Coincidence Avoidance and Formula1ng the Access Problem Sharon E. Berry <seberry@invariant.org> In this paper, I discuss a trivializa:on worry for Hartry Field's official formula:on of the access problem for mathema:cal realists, which was pointed out by Øystein Linnebo (and has recently been made much of by Jus:n Clarke-Doane). I argue that various aMempted reformula:ons of the Benacerraf problem fail to block trivializa:on, but that access worriers can beMer defend themselves by s:cking closer to Hartry Field's ini:al informal characteriza:on of the access problem in terms of (something like) general epistemic norms of coincidence avoidance. 1. Introduction In "Mathema:cal Truth" (1973), Benacerraf presents a dilemma which includes the following classic worry for realists about mathema:cal objects-what is some:mes called the access problem. He argues that a certain causal constraint on knowledge, together with mathema:cal realism, implies that human knowledge of mathema:cs would be impossible. Many philosophers have been deeply moved by something about this worry (and analogous concerns in related domains), even while rejec:ng the specific premises employed in Benacerraf's argument (Field 1980; Linnebo 2006). However, a sa:sfactory formula:on of this access worry has proved elusive. In the first half of this paper, I'll review Hartry Field's informal characteriza:on of the access problem as arising from realists' (apparent) commitment to a match between human beliefs and realist facts, which cries out for explana:on, and their (apparent) inability to provide such an explana:on. I'll then discuss various aMempts (by Field and others) to elaborate on this core idea by fixing on a single fact about human accuracy/reliability, , which the mathema:cal realist must explain. I will note that these formula:ons face a trivializa:on problem (pointed out by Øystein Linnebo and recently emphasized by the work of Jus:n Clarke-Doane and David Enoch) involving the apparent existence of explana:ons for 1 the (supposedly explana:on requiring) fact R which do nothing to address intui:ve access worries. And I R Clarke-Doane argues that access worries should be rejected because they can't be sa:sfactorily 1 formulated in a way that doesn't allow for trivializing response (2017). In contrast Enoch (2010) takes access worries seriously but argues that certain apparently trivializing responses can answer it. will argue that exis:ng aMempts to solve this trivializa:on problem (by changing or clarifying the fact R to be explained) fail. In the second half of this paper, I'll advocate s:cking much more closely to Field's original informal proposal when characterizing the access problem. Specifically, I'll argue that access worriers can reasonably state their concern (and reduce their confidence in realism un:l a solu:on can be found), without aMemp:ng to "cash this worry out" in various ways which have been presumed to be necessary in the literature. Specifically, access worriers needn't (and shouldn't) iden:fy the access worry with a mere demand to coherently explain some reliability fact R. They also don't need to provide a nontrivial conceptual analysis of the no:on of coincidence or an uncontroversially applicable criterion for being a coincidence. Instead, they should simply formulate the access problem as follows. A realist theory of some domain (such as mathema:cs or morals) faces an access problem to the extent that adop:ng this theory 2 would require posi:ng some "extra" coincidence (about the match between human beliefs and reality), 3 beyond those required by compe:ng, less realist, approaches to the same domain. This unambi:ous formula:on ar:culates the part of Field's proposal which almost everyone accepts, while avoiding the extraneous philosophical commitments (e.g., to a causal theory of reference, or a par:cular analysis of coincidence, or a single explanandum at issue in access worries) which have bedeviled previous proposals. Understanding the access problem in this way (i.e., via direct appeal to no:ons of coincidence and coincidence avoidance) has some other important advantages. For example, it lets us aMrac:vely explain philosophers' failure to find a single reliability claim R such that explaining R suffices to banish access worries. And it clarifies what goes wrong with certain intui:vely unsa:sfying trivializing responses to access worries, which explain away one apparent coincidence involving human accuracy by appealing More specifically, adop:ng this claim together with typical claims about the extent of human knowledge 2 regarding this domain seems to require posi:ng some such coincidence. Some might argue that not all coincidences cry out for explana:on, and only the laMer tell against a 3 theory in the way that gives force to an access worry. I'm not sure if that is correct, but I take no posi:on on this issue here. For the sake of brevity, I will con:nue to talk simply about coincidences with the understanding that I mean coincidences which cry out for explana:on. to another. I will conclude by responding to some objec:ons. First, one might object that we need to go beyond my unambi:ous formula:on of the access problem if we hope to resolve philosophical disputes over the access problem. However, I argue that we can charitably state, and even plausibly hope to solve, disputes about the access problem in philosophy of mathema:cs without providing a further conceptual analysis of the access problem (or even the relevant no:on of coincidence). Second, one might suggest (as Clarke-Doane (2017) appears to) that intui:ve dissa:sfac:on with trivializing explana:ons of human accuracy about realist mathema:cs and morals shows that our coincidence avoidance intui:ons become unreliable when applied to theories involving necessary truths. But I argue that this principle should be rejected because accep:ng it would require junking an important and apparently fruicul part of current mathema:cal prac:ce. 2. Field's formula1on of the access problem and trivializa1on worries 2.a Field's formulation of the access problem Let me begin by reviewing Hartry Field's approach to the access problem, and the trivializa:on worries which have arisen for it. In Realism, Mathema4cs and Modality (1989), Field suggests that we should think of the access problem for mathema:cal realists as arising from a challenge for the realist to "explain how our beliefs about [mathema:cal objects] can so well reflect the facts about them" in some internally coherent fashion. He notes that, "[I]f it appears in principle impossible to explain this [match between our beliefs and reality], then that tends to undermine . . . belief in mathema:cal en::es, despite whatever reason we might have for believing in them." I will develop and defend this core proposal in what follows. However, Field elaborates this core idea in a way that (I will suggest) raises concerns about triviality. He argues that realists are commiMed to holding that, "for most mathema:cal sentences" the following reliability R claim holds (we will discuss other ways of understanding this reliability claim below), and that some explana:on for the truth of R must be possible. R: Reliably, if mathema:cians accept that " ," then . φ φ φ Typical mathema:cal realists seem commiMed to accep:ng the above reliability claim. But, Field 4 suggests, it appears in principle impossible for the realist to give any sa:sfactory explana:on for . And this fact casts doubt on the truth of realism. This account of the access problem has obvious appeal. It has been used (with some minor modifica:ons) to ar:culate access worries concerning other domains like morals and metaphysical possibility. Unlike Benacerraf's original access worry, Field's formula:on does not depend on any 5 conten:ous assump:ons about causal constraints on knowledge. Furthermore, Field's formula:on appears to reveal an internal tension within the (typical) realist's total web of beliefs. It thereby vindicates the common intui:on that access worries are different from (and more troubling than) mere skep:cism . 6 I think Field is quite right that the mathema:cal realist faces strong epistemic pressure to explain R, and that dispelling the impression that they can't do so is a necessary condi:on for dissolving access worries. However, we will see that explaining R is plausibly not sufficient to answer access worries. So R Field (1989) writes that the Platonist's commitment to accep:ng this reliability claim is "beyond serious 4 ques:on." And Linnebo (2006) buMresses this idea by appealing to a connec:on between reliability and knowledge as follows. AdmiMedly, a thinker could have significant true mathema4cal beliefs without this kind of reliability. However, such a person would not qualify as having knowledge. For example, a "lucky fool" who decides whether or not to believe mathema:cal statements on the basis of a coin toss, and winds up with many true beliefs in this way, would (plausibly) not count as knowing these mathema:cal statements. However, the realist does take us to have knowledge. So they are commiMed to the stronger claim that we have reliable true belief, unlike the lucky fool. Just as it seems mysterious that our mathema:cal intui:ons match objec:ve facts about (say) platonic 5 mathema:cal objects or proof transcendent coherence facts, it can seem mysterious that our a priori intui:ons about goodness, beauty, or what Lewissian possible worlds exist match objec:ve facts. It's not just that the access worrier can't jus:fy their mathema:cal beliefs from indubitable premises 6 which the skep:c accepts, but that their account of human accuracy seems troubling from their point of view. Also, note that the Fieldian access problem seems to point out a tension within the (typical) moral realist's total web of beliefs (including, e.g., various uncontroversial scien:fic and historical claims, and the idea that we have many true beliefs about moral topics), not within moral realism itself. A realist could (in principle) avoid Field's access problem by denying that we have any true moral beliefs or knowledge, but this fact provides liMle comfort to any actual moral realists. although Field's core approach is right and his further argument highlights crucial issues, it would be a mistake to take the final step of iden:fying access worries with an inability to explain R. 2.b Safety and the trivialization problem To see why explaining R (the fact that "Reliably, if mathema:cians accept that ' ,' then ") seems insufficient to answer access worries, let's consider a few different ways of cashing this claim out. One popular approach (Clarke-Doane 2014, 2017) is to read R as demanding that our mathema:cal beliefs be "safe" in the sense that they could not have easily been wrong, i.e., mathema:cians' belief-forming methods would not have lead them to form false beliefs at any suitably close possible worlds. In all sufficiently close possible worlds if mathema:cians believe that then Another possibility, which Field men:ons as a fallback op:on, is to drop the appeal to reliability and simply say that the actual abundance of true mathema:cal beliefs and lack of false mathema:cal beliefs is something for which the realist owes us an explana:on. But if one takes either of these approaches, then (as Øystein Linnebo [2006] and Jus:n ClarkeDoane [2017] have separately noted) it seems like one can "trivially" explain the relevant form of reliability using other premises which the realist accepts, as follows: TRIV: Mathema:cians reliably believe truths because they reliably believe only those mathema:cal claims which can be validly derived from a certain collec:on of mathema:cal φ Rsafet y φ φ necessary truths 7 TRIV seems to explain the safety of our mathema:cal beliefs. For, as it's robustly the case that mathema:cians form mathema:cal beliefs entailed by , they will con:nue to form mathema:cal beliefs entailed by in all relevantly close possible worlds. And all proposi:ons entailed by proposi:ons in are necessary truths. Hence all these close possible worlds will be ones in which they con:nue to form mostly true mathema:cal beliefs, thereby explaining the safety of our mathema:cal beliefs. One might think of TRIV as explaining safety via the fact that if our mathema:cal methods are accurate, then those methods are necessarily so. And TRIV also seems to explain (at least in some sense) our possession of many true and few false mathema:cal beliefs (in the actual world), by poin:ng out that we arrive at mathema:cal beliefs by reasoning validly from true axioms. However, it is equally clear that ci:ng TRIV does nothing to assuage intui:ve access worries. This suggests that intui:ve access worries cannot be reduced to the need to explain either the safety of our mathema:cal beliefs or the fact that we have many true and few false mathema:cal beliefs. Now Field could, obviously, respond to this objec:on by denying that TRIV cons:tutes a genuine explana:on for R (or for our possession of many true and few false mathema:cal beliefs). And this idea has some prima facie aMrac:on. Σ Σ Σ Σ One might object that this version of Linnebo's trivializing explana:on doesn't account for our true 7 belief in the consistency of ZFC (or in the arithme:cal sentence CON(ZFC)). However, I think one can naturally extend the trivializing explana:on to explain our true belief in Con(ZFC) as follows. If we only need to explain R for most mathema:cal claims encountered in normal mathema:cal prac:ce, it suffices to let consist of ZFC plus all finite iterates of the CON operator (i.e., ZFC + CON(ZFC) + CON(ZFC + CON(ZFC)) . . .). Of course, it is probably true that, for some computable ordinals , we believe (where indicates itera:ng the CON operator many :mes). However, our inability to know which puta:ve computable ordinals are truly well ordered prevents this chain from con:nuing indefinitely. So one can give a similar explana:on for our accuracy about even claims derivable from these infinitary iterates of the CON operator. Namely, there is some computable ordinal (though not one who we recognize any descrip:on of as a computable ordinal) such that the -itera:on of CON applied to ZFC is both true and entails all the iterated CON sentences we accept. Σ α CONα(ZFC ) CONα α β β However, many readers (like Linnebo and Clarke-Doane) seem to have the opposite intui:on. And I think it is ul:mately hard to deny that TRIV provides some kind of an explana:on of R. For we can easily imagine nonphilosophical contexts where TRIV would cons:tute an excellent response to an explanatory demand: an anthropologist could explain why some newly discovered community is reliable about mathema:cs/ that is explain why the community reliably had so many true and so few false mathema:cal beliefs by showing that all their mathema:cal reasoning can be reconstructed in terms of some formal system and then no:ng that this system is sound. So, in the absence of further sharpening 8 of the intui:ve no:on of explana:on (something Field doesn't provide), it appears that TRIV does explain R and access worries cannot be reduced to the need to explain R. Note, also, that one cannot defend Field's account of the access problem by rejec:ng TRIV merely on the grounds that it assumes the theorems of are true in a realist sense (e.g., correctly describe the platonic objects) which philosophers pressing an access worry wouldn't accept because this doesn't prevent TRIV from being an internally coherent explana:on from the realist's perspec:ve of our accuracy about mathema:cs. One of the great benefits of Field's proposal was that it appeared to reveal an internal problem for realism, not just a skep:cal worry. Thus, it suffices for the realist to give an internally coherent explana:on. 2.c Interpreting R more demandingly One natural thought is to interpret the "reliability" invoked in Field's R more demandingly, and use this as a basis for rejec:ng TRIV. Suppose we grant that TRIV explains why there aren't any extremely close possible words at which mathema:cians' beliefs are massively false. If we read Field's reliability claim R more demandingly -as requiring mathema:cians to be accurate in a larger sphere of close possible words including somewhere they don't form beliefs via -then we can s:ll resist the claim that TRIV explains R. 9 It's not immediately obvious that the realist is commiMed to the truth of such a demanding Σ Σ Such explana:on would admiMedly be par:al, but that doesn't prevent it from being an explana:on. As 8 David Lewis (1986) notes, everything we give is a par:al explana:on: the accident occurred because of the bald :re, because of the driver's slipshod maintenance, etc. This corresponds to individua:ng our methods more broadly.9 version of R. Rigorously defending this approach would require arguing that the realist is commiMed to some specific and much higher degree of reliability, and I haven't seen anyone do this. 10 But I won't dwell on this hurdle, as I think a deeper problem is lurking. The problem is that we can imagine discoveries which would imply and (in a sense) explain even very modally robust agreement between human psychology and realist facts about something like math or morals, while s:ll leaving intui:ve access worries untouched. Thus, a more demanding interpreta:on of R is incapable of rescuing Field's elabora:on of his core intui:ons. For example, consider the classic moral realist, who takes our beliefs about permissible favori:sm toward rela:ves to be "robustly objec:vely correct" in a sense which implies that creatures apparently inclined to advocate and prac:ce a different degree of favori:sm would have false beliefs about morality (rather than true beliefs about some other no:on "shmorality" of equal metaphysical status). Moral realists of this stripe intui:vely face an access worry about the accuracy of our moral beliefs. Now 11 imagine such a moral realist aMemp:ng to address access worries by giving the following kind of explana:on of our accuracy about permissible favori:sm facts. EV-MOR: It is a robust fact that in all circumstances conducive to the evolu:on of intelligence, natural selec:on favors the trait of advoca:ng and valuing as being twice as generous with immediate family as with other individuals. Furthermore, it is morally correct to be (exactly) twice as generous with family, and this is a necessary truth. This story certainly seems to provide some kind of explana:on for our accuracy about moral facts in a very wide range of possible worlds, yet considering it does nothing to answer intui:ve access This is a version of the famous "generality problem" for reliablist epistemologies (Goldman and Beddor, 10 2016). Note that even imperfect moral accuracy (at a rate substan:ally beMer than chance) can give rise to 11 such an access worry. worries. This is not just because the genealogy of morals suggested above is probably false. For even if 12 we imagine that the evolu:onary/game theory part of EV-MOR were unques:onably true and gesng at a deeply reliable law of nature, considering EV-MOR would s:ll do nothing to address intui:ve access worries. Thus, Field's official formula:on of the access problem can't be rescued by increasing the level of reliability (in the sense of safety) which is to be explained. 2.d Sensitivity and counter-possible conditionals A different strategy for understanding the reliability claim in Field's R is to appeal to metaphysically impossible worlds. Employing metaphysically impossible worlds has liMle effect on safety. However, it does give 13 teeth to sensi:vity requirements (another popular way of thinking about reliability). Sensi:vity demands that if hadn't been true, we wouldn't have believed (i.e., in the closest possible worlds where isn't true, we don't believe ). Our mathema:cal beliefs are trivially sensi:ve if we interpret this requirement using regular Lewisian counterfactuals (because there are no possible worlds where they are false). However, demanding that realists explain sensi:vity at metaphysically impossible worlds promises to let us reject explana:ons like TRIV and EV-MOR. For the fact that mathema:cians reliably tend to accept proposi:ons derivable from certain necessarily true axioms doesn't appear to explain why, in metaphysically impossible worlds where these axioms are false, we would s:ll wind up having true φ φ φ φ A similarly unsa:sfying example explana:on can be developed in the case of mathema:cs. 12 EV-MATH: The only way for intelligence to evolve involves having a composi:onal language, and the only way that mathema:cs-like prac:ces ever arise involves fluke reusing the brain structures which compute gramma:cality to produce asser:ons about certain mathema:cal structures, and it just so happens that these correspond to the platonic mathema:cal objects which actually exist. As Jus:n Clarke-Doane points out, even if we allow that "impossible worlds" where mathema:cal facts 13 are different can in principle be relevant to truth condi:ons for counterfactuals, it would seem that these worlds would be very remote from the actual one. So it's not clear why explaining reliability should require showing that mathema:cians' beliefs would con:nue to express truths in these very remote possible worlds (2017). mathema:cal beliefs. Indeed, such explana:ons seem to suggest that if mathema:cs/morals had been different, then our beliefs would have been just the same (because these beliefs are shaped by unrelated evolu:onary/game-theore:c/anatomical considera:ons). However, this approach faces very serious problems. First, there are reasons for doub:ng that we have any coherent shared grip on the closeness rela:on for metaphysically impossible scenarios (aka "counterpossible condi:onals"). For example, if would s:ll sa:sfy the usual induc:ve defini:on? If not, how would things be different? Despite advances in understanding the logic of counterpossible condi:onals (Nolan 1997), we s:ll face significant uncertainty (or perhaps conceptual underdetermina:on) concerning the substan:ve closeness rela:on on impossible worlds. Given this 14 uncertainty, cashing out informal access worries in terms of a demand to explain counterpossible condi:onals doesn't seem very helpful. A second problem for this approach is that the counter-possible sensi:vity requirement seems to fail (or counter-possible sensi:vity seems hard to explain) in many cases which are intui:vely unproblema:c. For example, if bachelors were unmarred women rather than unmarried men, would we s:ll believe that bachelors are unmarried men? Presumably, there is no reason to doubt our knowledge 15 of bachelorhood facts, and this calls into ques:on this interpreta:on of the sensi:vity requirement above. 16 2 + 2 = 5 + Or the substan:ve closeness rela:on which would be relevant to this aMempt to formulate access 14 worries, if there is some kind of context dependence as David Lewis has suggested (1986b). Jus:n Clarke-Doane (2017) gives a somewhat more complicated example along these lines: If the facts 15 about what configura:ons of maMer cons:tuted a chair were different, would our beliefs be different? A third problem for cashing out the access worrier's demand in terms of any sensi:vity demand is 16 pointed out by Donaldson (2014). Imagine someone who forms the belief that none of her colleagues' loMery :ckets will win based merely on the fact that there are a million other :ckets in some loMery and only one that will win. Her beliefs may well not be sensi:ve: had one of her colleagues won the loMery she would have s:ll expected them to lose. Yet her accuracy will not be mysterious or coincidental or give rise to any kind of intui:ve access problem. Thus, explaining human accuracy in Field's sense should not require sensi:vity. 3. Linnebo and alterna1ve languages Now let us turn to a variant on Field's R suggested by Øystein Linnebo (2006). Linnebo discusses a version of the problem for cashing out R in terms of sensi:vity noted above. He then highlights a different kind of "counterfactual dependence of people's disposi:on to accept mathema:cal sentences upon those sentences being true" (566), which might be relevant to access worries. Specifically, he proposes that a good strategy for answering access worries could involve defending a metaseman:c claim along the following lines. If mathema:cal sentences (like " ) had not expressed truths, then mathema:cians wouldn't have accepted them. In terms of possible worlds, asserts that the closest possible worlds in which linguis:c differences ensure that the sentence " " expresses a falsehood are ones in which mathema:cians no longer accept this sentence. Thus, we can think of as spelling out the sensi:vity 17 requirement from the prior sec:on using counterfactuals about seman:c facts instead of metaphysically impossible worlds to spell out the sensi:vity requirement from the prior sec:on. Now might seem like a promising candidate for the reliability claim R in Field's formal proposal. For, intui:vely, TRIV seems bad because the connec:on between the two sides of the 18 explanandum look fortuitous. In many cases, one can dis:nguish this kind of fortuitous agreement by looking to counterfactual sensi:vity. But, as we have seen, a straighcorward counterfactual sensi:vity analysis runs into problems with metaphysically necessary claims. Thus, one might be inclined to turn to RMS 2 + 2 = 4" RMS 2 + 2 = 4 RMS RMS So, for example, mathema:cians in this world don't assent to this sentence in conversa:on or place it 17 into textbooks. Linnebo does not commit himself to this claim. He merely suggests that a good answer to access 18 worries could take the form of an explana:on for , not that any explana:on for why is true would suffice to answer access worries. RMS RMS Linnebo's linguis:c counterfactual for a more sa:sfactory account. However, Linnebo's counterfactual faces its own trivializa:on problem as well as an overdemandingness problem. The trivializa:on worry arises as follows. It's hard to be confident about what 19 the closest possible words at which " " doesn't express a truth look like-something which might already be cited as an inconvenient aspect of Linnebo's view. But to the extent that I grasp this no:on at all, it seems it might well be that the closest possible worlds where " " doesn't express a truth are ones where some superficial and recent change in language/orthography went differently. However, we can explain why mathema:cians at these worlds don't accept " " in an intui:vely unsa:sfying fashion just by ci:ng the principle that when linguis:c/orthographic changes are made, people adjust what sentences they endorse accordingly. For example, these closest possible worlds might well be ones where the transi:on from Roman numerals to Arabic numerals went differently so that the symbol " " was used to mean " in most of the western world, for note that the history of such worlds could be exactly like that of the actual world up to this orthographic change. And it seems imaginable that a rather small copying error (a Lewisian "minor miracle") propagated by a few monks at some key boMleneck in communica:on between the Arabic numeral and Roman numeral using mathema:cal communi:es could have produced such a difference in orthography (and, hence, in the meaning and truth value of the relevant sentence). 20 So plausibly we can explain why we wouldn't have accepted " " in a world where " " named "3" (in English) as follows. In these possible worlds, at the :me that the transi:on to Arabic numerals occurred, speakers were reliably disposed to confidently reject sentences using Roman numerals to express the proposi:on . Thus, the principle that (considered, confident) views aren't affected by changes in orthography explains why people in those worlds didn't accept " " immediately following the change in orthography, and simple iner:a explains why later genera:ons 2 + 2 = 4 2 + 2 = 4 2 + 2 = 4 2 3 2 + 2 = 4 2 3 + 3 = 4 2 + 2 = 4 I am indebted to Warren Goldfarb for forma:ve conversa:ons on this point.19 I don't know how plausible it is that just few transcrip:on errors of this type could have resulted in this 20 difference. But I take that detail not to maMer much for my argument because it would be bad enough if Linnebo's formula:on implied that had our choice of symbols (or words) been so highly con:ngent there would be no access problem or access worries would be trivially solvable. con:nued to reject them. Pusng this together gives us the following explana:on for . Plausibly, the closest possible worlds where "2 + 2 = 4" expresses a falsehood are ones where this is so just because of some change in orthography (e.g., where "2" is adopted as the name for "3" instead of 2, so "2 + 2 = 4" expresses the mathema:cal falsehood "3 + 3 = 4"). But such changes in orthography don't tend to change what proposi:ons people accept. So, given that people were disposed to reject 3 + 3 = 4 when working with 21 Roman numerals, they'd likely con:nue to reject it axer adop:ng (this modified version of the) Arabic numerals and intellectual iner:a could explain why later genera:ons would con:nue to reject it. This 22 explana:on is unsa:sfying because it explains peoples' accuracy about mathema:cs at a later :me simply by appeal to their accuracy about mathema:cs at an earlier :me, plus a principle of con:nuity regarding their beliefs. More generally, Linnebo's condi:onal seems to be poten:ally explicable via the "deeper," unsa:sfying explana:ons for human accuracy about mathema:cal/moral facts discussed in the previous sec:on. For instance, EV-MOR asserted that evolu:on and game theory determine that intelligent creatures are overwhelmingly likely to treat a certain amount of favori:sm as permissible, and that ra:o of permissible favori:sm also happens to be objec:vely correct. Now imagine discovering that evolu:onary and psychological mechanisms gave us moral sen:ments matching this game-theore:c ideal in a way that was very counterfactually robust. So, for example, smallish changes to the human evolu:onary environment would have made liMle difference to the moral sen:ments with which we RMS RMS That is, people iden:fy what sentences in the new system correspond to the sentences they accepted 21 in the old orthography and accept those in the new orthography. One might worry that because we oxen do change beliefs when they turn out to produce prac:cally 22 harmful results, the person responding to (this version of) the access problem is on the hook to explain why rejec:ng the sentence "2 + 2 = 4" doesn't cause harmful outcomes. However, one can respond to this concern by extending the explana:on to include the fact that rejec:ng the sentence "3 + 3 = 4" doesn't seem to lead to prac:cal difficul:es in the actual world and arguing that the similar inferen:al role played by "2 + 2 = 4" in a world with the orthographic change in ques:on suffices to explain the lack of prac:cal difficul:es as a result of rejec:ng "2 + 2 = 4" and, thus, explain why it would con:nue to be rejected in such a world. wound up. And a human raised in almost any environment where they could learn to talk, survive to adulthood, etc. would be very likely to form some concept with the ac:on-guiding role which we assign to permissibility and have similar intui:ons to the ones we do about how this concept applies. Learning that our moral sen:ments were robust in this way would make it very plausible that, if language had been different (so other moral sentences had expressed truths), we would s:ll have been disposed to accept the same moral proposi4ons (and hence, from the realist point of view) s:ll accepted true moral proposi:ons. Thus, it would provide a direct explana:on for Linnebo's counterfactual. But it would do nothing to assuage access worries. Assuming EV-MOR is true, the closest worlds where "helping friends twice as much as strangers is permissible" expresses a falsehood would plausibly be ones where our language is different (rather than our moral sen:ments) so that we don't accept this sentence (and Linnebo's metaseman:c variant on the sensi:vity requirement is sa:sfied). Thus, if EVMOR were true, it would plausibly explain (as well as Field's ) without answering intui:ve access worries. This caveat raises the issue of what kind of grip we have on these linguis:c counterfactuals at all. For example, if "there are dogs" had expressed a falsehood, what claim would it have expressed? Would it s:ll have expressed a true claim? There are many different scenarios where some sequence of symbols like " " fails to express a truth and it's not at all clear that the closest such worlds are ones in which " " even has anything to do with mathema:cs. This brings us to a second problem. The problem is that we can construct cases where some quirk of history ensures the falsehood of the counterfactual in a way that does nothing to generate an intui:ve access worry or any kind of problem with posi:ng knowledge. For example, it's been argued that medieval science oxen expected deep analogies between different domains, so that very different things (personality types, metals, planets, mythical Greek gods) which somehow par:cipated in the nature of Neptune would behave analogously. Imagine a possible world where analogous theories were developed for astrology and fledgling chemistry (and each had a special nota:on) such that there was a fairly simple correspondence between sentences expressing (supposed) truths of the astrological theory and those expressing (supposed) truths of the chemical theory in the year 800 CE. Now suppose that because of these RMS R 2 + 2 = 4 2 + 2 = 4 RMS analogies, some monas:c copying error swapped the symbols used to express chemical reac:ons and astrological claims so that " went from originally expressing an astrological claim (say, the proposi:on that male Leos and female Libras are roman:cally linked when Mars is entering Scorpio) to expressing the claim that it expresses in normal English. And suppose that chemistry and astrology developed separately in the years axer 800, with both con:nuing to enjoy great popularity. We can imagine a chemist who has (intui:vely) jus:fied beliefs about chemistry and unjus:fied beliefs about astrology. Plausibly, some of the closest possible worlds to this one where " fails to express a truth would be ones where this copying error never happened (rather than the very remote ones in which the chemical reac:ons proceed differently). In such worlds, the above sentence will express a widespread and a long-standing, but false, doctrine about astronomy, which our horoscopereading chemist also accepts. Thus, it won't be the case that had various chemical sentences not expressed a truth, she wouldn't have believed them. Yet intui:vely our chemist could qualify as having chemical proposi:ons actually expressed by these sentences. Thus, we seem to have a counterexample to . This final problem is only heightened if we try to avoid trivializing explana:ons (like the Roman numerals example discussed above) by strengthening our reliability requirements. For doing this only increases the risk of demanding too much, i.e., that Linnebo's condi:onal could fail for reasons (like the chancy chemical-astrological symbol swap) that do nothing to impugn our claims to knowledge of a given domain. Thus, there's no plausible interpreta:on of Linnebo's which lets us avoid both trivializa:on worries and appeal to a sensi:vity principle which we have independent reason for doub:ng. Stepping back for a moment, I think the core problem for this proposal is the same one that generates trivializing answers to the other formula:ons of the access problem above. The realist can almost always explain a given fact about human mathema:cal accuracy if they are allowed to assume -and use unexplained-every other fact about the match between human psychology and objec:ve mathema:cal reality which they believe in. But such explana:ons won't sa:sfy access worriers because "H+ + OH− ⇋ H2O "H+ + OH− ⇋ H2O RMS RMS RMS R doing this amounts to showing that the existence of one prima facie mysterious match between human psychology and objec:ve mathema:cal fact is unsurprising given the existence of another such match. In the current case, it seems that no explana:on which brutely appeals to the fact that people got mathema:cs right at some earlier :me, e.g., when we were using Roman numerals, cuts ice with regard to assuaging intui:ve access worries. Yet invoking such facts seems quite relevant and useful in explaining why the closest possible worlds where " " expresses a different (false) proposi:on are ones in which we don't accept " ." I read philosophers like Clarke-Doane as, in effect, sugges:ng that such trivializing explana:ons pose a dilemma for the access worrier. Either the access worrier abandons Field's ambi:on of loca:ng a tension within the mathema:cal realist's own worldview or they allow the realist to explain one 23 seemingly mysterious match between human psychology and objec:ve mathema:cal facts which they believe in by appeal to another (since belief in these other apparent coincidences is, axer all, part of the mathema:cal realist's worldview). Thus, it might seem that the access problem is, ul:mately, an illusion. However, we can :dily avoid both horns of the dilemma by rejec:ng the hidden premise that access worries are simply a maMer of realists' inability to explain some reliability fact . Below I will argue for the following picture (which we get by taking Field's informal version of the access problem seriously). Access Problems aren't a maMer of realists' inability to provide any explana:on for some fact. Instead, they arise from the interac:on between realists' intui:ons about what kinds of explana:ons certain facts cry out for (i.e., intui:ons about coincidence which they share with an:realists) and the (disappoin:ng) nature of the explana:ons the realist can provide. 2 + 2 = 4 2 + 2 = 4 R If the access worrier does this, they can reject trivializing explana:ons as employing premises which 23 beg the ques:on against skep:cs about mathema:cal realism like themselves. But this comes at a very serious cost. For it's no longer clear that they have located a problem for mathema:cal realism, as opposed to merely showing the possibility for internally coherent doubt about some por:on of the things which the mathema:cal realists believes, i.e., merely showing that mathema:cal realism is not indubitable (something nearly all contemporary philosophers would be happy to grant since the doctrine that knowledge requires indubitability is widely rejected). 4. A coincidence avoidance approach to access worries 4.a Field's core idea and general norms of coincidence avoidance In view of the problems for spelling out (or replacing) Field's explanandum R discussed above, I propose that we s:ck to Field's ini:al characteriza:on of the access problem in terms of general norms of coincidence avoidance, rather than trying to specify any single reliability fact, such that merely explaining this fact (from more general premises which the realist believes) would suffice to answer access worries. We should, instead, simply say something like the following. A realist theory of some domain of inves:ga:on (such as mathema:cs or morals) faces an access problem to the extent that accep:ng it commits one to posi:ng a certain kind of coincidental match between human beliefs and the facts about that domain, but prevents one from giving any explana:on which would remove this appearance of coincidence. A liMle more formally, a realist theory faces an access problem to the extent that: Combining this theory with uncontroversial claims about the extent of human accuracy about the domain in ques:on forces us to posit some coincidental match between human beliefs and belief-independent facts (a match which intui:vely "cries out for explana:on," but has no explana:on). When this holds, it would seem that we have a significant (if defeasible) reason to reject the realist theory in ques:on. Such theories are ceterus paribus undesirable in that they commit us to posi:ng an extra inexplicable coincidence: a match between human psychology and the realist's subject maMer, which cries out for explana:on but cannot be explained. Note that this cons:tutes an internal problem for advocates of the relevant realist theory. For the shared norms of coincidence avoidance which we draw on in phrasing access worries are themselves part of the realist's total picture of reality. Thus, (we can con:nue to say that) the realist faces an internal tension-in this case, a tension between their philosophical beliefs about some domain and their own sense of which kinds of correla:ons cons:tute an unaMrac:ve coincidence. Also note that, on the view 24 I'm advoca:ng, access worries only give us ceterus paribus reason to reject a given realist theory of some domain. If it turns out that all the alterna:ve views which avoid this access problem have worse flaws (as, e.g., formalist theories which have trouble capturing proof transcendent truth condi:ons and the role of math in the sciences plausibly do), this bullet might be worth bi:ng. While, strictly speaking, a theory has an access problem to the extent no sa:sfactory explana:on of the match between beliefs and belief-independent facts is possible, we can some:mes also speak loosely and say that a theory faces an access problem when it appears that no such explana:on is possible (though, to be pedan:c, it only apparently faces an access problem). When it no longer appears that no such sa:sfactory explana:on is possible, we would say that the (apparent) access problem has been solved or dissolved. Thus, classical aMempts to eliminate access worries like Modal-Structuralism, Quan:fier Variance, Quineinism, and Neo-Fregean view can be seen as aMempts to solve (or par:ally solve) the access problem as conceptualized above. 25 4.b Helpful consequences Formula:ng Field's access worry as an applica:on of more general norms of coincidence avoidance has two interes:ng and helpful consequences. First, this proposal iden:fies access worries with a holis:c problem with the realist's account and thus explains why (as noted above) they can't be dismissed by explaining one type of accuracy in terms of another, equally mysterious, type of accuracy. While philosophers like Clarke-Doane represent access worries as presen:ng new evidence, I think they 24 are-like mathema:cal arguments-making an a priori philosophical point (hence presen:ng facts which they think an ideal Bayesian agent would already have recognized rather than presen:ng new evidence on which such an agent would update). This difference may also help explain the different conclusions we reach about intui:vely unsa:sfying explana:ons like TRIV and EV-MOR. Modal-Structuralism (Hellman 1994), Quan:fier Variance (Hirsch 2010), and Neo-Fregeanism (Wright 25 1983) help answer access worries (as characterized above) by sugges:ng that (almost) any logically coherent mathema:cal posits would express truths and thus explaining how any coherent mathema:cal beliefs we have correspond to mathema:cal truths; of course, the issue of how we come to have coherent mathema:cal beliefs remains. For, on the view above, (dis)solving one's access problem requires removing the appearance that one is commiMed to posi:ng any extra coincidences. So we can allow that TRIV and EV-MOR do, in some sense, explain human possession of true beliefs but s:ll maintain that they are useless in addressing access worries because each makes salient appeal to an extra coincidence, which more defla:onary rival understandings of mathema:cal/moral prac:ce let us avoid. Specifically, TRIV explains our accuracy about realist mathema:cal facts by appealing to an unexplained coincidental-seeming match between our mathema:cal reasoning method (our acceptance of sentences in as something like mathema:cal axioms) and realist mathema:cal facts. And EV-MOR only explains our good intui:ons about morality by appealing to an unexplained match between game-theore:c op:mality and objec:ve moral facts. Second, this approach suggests an important way in which access worries can be a maMer of degree. While a philosophical theory either does or doesn't allow for an explana:on of or (and thus does or doesn't face an access problem), on this approach, one theory can be preferred to another as it requires accep:ng fewer coincidences. Because of this compara:ve element, we should not think of access worries as invoking an epistemic requirement to "consign to the flames" every theory that posits a coincidence (analogous to Hume's famous empiricist exhorta:on to reject all concepts that weren't suitably related to experience [2007)]). Instead, access worriers appeal to general norms in favor of reducing the number of coincidences one is commiMed to posi:ng, insofar as this is compa:ble with other epistemic goals. This is important and helpful because it means that, even if our knowledge of induc:ve generaliza:on raises an access problem in its own right (maybe even an insoluble access problem), we can s:ll invoke induc:ve generaliza:on to dispel our access worries regarding a domain like mathema:cs (as no rival theory would dispel the coincidence that the future seems to behave like the past). Thus, theories can suffer access worries to varying degrees depending on the number and implausibility of the coincidences they are commiMed to posi:ng. 4.c Do we owe a further analysis of coincidence? This way of understanding access worries can seem to require using an unacceptably imprecise no:on of coincidence avoidance. However, the same imprecise no:on already plays an important role in scien:fic Σ R RMS and philosophical reasoning. We clearly have a prac:ce of dis:nguishing certain parts of a theory as unaMrac:ve coincidences. And we take commitment to any such extra coincidences to be a (ceterus paribus) reason to disfavor a theory. Think of the kind of argument we might use to convince someone to stop believing in the Loch Ness monster. We generally wouldn't be able to derive the nonexistence of the monster from beliefs we share with the Loch Ness conspiracy theorist or locate a literal contradic:on within their beliefs. Rather, we would point out unaMrac:ve extra coincidences that the Loch Ness monster theory has to admit (the monster never shows up when someone has a really good camera, it only appears in pictures which could plausibly be faked, etc.) but can't elegantly explain. We would appeal to a kind of shared general epistemic norm, which says that one has ceterus paribus reason to avoid theories which posit certain kinds of (inexplicable) coincidences. What results isn't a deduc:on that the Loch Ness monster doesn't exist, but, rather, ceterus paribus reasons for disfavoring its existence. AdmiMedly, what makes something a coincidence is rather complicated. Coincidences aren't 26 just facts posited by a theory which would otherwise be assigned low probability given the rest of a theory. For example, any par:cular long sequence of outcomes of a coin toss is unlikely, but we don't take total theories of the world which include the results of past coin tosses to be commiMed to an extra unaMrac:ve coincidence. Nonetheless, sposng and rejec:ng such coincidences plays an important role 27 in scien:fic and commonsense reasoning, even when we can't appeal to anything like a general 28 Carnapian logic of induc:on. We might wish to have a :dy and uncontroversial criterion for when a theory counts as posi:ng extra coincidences. However, we are all commiMed to using this kind of reasoning all the :me on a "know it when you see it" basis. Thus, it seems reasonable to take these intui:ons about theore:cal badness at face value. See Lando (2016) and Bhogal (Forthcoming) for some examples of recent work on this project.26 The feeling of coincidence/crying out for explana:on seems related to an intui:on that some other 27 theory predic:ng the same things but with fewer dimensions of freedom should exist, but the ques:on of a priori theory plausibility is an infamously hard one and I won't speculate about this more here. For example, the clustering of the orbits of many trans-Neptunian objects has lead astronomers to 28 hypothesize the existence of a ninth planet orbi:ng beyond 200 AU (Wikipedia 2016). 5. Objec1ons 5.a Tractability Let me conclude by addressing two objec:ons. The first objec:on concerns the tractability of disputes concerning access problems. Many philosophers currently disagree about how much of an access worry various forms of realism about mathema:cs, morals, etc. face. In this paper, I have argued that cri:cs of mathema:cal/moral realism can reasonably ar:culate and press an access worry by appealing to shared intui:ve norms of coincidence avoidance while taking a "know it when we see it" astude to the relevant concept of coincidence, rather than providing any explicit theory of what it takes for something to be an unaMrac:ve coincidence. But one might fear that adop:ng this posi:on makes disputes about the access problem deeply intractable by lesng access worriers issue their challenge from an unassailable swampland of brute intui:ons without commisng themselves to any general theses which the realist could defend themselves by aMacking. However, I will argue that such pessimism is unwarranted because there are other credible ways in which debate about access worries can be carried out, and by which widespread philosophical agreement could plausibly be produced. On one hand, realists can reasonably hope to win over opponents by providing a suitable sample explana4on for our accuracy about the relevant domain which suffices to banish coincidences (or only employs coincidences which an:realists about the relevant domain are also commiMed to accep:ng). I propose such a story in "Not Companions in Guilt (2018) and "The Residual Access Problem (Forthcoming). Conversely, there are also credible paths to philosophical agreement that there is a genuine access problem for realism about a given domain. For example, a history of massive effort and con:nued failure to discover any plausible explana:on of a certain coincidence can itself gradually increase access worries on my account. Thus, this way of formula:ng the access problem provides a way for access worries to get worse and a way for them to get beMer. 5.b Coincidences involving necessary truths The second (and final) objec:on I want to consider concerns the reliability of our intui:ons about coincidences and coincidence avoidance in domains involving necessary truths. One might imagine the philosophers like Jus:n Clarke-Doane (2017) who have pressed trivializing responses to the access problem responding to my proposal as follows. They might allow the above 29 general point about the general legi:macy and usefulness of coincidence-avoidance intui:ons but suggest (perhaps partly on the basis of mathema:cal access worriers' failure to cash out their intui:ve appeals to coincidence avoidance in other terms) that something special goes wrong when we apply these intui:ons to evalua:ng whether mathema:cal realists face an access problem. Specifically, one can think of them as sugges:ng that (either) our coincidence-avoidance intui:ons about which correla:ons involving necessary truths "cry out for explana:on" are deeply unreliable, or that (appearances Clarke-Doane (2017) proposes that access worries cannot call into doubt the safety or sensi:vity of a 29 realist's beliefs if the realist can explain the safety and sensi:vity of her beliefs from other premises she accepts. However, I would argue the mere fact that a web of belief contains elements that imply/explain the safety and sensi:vity of some faculty/belief-forming mechanism, doesn't prevent this web of beliefs from having other features which call this safety and sensi:vity into doubt. For example, I might have a great story (involving op:cs, brain processing, etc.) about how using my eyes and memory provided me with many safe and sensi:ve beliefs about Jane's office, so this aspect of my total picture of myself may look great. But if my other beliefs imply that the air in Jane's room contains a hallucina:on-inducing drug which would interfere with this belief-forming mechanisms, this will give me reason to doubt both the truth of my beliefs about Jane's room and their safety and sensi:vity. My ability to provide a (so to speak) "locally" internally coherent explana:on for how my beliefs about Jane's room are safe and sensi:ve doesn't mean that I shouldn't doubt this safety and sensi:vity because, among other things, it doesn't imply that my total web of beliefs is free of tensions. And, on the picture I have painted above, access worriers feel something similar is going on with the realist who explains her moral reliability via EV-MOR or her mathema:cal reliability via TRIV. The premises which the realist believes and uses in EV-MOR or TRIV provide a good explana:on for our reliability about realist morals/mathema:cs if they are true. But this fact alone doesn't guarantee that other elements within her total web of beliefs (such as norms that we should minimize our commitment to posi:ng certain kinds of inexplicable coincidence) can't give her reason to doubt these premises. notwithstanding) all such cries for explana:on can be adequately answered by just by "stapling together" two unrelated explana:ons for each half of the coincidence (as these trivializing explana:ons do). However, I think this line is hard to maintain. First, trivializers haven't presented much reason for thinking that analyzing the no:on of coincidence avoidance in cases where both sides of the relevant coincidence are con:ngent truths is any easier. No substan:ve (informa:ve) analysis of what it takes for a con:ngent regularity to cry out for explana:on is widely accepted. And there are plenty of good paradigms for thinking about coincidence avoidance which apply equally to necessary and con:ngent regulari:es (e.g., one might relate coincidence avoidance to a preference for theories that have fewer degrees of freedom or a general scien:fic desideratum to favor theories that unify [Kitcher 1981]). Second, and more importantly, saying that our intui:ons about coincidence avoidance become incoherent when applied to necessary truths seems to conflict with exis:ng mathema:cal methodology. For mathema:cians seem to fruicully use explana:on seeking and coincidence avoidance intui:ons (including the intui:on that merely "stapling together" two unrelated, but modally robust, explana:ons for each half of an apparent coincidence is unsa:sfactory) to guide research (Baker 2009; Lange 2010). The history of John Conway's "Monsterous Moonshine" conjecture provides a drama:c illustra:on of this. It shows how discovering a rela:onship between pure mathema:cal facts which intui:vely "cries out for explana:on," and, then, seeking such an explana:on can lead to important discoveries even when a proof of both facts already exists. In this episode, mathema:cians no:ced that the same number- -appeared in two seemingly unconnected areas of mathema:cs. It appeared both as one of the dimensions of the monster group (the largest of the sporadic simple groups) and as the first nontrivial coefficient of the -func:on (an important func:on in number theory). Later mathema:cians discovered further that the second nontrivial coefficient of the -func:on was the sum of the first three special dimensions of the monster group. Despite the lack of prior reason to believe that there was any connec:on between these two areas of mathema:cs, the fact that these coincidences seemed to call out for explana:on mo:vated mathema:cians to hypothesize a connec:on and eventually discover one which lead to deep 196,883 j j mathema:cal insights. Mathema:cians thought there must be some further explana:on for the above regularity involving necessary truths on both sides (and they turned out to be right) (Klarreich 2017). So I think considering mathema:cal cases like the one above tells strongly against any sugges:on that our intui:ons about coincidence avoidance become (generally) unreliable when applied to necessary truths. Similarly, in philosophy, we seem happy to accept that avoiding coincidence in the sense of favoring theories that unify many explana:ons with few resources. And proponents of trivializing explana:on like Clarke-Doane haven't shown that there's any principled and theore4cally aKrac4ve line which carves off the specific intui:ons about coincidence avoidance and necessary truths which he wants us to be suspicious of (those driving access worries) from general methods of reasoning which are aMrac:ve and ubiquitous in philosophy and mathema:cs. Therefore, absent a stronger argument that such reasoning leads us astray, I don't see any reason to eschew its use. 30 6. Conclusion In this paper, I discussed a trivializa:on problem for Hartry Field's formal characteriza:on of the access Now one might further ask: Is there any mathema:cal precedent/analog for the access worrier's overall 30 sugges:on that norms of coincidence avoidance should mo:vate us to reject an antecedently aMrac:ve metaethical theory, such as realism? Can recognizing a mathema:cal regularity's cries for an explana:on ever make it ra:onal to reject a previously aMrac:ve theory? (Thanks to an anonymous reviewer for sugges:ng this ques:on.) Perhaps Gödel's idea that new axioms for set theory can be jus:fied by what they let us explain about known results suggests one possible example of such a scenario. He writes, "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 maMer whether or not they are intrinsically necessary, they would have to be accepted at least in the same sense as any wellestablished physical theory" (1947). For imagine a case where some proposed new axiom extending ZF set theory is known to be incompa:ble with some other axiom we are now moderately aMracted to (e.g., some large cardinal axiom are known to be incompa:ble with the axiom of Choice [Kunen 2017]). And suppose it turned out that (analogous to what we seem to find regarding the access problem) the new axiom explained many "coincidences" as striking as the magic moonshine example above, and we (somehow) had reason to think that we could not sa:sfyingly explain these coincidences if the new axiom was false. In this case, we'd seem to have a very strong form of the kind of explanatory benefit Gödel endorses. So I think (if one is sympathe:c to the Gödelian idea at all) it's quite conceivable that mathema:cians could reject an antecedently aMrac:ve principle on the basis of intui:ons about coincidence avoidance. However, I admit that it's hard to imagine what strong evidence that some axiom is necessary to explain some known regularity could look like. So I wouldn't be surprised if no such case can be found in the actual history of mathema:cs. problem for realist theories of mathema:cs, morals, and the like. I argued that various aMempts to fix this problem by beMer specifying a reliability fact, , which the realist is challenged to explain, fail. I then suggested a reason for this failure: all such "single explanandum" accounts (in effect) get the logical structure of the access problem wrong. They aMribute the access worrier an " " intui:on, that some par:cular apparent coincidence can't be explained by any realist account of human mathema:cal/moral accuracy (even a bump-pushing one). But what actually drives the access worry is a " " intui:on that (while it's usually easy to explain one mysterious predes:ned harmony by posi:ng another) every realist account of human mathema:cal/moral accuracy would leave some mysterious coincidence unexplained. Accordingly, I argued that access worriers would do beMer to s:ck closer to Field's informal statements. They should cash out access worries in terms of the realist's apparent commitment to some coincidence involving human accuracy about realist moral, mathema:cal, etc. facts. And they should reject demands for informa:ve further analysis of what qualifies as a coincidence. Because of the good work which this no:on of coincidence reduc:on already does in mathema:cs and the sciences, it is something to which all par:es in debate are commiMed. Finally, I noted that we don't need to go beyond this unambi:ous way of formula:ng access worries in order to resolve debate about them. Biography: Sharon Berry received her Ph.D. in Philosophy from Harvard and joined Oakland University axer postdoctoral research at Australian Na:onal University and the Polonsky Academy in Jerusalem. Her primary areas of research are epistemology and the philosophy of mathema:cs. She has published papers on the Benacerraf problem, a priori knowledge, modal metaphysics, and poten:alist set theory. R ∃∀ ∀∃ References Baker, Alan. 2009. "Mathema:cal Accidents and the End of Explana:on." In New Waves in Philosophy of Mathema4cs, edited by Otávio Bueno and Øystein Linnebo, 137–59. Basingstoke, UK: Palgrave Macmillan. Benacerraf, Paul. 1973. "Mathema:cal Truth." Journal of Philosophy 70: 661–80. Berry, Sharon. 2018. Not Companions in Guilt. Philosophical Studies 175 (9): 2285–308. Berry, Sharon. Forthcoming. The residual access problem. Bhogal, Harjit. Forthcoming. "Coincidences and the Grain of Explana:on." Philosophy and Phenomenological Research. Clarke-Doane, Jus:n. 2014. "Moral Epistemology: The Mathema:cs Analogy." Nous 48 (2): 238–55. Clarke-Doane, Jus:n. 2017. "What Is the Benacerraf Problem?" In Truth, Objects, Infinity: New Perspec4ves on the Philosophy of Paul Benacerraf, 2nd ed., edited by Fabrice Pataut. Switz.:Springer Interna:onal. Donaldson, Thomas Mark Eden. 2014. "If There Were No Numbers, What Would You Think?" Thought: A Journal of Philosophy 3 (4): 283–87. Enoch, David. 2010. "The Epistemological Challenge to Metanorma:ve Realism: How Best to Understand It, and How to Cope with It." Philosophical Studies 148 (3): 413–38. Field, Hartry. 1980. Science without Numbers: A Defense of Nominalism. Princeton, NJ: Princeton University Press. Field, Hartry. 1989. Realism, Mathema4cs, and Modality. Oxford: Blackwell. Gödel, Kurt. 1947. "What Is Cantor's Con:nuum Problem?" In Kurt Gödel: Collected Works, Vol. II, edited by Solomon Feferman et al., 176–87. Oxford: Oxford University Press. Goldman, Alvin, and Bob Beddor. 2016 (Winter). "Reliabilist Epistemology." In The Stanford Encyclopedia of Philosophy, edited by Edward N. Zalta. Hellman, Geoffrey. 1994. Mathema4cs without Numbers. New York: Oxford University Press. Hirsch, Eli. 2010. Quan4fier Variance and Realism: Essays in Metaontology. New York: Oxford University Press. Hume, David. 2007. An Enquiry Concerning Human Understanding and Other Wri4ngs. Cambridge: Cambridge University Press. Kitcher, Philip. 1981. "Explanatory Unifica:on." Philosophy of Science 48 (4): 507–31. Klarreich, Erica. 2017. Mathema:cians Chase Moonshine's Shadow. In The Best Wri4ng on Mathema4cs 2016, edited by Mircea Pi:ci. Princeton, NJ: Princeton University Press. Kunen, Kenneth. 2017. Set Theory: An Introduc4on to Independence Proofs, Vol. 102. Amsterdam: Elsevier. Lando, Tamar. 2016. Coincidence and Common Cause. Noûs 51 (1): 132–51. Lange, Marc. 2010. "What Are Mathema:cal Coincidences (and Why Does It MaMer)?" Mind 119 (474): 307. Lewis, David K. 1986a. "Causal Explana:on." In Philosophical Papers, Vol. II, 214–40. New York: Oxford University Press. Lewis, David K. 1986b. On the Plurality of Worlds. Malden, MA: Blackwell Publishers. Linnebo, Øystein. 2006. "Epistemological Challenges to Mathema:cal Platonism." Philosophical Studies 129 (3): 545–74. Nolan, Daniel. 1997. "Impossible Worlds: A Modest Approach." Notre Dame Journal of Formal Logic 38 (4): 535–72. Wikipedia. "Planets Beyond Neptune." Accessed October 27, 2016. hMps://en.wikipedia.org/wiki/ Planets_beyond_Neptune. Wright, Crispin. 1983. Frege's Concep4on of Numbers as Objects. Scotland: Aberdeen University Press.