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

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]

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?

A relevant related discussion: Was mathematics invented or discovered?

## Appendix

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

Bertuzzi: Graham, it sounds like if we could split the argument in two 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".

Craig. Well, I hope that our listeners have understood your alternative because, honestly Graham, 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, Graham, 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.

• As many have said, math is just a very precise form of language. Underpinning math is something else which seems to be the quality of minds, which is the logical distinction between is and is not. Obviously, as another user points out, take two rocks and 1+1=2. But how come we distinguish objects from the background? Commented Mar 29 at 19:31
• Comments have been moved to chat; please do not continue the discussion here. Before posting a comment below this one, please review the purposes of comments. Comments that do not request clarification or suggest improvements usually belong as an answer, on Philosophy Meta, or in Philosophy Chat. Comments continuing discussion may be removed. Commented Mar 30 at 14:22
• OP, you're asking about one of the 2 or 3 "greatest mystery" questions. Clarification: are you looking for a survey answer ("Here's a quick recap of what various historic & present major figures think about that fundamental mystery"), or, are you looking for an answer as such ? (IE, you you want someone to, you know, solve it and then pop on here and outline their solution. ie, the post here would become the single greatest document, in all of reality, in all of time.) I think on this site if someone (re-) asks "Ultra" questions, nature of the answer sought should be expressed. Commented Mar 30 at 15:04
• @Fattie I'm okay with both.
– user66156
Commented Mar 30 at 17:00
• I was about to Post much of what Corey said below, when I saw how Corey had got their first. Commented Mar 30 at 20:48

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.

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).

• And as we kept looking at theories of supernatural things, we found fewer and fewer good answers. The pond has about dried up, the fish needs to grow some legs :-) Commented Mar 29 at 10:52
• We live in a world whose geometry is intrinsically non-Euclidean, which has been conjectured since at least when Gauss measured three mountains having less than 180 degrees of interior angle and demonstrated in countless experiments since the 1920s. Euclidean geometry, though an apt approximation, is the fallacious version of reality. The most interesting part of your statement is that you said "no direct parallel" which is ironic given there are infinite parallel lines through a given point in contrast with Euclidean geometry. Commented Mar 30 at 23:00
• @user121330 if a triangle was measured on a spheroid planet, then the total angle should be greater than 180 degrees, but I doubt that such an exacting measurement could be made over 100 years ago. Commented Apr 1 at 0:34
• @user121330: As far as I know, experimental data suggests that the intrinsic (spatial) geometry of the (observable) universe is actually mostly Euclidean. en.wikipedia.org/wiki/… Commented Apr 1 at 0:39
• @tomasz The key word there is locally. At infinitesimal scales, we approximate everything as flat, but near large masses, space curves. To great approximation, the universe as a whole is also mostly flat, which is little surprise since the universe is mostly empty space. Commented Apr 1 at 4:30

Reality existed.

Math was invented, partly to describe and predict reality, a useful tool.

Calculus specifically is an example...

Isaac Newton (1642–1727) is best known for having invented the calculus in the mid to late 1660s (most of a decade before Leibniz did so independently, and ultimately more influentially) and for having formulated the theory of universal gravity — the latter in his Principia, the single most important work in the transformation of early modern natural philosophy into modern physical science.

Source: SEP - Isaac Newton

Think of it this way... cheetahs and roses and frogs all existed, long before we invented languages... then we invented languages and ran around sticking labels on everything... "cheetah", "rose", "frog"... and so on.

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

We made the language to describe and label and discuss the reality.

We made the math to describe and label the reality.

Nothing more mystical than that.

• Comments have been moved to chat; please do not continue the discussion here. Before posting a comment below this one, please review the purposes of comments. Comments that do not request clarification or suggest improvements usually belong as an answer, on Philosophy Meta, or in Philosophy Chat. Comments continuing discussion may be removed. Commented Mar 30 at 14:23

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:

• Right, it's like the Platonist argument for math being real turned around: math means God is real. Come on guys, stop messing around with words! Commented Mar 29 at 10:50
• @JackAidley no one said one cannot get involved in math without applications (or without obvious immediate applications). The point was about the reasons of their effectiveness in natural sciences Commented Mar 29 at 14:36
• @JackAidley this is meant as "worked upon in relation to natural sciences", not regarding pure math Commented Mar 29 at 14:46

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.

• Granted, it is difficult to envision a universe in which one couldn't count, which is a statement to the idea that it is difficult to envision a universe without the existence of Truth. The idea that an ensemble of particles is the least designed is not so easy to grasp. What should be one's criterion? Indeed, the fact that such systems manifest complex mathematical relationships is a testimony to design; why else should order emerge from randomly moving particles, i.e. the system was designed to be intelligible.
– user73419
Commented Apr 1 at 2:13
• The fact that the universe is lawful at all isn't obviously necessary, as in the Einstein quote in the OP. Your earliest statement "numbers are naturally used to count things" presupposes that there are "things" which are coherent enough and obey enough rules to be counted. OP's question is: "isn't it surprising that things are coherent enough that mathematics applies to them", which you seem to have sidestepped completely. In reducing to a universe of particles colliding at random, you are still assuming a lot of lawful structure! Commented Apr 1 at 13:43
• @PatrickStevens similarly, I was just thinking today that it is surprising that there is enough oxygen in the atmosphere that our fossil-fueled vehicles can function. There don't seem to be a lot of other good compact energy sources that don't need oxygen, yet the early Earth didn't have it in the atmosphere. I won't speculate on why things turned out this way. Commented Apr 1 at 22:56
• Great answer! If I was a theist I would focus more on how many laws stay the same everywhere and how in many but not all cases objects in the universe exhibit order rather than complete and utter chaos Commented Apr 12 at 8:21

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.

• To be fair, if one says “the world was made without a designer” then there is no apriori reason to think that it would be represented by mathematics, and hence the theist could argue that this is still unlikely. However, the general gist of your response points out something that is very interesting, that is that God can do anything. If He is omnipotent, this automatically makes any of His actions the most unlikely of unlikely since He has an infinite number of possible actions to choose from Commented Mar 29 at 12:03
• @Baby_philosopher The a priori reason to think the world would be representable by mathematics is that mathematics is simple and we apply Occam's razor to prefer simple hypotheses. Commented Mar 29 at 16:08
• Hm I’d have to think about that but fair point Commented Mar 29 at 16:33
• @JedSchaaf Naturalistic explanations have it better because they are inherently simpler, as they are just a few math formulas. Being simpler, their a priori probability is a lot higher, while still being able to produce the phenomena we actually observe in the universe. Commented Mar 30 at 1:04
• There is at least another hypothesis: the world was created/came into being by an unknown method, and mathematics still describes it. There is no need to postulate that mathematics created the world or that math was used in its creation. We could have a world created by a sentient teapot, or by pure chance, or anything else, and still figure out mathematical rules to describe it a posteriori. Essentially, since we derived math from the world we see around us, it doesn't imply anything about how that world was created. Commented Mar 31 at 16:46

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.

• "If God has complete free will, then His actions are random. " this is a non-sequitur. Having free will does not make your actions random. Hence the following statement is based on a faulty premise. Commented Mar 30 at 15:37
• @DikranMarsupial In the case of an omnipotent entity - or one that is at least only restricted from impossible actions - that is truly free in will, the result of the being's choice is non-deterministic. Which is another way of saying 'random.' If God's will is deterministic then it is not free. So no, it's not a non-sequitur. Commented Mar 31 at 5:32
• @Corey an omnipotent entity cannot be restricted from impossible actions or it wouldn't be omnipotent, that's kinda what omnipotent means. In any case, while we use deterministic as the opposite of random in some contexts, it doesn't apply here. The point here is that if this being's actions are not pre-determined, then the being can choose how to act. But that doesn't make the actions random! There is a will involved here, the being chooses, so "not predetermined" does not imply "random", it only implies "not determined". Commented Mar 31 at 16:42
• The definition of random isn't that nobody can predict. I can make a choice between two doors based on my own preference. Nobody forces me and nobody can predict my choice in advance except me, but my selection isn't random. That's the kind of thing I mean. As for your second point about omniscience, no argument there. The whole concept of omniscience is problematic. Commented Mar 31 at 21:20
• Corey is correct. If you have true free will, your actions are by definition random. The alternative is that they are determined by something else. But if they are determined, they are not free. I wish I could find who summarized this best but the quote was along the lines of “you are free to do what you wish, but you are not free to wish what you wish!” Commented Apr 6 at 7:13

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.

• If "God" is restrictedly defined (for the purpose here, as such deity could have other aspects or characteristics not relevant to this particular argument) of some fact of orderliness, and "not God" is similarly defined as random chaos, Craig's statements become perfectly coherent. Commented Mar 30 at 3:15
• @JedSchaaf I categorically deny your characterisaion of "not God" as random chaos. You're working with a false dichotomy. Commented Mar 30 at 4:36
• What's the false dichotomy between order and chaos? Craig calls whatever is producing order by the term "God." Under that definition, if "not order" is equivalent to "chaos," then "not God" is equivalent to "chaos" as well. If you're proposing some alternate way to generate order without something inducing/creating/influencing order to exist, I'd like to hear it. At this point (OP's question and your answer) in the argument, neither the God from any particular religion nor even any of the gods from any of the pantheons are being argued for; the choice of deity is a later part of the argument. Commented Mar 30 at 9:50
• @JedSchaaf Your comments are worth becoming an answer. The difficulty with all such arguments is the highly random(!) chaotic(!!) polysemy around the word "God". The issue is that a person like William Lane Craig is clearly going to be pushing for a very Christian God — we know this from other context! Whereas in arguments like this one, this morphs into a much more philosophically astute idea like Spinoza's Nature or Lao Tzu's Tao Commented Mar 30 at 10:49
• @Rushi Spinoza's God is non-sentient, non-caring, non-active. It's little more than a way of referring to the fundamental unknowns underlying reality. Since it's a non-conscious entity it's a pointless diversion when talking about anything to do with Christianity. Apologists occasionally divert to it when looking to slip out of a line of logic that they can't otherwise present an answer to. It's not on-topic here. Commented Mar 31 at 7:33

A brute fact.

I don't even see why the question of God's (non-)existence could emerge from recognition of 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 ?

• Exactly. God further complicates things Commented Mar 29 at 16:33

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.

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

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

• It is sad that arguably the most important possible word is poorly defined. Shouldn't we fix that? Commented Mar 31 at 23:51
• People have tried, but no one seems to agree. One of the most common definitions is an all-powerful, all-knowing, entirely good, eternally existing being. Note that this neither establishes any particular religion or belief, nor is it the only explanation for the unreasonable effectiveness of mathematics. Commented Apr 1 at 0:59
• @ScottRowe The definition of God has become more obscure as time goes by because every definition that isn't extremely broad gets picked apart. Christian apologists like Craig intentionally obfuscate and lie about their definitions because they know they can't defend them. And that's how we get Presuppositionalist Apologetics. Commented Apr 1 at 7:32
• @Corey "If I'd asked philosophers what they wanted, they would have said, 'Better definitions'." - Not Henry Ford - Maybe the inability to come to agreement about the biggest and most important thing in existence would hint to people that they were looking at a mirage? Wouldn't their own common sense tell them that? Some counter-force in the psyche must be stronger, there must be survival value in this idea. Commented Apr 1 at 10:11
• The problem with saying this is a "god of the gaps" argument is that it assumes a particular deity and then argues against that one, which is a strawman argument. All this particular argument shows is that "God" (however you want to define Thon) has the attribute of being the source of math. Commented Apr 1 at 19:32

Craig, Oppy: Premise... "mathematics is surprisingly applicable to physical reality, indicating God exists"

Which mathematics is it, that are supposed to be "surprisingly applicable to reality"?

Brute fact: We do not have mathematics that correctly describe reality. We have different mathematics that describe some aspects of different parts of reality. (domain of applicability). They seem to do this well. But both are known to be at least somehwat flawed or incomplete... and more importantly...

... our two "best efforts" (plural) to desribe physical reality... are not-cohesive. They very much do not reconcile.

They are seriously not-cohesive. They fail to reconcile by between 50 and 120 orders of magnitude. (that there is a range of theorized failure of 70 orders of magnitude is a sign unto itself).

Here from Wikipedia...

Depending on the Planck energy cutoff and other factors, the quantum vacuum energy contribution to the effective cosmological constant is calculated to be between 50 and as much as 120 orders of magnitude greater than observed, a state of affairs described by physicists as "the largest discrepancy between theory and experiment in all of science" and "the worst theoretical prediction in the history of physics"

Our two best theories are:

• The standard model of particle physics (Quantum mechanics, evolved to become Quantum field theory, a.k.a. QFT)
• The standard model of cosmology (relativity plus dark matter, plus dark energy, a.k.a. Lambda-CDM

They do not reconcile.

We do not have a "surprisingly applicable mathematical description of reality".

We have a disjointed mess, and literally dozens of efforts and attempts to fix it, none of which are thought to be "Oh this one is likely the one that describes actual reality"

From Wikipedia:

At present, there is no candidate theory of everything that includes the standard model of particle physics and general relativity and that, at the same time, is able to calculate the fine-structure constant or the mass of the electron.

Before "the surprising applicability of mathematics in describing physical reality" is accepted as a premise, it would be dilligent, prudent... a philosophical obligation... to request that "mathematics" be specified. Not accepted as a warm, fuzzy, generality....

... don't just say "mathematics"...

... provide the actual mathematics.

If said mathematics cannot be shown to exist, the "surprisingly applicable mathematics"... then all argument based on that underlying premise become null and void.

And I would call that "Case Closed".

Though, I will happily amend my answer, should the mathematics that are surprisingly applicable to physical reality" show up.

Also, Stanford Encylopedia of Philosophy still declares that Newton invented the calculus, for what that is worth.

Isaac Newton (1642–1727) is best known for having invented the calculus in the mid to late 1660s (most of a decade before Leibniz did so independently, and ultimately more influentially) and for having formulated the theory of universal gravity — the latter in his Principia, the single most important work in the transformation of early modern natural philosophy into modern physical science.

We humans do like to invent things.

• Very good point. And, we've already tried attacking the other side of the assertion - which God? So, it's pretty much a bust, as arguments go. But it is warm, fuzzy and appealing... We can invent anything, including stories about how we all came to be here and what we should be doing. Commented Apr 5 at 11:25
• @ScottRowe Thx. Twas my thinking... take a closer look at the other half of the proposed equation. Commented Apr 5 at 14:36

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.

• There are facts and there are facts. Both "the world is round" and "1 + 1 = 2" are facts, but they are different. One is contingent and only contingently brutish. The second is a priori necessary (whether due to a tautology or a necessary structure of the mind which may amount to the same thing) and is thus more despotic. The most dogmatic atheist can acknowledge things outside her understanding without pretending to understand it by labeling the unknown as "God." Commented Mar 31 at 22:47
• @Gerry How can a fact be both contingent and brute? And as a theist myself, I reject "God of the gaps" as far insufficient, as I believe that God is the ultimate reason for everything, not just the "gaps" in our knowledge. Commented Apr 1 at 19:24
• Your question was, if I recall, is math a brute fact of the world? Maybe I should ask you, what do you mean by "brute fact"? Commented Apr 1 at 20:12
• It's the OP's question, not mine. As I understand it, a "brute fact" is a fact that cannot be explained further; it simply exists. Wikipedia: Brute Fact Stanford Encyclopedia of Philosophy: Facts: Brute Facts Commented Apr 1 at 20:21
• Sorry for getting mixed up, if it's OP's question, then maybe he or she can tell us what is meant by "brute fact." As for your statement, "it [a brute fact] simply exists.," in what sense does a fact, brutish or otherwise, exist? I would also ask OP's, is "brute fact" redundant? Are there non-brute facts? Maybe OP added the word "brute" for color or emphasis and simply meant "fact". Commented Apr 1 at 20:29

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.

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 all 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.

• So truth is God. And God is love. Therefore truth is love. Yeah, semantics is dumb. Commented Apr 1 at 1:20
• @Corey No, the argument is not so "dumb" as it sounds. However, even the best argument accomplishes nothing philosophically serious. The last serious contribution to the corpus of arguments pro contra theism was made by Kant. Any argument which asserts that it has proven or disproven God's existence is mere rubbish, just as much so as the semantics you disdain.
– user73419
Commented Apr 1 at 1:31

This site's post and edit functionality is HORRIBLE!

This was properly edited and formatted. Now look at it! It will not let me space out the bottom part of my answer. I am really not interested in learning yet another editing tool. Standardize, please!

[Quote=Mark]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:

If God does not exist, the applicability of mathematics to the physical world is just a happy coincidence. The applicability of mathematics to the physical world is not just a happy coincidence. Therefore, God exists.[/quote] To me this is JUST hilarious! It cannot be 'serious' philosophy or perhaps it's too serious from a certain POV. I will explain. At the end.

[Quote=Mark]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][/quote] This justification is a bit weak. Just because ... someone deemed important ... stumbles when faced with a new or old but previously hand-waved argument as difficult to 'deal' with, means ... not so much. Ignorance and dismissal are very normal motivations. We should be more surprised if there was no stumbling, yes? We should be more suspicious is there was no stumbling, yes? Is confidence the only 'cool' thing. Prepare to be boarded then!

[Quote=Mark]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.[/quote] OK, is it just me, or does Craig Oppy sound like a generic name for the opening poster? I do not know the name, but, really?

[Quote=Mark]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,[/quote] And you throw a term like this, 'brute fact' without defining it. Does everyone know what a 'brute fact' is? Is it different than a 'fact'? Is a 'fact' different in some way to a belief? Why am I quoting the word 'fact'? The mistake, all mistakes, could be hidden within THAT ONE TERM. Faite attention!

[Quote=Mark]thus not requiring further explanations (as necessary things explain themselves).[/quote] See what you did there? Once you and FALSE onto any string of assertions, you are done. SO my previous comment was well suited. Nope. Define 'brute fact'. I promise you in that definition your error will be shown. It will not be hard to believe that it is an error. Something in you prefers certainty and that something is gravely mistaken (and cowardly I might add). The bold one wins points with you for having what you do not, courage! To put it in your terms, that would be my 'brute fact' (in this specific case - there is more)

[Quote=Mark]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.[/quote] Ugh! I think Capitalism cries out for an explanation as well as Democracy! But I think, I believe, I factualize, that I know the answer; and any of those efforts could partake as well of brute force, more is the pity. Let's chat with Socrates. 'Men of Athens (the universe) ... you are not wise!

[Quote=Mark]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?[/quote] All of these necessities you ascribe to God I suppose, or even mathematics, are NOT necessities. They are desires. By my own demand earlier I must define this horrid term, desire. Desire: The emotion given to describe the state or motivation in the universe often referred to as the meta-emotional concept of Chaos. Desire in state is referred to as freedom. This is the meaning of 'pursuit of happiness' in the Declaration of Independence. Desire is the left hand of the universal law that defines all reality. It represents Plato's forms, ideals, perfection itself, finally. But desire only intersects perfection at the single point of objective moral truth (the GOOD). Now let's see if the left hand knows what the right one is doing, shall we? Fear: The emotion given to describe an excited state arising as a consequence of matching a pattern from one's past. This can be confused with or relegated to the meta-emotional concept of Order. Fear is the right hand of the universal law that defines all reality. It represents logic, mind, thought, and all of Aristotle's Pragmatism in the face of impossible idealism. But fear only intersects perfection at the single point of objective moral truth (the GOOD). All your beloved math, the reason you pattern so much to have the order, the discipline, to ask such a question here, is because of fear. Fear separates and limits. It places delusional lines where none exist. Fear is ONE limit as choice approaches Truth(perfection). There is seen the USE, the PROPER use of your mathematics. Desire is another limit. But a good maths person should realize as a potentially 'brute fact' that limits are asymptotic in character. They do not arrive at the final point until WHATEVER infinity is. Is infinity perfection? Circle takes the square! And thank you Paul Lynde! So, your order fixation is NOTED. You and your maths! YOU! and your 'brute facts'
A fact is only a belief for which one has currently ascribed the full value of truth. Write that down. Inscribe it on your forehead in reverse worded relief, burned in. It will be service to the word, and believers in delusional limited fear. The mind is but one part in three. So there is mind and fear maybe. There is heart and desire maybe. What is this unmentioned third part. It is your wrongly chosen master. The confidence needed. It is anger. It is being. It is standing to ALL comers and accepting only belonging and unity. Anger demands that fear recede and the mystery of desire be faced with confidence of belonging! Anger demands that desire recede and the puzzles of awareness be properly hypothesized and navigated. This can only happen if one considers oneself worthy and not some needy thing that is insufficient unto itself (eg taken by desire) But anger, being the balance, representing a meta step towards wisdom, is STILL not perfect. It to is a limit. For laziness or lack of action is a thing, the sin of anger. A balance can be dead. That is not the good. The perfect GOOD of Socrates, encompassing all ideals of Plato and all doubt of Aristotle, is only and always a maximal balance of all three emotions.
Fear, Anger, and Desire are all that exists in this multiverse. They balance amid perfection to support only one thing, free will.
You are free to choose to believe what you will because of Perfection. Call it God if you choose. Call it 'All'. Call it you. Call it me. It is only everything.
Perfection is the clarion call within that system. It gives rise to desire and reflects worthlessness upon us in the guise of fear. We must use anger to balance both other emotions and only from that balance is wisdom's first light seen.
Thence we step upon the path of the GOOD, more and more right action amid balance. This is the Stairway to Heaven, the real one, and no window-dressing of mere human belief is required to exalt it. It is Truth.
**

• This site uses markdown, not BBCode or whatever it is you're trying to use. Commented Apr 1 at 1:12
• This is a very bold answer. About 75% of it, I think. Commented Apr 1 at 22:35