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#141 God and Abstract Objects

December 28, 2009

Dear Dr. Craig,

I have found your review of the pros and cons for the existence of abstract objects the most helpful and clearest I have encountered. Nevertheless, I have some questions and I would be very grateful for your help.

First, regarding the Uniqueness Objection to Platonism you write in Creation out of Nothing (171,4,11-12): "But if Platonism is true, there is a unique sequence of abstract objects that is the natural numbers." But why should this be so? There are many things one can order in a sequence of first, second etc. One can order horses by weight, people by income, soldiers by rank etc. Each of these ranks of concrete objects would have an abstract Platonic Form. While each of these Forms is unique as regards the object, none of them would be unique as regards their ordinal structure because they are ranked in the same way as first, second, etc. Presumably, Plato would understand this situation as a hierarchy of Forms with the Forms for the concrete objects subsumed under the single Form for ordinal structure. I suspect that you would reject my argument for the non-uniqueness of the ordinal structure of the natural numbers by arguing that the highest Form in this hierarchy is the unique sequence of abstract objects that is the natural numbers. Am I correct? I would counter on behalf of Plato that the highest Form of the natural numbers is not unique because it participates in the lower Forms. So I do not see that Platonism requires the sequence of abstract objects that is the natural numbers to be unique.

Second, also regarding the Uniqueness Objection to Platonism you write in Creation out of Nothing (171,4,5): "The internal properties of numbers are irrelevant to mathematics; only their relational properties rooted in that ordinal structure matter." But, if I am correct, then internal properties of numbers include: 'is divisible by' and 'is greater than.' External properties of numbers include such properties as: 'a number interests me' or 'I think about a number.' Given that internal properties include the property 'is greater than', this internal property entails the sequence of abstract objects that is the natural numbers. Would this affect your uniqueness argument?

In Creation out of Nothing (p. 173) you argue that Platonism must be rejected because it "posits infinite realms of being that are metaphysically necessary and uncreated by God." But in your website in the answer to Question 94 about Classifying Immaterial Objects, you write: "What I have come to realize since writing that chapter is that, even more shockingly, some abstract objects, if they exist, exist not merely in time and space but also exist contingently!" Am I correct in concluding that the contingent existence of some abstract objects still leaves the necessary existence of other abstract objects as a theological reason for rejecting Platonism?

With many thanks for your attention.

Jitse van der Meer

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Dr. craig’s response


Thank you, Jitse! (Are you related to the philosopher of science by the same name?) I presume you're referring to my chapter "Creatio ex Nihilo and Abstract Objects" in my and Paul Copan's book Creation out of Nothing (Baker, 2004). For readers who aren't familiar with that discussion, let me say briefly by way of background that the school of thought known as Platonism holds that in addition to concrete objects, which are things that can stand in cause-effect relations, there are also abstract objects, or things that cannot stand in such relations, things like numbers, sets, propositions, and properties. As I explain in my chapter, there are two typical objections posed to Platonism: (i) the epistemological objection, which says that in view of their causal isolation, knowledge of mathematical objects ought to be impossible on Platonism, which would leave us bereft of any mathematical knowledge; and (ii) the uniqueness objection, which states that virtually anything can fill the role of a mathematical object, so long as it stands in the right relations with other objects, since that's all that's needed for mathematical truth.

But, as I made clear, neither of these is my reason for rejecting the existence of abstract objects. Rather my misgiving is theological: Platonism compromises God's aseity (self-existence) and undermines the doctrine of creatio ex nihilo by positing beings which are self-existent and uncreated by God. It implies a sort of metaphysical pluralism according to which God is not the ground of being for everything other than Himself.

Now in your response to the uniqueness objection, you ask why the Platonist must be committed to the view that there is a unique sequence of abstract objects which is the natural numbers. You rightly note that even a collection of concrete objects can stand in the structural relations in which the natural numbers stand. (This is the so-called "Caesar problem" noted by Gottlob Frege in the 19th century: why couldn't Julius Caesar be a natural number?) This just is the point made by the uniqueness objector. Your proposal for circumventing this problem is known as structuralism, the view that there really is no unique series of objects that are numbers; rather numbers are just the places in the ordinal structure into which any objects can be slotted. So there really isn't an object which is the number 3, for example; rather there is just the third place in this structure which can be filled by any object. Structuralists thus espouse a novel and, I think, attractive approach to what mathematical entities are.

The original question then arises all over again with regard to these structures: are they merely useful fictions or do they actually exist? Structuralists who are also Platonists are called ante rem structuralists, "ante rem" indicating that the abstract structure exists ontologically prior to any concrete sequence of objects which is an instantiation of that structure.

My theological concern remains then unresolved by ante rem structuralism. For these structures seem to have an existence which is independent of God. The resolution of the uniqueness objection proposed by the ante rem structuralist does nothing to alleviate my objection to Platonism.

So my response to your proposal would not be the objection you suggest. Rather my objection would be the same as before: the existence of structures uncreated by God compromises divine aseity and creatio ex nihilo. (By the way, what passes for Platonism today shouldn't be identified with what Plato himself actually believed. For Plato, the Forms do not seem to be at all causally impotent but shape the world to be as it is. The debate over so-called abstract objects is actually a very recent development of contemporary philosophy which arose only in the late 19th century.)

Your second point misidentifies certain properties as intrinsic rather than relational. Properties like being divisible by or being greater than are precisely the sort of relational properties that the uniqueness objector has in mind. All you need for mathematics is these sorts of relational properties among objects; intrinsic properties don't matter. That is precisely the insight appropriated by the structuralist in advocating that mathematical objects just are the positions in some structure, regardless of what, if anything, occupies those positions.

My work on abstract objects and divine aseity is a work in progress, and, as you note, I've come to realize that not all abstract objects, if they exist, exist necessarily, the way that numbers and properties do, contrary to the usual assumption. Some abstract objects, if they exist, exist contingently. They might, therefore, be created by God, as so-called Absolute Creationists maintain. For example, by creating the earth, God can be said to also be the Creator of the abstract object which is the earth's equator. God's aseity and creatio ex nihilo are thereby preserved. Of course, there still remains, as you observe, the problem of abstract objects which seem to exist necessarily, if they exist at all.

But even worse, as I've come to see, there seems to be a kind of entity which we may call an uncreatable. This is the sort of thing that I noted in my aforementioned chapter when I raised the boot-strapping objection (or vicious circle objection) to Absolute Creationism. Uncreatables are objects which cannot be created by God because in order for them to be created they must already exist. For example, take the property of being powerful. That property cannot have been created by God because in order to create that property God must already possess that property! In order to create the property of being powerful, God must already be powerful. The Absolute Creationist is thus caught in a vicious circle.

So what to do? The would-be Christian Platonist cannot consistently say that God can be powerful without possessing the property of being powerful, for that is to embrace the nominalist position, which holds that properties are illicit reifications of grammatical nominative expressions (like sentence subjects, direct objects, objects of prepositions, etc.). If God can be powerful without possessing the property of being powerful, then what need is there of properties for anything? Why can't a dog be brown without possessing the property of being brown? Property talk seems to be just a useful and easy way of speaking, but it is bad metaphysics to reify such talk by unnecessarily postulating abstract objects which are the properties of things.

This difficulty of uncreatables, it seems to me, is the real heart of the problem posed by the existence of abstract objects. The Bible affirms that God through His Word is the Creator of all else that exists (John. 1: 1-3), an affirmation that assumed creedal status in the confession of the Nicene Creed that God is "the maker of all things visible and invisible." I therefore cannot square Platonism with Christianity.

- William Lane Craig