The Unreasonable Ineffectiveness of the Argument From the Unreasonable Effectiveness of Mathematics
The argument is mediocre
William Lane Craig has an argument from the unreasonable effectiveness of mathematics. The argument is as follows.
“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.”
Craig thinks that the argument is very persuasive. I find the argument remarkably unpersuasive.
While Craig is fond of deductive arguments, in my view, a better summary of the argument is abductive. Craig thinks that the best explanation of math being effectively applicable to the world is God made it so. I have several objections to the argument.
For one, the argument seems to commit the fallacy of understated evidence. Even if we think that the bare fact of mathematics being applicable to the world is evidence for theism, more specific facts about how it’s applied serve as evidence for atheism. Here are several of those facts.
Mathematics is highly complex—leaving most people unable to make meaningful contributions mathematically to the world. This is surprising. If theism were true, we’d expect anyone who works hard to be able to apply mathematics well. Yet sadly, that is just not the case. Lots of people are not able to make significant mathematical contributions.
If God was the cause of mathematics being applicable, we’d expect the applicability of mathematics to correlate with the beauty of the mathematics. However, there are many examples of very cool types of math that are not useful in the real world. Googology is one area of math that’s largely inapplicable—relating to making very big numbers. Googology is almost entirely useless, which would be surprising on theism.
As Craig says “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?”
Craig’s answer, however, is lackluster here. It just seems to be the fallacy of understated evidence. While maybe the fact that some math is applicable is surprising on naturalism, the fact that most of it is totally inapplicable is very surprising on a form of supernaturalism wherein God is making it applicable because he loves math.
We’d similarly expect mathematics to not be despised by lots of people, leading to some even committing suicide.
We’d similarly expect to be able to generate a nice elegant theory of everything—something like string theory. The fact that we haven’t been able to do this is evidence against the theory. Why would God make finding a theory of everything so damn difficult.
We’d expect math to accord with our common sense intuitions. For example, it seems like God would make the world Euclidian.
We’d expect math to be more effective at solving problems. For example, if God existed, it seems like we’d make mathematical models that would allow us to easily cure lots of disease.
We’d also expect mathematical models to be more precise in social sciences. For example, why wouldn’t God make it easy to create an economic “theory of everything.”
We’d also expect math to be easier to teach. For example, it seems a world wherein people could telepathically give others math information would be a much better world.
We’d expect most mathematical discoveries to be understandable outside of the Ivory tower of academia. For example, if there is virtue in understanding math, it seems God would make laypeople like me able to understand why fermat’s last theorem is true.
We’d also expect our mathematical reasoning not to be subject to the series of biases and heuristics that do in fact affect it.
We’d also expect it to be hard to use math knowledge to cause lots of harm, such as by making a bomb or a bioweapon.
Thus, God is a remarkably poor explanation for many facts about the applicability of mathematics.
Craig also spots another good objection, before dismissing it, writing
“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.”
This is, once again, the fallacy of understated evidence. If God existed and were using math to explain thing, we’d expect consciousness, for example, to be mathematically describable. The fact that it isn’t is evidence against God.
I also don’t think that naturalism has a difficult time explaining the applicability of mathematics. As Carrier argues , any complex system of physics must be mathematical. Mathematics just denotes the quickest way to describe a physical system. It’s not surprising that simpler theories (which are a priori more probable by virtue of their simplicity) would be able to be described mathematically.
This is not limited to things that are made by God. Other examples of things describable mathematically include
Probability
Morality
Economics
Biology
History
Features of Law
Programming
Linguistics
Thus, the effectiveness of mathematics is widespread and unsurprising. Mathematics is applicable throughout a wide range of things.
One last problem remains for the theist—the task of explaining why God would make math applicable in the first place. Oppy pressed Craig on this in their debate, and Craig lacked a good response. This is especially difficult if one takes the skeptical theist route. If we can’t predict that God wouldn’t cause millions of babies to die, can we really predict confidently how important he’d make calculus feature in the fundamental theory of everything?
Thus, theism is a terrible explanation of math being applicable. Naturalism has a very easy time explaining it. This is thus a terrible argument for God.