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William Lane Craig on the unexpected applicability of mathematics to nature

You might remember that Dr. Craig used a new argument in his debate with Lawrence Krauss in Melbourne, Australia.

My notes on the debate record it thus:

The unreasonable effectiveness of mathematics:

  • The underlying structure of nature is mathematical – mathematics is applicable to nature
  • Mathematical objects can either be abstract objects or useful fiction
  • Either way, there is no reason to expect that nature should be linked to abstract objects or fictions
  • But a divine mind that wants humans to understand nature is a better explanation for what we see

And now Dr. Craig has expanded on it in the Q&A section of his Reasonable Faith web site.

The question:

Dear Dr Craig

Firstly can I thank you for all your work. My faith in Christ has been enormously strengthened through studying your work in apologetics in particular and I have grown in confidence in my Christian witness.

My question relates to numbers and mathematics as a whole. On the Defenders podcast you state that as God is the only self-existent, necessary being, numbers and mathematical objects, whilst being useful, don’t actually exist as these too would exist necessarily and independently of God. If this is the case, how can it be that mathematics is so easily applied to the natural world? Surely if mathematics only existed in our minds, we would expect to see no correlation between it and how the physical world actually is?


United Kingdom

Excerpt from the answer:

As philosopher of mathematics Mary Leng points out, for the non-theistic realist, the fact that physical reality behaves in line with the dictates of acausal mathematical entities existing beyond space and time is “a happy coincidence” (Mathematics and Reality [Oxford: Oxford University Press, 2010], p. 239). Think about it: If, per impossibile, all the abstract objects in the mathematical realm were to disappear overnight, there would be no effect on the physical world. This is simply to reiterate that abstract objects are causally inert. The idea that realism somehow accounts for the applicability of mathematics “is actually very counterintuitive,” muses Mark Balaguer, a philosopher of mathematics. “The idea here is that in order to believe that the physical world has the nature that empirical science assigns to it, I have to believe that there are causally inert mathematical objects, existing outside of spacetime,” an idea which is inherently implausible (Platonism and Anti-Platonism in Mathematics [New York: Oxford University Press, 1998], p. 136).

By contrast, the theistic realist can argue that God has fashioned the world on the structure of the mathematical objects. This is essentially what Plato believed. The world has mathematical structure as a result.

This argument was also made by mechanical engineering professor Walter Bradley in a lecture he gave on scientific evidence for an intelligent designer. You can read an essay that covers some of the material in that lecture at Leadership University.


The physical universe is surprising in the simple mathematical form it assumes. All the basic laws of physics and fundamental relationships can be described on one side of one sheet of paper because they are so few in number and so simple in form (see table 1.1).

[…]It has been widely recognized for some time that nature assumes a form that is elegantly described by a relatively small number of simple, mathematical relationships, as previously noted in table 1.1. None of the various proposals presented later in this chapter to explain the complexity of the universe address this issue. Albert Einstein in a letter to a friend expressed his amazement that the universe takes such a form (Einstein 1956), saying:

You find it strange that I consider the comprehensibility of the world to the degree that we may speak of such comprehensibility as a miracle or an eternal mystery. Well, a priori one should expect a chaotic world which cannot be in any way grasped through thought. . . . The kind of order created, for example, by Newton’s theory of gravity is of quite a different kind. Even if the axioms of the theory are posited by a human being, the success of such an enterprise presupposes an order in the objective world of a high degree which one has no a priori right to expect. That is the “miracle” which grows increasingly persuasive with the increasing development of knowledge.

Alexander Polykov (1986), one of the top physicists in Russia, commenting on the mathematical character of the universe, said: “We know that nature is described by the best of all possible mathematics because God created it.” Paul Davies, an astrophysicist from England, says, “The equations of physics have in them incredible simplicity, elegance and beauty. That in itself is sufficient to prove to me that there must be a God who is responsible for these laws and responsible for the universe” (Davies 1984). Successful development of a unified field theory in the future would only add to this remarkable situation, further reducing the number of equations required to describe nature, indicating even further unity and integration in the natural phenomena than have been observed to date.

The whole paper that started this off is called “The Unreasonable Effectiveness of Mathematics”, and it is a must read for advanced Christian defenders. You can read the whole thing here.

8 thoughts on “William Lane Craig on the unexpected applicability of mathematics to nature”

  1. Here are some key quotes from the (secular) article you link to:

    “The enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious and that there is no rational explanation for it”

    “It is hard to believe that our reasoning power was brought, by Darwin’s process of natural selection, to the perfection which it seems to possess.”

    “It is, as Schrodinger has remarked, a miracle that in spite of the baffling complexity of the world, certain regularities in the events could be discovered.”

    “It is difficult to avoid the impression that a miracle confronts us here, quite comparable in its striking nature to the miracle that the human mind can string a thousand arguments together without getting itself into contradictions, or to the two miracles of the existence of laws of nature and of the human mind’s capacity to divine them.”

    “The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve.”

    Here are some additional quotes from other sources:

    “Mathematics, rightly viewed, possesses not only truth, but supreme beauty – a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show. The true spirit of delight, the exaltation, the sense of being more than Man, which is the touchstone of the highest excellence, is to be found in mathematics as surely as in poetry.” — Bertrand Russell

    “How is it possible that mathematics, a product of human thought that is independent of experience, fits so excellently the objects of reality?” — Albert Einstein

    “Math is our one and only strategy for understanding the complexity of nature” — Ralph Abraham

    Liked by 1 person

      1. Mathematicians universally acknowledge the elegance and simplicity of the basic formulas that describe nature. Indeed, they have a bias toward proposed solutions that are elegant. You can often hear them speak about their craft in the same way that artists and poets speak of their own.

        Liked by 1 person

  2. One of my Master’s degrees is in mathematics, and my PhD dissertation is highly mathematical, and I always felt something “supernatural” or “transcendental” when working with mathematics.

    Now, as a Christian, I know what that Something is.

    I also remember thinking what a miraculous set of coincidences macro-evolution was. Of course, such miracles occur in fairy tales too.

    Liked by 2 people

    1. I’m not a mathematician or the son of a mathematician, but I’ve always found it interesting to hear them talk about their craft. One was talking about how there was a point of learning at which the beauty of it all became apparent to him.

      In my view, the biggest challenge to evolutionary theory is in the realm of statistical probability of producing meaningful mutations — more specifically, in producing new genes. You’ve likely seen this before:

      Liked by 1 person

      1. Yes, I definitely agree with all of this, but I would just add that when it comes to macro-evolution, it is miracle piled on top of miracle ad infinitum, or practically so. At each stage of so-called “evolution,” you can insert the phrase “then a miracle occurs.” To me, it goes far beyond even producing new genes, difficult as that might be. It’s a fairy tale on every single level.

        To see that, just note that not only did Darwin’s finches remain finches, but their beaks remained beaks, even though the beak lengths varied. To extrapolate macro-evolution from that is far more ludicrous than me standing in an empty gym making a layup and somehow concluding that I would be the highest scoring player in NFL (not NBA) history. It’s not even the same sport, and the analogy goes much deeper from that. It is a colossal extrapolation error, the kind which an undergraduate engineering major would never make.

        Liked by 2 people

        1. I think the finch beak changes were not additions of genetic information, and may even have been mere selection of existing alleles. Even if you can eek out minor incidental improvements from existing structures (often by breaking something else), you are correct about how they stretch this to meet the astronomical improbabilities that must regularly occur to get from one species to another in the “tree of life.” To use another analogy, it’s like saying that since my son can jump higher than me, then it’s possible that one of my descendants could eventually jump to the moon.

          Liked by 1 person

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