God and the Unreasonable Effectiveness of Mathematics

This is the weekly Q & A blog post by our Research Professor in Philosophy, Dr. William Lane Craig.

Question

Dear Dr Craig,

I'm a Christian who really enjoys your work, and I was wondering if you might take the time to answer a quick query I have. I've asked a few questions before, but I've always discovered quickly afterwards that they've already been answered somewhere else on this gargantuan site. I've therefore looked very thoroughly, and am incapable of finding anything relating as an answer, so I'm coming to you.

I saw on one of your popular articles, in which you propose eight reasons for belief in God, that you think of the applicability of mathematics to the natural world a good reason for belief in God. Originally, I couldn't fault this, and I was stunned that such a good argument a) hadn't been created before, and b) wasn't more popular upon you creating it. It surprised me that papers, books, treatises and criticisms are not being written on what could be called 'the hard problem of mathematics'. I couldn't help but try to find answers to the problem, yet none of them succeeded,

However, as I thought about it, it occurred to me that the argument wasn't quite as convincing as it seemed at first to be. For starters, couldn't it be said that most of the mathematical world is not applicable to the natural world- some concepts, generally known only to thinkers at the highest level, have no application. Taking it further, couldn't you remark upon the many things that mathematics and science cannot explain (e. g. love, the soul, God), and couldn't this work as substantial evidence against such a theory?

And secondly, why not just dismiss the whole problem altogether, and say that the reason for the unnaturally neat correlation between the two is down to evolution: the ones to see the link between the two are the ones most likely to survive, given more knowledge of the world around them (I realise this is a debatable one, and I'm not entirely sure about it myself, but I feel that it could be a worthy reply.)

The reason why I feel the need to play-the-Devil's-advocate to such a great degree is simply that I can't see why such a good proof (to my eyes, pretty irrefutable evidence) hasn't entered the academic and public sphere. Even beyond my criticisms at the moment, I can't help but think that there must be a trick somewhere, and that somehow it can't be as good as it seems. Surely we'd all be Christians if there was no problem. However, this attitude was also mine to Alvin Plantinga's modal ontological argument, which I have since come to accept. If these problems are broken, I will have to conclude that this really just is as good as it seems to be. In which case, I think I'll begin to spread the word. Anyway, thank you for reading my question, and apologies if, despite my careful checking, I've still missed that you've answered it somewhere else.

Many thanks,

Josh

United Kingdom

Dr. William Lane Craig's Response

For those who are unfamiliar with the argument for God from the applicability of mathematics to the physical world, here is a simple formulation I have used:

1. If God does not exist, the applicability of mathematics to the physical world is just a happy coincidence.

2. The applicability of mathematics to the physical world is not just a happy coincidence.

3. Therefore, God exists.

I agree with you that this is an extremely persuasive theistic argument. Just look how Alex Rosenberg stumbles around it when I proposed it in my debate with him![1]

Leave aside for the moment the question of why this argument is not more widely used. Consider your two responses to the argument.

First, “couldn't you remark upon the many things that mathematics and science cannot explain (e. g. love, the soul, God), and couldn't this work as substantial evidence against such a theory?” A moment’s reflection shows that the question is misconceived. In this argument one isn’t trying to explain things by mathematics and science; rather one is trying to explain why mathematics, which is either about a causally unconnected, abstract realm or a merely make-believe realm, is so scientifically useful in describing the way in which the world works. Of course, there are things which are not mathematically describable, like love or persons, but the argument concerns those aspects of reality, chiefly studied by physics, which are.

But “couldn't it be said that most of the mathematical world is not applicable to the natural world?” Absolutely! The higher reaches of set theory and higher orders of transfinite arithmetic are incapable of being physically realized, and so the world does not exhibit such a structure. But the unresolved point remains: why does the physical world have the incredibly complex mathematical structure that it does have, so that mathematical calculations are applicable to it? Is this just a happy coincidence? Or the product of an intelligent designer?

Second, “why not . . . say that the reason for the unnaturally neat correlation between the two is down to evolution: the ones to see the link between the two are the ones most likely to survive.” This question is based on a confusion of epistemology and ontology, Josh. One is not trying to explain how or why we have mathematical knowledge but why the physical world is describable by mathematics, whether or not any intelligent life ever evolved or gained a store of mathematical knowledge. Suppose that the creatures most likely to survive are the ones who can discern mathematics’ applicability to nature. The question still remains: Why is mathematics’ applicability to nature there to be discerned in the first place? Evolution doesn’t address that question.

Now to the $64,000 question: why isn’t such a powerful argument more popular? Why aren’t “papers, books, treatises and criticisms . . . being written on what could be called 'the hard problem of mathematics’?” Well, they are, if by ‘the hard problem of mathematics,’ you mean the applicability of mathematics to the physical world. But these discussions are going on in the philosophy of mathematics, a tiny discipline to begin with, and one in which scarcely any Christian philosophers specialize. The only Christian philosopher of mathematics that I know of is Christopher Menzel at Texas A&M, and I’m not sure he’s written on the question. Then keep in mind that philosophy of mathematics has existed as a discipline really only since Gottlob Frege’s Foundations of Arithmetic (1884), and that anti-theistic positivism dominated philosophy until the later twentieth century when the renaissance of Christian philosophy began. To illustrate, the prominent German philosopher Bernulf Kanitscheider in his Natur und Zahl [Natur and Number} brushes aside the medieval view that mathematical entities exist in the mind of God as “no longer accepted in today’s secular world” and therefore “obsolete,” as well as incompatible with materialistic naturalism.[2]  Philosopher of mathematics Mark Steiner has written extensively on the problem and recognizes that the applicability of mathematics constitutes “a challenge to naturalism.”[3] Unfortunately, secular philosophers never seriously consider theism as an option, and there are almost no Christians working in the field to challenge them.

In his retrospective, “Christian Philosophy at the End of the 20^th Century,” Alvin Plantinga asks how Christian philosophers have done with respect to providing and developing arguments for God’s existence. He answers, “On the whole, I think, not well. . . . much more can and should be done. There are really a whole host of good theistic arguments, all patiently waiting to be developed in penetrating and profound detail. This is one area where contemporary Christian philosophers have a great deal of work to do.”[4] Plantinga mentions specifically arguments from sets and numbers, though he doesn’t mention the applicability of mathematics. The argument from applicability is mentioned, however, by Tyron Goldschmidt and Robin Collins in their essays in the Alvin Plantinga Festschrift Two Dozen (or so) Arguments for God.[5] So there’s no trick here. We just need more Christian philosophers and mathematicians.

Start spreading the word!
 

Endnotes

[1] See William Lane Craig and Alex Rosenberg, Is Faith in God Reasonable? Debates in Philosophy, Science, and Rhetoric,ed. Paul Gould and Corey Miller, Routledge Studies in the Philosophy of Religion (London:  Routledge, 2014); for a video recording (which does not, however, include Rosenberg’s final comments to our respondents in the book), see https://www.youtube.com/watch?v=bhfkhq-CM84.

[2] Bernulf Kanitscheider, Natur und Zahl:  Die Mathematisierbarkeit der Welt (Berlin:  Springer Verlag, 2013), pp. 92-93, 205.

[3] Mark Steiner, Mathematics as a Philosophical Problem (Cambridge, Mass.: Harvard University Press, 1998), p. 176.

[4] Alvin Plantinga, “Christian Philosophy at the End of the Twentieth Century,” in The Analytic Theist: An Alvin Plantinga Reader, ed. James F. Sennett (Grand Rapids, Mich.: William B. Eerdmans, 1998), p. 339.

[5] Two Dozen (or so) Arguments for God, ed. Jerry L. Walls and Trent Dougherty (Oxford: Oxford University Press, 2018), pp. 74, 91.

This Q&A and other resources are available on Dr. William Lane Craig's website.