Is the Applicability of Mathematics Necessary?

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

Question

Dear Dr. Craig,

Recently I listened to your argument from the applicability of mathematics on premier Christian radio as well as the debate you had with Alex Rosenberg. I'm not particularly knowledgeable when it comes to mathematics but I was intrigued by your comments on how the argument from the applicability of mathematics is similar to the argument from fine tuning. The argument as you put it in the Rosenberg debate was as follows:

1. If God did not exist, the applicability of mathematics would be a happy coincidence
2. The applicability of mathematics is not a happy coincidence
3. Therefore God exists

I have two questions regarding this argument. Firstly, concerning the first premise, could it be that happy coincidence is not the only alternative explanation to God for the applicability of mathematics. Could it be that, as with the fine tuning argument, the applicability of mathematics might be explained by necessity? For example, any possible world has to necessarily have some sort of mathematical explanation. It is difficult to conceive of a world that cannot be explained by mathematics. What would such a world look like? Since it would seem to be impossible for a world not to be explained by mathematics, therefore the applicability of mathematics to the physical universe is necessary. Although maybe here I've misunderstood the nature of abstract objects.

Secondly, the second premise, my question is really one of comprehension. If I understand the argument, the second premise is shown on the basis of the causal impotence of abstract objects. Have I rightly understood?

I really appreciate your work and it has been a great support and help to me in my growth as a Christian. I now regularly use your apologetics resources in talks to the youth of our church.

For Christ and his Kingdom

Tim

United Kingdom

Dr. William Lane Craig’s Response

The conundrum of the applicability of mathematics to the physical world is one that comes up continually in the professional literature and seems to be a matter of genuine bafflement. If the puzzle were resolved by simply saying that the physical world must of logical necessity have the mathematical structure it does, the question would have been dropped long ago. But both philosophers and physicists continue to express their bewilderment. Indeed, as I illustrated in the interview, we know that space (or spacetime) does not have to have the geometrical structure it does. Prior to Riemann, it was thought that Euclidean geometry was necessary, but the discovery of non-Euclidean geometries has made it a matter of empirical discovery what sort of geometry characterizes the physical world.

Now maybe the world had to have, of logical necessity, some sort of mathematical structure. I say “maybe”—for why couldn’t physical reality have just been a chaos? But maybe the world had to be describable by the theorems of elementary arithmetic like 2+3=5. But what about other fields of mathematics and the higher reaches of mathematics? It is the incredibly complex mathematical structure of the physical world, as revealed by natural science, that is surprising and calls out for explanation.

In that sense, the argument from mathematics’ applicability is akin to the fine-tuning argument, which states that the complex constellation of physical constants and quantities necessary for embodied, interactive life cries out for explanation. As you say, the explanatory alternatives are similar: necessity, chance, or design. Which is the best explanation?

The causal impotence of abstract objects (such as mathematical objects) is relevant, not to the second, but to the first premiss. One can’t say that the mathematical objects themselves somehow shaped the physical world to be as it is because mathematical objects, being abstract (or, alternatively, fictitious) have no causal powers. That’s why the applicability of mathematics, being non-necessary, appears to the naturalist as an amazing coincidence.

The warrant for the second premiss is that it seems improbable to attribute the complex mathematical structure of the world to chance. Design seems a more plausible explanation.

I have but scratched the surface of this intriguing question and hope to explore it further.