Mathematical Analogies: God's Ordinal Infinitude

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An advantage of mathematical thought is that exposure to advanced mathematical concepts can provide a strong, philosophically imaginative mental framework for other kinds of complex ideas.  This is especially the case with theological ideas relating to God.  Given God's metaphysically unique status as Being in itself, we can only imagine him analogically.  Any attempt to fully capture him falls short.  If we think we've understood Him, then our conception is not God.  And yet, this does not mean that our conception is totally unrelated to God.  Since we are created in His image and likeness, our intellectual relationship with God is something real and true, and our conceptions can be a true "projection," so to speak, of who He is. 

Here, we give a way to consider God's infinitude, using the mathematical study of infinities found in set theory.  One thing about this analogy, as we shall see, that sets it apart from others is that it contains within itself an additional analogy of its limitations.  Typically, we have to supply the limitations of an analogy in a manner exterior to it.

So what do we mean when we say that God is infinite?  We mean that there is no limit to him: i.e., that he is not finite.  He does not have "boundaries" like we do. Now this means two things in terms of creation: the first is that God permeates creation.  This doesn't mean that we are all a part of God (given his simplicity), but rather that he is present in all creation.  As the Psalmist says:

Where shall I go from your Spirit?
Or where shall I flee from your presence?
If I ascend to heaven, you are there!
If I make my bed in Sheol, you are there!
If I take the wings of the morning
and dwell in the uttermost parts of the sea,
even there your hand shall lead me,
and your right hand shall hold me.
If I say, “Surely the darkness shall cover me,
and the light about me be night,”
even the darkness is not dark to you;
the night is bright as the day,
for darkness is as light with you.

 

The second is that God is not bound by creation itself, but is beyond it.   This contrary to the notion of certain forms of Pantheism which identify God with the collection of creation itself.  As you shall see, this one will be a bit harder to grasp analogically.  But let's see how we can start.

Analogy 1.0: The natural numbers

The natural numbers have a set theoretic representation. Zero corresponds to the empty set $\{ \}$ , $1 = \{ \{ \} \}$, $2 = \{0, 1 \} = \{ \{\}, \{ \{ \} \} \}$, etc. In general, assuming that $0, 1, ..., n$ are represented as sets, we can just let $n+1 = \{0,1,2,..., n\}$. Now this is intuitively nice for us for two reasons: 

    1. The number of elements in the set "$n+1$" is $n+1$ itself, and 

    2. we tend to think of the natural numbers as comprising the numbers under it. 

So let's make a first attempt at an analogy: God isn't finite, but he contains within himself all finite things. In some sense, then, God is like the set of natural numbers itself. Call this set $\omega = \{0, 1,2, 3, ... \}$.   This analogy does not quite capture, as much as we would like, however, God's infinitude being totally "other", as we can in some sense comprehend the natural numbers as an increasing union of the finite sets.

Now this seems more like in the spirit of what you are trying to get at, so if we stop here with the recognition of the imperfection of analogies, we will be ok. 

Analogy 2.0:  Ordinal numbers and countability

But what if we could mathematically improve the analogy? Let's see how we can do this. The reason I don't like the natural numbers as an end point is that we human beings also contain them within our minds. In other words, "God's infinitude = ω" is too human for me. Note, for one, that we easily consider ω to be another set (in set theory, this assumption is the typical instantiation of what is called the "axiom of infinity", i.e., there exist infinite sets). That, and note that ω itself is comprised of the natural numbers. In some sense, ω behaves a lot like another number, just not a finite one. Then we can extend "numbers" beyond the natural numbers, for example: $\omega+ 1 = \{0,1,2,..., \omega\}$, $\omega+2 = \{0,1,2,..., \omega, \omega+1\}$, or something like $2\omega = \omega \cup \{\omega,\omega+1, \omega+2,...\}$.

We call these numbers "ordinals" in set theory. Using techniques for expanding ordinals (i.e., taking increasing unions like we did to get ω, or getting successors like ω+1), we can give constructions for things like $2\omega$ or $\omega^2$ , or $\omega^\omega$, $\omega^{\omega^\omega}$, and so on. There is also something funny about these ordinals though... it's that you're not really increasing the "size" of the set. By this, I mean that I can create a one-to-one correspondence between ω and, for example, ω+1 (by mapping 0 to ω and n > 0 to n - 1), or mapping ω to 2ω (for example, map even numbers 2n to n, and odd numbers 2n+1 to ω+n). In math jargon, this means that these sets all have the same "cardinality," and having a one-to-one correspondence with ω, or some subset of it, is called being "countable." Intuitively, this means that I can enumerate the elements sequentially as a human,  and assuming I had enough time could eventually reach any number inside my ordinal set that I wanted. So whatever God is in our improved analogy, he isn't countable, because he is beyond our own "infinity". It turns out that in set theory, we have ordinals that are uncountable sets, and we have several sizes of infinities, all surpassed by God.  For instance, the set $P(\omega)$, which is the set of all subsets of $\omega$, is not a countable set.

Analogy 3.0: Sets vs Proper Classes

But of course, we don't want to stop at any such ordinal, because we can increase that ordinal like we did with $\omega$. So how do we address this? It turns out that we can still envisage this incomprehensibility of God's infinity by just letting God's infinitude correspond to the collection of ALL ordinals in our set theoretic "universe". But wouldn't this just be another set subject to the same problem? Here's where it gets interesting. Such a collection is actually not a set, but rather what we call a "proper class". When Cantor was developing set theory, he thought sets were simply collections of things, but this led to some contradictions (see "Russell's paradox"). 

  So naive set theory had to be tweaked by adding something called the Axiom of Foundation.  Essentially, what this means is that you cannot have an infinite chain of relation of reverse inclusion.  Using technical terms, you can't have an infinite sequence $a_1, a_2, a_3, ...$ of sets such that for each natural number $n$, $a_{n+1} \in a_n$.  How does this imply that the ordinals are not a set?  This is proved using two facts: the first is that we can assume the axiom of foundation.  The second is that assuming this axiom we can actually determine that a set $S$ is an ordinal if it satisfies these conditions:

1.  It is transitive: for any elements $a,b,c \in S$ with $a\in b$ and $b\in c$, $a\in c $ as well.
2.  It is has the following trichotomy: for all $a,b \in S$ either $a\in b$, or $b\in a$, or $a = b$.
3. It is well-founded: each non-empty $A\subset S$ has a "smallest" element:  meaning, an element $x$ in $A$ such that every other $y$ in $A$ contains $x$.
   

Note that foundation implies that you can't have an alternating loop of inclusion with $a \in b$ and $b\in a$.  Then you would have an infinite sequence $a,b,a,b,a,b...$ that contradicts foundation.

 It's too complicated to explain how here, but basically, the collections of elements in our set theoretic universe have to be "small" in some sense. You can't, for example, have a "set of all sets." It takes some argumentation, but you also can't have an "ordinal of all ordinals" either.  Essentially, if there were such a set of all ordinals, let's call it $O$, then with some work, you can show that the set $O' = \{O\} \cup O$ is also an ordinal, so $O' \in O$.  This violates the axiom of Foundation, as now $O \in O' \in O$.

Analogy 4.0+: Expanding the set theoretic universe

 So now we have a mathematically cool/ sophisticated way of describing God's infinitude. God's infinitude transcends all categories of human understanding as the class of ordinals transcends the standard set theoretic universe. But I'm going to crash the party one more time to show, also mathematically, why this isn't enough. Set theorists like to expand our typical set theoretic universe by throwing in certain "large cardinals" whose existence can't be proved or disproved using our standard set theoretic axioms. Under such an extended universe, this class of ordinals can become a set. But then we are capturing God in our universe, which means that what we have analogized as God's infinitude ceases to be so.  While in this expanded universe our current class of ordinals is now a set, there class of ordinals in that universe is not a set in that universe.

Analogy ω.0: The futility is the analogy

The moral of the story is that the stronger the analogy becomes, the more elusive his infinitude turns out to be. The cool thing about this though, is that with set theory, we can actually experience for ourselves how this is, this coming closer to God and recognition of our own limitations at the same time.

Mathematical Analogies: Reasoning about the Trinity: part III

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 In this final installment on the Trinity and the Sierpinski Triangle, We show how some of the principles  and techniques described in part two illustrate in some interesting ways the orthodox theology on the Trinity, particularly in cases where the doctrine holds certain ideas seemingly in tension with each other together. 

Previous posts for part I and part II are here and here (include hyperlinks).  

Let's start with the initial observation about the Triangle's self similarity.   Observe that the cyan, yellow, and magenta portions of the Triangle are themselves full Triangles, and recall that the Triangle is the minimal closed fixed set of the following Iterated Function System $\{f_1, f_2, f_3\}$ with 

\[ f_1(x,y) = \bigg(\frac{x}{2}, \frac{y}{2}\bigg) \]  \[ f_2(x,y) = \bigg(\frac{x}{2}+ \frac{1}{2}, \frac{y}{2} \bigg) \] \[ f_3(x,y) = \bigg(\frac{x}{2} + \frac{1}{4}, \frac{y}{2} + \frac{\sqrt{3}}{4} \bigg). \]

  We can initially analogize the self similarity in the yellow, magenta, and cyan Triangles to the fact that the three persons, the Father, Son, and Holy Spirit, are themselves fully God, while not being each other, and by the self similarity of the Triangle, when put together, we see that they form one Triangle as well.  The weakness of this, of course, is that we can just as easily say there are three Triangles, but we can't say there are three Gods. 

Is there a way to strengthen the analogy to at least partially avoid the problem?  One way, I think, lies in allowing ourselves to consider the multiple ways we can talk about the "threeness" of the Triangle.

  Recall that the Trinity consists in three persons in one God.  But we cannot think of the persons as parts.  Recall also that the concept of "person" answers a different question than that of nature.  Persons answer who God is, while substance or nature refer to what God is. Thinking of God according to his persons  is thus a different mode of thinking according to his substance or nature.  There is a sense in which this is also true of an Iterated Function System and its fractal.  We can study the Sierpinski triangle and its properties according to the fractal itself; it can be described, after all, as a compact set of points in the coordinate plane.  We can also think of the Triangle in terms of the three functions $f_1, f_2$, and $f_3$ which generate it.  The functions work together in coordination to generate the resulting figure. Alone, they don't mean much, but together, they define the Triangle itself.

God cannot be God except as Trinity. When we baptize, we do not say "I baptize you in the names (plural) of the Father, Son, and Holy Spirit," but rather in the "name" (singular). In Hebrew, there was a strong connection between something's essence and its name (hence the philosophical significance of God's response to Moses of "I am who am" when asked what his name was).  In light of this, Christians will see hints of the Trinity in the Old Testament as well.  Consider, for instance, the first few verses of Genesis: 

In the beginning, God created the heavens and the earth. The earth was without form and void, and darkness was over the face of the deep. And the Spirit of God was hovering over the face of the waters. And God said, “Let there be light,” and there was light.

 Traditional Christian Theology looks at these verses as presenting God as Trinity: God the Father as creating the world in his person, the Son, also known as the Word of God, who comes from the Father making creation happen (see also John 1:1), and finally the Spirit of God, hovering over the chaotic waters to bring life. 

When we focus on any one of the three persons as God, we also encounter the other two persons.  When the Father creates, the Son and the Spirit are creating.  When we see the Son, we encounter the Father and are filled with the Spirit.  The Holy Spirit who moves us, points not to itself, but to the Father and the Son.  To encounter one is to encounter all, as if in a never ending dance.  In pondering this, Christians, especially in the East, called this folding back of the Father, Son, and Spirit on each other as a perichoreisis. 

 This is much like how when we see $f_1$'s "contribution to the formation of the Sierpinski Triangle, we don't just see $f_1$; we see the Triangle again, which itself is generated by $f_2$ and $f_3$ as well.   Again, let $T$ be the set of points of the Triangle.  Then we have 

\[ f_1(T) = f_1(f_1(T) \cup f_2(T) \cup f_3(T) ) =  f_1(f_1(T)) \cup f_1( f_2(T)) \cup f_1(f_3(T) ). \]

That is, looking at $f_1$, you will also see $f_2$ and $f_3$. 

Pope (now emeritus) Benedict XVI, in talking about the Trinity, considers the persons themselves to be relations.  The Son is not Son without a Father, and the Father is not Father without the Son.  The Spirit (meaning breath) can't be Breath without one Breathing.  While people might think of the persons like the corners of a simple triangle, I wonder if Benedict likely saw the persons more akin to the sides connecting the corners.  At least, when I read him, that was the picture that came to mind.

In a sense, $f_1, f_2, $ and $f_3$ provide the "personality" of the Triangle, while the actual fractal points more to  its "substance." The functions are themselves mathematical relations, taking the triangle and mapping it to the points that reveals the particular contribution of the function, but in such a way that is in concert with the other functions.  These relations, understood as sets themselves, are also isomorphic copies of the Triangle, each distinct, but bearing the marks of the other persons in the works commonly assigned to each.  It isn't quite right to compare the persons to the self similar parts of the Triangle, therefore.  These self-similarities arise from certain special relations (i.e., functions), which in the end only make sense when seen in concert together, but each encapsulating the fullness of the Triangle. 

Welp, that's it, y'all!  Obviously we can't stretch the analogy too far, and even an enhanced one like this falls far short of the real thing.   The Trinity is quite gnarly, after all; it is hard to capture one part of our understanding of it without at the same time imaginatively letting go of an equally important part. Having said that, fractals like the Sierpinski Triangle are themselves are pretty gnarly objects.  My hope is that the gnarliness of latter could provide some fun, light hearted reflection on that of the former.

Reasoning mathematically about the Trinity, part II

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In part I of Reasoning about the Trinity, we talked a bit about the doctrine of the Trinity and the philosophical concepts sustaining  its contents.  We also looked as common analogies used to help people partially understand the Trinity, and ended with a brief exposition of the Sierpinski triangle as a possible more "mathematical" analogy with less weakness, at least, than typical, empirically based analogies. In part II, we approach the Sierpinski Triangle (from now on, I will shorthand this as just the Triangle) using some mathematical tools. 

Let's begin! The Triangle is an example of what we call fractals.  While there are multiple mathematical definitions of fractals, the gist is that these are objects which can be partitioned into mirror images of themselves; in short, they have self-similarity. The Triangle itself is made up of three copies of itself  (look at the picture in the corner, and see how the cyan, yellow, and magenta portions are themselves Triangles), and each such copy is made of three copies of itself, and so on.  

An interesting thing about fractals like the Triangle is that they can be described as minimal fixed set of Iterated Function Systems (IFS).  Not all sets commonly called fractals can be generated by IFS's but there are many such fractals (for example, the Mandelbrot set), including several familiar ones, which can be described in this way.  IFS's are finite sets of contractive functions on some metric space (for example, the real numbers or the Cartesian plane).  For the mathematical novice:  by contractive, I mean that the functions take any two points a given distance, and return two points that are even closer together.  For an IFS with functions $f_1, f_2,..., f_n$, by fixed set,  I mean a closed set $A$ such that if I take all the points $ y$ such that $y = f_i(x)$ for some $x$, I end up with $A$ again.  It turns out that the Triangle is the smallest fixed set for functions $f_1, f_2, f_3$ over real number pairs with 

 

\[ f_1(x,y) = \bigg(\frac{x}{2}, \frac{y}{2}\bigg) \]  \[ f_2(x,y) = \bigg(\frac{x}{2}+ \frac{1}{2}, \frac{y}{2} \bigg) \] \[ f_3(x,y) = \bigg(\frac{x}{2} + \frac{1}{4}, \frac{y}{2} + \frac{\sqrt{3}}{4} \bigg) \]

One thing to note is that each function "contributes" to a portion of the fractal. Suppose that the Triangle above is in a coordinate plane, and the bottom left point of the cyan Triangle is the origin $(0,0)$. If you take the points in the Triangle and pass them through $f_1$, you shrink everything down to the bottom left corner, which takes the whole triangle to the cyan Triangle. $f_2$ shrinks the Triangle and moves it to the right, mapping it onto the magenta Triangle, and then $f_3$ shrinks it and  moves it diagonally upwards and rightwards where the yellow Triangle is.  If we call the set of points in the Triangle $T$, it's easy to see that $f_1(T) \cup f_2(T) \cup f_3(T) = T$, making the set $T$ a fixed point for the IFS $\{f_1, f_2, f_3\}$.

This is by no means the only such fractal that can be described this way.  For instance, If you have functions $g_1,g_2,g_3,g_4$ with 

\[ g_1(x,y) = (0.85x +0.04y,-0.04x + 0.85y +1.6) \]

\[ g_2(x,y) = (0.2x - 0.26y, 0.23x +0.22y + 1.6) \]

\[g_3(x,y) = (-0.15x +0.28y, 0.26x +0.24y +0.44) \]

\[g_4(x,y) = (0, 0.16y ) \] 

You get what is known as the Barnsley fern. The individual functions also contribute to the whole fern, and the pictures are color coded in such a way that corresponds to each function in the system.  Assuming the origin is a bit below the bottom stem of the fern, $g_1$ gives the main structure of the fern. You shrink it just a bit, rotate it, and move it up, mapping it to the cyan portion.  Meanwhile, $g_2$ and $g_3$ shrink it significantly with some rotation and distortion, and shift it off, generating the yellow and magenta leaves of the fern.  Finally, $g_4$ flattens it vertically into a line, shrinks the line and shifts it upwards, mapping the fern onto the green stem.  Again, the fern is a fixed point of the IFS $\{ g_1, g_2, g_3, g_4 \}$.

Let's show one more.  We can actually generate some interesting fractals with IFS's containing only two functions.  For instance, let 

\[ f_1(x,y) = (0.2875x +  1.575, 0.5875y + 1.6) \]

\[f_2(x,y) = (0.603x +0.5591y +0.3880, -0.5864x +0.6324y -0.1549) \]

You get this cool fractal:

 (Yes, these are computed using some software and much experimentation). Using what you know of distortions (rotations, skews, translations, etc.) and high school geometry and algebra, try to figure out which portion $f_1$ and $f_2$ go to. 

In fact, we can use this approach to figure out, for a a given IFS, what the associated fractal is.  Start with the origin point (0,0) (or some other point, if you want),  let $G_1 = \{ (0,0) \}$, and apply each of the functions to it to get some set

\[G_1 = \bigcup_i f_i(G_0). \] 

Then do the same thing again, except instead of just with (0,0), you do it to your newly generated set $G_1$, and let $G_2 = \bigcup_i f_i(G_1)$.  Keep repeating this, and the closure of the limit is your fractal.  The repetition also should help you see why these fractals have self similarity properties.  As you can see, each of the functions takes the fractal and maps it to a smaller and possibly distorted version of itself.  Though the individual functions are not each other and only form a part of the function system, their manifestations, so to speak, in the IFS, result in complete copies of the fractal.

Thanks for reading! Part III will describe how these tools can help us to visualize certain doctrines seemingly in tension with each other at the same time.