This is the weekly Q & A blog post by our Research Professor in Philosophy, Dr. William Lane Craig.


Hello Dr. Craig,

I have recently become interested in your work on abstract objects. I have a quick question regarding the Fregean argument for mathematical platonism. The argument concludes that mathematical objects exist because they are referred to by singular terms. For example, "3 is prime" is a true, simple sentence in which "3" is a singular term referring to an abstract object.

So, does the claim that abstract objects exist mean anything other than that they can be referred to by singular terms? I don't see how it could, since they have absolutely no impact on the world. But if that's the case, their "existence" seems to be more about the function of a word than anything to do with ontology.



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Dr. William Lane Craig’s Response

Dr. William Lane Craig

Just this morning, Ander, I was revising my chapter on neutralism as a response to the challenge posed by Platonism to the doctrine of divine aseity. I hope that general readers will forgive me for taking a question which connects with my current work in so insightful a way!

For those who don’t know the name of Gottlob Frege, he was a 19th century German mathematician with a strong philosophical bent who was the fount of the contemporary debate over the reality of abstract objects like numbers, sets, and other mathematical objects. As my colleague Paul Gould explains in his Introduction to the recent book Beyond the Control of God? Six Views on the Problem of God and Abstract Objects (Bloomsbury, 2014), this is a debate in which the Christian has an important stake, since the existence of such abstract objects threatens God’s position as the sole ultimate reality, the Creator of everything apart from Himself.

The argument which Frege gave for the existence of such objects remains the most influential today, namely, that our ability to refer to such objects in statements we take to be true requires that those objects exist.

Now at one level such an argument seems to be a preposterous attempt to read metaphysics off of language. Are we similarly to seriously maintain that true statements like “Your lack of understanding is disheartening” or “The whereabouts of the President are unknown” or “Wednesday is the day of the faculty meeting” commit us to mind-independent objects like lacks, whereabouts, and Wednesdays? That seems crazy.

But, in fact, as you quite astutely observe, Ander, Frege himself had a very thin concept of existence. In saying that there are objects correlated with such words, he meant merely that there are semantic objects of such words, that is to say, the words in question are singular terms. They pick out a particular thing to talk about. You’re right on target when you say of such objects that “their ‘existence’ seems to be more about the function of a word than anything to do with ontology.” Such objects just function as the referents of singular terms.

So on the contemporary scene, it is important to distinguish between what has been called “lightweight Platonism” and “heavyweight Platonism.” Contemporary Platonists who remain faithful to Frege’s understanding are lightweight Platonists. This sort of Platonism does not challenge divine aseity, I believe, because it does not make a metaphysical commitment to the reality of abstract objects, as heavyweight Platonism does.

To illustrate, John Burgess is a prominent Platonist who argues for the existence of mathematical objects. But do such objects really exist or are they just the semantic referents of mathematical words? Look carefully at what he says,

One very traditional sort of way to try to make sense of the question of the ultimate metaphysical existence of numbers would be to turn the ontological question into a theological question: Did it or did it not happen, on one of the days of creation, that God said, ‘Let there be numbers!’ and there were numbers, and God saw the numbers, that they were good? According to Dummett, and according to Nietzsche—or my perspective on Nietzsche—this is the only way to make sense of questions of ontological metaphysics. . . . I myself believe, like Russell, that analytic atheism [the thesis that theological language is meaningless] is false, and suspect, contrary to the Australians, that the Nietzsche-Dummett thesis is true. If as I believe the theological question does make sense, and if as I suspect it is the only sensible question about the italics-added real or capital-R Real existence of numbers, then I would answer that question in the negative; but then I would equally answer in the negative the question of the Real existence of just about anything.[1]

It’s clear that for Burgess a theological perspective — “the only way to make sense of questions of ontological metaphysics”—yields a clear, negative answer to Burgess’ question about the metaphysical existence of numbers. He rejects what he calls “capital-R Realism” in favor of a much weaker “realism” which amounts to just “a willingness to repeat in one’s philosophical moments what one says in one’s scientific moments, not taking it back, explaining it away, or otherwise apologizing for it.”[2] This weak realism does not presume to tell us “just what God was saying to Himself when He was creating the universe.”[3]

If mathematical and other abstract objects exist only in the weak Fregean sense of semantic objects, then no threat to divine aseity springs from their corner. It also follows that heavyweight Platonists have more work to do if they are to prove that such objects really exist.

This post and other resources are available on Dr. William Lane Craig's website:

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[1] John P. Burgess, “Mathematics and Bleak House,” Philosophia Mathematica 12 (2004): 30-1. I suspect from his remarks that Burgess probably denies the existence of composite, physical objects as well.

[2] Ibid., p. 19.

[3] Ibid.