August 05, 2012
The Applicability of Mathematics
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?
I can’t resist taking questions pertinent to my current work on God and abstract objects! Your question, Michael, concerns what the great physicist Eugene Wigner called “the uncanny effectiveness of mathematics.” How is it that a theoretical physicist like Peter Higgs can sit down at his desk and on the basis of certain mathematical equations predict the existence of a particle and field which nearly a half century later the experimental physicists go out and discover? Why is mathematics the language of nature?
Whether one is a realist or an anti-realist about mathematical objects, I think that the theist enjoys a considerable advantage over the naturalist in explaining the uncanny success of mathematics.
Take realism first. 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.
Now consider anti-realism of a non-theistic sort. Leng says that on anti-realism relations said to obtain among mathematical objects just mirror the relations obtaining among things in the world, so that there is no happy coincidence. Well and good, but what remains wanting on secular anti-realism is an explanation why the physical world exhibits so complex and stunning a mathematical structure in the first place. Balaguer admits that he has no explanation why, on anti-realism, mathematics is applicable to the physical world or why it is indispensable in empirical science. He just observes that neither can the realist answer such “why” questions.
By contrast, the theistic anti-realist has a ready explanation of the applicability of mathematics to the physical world: God has created it according to a certain blueprint He had in mind. There are any number of blueprints He might have chosen. Philosopher of mathematics Penelope Maddy asks,
can the Arealist account for the application of mathematics without regarding it as true? . . . contemporary pure mathematics works in application by providing the empirical scientist with a wide range of abstract tools; the scientist uses these as models—of a cannon ball’s path or the electromagnetic field or curved spacetime—which he takes to resemble the physical phenomena in some rough ways, to depart from it in others. . . . The applied mathematician labors to understand the idealizations, simplifications and approximations involved in these deployments of his abstract structures; he strives as best he can to show how and why a given model resembles the world closely enough for the particular purposes at hand. In all this, the scientist never asserts the existence of the abstract model; he simply holds that the world is like the model is some respects, not in others. For this, the model need only be well-described, just as one might illuminate a given social situation by comparing it to an imaginary or mythological one, marking the similarities and dissimilarities (Defending the Axioms: On the Philosophical Foundations of Set Theory [Oxford: Oxford University Press, 2011], pp. 89-90).
On theistic anti-realism the world exhibits the mathematical structure it does because God has chosen to create it according to the abstract model He had in mind. This was the view of the Jewish philosopher Philo of Alexandria, who maintained that God created the physical world on the mental model in His mind.
The theist—whether he be a realist or an anti-realist about mathematical objects—thus has the explanatory resources to account for the mathematical structure of the physical world and, hence, for mathematics’ applicability, resources which the naturalist lacks.