An infinite series is a sum of infinitely many numbers or terms, related in a given way and listed in a given order. They can be used to calculate the values of irrational numbers, like pi, out to trillions of decimal places. Or to calculate values of trigonometric and exponential functions. And of greatest interest, they can be used to see non-obvious relations between different areas of mathematics like integers, fractions, irrational numbers, complex numbers, geometry, trigonometric functions, and exponential functions.
I’d like to start things off with a joke. A math joke.
An infinite number of mathematicians walk into a bar. The first orders a beer, the second orders half a beer, the third orders a quarter of a beer, and so on. After the seventh order, the bartender pours two beers and says, “You fellas ought to know your limits.”
OK, I tried to start things off in a cute way. Anyway, the joke is that if they keep following that pattern to infinity the total approaches the equivalent of two beers. That is the limit of the series. In mathematical terms this is an infinite series. An infinite series is a sum of infinite terms. With the series in the joke the series is:
1 + 1/2 + 1/4 + 1/8 + 1/16 + … = 2
Each term in the series is half the previous term. And if you continue this out to infinity (whatever that means) it ends up adding up to to 2.
Infinite series can be either convergent or divergent. The series just mentioned is convergent because it adds up to a finite number. But others end up blowing up to infinity. And those are divergent.
Convergent series fascinate me. Their aesthetics remind me of cyclopean masonry, famous among the Inca, where all the stone pieces fit together perfectly, as if part of a single block. These series reveal infinite structure within numbers.
I’d like to share some of my favorite examples of infinite series. The previously mentioned series adding up to 2 is interesting. It shows infinite structure within this simple integer, 2. But I’m especially interested in infinite series for irrational numbers. Irrational numbers, like pi, e, and logarithms, have infinite, non-repeating, decimal places. How can we find the values for all these digits? This is where infinite series became extremely useful.
Pi, approximately 3.14159, is the ratio of circumference to diameter. How can we find its value? Maybe make a really big circle and keep on making measurements with more and more accurate tape measures? No, that won’t go very far. Fortunately, there are infinite series that add up to pi or ratios of pi. And what’s fascinating is that these series seem to have no obvious relation to circles, diameters, or circumferences. Here are some of those series:
1/1^2 + 1/2^2 + 1/3^2 + 1/4^2 + … = π^2/6
1/1 – 1/3 + 1/5 – 1/7 + 1/9 – … = π/4
You look at something like that and wonder. What on Earth is pi doing there? Where did that come from? Irrational numbers, by definition, cannot be expressed as fractions. But they can be expressed as infinite sums of fractions. You can get an irrational number like pi just by adding up these simple fractions. At least, by adding them up an infinite number of times. Of course, we can’t actually do that. But we can still add up many, many terms. And with computers we can add up millions of terms to get millions of digits. But different series will converge on the accurate value for a given number of digits more quickly than others. If we’re actually trying to get as many digits as possible as quickly as possible we’ll want a quickly converging series. A great example of such a modern series for pi is the Ramanujan series:
This series computes a further eight decimal places of pi with each term in the series. Extremely useful.
But back to the earlier question. What is pi doing here? How do these sums, that don’t seem to have anything to do with the geometry of circles, spit out pi?
Let’s just look at that series that adds up to pi/4. This is known as the Leibniz formula. pi/4 is the solution to arctan(1), a trigonometric function. In calculus the derivative of arctan(x) is 1/(1+x^2). 1//(1+x^2) can also be represented by a power series of the form:
1/(1+x^2) = 1 – x^2 + x^4 – x^6 + …
Since this is the derivative of arctan(x) we can integrate it and get a series that is equivalent to arctan(x).
arctan(x) = x – x^3/3 + x^5/5 – x^7/7 + …
This is quite useful. Since it’s a function we can plug in different values for x and get the result to whatever accuracy we want by expanding out the series as many terms as we want. To get the Leibniz formula we plug in 1. So arctan(1) is:
1/1 – 1/3 + 1/5 – 1/7 + 1/9 – … = arctan(1)
And we already know that arctan(1) is equal to pi/4. So this gives us a way to calculate the value of pi/4 and, by just multiplying the result by 4, the value of pi. We can calculate pi as accurately as we want by expanding the series out as many terms as we want to. Though in the case of pi, we’ll do this with a faster converging series like the Ramanujan series. But I picked the Leibniz series as an example because it’s easiest to show why it converges on pi, or specifically pi/4. You can see a little bit here how different areas of mathematics overlap: geometry relating to trigonometry, calculus, and infinite series. Steven Strogatz has made the point that infinite series are actually a great way to see the unity of all mathematics.
So that’s pi. Let’s look at some other irrational numbers that can be calculated by series.
To calculate e, approximately 2.71828, we can use the following series:
1/0! + 1/1! + 1/2! + 1/3! + 1/4! + … = e
Or to calculate ln(2), approximately 0.693, we can use the following series:
1/1 – 1/2 + 1/3 – 1/4 + … = ln(2)
I find these similarly elegant in their simplicity. We’re just adding up fractions of integers and converging to irrational numbers. I think that’s remarkable.
We saw earlier with the arctan function that it’s also possible to write a series not only with numbers but with variables so that the series becomes a function where we can plug in different numbers. There are series for sine and cosine functions in trigonometry. That’s good because not all outputs from these functions are “nice” values that we can figure out using other principles of geometry, like the Pythagorean Theorem. What are some of these “nice” values? We’ll use radians, where pi/2 radians equals 90 degrees.
These are angles of special triangles with angles of 45, 30, 60 degrees, where the ratios of the different side lengths work out to ratios that we can figure out using the Pythagorean Theorem. But these are special cases. If we want to calculate a trigonometric function for other values we need some other method.
Fortunately, there are infinite series for these functions. The infinite series for the sine and cosine functions are:
sin(x) = x – x^3/3! + x^5/5! – x^7/7! + …
cos(x) = 1 – x^2/2! + x^4/4! – x^6/6! + …
Again, we have this kind of surprising result where we get trigonometric functions just from the sum of fractions of integers and factorials, which don’t seemingly have much to do with each other. Where is this coming from?
These trigonometric functions are infinitely differentiable. You can take the derivative of a sine function over and over again and no matter how many times you do it the result will be either a sine function or a cosine function. Same for the cosine function. They just keep circling back on themselves when we differentiate them. These series come from their Taylor series representations, or specifically their Maclaurin series representations. The Maclaurin series for a function f(x) is:
What happens if we apply this to sin(x)? Let’s take repeated derivatives of sin(x):
First derivative: cos(x) Second derivative: -sin(x) Third derivative: -cos(x) Fourth derivative: sin(x)
So by the fourth derivative we’re back where we started. And so on. What are the values of these derivatives at 0?
sin(0) = 0 cos(0) = 1 -sin(0) = 0 -cos(0) = -1
So the terms with the second and fourth derivatives will go to 0 and disappear. The remaining terms will alternate between positive and negative. In the case of sin(x) each term in the resulting series will have x raised to odd integers divided by odd factorials, with alternating signs. The result being:
sin(x) = x – x^3/3! + x^5/5! – x^7/7! + …
And for cos(x) it will be similar but with x raised to even integers divided by even factorials, with alternating signs. The result being:
cos(x) = 1 – x^2/2! + x^4/4! – x^6/6! + …
Using these series we can now calculate sine and cosine for any value. Not just the special angles of “nice” triangles.
The Maclaurin series also gives an infinite series for another important infinitely differentiable function: the exponential function e^x. The derivative of e^x is just itself, e^x, forever and ever. So in this case the Maclaurin series is quite simple. No skips or alterations.
e^x = 1 + x + x^2/2! + x^3/3! + …
We already saw one solution to this equation where x is set equal to 1, which is simply the number e.
These three series – for sin(x), cos(x), and e^x – allow us to see an interesting relation between exponential functions and trigonometric functions. The series for e^x has all the terms from the series for both sin(x) and cos(x). But in the series for e^x all the terms are positive. Is there a way to combine these three? Yes, there is. And it will connect it all to another area of mathematics: complex numbers. Complex numbers include the imaginary number i, which is defined as the square root of -1. The number i has the following properties:
i^2 = -1 i^3 = -i i^4 = 1 i^5 = i
And the cycle repeats from there. It turns out that if we plug ix into the series for e^x all the positive and negative sines work out to match those of the series for cos(x) and i*sin(x). With the result that:
e^(ix) = cos(x) + i * sin(x)
To make things really interesting let’s also bring pi into this and substitute pi for x. In that case, cos(π) = -1 and sin(π) = 0. So we get the equation:
e^(iπ) + 1 = 0
Steven Strogatz said of this result:
“It connects a handful of the most celebrated numbers in mathematics: 0, 1, π, i and e. Each symbolizes an entire branch of math, and in that way the equation can be seen as a glorious confluence, a testament to the unity of math. Zero represents nothingness, the void, and yet it is not the absence of number — it is the number that makes our whole system of writing numbers possible. Then there’s 1, the unit, the beginning, the bedrock of counting and numbers and, by extension, all of elementary school math. Next comes π, the symbol of circles and perfection, yet with a mysterious dark side, hinting at infinity in the cryptic pattern of its digits, never-ending, inscrutable. There’s i, the imaginary number, an icon of algebra, embodying the leaps of creative imagination that allowed number to break the shackles of mere magnitude. And finally e, the mascot of calculus, a symbol of motion and change.”
He also said, speaking of infinite series generally:
“The most compelling reason for learning about infinite series (or so I tell my students) is that they’re stunning connectors. They reveal ties between different areas of mathematics, unexpected links between everything that came before. It’s only when you get to this part of calculus that the true structure of math — all of math — finally starts to emerge.”
I think we can see that effect in some of the relations between some of my favorite infinite series that I’ve shared here.
Having looked at all this I’d like to make a couple philosophical observations.
One is on the possibility of objectivity in mathematics, mathematical realism, or mathematical platonism. Infinite series enable us to calculate digits for irrational numbers, which have infinite digits. We find the values millions and billions of decimal places out and we will always be able to keep going. Last I checked, as of 2021 pi had been calculated out to 62.8 trillion digits. What of the next 100 trillion digits? Or the next quadrillion digits? Well, I think that they are already there waiting to be calculated, whether we end up ever calculating them or not. And they always have been. Those 62.8 trillion digits that we’ve calculated so far have been there since the days of the dinosaurs and since the Big Bang. There’s a philosophical question of whether mathematical conclusions are discovered or created. You can tell I believe they’re discovered. And part of the reason for that is because of these kinds of calculations with infinite series. No matter how deep into infinity you go there’s always more there. And you don’t know what’s there until you do the calculations. You can’t decide for yourself what’s there. You have to do the work to find out and get the right answer. Roger Penrose had a similar line of thinking with the infinite structure of the Mandelbrot set.
Now, I do think there’s a certain degree of human activity in the process. Like in deciding what kinds of questions to ask. For example, geometry looks different whether you’re working in Euclidean, hyperbolic, or elliptic geometry. Answers depend on assumptions and conditions. I like a line that I heard from Alex Kontorovich: “The questions that are being asked are an invention. The answers are a discovery.”
The other philosophical question is: What does it actually mean to say that an infinite sum equals a certain value or converges to a certain value? We can never actually add up infinite terms. Nevertheless, we can see and sometimes even prove where a convergent series is headed. And this is where that concept of limits comes up. I don’t know how to answer that question. There are different ways to interpret that. Presently, the way I’m inclined to put it is this: The limits of infinite series are values toward which series tend. They never actually reach them because infinity is not actual. But the tendency of an infinite series is real, such that, as you continue to add up more terms in the series the sum will continue to get closer to the value of convergence.
The primacy of Logos entails that all things are intellectually structured, that the order of reality is rationally intelligible. All things come to pass in accordance with the Logos (Heraclitus) and all things were made through the Logos (John 1:3). This episode looks at Logos through the thought of Benedict XVI, who calls Logos the intellectual structure of being. The implications are significant to our general understanding of reality. Even more significant is the identification of the Logos with the person Jesus Christ.
My deep dive, theological and philosophical topic of study lately has been the Logos, a term that in Christian scripture is used to describe Jesus Christ. The Gospel of John opens, speaking of Christ, saying:
“In the beginning was the Word, and the Word was with God, and the Word was God. He was in the beginning with God. All things were made through Him, and without Him nothing was made that was made.” (John 1:1-3)
The word translated here as “Word” is Logos (Λόγος) and it’s often referred to in theology by just using the Greek term, which I’ll do in this episode. As I did with the episode on Hell, I’ve been reading the scriptures, early Church Fathers, and theologians throughout history to learn what they’ve had to say on the subject. There’s a lot there and I’d love to share it all at some point. But I eventually decided to break it down into more manageable parts.
Part of the reason for this, beyond just deciding to make it easier on myself, is that I came across a fantastic treatment of the Logos by Joseph Ratzinger, later Pope Benedict XVI, in his 1968 book Introduction to Christianity. Calling it an “introduction” may be a stretch, but regardless the book is excellent and I found that he put into words pretty much exactly what I wanted to say and understand about Logos. So with this episode I’d like to focus on Benedict’s theology of Logos in that book. And maybe, hopefully, go into more of a historical study later in another episode.
What is Logos? My own definition is that the Logos is the rational order of all things. Benedict refers to Logos as “intellectual structure”; and I love that. I’ve been trying to work out my own philosophy of structure so that description definitely gets my attention.
Logos has been a term of philosophical significance going back to at least Heraclitus (535 – 475 BC). Heraclitus said all things come to pass in accordance with the Logos (γινομένων γὰρ πάντων κατὰ τὸν λόγον, ginomenon gar panton kata ton logon). Or put another way, the Logos is what makes all things come to pass in the way that they come to pass. This is quite similar to the statement in the Prologue to the Gospel of John:
“All things were made (egeneto) through Him, and without Him nothing was made that was made.” (John 1:3)
Both Heraclitus and the Gospel writer used the word γίνομαι (ginomai): to become, to happen. All becoming, all happening, proceeds only in accordance with and not without the Logos.
As someone who is always trying to understand things I find this concept very interesting and important. And it’s a subject in which my scientific, philosophical, and theological interests converge. Every “why” and “how” question ultimately falls under the umbrella of the Logos. If you ever think about why something is the way it is or works the way it does you are pursuing the Logos. To one degree or another, everyone is pursuing the Logos. But what I find especially useful about it is that as a concept it serves as a way to unify and systematize our thinking about all these things.
Another way of describing Logos that I read recently and quite liked is as “an interlocking system of ideas”. I read that one from Edward Feser (Five Proofs of the Existence of God, 104-105, 110). That is not by any means an exhaustive description of what the Logos is. But it’s one description. An interlocking system of ideas. If you think about everything that happens what is it that makes it all happen the way it does? Just thinking of it from a purely scientific, physicalist, closed-system perspective. Laws of physics? Constraints of logic? I think that’s right. We often predict the evolution of a system over time with some set of starting and boundary conditions. Such a system is governed by an interlocking system of ideas.
So now let’s look at Benedict. First quote, he says:
“Christian faith in God means first the decision in favor of the primacy of the logos as against mere matter. Saying “I believe that God exists” also implies opting for the view that the logos—that is, the idea, freedom, love—stands not merely at the end but also at the beginning, that it is the originating and encompassing power of all being.”
Let’s break this down. Benedict is starting with the philosophical idea that Christianity holds in common with Hellenistic philosophies like Platonism, Stoicism, and Neo-Platonism. It’s a metaphysical stance. What is the nature of reality? Is it strictly material? Does reality consist only of “atoms and the void”, in the words of Democritus (c. 460 – c. 370 BC)? This would be what Benedict calls “mere matter”. The alternative is “the primacy of the logos”. Matter exists and it’s important. But it’s secondary, not primary. Logos is primary. Matter is derivative of and secondary to Logos.
Benedict calls Logos “intellectual structure”. So this means that prior to all matter there is an intellectual structure undergirding it. Why should we think this?
Let’s look at three quotes on this:
“In the old Pythagorean saying about the God who practices geometry there is expressed that insight into the mathematical structure of being which learns to understand being as having been thought, as intellectually structured; there is also expressed the perception that even matter is not simply non-sense that eludes understanding, that it too bears in itself truth and comprehensibility that make intellectual comprehension possible.”
“The intellectual structure that being possesses and that we can re-think is the expression of a creative premeditation, to which they owe their existence.”
“All our thinking is, indeed, only a rethinking of what in reality has already been thought out beforehand.”
Put briefly we ought to consider reality as intellectually structured because we can intellectually understand it. Not entirely of course. But everywhere we look, the more we look at the universe we find intelligible structure. It’s not just utter chaos but reality is actually something that we’re able to make sense of.
So the order of reality is rational and intelligibile to us. But is it more than that? Is there some kind of rational intelligibility per se. Say that there were no intelligent beings in the universe at all. Would it still make sense to call it a rational order or rational and intelligible? Maybe. Consider three possibilities:
1. The rationality of reality is a conditional property, conditional on there being intelligent beings in reality. 2. The rationality of reality is independent of any intelligent beings. 3. The rationality of reality is the rationality of a mind that grounds reality.
In the first option the rationality of reality is a conditional feature, a feature that reality would have if certain conditions are met, even if they are not otherwise actualized. Something of the form:
1. IF there are intelligent beings in reality. 2. AND IF any existing intelligent beings obtain some degree of accurate understanding of reality. 3. THEN such intelligent beings will find reality to be intelligible and rational.
This is probably the option that seems most immediately plausible and straightforward.
The second option moves away from a subjective understanding of rationality to an objective understanding. This gets a little tricky because just by talking about rationality, or anything, we’re using language and terms that are human constructions. How can you talk about something independent of thought and first-person experience when the very terms we are using to talk about it are creations of thought and first-person experience? There’s a passage from Richard Rorty that I find helpful here:
“We need to make a distinction between the claim that the world is out there and the claim that truth is out there. To say that the world is out there, that it is not our creation, is to say, with common sense, that most things in space and time are the effects of causes which do not include human mental states. To say that truth is not out there is simply to say that where there are no sentences there is no truth, that sentences are elements of human languages, and that human languages are human creations. Truth cannot be out there – cannot exist independently of the human mind – because sentences cannot so exist, or be out there. The world is out there, but descriptions of the world are not. Only descriptions of the world can be true or false. The world on its own – unaided by the describing activities of human beings – cannot.” (Rorty, Contingency, Irony, and Solidarity)
This is a very fine distinction and it can be easily confused. But it’s about the best explanation I’ve found of the problem at hand: how to talk about reality as it is independent of language, how to think about reality as it is independent of thought. Immanuel Kant’s concepts of noumena and phenomena are also useful for those familiar with them, otherwise they require some significant introduction.
So how would reality be rational per se, independent of any intelligent beings? One way of understanding rationality is consistency. For intelligent beings instrumental rationality is consistency between actions and intentions. But apart from intelligent beings we could think of consistency between states of affairs. At a most basic level, noncontradiction. For some state of affairs, S, it won’t be the case that both S and not-S.
That can get a little complicated and require different levels of states of affairs. For example, in quantum mechanics we see superpositions of states in which it might seem that both S and not-S could indeed be the case. But I think that’s resolvable by understanding any such superpositions of states as merely part of a higher-level state. In the case of a quantum superposition the relevant level of consistency would not be the superposed states, or eigenstates, but the probability amplitude of the quantum system. For more on that see my earlier episode on quantum properties (Quantum Properties, 5 Oct 2020). That might actually give reason to think that the rationality of reality is intrinsically independent of the thought and first-hand experience of intelligent beings. Because intelligent beings might mistakenly see inconsistency where there is actually unseen consistency and rational order.
The third option is that the rationality of the rational order is actually the rationality of a rational mind. Such a mind would be very unique, absolutely unique actually. This would be a mind grounding all reality. This is the view taken by Pope Benedict and many Christian theologians, that the rational order, the Logos, is the intellectual activity of God.
Benedict proposes a crucial parallelism between God’s thought and our own. All thinking is rethinking, “a rethinking of what in reality has already been thought out beforehand”. What people re-think is “the expression of a creative premeditation, to which they owe their existence”. The Creation was an act of intellect and speech, something attested in scripture, both in Genesis and in John. How did God create?
וַיֹּ֥אמֶר אֱלֹהִ֖ים יְהִ֣י אֹ֑ור וַֽיְהִי־אֹֽור׃
καὶ εἶπεν ὁ Θεός· γενηθήτω φῶς· καὶ ἐγένετο φῶς.
“Then God said, ‘Let there be light’; and there was light.” (Genesis 1:3)
God said (vayomer elohim). God’s creative activity was an act of speech. That creation was an act of intellect was a view also held by the second century Jewish philosopher Philo of Alexandria (25 B.C. – 50 A.D.) Philo was very educated in both Jewish and Greek learning. Notably in his writings he makes frequent use of a Platonic idea of the image (εἰκών, eikon), which in Platonic thought is a form after which all things are patterned. Philo understood the God of Israel to have had these kinds of images in His intellect. For example, in his commentary On the Creation:
“Now we must form a somewhat similar opinion of God, who, having determined to found a mighty state, first of all conceived its form in his mind, according to which form he made a world perceptible only by the intellect, and then completed one visible to the external senses, using the first one as a model (ἐκείνῳ, ekeino). As therefore the city, when previously shadowed out in the mind of the man of architectural skill had no external place, but was stamped solely in the mind of the workman, so in the same manner neither can the world which existed in ideas (τῶν ἰδεῶν κόσμος, ton ideon kosmos) have had any other local position except the divine reason (τὸν θεῖον λόγον, ton theion logon) which made them.” (De Opificio, 19-20)
Philo imagines the process of Creation as an act of planning things out and that the “location”, so to speak, of this planning out was in the divine reason (τὸν θεῖον λόγον, ton theion logon). It is this “creative premeditation” that Benedict understands us to retrace whenever we come to understand the workings of reality in the universe in our own minds. We can retrace the divine thought in our own minds because there is divine thought there to retrace.
Let’s look again at the three possible understandings of the rational order of reality:
1. The rationality of reality is a conditional property, conditional on there being intelligent beings in reality. 2. The rationality of reality is independent of any intelligent beings. 3. The rationality of reality is the rationality of a mind that grounds reality.
All three can be interpreted theistically, i.e. in terms of God. We can think of the first two options, especially the second, as describing a kind of “God of the philosophers”. A God who is not especially personal or much like God as we read about Him in the Bible. More like an abstract principle.
The third view is Philo’s and also that of Judaism and Christianity. Philo shared many ideas about God, or the Logos, as held by the Hellenistic philosophies of his day, like Stoicism, Platonism, and Neo-Platonism. But he also affirmed more than this. Philo held the third view, that the rationality of reality is the rationality of a mind that grounds reality. The God of Israel is more than the God of the philosophers.
Pope Benedict stresses this point:
“It becomes easy to see the barrier to equating the ‘God of faith’ and the ‘God of the philosophers’ constituted by a narrow and insufficiently pondered concept of person.”
“The mathematician discovers the mathematics of the cosmos, the being-thought-ness of things; but no more. He discovers only the God of the philosophers.”
“Because in its investigations it [physics] abstracts, in accordance with its nature, from the aesthetic feeling and from the moral attitude, questions nature from a purely mathematical point of view, and consequently can also catch sight only of the mathematical side of nature.”
I think it’s quite remarkable and significant that “the mathematician discovers the mathematics of the cosmos, the being-thought-ness of things”. Let’s not downplay that. That’s a big deal. That kind of intellectual vision of transcendent, immaterial realities is an important breakthrough. It was something along these lines that helped Augustine of Hippo (354 – 430) in his intellectual development and eventual conversion to Christianity, when he read “certain books of the Platonists” and “having read then those books of the Platonists, and thence been taught to search for incorporeal truth” he saw the “invisible things, understood by those things which are made.”
That’s important. But it doesn’t get us all the way to the personal Father, Son, and Holy Spirit. Disciplines do what they are intended to do. You find what you’re looking for. Many phenomena lend themselves to mathematical description and modeling. Models don’t intend or claim to capture everything. They’re maps not replications. As Benedict says, aesthetic feelings and moral attitudes don’t factor into the investigations of physics because they don’t have to. We all know that they are there. But for some tasks we don’t have to consider them. But for other tasks we do. For other tasks aesthetic feelings and moral attitudes are central. And so thinking about them requires different techniques that don’t abstract away from them. Mathematics and the sciences do bring us to God. It’s a kind of communion. But there is further communion to be had, to recognize the personal God we find in scripture and in Jesus Christ.
Benedict makes an important connection between intellectual structure and the act of intellect that can be understood to underlie it:
“The world is objective mind; it meets us in an intellectual structure, that is, it offers itself to our mind as something that can be reflected upon and understood. From this follows the next step. To say ‘Credo in Deum—I believe in God’ expresses the conviction that objective mind is the product of subjective mind and can only exist at all as the declension of it, that, in other words, being-thought (as we find it present in the structure of the world) is not possible without thinking.”
This is an interesting idea. I don’t think it’s exhaustively proven here but it makes a lot of sense in my opinion. Benedict connects three ideas:
1. Intellectual structure 2. Objective mind 3. Subjective mind
Intellectual structure is the intelligibility of reality. We can think about it and make sense of it in our intellect. There is at least an “as-if” quality to the intelligibility of reality. It is structured “as if” rationally constructed. Let’s call this objective mind. But is it just that? Or is there actually a subjective mind behind the objective mind that is actually doing the thinking, so that it is not merely thinking “as if” but thinking “in-fact”? That’s the move Benedict makes.
Benedict then takes this idea and places it in the context of three major metaphysical systems. Metaphysics concerns the ultimate nature of reality; the question of all questions:
“The question to which everything finally leads could be formulated like this: In all the variety of individual things, what is, so to speak, the common stuff of being—what is the one being behind the many ‘things’, which nevertheless all ‘exist’?”
Benedict then looks at three possible answers to this question; three metaphysical systems:
1. Materialism 2. Idealism 3. Christianity
“The many answers produced by history can finally be reduced to two basic possibilities. The first and most obvious would run something like this: Everything we encounter is in the last analysis stuff, matter; this is the only thing that always remains as demonstrable reality and, consequently, represents the real being of all that exists—the materialistic solution. The other possibility points in the opposite direction. It says: Whoever looks thoroughly at matter will discover that it is being-thought, objectivized thought. So it cannot be the ultimate. On the contrary, before it comes thinking, the idea; all being is ultimately being-thought and can be traced back to mind as the original reality; this is the ‘idealistic’ solution.”
What Benedict has been saying up to this point, making the case for the intellectual structure and mind behind all reality fits quite well with idealism. But he will argue that idealism is not the final stop. There is more to reality than just idea:
“The Christian belief in God is not completely identical with either of these two solutions. To be sure, it, too, will say, being is being-thought. Matter itself points beyond itself to thinking as the earlier and more original factor. But in opposition to idealism, which makes all being into moments of an all-embracing consciousness, the Christian belief in God will say: Being is being-thought—yet not in such a way that it remains only thought and that the appearance of independence proves to be mere appearance to anyone who looks more closely. On the contrary, Christian belief in God means that things are the being-thought of a creative consciousness, of a creative freedom, and that the creative consciousness that bears up all things has released what has been thought into the freedom of its own, independent existence. In this it goes beyond any mere idealism.”
This is a fascinating idea. This is what moves Christianity beyond Platonism. In a sense, it brings us back to the material world, but with a renewed and richer understanding of it; no longer “mere matter”. The material world and, what is especially important, our physical bodies are not just illusions or mental projections. They are real. Logos, the divine intellect, is generative not only of all things but also of more intellects with powers to think on their own. And Benedict actually makes this kind of freedom foundational:
“In the Christian view what supports it all is a creative freedom that sets what has been thought in the freedom of its own being.”
“At the beginning of all being it puts not just some kind of consciousness but a creative freedom that creates further freedoms. To this extent one could very well describe Christianity as a philosophy of freedom. For Christianity, the explanation of reality as a whole is not an all-embracing consciousness or one single materiality; on the contrary, at the summit stands a freedom that thinks and, by thinking, creates freedoms, thus making freedom the structural form of all being.”
What I see as one of the essential features of Christian thought is an insistence on the reality, importance, and goodness of matter and physical bodies. This is especially evident in the doctrines of Incarnation and resurrection. In the Incarnation Christ became human with a material body. Not only that, but after he died he was resurrected and became embodied again. And that is our ultimate destiny as well. We humans are physically embodied beings. Our bodies are not meant to be escaped and transcended. Our bodies are essential to who we are. And so our salvation necessarily consists in our eventual resurrection as individual, embodied beings. We have our own personal identities and freedom. Benedict places this kind of individuality and freedom, along with Logos, into a primary position:
“If Christian belief in God is first of all an option in favor of the primacy of the logos, faith in the preexisting, world-supporting reality of the creative meaning, it is at the same time, a belief in the personal nature of that meaning, the belief that the original thought, whose being-thought is represented by the world, is not an anonymous, neutral consciousness but rather freedom, creative love, a person.”
“Accordingly, if the Christian option for the logos means an option for a personal, creative meaning, then it is at the same time an option for the primacy of the particular as against the universal. The highest is not the most universal but, precisely, the particular, and the Christian faith is thus above all also the option for man as the irreducible, infinity-oriented being. And here once again it is the option for the primacy of freedom as against the primacy of some cosmic necessity or natural law.”
There are three things here that Benedict gives primacy:
1. Logos 2. The Particular 3. Freedom
This brings together some concepts that at first glance seem contrary: order and freedom. On the one hand reality is rationally ordered. Things happen according to laws and patterns in ways that are intelligible. On the other hand reality is replete with animate life, both at its foundation in God and in its products in intelligent created beings. Benedict proposes that these ideas in apparent tension are not only compatible but mutually necessary:
“Moreover, it can be shown that the first option—for the primacy of the logos as opposed to mere matter—is not possible without the second and third, or, to be more accurate, the first, taken on its own, would remain mere idealism; it is only the addition of the second and third options—primacy of the particular, primacy of freedom—that marks the watershed between idealism and Christian belief, which now denotes something different from mere idealism.”
The implications from this are far-reaching and recur back to the foundations and redefine it. The Logos of Christianity, because it is coupled with the particular and with freedom, is radically distinct from the logos of Stoicism and other Hellenistic philosophies.
“But if the logos of all being, the being that upholds and encompasses everything, is consciousness, freedom, and love, then it follows automatically that the supreme factor in the world is not cosmic necessity but freedom. The implications of this are very extensive. For this leads to the conclusion that freedom is evidently the necessary structure of the world.”
Again this paradox. Freedom and necessity would seem not to fit well together. But Benedict calls freedom “the necessary structure of the world”, working freedom into necessity. Freedom is not only possible but absolutely necessary.
“The Christian sees in man, not an individual, but a person; and it seems to me that this passage from individual to person contains the whole span of the transition from antiquity to Christianity, from Platonism to faith.”
Let’s review some of the transitions Benedict outlines in all this. I see three transitions, each with three parts.
First, there’s the transition of:
Intellectual structure → Objective mind → Subjective mind
Second, there’s the transition of:
Materialism → Idealism → Christianity
And third, there’s the transition, or mutual primacy of:
Logos → The Particular → Freedom
These are all related and follow a transition, as Benedict describes it, from Platonism to faith.
Conversion to Christianity is a work of the Holy Spirit, something that’s not simple to characterize. Jesus said it’s like the wind, that “blows where it wishes” where you “cannot tell where it comes from and where it goes” (John 3:8). But as best we can discern, it can manifest in both emotional and intellectual forms. Or sometimes even as silence (hesychia, ἡσυχία). But the intellectual form is one manifestation of the Spirit. And it’s one that I focus on a lot.
As far as I know there aren’t too many people calling themselves Platonists these days but I think many technically and intellectually inclined people hold Platonist ideas, even if they don’t think of them in that way. And I’m interested in working with that. I see Platonism as a kind of “gateway drug” to Christianity. At least for the intellectually inclined. So as a ministry let’s see if we can convince people of Platonism first, appealing to the intellect, and then move from there to Christianity, through these transitions Benedict lays out.
In a way these concepts kind of work in a cycle, getting us back where we all start off. Before we ever get too sophisticated it’s natural to think of a world filled with free, thinking individuals whose decisions have meaning and value. Later we come to notice, or are taught about, the regularity of nature. And that regularity is extremely useful. We can do a lot of stuff with science and technology using our understanding of the order and regularity of things. That’s the value of empirical methods: experiments and data. All important parts of a material perspective. But there’s more to be gained and appreciated by still further abstraction, abstracting away the material components of systems to the patterns and structures themselves. Mathematical structures and algorithms have even more generalized utility and power. And their immaterial, substrate-neutral natures are especially amenable to the mental. It’s a perspective at the level of ideas. This is, in a sense, the goal of all education, to get us to be able to comprehend the intelligible structure of things. And I think that’s right, as far as it goes. It’s basically Plato’s model for education outlined in the Republic. But from an existential perspective there’s still more. And that is to consider, not just the ideas themselves but also the thinkers thinking and producing these ideas. Both the one great mind, the Logos, thinking all of reality into being and the multiple created beings living in the created order. And that brings us back to a world filled with free, thinking individuals whose decisions have meaning and value, as measured by the standards of the transcendent Logos. It ends up in much the same spot but picks up a great deal of insight through the process. More of an ascending helix than a circle.
Like Pope Benedict I take this all the way out to Christianity. But in my intellectual activity and conversations I tend to occupy the idealist space, being interested in both the transition into it and through it. I think a lot of intellectuals find materialism convincing. And I’m interested in the move from materialism to idealism, how to make a case for it, and to understand the difficulties and problems with making that move. I think that’s very interesting and challenging. The realm of intellectual structure is where a lot of exciting and interesting stuff is happening. More difficult but even more important is the move from idealism to faith. To understand Logos not only as order and intellectual structure but also as the person Jesus Christ.
The Neoplatonists were a group of philosophers active in the 3rd and 4th centuries A.D. Their most important “big idea” of significance to the history of philosophy and theology was that all things ultimately originate from a single source, called “The One”. This idea had important affinities with Christian monotheism. Neoplatonists believe in “The One”, a single source underlying all things. Jews and Christians, and later Muslims, believe in one God, who is the source of all things and sustains all things. There is significant theoretical overlap. This episode covers an argument developed by Edward Feser that uses ideas from the Neoplatonists to argue for “The One”.
I’ve been studying some works of the Neoplatonists, a group of philosophers active in the 3rd and 4th centuries A.D. Their most important “big idea” of significance to the history of philosophy and theology was that all things ultimately originate from a single source, called “The One”, τὸ ἐν (to hen) in Greek. A key figure in this movement was Plotinus, who lived from 204 to 271 A.D. In his book The Enneads he said:
Δεῖ μὲν γάρ τι πρὸ πάντων εἶναι – ἁπλοῦν τοῦτο
“Standing before all things, there must exist a Simplex” (Plotinus, The Enneads, 5.4.1)
“Anything existing after The First must necessarily arise from that First, whether immediately or as tracing back to it through intervenients; there must be an order of secondaries and tertiaries, in which any second is to be referred to The First, any third to the second.” (Plotinus, The Enneads, 5.4.1)
This idea of there being a single originating source for all things had important affinities with Christian monotheism, whose early form was contemporary with Neoplatonism in the Greco-Roman world. Neoplatonists believe in “The One”, a single source underlying all things. Jews and Christians, and later Muslims, believe in one God, who is the source of all things and sustains all things. There is significant theoretical overlap. And interestingly enough, Neoplatonist philosophy provided important technical and conceptual intellectual resources that Christian theologians could draw on as Christian theology and Christian thought came to be quite highly developed. It was actually in the context of Christian philosophy that I came across an interesting argument for “The One”, the argument that I want to share in this episode.
This exact presentation of the argument is from Edward Feser [1], a Catholic philosopher. He calls this a Neoplatonic argument for the existence of God. Not because it’s taken verbatim from any particular Neoplatonist in history. But because it draws on important Neoplatonist ideas, particularly of “The One”, or what Feser calls “the absolutely simple or noncomposite cause”.
Before sharing the argument I’ll go over a few of the concepts involved. One of the terms in the argument is that the things of our experience are composite. That they are composite means that they are composed of parts. This is the concept that will drive the argument. The key Neoplatonist conclusion will be in the 9th term of the argument, that the existence of each of the things or our experience presupposes an absolutely simple or noncomposite cause. This is “The One”. The terms following this then get into the nature of this absolutely simple and noncomposite cause.
Another important term in the argument, the 22nd term, is that everything is one of 4 kinds of things:
1. A mind 2. A mental content 3. A material entity 4. An abstract entity
Feser gets this idea from William F. Vallicella [2]. This premise isn’t explained in the argument below but it’s an interesting idea and something I want to study in more detail.
Another point to note is that to extend this argument for “The One” into an argument for God, in terms 36 and 37 of the argument Feser defines God as one having the following attributes:
simple or noncomposite
unique
immutable
eternal
immaterial
a mind or intellect
the uncaused ultimate cause of everything other than itself
purely actual
perfect
omnipotent
fully good
omniscient
In the course of the argument Feser gives reasons why “The One” must possess these attributes. These attributes are consistent with those God is understood to have in classical theism. But it is also worth noting that God as described in the Bible and in Christian doctrine has additional attributes that are not covered in this argument. For example, being the same God who made a covenant with Israel and who became incarnate as Jesus of Nazareth. In other words, the argument makes a case for theism generally, not for Judiasm, Christianity, Islam, or any theistic religion in particular.
I don’t think there’s any contradiction between the attributes described for God in this argument and the attributes of the God of Israel and Jesus of Nazareth. But demonstrating that the two are equivalent requires additional steps that this argument does not cover.
So now the argument:
The things of our experience are composite.
A composite exists at any moment only insofar as its parts are combined at that moment.
This composition of parts requires a concurrent cause.
So, any composite has a cause of its existence at any moment at which it exists.
So, each of the things of our experience has a cause at any moment at which it exists.
If the cause of a composite thing’s existence at any moment is itself composite, then it will in turn require a cause of its own existence at that moment.
The regress of causes this entails is hierarchical in nature, and such a regress must have a first member.
Only something absolutely simple or noncomposite could be the first member of such a series.
So, the existence of each of the things of our experience presupposes an absolutely simple or noncomposite cause.
In order for there to be more than absolutely one simple or noncomposite cause, each would have to have some differentiating feature that the others lacked.
But for a cause to have such a feature would be for it to have parts, in which case it would not really be simple or noncomposite.
So, no absolutely simple or noncomposite cause can have such a differentiating feature.
So, there cannot be more than one absolutely simple or noncomposite cause.
If the absolutely simple or noncomposite cause were changeable, then it would have parts which it gains or loses–which, being simple or non-composite, it does not have.
So, the absolutely simple or noncomposite cause is changeless or immutable.
If the absolutely simple or noncomposite cause had a beginning or an end, it would have parts which could either be combined or broken apart.
So, since it has no such parts, the absolutely simple or noncomposite cause is beginningless and endless.
Whatever is immutable, beginningless, and endless is eternal.
So, the absolutely simple or noncomposite cause is eternal.
If something is caused, then it has parts which need to be combined.
So, the absolutely simple or noncomposite cause, since it has no parts, is uncaused.
Everything is either a mind, or a mental content, or a material entity, or an abstract entity.
An abstract entity is causally inert.
So, the absolutely simple or noncomposite cause, since it is not causally inert, it is not an abstract entity.
A material entity has parts and is changeable.
So, the absolutely simple or noncomposite cause, since it is without parts and changeless, is not a material entity.
A mental content presupposes the existence of a mind, and so cannot be the ultimate cause of anything.
So, the absolutely simple or noncomposite cause, being the ultimate cause of things, cannot be a mental content.
So, the absolutely simple or noncomposite cause must be a mind.
Since the absolutely simple or noncomposite cause is unique, everything other than it is composite.
Every composite has the absolutely simple or noncomposite cause as its ultimate cause.
So, the absolutely simple or noncomposite cause is the ultimate cause of everything other than itself.
If the absolutely simple or noncomposite cause had potentialities as well as actualities, it would have parts.
So, since it has no parts, it must have no potentialities but be purely actual.
A purely actual cause must be perfect, omnipotent, fully good, and omniscient.
So, there exists a cause which is simple or noncomposite, unique, immutable, eternal, immaterial, a mind or intellect, the uncaused ultimate cause of everything other than itself, purely actual, perfect, omnipotent, fully good, and omniscient.
But for there to be such a cause is just what it is for God to exist.
So, God exists.
There’s a lot to say about this argument, especially about objections to it. But here I just wanted to share it, as something I came across and found interesting. I will say for it though that I actually do find it quite persuasive.
Notes:
1. Edward Feser. Five Proofs of the Existence of God. 2017
2. William F. Vallicella. A Paradigm Theory of Existence: Onto-Theology Vindicated. 2002. p. 255
In scripture Christ is called “the image of the invisible God”. As such Christ is supremely important to our access to and understanding of God. As God made his goodness pass over Moses and declared his graciousness and compassion, Christ shows us the Father in his words and in his works. Three important philosopher-theologians: Origen of Alexandria, Augustine of Hippo, and Thomas Aquinas developed theories to explain the process of coming to see and know God in a form of “intellectual vision”.
One of the challenges of religious belief in modern times is that many of the things we are supposed to believe in and refer to in our religious practices are not things that we see or sense with our physical senses. God, for example. We might ask, as many have asked, “Why doesn’t God just show himself to everyone?” Wouldn’t that clear up a lot? Why does God have to be so seemingly “hidden” and why is faith, whatever its purported virtue, even necessary? Some of the explanations for this may sometimes seem kind of forced and unsatisfactory. I think this is something that has to be addressed. And I actually think it’s quite an interesting subject. Not just in terms of apologetics, justifying belief in God or any number of other things we don’t sense with the physical senses. But also interesting just as a way of thinking about the nature of reality and the kinds of things that make it up and undergird it. It gets into some very interesting theological and philosophical issues.
In studying this question my main sources for insight have been the scriptures and a set of important Christian philosopher-theologians. Three in particular:
1. Origen of Alexandria (184 – 253) 2. Augustine of Hippo (354 – 430) 3. Thomas Aquinas (1225 – 1274)
What’s interesting to me in looking at these three is their continuity and consistency. We might be tempted to think sometimes that people who lived hundreds or thousands of years ago must have been less intelligent, educated, or sophisticated than we are in modern times. A few minutes of reading Aristotle can very quickly dispel that notion. And the same goes for these three. In most points I find that their ideas about God, and most things, are the most well-reasoned of any you could find from anyone, ancient or modern. Very few people today, believers or not, have thought about God as rigorously or deeply as they did. And when we look at God as found in scripture and explained systematically by these philosopher-theologians it makes more sense why the world is the way it is and why we stand in relation to God in the way that we do.
Let’s look first at some scriptures. There are many scriptures that talk about seeing or not seeing God. Some examples:
John 1:18 – “No one has seen God at any time.”
Matthew 5:8 – “Blessed are the pure in heart, For they shall see God.”
Exodus 33:11 – “So the Lord spoke to Moses face to face, as a man speaks to his friend.”
Exodus 33:20 – “You cannot see My face; for no man shall see Me, and live.”
Isaiah 6:1,5 – “I saw the Lord sitting on a throne, high and lifted up, and the train of His robe filled the temple… So I said: Woe is me, for I am undone! …For my eyes have seen the King, The Lord of hosts.”
Many passages about seeing God but apparently not all consistent. What to make of that? It could be that the passages are simply inconsistent. A consequence of the texts being authored and compiled by different authors and redactors. It’s certainly the case that it was compiled by different authors and redactors. So that’s one possible explanation. But I think there are more theologically interesting explanations. Apparent contradictions often have a lot of potential to provoke interesting philosophical insight. Whether or not that moves us closer to or further from truth is another question. It can do either. But I think in what follows it moves us closer to truth as well as philosophical insight.
The first of our philosopher-theologians, Origen of Alexandria, in his book On the First Principles (Περὶ Ἀρχῶν), makes an important distinction between things that are (1) not seen and (2) invisible:
“For the same thing is not to be understood by the expressions, ‘those things which are not seen,’ and ‘those things which are invisible.’ For those things which are invisible are not only not seen, but do not even possess the property of visibility, being what the Greeks call asomata, i.e., incorporeal; whereas those of which Paul says, ‘They are not seen,’ possess indeed the property of being seen, but, as he explains, are not yet beheld by those to whom they are promised.”
I think this is a very useful distinction that helps address some of the bafflement over why God would purposely conceal things from us. With this distinction we can see that in at least some cases it may not be that God is purposely concealing things that we would otherwise be capable of seeing, but rather that some things are just not visible by nature. Origen says of John 1:18.
“Moreover, John, in his Gospel, when asserting that ‘no one hath seen God at any time,’ manifestly declares to all who are capable of understanding, that there is no nature to which God is visible: not as if, He were a being who was visible by nature, and merely escaped or baffled the view of a frailer creature, but because by the nature of His being it is impossible for Him to be seen.”
This pertains specifically to physical sight and the physical senses. No one sees God with physical sight. But the same verse says that “The only begotten Son, who is in the bosom of the Father, He has declared Him.” The only begotten Son declares God to us. So we do have access to God but it is through other means than physical sight and senses.
Now, there are cases where deliberate concealment serves some instrumental purpose. Jesus apparently spoke almost entirely in parables, to the point that when he did speak directly it was very unusual. For example, in John 16:29 his apostles say: “See, now You are speaking plainly, and using no figure of speech!” But that was more the exception. Jesus deliberately made his teachings a challenge for his disciples. “Because it has been given to you to know the mysteries of the kingdom of heaven, but to them it has not been given. For whoever has, to him more will be given, and he will have abundance; but whoever does not have, even what he has will be taken away from him. Therefore I speak to them in parables, because seeing they do not see, and hearing they do not hear, nor do they understand.” (Matthew 13:11-13) There are many potential reasons for this form of teaching. One is just the benefit of Socratic “midwifery”. Students sometimes learn things better when they have to work harder for them. So there is that. But I don’t think deliberate concealment is applicable everywhere. There are things unseen. But there are also things that are intrinsically invisible.
So there’s that distinction; between the merely unseen and the intrinsically invisible. But what kinds of things are intrinsically invisible? Would this commit one to belief in supernatural things? In a certain sense I’d say, “yes”, which might be off-putting if you lean more secular or have more secular commitments. But I’d also say that most people tacitly assume or take for granted certain intrinsically invisible things anyway; things that are beyond just those things that subsist in the natural world. Even if we don’t realize it.
One example is abstractions. We make use of abstractions all the time. Some examples are: quantity, quality, relation, causality, possibility. We use these kinds of abstractions to make raw sense data intelligible. For example, we project causation onto events. When one billiard ball moves toward another, comes into contact with the second, and then the second billiard ball starts moving, we say that the first billiard ball caused the second billiard ball to move (by collision and transfer of momentum). That makes sense but we don’t actually see that causation. We see events and those events are only intelligible to us in terms of causation. But we don’t physically see the causation itself. We only “see” it in the intellect. The sciences are essentially projects of characterizing non-physical structures, laws of nature, to explain the data of empirical observations and experiments. We don’t want just isolated data points. We want to be able to describe relations and make predictions.
This way of thinking about the world is by no means obvious. Augustine of Hippo described in his autobiography, Confessions, how he had a very hard time understanding non-material entities. He “could not imagine any substance, but such as is wont to be seen with [the] eyes.” This made it difficult for him to think about God.
“But what else to conceive of Thee I knew not… I was constrained to conceive of Thee… as being in space, whether infused into the world, or diffused infinitely without it. Because whatsoever I conceived, deprived of this space, seemed to me nothing, yea altogether nothing, not even a void, as if a body were taken out of its place, and the place should remain empty of any body at all, of earth and water, air and heaven, yet would it remain a void place, as it were a spacious nothing.”
This is a very natural way to see the world. And I think it’s the way most people think of the world today and even the way we are educated to think. The modern outlook is very materialist or physicalist. Materialism and physicalism are defensible positions. But they’re not the only defensible positions. And I don’t think they hold up very well to extensive philosophical scrutiny. And it’s that kind of philosophical scrutiny that ultimately led Augustine to think past his materialism. He encountered this in the work of the Platonists:
“Thou [God] procuredst for me… certain books of the Platonists.”
“But having read then those books of the Platonists, and thence been taught to search for incorporeal truth, I saw Thy invisible things, understood by those things which are made.”
This is the essence of the process of the natural sciences that we go through even if without thinking about it. We come to understand incorporeal truths “by those things which are made”. We infer causation from the observation of events. We develop theories about laws of nature from empirical data. In Platonist thought this is movement along Plato’s “divided line”, an analogy he introduced in the Republic, moving from visible things to intelligible things.
In Plato’s thought this process of intellectual ascent has a single ending point, which he calls “the Form of the Good” [ἡ τοῦ ἀγαθοῦ ἰδέα (he tou agathou idea)]. Plotinus (205 – 270) called this ultimate principle “the One” [το ἕν (to hen)]. Augustine of course, being Christian, just called this God. The intellectual ascent ultimately leads to the Christian beatific vision, The immediate knowledge of God which the angelic spirits and the souls of the just enjoy in Heaven.
Both Origen and Augustine have theories of a certain form of vision that is distinct from physical vision. For Augustine this is his notion of “intellectual vision”. Origen describes it as a kind of seeing as knowing. In On the First Principles Origen says:
“It is one thing to see, and another to know: to see and to be seen is a property of bodies; to know and to be known, an attribute of intellectual being… [the Son] did not say that no one has seen the Father, save the Son, nor any one the Son, save the Father; but His words are: ‘No one knoweth [ἐπιγινώσκω (epiginosko)] the Son, save the Father; nor any one the Father, save the Son.’ (Matthew 11:27) By which it is clearly shown, that whatever among bodily natures is called seeing and being seen, is termed, between the Father and the Son, a knowing and being known, by means of the power of knowledge, not by the frailness of the sense of sight. Because, then, neither seeing nor being seen can be properly applied to an incorporeal and invisible nature, neither is the Father, in the Gospel, said to be seen by the Son, nor the Son by the Father, but the one is said to be known by the other.”
Augustine develops a similar idea, what he calls “intellectual vision”. In his book On the Literal Meaning of Genesis (De Genesi ad litteram) Augustine distinguished between three sorts of vision.
“These are the three kinds of visions… The first, therefore, let us call Corporeal, due to the fact that it is perceived by the body and revealed by the body’s senses. The second, let us call Imaginative; whatever is not truly of the body, and yet however is to some extent, it is said imagination correctly already: and in any case it is not material, it may be however similar to the body, is in the image of the absent body, nor is the gaze distinguished from itself for that purpose. The third is called Intellectual, from intellect, due to the fact that it is mental, of the mind.” (On the Literal Meaning of Genesis 12.7.16)
Augustine demonstrates the use of these three kinds of vision by giving an example of three levels at which a person can understand the commandment to “love your neighbor as yourself.” (Matthew 22:39) He says:
“Here in the reading of this one command, ‘love thy neighbor as yourself’ (Matthew. 22, 39), occur three types of vision: one is of his eyes, which appears in the writing itself; another is of the human imagination in which one’s neighbor is thought of in his absence; and third of these, which is love as such, as seen by the intellect.”
This is a very simple example from which Augustine demonstrates multiple layers of understanding. And this is similar to the levels of ascent in Plato’s divided line with likenesses of visible things, visible things, and the ideas abstracted from them, visible to the intellect. In reading a text there is the physical visual sensation in which we see the ink imprinted on a page. But while reading that we also think about what things the ink refers to, which is a more sophisticated level of understanding. And finally we can gather general principles from the particular thoughts generated by the text. In the case of this example: from ink, to the thought of one’s neighbor, one eventually thinks about the general principle of love itself. And love as such has no visible image. It is understood thoroughly by the intellect, no doubt accompanied by corresponding sentiments. Augustine says of the things seen in intellectual vision that they “have no images resembling them. The objects of intellectual vision are perceived proprie (“in their own nature”), not imaginaliter (“through a representation”).
Virtues like love are important objects of intellectual vision. And intellectual vision of the virtues is closely tied to the intellectual vision of God. Augustine says:
“This spiritual nature, therefore, in which not the bodies, but similarities to bodies are expressed, having visions of an inferior variety, as that of the mind, even the light of intelligence… they do not have any similar material forms; even as the mind itself and all good dispositions of spirit to which they are opposed in their vices, which are correctly condemned and are even condemned in men. To what end is the intellect to be understood, except truly in some other way? And thus love, joy, peace, longsuffering, kindness, goodness, faith, gentleness, self-control, and so forth, of such by which he is drawn near to God (Galatians 5:22-23) and God himself, from whom all things, through whom all things, in whom all things (Romans 11:36).” (On the Literal Meaning of Genesis 12.24.50)
The idea here being that the mind being able, through intellectual vision, to understand important moral abstractions like the virtues is also able, in a similar way, to eventually understand God, from whom these moral virtues emanate. There’s an interesting example of this kind of vision in Exodus.
In Exodus 33 it says that, “The Lord spoke to Moses face to face.” Not only that but, “as a man speaks to his friend.” They were in close proximity, both in space (of some kind) and in their regard for one another. It is repeated, several times in this chapter, that Moses had found “favor”, חֵן (chen), in the eyes of the Lord. And this favor is repeatedly mentioned as the reason that the Lord grants Moses’s entreaties.
Whatever had happened in verses 11 Moses requests even more. In verse 18 he says: “Please, show me Your glory.” Your כָּבוֹד (kabod). God’s response is interesting. He says: “I will make all My goodness [טוּב (tub)] pass before you, and I will proclaim the name of the Lord before you. I will be gracious to whom I will be gracious, and I will have compassion on whom I will have compassion.” (verse 19) Moses asks to see God’s glory (kabod) and God responds that he will see his goodness (tub). God also declares his capacity to be gracious [חָנַן (chanan)] and to be merciful [רָחַם (racham)].
We see more of this in the next chapter, Exodus 34, in which God proclaims the name of YHWH, saying: “The Lord, the Lord God, merciful [רַחוּם (rachum)] and gracious [חַנּוּן (chanun)], longsuffering, and abounding in goodness [חֵסֵד (chesed)] and truth, keeping mercy for thousands, forgiving iniquity and transgression and sin, by no means clearing the guilty, visiting the iniquity of the fathers upon the children and the children’s children to the third and the fourth generation.” (Exodus 34:6-7) Traditionally this act of self-revelation is known as the Thirteen Attributes. This prayer, the Thirteen Attributes of Mercy, is recited in times of crisis to beseech God to show mercy. It contains thirteen Names and descriptions of God, all of them referring to God’s compassion in various situations. In this remarkable theophany, self-revelation of God, to Moses what we have recorded is a revelation of attributes.
Isn’t that interesting? In what way would Moses, or any other person, perceive these kinds of attributes? Mercy, graciousness, longsuffering, goodness, truth, forgiveness, justice? I think Augustine’s theories make sense here. We could think about them at the three levels of vision: corporeal, imaginative, and intellectual. There are the words for these attributes, taking physical form as ink on a page, pixels on a screen, or sound waves in the air. Then we can imagine, or maybe remember, particular examples of these attributes in individuals, maybe people we know, or people in the scriptures, like especially Jesus Christ. And then we can perceive in the intellect the attributes as such: mercy, graciousness, longsuffering, goodness, truth, forgiveness, justice. And when we do this we are closer to perceiving God himself.
Full perception of God would seem to be beyond our possibility in mortality. The third of our philosopher-theologians, Thomas Aquinas, talks about this in his Summa Theologiae. This comes up in Question 12, Article 11: “Whether anyone in this life can see the essence of God”. Aquinas concludes:
“God cannot be seen in His essence by a mere human being, except he be separated from this mortal life. The reason is because, as was said above, the mode of knowledge follows the mode of the nature of the knower. But our soul, as long as we live in this life, has its being in corporeal matter; hence naturally it knows only what has a form in matter, or what can be known by such a form. Now it is evident that the Divine essence cannot be known through the nature of material things… This can be seen in the fact that the more our soul is abstracted from corporeal things, the more it is capable of receiving abstract intelligible things. Hence in dreams and alienations of the bodily senses divine revelations and foresight of future events are perceived the more clearly.”
This is consistent with the statement by God in Exodus that:
“You cannot see My face; for no man shall see Me, and live.”
What followed after this is (maybe) an intriguing illustration of the partial but necessarily incomplete vision of God that a human may have in mortality. God says:
“Here is a place by Me, and you shall stand on the rock. So it shall be, while My glory passes by, that I will put you in the cleft of the rock, and will cover you with My hand while I pass by. Then I will take away My hand, and you shall see My back; but My face shall not be seen.”’ (Exodus 33:20-23)
God says that Moses will see his back [אָחוֹר (achor)] but not his face [פָנִים (panim)].
What’s going on here? It would seem that Moses’s experience of seeing God must be limited. We might ask, as we asked at the beginning of this episode, is this limitation due to God deliberately withholding the complete vision? Or is it just an intrinsic limitation of the nature of the thing being revealed? To use Origen’s distinction, is the face, panim, of God merely “unseen” or is it actually “invisible”, not capable of being seen physically?
Rabbi Sforno (1470 – 1550) commented on this verse saying (in God’s words): “Your inability to see what you would like to see is not due to My depriving you, personally, of such an experience, but is rooted in man’s inability to see such things unless you had died first, as an eye of flesh and blood cannot see such things. You would be fatally blinded before understanding anything you would see.”
Both Aquinas and Sforno hold that no human can see the essence of God on this side of death. Sforno says “an eye of flesh and blood cannot see such things”. And Aquinas says that “Divine essence cannot be known through the nature of material things.” Knowledge of the divine essence can be approximated, as Aquinas says, “the more our soul is abstracted from corporeal things”. But full knowledge and intellectual vision of God can only be received after physical death, which is why, like Moses, no mortal person can see God and live.
The issue of God and images is prominent throughout the Torah. Making images to worship, even images of the Lord God interestingly enough, are strictly forbidden. The golden calf may have been an attempt to make an image of the Lord God himself. But that didn’t make it any less egregious. The Israelites are given the commandment:
“You shall not make for yourself a carved image [פֶסֶל (pesel)]—any likeness [תִּמוּנָה (temunah)] of anything that is in heaven above, or that is in the earth beneath, or that is in the water under the earth; you shall not bow down to them nor serve them.” (Exodus 20:4-5)
We find an expansion on this commandment in Deuteronomy:
“Take careful heed to yourselves, for you saw no form [תִּמוּנָה (temunah)] when the Lord spoke to you at Horeb out of the midst of the fire, lest you act corruptly and make for yourselves a carved image in the form of any figure: the likeness of male or female, the likeness of any animal that is on the earth or the likeness of any winged bird that flies in the air, the likeness of anything that creeps on the ground or the likeness of any fish that is in the water beneath the earth. And take heed, lest you lift your eyes to heaven, and when you see the sun, the moon, and the stars, all the host of heaven, you feel driven to worship them and serve them, which the Lord your God has given to all the peoples under the whole heaven as a heritage.” (Deuteronomy 4:15-19)
The text here is primarily concerned with the ethical issue of how the Israelites are to conduct themselves, what they should and should not do. But there’s an interesting hint here to a metaphysical matter as well that underlies the ethical. “For you saw no form [תִּמוּנָה (temunah)] when the Lord spoke to you at Horeb out of the midst of the fire.” It’s not just the case that the Israelites shouldn’t make carved images of the Lord; based off of some image that was present but forbidden for them to copy. There wasn’t even an image there. It wouldn’t even be possible in principle for them to make an image of the Lord because there was no temunah.
This is interesting to compare with a couple verses in the first chapter of the Bible:
“Then God said, ‘Let Us make man in Our image [צֶלֶם (tzelem)], according to Our likeness [דְּמוּת (demuth)]; let them have dominion over the fish of the sea, over the birds of the air, and over the cattle, over all the earth and over every creeping thing that creeps on the earth.’ So God created man in His own image [צֶלֶם (tzelem)]; in the image of God He created him; male and female He created them.” (Genesis 1:26-27)
Human beings are made in the image of God. Notably by God. The Hebrew words used here are not the same, but I think the concept is similar enough. In what way are humans created in the “image of God”, in the [צֶ֥לֶם אֱלֹהִ֖ים (tzelem elohim)]? One interesting thing about this tzelem is that it has both male and female manifestations. A lot of possibilities here but one thing I’ll observe is that attributes like humanness, maleness, femaleness, along with virtuous attributes like mercy, graciousness, longsuffering, goodness, truth, forgiveness, justice are abstract concepts that in the philosophies of Augustine and Aquinas are understood most fully by the intellect, rather than through physical form or the image of physical form.
Aquinas developed his theory of knowing and seeing God extensively in his Question 12 of the Summa, Prima Pars: “How God is Known By Us”. First, Aquinas affirmed that it is indeed possible for a created intellect to see the essence of God:
“Since everything is knowable according as it is actual, God, Who is pure act without any admixture of potentiality, is in Himself supremely knowable. But what is supremely knowable in itself, may not be knowable to a particular intellect, on account of the excess of the intelligible object above the intellect.”
Aquinas says that God is not only knowable but “supremely knowable”. God is “pure act”. Here he’s making use of the Aristotelian concepts of potentiality and actuality. Aristotle understood God to be supremely and recursively intellectual, as “a thinking of thinking” [νοήσεως νόησις (noeseos noesis)]:
Quoting from the his Metaphysics:
“Hence it is actuality rather than potentiality that is held to be the divine possession of rational thought, and its active contemplation is that which is most pleasant and best. If, then, the happiness which God always enjoys is as great as that which we enjoy sometimes, it is marvellous; and if it is greater, this is still more marvellous. Nevertheless it is so. Moreover, life belongs to God. For the actuality of thought [νοῦ ἐνέργεια (nou energeia)] is life, and God is that actuality; and the essential actuality of God is life most good and eternal. We hold, then, that God is a living being, eternal, most good; and therefore life and a continuous eternal existence belong to God; for that is what God is.” (Metaphysics 12.1072b)
“The actuality of thought is life, and God is that actuality.” But what is the subject of that thought? Aristotle says: “Mind thinks itself, if it is that which is best; and its thinking is a thinking of thinking (noeseos noesis)”. (Metaphysics 12.1074b)
This being the case, God is supremely intelligible by the intellect. Whatever limitations human beings have to seeing the essence of God in their intellect is due to limitations in the capacities of their intellects, rather than in the intrinsic intelligibility of God. Aquinas says:
“But what is supremely knowable in itself, may not be knowable to a particular intellect, on account of the excess of the intelligible object above the intellect; as, for example, the sun, which is supremely visible, cannot be seen by the bat by reason of its excess of light.”
Not only bats for that matter. Even we humans cannot see the sun directly. But that’s not because it’s not invisible, but rather because it’s too visible. It produces more light than we can handle. That’s a physical analogy. In the case of God the analogy is to intellectual visibility, or intelligibility. If we can’t see God with our intellectual vision it’s not because God is intrinsically unintelligible but rather because God is too intelligible, of greater intelligibility than we are able to understand. Nevertheless, Aquinas does think it is possible for created intellect to see God’s essence. It just needs the aid of divine grace.
In Question 12, Article 4 Aquinas responds to the question of “Whether any created intellect by its natural powers can see the divine essence”. Aquinas concludes that:
“It is impossible for any created intellect to see the essence of God by its own natural power… To know self-subsistent being is natural to the divine intellect alone; and this is beyond the natural power of any created intellect; for no creature is its own existence, forasmuch as its existence is participated. Therefore the created intellect cannot see the essence of God, unless God by His grace unites Himself to the created intellect, as an object made intelligible to it.”
Aquinas is appealing to the fundamental ontological difference between created things and their creator. He says, “To know self-subsistent being is natural to the divine intellect alone.” God is the only self-subsistent being, whose existence is not derived from any other thing. For created beings it’s not natural to know this kind of self-subsistent being. But this kind of knowledge can be given by grace.
So Aquinas is quite optimistic about the possibility of a created intellect seeing the essence of God, with the aid of divine grace. What is the nature of that seeing? Is it with physical sight? Here Aquinas is consistent with both Origen and Augustine. Consistent with Origen’s concept of God’s physical invisibility.
In Question 12, Article 3 Aquinas addresses the question of “Whether the essence of God can be seen with the bodily eye?” He says:
“It is impossible for God to be seen by the sense of sight, or by any other sense, or faculty of the sensitive power. For every such kind of power is the act of a corporeal organ.”
Aquinas makes the case that the kind of sight in which the created intellect may see the essence of God is a sight of some other kind, other than physical sight. He uses the example of Ephesians 1:17-18.
“Likewise the words, ‘Now my eye seeth Thee,’ are to be understood of the mind’s eye, as the Apostle says: ‘May He give unto you the spirit of wisdom… in the knowledge of Him, that the eyes of your heart’ may be ‘enlightened’”.
Wisdom [σοφία (sophia]) and knowledge [ἐπίγνωσις (epignosis)] may enlighten the “eyes of the heart” [τοὺς ὀφθαλμοὺς τῆς καρδίας (tous ophthalmous tes kardias)]. In talking about “eyes of the heart” we’re clearly speaking of something other than physical vision. It’s seeing in another way.
Speaking further of this “intellectual vision” Aquinas says:
“The sense of sight, as being altogether material, cannot be raised up to immateriality. But our intellect, or the angelic intellect, inasmuch as it is elevated above matter in its own nature, can be raised up above its own nature to a higher level by grace.”
“We have a more perfect knowledge of God by grace than by natural reason. Which is proved thus. The knowledge which we have by natural reason contains two things: images derived from the sensible objects; and the natural intelligible light, enabling us to abstract from them intelligible conceptions.”
Bringing this is all to conclusion I’d like to look at examples from scripture in which we are able to see God through His self-revelation in Christ. Paul says that Christ is “the image of the invisible God [εἰκὼν τοῦ Θεοῦ τοῦ ἀοράτου (eikon tou theou tou aoratou)]” (Colossians 1:15). That seems almost like an oxymoron. How can there be an image of something that’s invisible? I think that here again, as in many instances before, we have to think about the possible different meanings of the words so that they can make sense. Two important terms here are image [εἰκών (eikon)] and invisible [ἀόρατος (aoratos)]. Starting with the invisible, we could say that there are aspects of Christ that are physically visible and others that are physically invisible. Christ’s body is certainly visible. But since Christ is God he also has divine attributes that are invisible, just like those of the father. The senses in which Christ is an image are quite rich. Certainly he is a physical image in his body. Paul says that in Christ “dwells all the fullness of the Godhead bodily” [σωματικῶς (somatikos)] (Colossians 1:9). Jesus himself also says:
“I am the way, the truth, and the life. No one comes to the Father except through Me.” (John 14:6)
and
“He who has seen Me has seen the Father; so how can you say, ‘Show us the Father’? Do you not believe that I am in the Father, and the Father in Me? The words that I speak to you I do not speak on My own authority; but the Father who dwells in Me does the works. Believe Me that I am in the Father and the Father in Me, or else believe Me for the sake of the works themselves.” (John 14:9-11)
This can certainly pertain to Christ’s body. But Jesus also points to his words [ῥήματα, (rhemata)] and to his works [ἔργα (erga)]. Not only Christ’s body, which most of his do not physically see, but also his entire way of life, his words and his works, as recorded in the scriptures, show us the Father. And those certainly are available to us.
In Matthew Jesus says:
“Take My yoke upon you and learn from Me, for I am gentle and lowly in heart, and you will find rest for your souls.” (Matthew 11:29)
In inviting us to learn from Him Christ points to His attributes, that he is “gentle and lowly in heart”. Recall how God revealed Himself to Moses when Moses had asked to show him his glory. God made his goodness pass before him and proclaimed his graciousness, compassion, longsuffering, truth, forgiveness, and justice.
Augustine proposed that we could move up through levels of vision from the corporeal to the imaginative to the intellectual. If there’s something to that I’d propose that the most fruitful way to do this is through Jesus Christ. In my Christo-centric theology Christ is always the Way [ἡ ὁδὸς (he hodos)].
So let’s return to the question at the start of this episode. Why doesn’t God just show himself to everyone? Wouldn’t that clear up a lot? Why does God have to be seemingly “hidden” and why is faith, whatever its purported virtue, even necessary?
We see that the understanding of traditional Christianity, both in the scriptures and in the history of theology, is that seeing God is necessarily a different kind of seeing than that of physical sight. This is a consequence of God’s intrinsically invisible nature. As Origen said, it’s not that God could be seen physically and simply decides to hide Himself from us. Rather seeing God is a process of intellectual vision, with what Paul calls the “eyes of the heart” [τοὺς ὀφθαλμοὺς τῆς καρδίας (tous ophthalmous tes kardias)]. So how is this done? At the end of the day it comes down to basic Christian practice. Reading of scripture, prayer, Christian fellowship in the Church, and all the sacraments. All this theoretical background isn’t necessary to engage in the process. But if the question over why God doesn’t reveal Himself to everyone physically has bothered you it could be helpful. I find it helpful and think others may find it helpful, even though it’s quite demanding. We have the scriptures and we can read about Jesus Christ, his life, words, and teachings. As we read these words and think about them and put them into practice they will expand our understanding, so that we can grasp the fullness of these attributes in our intellect. This is the Way to the Father, always through Christ, eventually to be able to see, with the eyes of the heart, the very essence of God.
Todd Decker 