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#305 God and the Applicability of Mathematics

February 17, 2013

Dr. Craig,

In your recent debate with Dr. Rosenberg, you bring to the table two new arguments (at least that I've never seen you propose before). I am enamored with the argument against naturalism based on intentionality. My question regards the argument against naturalism based on the applicability of mathematics.

Isn't it the case that mathematics could, and in my opinion does seem to be, just a useful fiction as you mentioned in your debate? You say something along the lines of "this wouldn't explain how nature seems to be written in the language of mathematics". Isn't it also the case that if mathematical concepts are useful fictions, then they would describe (accurately if well thought out) the universe as apprehended by our perceptions? Shouldn't we expect that our useful fictions would be useful precisely because they accurately describe our observations?

I have thought that perhaps I am missing the point of the argument though. Perhaps it is the case that you aren't saying God must exist because our useful fictions, particularly those of mathematics describing reality, would just be happy coincidence. Indeed, what kind of coincidence would it be that our tools were designed for the purpose they serve? Perhaps you are making the point that without God the universe wouldn't necessarily exhibit these extremely logical properties.

Maybe I'm just completely wrong headed on this. Could you please set me straight?

Keep up the great work for God,


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Dr. craig’s response


Thanks for your kind comments, Brad! I was glad to be able to include two new arguments for the reasonableness of belief in God in my debate with Alex Rosenberg.

I, too, like very much the argument from intentional states of consciousness. Alvin Plantinga’s article “Against Materialism” had already convinced me that physical objects do not exhibit intentionality 1. But it was reading Rosenberg’s book, where he presents the same sort of considerations against physical objects’ exhibiting intentionality, that persuaded me to put this into the form of a theistic argument. Since God is an unembodied mind, the existence of minds fits much better into a theistic worldview than into a non-theistic worldview. Rosenberg’s conclusion that intentional states of consciousness do not exist is, given our experience, patently false. His answer that my experience of intentionality is an illusion is self-defeating. For since an illusion is itself an intentional state (we have an illusion of something), it is self-referentially incoherent to say, as Rosenberg does, that the experience of intentionality is illusory. An illusion of intentionality implies intentionality. Given the reality of intentional states of consciousness, theism seems much more probable than atheism.

But I digress. Your question is about the argument from the applicability of mathematics to the physical world. Question of the Week #277 is about the only place where I have reflected on this question, and I refer you there. Again, it was reading Rosenberg’s own book that prompted me to put this into the form of the theistic argument. For mathematics lies at the foundation of physics, at whose altar Rosenberg bows. Given his scientism (epistemological naturalism), he cannot dismiss applied mathematics as illusory. Rosenberg also emphasizes that naturalism simply cannot tolerate cosmic coincidences. But then what explanation can the naturalist offer for why mathematics is applicable to the physical world, that is to say, for why the physical world is imbued with the complex mathematical structure that physics discovers. Naturalism founders in this regard, whereas theism has an easy answer: God created the universe on the mathematical structure that He had in mind.

Now I’m inclined to agree with you that mathematical objects like numbers, sets, functions, and so forth are just useful fictions. That is to say, there are no mind-independent objects such as numbers. I do not think that this implies that mathematical statements, whether pure or applied, are not true. But that is a debate for another day. The point is that I am an anti-realist when it comes to mathematical objects. The anti-realist, however, still faces the question of the uncanny effectiveness of mathematics. Why does physical reality exhibit this incredibly complex mathematical structure? You say, “if mathematical concepts are useful fictions, then they would describe (accurately if well thought out) the universe as apprehended by our perceptions.” Of course! That’s what it means to be useful. But what we are asking is why these concepts are so useful. Why is it that physical reality exhibits this complex mathematical structure, so that mathematical concepts are applicable to it and thus useful in describing it? The happy coincidence is not that our useful fictions help us describe reality but that these fictions are useful in the first place!

So I think you are on target when you say “without God the universe wouldn't necessarily exhibit these extremely logical properties.” That’s right; without God that would be just a happy coincidence—which naturalism cannot tolerate, but which evaporates on theism.


  • [1]

     Alvin Plantinga, “Against Materialism,” Faith and Philosophy 23 (2006): 3-32.

- William Lane Craig