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#688 Explaining the Applicability of Mathematics

June 28, 2020

In the debate on the mathematical argument for God hosted by “Capturing Christianity,” and at 1:12:00 into the debate, Dr. Oppy asked Dr. Craig this question:

“Could God have freely chosen to make a physical world in which it was not the case that mathematical theories apply to the physical world because the structure of the physical world is an instantiation of mathematical structures described by those mathematical theories? [Could God have] freely chosen to make a world in which that was not the case?”

There are two options, right? If not, then it seems that what you’re going to end up saying is that it’s necessary that there be a physical world, mathematical theories apply, which means that you end up agreeing with what the naturalist said, right? That would be the explanation.

On the other hand, if it’s true, then it looks as though it’s just an outright contingency that mathematical theories apply to the physical world for the reason given, because it’s brutally contingent that God chose to make this world rather than other worlds that he could have made instead; we don’t have an explanation, right? When we get to free choice, and you think, “Why this rather than that?”, there’s no explanation now to be given of why you ended up with one rather than the other.

So, it looks as though either you’re going to accept necessity or you’re going to end up with ultimately “It’s a brute contingency,” which was the problem, that was the thing that was objectionable.”

You responded that you have no problem with God’s having made free choices that are ultimately inexplicable and that the theory still has greater explanatory depth than that which Oppy offered.

First, I have more of a request than a question. Could Dr. Craig expand on his reply to Oppy’s first contention, that “it looks as though either you’re going to accept necessity or you’re going to end up with ultimately ‘It’s a brute contingency,’ which was the problem”? Later, Oppy asked Dr. Craig if he was happy to replace the term “happy coincidence” with “brute contingency” in his formulation of the mathematical argument. This question was raised at 1:29:16. He notes that the disagreement between himself and Dr. Craig is the first premise. He thinks that the applicability of mathematics to the physical world is a brute fact, not a brute contingency.

Second, I do have a question. Regarding Oppy’s contention that the theist is also saddled with the problem of brute fact or brute contingency, could the same objection be raised against the Kalam or the fine tuning argument? Thank you for your time!


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Dr. craig’s response


Thank you for your thoughtful question, Weston! In the section you quote, Oppy presents the theist with a dilemma, either horn of which is thought to be unacceptable. Was it necessary that God create this world or could God have chosen to create a world in which the physical phenomena are describable by different mathematical laws?

The answer to this question is so obvious that to ask it is to answer it. Since the theist believes that God is a personal agent endowed with freedom of the will, of course He could have chosen to create a physical world characterized by different laws of nature, in which case physical phenomena would have had a different mathematical description. He could, for example, have chosen to create a world in which the physical phenomena were describable by Newtonian physics rather than relativistic or quantum physics.

So the only question is: what’s the matter with that? Oppy alleges that in that case no explanatory advance has been made, and we’re right back to where we started from: brute contingency. But Oppy is wrong. On theism, the applicability of mathematics to the physical world is a contingency, but it is not a brute contingency (a “happy coincidence”). It has an explanation in the free decision of a transcendent, personal Designer. What is, or may be, brutely contingent is God’s freely choosing A rather than not-A. But the applicability of mathematics to the physical world has an explanation on theism that naturalism cannot give. Theism thus enjoys greater explanatory depth than naturalism, which is an important theoretical virtue.

This same issue comes up in my debate with Eric Wielenberg on the best explanation of objective moral values and duties. Like Oppy, Wielenberg claims that moral values are brute necessities and that theism likewise must finally reach some brute explanatory ultimate. What I say there in response to Morriston and Huemer may be of help to you here:

Underlying my approach in this debate is the deep-seated conviction that explanatory depth is a theoretical virtue in ethics, just as it is in physics or mathematics. A theory which provides an explanation or grounding of ethical principles is superior to one which adopts what has been called a ‘shopping list’ approach, whereby one simply helps oneself to the principles one needs without any attempt at explanation. I resonate with the words of Shelly Kagan:

An adequate justification for a set of principles requires an explanation of those principles—an explanation of why exactly these goals, restrictions, and so on, should be given weight, and not others.  Short of this, the principles will not be free of the taint of arbitrariness which led us to move beyond our . . . ad hoc shopping lists. . . . Unless we can offer a coherent explanation of our principles (or show that they need no further justification), we cannot consider them justified, and we may have reason to reject them. . . . This need for explanation in moral theory cannot be overemphasized.[1]

My claim is precisely that Divine Command Theory is explanatorily superior to Godless Normative Realism and therefore the better theory.

Morriston and Huemer are quite content with an ethical theory that has no explanatory depth because, after all, explanations have to stop somewhere. Well and good; but as Kagan points out, this gives “no license to cut off explanation at a superficial level.”[2] Compare mathematical theory.[3] As philosopher of mathematics Penelope Maddy explains, explanatory depth is one of the most important theoretical virtues in mathematics.[4] Despite the broadly logical necessity (and self-evidence) of arithmetic truths such as 1<3, 2+2=4, 6-1=5, and so on, mathematicians would never be content with a theory that simply postulates an infinite blizzard of such truths. Rather they seek a theory in which such truths may be derived from explanatorily prior axioms, such as the Peano Axioms or, even better, set theoretic axioms such as lie at the basis of ZFC axiomatic set theory. Indeed, the singular accomplishment of axiomatic set theory is its amazing ability to provide a basis for the derivation of the whole of classical mathematics from a handful of axioms. The reason set theory is so prized is because of its astonishing explanatory depth. . . . But mathematicians would laugh at the idea that a mathematical theory which just postulated an infinity of arithmetic truths without explanatory depth would be a serious competitor to Peano arithmetic or Zermelo-Fraenkel set theory. Thus, Huemer’s appeal to arithmetic to justify adopting an ethical theory with no explanatory depth backfires.

Not only does theism have greater explanatory depth in accounting for the applicability of mathematics to the physical phenomena, but as in ethics, the theist’s explanatory ultimate is more satisfactory that the naturalist’s explanatory ultimate. Given the a priori nature of mathematics and the causal impotence of mathematical objects, it is astonishing that physical phenomena happen to have the elegant mathematical structure that they do. That’s why on naturalism the applicability of mathematics is just a happy coincidence, a fortunate accident. But it is not astonishing that a personal agent should freely choose A rather than not-A. As al-Ghazali remarked, it is the essence of free will to be able to distinguish like from like. On theism the explanatory ultimate is a personal agent endowed with freedom of the will, which is of a different order than the physical world, which is neither animate nor an agent.

Now, as you note, Oppy’s final view is that the mathematical description of the physical world is metaphysically necessary. As you nicely put it, it is “a brute fact, not a brute contingency.” I find this claim to be outlandish. Are we seriously to think that the world could not have been described by Newtonian physics rather than relativistic physics? By contrast on the theistic view the mathematical description of the physical world is a contingency, but not a brute fact.

With respect to the fine-tuning argument Oppy might similarly say that the fine-tuning of the constants and quantities of nature’s laws are metaphysical necessities, which makes his position even more extreme, and that on theism fine-tuning is brutely contingent due to God’s free choice, which is to repeat his error here. It’s not clear to me that similar questions arise with respect the kalām cosmological argument, though I’m certainly open to instruction.


[1] Shelly Kagan, The Limits of Morality (Oxford: Clarendon Press, 1989), p. 13.

[2] Kagan, Limits of Morality, p. 14.

[3] A similar point could be made with respect to physical theory (David Lewis, Counterfactuals [Oxford: Blackwell, 1973], pp. 72-77). On a Mill-Ramsey-Lewis or “best systems” approach to natural laws, one does not simply posit a flat layer of natural laws but constructs a hierarchy of explanatorily prior laws to account for lower level laws. Mathematics is the more interesting analogy here because of its necessity and self-evidence.

[4] Penelope Maddy, Defending the Axioms:  On the Philosophical Foundations of Set Theory (Oxford:  Oxford University Press, 2011), p. 82. In particular she points to set theory’s ability to systematize and explain number theory and geometry/analysis (Penelope Maddy, “Believing the Axioms II,” The Journal of Symbolic Logic 53/3 [1988]: 762). Thus, Huemer’s claim that if 2<3 has an explanation, “it would be in terms of some other arithmetical fact that is similarly obvious and itself has no explanation” is demonstrably mistaken.

- William Lane Craig