This is a Q & A blog post by our Visiting Scholar in Philosophy, William Lane Craig.
Hello Dr. Craig,
Firstly, Thank you for the incredible contributions you’ve made in advancing the intellectual respectability of Christianity.
Lately, I’ve been pondering the argument from the applicability of mathematics. After watching and reading much of your work on the subject, I think I’ve finally come to be at peace with it after previously not having been so. I’ve been able to overcome the usual objection that the function of mathematics is necessary, but one question still gnaws at me.
It seems to me that the argument necessarily implies that whether or not mathematics is applied to the universe is a contingent matter; indeed, this is what is so surprising. If so, then what would a universe to which mathematics has not been applied be like? That is, one in which, say, the laws of physics are not so elegantly expressible in terms of mathematics, or do not conform very well to it; what would such a world be like? Would it be an unintelligible chaos? Would physics still operate, but be unable to be deciphered and expressed by mathematics? Or would such a universe simply have less things to which mathematics can be applied (perhaps it lacks motion, for instance, and thus lacks a certain domain that mathematics can express)?
William Lane Craig’s Response
It is a contingent matter whether or not mathematics is applied to the physical universe! The easiest way to see this is to realize that it is contingent whether there is a physical universe at all. There might have been no physical phenomena whatsoever, since creation is a freely willed and, hence, contingent act of God.
But even if there were a physical reality, it’s not obvious that it had to be mathematically describable. Couldn’t there have been a chaos? No less a figure than Albert Einstein thought so. He wrote:
“One should expect a chaotic world which cannot be grasped by the mind in any way. One could (yes one should) expect the world to be subjected to law only to the extent that we order it through our intelligence ... By contrast, the order created by Newton’s theory of gravitation, for instance, is wholly different. Even if the axioms of the theory are proposed by man, the success of such a project presupposes a high degree of ordering of the objective world, and this could not be expected a priori. That is the ‘miracle’ which is being constantly reinforced as our knowledge expands.”
The question continues to be a matter of debate. In 2015, the Foundational Questions Institute, an independent non-profit organization devoted to the exploration of questions at the foundations of physics and cosmology, sponsored an essay contest, “Trick or Truth: The Mysterious Connection Between Physics and Mathematics,” inspired by Wigner’s article, aimed at addressing the question: Why does there seem to be a mysterious connection between physics and mathematics? In his contribution, philosopher of physics Tim Maudlin maintains that even the applicability of elementary mathematics like arithmetic and geometry requires explanation.
But all this is really somewhat beside the point. For Wigner’s argument did not have to do with the applicability of the necessary truths of elementary mathematics. Rather his entire concern was with the elegant mathematics that come to expression in the laws of physics. Those physical laws are not logically necessary, but contingent. The universe did not have to be describable by the awesome and amazingly accurate laws discovered by the mathematical physicist. In describing the a priori nature of mathematical inquiry, especially of the mathematics that is so valuable in physics, Wigner writes:
"Whereas it is unquestionably true that the concepts of elementary mathematics and particularly elementary geometry were formulated to describe entities which are directly suggested by the actual world, the same does not seem to be true of the more advanced concepts, in particular the concepts which play such an important role in physics. ... Most more advanced mathematical concepts, such as complex numbers, algebras, linear operators, Borel sets — and this list could be continued almost indefinitely — were so devised that they are apt subjects on which the mathematician can demonstrate his ingenuity and sense of formal beauty.
By using as his examples laws of nature which are fearsomely advanced mathematically, Wigner already forced the question to a higher plane.
According to Mark Steiner in his fine book The Applicability of Mathematics as a Philosophical Problem, physicists see no difficulty in the applicability of arithmetic to the world, since this is just a matter of logic, not physics; rather they concentrate upon the seemingly miraculous appropriateness of physically meaningless concepts like matrix algebra or Hilbert spaces for quantum mechanics. It is the burden of Steiner’s book to provide numerous examples of the applicability of mathematical concepts that cannot be physically instantiated. Some of his examples are the same ones to which Wigner already appealed, such as the descriptive applicability of analytic functions of complex variables, Heisenberg’s utilization of matrix mechanics in his classical equations, a procedure for which, Steiner says, “there is no physical rationale” and which replaces all the variables by matrices “which have no physical meaning,” and the descriptive applicability of the Hilbert space formalism to quantum mechanics, which Steiner calls “physically unintelligible.” So even if the physical universe had to have some mathematical structure, that fails to address the question raised by Wigner.
Islami and Wiltsche, no friends of philosophical theism, conclude, “Confronted with this situation, two options seem to be available: either we admit that the situation is indeed highly mysterious; or we abductively infer that there must have been some pre-established harmony.” Wigner settled for the first option. Steiner advocates the second, arguing that we should accept that ours is a “user-friendly universe,” one that is specially attuned to the workings of the human mind and hence to mathematical reasoning. Islami and Wiltsche rightly muse, “Steiner’s ‘anthropocentrism’ is reminiscent of the old rationalist idea that it is part of God’s creation to have made the world and our minds to fit like hand to glove.”The great physicist Paul Dirac characterized the situation as follows: “One could perhaps describe the situation by saying that God is a mathematician of a very high order, and He used very advanced mathematics in constructing the universe.”
- William Lane Craig
 Albert Einstein, letter to Maurice Solovine, March 30, 1952, in Albert Einstein, Letters to Solovine, with an Introduction by Maurice Solovine, trans. Wade Baskin (New York: Philosophical Library, 1987), pp. 132–33. I’m indebted to Melissa Cain Travis for this reference.
 Anthony Aguirre, Brandon Foster, and Zeeya Merali, eds., Trick or Truth?: The Mysterious Connection Between Physics and Mathematics, Frontiers Collection (Cham, Switz.: Springer, 2016).
 Tim Maudlin, “How Mathematics Meets the World,” in Trick or Truth?, pp. 92-94. Similarly, mathematician Richard Hamming is amazed that numbers serve us to count things in reality (R. W. Hamming, “The Unreasonable Effectiveness of Mathematics,” American Mathematical Monthly 87/2 [February, 1980], p. 84). Philosopher of physics Arezoo Islami takes Hamming’s amazement to be quite appropriate: “Were it not the case that bodies were reasonably stable, we wouldn’t be able to abstract numbers and arithmetic would lose its application” (Arezoo Islami, “The Unreasonable Effectiveness of Mathematics: From Hamming to Wigner and Back Again,” Foundations of Physics 52 : 72ff).
 Eugene Wigner, “The Unreasonable Effectiveness of Mathematics in the Natural Sciences,” in Communications in Pure and Applied Mathematics 13/1 (New York: John Wiley & Sons, 1960), pp. 2-3.
 Mark Steiner, The Applicability of Mathematics as a Philosophical Problem (Cambridge, Mass.: Harvard University Press, 1998), pp. 15, 27.
 See Steiner, Applicability of Mathematics, pp. 36-40, 95-97, 102.
 A. Islami and H. A. Wiltsche, “A Match Made on Earth: On the Applicability of Mathematics in Physics,” in Phenomenological Approaches to Physics, ed. H. A. Wiltsche and P. Berghofer, Synthèse Library 429 (Cham, Switz.: Springer, 2020).
 Islami and Wiltsche, “Match Made on Earth.” Right; it really was a match made in heaven!
 P. A. M. Dirac, “The Evolution of the Physicist’s Picture of Nature,” Scientific American 208/5 (May, 1963), p. 53.