#844 The Applicability of Mathematics Once More

Dr. 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.[i]

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?[ii] In his contribution, philosopher of physics Tim Maudlin maintains that even the applicability of elementary mathematics like arithmetic and geometry requires explanation.[iii]

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.[iv]

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.[v] It is the burden of Steiner’s book to provide numerous examples of the applicability of mathematical concepts that cannot be physically instantiated.[vi] 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.”[vii] 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.”[viii] 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.”[ix]