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Why Abstract Objects Pose A Nasty Problem for Christian Theists

Philosophers have some pretty good arguments for the existence of abstract objects -- immaterial, timeless, spaceless, acausal entities that aren't concrete, such as propositions, properties, possible worlds, numbers, sets, and the like. When I was a Christian and an aspiring apologist, I was prodded to think (by apologist philosophers like J.P. Moreland and Alvin Plantinga) that abstract objects posed a nasty problem for non-theistic views of the world, such as naturalism (the view that the natural world is all there is). I also thought that such immaterial entities could best be explained in terms of God. For Christian philosophers have traditionally taken them to be (roughly) thoughts in the mind of God. Actually, there are a variety of views about the way in which abstract objects are taken to depend on God, but all such views can be classified as versions of what is known as 'theistic activism' -- the view that abstract objects depend on God in one way or another. In light of these sorts of considerations, Christians often use the existence of abstract objects to support theism and critique naturalism. The line of reasoning can be put in any number of ways, but here's a common one (although I seldom hear its proponents make the logic of the argument explicit):
If you deny the existence of God, then the most plausible alternative view for you to take is the view that the physical world is all there is. For if you thought that non-physical things existed as well, then you'd have to say that they arose from the physical. But nothing but physical entities can arise from the physical; therefore, you'd have to posit something immaterial, and very much like a god, to explain such things. Unfortunately, it's just not true that the physical world is all there is. For there are good reasons to think that abstract objects exist, such as numbers, propositions, possible worlds, moral values, etc. The existence of such things are crying out for explanation, and yet your naturalistic view of reality can't explain them. On the other hand, theism can handle them quite naturally. For God, you see, is an immaterial object, and so his nature bears the required sort of affinity with abstract objects to be able to cause, or in any case explain, their existence. Now a plausible and natural way to account for the relation between God and abstract objects is that of thoughts to a thinker; that is, as divine concepts -- they are the architecture of God's mind, as it were. For concepts, like other sorts of abstract objects, are immaterial. Furthermore, many abstract objects, such as propositions, are inherently representational, as are thoughts and concepts. Therefore, since theism can explain abstract objects quite naturaliiy, and naturalism cannot, abstract objects confirm theism and disconfirm naturalism.
Unfortunately, this argument is pretty terrible. For it turns out that (i) theistic activism is prima facie incoherent, and (ii) the argument relies on the dubious assumption that non-theists should adopt an extremely crude form of materialism. Let's discuss (i) and (ii) in turn.

Regarding (i): theistic activism is incoherent:
If you read the recent philosophical literature on theistic activism, you quickly realize that abstract objects actually pose a very nasty problem for Christian theism. To see this, consider a fairly recent and more-or-less standard critique of theistic activism by philosopher Matt Davidson (his paper is entitled, appropriately enough, "A Demonstration Against Theistic Activism". The paper is online -- you can google it). Here's my gisty summary of his argument:
God can't be the cause of abstract objects, for *being omnipotent* is both an abstract object and one of God's essential properties. If so, then it must exist and be instantiated before God can do anything at all. But God can't create and instantiate his own essential properties, for that would require him to be causally prior to himself, and that's wacko (and you can just forget about the Thomistic solution of collapsing the essence/existence distinction for God). But if at least some abstract objects aren't due to God's causal activity, then theistic activism is unmotivated.
Furthermore, most philosophers who accept the existence of abstract objects also think that they exist of metaphysical necessity -- that is, they cannot fail to exist. Or to put it another way, they exist in all possible worlds. Why do they think this? For a number of reasons. Here a two. First, since abstract objects seems to be timeless, spaceless, and acausal, then it would seem that they are immune to the conditions of concrete existence that render the latter contingent (e.g, if they're timeless, then they neither come to be nor pass away; if they're acausal, then they seem immune from things causing them to come to be and pass away, etc.) Second, at least some properties of many sorts of abstract objects seem to hold of logical necessity. So, for example, suppose you are a philosopher who is a realist about numbers -- you think that numbers exist and are abstract objects. Then since it's not just true, but necessarily true that 1+1=2, it's true in all possible worlds that 1+1=2. If so, then it's natural to think that numbers and mathematical propositions exist in all possible worlds (otherwise, there might be a possible world in which '1+1=2' is false). But if so -- and here's the punchline -- abstract objects don't need an explanation for their existence in terms of something beyond themselves. For they can't fail to exist; if the reason why abstract objects exist is because it's metaphysically impossible for them to fail to exist, then one can hardly ask for a better reason for their existence than that (if not, then God is in trouble!).

So it turns out that if you look closely at the doctrine of theistic activism, it turns out to be prima facie incoherent: (a) God's causal activity is necessarily dependent on the prior existence of at least some abstract objects (e.g., the property of being omnipotent), and (b) abstract objects exist of metaphysical necessity, in which case they need no explanation anyway -- Indeed, they can't have an explanation (as we've just seen with an attempt to explain them in terms of God). But if that's right, then theism, no less than crude forms of materialism, can't explain the existence of abstract objects.

Of course, a theist can avoid the problem by just getting rid of the idea that abstract objects depend on God for their existence. After all, as we've just seen, the reasons philosophers have for thinking that they exist at all are equally reasons for thinking that they're necessary beings -- they exist of absolute necessity. If so, then they don't need an explanation in terms of something beyond themselves. So the theist can just say that abstract objects are necessary beings. And if they hold to a traditional doctrine about God at least as ancient as Anselm -- viz., that God is a necessary being -- then they can say that although God is just one of the infinitely many necessary beings, he's nonetheless unique and special in the sense that he's the only one among the infinitely many necessary beings that's a concrete, substantial being. However, they may not like this, since it diminishes the doctrine of God as absolutely sovereign and the creator and sustainer of everything else that exists; for on this revised account, God is neither the creator nor the sustainer of abstract objects.

On the other hand, if they hold (as, e.g., Christian philososopher Richard Swinburne holds) that God isn't a metaphysically necessary being, but rather a factually necessary being -- i.e., that there are possible worlds in which God does not exist, but given that he does exist, he's eternal, all-knowing, all-powerful, the creator and sustainer of all else that exists (except for abstract objects) -- then God's greatness seems to be a bit diminished by the fact that abstract objects have a greater kind of existence than God, viz., metaphysically necessary existence.

In either case, though, theists do not have a piece of evidence for theism and against naturalism with the existence of abstract objects. For abstract objects (a) are necessarily existent entities, and thus need no explanation (indeed, this is so even if one accepts the Principle of Sufficient Reason), and (b) theism cannot -- logically cannot -- explain abstract objects in terms of the causal activity of God. What about non-theists, though? Don't abstract objects render their view of reality hopelessly implausible? This brings me to my last point.

Regarding (ii): the argument relies on the dubious assumption that non-theists should adopt an extremely crude form of materialism:
Contrary to what the argument asserts, abstract objects do not pose a problem for non-theists in the least. This is for at least two reasons. First, as we've already seen, if abstract objects exist, then there's excellent reason to think they're necessarily existent entities -- i.e., it's impossible for them to fail to exist. If so, then there's no need to postulate an explanation of their existence. But second, non-theists aren't commited to a crude form of materialism. They need not be commited to the view that the material world is all there is. Rather, they can happily grant the existence of abstract objects. Let me explain this by returning to the argument for a god from the existence of abstract objects.

Recall that a key premise of the argument was that if theism is false, then one must account for everything in terms of physical objects. And the argument for that premise was that only the physical could arise from the physical. But now we can see what's wrong with this inference (at least one of the things). For the non-theist need not explain the existence of non-physical, asbtract objects in terms of the physical if the latter never "arose" at all, but rather are timeless, spaceless, acausal, eternal, necessarily existent entities. A non-theist can hold that all contingent reality is physical, or arose from the physical, all the while serenely granting the existence of abstract, immaterial entities that exist of metaphysical necessity. For again, (i) if abstract objects exist of necessity, then they need no explanation, and (ii) God cannot explain the existence of abstract objects. Thus, the existence of abstract objects pose no problem at all for the non-theist.

To conclude: to the theist who asks me how I explain the existence of abstract objects, I say, "you're falsely assuming that abstract objects need an explanation, as well as that non-theists can only plausibly accept a crude form of materialism. But neither assumption is correct. As to the first assumption, abstract objects can't fail to exist if they exist at all, in which case they're in no need of explanation in terms of something beyond themselves. As to the second, and relatedly, non-theists aren't commited to crude materialism, especially if abstract objects exist of necessity, and thus need no explanation -- much less of an explanation in terms of the material world. But to turn the tables, how can you account for abstract objects? For if you take properties to be abstract objects, then you can't plausibly take them to be explained in terms of the causal activity of God. For God's ability to cause anything is posterior to the existence of at least some properties -- most saliently, in this case, the property of being omnipotent. And if some abstract objects don't depend on the causal activity of God, then what principled grounds can be offered for saying that any must so depend on him? (And if that's right, then what happens to the docrines of absolute creation and sovereignty?) So the argument seems to turn itself on you; abstract objects aren't puzzling in the least for non-theists; they are, however, for theists."

A Short Bibliography on Theistic Activism
(It should be noted that every philosopher below is a Christian theist)

Bergmann, Michael and Jeffrey Brower. “A Theistic Argument Against Platonism (And In Support of Truhmakers and Divine Simplicity)”. Oxford Studies in Metaphysics 2 (2006), 357-386. Available on line here.

Davidson, Matt. “God And Other Necessary Beings” (entry at the Stanford Encyclopedia of Philosophy, available here. See also the bibliography at the end of the article for further readings)

-“A Demonstration Against Theistic Activism”. Religious Studies 35 (1999), pp. 277-290. Available online here.

Morris, Thomas V. and Christopher Menzel. “Absolute Creation”. American Philosophical Quarterly 22 (1985), pp. 353-362.

Plantinga, Alvin. Does God Have A Nature? (Milwaukee: Marquette University Press, 1980).

-“How To Be An Anti-Realist” Proceedings and Addresses of the American Philosophical Association, 1982, pp. 47-70.

-“Two Dozen (Or So) Theistic Arguments” Available online here.


Chad said…
Your representation of the argument from abstract objects is a bit dense. Both (i) and (ii) are dispensable to a good argument for God based on abstract objects. Briefly to boot, it is doubtful that most reasonable presentations of the argument from abstract objects would maintain something as bold as (i). Rather, they’d most likely mention so-called ‘arguments from queerness,’ which highlight not inconsistency per se between naturalism and abstracta, but tension. As such, broader issues would surface, such as which theory most plausibly incorporates and accounts for abstracta. Moreover, a deductive outline of the argument from abstract objects to God can be given that doesn’t rely on (i).

Lastly, (ii) just seems false. Your description of theistic activism is wrongheaded. Theistic activism is not “ that abstract objects depend on God in one way or another,” but “the view that a divine intellectual activity is responsible for the…Platonic realm of necessity as comprising necessary truth as well as necessarily existent objects,” in a way that has “Abstract objects ‘depend on God as their cause’” See Thomas V. Morris and Christopher Menzel, “Absolute Creation,” in Thomas V. Morris (ed.) Anselmian Explorations: Essays in Philosophical Theology (University of Notre Dame Press, 1987), p. 168, 162, 164, respectively. As such, theistic activism is a very distinct view of the relation between God and abstracta. All of the critical literature recognizes this and proceeds on those grounds. Curiously, the very articles you cite a clear about this, and the one by Davidson in particular even sketches a possible alternative to theistic activism. Furthermore, Davidson is not alone in his dual project of rejecting theistic activism and proposing an alternative. See also Greg Welty, An Examination of Theistic Conceptual Realism as an Alternative to Theistic Activism (Oxford, 2000). So while it is true that “there are a variety of views about the way in which abstract objects are taken to depend on God,” it is patently false that “all such views can be classified as versions of what is known as 'theistic activism'”.

If the argument from abstract objects is a failure, it seems to be that you haven’t given us the reasons to think so.
exapologist said…
Hi Chad,

Ah, you even have the *snarkiness* I used to have as a Christian. The case for being counterparts builds...

It's true that Morris and Menzel give a narrower characterization of theistic activism than the one I gave, but I wanted to broaden the definition so as to an account for Davidson's characterization of the view in his "A Demonstration Against Theistic Activism":

"Some have contended that (necessarily existing) abstracta depend on God for their existence and natures (their essential properties). Let's call such a view theistic activism." (Davidson, 1999, p. 1)

Thus, on Davidson's broader characterization here, theistic activism is characterized in terms of just an asymmetric dependence relation between God and the existence and nature of abstracta.

You're surely right that it's possible to give an argument for theism from abstract objects that use a different line of reasoning, but here I'm specifically addressing a popular line of thought I think is at least implicit in two apologists in particular: Moreland and Vic Reppert. When I was a grad student at Talbot, I remember Moreland discussing this sort of argument in his Metaphysics I class. He also mentions it in his "intermediate level" apologetics book, Scaling the Secular City (e.g., as a piece of evidence against naturalism), but also in introductions and chapters to books he contributes to, such as the Intro. to Jesus Under Fire, and, I think, the intro., or his chapter, in The Creation Hypothesis). Is it in Philosophical Foundations for a Christian Worldview? I can't remember off-hand.

But Welty's theistic conceptual realism seems to me to fare no better. I think he fails to beat back Loux's criticism, which the former discusses at the end of his Master's thesis. Loux objects that such an account gets the order of explanation backwards: God can do x because x is possible; it's not that x is possible because God can do x. Welty dismisses this criticism too quickly. The deep point here is that metaphysical possibility is an *intrinsic* feature of entities, but accounts like Welty's entail that possibility is *extrinsic* to an entity -- it depends on what God can do. That's just obviously wrong, I think.

In any case, I think that people like Moreland and Bonjour have given plausible responses to Beneceraff's classic objection to knowledge of abstracta (see Moreland's Universals, and Bonjour's In Defense of Pure Reason, as well as his exchange in the PPR book symposium on it). Thus, I see no good reason to move from straight Platonism to theistic conceptualism.
Chad said…
It's true that Davidson gives a narrower characterization of theistic activism than Morris and Menzel’s, but Davidson’s article is clearly targeting Morris and Menzel’s. But even on Davidson's broader characterization of theistic activism as “an asymmetric dependence relation between God and the existence and nature of abstracta”, it remains the case that not all views “about the way in which abstract objects are taken to depend on God” are explicated in terms of “an asymmetric dependence relation between God and the existence and nature of abstracta.” Theistic Conceptual Realism avoids this (at least post creation; sans creation, however, TCR would perhaps have to deal with that problem), for example. Thus, it is still misleading to say “all such views can be classified as versions of what is known as 'theistic activism'”. The critiques of Davidson and others you mention in (i) are therefore welcome contributions to other views relating God and abstracta.

So where does this leave your critique of the argument for God from abstract objects? As I said, (i) isn’t likely to be a premise in more sophisticated versions of the argument (though I do realize, pace Moreland, Reppert, et. al., that this is a common way of advancing the argument). Much of what you outline in (ii) also seems avoidable for a good argument from abstract objects. But you do mention one objection in (ii) that I think would apply to all of the views “about the way in which abstract objects are taken to depend on God”, and that is that abstract objects, as metaphysically necessary beings, “don't need an explanation for their existence in terms of something beyond themselves.”

I appreciate the reference to Swinbrune’s distinction between metaphysical and ontological (perhaps what you referred to as factual) necessity, but unfortunately I think you’ve missed an important aspect of it. Explanation or dependency for metaphysically necessary beings is not ruled out tout court. In fact, as Swinburne defines it, if something exists of metaphysical necessity, it is possibly admitting of an eternal cause—i.e., explanation. To see this, all you need to do in conceive of a dependence relation between two metaphysically necessary beings. As such, the existence of metaphysically necessary beings is readily acknowledged as possibly ab alio. Leftow says metaphysically necessary beings “derive their specious plausibility from insufficiently precise understandings of alethic necessity. According to currently popular semantics, “x exists necessarily” asserts only that x is to be found in every possible world. It entails nothing at all about why this is so; it leaves open the question of whether there may be some cause or causes which account for this” (“A Leibnizian Cosmological Argument” Philosophical Studies 57 [1989], p. 137). On the other hand, it is ontologically necessary beings which “don't need an explanation for their existence in terms of something beyond themselves.” So only if abstracta are ontologically necessary does it follow that “they need no explanation…. Indeed, they can't have an explanation,” for existence a se is not a necessary characteristic of metaphysically necessary beings. I leave it to your intuitions to decide whether ontological or metaphysical necessity is greater.

But what about Loux’s objection? I must admit being somewhat sympathetic to this objection, and also felt it received under-deserved attention in Welty’s masters thesis—and so it was gratifying to see him give it a bit more attention in his D.Phil thesis (2006) on the same topic.

But I don't think it succeeds. As Welty points out, quoting Loux, “To suggest ‘that we need an account of possibility that is independent of what God could or could not have done’ implies that genuine possibility for any created world is independent of divine power, but this is precisely what is to be proved.” Indeed, if abstracta are intimately grounded in God, as the defender of theistic conceptual realism would hold, then it should not be surprising that God’s omnipotence should be the standard for an acceptable theory of modality. Here are some questions which would help bring the objection’s force back: why must an acceptable theory of modality be reducible to nonmodal notions? Why must an acceptable theory of modality be more fundamental than omnipotence? Though forceful, Loux’s objection can be sufficiently beaten back.

What reasons are there for preferring theistic conceptualism over Platonism? No doubt it would be the task of the theistic conceptualist to meet contemporary defenses of platonism. But in fact I think this can and has been done. See for example Colin Cheyne’s Knowledge, Cause, and Abstract Objects: Causal Objections to Platonism (Springer, 2001), or the argument that platonism commits us to the existence of an actual infinite. This is to say nothing of what other reasons we might have for preferring conceptualism over Platonism. So again it seems to me that you have not done much against the argument for God from abstract objects.

As a side note, I really apologize if I came off as snarky! I really enjoyed your post and appreciated your response equally as much!
wissam h said…
@Ex Apologist,

I think you'd enjoy reading Kenneth Boyce's paper: A Response to Bergmann and Brower.

He argues that Bergmann and Brower's argument doesn't refute theistic activism.

exapologist said…
Hi Wissam,

Thanks for the pointer. I'll have to give it a close read, but I'm antecedently inclined to think that theism is compatible with platonism. My own concern is not primarily with compatibility. It is rather with whether theism is good (or even decent) evidence for theism. I'm still inclined to think it isn't. I myself lean toward platonism about propositions, properties, possible worlds, and the like. I think they're eternal, uncreated, existentially independent, necessarily existent beings that God (if a god should exist) did not create. Tacking God on to explain the data related to abstracta seems to needlessly complicate one's ontology, or so it seems to me.

wissam h said…
I'm actually inclined to believe that platonism poses a problem for theism to some extent, given what I read about the subject (esp. Bergmann & Brower, and others.) You have also argued that in this post. Did you change your mind on the matter?

I haven't read the alternative theories theists provide to reconcile abstracta with God, but I think they provide some sort of conceptualism.
James Urban said…
Yablo's figuralism, problem solved.
Luis Takahashi said…
Hello, exapologist

As Chad pointed above, it is difficult for the naturalist to account for the existence of abstract objects just by appealing to metaphysical necessity. A metaphisically necessary being is still in need for an explanation. Thomists would argue that God is a metaphysically necessary being, and they defined God as a being whose essence is existence, which therefore explains why God exists. As I recall from reading your blog, you defend a version of the PSR that the naturalist might safely agree with. That being said, I still don't know how a naturalist can avoid a materialist world at all.
exapologist said…
Hi Luis,

There is a standard explanation amongst philosophers that I find persuasive: Just as in theoretical physics, we posit theoretical entities and attribute the relevant features to them when they are needed to explain the relevant data. Abstract objects are posited to explain a wide range of data, and they are posited to have necessary existence to do this adequately. So, for example, mathematical truths are true in all possible worlds, and thus we posit the mathematical facts that make them true to exist in all possible worlds.

One might further ask what makes them necessary, but I'm not sure I can make sense of that question: how could it be otherwise that, say, 1 + 1 = 2? Furthermore, those who aren't modal skeptics might argue that it's prima facie conceivable that God doesn't exist, while it's not prima facie conceivable that 1+1 is possibly 3. Therefore, if conceivability is prima facie evidence of possibility, then we have a positive reason to doubt that it's possible for God to ground the necessity of mathematical truths.


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