The problem of the many threatens to show that, in general, there are far more ordinary objects than you might have thought. I present and motivate a solution to this problem using many-one identity. According to this solution, the many things that seem to have what it takes to be, say, a cat, are collectively identical to that single cat.
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I need some mereological terminology. I take parthood as a primitive relation. And I stipulate the following regarding its behavior: parthood is reflexive (for all x, x is a part of itself), anti-symmetric (for all x and y, if x is a part of y and y is a part of x, then x and y are identical), and transitive (for all x, y, and z, if x is a part of y and y is a part of z, then x is a part of z). Something is a proper part of another thing iff the first is a part of the second, but they are not identical. Some things overlap iff they have a part in common. Some things are disjoint iff they do not overlap. Some things compose another thing iff they are all parts of it and every part of it overlaps one of them. Something is a fusion of some things iff they compose it.
I have presented the problem in terms of atoms. I don’t make any assumptions about the ultimate architecture of matter. Perhaps there are smallest parts of matter. Or perhaps there is gunk, i.e. perhaps every part of matter has further parts. Nothing I say turns on any of this.
The problem of the many, as presented here, comes from Unger (1980). Lewis (1999) presents a problem in which it is indeterminate whether certain things are part of a cat. Sometimes this is presented as the problem of the many, or as a version of the problem of the many. It’s not obvious that they’re the same problem. Unger’s problem isn’t presented in terms of vagueness, for instance, and Unger (2006, pp. 369–370) argues that the problem arises even without vagueness. See Jones (2010) for an extended discussion of whether the problems are the same. I return to discussion of Lewis’s problem at the end of Sect. 4 in addressing an objection.
Philosophers disagree about what objects are. Some philosophers (e.g. Lewis 1986) hold that objects of familiar kinds are temporally extended; proponents of perdurantism will hold, roughly, that there is a cat on the mat when there is a “time slice” of a cat on the mat. They hold, further, that the time-slice itself is not itself a cat. There is a cat on the mat because there is a temporally extended object that is a cat that has a time-slice as a part that is on the mat. Others hold that objects are not temporally extended. There are two ways to hold this. One way is to hold that persisting objects are wholly located at different times; this is endurantism. Proponents of this view (e.g. Haslanger 1989; van Inwagen 1990) hold, roughly, that there is a cat on the mat when the cat is wholly located on the mat, i.e. when one of its many locations it has at different times is on the mat. Another way to hold that objects are not temporally extended is to hold that objects are instantaneous; proponents of stage theory (e.g. Sider 1996, 2001a; Hawley 2001) will hold that there is a cat on the mat when an instantaneous cat is on the mat. I discuss perdurantism and stage theory in more detail in Sect. 3.
These different views make it somewhat difficult to state the problem in full generality. Different views require stating the problem somewhat differently. I have found it best to state the problem in one way and note that others can understand my terminology in a slightly different way, if they so desire.
Proponents of the second and third views can adopt the problem and solution I propose as stated. Proponents of perdurantism, however, will need a slightly different statement of the problem and of the solution. I suggest they hold that, in those cases in which the problem arises, the many candidates for being time slices of objects of kind K are collectively identical to a single time-slice of an object of kind K, and that there is just one such K in the vicinity. Proponents of this view should read “object of kind K” as something like “time-slice of object of kind K.” For the perdurantist, the problem of the many will also arise temporally; I don’t discuss this in any detail in the paper, mostly because it would require too much space. However, the solution I propose generalizes, mutatis mutandis.
The problem does not arise for those objects that don’t admit of such leeway. Perhaps molecules and chemical atoms are like this. The problem also does not arise for those kinds of objects, if there are any, where we don’t take it that there is just one of them where we take there to be just one.
If mereological nihilism, the thesis that there are no composites, is true, then abundance is false. Some might deny that there are any such things by accepting mereological nihilism, the view according to which there are no composite objects, i.e. no objects that have any parts besides themselves. Unger (1980) uses the problem of the many to motivate this position. See Rosen and Dorr (2002) and Sider (2013), for instance, for defenses of mereological nihilism. The statement of the problem in this paper presupposes that the true theory of the conditions under which composition occurs is not nihilism and that composition occurs often enough that there are multiple K-candidates in at least some situations. This presupposition could, perhaps, be dispatched with; see Jones (2013) for discussion.
Baxter (1988) advances such a view. Lewis (1991), according to van Inwagen (1994), holds that identity and composition are merely analogous. Bøhn (2011) and Bricker (2016) read Lewis (1991) as holding that many-one identity and one–one identity are instances of a more general form of identity. Wallace (2009, 2011a, b, 2014) defends the view that composition is a form of identity. Bøhn (2009, 2014) defends the view I accept in this paper, which is that identity and composition are the very same relation. Many of the arguments in this paper could be run with Wallace’s account of composition-as-identity, however.
This example is taken from Bøhn (2009, p. 7).
See van Inwagen (1994, p. 210ff.), for instance.
You might think that this is a counterpossible. But counterpossibles aren’t all trivially true.
Importantly, they are not distributively identical to the single cat; that is, it is not the case that each candidate is individually identical to the single cat. Instead, all of the candidates, taken together, are identical to the single cat.
One might think that the fusion of all of the candidates is the best candidate for being the cat; it has a feature that no other candidate has, namely being the fusion of all of the candidates. Although it has a special feature, at this point it is appropriate to ask why that feature should matter. The goal isn’t merely to find some feature that a candidate has and that no other candidate has—that’s very easy since each candidate has the property being identical to that very candidate—but to find some reason to think that that very feature matters.
See Lewis (1986, pp. 203–204) for the classic statement of the problem.
See, again, the discussion of Sect. 2.1.
Korman (2015), for instance, adopts a much more expansive form of conservatism. On his view, not only are we largely correct about what ordinary objects there are, but folk ontology is correct that extraordinary objects do not exist.
See Lewis (1986, pp. 202–204) for the classic defense of perdurantism.
The standard definition is: x is a proper temporal part of y during temporal interval (possibly instantaneous) T, iff x exists at, but only at, times in T, is part of y at every time during T, at every time in T overlaps everything that is a part of y during T, and x ≠ y. In the case where T is an instant, x is an instantaneous proper temporal part of y.
On this view, de re temporal predicates are what Noonan (1991) calls “Abelardian” predicates. Note that one need not adopt a counterpart-theoretic account of de re temporal claims in order to make use of the idea that predicates are Abelardian; see Noonan (1991, p. 191) and Lewis (1986, p. 248ff.) for discussion.
Here’s Wallace’s (2014, p. 117) definition: “x is a world-bound modal part of y at a world w iff (i) x exists at, but only at, w, (ii) x is part of y at w; and (iii) x overlaps at w everything that is part of y at w.”.
Or, perhaps, there are two different problems that are called the problem of the many. See note 3.
The following objection comes from an anonymous referee of this paper. They note that similar problems arise for the location and mass of the cat, as well. The solution I offer below seems to generalize to these cases. Thank you to this referee for pushing me on this objection and urging me to be clearer about the costs of my response.
This general response seems to work for the version of the problem as it arises for the cat’s location and mass.
See Korman (2015, pp. 29–30) for discussion of this sense of begging the question.
The view that borderline parts are impossible is controversial, but it’s not unheard-of; at least one philosopher expresses preference for such a view in print. Ned Markosian (2014, p. 82). endorses regionalism, the view that “[n]ecessarily, for any xs, there is a y composed of those xs iff there is a region, r, and an object, z, such that r is the fusion of the regions occupied by the xs and z occupies r.” Markosian (2014, pp. 87–88) notes that, if regionalism is true, then if it can be indeterminate whether an object is located at a particular region, then it will be indeterminate whether that object has such-and-such as a part. He surveys three possible responses; the response Markosian expresses a preference for is that it is impossible for it to be indeterminate where an object is located. He notes that, given regionalism, this entails that it is impossible for it to be indeterminate whether something is a part of another thing.
For a similar proposal, see Jónsson (2001). While Jones holds that each of the candidates constitutes the cat, Jónsson holds that it is only collectively that the candidates constitute the cat.
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I benefitted greatly from comments on ideas in this paper from Mike Bertrand, Ethan Brauer, Ben Caplan, Sam Cowling, Cruz Davis, Eric de Araujo, Carolyn Garland, Jared Henderson, Julia Jorati, Teresa Kouri Kissel, Erin Mercurio, Adam Murray, Eileen Nutting, Jonathan Payton, Craige Roberts, Neil Williams, three anonymous referees for this journal, and numerous attendees at the following conferences: Filosofidagarna 2017, The 2017 Pittsburgh Area Philosophy Colloquium, the 2018 meeting of the Society for Exact Philosophy, and the 2018 Canadian Philosophical Association Annual Congress.
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