge of logical coherence claims: knowledge that it is coherent for objects to satisfy some mathematical description and logically necessary³⁵ for objects satisfying this description to also satisfy some sentence $\phi$. For example, if $PA_{2}$ is a sentence in second order logic which uniquely describes the the natural numbers and $\phi$ is a sentence in the language of arithmetic, then the modal structuralist can render the intended meaning of $\phi$ as ‘it is logically possible that $PA_2$ and logically necessary that if $PA_2$ then $\phi$’. Similarly, Fictionalists say that mathematical existence statements may all be literally false, but take the correctness conditions for asserting $\phi$ in the fiction to be essentially the same as the truth conditions given by Hellman.³⁶

Now one might object that mathematicians don’t make explicit stipulations introducing new mathematical structures, or accept any view within the Structuralist Consensus for use in reasoning about which mathematical stipulations can succeed. Thus, one can’t appeal to the idea of mathematicians explicitly reasoning as above to answer access worries. However, (I claim that) if the structuralist consensus is true, then then mathematicians plausibly gain true beliefs by introducing mathematical structures in a way that is endorsed the structuralist consensus, even though they don’t accept any particular theory about why introducing mathematical structures in the way that they do is acceptable.

As I highlighted in Sect. 3.1, contemporary mathematical practice seems to allow mathematicians significant freedom to introduce new kinds of mathematical objects, such as complex numbers, sets and the objects and arrows of category theory. If this process is actually reliable (as per the structuralist consensus), then it seems plausible that mathematicians have defeasable default warrant (of the kind advocated in Field 2005 and Berry 2013) for using it, and can gain knowledge from it without having an explanation for why it’s reliable, much as with other belief forming methods such as sight or logical deduction.³⁷ (Though maybe if mathematicians do get worried they have to answer access worries, but can do so by accepting the philosophical arguments I have proposed).

4 The access problem for logical coherence

4.1 The question

So far I’ve tried to motivate the idea that one can plausibly explain our accuracy about mathematics in terms of accuracy about (a suitably powerful notion of) logical coherence. But does this help with access worries?

Many philosophers have felt that (some kind of) ‘logical knowledge’ is somehow specially unproblematic or immune to access worries. But, it does not follow from this that the kind of knowledge needed to explain mathematical accuracy counts as logical knowledge in the relevant sense.³⁸ Thus, we can’t simply rely on these intuitions to address residual access worries about the kind of logical knowledge needed to explain our mathematical accuracy.³⁹

One strategy for addressing this worry has been to conceptually analyze the notion of logic.⁴⁰ Happily however, we don’t need to settle this vexed question to evaluate companions in guilt arguments. For our purposes it doesn’t matter whether modal logic, second order logic, or logical coherence are really logic. What’s needed for our argument is just for there to be some notion of logical coherence/knowledge which does the needed work, i.e., some notion such that one can plausibly explain accuracy about mathematics largely in terms of accuracy about it and its access problem looks relatively tractable.

The rest of this section will be devoted to suggesting that access worries about logical coherence (suitably understood) are tractable. I will defend the intuition that our knowledge of facts about logical coherence (even the logical coherence of second order statements) creates less of an access problem than moral realist knowledge initially appear to, by discussing some promising mechanisms for explaining our accuracy about logical coherence.

4.2 Some mechanisms and a toy model

To appreciate the scope of what needs to be explained, note that merely using the correct introduction and elimination rules for first order logic does not allow one to recognize the positive fact that a scenario is logically coherent. For example, first order introduction and elimination rules don’t allow one to recognize that it would be logically coherent for there to be two distinct things $(\exists x) (\exists y) \lnot x=y$.⁴¹

So, imagine people who begin with the ability to use standard first order language to describe their macroscopic physical environment (something which people on all sides of companions in guilt worries are willing to take for granted). Further suppose that they have some concept of scenarios being logically coherent or not, i.e., possible or impossible with regard to ‘the most general constraints on how any objects can be related by any relations’⁴², ⁴³ but have little or no knowledge that anything is logically possible or impossible. How could they non-miraculously acquire sufficiently powerful methods of reasoning about logical possibility to vindicate contemporary mathematics?

I want to suggest a story along the following lines. As inquirers, we attempt to predict and explain the behavior of concrete objects. There are more and less economical ways of doing so.⁴⁴ When we are dealing with sufficiently diverse and plentiful collections of concrete objects, the most economical explanations for regularities may well appeal to a combination of general principles which constrain how any objects can be related by any relations, and specific physical or metaphysical laws whose application is restricted to certain particular kinds of objects or relations.

I will suggest that pressure to efficiently predict what is practically possible in situations of evolutionary interest can help explain how creatures like us could have gotten correct methods of reasoning about logical possibility. One can think about our general methods of reasoning about logical possibility as being susceptible to improvement and correction by the world in three ways.

On the one hand, pressure to acknowledge facts about concreta encourages us to accept the logical coherence of certain things. For example, imagine that you aren’t sure whether the first order state of affairs described by some mathematical hypothesis involving relations $P$, $Q$, and $R$ is logically possible. If I then point out that the relations of friendship, nephew-hood and having been in military service together apply in just this way to the royal family of Sweden, this will cause you to accept that the scenario in question is logically possible.

On the other hand, our need to elegantly explain regularities in the world creates pressure to conclude certain states of affairs are logically incoherent. Suppose, for example, that someone thought it was logically coherent for $9$ items to differ from one another in which of three properties they had, e.g., for $9$ people to choose different combinations of sundae toppings from a sundae bar containing three toppings. This person would have to explain the striking law-like regularity that, regardless of the type of items and properties in question, we never wind up observing more than $8$ such items. They might postulate new physical regularities to explain why apparently random processes of flipping three coins never generated the forbidden $9$th possible outcome. However, this explanation (or some analogous one) would have to apply at every physical scale we can observe, from relationships between the tiniest particles to relationships between planets and stars (as well as to less concrete objects like poems and countries). A much more elegant explanation is that the unrealized outcome isn’t logically coherent. Recognizing that the forbidden 9th outcome is forbidden in all possible domains is much more efficient than hypothesizing separate laws prohibiting it in each specific situation (and thus there is pressure to do so).

Accordingly, we can think of facts about what’s actual as simultaneously a useful source of data about what’s logically possible, physically possible, chemically possible, etc. We try to efficiently predict what will happen by patching together laws with different levels of generality. Though we face an in principle choice about whether to explain any specific regularity in terms of logical necessity vs. physical law, metaphysical necessity or mere ceterus paribus regularity, patterns in our experience can still motivate attributing a noted regularity to logical necessity rather than physical law. For, as noted in the case above, if the right explanation for some regularity is that it holds as a matter of logical necessity, we should expect to see that all substitution instances of it (i.e., all sentences with the same logical structure) are true, whereas we would expect the opposite if some principle holds as a matter of merely metaphysical necessity or physical necessity.

Finally, one should note that the pressures mentioned above don’t exist in isolation. Rather, the resulting beliefs (and inference methods) will be further corrected by interaction with one another. If one accepts the above story about how we could have gotten some initial ‘data points’ about logical possibility from our knowledge of the concrete world, one can then appeal to familiar processes of reflecting on our beliefs and recognizing when they conflict or cohere with one another to explain further improvements in our accuracy.

Once some methods of reasoning come to strike us as initially attractive via the two mechanisms above, we can arrive at new more powerful laws (just as we do in the sciences) by considering how they unify and explain these methods of reasoning. For example, in mathematics we can reliably add new axioms by choosing principles which unify and explain the mathematical beliefs which we already have Koellner (2010). As Gödel puts it, “There might exist axioms so abundant in their verifiable consequences, shedding so much light upon a whole field, and yielding such powerful methods for solving problems... that, no matter whether or not they are intrinsically necessary, they would have to be accepted at least in the same sense as any well-established physical theory” (Gödel 1947). If this is true, then it also seems plausible that one could reliably expand an initial collection of good methods of reasoning about logical coherence in the same way. Moreover, when we make incorrect generalizations these can be corrected by coming into conflict with well-entrenched and concretely motivated general principles.

Note that the kind of elegant generalization which we see in the sciences (and which I want to invoke) goes beyond simple inferences like ‘the sun rose every day for the past billion years, so it will rise tomorrow.’ It can include seemingly astonishing leaps, like when astronomers go from observations of points of light in the night sky to a whole model of how the planets are arranged.

Also note that our deployment of these general principles can involve thinking about how it would be logically possible for objects satisfying one description, e.g., $PA_2$, to exist within a larger universe—and that such reasoning can lead one to new conclusions about the original structure. For example, general principles might cause you to believe that it is coherent for there to be a copy of the natural numbers inside of a structure satisfying $ZFC_2$ and, by applying generally valid inference methods (developed as above) to this structure, thereby conclude that the natural numbers satisfy the Paris-Harrington principle (Kaye 1991). Since there are models of the first order version of $PA_2$ in which the Paris-Harrington theorem is false, such reasoning can be thought of as working out the second order consequences of $PA_2$.

Once we have good methods of reasoning about logical coherence we can use these to recognize logically coherent (putative) mathematical structures.

Admittedly a tiny issue remains. The examples of initial data used above all involved first order states of affairs, and thus plausibly generated good principles for reasoning about which such states of affairs are logically possible. But what about modal knowledge involving second order claims—which we need to account for knowledge of the logical coherence of second order states of affairs like $PA_2$ as well?

If we could presume some initial knowledge about (actual) concrete second order states of affairs, then we could feed it in to the generalization mechanism above to get good methods of reasoning about which second order states of affairs are logically possible. One might say that we get pressure to recognize that there is (something like) a second order collection X of some first order objects (say, the cats) satisfying $\phi$ from noticing that the white cats $\phi$, or that it is physically possible to paint some cats white so that the white cats would satisfy $\phi$. And we can get pressure to recognize that there isn’t any second order collection X of some objects which satisfies $\psi$, by way of this being the most elegant explanation for why it is physically impossible to paint some of the cats white so that the white cats $\psi$.

Developing this idea is not trivial.⁴⁵ However, since my goal here is just to argue that (contra companions in guilt arguments) one can rationally think the ‘structuralist’ program of reducing mathematical access worries to access worries about logical possibility and then solving the latter is more promising than Kantianism rather than to complete this program, I won’t say more about these issues here.

5 Objections

5.1 Logical coherence and large collections

I will conclude by considering some objections.

First, one might worry that the above story can’t account for our apparent knowledge of facts about logical coherence (and necessity) involving large infinite collections.

An objector might allow that one can explain our accuracy in reasoning about countably infinite collections as above. But capturing intuitively correct truth conditions for statements of set theory (via the structuralist consensus) requires evaluating claims about the logical coherence of scenarios involving uncountably many objects. Thus, one might worry that principles of reasoning which are shaped to elegantly predict and explain what is logically coherent for finite and countably infinite collections cannot account for the degree of logical (and hence mathematical) knowledge which we actually have.

A critic might advance the following analogy: saying that elegant generalization from facts about finite and countable collections (such as physical objects and segments of space) yields principles which accurately describe what is logically coherent for some of the larger collections considered in pure mathematics is like saying that inference to the best explanation plus observations of birds in New Mexico explains our possession of true beliefs about birds in Canada as well. Presumably, in the ornithological case, we need to go gather more data in order to get many true beliefs about birds in Canada. But, in the mathematical case, we can’t gather more data. Thus, our apparent possession of substantial true beliefs about what is logically coherent for larger infinite collections remains mysterious on the story I have sketched above.

I want to respond to this worry by accepting the analogy about birds above, and saying that it fits the current state of human knowledge with regard to facts about the higher infinite rather well. Even in the case of birds, we can arrive at some true beliefs about birds in Canada just by inference to the best explanation from the facts about the birds in New Mexico. If we discovered tomorrow that some new island which had never yet been visited by explorers contained birds, I think we would reasonably expect many facts to carry over: any birds on that island would breathe oxygen, that they would have hollow bones etc. Our expectations about birds on this island would just be more sparse and less confident than our beliefs about birds in locations that we have observed.

But, this is just what happens with our beliefs about logical coherence and large collections: as one moves from logical coherence facts concerning finite collections to those concerning countably infinite collections (like the natural numbers), and then uncountable collections (like the sets) our beliefs do get more sparse and less confident. For example, the continuum hypothesis⁴⁶ (CH) is a fairly simple statement involving sets of (relatively) small infinite size, yet it is known that both the truth and the falsity of CH are compatible with ZFC. Our beliefs about what large infinite collections of objects and relations are logically coherent are also frequently less confident than our beliefs about what finite and countable collections of objects are logically coherent. Sociologically, mathematicians are frequently much more confident in their claims about numbers, sets of numbers and sets of sets of numbers than in claims about higher set theory.⁴⁷

Thus, I think this last worry points to something that is an attractive feature rather than a flaw of the account at hand: it explains why we have relatively sparse beliefs about what’s logically possible with respect to large collections, and hence relatively sparse beliefs about the corresponding facts concerning higher set theory.

5.2 Contrast with Quine

There is obviously a certain similarity between my proposal and Quine’s famous empiricism about mathematics (if the latter is read as an answer to access worries as well as a claim about what justifies mathematical beliefs). For example, we both invoke dealings with concreta as part of a (potential) explanation for human accuracy about mathematics. So one might wonder whether existing (fairly persuasive) objections to Quine’s proposal also apply to my proposal. I will now attempt to answer this question by reviewing some important ways in which my proposal differs from Quine’s and how those differences allow me to avoid some of the most troubling objections to Quine’s approach.⁴⁸

First, where Quine’s proposal takes dealings with concrete objects to push us to recognize the existence of the particular mathematical structures which we use in the sciences, my story takes dealings with concrete objects to push us to accept correct general inference methods which can be used to derive the logical possibility of the structures we use in mathematics. Because my story makes the relationship between scientific and mathematical beliefs indirect in this way, it naturally avoids the ‘problem of recreational mathematics’ that besets Quine, i.e., the fact that we seem to know things about mathematical objects which are scientifically useless (like sets in the higher reaches of set theory).

My story also allows for the fact that (as emphasized by Friedman 2001) even in cases where mathematical structures do get quantified over in physical theories, mathematicians appear to acquire significant knowledge of these mathematical objects before a use is found for them in physics. And it makes good sense of the apparent cavalierness of both physicists and mathematicians with regard to positing new mathematical structures.⁴⁹ For one can say that (in such cases) mathematicians and physicists are usually already convinced of general methods of reasoning which let them derive the logical possibility of suitable structures (due to prior experiences and perhaps selection on an evolutionary time scale), and, if one of the views in the structuralist consensus is true, this is enough for them to correctly⁵⁰ use such a structure.

Second, where Quine’s story appeals to continuing indispensability mine appeals to past usefulness. If (as Field argues in the case of Newtonian Mechanics Field 1980) all quantification over mathematical structures in physics is ultimately dispensable, this would be a problem for Quine’s empiricism but not for my proposal. All that is necessary for my story to work is that recognizing the logical possibility or impossibility of various claimed patterns of relationships between concrete objects was practically useful at whatever time our dispositions to reason about logical possibility were formed. Third, while Quine says mathematical knowledge is empirical, my explanatory story is entirely compatible with mathematical knowledge being a priori.⁵¹

5.3 The Kantian program

Finally, one might accept that the answer to access worries about mathematics sketched above looks promising, but maintain that an analogous approach to moral knowledge looks equally promising.

I’ve suggested that if we employ a rich notion of logical coherence (treating second order quantifiers as logical vocabulary and letting coherence require more than mere syntactic consistency) then it is plausible that mathematical accuracy can be explained in terms of logical accuracy—in a way that helps with access worries.

But an objector might point out that an exactly analogous proposal for morals has been tried by Kant and his followers—and argue that this proposal looks equally promising. One can think about Kant’s appeal to the categorical imperative as a way to explain our capacity for moral accuracy by appeal to something like logical coherence (practical rationality/coherence in one′s maxims). And one can see Kant-influenced contemporary philosophy of action’s many attempts to move beyond the desire satisfaction model of rationality as endeavoring to enrich and develop the relevant notion of logical/practical coherence in a way that would let this explanatory strategy succeed.⁵²

Admittedly, the kind of practical coherence the Kantian takes to be sufficient for moral accuracy is not logical coherence (and certainly not the kind of logical coherence discussed in Sect. 4.2). But the defender of moral realism might try to replace logical coherence with something else, like Scanlonian reasons for action, and tell a similar story. However, for that approach to succeed, they must offer a promising account of how pressure to get practical reasoning right could give us something like a general ability to detect Scanlonian reasons to the extent necessary to explain accuracy about morals.

I admit that if this loosely Kantian approach of reducing moral knowledge to something like logical knowledge⁵³ and the structuralist program of so reducing mathematical knowledge looked equally promising, the companions in guilt argument would be defensible. However, I don’t see any powerful argument that these two philosophical research programs are equally promising.

First, Kant’s idea that morality can be reduced to a kind of logical coherence is (already) wildly controversial, much more so than the idea that something like one of the structuralist consensus approaches discussed above could work. The plain fact is that, as discussed in Sect. 3.3 above, it sure looks like an artificial intelligence bent on maximizing the number of paper clips in the universe or Hume’s sensible knave could be logically coherent but wrong about realist morals. And the history of Kantian research since 1781 hasn’t done much to shift this impression.

Second, note that even if we accepted such a reduction, not every way of explaining morality in terms of something like logical accuracy would be helpful in addressing access worries. Even if this first part of the Kantian program succeeds, you can’t just expand your notion of logic arbitrarily without losing the intuition that ‘the access problem for logic’ is tractable. For example, consider a Scanlonian picture where reasons to do things (including believe things, treat people in certain ways, reject certain kinds of treatment etc.) are treated as fundamental (Scanlon 2014).⁵⁴ Suppose that you say that a general faculty for ‘detecting reasons’ plays a similar role in explaining our moral knowledge to the one which a faculty for reasoning about logical coherence plays in explaining our mathematical knowledge. Even if some version of Scanlon’s idea can succeed in explaining accuracy about morals by appeal to such a faculty, this would not suffice to banish access worries.

For a question remains about how we could have gotten a faculty which detects the (objective, realist, disagreement-criterion satisfying) reasons for rejecting certain kinds of treatment and policies which Scanlon posits. There might be pressure on someone to evaluate reasons to believe in a way that tends to produce true descriptive beliefs in the evolutionary (or meme-evolutionary) environment. And there might be pressure towards accurate means-ends reasoning, ‘If there is reason for me to $\phi$, and the only way for me to $\phi$ is to $\psi$, then I should $\psi$’.

But why should selection for good reason detecting faculties (in the sense above) also give us faculties that correctly detect Scanlon’s supposed reasons for treating people a certain way? This seems prima facie far more mysterious than the idea that, in getting that rules which elegantly predict what statements are logically coherent for small collections, we should also get rules which also tend to yield true verdicts about large collections (insofar as they say anything about these collections). It is far from clear that, in getting a reasons-detecting faculty which is good for means-ends rationality, we would also be likely to get one that tracks reference magnetic facts about permissibility.⁵⁵

6 Conclusion

In this paper I have attempted to defend the position that access worries for mathematics are less severe than access worries for morals – at least relative to the current state of philosophical play. Specifically, I have argued that certain core intuitions necessary for something to count as moral/mathematical realism make it far easier to say that nearly all alternative ‘logically coherent’ mathematical practices would have been right about something else, than it would be to take an analogous position on alternative logically coherent moral practices.

Thus, there is an attractive strategy for answering access worries about mathematical realism which seems to have no (comparably plausible) analog for defending moral realism. Admittedly, this point alone does not suffice to show that the access problem for mathematics is worse than that for morals; maybe current appearances regarding the prospects for Kantianism in morals and/or the ‘structuralist consensus’ regarding mathematics are wrong. Or maybe there is some other, even better, strategy for addressing access worries about morals (unrelated to exploiting a connection to logic) which has no analog for mathematics.

But it does defend the current position of many philosophers, who accept mathematical realism while rejecting moral realism on the basis of access worries. Contrary to the companions in guilt argument, no hypocrisy is needed to take different positions on moral and mathematical realism in this way. One only needs to combine a fairly common assessment of the of the relative health of two different philosophical research programs (Kantianism about ethics and the structuralist consensus about mathematics) with pessimism about finding some radically different solution to access worries for realists about morality.

The problem of logically coherent versus ‘conservative’ stipulations

A well known (Field 1989; Uzquiano 2006; Eklund 2009) issue for all the ways of reducing access worries about mathematics to access worries about logic discussed above, is that (separately) coherent posits which attempt to introduce new abstract objects can conflict with one another, or imply implausible constraints on the behavior of other objects not mentioned in these posits.

For example, posits saying (in the usual Fregean way) that there are exactly four things and exactly five things in the universe are both logically coherent, but not compatible with one another. Also, if either of them were taken to be true, this would impose constraints on the size of the non-mathematical universe in a way that we expect mathematics not to do.

Now, this problem is not so big as it initially seems. For we can easily reconcile these facts with the idea that knowledge of logical coherence is all we need to get pure mathematics right, by saying that what needs to be coherent are total bundles of mathematical stipulations (characterizing the various mathematical structures we are talking in terms of) and saying that stipulations characterizing pure mathematical structures must have their quantifiers restricted to some collection of mathematical objects (so that they cannot imply any logically contingent constraints on the non-mathematical objects).

For, it is already common mathematical practice to describe structures like the natural and real numbers via principles whose quantifiers are (implicitly) restricted to the mathematical structure being defined by these principles (or to this structure plus some old mathematical structure which it is supposed to be extending). For example Peano Arithmetic is often formulated to include the claim that everything has a successor, by people who would actively deny that (say) the color blue or the Great Barrier Reef has a successor. When we restrict our attention to logically possible mathematical postulates which (like $PA_2$) are quantifier restricted to the structure you are stipulatively introducing, mere logical coherence does ensure acceptability in this sense. For it will always be logically possible to supplement whatever objects you are currently talking in terms of with a model of this logically coherent system.

Admittedly, when philosophically investigating applied mathematics, we often want to appeal to impure mathematical structures (like sets with ur-elements or functions from physical object to numbers) so we will need to allow characterizations of mathematical structures that quantify over (at least some) non-mathematical objects. Some options for a notion of conservativity (which prevents impure mathematical objects from inappropriately constraining the possibilities for the non-mathematical world) have been considered in the recent-ish literature - such as Field (1989)’s story about which mathematical fictions are acceptable, and Rayo (2013)’s development of this story in an ontologically realist but meta-ontologically deflationist spirit.⁵⁶ Adopting any of these proposals lets us retain the needed claim that the knowledge needed to make acceptable mathematical postulates is still basically logical knowledge. Accordingly, I ask readers to take my talk of ‘logically coherent extensions’ of our mathematical practice above to refer to ones that are either quantifier restricted to mathematical objects, or at least satisfy (the right way of filling in) the conservativity requirement above.