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
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.