Is the surprising applicability of mathematics to the physical world a brute fact, or something crying out for a theistic explanation?

Question

William Lane Craig proposed the following argument for God's existence:

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]

Source: #608 God and the Unreasonable Effectiveness of Mathematics | Reasonable Faith

Subsequently, this argument became the topic of a debate between Willian Lane Craig and Graham Oppy: Does Math Point to God? William Lane Craig + Graham Oppy.

Among the many things Oppy said, one of his main rebuttals focused on asserting that, even if it's in fact the case that mathematics can be applied effectively to the physical world, this applicability can be postulated as a necessary brute fact, thus not requiring further explanations (as necessary things explain themselves). On the contrary, Craig kept on insisting that the surprising applicability of mathematics to the physical world cries out for an explanation, meaning that such an explanation is God, who must have been the responsible for intelligently designing the universe using mathematics.

Is it okay to postulate that the applicability of mathematics to the universe is a brute fact? Or is this something that, as Craig asserts, cries out for an explanation?

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A relevant related discussion: Was mathematics invented or discovered?

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Appendix

The transcript of part of the exchange between William Lane Craig and Graham Oppy (from t=55:05).

Bertuzzi: Gram, it sounds like if we could split the argument into stages, stage 1 being about whether or not math does apply or does have this sort of uncanny applicability to the universe, if we labeled that stage 1 and then stage 2 is how do we explain this, is it evidence for theism, it sounds like you're wanting to go back to stage 1 and say "well I don't really know if mathematics does have this uncanny applicability".

Oppy: Right, so that's all I've argued about so far. But let me say something about the argument, right, because I think that it's not true that naturalists have no resources here. So suppose it's true that there's this fit between mathematics and physical structure, right, of the kind that we're imagining. There are versions of naturalism that can explain this in a very straightforward way. And so one of the versions of naturalism can do this is one that I've been playing around with for about a decade now. And so let me give you the kind of tenants of the theory that you need in order to explain the effectiveness. When I get to the end of it you may think it's, I don't know, disappointing that it turns out that this is the way the explanation goes but it's definitely an explanation. So, start with this. A theory of modality. So, every possible world shares some initial history with the actual world, diverges from it only because chances play out differently. So that's all the possibilities there are. The only possibilities that you need really of a chance. Only talking metaphysics here, we're not talking doxastic possibilities or epistemic possibilities, we're talking metaphysical possibilities. So that's all the possibilities that there are. The laws are necessary, the boundary conditions are necessary. This is true and it doesn't matter whether we're thinking about one universe or many universe model. So we're supposing that where contingency comes in is in the outplaying of chances, that's the only place that contingency comes in. We suppose also--and this is the only kind of new assumption that we're going to make to go along with the kind of metaphysical picture that we've already outlined which is going to be a naturalistic picture--that the laws and the boundary conditions are amenable to mathematical formulations. On that assumption and given the other assumptions it just turns out that it's necessary that that's the case. It couldn't possibly have failed not to be so. Now adding a couple of other things that I don't really need just but that are also part of this picture that I developed when I was thinking about the origins of the universe (it had nothing to do with the applicability of mathematics), there's no explaining why something's necessary. Once you get to the postulation of necessities you've reached the end of the explaining that you can do. And last of all, if you've got a non-modal claim P net and you believe it, you accept that necessarily P, then it's being necessary that P explains why P. Okay, so now, given that, we have an explanation for the effectiveness of mathematics, which is that it had to be. Because it had to be so. And it just falls out of the picture. Now that's a naturalistic story that has an explanation. You might not like the explanation but at least for me it comes for free, from things that I've said elsewhere.

Craig. Well, I hope that our listeners have understood your alternative because, honestly Gram, I think it takes you more faith to believe that than it does to believe in God! The claim for example that the mathematical formulation of the physical world is necessarily true, that just doesn't seem to be correct at all. There might have been no physical universe whatsoever, in which case mathematics would not be applicable, because there would be no physical universe. Or there might have been a sort of chaos. Albert Einstein wrote to Maurice Sullivan in 1952:

"One should expect a chaotic world which cannot be grasped by the mind in any way. One could, yes, one should expect the world to be subjected to law only to the extent that we order it through our intelligence. By contrast, the order created by Newton's theory of gravitation, for instance, is wholly different. Even if the axioms of the theory are proposed by man, the success of such a project presupposes a high degree of ordering of the objective world and thus could not be expected a priori. That is the miracle which is being constantly reinforced as our knowledge expands."

So even so great a mathematical physicist as Einstein thought that it was a contingent matter that the world should exhibit this sort of mathematical order. That we should have expected, on the contrary, a chaotic world.

Bertuzzi: Well, let's get a response from dr. Oppy and then we'll move to some Q&A. So unfortunately we do have to move on.

Oppy: So when you talk about expectation, you may be talking about something epistemic or doxastic. I was talking about metaphysics. I was doing metaphysics and and my claim is that this is the best metaphysical theory. I'm not saying that it's true a priori. I'm saying that it's the best metaphysical theory when you take everything into account.

Craig: Can you specify, Gram, for us in a sentence or two why is it the best metaphysical theory in your view?

Oppy: Because if you think about the goal of theorizing, what you're trying to do is strike the best balance between minimizing all of your theoretical commitments and maximizing the explanation that you can do. And I think that this theory strikes that sweet spot. That's the reason. But there's a lot of data and there are hundreds of data points that you have to think about if we're going to compare this theory say with a theistic theory so I've written elsewhere at considerable length about why I think that you should prefer the naturalistic story to the theistic story. It just turns out that the naturalistic story, so, because this is the point, when you are formulating your theory, you said naturalist just have no explanation. That's not true, here is a naturalistic theory that does have an explanation. And what needs to be argued is about which one is the better theory, and that's not something that's settled by these considerations. It's settled by general considerations.

Craig: Okay, it didn't sound very explanatory to me. But we'll leave it at.

Oppy: Well, do you think that you can't explain why something's the case by pointing out that it's necessary? Because that's all that's going on here.

Craig: Yeah, I mean, it's really a way of avoiding explanation by just begging the question and assuming that it's necessarily the case. And that is implausible and certainly not incumbent or there's nothing that would lead us to think that that's true.

Oppy: So that's not right though. We've got two theories and we're comparing their virtues. The theories are what they are, they say what they say. It turns out that on this naturalistic theory there is an explanation and the explanation is that this stuff is necessary.

Answer 1

The biggest issue seems to be that Craig implies that mathematics is entirely disconnected from the physical world. But maths emerged from our understanding of physical world. We saw that when you put 1 thing and another 1 thing together, you get 2 things. So it's entirely unsurprising that we came up with 1+1=2. It would've been quite surprising had we come up with 1+1=3, and that instead would've been something that "cries out for an explanation".

There are also non-Euclidean geometries and other things in maths that may not have any direct parallel in the physical world. We have spent some time trying to figure out the "right" answers in maths. Heck, people took some time to come up with the concept of zero and negative numbers.

So I'd say roughly what I'd say about the "surprising" accuracy of science: we came up with the correct or accurate answers because we kept looking until we found the correct or accurate answer.

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Our ability to accurately model reality can be explained by this providing a survival benefit that evolution selected for.

Organisms that model reality inaccurately, and act based on those models, would be less likely to be able to consistently act in a way that's beneficial to their survival. If you don't see that a tiger views you as a juicy steak, and that having a few bites taken out of you would be bad for you, then you're probably less likely to successfully reach the other end of an encounter with a tiger, and thus potentially go on to make babies.

Side note: this selection pressure is less present at large scales (e.g. global problems) and things that are beyond what we can observe, because we mostly evolved to deal with our immediate surroundings. So this explains why there can be much disagreement about those things. But the principles that we developed based on our immediate surroundings extend beyond that (which explains, for example, why there are things that almost all scientists agree on, and why their predictions tend to be very accurate, even if that agreement isn't as universal among the general public).

There are also various biases that may lead to incorrect modelling of reality, because that provided some other survival benefit. And accurately modelling reality wouldn't just be the result of one single mutation - it would be the complex interplay of many mutations. There's certainly a lot to say about this topic (which scientists research, albeit not necessarily directly), but for a non-scientist understanding, it should suffice to say "accurately modelling reality provides a survival benefit".

This is, of course, explaining things from a physicalist point of view, but that does roughly seem to be the question at hand, because it's a theist trying to conclude that this would be surprising under atheism (although there are also other worldviews consistent with atheism).

Answer 2

My three cents.

It has been claimed that the effectiveness of mathematics on physical reality is anything but unreasonable.

The reason is twofold. First many theories and areas of mathematics were initiated and formulated precisely as a result of and through physical investigation (eg Vectors,Calculus by Newton/Leibniz). Second is that mathematical theories which find no application become obsolete and not worked upon (at least as far as natural sciences are concerned), whereas physical theories then are formulated based on mathematical theories which are found applicable. Through sustained use of that theories which have many applications get used and referenced more and more whereas all the remaining theories become obsolete, giving the impression of an unreasonable effectiveness.

PS. It has to be noted that pythagoreans were actually not doing mathematics but rather (symbolic) physics. For example, the pythagorean motto "number rules the world" is a physical statement about physical relationships expressed symbolically not a mathematical one (eg harmony of the spheres).

References:

1. The Unreasonable Effectiveness of Mathematics in the Natural Sciences
2. On "The Unreasonable Effectiveness of Mathematics in the Natural Sciences"

Answer 3

The current answers are focused on whether or not math was invented but that doesn’t fully address your question.

Oppy’s argument applies even if mathematics was not just an invention to describe reality but rather was a fundamental part of reality. He himself points this out despite not being sure that mathematics really is a coincidence.

Let’s assume that math is fundamental. Then, it still does not follow that it is a coincidence. The concept of a “coincidence” only applies if other states of affairs were possible. But there is no such evidence; hence Oppy postulates that the applicability of mathematics to the real world may be necessary. If it is necessary, there are no other possibilities.

God does not escape this problem. If God has complete free will, then His actions are random. This implies that His decision to make mathematics apply to the real world becomes a brute coincidence. But that is the very thing William Craig complains about. If His actions are determined, then His action to apply mathematics to the real world becomes necessary. This then becomes equivalent to Oppy’s postulate. The difference though is that Oppy’s postulate doesn’t have the unneeded ontology of God, making it simpler.

This highlights the general problem of postulating God or really postulating anything as an explanation without reason. It is only advantageous to postulate something as an explanation if one has prior evidence for it. If one doesn’t, there is no advantage.

Sure, if my friend steps outside for one second and gets hit by a lightning bolt, it would be a pretty unfortunate coincidence. If a devil existed specifically to harm my friend and controlled lightning, then this wouldn’t be a coincidence. But does this mean we now have evidence of this kind of devil? Nature being constructed in a way such that it happens to create a lightning bolt to kill my friend right at that time explains it equally well.

Answer 4

The argument presupposes that the relevance of mathematics to physics is remarkable. However, if you spend any time reflecting on that, you should readily conclude that it is not remarkable at all.

To begin with, numbers are naturally used to count things. I cannot imagine any sort of universe in which numbers were not applicable for the counting of objects, so no divine intervention is needed to explain that foundational aspect of the effectiveness of mathematics.

Then numbers can be used naturally to compare properties- is this river wider than that, is this apple bigger than that, does this change happen more often than that change, and so on. Simply by counting and comparing, you arrive straightforwardly at ideas such as units and ratios. Again, it is hard to see how any universe, however designed, would not be amenable to that kind of simple numeric assessment.

There are not that many fundamental aspects of physics that characterise the classical world around us. They include conservation laws, Newtwon's laws, the property of charge, the fact that the forces associated with charge and gravity fall away with distance, and so on. With relatively few principles of that sort, plus the ideas of countable units and ratios, most of the supposedly unreasonable effectiveness of mathematics to classical physics follows inevitably.

Once you get into the quantum realm, the mathematics becomes much more complicated, but it can all be boiled down to a relatively small number of basic characteristics.

Then consider statistical mechanics, a very mathematical branch of classical physics that deals with what? With randomly moving featureless particles- perhaps the least 'designed' subject you can imagine. In a universe that consisted only of particles colliding at random, the behaviour of the particles would still manifest complex mathematical relationships. That seems to me to put the final nail in the coffin of the idea that anything that can be described with complicated mathematics must have had a designer.

Answer 5

So wait a second. The two alternative hypotheses are:

1. God created the world and made physics work by math because he is generous to human physicists and wanted them to be able to use math.
2. Simple, mathematical laws of physics created the world, and mathematics is useful for physicists because that's how the laws are inherently.

If simple, mathematical laws of physics created the world, then of course the world would be susceptible to mathematical investigation!

But if God created the world, then he had many options, and could have made a world that was not susceptible to mathematical investigation. He could have been generous to physicists in a different way by, say, allowing them to predict what would happen to a moving block based on what would rhyme best in a poem, or any other method. Or he could have chosen not to be generous to physicists. Certainly there are many miserable groups of people that God was not generous towards!

So, the chance that the world is susceptible to mathematical investigation is far more likely under the "simple laws created the universe" hypothesis than under the "God created the universe" hypothesis.

Comparing the plausibility of the competing "brute" facts, the brute fact of God is far less likely and far harder to accept a priori than the brute fact of a universe that runs on simple mathematical laws. That's because of Occam's razor: simple hypotheses are exponentially favored.

Nothing else needs to be said. We can express the above using the odds form of Bayes' theorem. Let G = "God created the universe," L = "Simple mathematical laws created the universe," M = "math works for describing the universe."

Odds of G given M = (prior odds of G) * (Bayes factor)

The prior odds of G are very low by Occam's razor, because God cannot be described by a concise formula.

The Bayes factor is P(M | G) / P(M | not G). This is fairly low, maybe 0.1, because as described earlier, God could have been generous to physicists in many ways other than making Math Work for them.

Odds of L given M = (prior odds of L) * (Bayes factor)

The prior odds of L are far higher than the prior odds of G, because by definition the laws can be described by a simple formula, so they are greatly favored by Occam's razor. I cannot overstate how important simplicity of a hypothesis is. In mathematical formulations of Occam's razor, every additional bit of description length makes the hypothesis exponentially less likely.

And the Bayes factor here, P(M | L) / P(M | not L), is also a lot higher than the Bayes factor for the God hypothesis. We would definitely expect physicists to be able to analyze the world with simple math, if it inherently just works by simple math.

So, odds of L given M are a lot higher than odds of G given M.

Answer 6

No, the applicability of mathematics to the physical world is not surprising - to me at least, and evidently to many others - with or without explanation, theistic or otherwise. Nor is it necessarily a brute fact.

The more interesting question is why William Lane Craig claims any surprise at all.

Well, mildly interesting if you've never listened to Craig talk before.

Craig's argument is based on his presuppositions and some bald assertions, not on a rigid logical foundation. Specifically, Craig believes that the only possible alternative to a God-created universe is one that is created from pure randomness, and that if it is sourced from randomness then it must retain randomness as a core feature under all possible conditions.

This presupposition is concealed in his first premise:

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

If I assume that God's existence is a necessary precondition for the universe to act in predictable ways then this is self-evident. Lacking that perspective, the premise needs a lot of support to demonstrate validity. Rather than attempt to justify his position however, Craig simply states it as a brute fact.

Neither Craig's argument nor his other public statements provide rigid logic to validate this premise. Nor does he manage to sufficiently address any of the possibile alternate reasons why math may be applicable to the physical world. In his discussion with Oppy he was presented with one possible answer, but he attempted to poison this possibility with the following statement (from your quote):

Craig: Yeah, I mean, it's really a way of avoiding explanation by just begging the question and assuming that it's necessarily the case.

This is the height of hypocrisy, since Craig has claimed necessity for decades without sufficiently justifying that claim. Here's an example (from the page linked above):

What God has that we don't, then, is the property of necessary existence. And He has that property de re, as part of His essence. God cannot lack the property of necessary existence and be God.

It's a bit old, and I have no doubt that he's moved on significantly in the last 15 years, but this is a transparent attempt to simply define God into necessity, much as previous apologists have attempted to define God into existence.

He continues later in the article with this:

So is it logically possible that God not exist? Not in the sense of metaphysical possibility! There is no strict logical contradiction in the statement "God does not exist," just as there is not a strict logical contradiction in saying "Jones is a married bachelor," but both are unactualizable states of affairs. Thus, it is metaphysically necessary that God exists.

Note that there is in fact a strict logical contradiction entailed by the pairing of "married" with "bachelor" which invalidates this example, but since Craig believes that it is equally contradictory to talk about God not existing he sees the statements as equivalent. Regardless of his opinion on the matter however, placing a 'thus' statement here is unwarranted since he does not provide any argument. Nor does he in any part of the article. It is a simple bald assertion.

Answer 7

No one's really given what I think is the best answer yet.

First of all, let's grant Premise 2 here. I think it is absolutely correct that mathematics is surprisingly applicable to the physical world. More than that, the physical world can be described by simple, elegant mathematics.

It's crazy that the world can be described by simple, regular laws. That does cry out for an explanation. I don't know how people can deny that.

But the idea that this proves God is not accurate. The problem is that this is a "God of the Gaps" argument. The argument is "we don't know why the world seems to be based on math, therefore God." That does not follow.

In other words, Premise 1 has not been established. There could be other explanations besides God. He would have to show that there could be no other explanation for the way the world is based on math, besides God.

Not only that, but he, like many Christian apologists, equivocates on the word God. God can mean many things, but he's arguing for a certain type of God - a personal god, an all-powerful god, a loving god, and so on.

For all we know, this world could have been created by the cosmic equivalent of mad scientists. Or by some sort of mathematical process. Or something we don't even have the language for. We obviously don't fully understand how the universe was created, but that doesn't mean we can come to the conclusion that it was God - unless you define God to mean whatever you want. The big problem with discussion about God is that the word God is so poorly defined.

Check out Wikipedia on "god of the gaps":

https://en.wikipedia.org/wiki/God_of_the_gaps

Answer 8

A brute fact.

I don't even see why the question God's (non-)existence could emerge from the usefulness of an idealized framework like math to the physical world that we live in.

Even if you envision a God whose 'highest' activity would be the subtle weaving of matter, space and time you would still have to admit that the same world could have emerged without any such guiding hand.

And why should a God be overly concerned with physics or mathematics when our hardest challenges in life are usually economic or moral ones ?

Answer 9

Mathematics is the study of assumptions, and the consequences of those assumptions. If, then. If x is a real number and x^2 + 6 = 5x, then x=2 or x=3.

The natural numbers have applications to the physical world because many things come in discrete lumps. The real numbers have applications to the physical world because many properties appear continuous. Graph theory has applications to the physical world in many ways, one of which is the study of causality. Many areas of mathematics are not applicable to the physical world, such as the study of inaccessible cardinals, because no aspect of the physical world appears to resemble them.

If the world behaved drastically differently, such that our applied mathematics was pretty much useless, any inhabitants of that world would use different mathematics to reason about it. If mathematics were entirely inapplicable to the world, inductive reasoning would have to be meaningless and causality couldn't exist.¹

Can life or thought even exist without induction or causality? Well… that's a different question. But certainly, life as we know it couldn't, so we can probably invoke the anthropic principle without too much controversy.

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¹: I just reasoned logically about a hypothetical world where logic doesn't work. … My head hurts.

Answer 10

From the transcript in the question (emphases mine):

Oppy: ... We suppose also--and this is the only kind of new assumption that we're going to make to go along with the kind of metaphysical picture that we've already outlined which is going to be a naturalistic picture--that the laws and the boundary conditions are amenable to mathematical formulations.

Oppy has assumed at the outset that he's going to offer a naturalistic explanation (italics), and then simply added the assumption that math works effectively in the physical world (bolded). That's a circular argument if he's trying to show that there is a naturalistic explanation for the effectiveness of math to the physical world.

And Craig calls him out on it:

Craig: Yeah, I mean, it's really a way of avoiding explanation by just begging the question and assuming that it's necessarily the case. And that is implausible and certainly not incumbent or there's nothing that would lead us to think that that's true.

Now, a circular argument (a.k.a. a tautology) can be a useful tool, as it's one of the only ways we can work backwards to the beginning point of logical arguments. But when determining whether one tautology or a different one or even one of the other ultimate sources (infinite regress or dogma) is the "best fit" for a given situation, we cannot use those sources directly as part of the determination. We have to use other means, for example, extrapolating to see whether such an assumption results in a self-contradiction or some other obviously false conclusion, or sensing whether the assumption matches with some aesthetic or moral principles with which we agree.

As for the OP's question:

Is the (surprising) applicability of mathematics to the physical world a brute fact or something that cries out for a (theistic) explanation?

A "brute fact" is a dogma. A "theistic explanation" is also a dogma. Arguments about whether the applicability of mathematics to the physical world is "surprising" relate to the question whether mathematics is invented or discovered, and are perhaps outside the scope of this question.

The real difference between them is where the dogmatic step is taken. If God does not exist, then quite a number of "brute facts" must be dogmatically accepted as independent existences. If God does exist, then He is the only "brute fact" to necessarily exist, as all others can be placed in dependence on Him.

Answer 11

The brute fact that mathematics can be applied to the universe is a statement to the effect that the real objective existence of truth is necessary. Truth is the necessary basis of the existence of the universe, without it, nothing could exist.

Saint Augustine nicely relates truth to the existence of God in his $De Libero Arbitrio$, where the dialog attempts to convince the reader that if you can accept that, that which is beyond man is God, then truth is beyond man; therefore truth is God.

Since truth is necessary for the existence of the universe, then Augustine's argument is not unappealing. This is, at least, one of my favorite ways of thinking about theism, however, I think that, to be truly rigorous, arguments for and against theism are only really good if they argue from probability, i.e. one can stack evidence, but one cannot deductively establish or abolish the existence of God. This much has been proven by Kant. Thus, all deductive arguments proving or disproving theism mere toys for thought.

Answer 12

Null. Your question,

Is the (surprising) applicability of mathematics to the physical world a brute fact or something crying out for a (theistic) explanation?

only addresses two options. It does not entertain other options. The word "surprising," has had different meanings to different people. The word "theistic," has had different meanings to different people. Therefore, using any of these conditions the answer is not available and thus null.

As to the stability of math: It has been recorded that the sun was stopped in the sky and the moon was stopped in the sky "about a whole day." Reference: Book of Joshua, Chapter 10, Verses 1 - 14, King James Bible.

Is it okay to postulate that the applicability of mathematics to the universe is a brute fact?

"Brute," no. "applicability of mathematics to the universe," known records report that it is not.