Lowe on Metaphysical Nihilism
February 11, 2014 — 15:24

Author: Kenny Pearce  Category: Existence of God Prosblogion Reviews  Tags: , , , , ,   Comments: 9

Like several other contributions to The Puzzle of Existence, the essay by the late E. J. Lowe is devoted to the question whether there might have been nothing. Lowe calls the view that there might have been nothing ‘metaphysical nihilism,’ and he offers an argument against a certain version of it.

Lowe’s paper begins with some very helpful context-setting. In 1996, Peter van Inwagen had argued that there is a possible world which was ’empty’ in the sense of containing only abstract objects, and no concrete objects. However, according to van Inwagen, out of the infinitely many possible worlds, only one is the empty world, hence the probability that the empty world is actual is 0.

Van Inwagen’s argument was published together with a response by Lowe. In his article in Puzzle, Lowe summarizes his earlier argument as follows:

Some abstract objects exist necessarily, and so exist in every possible world. But all abstract objects depend on there being concrete objects – although not necessarily the same concrete objects in every possible world. Hence, concrete objects exist in every possible world, even if there is no necessary concrete being (182-183).

Lowe’s key examples of necessarily existing abstract objects were numbers, which he then thought were required to ground arithmetic truths. One important difficulty of Lowe’s understanding of the grounding of abstracta in concreta was that it required him to deny the existence of the null set and the number 0.

After 1996, two things happened: a number of objections to Lowe’s argument were raised, and (for reasons independent of those objections) Lowe stopped believing in numbers. The current paper reformulates the argument in a way that relies only on universals and ‘impure’ sets. Lowe’s argument is, essentially, that universals depend ontologically on their instances, and sets depend ontologically on their members, but there can be no cycles or regresses of ontological dependence, hence, if there are abstracta, there must be concreta. Lowe continues to reject the existence of the null set, and consequently of all ‘pure’ sets, on grounds that the null set cannot be properly grounded.

My key worry as I was reading the paper concerns a shift in Lowe’s characterization of van Inwagen’s position, against which he is supposed to be arguing. At the beginning of the paper, he describes van Inwagen as arguing for the existence of a world empty of concreta, but conceding that that world still contains abstracta (182). But later in the paper, Lowe characterizes van Inwagen as arguing that “there is an ’empty’ world in which there exist no concrete objects but abstract objects do exist” (187). At this point, Lowe makes it clear that he is no longer arguing that there must be concrete objects, but only that there can’t be abstracta without concreta. In other words, what was earlier a concession of van Inwagen’s has become part of van Inwagen’s thesis.

To his credit, Lowe explicitly addresses this worry in the very last paragraph of the paper. However, his response is concessive: he is indeed no longer arguing that there must be concreta. “Doesn’t that significantly reduce the metaphysical significance of the argument?” he asks rhetorically (194). I certainly think so. However, Lowe is certainly right that the fact that this argument is of lesser metaphysical significance than the argument he once tried to offer does not mean that the present argument is not interesting or significant. If Lowe is right about the ontological dependence of abstracta on concreta and the well-foundedness constraints on ontological dependence, there are wide-ranging metaphysical consequences.

(Cross-posted at blog.kennypearce.net.)

  • I don’t know Lowe’s earlier argument. But I am puzzled about what concrete objects numbers would be dependent on.

    Off-hand I can see three different stories.

    1. Numbers are dependent on the mind of a being that conceives all the numbers.

    And that’s fine with me–that leads to theism, in the way Leibniz notes, and I wouldn’t think this would be Lowe’s move.

    2. The number n is dependent on there being n concrete things.

    But this one, plus the necessity of the existence of numbers, would yield not just the claim that there must be something concrete, but that there must be infinitely many concrete things. And that’s implausible.

    3. The number n is dependent on there being something concrete that has n different properties.

    But this one, plus the necessity of the existence of numbers, would yield that there must either be infinitely many things or something with infinitely many properties. And on sparse views of properties, this is implausible, unless perhaps one thinks God has infinitely many properties.

    February 17, 2014 — 8:37
  • Based on this article, I’m pretty sure Lowe’s earlier view was your 2. The problem you indicate with that did occur to me, and I don’t know how Lowe dealt with it (on his earlier view), but I got the impression that his response to this sort of thing was not very satisfactory and that this was among his reasons for ultimately rejecting the earlier view.

    February 17, 2014 — 10:39
  • There would also be the problem that some of the reasons for thinking that there are numbers are reasons for thinking that there are higher cardinalities. But then we get even more trouble: for any cardinality k, we have to have k concrete objects. And hence there is no set of all concrete objects.

    February 18, 2014 — 7:49
  • Ty

    This is answered by Lowe, especially in chapter 12 of ‘The Possibility of Metaphysics’. Lowe’s earlier view was that numbers are universals—in particular, kinds of sets. Since a concrete object is distinguished from its unit set, with just one concrete object, there’ll be an infinite number of sets—the set of the object, the unit set of that set, the set of the object and the unit set of the object, etc. I think–I’ll have to check again in his contribution to Szatkowsky’s ‘Ontological Proofs Today’–Lowe was eventually more sympathetic towards something like your first proposal, and developed an ontological argument on the basis of that.

    February 18, 2014 — 14:09
  • So the thought is that you just need *one* concrete thing, and then you get infinitely many sets. I missed that possibility.

    February 20, 2014 — 7:45
  • Here’s another kind of problem with this proposal: the Benacerraf dilemma. It would be very hard, I think, to think of the Lowe proposal as giving *the* numbers. For one, if all beings are contingent, then different worlds will have different numbers. And there surely won’t be a canonical concrete being to use in the construction anyway, so even within one world there will be multiple candidates. So in the dilemma, one must take the horn of saying that there is no such thing as the numbers, but there are things that play their role. But if we say that, then the pull of the modal thought that there *must* be numbers diminishes significantly. With numbers identified by their functional role, structuralism is the natural philosophy of mathematics, and now arithmetical theorems are necessary conditionals like “If there are xs such that the Peano axioms hold of them, then …”. Of course, one can, and should, still ask for the grounds of the necessity of the conditionals, but that’s a different argument than the one from abstracta.

    February 20, 2014 — 7:53
    • Ty

      I think Lowe’s original argument won’t work for other reasons. If I recall correctly, Lowe prefers reducing numbers to universals rather than to sets in part because of the problem raised by Benaceraff. Of course, the concrete beings and, hence, the sets in the different worlds might be different. I’m not sure why there couldn’t be the same number universals though?

      February 20, 2014 — 10:13
  • The impression I got was that there was supposed to be a universal unity, from which both numbers and sets were derived. You need one concrete thing (anything will do) in order for the universal unity to exist, but once you’ve got that you’ve got all the sets and numbers (except 0 and the null set, in which Lowe does not believe).

    On his new view, I think the universal unity is still playing an important role in grounding the arithmetic and set-theoretic facts, even though he no longer believes that there are such things as numbers and sets.

    February 20, 2014 — 10:20
    • Here’s an intuition in the spirit of essentiality of origins. Assume eternalism. If S is the set of all the entities that ground an entity E, then E could not exist if *none* of the members of S existed. There may be a possible world where Spain has very few people and land in common with the actual world’s Spain. But there is no world where Spain has no people or land ever in common with the actual world’s Spain. (I think it’s the initial ones that count here, but I don’t need that.)

      If so, then a Unity that is grounded in x will differ from a Unity that is grounded in y, if x is different from y.

      February 25, 2014 — 10:12
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