Plantinga’s Abstract Objects Argument
June 22, 2012 — 23:21

Author: Andrew Moon  Category: Existence of God  Tags: , , , ,   Comments: 19

In Where the Conflict Really Lies, which James Beebe has nicely reviewed, Alvin Plantinga writes,

But numbers and sets themselves make a great deal more sense from the point of view of theism than from that of naturalism. Now there are two quite different but widely shared intuitions about the nature of numbers and sets. First, we think of numbers and sets as abstract objects, the same sort of thing as propositions, properties, states of affairs and the like… On the other hand, there is another equally widely shared intuition about these things: most people who have thought about the question, think it incredible that these abstract objects should just exist, just be there, whether or not they are ever thought by anyone. Platonism with respect to these objects is the position that they do exist in that way, that is, in such a way as to be independent of mind… But there have been very few real Platonists, perhaps none besides Plato and Frege, if indeed Plato and Frege were real Platonists (and even Frege, that alleged arch-Platonist, referred to propositions as gedanken, thoughts). It is therefore extremely tempting to think of abstract objects as ontologically dependent upon mental or intellectual activity in such a way that either they just are thoughts, or else at any rate couldn’t exist if not thought of. (287-288)

I am inclined to think that there are numbers and that they are abstract objects, but I don’t have the second intuition that they must be thought. Is there something I’m missing? I do have the intuition that contingently existing objects must have a cause for their existence, but I don’t have the intuition that abstract objects must be thought, which, if they exist, necessarily exist.
Maybe somebody could help motivate this intuition for me? Or is this intuition not very widely shared (contra Plantinga’s remark)?

  • Andrew: Good question. I share your puzzlement about the view Plantinga defends. While I don’t think that all abstract objects exist necessarily (the set {Mars} exists only contingently, just like its only member), I think that infinitely many abstract objects do exist necessarily. As far as I can see, to say that those objects don’t exist independently of God’s thoughts is to regard the subjunctive (or counterfactual, or counterpossible) conditional (C1) “If God didn’t exist, those objects wouldn’t exist” as nontrivially true. (If we regard it as only trivially true, it’s no more trivially true than “If God didn’t exist, those objects would (still) exist.”) But I can’t see how to understand the nontrivial truth of C1 if this conditional is also nontrivially true: (C2) “If God didn’t exist, C1 wouldn’t be true.” A vicious regress seems to threaten, or something else is amiss that I can’t put my finger on.

    June 23, 2012 — 7:47
  • Patrick Mefford

    Hi Andrew,
    I had similar reaction when I read that passage, my biggest hang up is that being acausal strikes me as something necessary for abstract objects and being mind dependent conflicts with that.

    June 23, 2012 — 9:17
  • Mike Almeida

    The intuition might have to do with the knowledge of non-concrete objects. If there are no uninstantiated properties and there are no sets with non-existing members, then sets and properties are concrete objects. They stand in causal relations to those who know about them. Maybe look at E.J. Lowe for a contemporary metaphysician who has these intuitions. But the theist can, in a way, eat and have his cake. Properties and the like are at least intentional objects, so they are concrete, but they are not necessarily instantiated, since they’re in the mind of God.

    June 23, 2012 — 10:10
  • Andrew Moon

    Right, sets probably don’t exist necessarily (I was sloppy). I don’t think of these issues in terms of counterfactuals because of the problems that arise when the antecedent might be necessarily false (in the case that God exists). Anyway, I do understand the notion of two necessarily existing things, where one depends on the other. E.g., God’s belief that God exists, and God both necessarily exist, but the belief depends on God and not the other way around. And it seems to me that one depends on the other in this case. But I don’t have the intuition here.
    Ah, yeah, the acausality does add another problem. However, it’s not clear to me that numbers couldn’t depend for their existence on God (despite the fact that they’re supposed to be acausal). It’s that when I try to examine how things seem to me, my seeming goes neither way.
    I wasn’t sure how what you said was supposed to motivate the intuition. But I agree that there is the problem of how we could know about abstract objects if they are acausal.

    June 23, 2012 — 16:51
  • Steve Maitzen

    I do understand the notion of two necessarily existing things, where one depends on the other. E.g., God’s belief that God exists, and God both necessarily exist, but the belief depends on God and not the other way around. And it seems to me that one depends on the other in this case.
    Andrew: I’d be interested to know how you analyze that dependence relation if not in terms of some nontrivially true conditional. Is it maybe a relation of logical priority: God (or God’s existence) is logically prior to God’s belief that God exists? Is it like the relation that {Mars} bears to Mars: neither exists without the other, but the former (somehow!) depends on the latter?

    June 23, 2012 — 18:45
  • Andrew Moon

    Well, I don’t analyze the dependence relation at all. And I’m not sure what “logical priority” means.
    Here’s my thinking. It seems clear that God’s beliefs, whether contingent or necessary, depend on God. Furthermore, those beliefs of God that necessarily exist will depend on God in the same way the beliefs that contingently exist do. So, I hope that makes it clear to you that it at least makes sense for God’s belief that God exists (a belief that exists in all worlds) depends on God. And I think it is in a similar way that numbers might depend on God’s mind. (This is insofar as it makes sense for abstract objects to depend on a concrete object like God.)

    June 24, 2012 — 16:53
  • Steve Maitzen

    I hope that makes it clear to you that it at least makes sense for God’s belief that God exists (a belief that exists in all worlds) [to depend] on God.
    Andrew: Thanks for your patience. I can try making sense of it in terms of explanation or presupposition:
    (E) God’s necessary existence at least partly explains God’s necessarily believing that God exists, but not conversely.
    (P) God’s necessarily believing that God exists presupposes God’s necessary existence, but not conversely.
    My problem is that paradigms for the concepts of explanation and presupposition imply either (1) a nontrivially true subjunctive conditional or (2) a one-directional entailment (or both); it’s hard to make sense of (1) when the antecedent is impossible and hard to make sense of (2) when the antecedent and consequent are true in exactly the same worlds. So unfortunately I’m no further ahead.

    June 25, 2012 — 7:04
  • 1. Abstracta can be mind-dependent and acausal. It could be that an object B is grounded in one or more objects with causal powers, with B itself not having any causal powers. For instance, maybe, wealth has no causal powers. (Any story about wealth causing something is elliptical for a real causal story.) But wealth is grounded in our practices, and our practices are at least partly grounded in our attitudes and activities, and our attitudes and activities certainly have causal powers.
    2. But I don’t see why abstracta should be acausal. I have never seen a good argument for why a property or a proposition or a number should be acausal. First of all, Plato who introduced the idea of abstracta didn’t seem to think they were causal–that’s evidence that acausality isn’t a part of their concept. Second, I think of abstracta primarily as something posited to solve some philosophical problems. But positing acausality does nothing to contribute to the solutions to the problems. On the contrary, it is the major contributor to the epistemological problems with Platonism.
    3. That said, I do share Plantinga’s intuition. I find abstracta hard to believe in, especially sets, then to a slightly lesser extent numbers, and then to a somewhat lesser extent properties. But given a reduction to divine thoughts I feel much better about it. So for me the argument is something like:
    (a) positing abstracta solves all sorts of problems (or so it’s said)
    (b) but it is weird to think that there exist sets, numbers and properties in the way modern Platonists say there do
    (c) divine thoughts can fill the theoretical roles of sets, numbers and properties
    (d) so, we have reason to think God exists and has thoughts that do fill the theoretical roles of sets, numbers and properties.
    The thought behind (b) is basically a nominalist intuition…

    June 25, 2012 — 8:07
  • Steve:
    I think there is little hope in analyzing explanation (or much of anything else!) in terms of subjunctive conditionals. One gets all the finking problems under the head of the “conditional fallacy”. It’s hopeless.
    And there are plenty of cases where one necessary truth explains another.
    Here are three representative cases.
    Science: Why does water have oxygen atoms in it? Because water is H2O.
    Ethics: Why is torturing the innocent always wrong? Because torturing the innocent fails to respect their dignity.
    Mathematics: Why is 27 not a prime? Because 27 is divisible by three.

    June 25, 2012 — 14:18
  • Heath White

    My suspicion is that the average person on the street finds Platonist abstracta VERY hard to believe in, but that analytic philosophers have been force-fed enough Frege & co. that they have lost their ordinary intuitions about this. (Generalizing unfairly, my impression is that continental philosophers think a non-intuitive amount of things are socially constructed.) When I have asked my students whether the number 3 exists, the answers I get are confused.
    I used to be a sort of conceptualist about numbers and properties. I now tend to think numbers are something like the structure of any possible sequence of items. My intuitions can perhaps be motivated by something like this: we wouldn’t have any concept of numbers unless we had something to count, and we wouldn’t have any concept of blueness unless we had blue things. So those numbers and properties (etc.) are in some way derivative or epiphenomenal with respect to the concreta.

    June 25, 2012 — 15:26
  • Dianelos Georgoudis

    If X truly exists independent of mind then it can’t possibly be known. But then it makes no sense to say that X exists in the first place. Or, to put it differently, it is a waste of time to wonder whether a mind-independent X exists or not.

    June 25, 2012 — 17:04
  • Steve Maitzen

    Alex: Nice examples. Thanks. Although in each case the explanandum and the explanans are (taken to be) logically or metaphysically equivalent — i.e., true in exactly the same worlds — in cases 1 and 3 the explanans is analytically sufficient but not analytically necessary for the explanandum. Or so it seems to me. (I’m not sure what to say about case 2.) Anyway, getting further into the niceties of explanation would sidetrack Andrew’s thread, so I’ll try to return to his topic.
    Could it be that all necessarily existing abstracta depend on God? I take it that Plantinga rejects as incoherent the notion that the Law of Noncontradiction (LNC) depends in any way on God for its truth (I’m thinking back to what I recall from Does God Have a Nature?). He’s right. On the last page of “How to Be an Anti-Realist,” Plantinga writes, “It is thus not the case that a proposition is true because God believes it. On the other hand it is the case, I think, that a proposition exists because God thinks or conceives it.” He doesn’t say why one and not the other. Why think that (say) LNC depends on God for its existence but not for its truth? Is Plantinga assuming that because we and God can think (of) propositions, it must be that propositions are thoughts?

    June 26, 2012 — 7:52
  • “Why think that (say) LNC depends on God for its existence but not for its truth?”
    Suppose we think of the target audience for Plantinga’s argument as people with nominalist intuitions. Now the nominalist thinks that LNC is true but doesn’t exist. That sounds self-contradictory, but of course the nominalist may have a story to tell, say one involving a deflationary theory of truth. Well, if the idea that LNC is true but doesn’t exist is among the conceptual possibilities, we might likewise suppose that the idea that LNC doesn’t depend on God for its truth but does depend on God for its existence is among the conceptual possibilities.
    In fact, I think there is nothing that strange about the truth-grounds and existence-grounds of a proposition falling apart. Take the proposition that Steve exists. Its truth is grounded in Steve. But its existence isn’t grounded in Steve.

    June 26, 2012 — 8:06
  • Steve Maitzen

    Thanks, Alex, but I don’t find the talk of “grounding” illuminating. If I try to understand the ontological relation between Mars and {Mars} by saying that the former grounds the latter but not conversely, I don’t think I’ve thereby improved my understanding of that relation. Still less does it help me in the case of necessarily existing things — to say, for example, that 2 grounds {2} but not conversely. Is there a good explanation and defense of grounding that you can recommend?

    June 26, 2012 — 18:51
  • Andrew Moon

    Thanks for helping make clear what Plantinga was after.
    I don’t think I have a further explication of the dependence or grounding explanation. I think that I understand it, though, even though I don’t think I’d explain it by way of counterfactuals. That’s all I’ve got. =P

    June 27, 2012 — 22:45
  • Cruz Davis

    Gideon Rosen has an interesting paper defending grounding that you can find here:
    He takes grounding to be a primitive but well-behaved notion that is necessary to do a decent amount of basic work in many different realms of philosophy.
    Fabrice Corriea and Kit Fine also have some interesting work on grounding.
    There is also some interesting work on non-modal notions of dependence that you can find in Fine’s paper ‘Ontological Dependence’ and then Koslicki has a couple of papers that she is working on that seem interesting.

    June 29, 2012 — 14:34
  • Steve Maitzen

    “Primitive.” I was afraid of that. But thanks for the references! I’ll have a look at the Rosen and Fine papers.

    June 30, 2012 — 8:07
  • Cruz Davis

    What kind of notions would you be more comfortable with as primitives, if grounding isn’t one?

    June 30, 2012 — 14:18
  • Steve Maitzen

    Cruz: Like anyone who doesn’t confidently grasp a particular notion, I’m disappointed to be told that the notion is “primitive.” In general I dislike appeal to primitives, but particularly when I’m not in the club of those who say they understand them.
    Primitives cause me the least discomfort when I can see how it might be impossible, or fruitless, to analyze them in terms of other notions. So truth might be one such primitive; maybe the law of noncontradiction is another.

    July 1, 2012 — 11:58
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