Truth Unites - Can Math Prove God? The Argument From Eternal Truths

Episode Date: December 9, 2024

Gavin Ortlund lays out the argument from eternal truths for the existence of God. At 1:02:10 I meant to say "objects," not "objections." Math is precise but I'm not. Truth Unites exists to promote gospel assurance through theological depth. Gavin Ortlund (PhD, Fuller Theological Seminary) is President of Truth Unites and Theologian-in-Residence at Immanuel Nashville. SUPPORT: Tax Deductible Support: https://truthunites.org/donate/ Patreon: https://www.patreon.com/truthunites FOLLOW: Instagram: https://www.instagram.com/truth.unites/ Twitter: https://twitter.com/gavinortlund Facebook: https://www.facebook.com/TruthUnitesPage/ Website: https://truthunites.org/

This video is an argument for the existence of God from eternal truths, like mathematical truths, for example. I'm so excited to share this video. It's probably my favorite video that I've ever prepared for just from how much I learned. Hopefully you'll feel, hopefully you'll feel something of the same excitement and joy about this. If I have one hope for this video, it's not necessarily that someone who's dead set against believing in God will be forced to the other side. Maybe someone who's in the middle could be influenced by it. Maybe somebody could be influenced like that. I would say the main hope is comfort and joy for those who are believers or who are open to believing to kind of push, nudge them along just to invite people to see just, oh, the only word I can use

in the English language is the enchantment. I'm thinking about the nature of God and how it explains so much. I've read a lot of books to make this video, lots of research, condensed it all down to a compact script that I've written out to be as organized as possible. So I would love to just invite people to carve out an hour or so and watch this video or listen to the podcast. I hope it'll be worth your time. The argument from Eternal Truths goes back to Augustin and Leibniz, as I'll document in the second half of this video, drawing both of them in. But today it's a lesser-known argument for God, sometimes seen as kind of a weirder argument. I've become convinced it's true, it works. I think it has real value. I think it deserves wider consideration. I hope this video

will help promote it more out there for people to consider and work with. And I will say that of the various arguments for God's existence, I find this one particularly intriguing and even emotional, as you will see. The argument typically involves more than just math. So other entities from what we call abstract objects, logical truths, universals, propositions, sets, functions, etc. All that is often involved, we're for this video going to focus just on mathematical truth. And I like to state this argument as an abductive argument, meaning an inference to the best explanation. So this will come out in basically two halves of this video, the two movements of this video. First, we'll highlight the nature of math, and will describe six of its characteristics, and then we'll highlight

the nature of theism and highlight six of its characteristics that seem particularly apt or fitting as an explanation for mathematical truth. You can see those on the screen. Yes, I worked hard on the alliteration. Really hard to, like I said, try to be organized. If you stick to the end, we'll go through five objections briefly, and then I'll talk about how this relates to the doctrine of the Trinity. The overall conclusion here is that theism, or something like theism, provides the best explanation for the ontology of math, the word ontology meaning the being of math. But I think that theism beats out some of the alternatives by a wider margin than others. So from the perspective of this argument, you might say that theism will barely edge out deism.

Outperform pantheism, but not by a wide margin, but it's going to beat the pants off of naturalism. Naturalism is the philosophy that everything boils down to physical causes and properties. There's nothing supernatural. As we will see, that worldview seems particularly weak at explaining math. Now, if you wanted to state this argument deductively, you could do so like this. I'm not making this argument, but I'm just for conceptual clarity. I'll put it up. You could do so like this.

You could just say, if eternal truths, then God, there are eternal truths, therefore God. but I'm going to be doing the more abductive approach that I just mentioned. Two quick notes, then we'll dive right in. First, I'm going to use the phrase eternal truths, because this is the customary label for this argument, the argument from eternal truths. Some people prefer to speak of necessary truths. Other people like different language. For the moment, you don't have to have any view on that. We'll unpack that a little bit as we go and explore what is math. Second preliminary comment, this is an a priori argument, meaning it doesn't depend upon any particular experience or observation of the world around us, it's based on rational reflection. And I think the

possibility of a priori arguments itself is kind of fascinating. You know, can we know things by sheer reason, apart from experience? And I have the intuition that if God exists, it would be fitting that he would be known by a priori arguments as well as other means. And I think that bolsters the stakes of this argument a little bit. The other major a priori argument is the ontological argument, which I've done a video on. I also just did a dialogue with Joe Schmidt, who's a great philosopher. It was an honor for me to talk with him on this argument. I'm going to release that on my channel as well. Interestingly, Robert Adams, who's a very good philosopher, says in his book on Leibniz, that the argument from eternal truths is more persuasive than the ontological argument,

and he calls it the most promising of all a priori arguments for the existence of God. So that's interesting. Ultimately, you could think of this video as a theological, explanation of the ontology of math. We're saying, what is math? Well, it looks like God provides a great explanation for that. So let's dive in. First, we'll explain the nature of math before we get to theism. What is math? We often take math for granted. Tyrone Goldschmidt has an essay in the two dozen or so arguments for God book. Have you read this book? Inspired by Album Plantiga, great book. Really high-level stuff. You feel like you're in kindergarten when you're reading it. It's like, wow, okay, taking you going pretty deep. The first three chapters of this book, by the way,

are all related to this argument. But before building his own argument, Goldschmidt says, I went through years of math class without stopping at all to ask what kind of thing numbers are if they are so much as things at all. I think most of us can relate to that. We just tend to assume math. We don't necessarily think about it too much because it seems so basic to reality. For example, If you were to, boy, believe me, I understand that I'm going to risk really nerding out by even making this video on this topic. But you could think of it, because we're going to ask a lot of really abstract questions. But if you could just think like, you know, could reality be a numerical? No numbers. And as soon as you think about that, you realize very quickly, it's hard to imagine what that would be.

If there's any objects whatsoever, it just seems any finite reality seems to require math. David Oderberg talks about this. He wrote this amazing book we're going to talk about again later, called. real essentialism. And he basically says if there's any objects at all, then it would seem essential that there's going to be numbers. Numbers seem inherent to reality. And this is what a lot of philosophers say. Edward Lowe, another great metaphysician, says if anything exists, numbers exist. So a lot of us have that intuition. It's like, well, of course numbers are out there. On the other hand, if you stop to think about it, math is really intriguing. Why are these particular numbers

and equations so basic to reality. See, we've got to ask this. In philosophy and especially metaphysics, you ask these really basic questions, and they're interesting. One introduction to math puts it like this. If you do mathematics every day, it seems the most natural thing in the world. If you stop to think about what you were doing and what it means, it seems one of the most mysterious. The word mysterious, it's going to come up a lot in this video as we describe math now, as well as the word mystical. Why is that? Well, let's dive in. Let's work through six features of math. By the end of this, by the end of these six, I hope, again, I hope this video will be worth your time. It's really interesting to think about this. I hope by the end of it, you'll have a sense of why we call

math mystical. And I hope it'll even feel, I mean, I'm going to put it this strong. Again, at the risk, I realize this is really nerding out here. But I'm going to put it this strong that to think about the nature of math feels like walking through the wardrobe into Narnia. It's enchanting. Let's explain this. Let's work through these six aspects of math. Its objectivity, its independence, its exactitude, its interconnectedness, its usefulness, and its excitement. First, objectivity. The experience of math, this is the most important point. The experience of math seems to suggest it has a kind of objectivity, that is to say, its truth seems to persist, irrespective of our awareness of it, our engagement with it, it just seems like it's out there.

One introductory text to the philosophy of math puts it like this. The activity of mathematical research forces a recognition of the objectivity of mathematical truth. The Platonism of the working mathematician is not really a belief in Plato's myth. It's just an awareness of the refractory nature, the stubbornness of mathematical facts. I like this word stubbornness, you know. I like to put it like this. Two plus three equals five seems to be true regardless of whether you like it or not, whether you've traveled to Mars or not, or you're on planet Earth. Two plus three still equals five. You get in a time machine, you go a billion years into the future, a billion years into the past.

It seems like two plus three will still equal five. It seems to be this objective truth, irrespective of your location, your perception, et cetera. Now that is sometimes disputed, as we will discuss. but most mathematicians, certainly most just human beings, understand the objectivity of mathematical truth to entail some kind of mathematical realism. This is the term I used to describe simple view, mathematical truth exists independently of human minds. You can summarize the core intuition of mathematical realism by saying doing math is not like being an architect who builds from scratch, it's like being an archaeologist who excavates what is already there. Mathematical truth is out there, fixed, external, and available for discovery.

Now, again, this is sometimes disputed. You can see a lot of different views on the market today about the sort of metaphysical underpinnings of math. Various different kinds of anti-realism have become, they're kind of trendy in certain circles. One non-realist mathematician puts it in this poem, basically using the image that the path is not there until you walk. You make the path by the walking. It's an image for an anti-realist view of math. We're sort of constructing. But it's interesting to note how rare such views continue to be. They're less popular across the board. Some species of mathematical realism is the overwhelming

view, certainly of human beings, and even among working mathematicians, one older study estimates that about 65% of working mathematicians are realists, and they suggest it's actually much higher in practice. And it jokes, the typical working mathematician is a Platonist on weekdays and a formalist on Sundays. Formalism is one of the dominant kinds of anti-realism. Another introductory text to math contrasts Platonism with formalism and says, Platonism is dominant, but it's hard to talk about it in public. Formalism feels more respectable philosophically, but it's almost impossible for a working mathematician to really believe it. Later, this book says, an inarticulate half-priculate, conscious platonism is nearly universal among mathematicians. By the way, note on terminology,

the word platonism can be used in different contexts. In these quotes I've just given, it's basically denoting a kind of realist view. Sometimes it's used as a stand-in for mathematical realism. Other times it's used as a species of mathematical realism. Later on in this video, that word will come up in a different context, but just here we're talking about a realist view of math. So the question is, why is mathematical realism such a persistent? view. Realism is often perceived as the most intuitive view, since the objective existence of mathematical truths is often assumed in the development and application of mathematical theories, and it just is the way it seems, right? On such a view, you can explain the powerful sense

of discovery that mathematicians often have in the experience of mathematical progress. Here's one of my favorite quotes I'm going to share in this video from Paul Erdash. He's a Hungarian mathematician. He used to refer to the book as the place where God writes out the elegant proofs of each mathematical theorem. And he'd talk about, you know, discovering a new math. Math proof is like God showing you a page in the book and you're reading the book. And he called the book S-F's book. S-F stands for Supreme Fassist, his nickname for God. So this is a very powerful image of a realist view of math, right? The book is out there and we discover the new page of the book. The British physicist Roger Penrose describes the discovery of a fractal called the Mandelbrot set and makes the comparison

with Mount Everest. Like Mount Everest, the Mandabroft set is just there. This is what you see so much as this is what overwhelmed me as I was reading about the philosophy of math, hearing the testimony of mathematicians, this experience of discovery and the excitement of it. And you see this so much. Penrose, spoke of doing math is like being guided to eternal self-existing truths. Put this quote up on the screen, you can read that if you want. That's a powerful image. I mean, you can see with the language of being guided to eternal truths why a word like mystical is going to come up, but just wait. We're just getting started here. Penrose wrote a follow-up book, and he suggested that there's three distinct realms of reality, the physical, the mental, and the platonic. So the platonic realm involves

abstract truth like math and logic. The material world is the physical world. The mental world is our experience of consciousness. And he emphasizes the mysteriousness of how these different realms can interact with each other. And interestingly, he emphasizes the primacy of the platonic realm over these other two. And according to Penrose, the platonic realm is just as real as the other two. He says it might seem like a rag bag of concepts that we conjure up, but its existence is solid By the way, I looked up the word ragbag. It's a real word. And it means a bag of, maybe it's British, I don't know.

It means like a bag of scraps. Okay, so listen to this quote. I'm going to take the risk in this video, making number one a really long, abstract video, but number two, reading a lot of quotes, but every quote has been chosen carefully. Listen to what he's saying here. What right do we have to say

that the platonic world is actually a world that can exist in the same kind of sense in which the other two worlds exist? It may well seem to the reader to be just a rag bag of abstract concepts that mathematicians have come up with from time to time. Yet its existence rests on the profound, timeless, and universal nature of these concepts and on the fact that their laws are independent of those who discovered them. The ragbag, if indeed that is what it is, is not of our creation.

The natural numbers were there before there were human beings or indeed any other creature here on earth and they will remain after all life has perished. Kurt Gertel. So that's a very obviously, I mean explicitly, clearly a realist view. Kurt Gertl, the famous 20th century logician and mathematician, said our mathematical intuition should be as confident as our sense perception. You can read that quote on the screen. So in other words, if Kurt Gertl is right, you should be as confident that 7 is the square root of 49 as you are confident of the air in your lungs. Now, one reason for accepting a realist view of math is its ability to explain how the same

mathematical truths are capable of being shared by different minds. They are intersubjectively available. Anybody can go out and discover that two and five are the only prime numbers that end in two or five, and that discovery can be shared by multiple people. Ed Fraser says, when you think about the Pythagorean theorem, and I think about the Pythagorean theorem, that's A squared plus B squared equals C squared. We are each thinking about one and the same truth. It's not that you're thinking about your own personal one and I'm thinking about mine. In fact, the objectivity of math means that not only can multiple people share its knowledge,

but even mathematical truths that are forgotten can be rediscovered in the future. And if there are multiple independent alien civilizations, they could all independently discover the exact same mathematical truths and potentially even use them to communicate like a language. This is what happens in the novel Contact. If you remember this movie, they use prime numbers to communicate with the human beings. And if you read the book, there's a hidden code inside the number Pi that has special significance, which is really interesting. You know, you can read the book about that. Basically, pi is the mathematical constant. That's the ratio of a circle's circumference to its diameter. It equals approximately 3.14159. If you remember this number from like a geometry class and it goes on repeating.

And so the book is saying there's a code hidden in that number. It's kind of interesting. So that's objectivity. Second of all, math appears to have not only a kind of objectivity, but independence. specifically independence from physical spacetime reality. So in other words, I like to use the thought experiment to explain this. Suppose that the physical universe were to collapse into non-being, would it still be the case that 2 plus 3 equals 5? Most people, again, not everybody, but most people would say yes. And they would kind of say, well, you know, what else could 2 plus 3 equal other than five. You might even have the intuition that two plus three is going to equal five, and that's going to go merrily on its way, regardless of whether universes pop in and out of being.

Who cares about universes? You know, two plus three just equals five. You don't need a physical realm for that to be instantiated. And that idea that two plus three equals five is somehow independent of physical reality is really significant because it raises the question of where does math get this kind of fierce, persistent kind of truth. I'll never forget when this problem was impressed upon me. I was reading Thomas Nagel. If you know Thomas Nagel, great philosopher, he's not a religious person, but he is talking about the problem of consciousness, and he gets to the mind-body problem, and he basically says, you know, physical science cannot explain the whole of reality, because consciousness is not reducible to strictly material processes.

and he says, and I'll never forget getting to this sentence, it was a light bulb moment for me that led me to the problem of issues related to math. He says, there seemed to me to be two very different kinds of things going on in the world. The things that belong to the physical reality, which many different people can observe from the outside, and those other things that belong to mental reality, which each of us experiences from the inside in his own case. So you have this kind of two different realms. Now, the question that immediately, that came up for me when I read that sentence is, well, why would that mental realm even be there? You know, how did that come into being? How does the physical produce the mental? I told you this

was going to get abstract. Now, that question applies not only to the problem of consciousness and our mental thoughts, what Nagel is talking about and what Penrose calls the mental realm, but also the abstract truths that our minds can intercept what Penrose calls the platonic realm. And the same question comes up is just where does this realm come from? What's it doing here? If the realm of math seems to possess this kind of independent reality from physical objects, we get invited to kind of compare the two and say how, you know, let's try to describe the differences between material reality and mathematical reality. And here's my absolute favorite quote I'm going to put up. This is from, he's a Belgian mathematician. I'm going to pronounce his name Sylvan Capel. I hope I'm

not mispronouncing that. Maybe it's Capel. But let's just got to pick one, because I'm going to mention him a lot. He's a really good mathematician at NYU. This is my favorite quote, even more than the Erdosh quote about the Supreme Fascist. Like Penrose, Capul thinks that math and physical objects are kind of these two different worlds. Here's how he describes them. All mathematicians live in two different worlds. They live in a crystalline world of perfect platonic forms, an ice palace, but they also live in the common world where things are transient, ambiguous, subject to vicissitudes. Mathematicians go backward and forward from one world to the other.

Now leave this quote up for a second, and note what I underlined. There's the words perfect and crystalline to describe one world, in contrast to the words ambiguous and subject to vicissitudes to describe the other. and what that highlights is how qualitatively different these two worlds are, which is also implicit in the reference to going back and forth between them. So on the one hand, we have this coarse, rough, variegated realm of material objects. On the other, we have the ice palace, this perfect, clean world of mathematical objects. Math has this particular feel to it. I like that image of an ice palace. It communicates something of this.

It feels firm and hard-edged. We're going to come back to this in a second when we talk about mathematical exactitude. But first, note the other word. I'll put this back up that Capel uses that I'll put in red font here, the word transient. This highlights another difference between the world of materiality versus the world of math. Physical objects are constantly changing, fading in and out of being. Everything is in flux. Mathematics seems permanent.

it's enduring, it's durable, it's stable. Again, the ice palace. I like to think of math as an invisible castle, rising up all around us. So, but it's invisible, so you don't see it with your eyeballs, but it's there. And you can see it with your mind by thinking. And it's just as real, just as objective. You can count on it to be consistent, for example. Just like if there's a physical castle, you're not going to think it's just going to start flying around. So also, math, is this permanent, stable truth all around us. Now, the classic view of math, going back to Pythagoras and Plato, as well as Christians like Augustine, says that mathematical truths are eternal.

Plato said the knowledge at which geometry aims is the knowledge of the eternal. And that would be true for arithmetic as well, like 2 plus 3 equals 5. The intuition is 2 plus 3 equals 5 has been true forever. Now, if you agree with that or not, we can at least say math seems to have this kind of robust objectivity that is independent of physical reality. That's certainly the most basic intuition. Now, though, there are some wrinkles here. One wrinkle is that you do find mathematicians talking about modular arithmetic, where, or they'll talk about mathematical truth being, having context to it. Boy, I'm going to talk a little bit about a book here that uses the imagery of

trees, how trees can bend in the wind, but they're still planted in the ground. So they have rigidity and stability, but they can move a little. So, you know, you get into some kind of those discussions, but even there, there's still permanence. They're planted in the ground, the trees. So mathematical truth has this kind of objectivity and this kind of independence that we intuit. Third, it has an exactitude. It's not only permanent, it's precise. Not only would two plus three equal five seem to continue, did I say that right? Two plus three equals five. seem to continue apart from physical reality. But it seems like 2 plus 3 equals 5 will never fade into 2 plus 3 equals 4.999.999.

2 plus 3 equals 5 doesn't usually mean 5, but every now and again it means 5.000000 1.2 plus 3 equals 5 always equals exactly 5 and nothing else. It has this unbelievably consistent laser-like precision. And again, that's extremely different from physical reality. This is why we call it the Ice Palace. And so it raises the question of why should that be? You know, why doesn't 2 plus 3 eventually fade into 4.999999? What sustains that level of laser-like precision?

And this is important because the precision of math is necessary for its usefulness. The fact that math is so precise and so predictable and so consistent is what enables us to rely on it for making complicated predictions and developing the technology that we have. We couldn't send people to the moon if math was imprecise. The usefulness of math, it's really fascinating because sometimes even just the simplest proof or theorem will have an unbelievable range of applicability. So I mentioned earlier, or I think Ed Phaser quoted talking about the Pythagorean theorem. This is A squared plus B squared equals C squared. It's like, that's just this tiny little truth, you know, a third grader. You can draw out a triangle and explain it.

But the things you can do with that, it's like math gives you, it's like this tiny little tool, but you can move mountains with it. The precision and usefulness of math is just amazing. And it's amazing how consistently and precisely math, maps onto physical reality, which when you think about it, why should we assume that? Why should mathematics, these two realms, you know, the course realm and the ice palace, why should they fit together like two puzzle pieces? Why should ideas and objects work together so consistently? That's not obvious that it should be that way. I'm going to say more about that when I get to

math, math youth, youth, I can't talk today, math, math youth, yeah. I can't talk today. Maths youth. Man, I'm going to leave this in the video just for fun. This is to see, this is a long video, so it's to see if you're paying attention. Can I try it the third time? Maths usefulness. Wow. Why is that so hard to say? I don't know.

I got a lot to cover in this video. I could ask you to bear with me. All right. So we're going to get to that. That's characteristic five of math. But here, let me bring in the fourth characteristic next, because this is going to be relevant to that. That's the interconnectedness of math.

In her book, the joy of abstraction, which I'm going to talk about more in a second. Eugenia Chang points out, and I'll put up the picture I took from this page, early on in the book, different mathematical disciplines all form one interconnected web of truth. So the different disciplines of math all relate to each other with harmony, such that mathematical truth forms a coherent, intelligible whole. Once again, that is necessary for its usefulness. That needs to be the case for us to put it to work. And just as it's surprising that mathematical truth and physical reality fit together so perfectly,

so it is comparably surprising that the different disciplines of math, arithmetic, calculus, trigonometry, geometry, et cetera, all work together so harmoniously. Why should that be? That leads to the fifth point here, and that's the usefulness of math, which draws from these previous points. The fact that mathematical truth works with the different disciplines work in harmony with the each other, they apply so consistently to the physical universe. This is incredibly surprising. Albert Einstein once delivered a lecture in which he marveled on this, quote, An Enigma presents itself which in all ages has agitated inquiring minds. How can it be that mathematics, being, after all, a product of human thought, which is independent of experience,

is so admirably appropriate to the objects of reality? Back in 1960, Eugene Vigner put forth a famous argument that the enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious and that there is no rational explanation for it. Vigner spoke of the applicability of math to physics as a kind of miracle and as a gift for which we should be grateful. You can read this quote if you want. He says, even if we remain baffled at why this is the case, we should be grateful that it is the case. An amazing statement. Roger Penrose calls this uncanny usefulness of math, mathematical fruitfulness, and he basically emphasizes the more you learn about math, the more remarkable it is. By the way, parenthetical pause coming off of my

notes here just to say this, if you're a working mathematician and you watch this video, I hope this could get sent to people who are mathematicians. I'm not actually good at math. I've never enjoyed math. I've gotten into the philosophy of math as an adult, but I wasn't like really enjoying math when I was a kid. But if you're a working mathematician, I would really value your thoughts in the comments about just how you experience all of this. Because what Penrose, because I'm trusting the testimony of others, what Penrose is saying is, the more you get into math, the more you feel, just how amazing it is that it's so useful. And Penrose is citing Einstein's theory of general relativity as a prime example.

And he says, Einstein was not just noticing patterns in the behavior of physical objects. He was discovering a profound mathematical substructure that was already hidden in the very workings of the world. And that word hidden there in the last sentence emphasizes that this remarkable congruence between math and physical reality becomes more detectable as we learn more about both realms. You study physics, you study math, and you see, oh, there's something hidden. They're connecting at points. But it was hidden before we studied it.

So progress reveals more of this. It's interesting. And this raises the question, of course, of how it's that way. Why is it that way? Why is there that connection? And many scientists and mathematicians speak of this mystery in religious terms, albeit sometimes metaphorically. You saw Vigner speak of it as a miracle. Paul Dirac says God used advanced mathematics and constructing the universe.

Even if this language is metaphorical, you can appreciate why people are saying things like this, or why Erdash will speak of the book written by the Supreme Fascist, or Penrose will speak of being guided mystically to the eternal truths, or Capo, speaking of the Ice Palace and so forth, but we're not done yet. All of this snowballs and leads us to the last and most intriguing quality of math, and that's the excitement of mathematical experience. People often think of math as a very analytic, rigid, left-brain activity, and there's some truth to that, of course. But one of the things you discover is you get further into the literature of the philosophy of math is how, the more you go into math, you also get into

right brain territory and creativity, intuition, flexibility, and you discover much that is beautiful and exciting. So let me talk about this 2003 book by Eugenia Chang called The Joy of Abstraction. This is a really cool book just from the standpoint of the philosophy of education, how you take really abstract ideas and popularize them. It's a great book from that standpoint, how it goes about things. But it's making the point that math is a right-brained activity It's emphasizing that math is relevant to all of life, even like your relationships. And the specific focus of this book is on category theory, which is kind of a very general theory of math, sometimes called the mathematics of mathematics.

And it's talking about these very high levels of abstraction and so forth. And it's pretty fascinating and mind spinning. But let me just read you the first few sentences of the whole book, just to make this very basic point. There's so much more we could say. But she says, she starts off the whole book saying abstract mathematics. brings me great joy. It is also enlightening, illuminating, applicable, and indeed useful, but for me, that is not its driving force. For me, its driving force is joy. So, do you hear what

she's saying there? There's joy in math, and the joy is not reducible to its usefulness. Its value is more than just being a tool or an instrument. And this is an extremely common testimony of mathematical experience. So my favorite movie, for example, is a beautiful mind. Several years ago, my wife bought me the book that is based on for Christmas. I haven't even finished it, but I've read a bunch of it. And this is the story of John Nash, who's a mathematical genius at Princeton. And one of the main themes that comes up in this book repeatedly is the beauty of math. At one point, a fellow student is entranced by one of his ideas, not because of its application, but because of its elegance and its beauty. Paul Erdog,

considered the beauty of math so obvious that it needs no explanation. He says, why are numbers beautiful? It's like asking why is Beethoven's Ninth Symphony beautiful? If you don't see why, someone can't tell you. I know numbers are beautiful. If they aren't beautiful, nothing is. Wow, what a sentence there. If math isn't beautiful, nothing is. Now, that might seem over the top, but some of you watching this, especially this is why I'm interested in the testimony of mathematicians, maybe you've experienced something like this. This is why I'm willing to risk nerding out here talking about this because I actually have the similar intuition that there's just something mystical and enchanting about the Ice Palace. It's just beautiful. It's hard to articulate why. It's like

trying to say, well, why is music beautiful? You can give a theory, but it's hard to put it into words, right? Another mathematician draws a similar comparison with music, saying, you know, you can kind of get this intuitive feeling of its beauty. Some mathematicians go so far as to describe the experience of beauty and excitement in mathematical progress as a kind of transcendence. So Luke Ferry, writing from a secular humanist perspective, describes the objectivity of math that we've discussed here, but then note what he says next. He says, I can do nothing about it, two plus two equals four, and this is not a matter of taste or subjective choice. The necessities of which I speak impose themselves upon me as if they come from somewhere, from elsewhere.

And yet, here it is. It is inside myself that this transcendence is present and palpably so. So you hear what he's saying. He's saying, the math is out there, the invisible castle, the book written by SF, the Ice Palace, etc. Penrose's separate realm and so forth, but it's also inside me. And I can relate to it, and the experience of it is exciting. In Paul Davies' book, The Mind of God, he discusses the experience of euphoria,

after mathematical breakthrough that some people go through, and remember him describing Fred Hoyle, who had this experience while hiking in the Scottish highlands, all of a sudden insight into the mathematics involved in a cosmological theory of electromagnetism hits him, and he said it felt as if a huge brilliant light had suddenly been switched on. We're going to talk about divine illumination later as, oh boy,

that's going to sound crazy now, but we'll get there. This book, Truth and Beauty, describes a lot of these experiences. It talks about the 19th century French mathematician Gabriel Lame, speaking of getting goosebumps. Probably Lom is the name there, getting goosebumps, while reading about fellow mathematicians' treatment of modular functions. It talks about the 20th century British mathematician George Watson, comparing mathematical discovery to the thrill of seeing art.

It talks about the 20th century in German theoretical physicist, Werner Heisenberg, feeling giddy upon his discovery of quantum mechanics and expressing gratitude and talking about its beauty and so forth. I'm not trying to go over the top here, but I'm just trying to say this is a common experience in upper-level math of this sense of transcendence and just something mystical that we're coming across. We could come up with other examples of that. We'll come back to talk about this further when we get to divine illumination, the sixth point in the second half of the video. but the net effect of all of this is just to encourage us to consider that math seems to have not just a robust objectivity, but a very particular and curious kind of objectivity. Math really is a curious and intriguing phenomenon, and it begs the question, how do we understand this?

Now, before I get to theism as an explanation, this might be my longest video ever, let me just share personally of how this has come into my own thinking, okay? You don't have to agree with what I'm about to say to accept some of the previous points. This is just personally for me. And this is going to be called the what is Iraq thought experiment. So here's so for me, I've found math useful and illuminating not just in terms of like science and technology, but for philosophy and for basic questions of worldview. So I would say that after thinking about the nature of math, you might even look like a rock,

look at a rock differently. Now, this is inspired by the amazing book by Oderberg called Real Essentialism. This is a book that's rehabilitating essentialism using the Aristotelian notion of substantial forms. Essentialism is belief in essences. Things have an essence. An essence is an objective metaphysical principle that determines the classification and definition of a thing.

So this is a very abstract book of metaphysics. But it's interesting because, you know, this is something I've thought about is like, If you pick up a rock and you're holding the rock in your hand and you ask the question, what is the essence of this rock? It turns out that's a harder question to answer than you expect, and math becomes relevant to it. So here's the thought experiment. Suppose you're a naturalist, you believe in only physical things. And within that framework, you're trying to explain, what is the essence of a rock? And you're struggling. Because you're asking, what is it that makes for a qualitative distinction that demarcates the rock from the other physical

reality around it. And you realize this is not easy to answer. This is a bit slippery because we never experience reality just as generic reality. At the physical level, reality breaks down into smaller and smaller units. So it's hard to locate qualitative distinctions of ontological order or rank. It's difficult to say why is this rock robustly its own thing. It's a bundle of different things, but what is it that makes it itself in a non-superficial way? Again, I realize this is too abstract for some people to appreciate, but I've thought about this, you know? So suppose that, if I, like, for example, if I was a naturalist, if I only believed in physical things,

I would struggle to articulate why anything has an essence. All of reality would start to blur and in a terrifying, terrible way. That's just my, again, you don't have to agree with me on this little thought experiment, but here's the thought experiment. So now you're wondering, what's the essence of the rock? Now you see an angel. And you don't think you're hallucinating, and so you're no longer a naturalist. You think, well, there's at least some supernatural things like that angel.

Now the question is, how do you look at the rock differently? Because if angels exist, that has implications for your whole metaphysics. Because the essence of an angel is not reducible to its internal structure or its physical composition. It's a spiritual being, but it has a finite essence. So that raises up the question of, what about other things having a more stable identity? And, you know, so now you might look at the rock and say, okay, I don't need to define this rock's essence in terms of its physical properties only. Now here's the challenge that was raised for me by reading Odebriggs book. Why do you need an angel to make that transition in your

thinking when you already have numbers? Because numbers can do the same thing. Numbers themselves imply this rich ontology. Take any number, say the number 11. The number 11 has an essence that's not reducible to any kind of physical composition or internal structure. So what that tells you is, okay, you can't reduce the essence of things to their physical composition or internal structure. And this pushes against the bias for the quantitative over the qualitative that's often present in the natural sciences. If you're starting to think along these grooves, you can even speak of ontological ranks and an ontological hierarchy. Math has lots of implications for this kind of stuff. For me, that's

thrilling. Having considered what I regard as the barren wasteland of naturalism, I experience math and essentialism as this tremendous relief and comfort. Like there's more out there. It does feel like stepping through the wardrobe into Narnia. Now again, you don't have to agree with me on that particular thought experiment. It's a little bit testimonial. But hopefully, I'm trying to raise the, I'm trying to help people feel all that's sort of involved as you start to think about the nature of math. So now we ask the question, second half of the video. I'm at 42 minutes on my clock here. So who knows? This will be a little shorter, but we'll probably go about an hour, hour 20 total in this video. If you stay to the end, again, we'll talk about the Trinity.

Hopefully that's incentive to hang in there. So this is the question for the second half. Why does God provide a good explanation for math? And the argument I'll make here is not necessarily you have to believe in God per se from this argument, some might try to do that. I'd say rather that the specific and curious features of math we have observed are particularly well explained by theism or something like theism, if not traditional theism, something in that ballpark that can give you a kind of metaphysical context to make sense of math as we experience it. Before I make the argument, let me explain the particular theory that I'll be utilizing for how God relates to math, and that is divine conceptualism. You'll see why it's important we say this up front.

This is the idea that mathematical truths and other abstract objects as well are constituted by the thoughts or ideas of God. God's intellectual activity constitutes math. And this is, I would say, the general historic Christian view. William Lane Craig notes that conceptualism is historically the mainstream view among Jews, Christians, Jews, and Muslims. By the way, his book, God over all is a really helpful. introduction to the question of math and God's asseity, meaning self-existence, even though Craig is not a proponent of divine conceptualism per se. But divine conceptualism does have a lot of Christian philosophers who advocate for that view today, like Alvin Plantaga,

if I understand him correctly. So you can envision some of the main options here. On the one side, you might think of a strict Platonism, where abstract objects exist independently of God. On the other side, you might think of a nominalism, which denies that abstract objects exist at all. It sees them as just useful fictions. And then in the middle, we'd put divine conceptualism. Now, of course, there's other views. There's mediating options. This gets complicated.

But I'm just trying to give a brief orientation here to start with. And if you want to get a good defense of divine conceptualism, see Greg Welty's really helpful work in this. This is a great book, really useful book. These multi-author books can be really helpful. This is Beyond the Control of God, edited by Paul Gould. It's on six views on the problem of God and abstract objects. And so that book could introduce you to some of the issues in this debate. By the way, when I talk about strict Platonism here,

trying to be clear, I'm talking about a non-theistic Platonism, since there's various, you can think of actually like theistic modifications of Platonism. That's kind of how I'd conceptualize my view. but so, you know, again, the word Platonism is used differently here. But let's put up the visual again. The challenge with a full-blown Platonism on the left side here is that it imperils God's aseity. Here you have this whole panoply of entities that somehow are independent of God.

That won't do it all. Traditional theism wants to say God is assay from himself and everything else comes from God. But the challenge on the other side with nominalism is that there seem to be good arguments that abstract objects are more than just useful fictions. They seem to have a kind of robust objectivity. So the sentence that I wrote out at the end of this book when I finished it, it's like this, my overall summative conclusion was, I would say, locating abstract objects like math as God's thoughts explains both their robust objectivity, since they're God's thoughts,

as well as maintains God's Aseity, since they're God's thoughts. We'll come back to that at the end when we consider objections. But God's thoughts don't compromise his uniqueness. God is allowed to think without that being some external thing to himself. So, but because they're the thoughts of an infinite eternal mind, that can account for the nature of math as we experience it. So I'm just explaining why I'm going to be articulating this in devise— in the categories of divine conceptualism. That's an introduction. Hope I didn't take too much time on that. I do want to add, though, if you, there are other Christian views, and you could translate my

argument here, the argument from eternal truths, into different frameworks, even though I'm going to be articulating it in this framework of divine conceptualism. All right. So, why should math suggest God? Why is God a fitting explanatory framework for math? Let's work through historically where we see this argument, the classic statement of the argument from Eternal Truth comes from Augustine of Hippo in his book on Free Choice of the Will. And by the way, the notion that math exists as divine ideas, that does not start with Augustine. That goes back to the intersection of Hellenistic philosophy and Jewish thought. So if you think of the era of what's called Middle Platonism, this is like from the first century BC, up until about the third century AD,

You think of people like Philo, the Jewish thinker. You can find this idea of math and abstract truths as the thoughts of God there. But Augustine, to my awareness, is the first one to turn this into an argument for the existence of God. This is in book two of this work. There's a dialogue with a character named Evodius, and he's distinguishing bodily perception from the knowledge of reason. And to give an example of what is known by reason, rather than bodily perception, he talks

a lot about numbers. And at one point, Vodias says, but seven and three are ten, not only now, but always. There has never been a time when seven and three were not ten, nor will there ever be. So that which is perceived by bodily perception can change, but math, which is perceived by reason, does not change. It's described as an incorruptible activity, and therefore it's available to anyone by means of reason. to use our previous categories, its objectivity leads to intersubjective availability. And then this discussion leads to the conclusion that this mathematical truth, discovered by reason, is greater than our minds, rather than equal to or inferior to them.

And basically, he says, we discover math. We don't construct it. You can read this quote on the screen. Ultimately, Augustine then identifies this truth that is greater than our minds, this mathematical truth we discover with God himself. God is the truth. Okay, more on Augustine toward the end. Later, though, in the modern era, you see this argument developed by the philosopher Gottfried Leibniz. And you can find a little longer version of this argument in his monodology, which is in this little book published by Hackett, where he weaves the argument from eternal truths, together with

an ontological argument and a cosmological argument elsewhere, more briefly, he just puts it like this. this is a good summation. Quote, if there were no eternal substance, there would be no eternal truths. And from this to, God can be proved, who is the root of possibility for his mind is the very region of ideas or truths. So you note here these two qualifying phrases for God. At the very end, there's a kind of spatial metaphor for God's relation to ideas and truths. He's the region in which they exist. Just like North America is the region in which the Mississippi River exists. So the mind of God is the region of truths and ideas. Then he also calls God the root of possibility. So for Leibniz, the ideas and truths, he has in mind here are threefold.

It's possibilities, eternal and necessary truths, and essences or ideas. So there's a lot more going on in Leibniz we can explore. He calls these possibilities non-existent possibles. And so you get into all kinds of stuff with modal metaphysics. If you're interested in that, if you're interested in philosophy, be aware of all that. Here we're just going to focus on the mathematical side of things. And basically what Leibniz says is, it is true and even necessary that the circle is the largest of isoparometric figures, even if no circle really existed. Likewise, even if neither I nor you nor anyone else of us existed. Indeed, even if none of those things existed which are contingent or in which no necessity is understood, such as the visible world and other similar things.

since therefore this truth does not depend on our thought, there must be something real in it, and since that truth is eternal and necessary, this reality too that is in it, independent of our thoughts, will be from eternity. So an isoparometric, this means having the same perimeter. So if a circle and a square have the same perimeter, the circle will be larger. This is a truth of geometry that he's referencing there. Now, Leibniz gives an important clarification that we're going to come back to in the objections. He says, we should not imagine as some do that since the eternal truths depend on God, they are arbitrary and depend on his will. This is true only of contingent truths whose principle is fitness or the choice of the best, but necessary truths depend solely on his

understanding and are its internal object. So here, Leibniz is reacting against Descartes, and this fits with divine conceptualism. Math depends on the thoughts of God, not upon the will, of God, more on that again, and when we consider the arbitrariness objection, but just be aware of that. Now, look, I'm not developing or defending Leibniz's argument or Augustine's. I'm trying to give a sketch of them, so you're aware of them, then we'll get into this. This is historical backdrop. But what we can just observe here is Augustine and Leibniz are just two examples of an articulation of a common pre-modern human intuition. Okay. Historically, this was the general human view of math. it somehow is intersecting with supernatural metaphysics.

So, in other words, this argument doesn't just pop up here and there arbitrarily in obscure works of philosophy. Most human beings have had this perception of math is somehow situated in metaphysics of some kind of theism. And the struggle to account for the ontology of math is actually more of a modern Western development. Here's how Ruben Hirsch puts it, even though he ultimately ends up arguing that math is constructed by human minds in a way. He says, the present trouble with the ontology of mathematics

is an after effect of the spread of atheism. In other words, it's only in the modern West, as atheism has spread that we now struggle to understand what math is. Listen to this metaphor he gives. This is powerful. Most mathematicians and philosophers of mathematics continue to believe in an independent, immaterial abstract world, a remnant of Plato's heaven, attenuated, purified, bleached with all entities but the mathematical expelled. Platonism, without God is like the grin on Lewis Carroll's Cheshire cat. The cat had a grin. Gradually the cat disappeared until all was gone except the grin. The grin remained without the cat. So you can see this is a powerful image. I mean, he's saying Platonism without God is like the grin

without the cat, which is a way of saying that basically it's extremely mysterious and hard to make sense of. And lots of philosophers today have this same intuition that basically mathematical truth just doesn't, is not well situated within the broader metaphysical assumptions of naturalism. Why should a finite space-time universe in constant flux produce this mental realm of truths characterized by permanence and independence and exactitude and so forth? Where does the ice palace come from and why is it here? You know? And so, and this is already sort of, a little bit in some of what we've already recounted. You know, we saw Paul Dirac say God used math to construct the universe. We think about Erdosh's book written by the Supreme

Fascist, Vigner talking about miracle and gratitude, Penrose, you know, being guided to eternal truths, so on and so forth. And this is what a lot of philosophers argue. So Alvin Plantanga puts it like this, speaking of numbers and sets. Most people who have thought about the question think it incredible that these objects should just exist, just be there, whether or not they are thought of by anyone. It is therefore extremely tempting to think of abstract objects as ontologically dependent upon mental or intellectual activity in such a way that either they just are thoughts or at any rate couldn't exist if not thought of. So the intuition seems to that work seems to be like this, that certain features of the intellectual realm like math, they're less arbitrary if they're sustained by some kind of mental activity.

So let's flesh this out a little bit and say, if theism were the case, why would that be a particularly apt framework to explain the nature of math? Let's go through six characteristics. God's eternity, God's infinity, God's unity, God's intelligence, God's agency, and God's illumination. First, God's eternity. So traditional theism posits that God is an eternal mind. That would obviously have incredible explanatory power for math as we experience it.

If you recall our discussion of the permanence and independence of math and our curiosity about whether there is some kind of source for math apart from physical space time, well, if theism is true, we're just allowed to trust fully this common intuition about mathematical realism. We can say that basically Plato was just literally correct, that geometry is eternal. that would explain why it's so difficult for us to imagine two plus three equals five coming into being at some point. Mathematical truth would have this kind of apparent permanence and objectivity simply because it is permanent and objective and even everlasting. Now, you could posit some other kind of source other than God for this as well, but any candidate in order to compete with

theism is going to need to honor the principle of proportionate causality, according to which whatever's in the effect must be somehow related to the cause. The cause has to be sufficient to produce the effect. So if you do come out thinking that mathematical truth is eternal, then a non-eternal cause won't do. That would be like saying Abraham Lincoln founded ancient Rome. It's like, no, the effect cannot vastly predate the cause like that. So if there's eternity in the effect, then theism provides a really robust explanation by positing eternity in the cause. Second, God's infinity. The quantity of mathematical truths is infinite. For example, there's infinite numbers. Therefore, this would be very well explained by an infinite mind. No finite mind could ever house an infinite number of truths. A finite mind couldn't be the region that they inhabit to use Spinoza's image. They would spill over. It would be like a 10 ounce glass trying to hold three gas. gallons of water or something like that. But if there's an infinite source for mathematical truth, this would explain not only its size, but how it can be intersubjectively available in the universal way that it is. You know, why is it that this invisible castle is everywhere available for discovery by anyone? Any form of human conceptualism will fail because, among other issues, human minds are finite, but divine conceptualism, where you have an

infinite mind can explain this objective existence of math. Here's how Greg Welty puts it. Objectivity is secured by there being just one omniscient and necessarily existent person whose thoughts are uniquely identified as abstract objects. There is no reason why divine thoughts cannot supply the requisite objectivity by being the objects and reference of these human attitudes and verbs. We humans would be taking up attitudes to propositions which exist independently of our cognitive faculties. By the way, Greg Welty's contribution to this book is really great as well. So we're just going along here. We're just, you know, abductive reasoning. We're trying to say, look, here's a possible framework. Wow, this has some real reach. It has some explanatory power.

Eternity in the effect, eternity in the cause. Infinity in the effect, infinity in the cause. Now here's where it gets interesting. Third, God's unity. So traditional theism says God is one. He's a singular source for mathematical truth. That would perhaps explain the unity and coherence we find in math itself. A strict Platonism is ontologically bloated. That just means it posits this vast array of diverse entities as the source of mathematical truth. Now, not only does that potentially have issues with Occam's razor, not only is it, for many of us a little bit counterintuitive,

but it seems less equipped to explain the unity of mathematical truth. It seems more parsimonious to envision, again, what's in the effect must be in the cause. Unity in the fruit, unity in the root. If the source of math is a vast array of diverse entities, why should math itself form a coherent whole? What is it that would create that unity and that harmony and that order between the different disciplines of math, for example? The whole arena of math gives the impression that there's some one thing standing at its foundation. Just like the unity and coherence of a novel suggests a singular author rather than a bunch of different people writing independently of each other. I drew this out on a sheet of paper just to make it clear.

Essentially, what I'm trying to say is this kind of makes sense that the blue arrows can explain the red arrows. In other words, mathematical disciplines have harmony with each other because they have a common foundation. Okay. So the flow of thought thus far is like this. We're saying math needs some kind of source or explanation. We're trying to think what could the best one be. An eternal source explains its seeming permanent, independent, objective nature. An infinite source explains its infinite nature and objectivity as we experience it. A singular source works better, works well to explain its unity and coherence. Fourth, God's intelligence stands as a good a good piece of the pie here in terms of the explanatory framework. So what I mean here is not just

that God is smart, but that God has a mind that is capable of active thought. So being eternal and infinite and singular doesn't do much without this. It takes the capacity for thought and ideas. So if you recall planting us language and the word activity in the phrase mental or intellectual activity. God's thinking has a rich explanatory power because numbers and other mathematical objects are abstract thoughts. They're abstractions. One way to define math is abstraction plus logic. So if mathematical objects are basically abstractions, it seems intuitive that they're abstractions from something. Abstract objects have a kind of dependent existence on what they abstract from. Leibniz, in writing a preface for another book in 1670, put it in these categories.

Concrete objects are truly things. Abstract objects are not things, but modes of things. By its very nature, mathematical truth as abstract doesn't seem self-explanatory. Abstractions make the most senses depending on something from which they abstract. So David Oterberg points this out that numbers can't be abstractions from themselves because then their individuation conditions would be circular. So they need something to abstract from. He says if numbers must be abstractions from something other than themselves, there must be something other than numbers in any world in which there are numbers.

Yet nothing contingent might have existed. But we cannot simply regard numbers in a world without contingent objects as abstractions from other non-numerical abstractions, since these two are abstractions from things other than themselves and cannot be abstractions from numbers on pain to be their circularity or incoherence. So this necessitates his conclusion. He says if there are necessary things such as numbers and other logical and mathematical objections, there must be things other than numbers from which the numbers are

abstracted. But again, the reality from which numbers are abstracted can't be contingent since the numbers are not contingent. So here's his conclusion. We need something else, something necessary and particular that it is a very, that it is a is, by that very fact, not a number or any other logical or mathematical object, the only plausible candidate for such a being, as far as I can tell, one whose essence is his existence, is God. Hence, either there might be nothing whatsoever, or if the numbers must exist, then so must God.

Anyone committed to the necessary existence of numbers or other logical or mathematical objects must countenance the existence of God. Now, there's a lot that goes into that conclusion for Odeburg. But the one piece that concerns us here is this intuition that the abstract needs the concrete. Abstraction doesn't explain itself very well. And the most fitting kind of source for abstract truths is therefore something like God. God is an intelligent subject capable of concrete thought. This is the appeal of divine conceptualism. By envisioning math as the thoughts of God, they're not left in the realm of mere abstraction. So if Theism, is true than what we experience as abstract objects. Here's a thought. Again, to me, this sense chills down

my spine to think, it's not abstract to God. God is concrete. He's a subject. Therefore, his thoughts are concrete. What we experience as abstract truths, he sees as concrete. Think of it like this. In his divine mind and intelligence, God sees the original two plus three equals five. Again, it's pretty, I won't riff. You can tell I find this interesting. Fifth characteristic, God's agency. God is understood to be a personal agent capable of thought and action. Philosophers often distinguish between event causation and agent causation.

Event causation is like billiard balls striking each other. Agent causation is like your phone dings because your friend texted you. The personal element involved in agent causation has tremendous utility for explaining math, because in itself, the realm of mathematical truths is causally inert. It's hard to see how it could have any meaningful connection to the real world of concrete objects. Remember Penrose talking about these three different realms? One of his points is it's mysterious how they can interact with each other at all. On the other hand, if these ideas actually exist in the mind of a personal agent, then their meaningful

relationship to the concrete realm can be explained as a function of his personal activity. The relationship between math and the physical universe that Einstein called an enigma, Vigner called a miracle, no longer is it that surprising because we think God made the physical universe. So it's not at all shocking that his thoughts would bear their mark on it. It would be like, you know, as mundane as the blueprint of a building having resemblance to the building itself. It's not a miracle, it's just exactly what you would expect. And this would help us understand another aspect of abstract truths like math and especially like propositions. And that is they seem to have a kind of intentionality to them, a kind of aboutness.

They act suspiciously like thoughts. And if they're divine thoughts, that can explain that. Here's how Lorraine Keller puts it. Truth involves representation. Something is true only if it represents reality as being a certain way, and reality is that way. But representation is a function of minds. So truth is mind dependent. Yet there are truths that transcend the human mind. For example, eternal truths. So there must be a supreme mind with the representational capacity to think these transcendent truths, therefore a supreme mind like God exists. So she's talking about propositions here. But the reasoning has some analogy here because basically there's

same kind of reasoning. It's like, think of it like this. Eternity in math, eternity in the source of math, infinity in math, infinity in the source of math, abstractness in math, concreteness in the source of math, applicability to physical reality and intentionality in math, agency in the source of math. I'm just trying to summarize. I think I skipped the third one. That's unity in math, unity in the source of math. Trying to just me clarify as we go here. So those five steps favor a source of math that looks something like an eternal, infinite, singular, intelligent, personal agent. It's not a proof of God, because this is focused more on certain qualities of God. And we're saying this is one powerful explanatory framework. We've

not gone along and ruled out every other possible one, nor have we insisted you have to think of this, but it's suggesting something like that. Again, it doesn't really prove God per se. Like, you know, we haven't talked about all of God's attributes like goodness and mercy or even something like omnipotence. Okay. This is why we're saying God or something like God. Deism could possibly fit the bill here as well. Pantheism could maybe work, but it would seem to struggle to explain the distinctness and independence of mathematical truth. A strict Platonism doesn't seem very good to explain math because, again, it's ontologically bloated, and it has an inferior explanation for mathematical unity.

Naturalism definitely seems to be near the rear of the pack of explaining math, because it can explain so little of it. It leaves so much mysterious. But there's one last step that's admittedly maybe less forceful, but maybe more fun. So that's divine illumination. Another potential reason why God fits the bill of explaining math, so to speak, is it would explain the mystical and transcendent experience of math, this quality of mathematical experience, this sense of transcendence.

There are epistemological and existential implications to having a theological account of math. Simply put, put it like this. So we've been saying all these, you know, five things like eternity in the source, eternity in the cause. Infinity here, infinity there, this kind of reasoning, the principle of proportionate causality and so forth. Think of it like this here. Beauty and glory in the source, beauty and glory in the cause. In other words, a divine account of the ontology of math would explain why it's so exciting. If you're a theist, then when you're doing math, you're potentially thinking God's thoughts. you're not making the path, you're walking in a path that he's been seeing forever.

So just as to move about in physical space is to encounter what God has made, to discover mathematical truths is to encounter what God has thought. And in this connection, it's interesting to remember that the classical way of understanding mathematical experience is divine illumination. So for Augustine, for example, the perception of mathematical truths is a kind of participation in divine light. To perceive that the three angles of a triangle always and necessarily must equal 180 degrees is to be illumined by God. One's mind is seeing something that God's mind has always seen. One is sort of stepping into the light. In his confessions, Augustine puts

this more personally and emotionally. We're almost toward the end here, but listen to this. Eternal truth and true love and beloved eternity, you are my God. When I first came to know you, you raised me up to make me see that what I saw is being and that I who saw am not yet being. And you gave a shock to the weakness of my sight by the strong radiance of your rays and I trembled with love and awe. So you see the distinction here, I who am not yet being, Augustine is making an ontological distinction. There's being and not yet being, the bright world of permanence and the shadowy world of becoming. the ice palace out there, the shadowy realm here. To exist in the realm of becoming and encounter the world of permanence,

to see the ice palace is like stepping out of the shadows and into the light. It's shocking and dizzying. Augustine has to be raised up to experience this. It's a kind of divine illumination. You know, the radiance of God's rays is dazzling him. He's trembling with love and awe and so forth. That's, I don't know how else to say this, except if you felt that, you know it. You know that feeling. Like when you're listening to music

or when you see the beauty of ideas. And it is, what a framework to have to see that as a divine illumination. Some take this as kind of out there. But consider the words of Robert Adams. And he's a good philosopher. He says an epistemology of divine illumination is not a silly theory. I think, in fact, it may be the best type of theory available to us for a explaining the reliability of our supposed knowledge of logic and mathematics, since the alternative type of theory most salient for us in terms of natural selection does not obviously explain our aptitude for the higher reaches of those subjects, which was of no use to our ancestors on whom the selective pressures were supposedly operative. This seems very intriguing. You know, in other words,

he's saying it's not obvious how the evolutionary process is going to give us reliability in upper-level math, because that doesn't have much survival value, more to say about all that. And of course, you don't have to accept that particular point to see the argument as a whole here. So this sums it up. These six characteristics of math, we're getting a picture of math, are saying, Theism fits the bill really well. Let's consider a couple of objections. Just briefly, not even giving a lengthy response, but just flagging these so you know some of the potential responses and so you can keep thinking about it. All of this is sort of introductory, even though it's a long video. It's just sort of putting this out on the table. Obviously, there's a lot more work still to do here.

The first concern, though, is arbitrariness. Does this make math arbitrary? You know, if we say math is dependent upon God and explained by God, could God have made two plus three not equal five, but equal 125? And part of the answer here is to recall what Leibniz said about the truth of math, not depending on God's will, but on God's mind. But you still might wonder, I mean, couldn't God have thought it differently? So it's important to clarify here that the nature of math's dependence on God is not causal but constitutive. God's thoughts don't cause math, they constitute math. Math's dependence on God, in other words, doesn't obliterate its own intrinsic rationality. As Robert Adams puts it, the fact that the ontology of math is grounded in God does not take away from the role of its own content. in determining its truth or falsity.

Now, some have concerns here about divine omnipotence, and they might say, well, look, if you're saying that God couldn't have made 2 plus 3 equal 11, then you're threatening God's omnipotence. But I don't think that's necessarily true. We already have to qualify God's omnipotence in lots of different ways. We have to say, God is all powerful, but that doesn't mean he can cease to exist or he can sin or he can cease to be God and so forth. And those don't threaten divine omnipotence, and neither, neither. threaten divine omnipotence, these kinds of mathematical and logical challenges, just as God

must act in accord with his moral nature, and that doesn't threaten his power, so he must act in accord with his rational nature, and that doesn't threaten his power. But that's not really a restriction upon God. This is more a function of our language, and Anselm addresses that in the book Proslogion, chapter 6 through 8, if you want more on that. The second objection that comes up is about circularity here. I'm just trying to flag these, I guess, you know, people might be thinking about this. So we've been saying math depends upon God,

but lots of people, like Bertrand Russell, have argued that God depends upon math and logic in various ways, like the law of non-contradiction, for example. So basically the thinking here is that without the law of non-contradiction, you can't distinguish true and false, and therefore you can't say that it's true that God exists or that God is good or something like that. Therefore, God depends upon the law of non-contradiction.

But I think, put it like this, the circularity here is not at all vicious because the way math and logic depend upon God is very different from the way that God interacts with them. Strictly speaking, I wouldn't call that a dependence. We could say that math and logic have a kind of asymmetrical dependence upon God. They depend upon God in a very particular way. I'll put up how Robert Adams caches this out. But again, if you just think that math and logic are God's thoughts, you know, from our vantage point, we can't account for God apart from these things, but that doesn't mean there's something other than God that is limiting him or something like this.

They're just God's thoughts. I mean, you know, we have to remember we're thinking analogically, but I'll see more about this because this comes up with this third objection. So the third worry is, what about divine simplicity? What about classical theism? Does divine conceptualism, if there's a plurality of thoughts in God, does that threaten his simplicity? Lots of people raise this concern. But, you know, just the first thing to observe is that divine conceptualism and divine simplicity are both traditional views. They're both majority views in the Christian tradition. And very few people felt a problem with these two. So it would be surprising if there was a big problem here and just nobody saw this. I think the short answer here is just to say God's thoughts are not parts.

Divine simplicity is denying ontological composition in the being of God. It doesn't deny that we can speak of God's eternal thoughts, just as we can speak of God's eternal love. The motivation for divine simplicity is trying to protect God's aseity. But God's thoughts are not some separate external entity. Even in an analogous way, we have to remember we're thinking analogically about God here, but if you were to say, you know, John is in the living room all by himself, no one would be contradicting you if they said, yeah, but his thoughts are there too. It's like, well, yeah, if he's there, his thoughts are there, right? Now, even understanding this is analogically related to God, we can see the same kind of problem here. God's, God is allowed to think. God's thoughts

are not some external threat to his supremacy and uniqueness and so forth. In the Sumo Theological, Thomas Aquinas addresses this. And he follows Augustine in affirming a plurality of ideas in the divine intelligence. And he addresses this concern that says it's not repugnant to the simplicity of the divine mind that it understand many things, though it would be repugnant to its simplicity were his understanding to be formed by a plurality of images. So Thomas argues that the ideas of God are that which he knows, the objects of knowledge. They're not images by which he knows, other things. They're just what God sees, what God knows, what God thinks. And so they're not shaped by anything external to God. And what Thomas goes on to do basically is ground God's ideas in his self-knowledge.

So God knows his own essence, and he knows his own essence in every mode that it can be known, including the mode that creatures know it by way of participation. So God's knowledge of all things is through his own essence. And that entails that God's knowledge of his essence is constituted in part by a knowledge of his own power. And that implies that God knows the full range of possible worlds. And that's grounded in the knowledge of his own essence. Now, there's a lot more to say about all that, but I'm trying to at least sort of flag this start to chart out some pathways of how you might think about this, things to be aware of.

And there's different ways you could articulate this, but at the end of the day, God's thoughts are not a threat to his supremacy and his simplicity and so forth. Here's a fourth concern is, I'll use the label Mnongianism, which is a particular philosophy of existence that says some things have a kind of shadowy existence. And the concern here is, I've seen people respond to this like this saying, doesn't this argument depend upon a specific philosophy of existence like that, we've been talking about how mathematical truth exists. And some people want to push back on that and say, well, it can be eternally true, but not eternally existing. But I would just say, speaking of math as existing, it's just a matter of language. We don't have to assume any particular

theory of existence for this argument to get off the ground. We still need to explain its truth. So in other words, it's not mere existence that calls for an explanation. It's the whole of what math is that calls for an explanation. So the argument doesn't require one specific philosophy of existence. Whatever philosophy of existence you have, you need to account for the whole reality of what math is. Final objection, which is not a very good one, but I'll just mention this because I've heard people bring this up,

is why are so many mathematicians atheists? Now, I would say two things. First of all, actually, they're not. It's amazing how much in science and physics and math, you find various, I mean, Paul Davies, a guy I quote all the time, there's a lot of agnostics who are kind of like, yeah, it looks like there's something out there. But the other thing is, though, this is somewhat irrelevant because there's a difference between skill in doing math versus reflecting on the nature of the ontology of math. Leibniz, this is a great quote from him, he says, an atheist can be a geometer, but if there were no God, there would be no object of geometry. And it is true.

it seems like that reflecting on the nature of math and actually doing math are very different skill sets. One is highly abstract. The other is often not highly abstract. Okay, final point on the Trinity. In this is, because people might have wondered about this, you know, it's like, well, we're talking about math depending upon God, but if you, if you're a Christian and you believe in the father, the son, and the Holy Spirit as the one God, how does that play in? In the Christian tradition, the nature of mathematical truth has been an especially interesting point. Since the Trinity entails that in some sense numbers are eternal in the very being of God, not only in the mind of God.

Now again, we need to be very careful about how we understand this and how we articulate this, but we can say that for the Trinitarian Theist, reality is in some sense intrinsically numerical, not just in God's thoughts, but because in some sense, God is both three and one. and so you can say that threeness and oneness are predicated of what is most basic primal reality. And historically, Trinitarian theists consider mathematical truth to be ultimately rooted in the Word or the Logos of God through whom everything was created. You see, Colossians 1 and John 1, it's through the Son of God or the Word of God that everything seen and unseen was made.

So in Christian theology, the second member of the Trinity is associated with the third or the reason or the speech of God the Father, while also being himself fully divine. And so drawing from passages like John 1 and Colossians 1, Christians have historically spoken of the Son of God as the one in whom are these archetypal truths by which the world was built, such as archetypal truths like math. Here's how Pope Benedict put it. He's talking about the church's use of music and liturgy. I like this quote.

He says, The mathematics of the universe does not agree. exist by itself. It has a deeper foundation, the mind of the creator. It comes from the Logos, in whom, so to speak, the archetypes of the world's order are contained. The Logos, through the spirit, fashions the material world according to these archetypes. All right, let me know what you think in the comments. Thank you for watching to the end. I know this was a long video. I know that it had rabbit trails, even though, actually, as I finish, if I'm conscious of one thing here at the end of recording this on a Friday afternoon. It's how much this was just brief. They're so, it's vulnerable making videos because you realize how vulnerable you are to criticism.

A smart philosopher can come and pick this point and pick that point and so on and so forth. So I'm doing this as a labor of love, hoping it's comforting and nourishing and interesting to people and inviting to further thought. I realize there's a lot of loose ends. I just kind of did a drive-by overview of this argument. But thanks for watching. If you did make it all the way to the end, let me know in a comment. I'll heart your comment. I'll read the comments this video very carefully because I put a lot of work into this video. I am well aware a video like this gets a lot less views. That's fine with me. It was so worth doing it just for what I learned. I hope you enjoyed it as well. If you'd help me share this video, get it out there,

send it to somebody who's interested in math. That would mean a lot to me because I really would love to rehabilitate this argument. I'd love for this argument to get greater consideration. I'd love for others to come along and strengthen what I've done here and add on to it and build on to it. A lot more work needs to be done on this. I think math really is this durable thing. It's like last thing I'll say, there's a quote from John Adams about how facts are stubborn things. You know, truth is durable. It's like you come back to it again and again and again. That's what math has been for me. You know, I just, when all reality feels slippery, you come back to it again and again and say, there is stable truth out there. Two plus three

just equals five. And that does have broad implications. And it is interesting. Truth Unites has an Instagram account. Did you know that? If you're interested in following me on Instagram, check it out there. If you'd like to support Truth Unites, I do appreciate that a lot. The main thing is prayer.

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Let me know what you think in the comments. We'll see you next time, everybody.