The following is an analysis of drcraigvideos' essay, "God and Mathematics." Video content is highlighted in red, followed by my rebuttals in black.
Why does mathematics work?
Think about it. Mathematical entities like numbers, sets, and equations, are nonphysical and abstract. They can’t cause anything. Yet, for some reason, the physical universe operates mathematically. As Galileo put it, “The book of nature is written in the language of mathematics.”
Why does mathematics work? That question honestly makes about as much sense as asking “Why does English work?” It works because we built it to work. And if it didn’t work, then we would just throw it away and invent something else that did. You even said it yourself: Math is a language.
Every now and then, I have to remind myself what exactly Christian apologists are trying to prove. According to mainstream Christian theology, God is supposed to be the living embodiment of absolute goodness, power, and love. We’re talking about a being that desperately wants to build a deep, personal relationship with all of us and who will have no choice but to condemn us to hell if we fail to reciprocate. It therefore stands to reason that, in principle, this ought to be trivially easy. It’s like trying to prove the existence of the moon, where all you should have to do is look up into the sky and immediately see it for yourself. But instead, what we really get is a bunch of obtuse philosophical rhetoric: God exists because… math works? The very nature of the argument practically reeks of its own desperation.
Setting all that aside, it’s nice to see apologists finally breaking away from the usual “Top Five” arguments for God’s existence. And since this is a ReasonableFaith video, it seems fair to assume that Dr. William Lane Craig is the brain behind it. You can therefore imagine my disappointment as a PhD-level philosopher completely fails at some of the most basic principles in philosophy itself. For example, ask yourself right now: what does it mean to say that math “works?” Sure, we all have a vague sense of what this is might entail, but nowhere in the video does Craig elaborate in detail. Furthermore, Craig makes no effort to explain what he means by “mathematics," either. That is to say, what is mathematics, really? Where does it come from? How does it work? He knows perfectly well that there are several competing schools of thought on these questions, and we have no way of knowing which one is being assumed for the sake of this argument. It’s such a blatant oversight, too, that I can’t help but feel as if he’s doing it on purpose. After all, what better way to sneak in a bunch of dubious presuppositions than by deliberately avoiding any hard definitions? So before entertaining a second more of this presentation, we need to establish the ontological status of mathematics.
Mathematics, as properly understood by the majority of relevant professionals, is nothing but a body of formal languages and deductive systems built on axioms and rules of inference. Logic, numbers, sets, algebra, geometry, and everything else in between was specifically invented by human beings. The inventive process itself is also trivially easy to demonstrate, and there really isn't any significant debate over this stuff in modern textbooks.
Notice also how Craig himself all but admits to this view outright through his implicit endorsement of Galileo’s quote: The Book of Nature is written in the language of mathematics. However, while Galileo was arguably being poetic, Craig apparently wants us to interpret this comment literally---almost as if he thinks that mathematics is somehow interwoven into the very fabric of space and time. Yet the moment we adopt the view that math is a language, it immediately becomes obvious how mathematics can seemingly “work” so well. Given that a universe exists, such existence is necessarily manifest to us in our sense experience. We human beings thus devised language in order to communicate that shared experience with each other, and we can use that language to formulate testable empirical predictions. Mathematics, then, is the term we give to various systems of formal languages designed to enhance precision and mitigate ambiguity. Nothing about this process either says or implies anything about supernatural entities, which means any argument to that effect is already wrong before it even had a chance to begin.
But who knows? Maybe I’m jumping the gun. So let’s see how a professional PhD philosopher with his own crack animation team thinks he can derive God’s existence from mathematics.
Scientists do not use mathematics merely as a convenient way of organizing the data. They believe that mathematical relationships reflect real aspects of the physical world. Science relies on the assumption that we live in an ordered universe that is subject to precise mathematical laws. Thus, the laws of physics are all expressed as mathematical equations. – Paul Davies
Okay, let’s just unpack this a little bit.
For starters, Paul Davies is not the holy patron saint of all science and engineering. Just because the guy happens to be a renowned cosmologist, that does not mean he speaks for all scientists at all time in all disciplines. If anything, Dr. Davies probably has little formal training in the foundations of mathematical logic, number theory, proof theory, or the philosophy of mathematics. Contrary to popular opinion, these are not the kinds of topics that physicists are required to learn for their profession. He therefore does not carry any official authority on the subject, and there are find plenty of other renowned scholars who take issue with this kind of sentiment. In fact, if Craig had actually bothered to read the very article from which this quote was derived, then he would have found that Paul Davies himself openly questioned its validity. He only made this assertion as a rhetorical springboard from which to explore how it might possibly go wrong [1].
Barring that little bit of outright quote-mining, what exactly do you think it means to say that “mathematical relationships reflect real aspects of the physical world?” It sounds cool and all, but it’s kind of ambiguous, don’t you think? I mean, I can describe the world using ordinary English propositions, and those propositions presumably “reflect” reality in some sense, don’t they? It's a classic philosophical concept known as the correspondence theory of truth. But that doesn’t have to mean the English language itself is metaphysically ingrained into the underlying structure of the cosmos. So why is Dr. Craig insinuating as much for mathematics? Mathematical propositions, for all their technical precision, are still just propositions, which means there is nothing magical or surprising about their capacity to describe our experience. That’s what all language is supposed to do.
Another weird statement that bears scrutiny is this idea that we live in an “ordered universe.” Because last time I checked, there were plenty of aspects about our universe that appear to be governed by utter chaos and randomness. It’s ironic, too, because we even have our own mathematical languages for describing this stuff as well---or did I just imagine all those college courses I took on chaos theory and probability? The very idea that mathematics and “order” go hand in hand is simply false.
It seems to me that when physicists like Prof. Davies speak of an “ordered universe,” they’re usually trying to say that we live in a determined universe---that is to say, a universe where present initial conditions can be used to predict some future state. How else is a mathematical theory supposed to “work” in this context? It means that we can use it to predict the future out of some given collection of initial conditions. It’s hardly an assumption, either, given that human beings have watched it happen since the dawn of time. We also know that this principle isn’t even universal, either, because subatomic particles are demonstrably random in their fundamental behavior.
It’s funny that Craig would try to build his case on this kind of premise, given that Christian theology requires him to accept the doctrine of libertarian free will. Yet as any freshman philosopher knows, the very definition of free will is antithetical to the concept of a deterministic universe. I therefore have no idea what Craig could possibly hope to accomplish with this premise, given that it only serves as evidence against his own theology.
For example, Pythagoras discovered that when a vibrating string is shortened by half, it plays the same note one octave higher. Isaac Newton’s observations led to his discovery of the law of gravity---a mathematical relationship expressed as a simple equation that enabled us to enter the space age. Mathematics enabled astronomers to pinpoint the location of a previously undiscovered planet, and James Clerk Maxwell used mathematics to predict the existence of radio waves. Albert Einstein, working with theoretical mathematics, developed 50 years earlier, formulated his general theory of relativity; a pillar of modern physics. His calculations were later confirmed during a solar eclipse, when Arthur Eddington observed light from distant stars bending around the sun. Then Peter Higgs used mathematical equations to predict the existence of an elementary particle. It took 48 years, billions of dollars, and millions of man-hours for experimental scientists to finally detect the Higgs boson.
Do you remember how, barely one minutes ago, Craig himself just told us that “mathematical relationships reflect real aspects of the physical world?” So why on Earth is he appealing to known falsehoods to make this case? Newton’s theory of universal gravitation is wrong! It does not accurately describe the behavior of gravity, and we have known this for over 100 years. It is only approximately valid under certain special conditions. It does not predict time dilation, it does not predict gravitational waves, it does not predict gravitational lensing, and it does not predict the precession of Mercury’s orbit. It was Einstein’s theory of relativity, in fact, that specifically replaced Newton’s laws, and even that theory is known to be flawed as well! So in what logical universe does it make any sense to cite a bunch of flawed, mutually incompatible theories as evidence that mathematics reflects reality?
I also find it hard to ignore the obvious fallacy of cherry-picking that's going on here. Just because some mathematical theories have allowed us to make reliable experimental predictions, that does not mean all of them can do it. For every successful mathematical theory, there are dozens, if not hundreds, of wrong theories that were mathematically formulated along the way. So what exactly does that tell us about the nature of mathematics when there are far more unsuccessful theories than there are successful? Does Craig know nothing of Ptolemaic astronomy, with its geocentric solar system and convoluted epicycles? Does he know nothing of the luminiferous aether? Or the Galilean coordinate transformation? Or the ultra-violet catastrophe? Or the vacuum catastrophe?
Again, you have to remember that math is just a language. I can use mathematics to describe the universe as it is, or I can use mathematics to describe a completely fictitious universe that defies all logic. Some mathematical theories are simple and elegant, while others are computational nightmares of ill-posed instability. Sometimes I want to describe perfectly ordered systems obeying absolute determinism, and sometimes I want to describe something totally unpredictable and random. Math does it all, which is why you cannot infer anything about God from this sort of premise.
How do we explain the astonishing applicability of math to the physical world?
Let me show you something:
“It is raining outside.”
Notice how I just “applied” a perfectly non-mathematical English sentence to the physical world. Yet, for some very strange reason, I don’t see Christian apologists scrambling from their ivory towers to derive God’s existence from it. So let’s not delude ourselves under the pretentious fantasy that something magical is going on here. The world is manifest to me through my sense experience, and I can use language propositions to describe that world accordingly. If that world is deterministic in any philosophical sense, then presumably I can use propositions to construct models, and those very same models can ultimately allow me to formulate reliable empirical predictions. The only thing that math adds to this situation is a capacity to construct very specific propositions within a formal structure. For example,
“It is raining at a rate of 2 inches per hour, and tomorrow it will peak at one more inch per hour.”
This is all it means to “apply math to the physical world.” The math itself did absolutely nothing, excect to give a formal structure to my propositions. Whether or not those propositions correspond with reality, however, is an entirely separate issue. You could be the greatest theoretical physicist of all time, solving the Schrodinger equation at some black hole event horizon, and this would still capture the fundamental essence of your work. A scientific model is only as good as its ability to make testable empirical predictions, and it really helps to have a formal language in which to express them.
In 1960, the Nobel Prize winning physicist and mathematician, Eugene Wigner, published an article that stunned the scientific community entitled, “The Unreasonable Effectiveness of Mathematics in the Natural Sciences.” Wigner concluded that the effectiveness of mathematics is “a miracle which we neither understand nor deserve.”
This is the third time, in barely two minutes, that this video has directly quoted some physicist to support its argument. I find it incredibly disingenuous, too, because William Lane Craig does not give a damn about the philosophical opinions of professional scientists. Take, for example, the famous cosmologist, Sean Carroll, who publicly argued that “God is not a good theory” [2]. Or better yet, consider the world-renowned physicist and mathematician, Stephen Hawking, who famously argued that “…there is no God. No one created the universe, and no one directs our fate.” [3]. The scientific community overwhelmingly agrees that God is terrible for science because God doesn’t explain anything. He cannot be measured, He cannot be experienced, and He offers no empirical predictions. Yet despite all of this, Dr. Craig appears totally unswayed by the scientific consensus over this particular issue. So why on Earth is he suddenly so eager to rely on their authority now when it comes to this philosophical quirk of mathematics?
It’s funny, too, because he’s just so ham-fisted about it. The scientific community was absolutely not “stunned” by Wigner's article, because there are dozens of scholarly publications specifically refuting his conclusions. If anything, most of the scientific community has no idea of its existence. Scientists, engineers, and mathematicians generally have very little regard for philosophy, and most of them would rather just focus on their research than waste time with such vapid, naval-gazing nonsense. Eugene Wigner’s personal inability to render an explanation for some obscure phenomenon does not automatically prove the existence of magic. Contrary to what Dr. Craig would have you believe, Eugene Wigner is not the high and mighty lord of science, either. Just because the guy happened to win a Nobel Prize in physics, that does not mean the rest of the scientific community must unilaterally bow to his conclusion over the existence of miracles.
Why is mathematics so effective? Philosophers who address this question fall into two camps: Naturalists, who believe that all that exists concretely is space-time and its physical contents. They exclude supernatural causes. And theists, who believe in a God who created the universe.
This is almost childish in its over-simplification, and Craig knows it. There is a huge variety of philosophical nuance on this subject, including Platonism, logicism, formalism, structuralism, intuitionism, and who knows how many other “isms.” I would even agree that most of these views are arguably terrible, too, but you can't just lump them all together under some catch-all umbrella like "naturalism." It's an absurd dichotomy that condenses the entire issue into a choice between “God did it” and “God didn’t do it.” So let me just respond to that with a simple question:
When in the history of science has the assumption of a supernatural explanation ever improved our understanding of anything? By definition, a “supernatural cause” cannot possibly serve as a coherent model because supernatural entities cannot be experienced. They don’t follow any causal relationships, they defy all logical description, they cannot be tested, they cannot be measured, and they make no empirical predictions. They serve no scientific of philosophical purpose, other then to put an end to all further investigation. “God did it” is therefore the most worthless explanation it is possible to render in the field of science and philosophy.
Naturalists cannot provide a reasonable explanation for why mathematics applies to the physical world. It’s just a happy coincidence. But this is no explanation at all.
I have never once encountered a single scientist, engineer, mathematician, or philosopher, who thinks it is all just a “happy coincidence” that mathematics works. That statement is nothing but a ludicrous strawman fabricated entirely out of Dr. Craig's imagination. There are plenty of detailed explanations scattered all throughout the philosophical literature, and I even summarized one of more respectable theories in this very analysis. Craig is either flat-out lying when he says this, or he is inexcusably ignorant on this subject.
At most, naturalists can say that it’s not surprising that math applies to the world because the world itself just happens to have a mathematical structure. So of course mathematics applies to it.
Isn’t it convenient how Dr. Craig thinks he can just categorically dismiss every naturalistic theory of mathematics in existence with little more than the casual wave of his hand? Furthermore, it is entirely misleading to say that the universe has a “mathematical structure.” As we have already established, math is a language. Human beings can use language to describe the universe, and it is only that language which may or may not have the mathematical structure.
But this explanation is unsatisfactory for two reasons. First, a great deal of mathematics in science cannot be physically realized. For example, imaginary numbers and infinite dimensional spaces. Although these concepts are useful, physical reality cannot possibly have the structure they describe. And second, this answer still doesn’t explain why the physical universe has such a stunningly elegant mathematical structure.
This entire paragraph is just confusingly bizarre to me. Remember that Craig has just gone through great lengths to emphasize the "mathematical structure" of the physical universe. Then, for no apparent reason, he completely undermines this premise by suddenly denying the physicality of mathematics. Only really, he says that "a great deal" of mathematics cannot be physically realized, apparently implying that some of it can. He admits openly that some mathematical concepts are apparently useful, as opposed to actual, only to then reiterate how "stunningly" actual it all seems to be. I therefore have no idea what in the hell Dr. Craig is babbling about. What does it mean for math to be physical vs non-physical? Is "structure" a physical thing? How do I tell the difference between the physically realizable math and non-realizable math?
Once again, we can solve all of this confusion by just accepting that math is a language. It is not some objectively real thing unto itself---it is something you do. Nouns, verbs, and adjectives cannot be "physically realized," either, unless you count their everyday usage in conversation. Yet, as far as I can tell, apologists are not awestruck with the basic functionality of English grammar. I therefore see no good reason for them to get bent out of shape when math does the exact same thing.
By contrast, for theists, mathematics works so well in the physical world, because God has chosen to create the world according to the plan he had in mind.
This is a textbook example of the classic logical fallacy known as the God of the Gaps. It's essentially just a glorified argument from ignorance, and it looks like this:
- Naturalists cannot explain how math works.
- Therefore, God did it.
This is the kind of argument that a freshman-level philosophy student would use on his mid-terms, only to get a note from the professor that says “come see me after class.” It’s the kind of thing that really makes me wonder how a guy like William Lane Craig could possibly earn his PhD. Do religious philosophy departments just hand these things out or something? It makes no difference how wrong you think the present collection of naturalistic theories may be. Unless you have a compelling, evidence-based argument of your own, then you don’t get to be right, either. So what possible evidence could Craig hope to provide in defense of his proposition that God just did it that way?
The first century philosopher Philo of Alexandria offered this analogy:
When a king wants to build a city, a trained architect first designs in his mind a plan of all the parts of the city that are to be completed. Then he begins to construct the city out of stones and timber, looking at the model and ensuring that the material objects are built according to the plan. Mathematics and physics work so well together because the same mind that designed the universe on a mathematical model also built the universe on the same mathematical model.
And that's it. That's his entire supporting argument: a single appeal to analogy! The universe appears mathematical because a king would never build a city without planning it, first---as if nothing in the natural world could ever exhibit any kind of empirical predictability without the express premeditation of some supernatural agent. Therefore, that one guy was right. Mathematics is a miracle, and God exists. Checkmate, naturalists.
It's bad enough that this is a classic God-of-the-gaps argument, but I think what annoys me most of all is how there technically isn't even any gap to fill. We know what math is, we know where it comes from, we know how to use it, and we know why it works. However, even if we didn't know any of that, and it was all just a huge cosmic mystery, then it still wouldn't justify the conclusion of miracles. Craig's appeal to supernatural God-magic is just the laziest possible explanation one can give in science and philosophy.
Thank you for reading.
Notes:
- “The felicitous symbiotic relationship between mathematics and science has flourished for several centuries, during which time each discipline has enriched and stimulated progress in the other. Most people assume this cozy arrangement will continue. But will it? Can we expect that the laws of nature will inevitably be expressible in straightforward terms?”
- Sean Carroll, “God is not a good theory,” [link]
- Stephen Hawking, “Brief Answers to the Big Questions,” Bantam Books, 2018