Choice and Determinism part I: Mathematical takes on choice

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  The following series of posts deal with the dichotomies of choice and determinism in math, as well as the corresponding theological dichotomy of free will and divine providence.  How does each field deal with the dichotomy? Is there even a real dichotomy between the two?

The Axiom of Choice

When I was an undergrad, my math program required us to write an undergrad thesis in order to graduate.  It could be some original research or a more in depth exploration of a mathematical topic.  At the time, I had no idea what to research, but I hoped to become a logician.   My very cool (and unknowingly prescient) advisor suggested an old school theorem which incorporates ideas from both Logic and Analysis (my eventual field), as well as a smattering of Group theory, known as the Banach-Tarski theorem.  I'm glad I did it, as it was quite helpful in learning both how to read academic papers and texts at a higher level, as well as introducing me to some basic concepts in graduate level Mathematics, especially as my own undergraduate program was rather small.

To put it in layman's terms, the Banach-Tarski theorem states that the unit ball in three dimensional Cartesian space $\mathbb R^3$ can be partitioned into finitely many (very weird) sets (as small as 5 sets), and by rotating and shifting these sets alone, reconstruct two unit balls out of that one.  It sounds pretty magical, but what is at work is how these sets are constructed to begin with... or rather not.  In fact, they can't be "constructed" in any "humanly conceivable" way.  The key principle in finding such sets is the Axiom of choice.  One formulation of it goes something like this: Suppose you have a set $A$ whose elements are non-empty disjoint sets. $(A_i)_i$ (just think of the $i$'s as elements in some indexing set $I$).  Then there is a set that contains exactly one element in each of the $A_i$'s in $A$.  

 Seems plausible, right?  But there is something sneaky about this.  First of all, when you employ the Axiom of Choice at its fullest potential, this set $A$ can be of arbitrary size.  If $A$ is finite, you don't even have to use choice, but if it's infinite, even countably so, then some amount of choice is may be necessary.  Furthermore, such a set created by choice is what mathematicians folklorically  call "non-constructive."  We are just told that such a set exists, but we don't really know how to build it.  Intuitively, it's because we as humans tend to reason in discrete steps, but the construction can take uncountably many steps if the indexing set $I$ mentioned above is uncountable. 

Here a nuance must be made clear, not all representative sets require choice.  For example, suppose that for my collection $(A_i)_i$, every set $A_i$ is a subset of the natural numbers indexed by $i$.  Then I can easily just pick the smallest number.  But this no longer requires choice, as the "smallest number" in a set of natural numbers is something one can theoretically distinguish beforehand. 

Oddities of Choice: non-measurable sets

 So when does the axiom of choice non-trivially appear? A common example where a student might first witness a funny outcome of choice is in the construction of non-measurable sets.  But to understand this, we must first understand what a measure is. To illustrate the properties of a measure, let's work on the unit interval $[0,\infty]$.  In set theory, we might consider the cardinality of a set to be its size, but $[0,1]$, and any other interval inside it, has uncountably many points.  Yet for us, the interval is still "small", geometrically speaking.  If we measure it, we get, well a line of length 1.  More generally, a measure is a more "geometric" way to assign size to subsets of a set. Formally speaking, it is a (partial) function $\mu$ whose inputs are (as we will see, certain) subsets of the unit interval, and whose output is some nonnegative number. 

On the onset, we want this measure $\mu$ to behave in intuitively nice ways for us.  For instance,  if we have an interval $[a,b]$, we want $\mu$ to correspond to the length of the interval: $\mu([a,b]) = b - a$.    It's also ok for non-empty sets to have measure 0.  This also works with our intuitions; after all, what is the length of a single point in the real line? We also can work beyond the unit interval, picking a larger set such as $\mathbb R$ or $\mathbb R^n$.

   Now, we want some ground rules for how such a measure works.

1. Monotonicity: if $A$ and $B$ are measurable sets with $A \subseteq B$, then $\mu(A) \leq \mu(B)$.

2. Countable additivity: if $(A_i)_i$ is a sequence of mutually disjoint sets, then 

\[ \mu \bigg(\bigcup_i A_i \bigg) = \sum_i \mu(A_i) \]

(Note: you can have sets of infinite measure.  In  such a case for some measurable set $A$, we just let $\mu(A) = \infty, $ but don't think about set theoretic sizes of infinity)

3. Closed under countable unions, intersections, and complements.

With those requirements above, we've got a pretty good rigorization of size in a geometric, and not merely set theoretic sense.  Observe also that we don't care much about countably many points.  In our particular case, the measure of an open interval $(a,b)$ is the same as that of $[a,b]$.  Also, any countable set has measure 0.  This can get pretty strange for some people, since it implies, for example, that the set of rational numbers in $[0,1]$ has measure 0.  It helps to think of such sets as "dust" in comparison to the intervals, however, even though the dust seems to be everywhere.

But is every set measurable?

It turns out this is not the case.   Going back to the Banach-Tarski theorem, for instance the weird sets partitioning the unit ball have to be non-measurable sets.  Here, we will give a different, earlier example of a non-measurable set that has some of the flavors of the Banach Tarski partitions. It is called the Vitali set, after the mathematician who discovered it. It's described in the following way:  define the following relation $E$ on $[0,1]$ by

\[ s E t \text{ iff } s - t \text{ is rational } \]

In other words, if the difference between two numbers is rational, we'll say they are related.  Now, for a given $s \in [0,1]$, let $E_s$ be the set of all $0 \leq t \leq 1$ so that $s E t$.  $E$ is a special type of relation: an equivalence relation, as it is reflexive, transitive, and symmetric.  Based on this, we can show that all the distinct $E_s$'s a partition of $[0,1]$, and given any $E_s$ and $E_t$, either they are disjoint or they are the same set (for the mathematically curious who aren't familiar with this construction, try to prove why this is the case using the definition of equivalence relations!).  Also, each $E_s$ is countable in size, as there are countably many rational numbers, and $E_s$ is defined to be all the $t$ in the unit interval that are a rational distance away from $s$.  Since there are countably many elements in each $E_s$, there are uncountably many distinct $E_s$'s (again, if you haven't seen this before, try to reason why this must be the case).

Here is where we use choice: for each distinct $E_s$, choose  exactly one "representative" element $s'$, and put it into a set $V$.  it doesn't matter which, any single element will do.  It turns out that this $V$ is not measurable.  to see why this is the case, we first observe the following:  for every rational $q \in [0,1)$, let $V+q = \{ (s +q) \text{ mod 1}: s\in V$.  Basically, $V+q$ is a modular translation  of $V$: shift all the elements of $V$ by $q$, and subtract 1 from any of the resulting elements greater than 1.  Three observations come to mind:

1. All the $V_q$'s are mutually disjoint. This follows from the property that $V$ contains exactly one element from each of the distinct $E_s$'s.

2. The union of all the $V_q$'s is the unit interval: for any $r \in [0,1]$, find $s \in V$ such that $r-s$ is rational.  Then $r$ itself is in the set $V+(r-s)$.

3.  If $V$ is measurable, then all the $V_q$'s have the same measure.  Think of it this way:  split $V$ into two sets: $V \cap [0,1]$ and $V \backslash [0,1]$.   These sets should then also be measurable.  $V+q$ shifts $V\cap [0,1]$ by $q$, and shifts $V\backslash [0,1]$ back by $1-q$.  These are just translations, and translations preserve measures.

So if $V$ is measurable, then either $\mu(V) = 0$ or $\mu(V) > 0$.  Seems obvious enough.  But it can't be either, because of the countable additivity property for measures.  If it were, then because the $V_q$'s partition $[0,1]$ the sum of the $\mu(V_q)$'s should be 1.    But a series of infinitely many zeros is still zero, so $\mu(V) > 0$.  Yet a infinite sum of $\mu(V)$ is also infinite. In other words, assuming that $V$ is measurable leads to a contradiction.  So we must conclude that $V$ is not measurable. 

But what is Choice anyway?

Now though such a set $V$ may exist, we can only show it by assuming the axiom of choice; in order to construct $V$, we had to choose an arbitrary element from each of the $E_s$'s.  Also, we have no idea what this set "looks" like.  It's not a constructed set in the sense that there are clear steps on how we distinguish certain representative points.  The arbitrary nature of the selection is why the logician Bertrand Russell was said to state about the axiom of choice that you don't need it to select one shoe for each pair when there are infinitely many pairs of shoes (just pick the left one!), but that you need it to select a sock from each pair of infinitely many socks.

One thing that is philosophically, and perhaps ideologically, interesting about this is that the act of calling this procedure "choice" says something about those who came to use or analyze the axiom and call it the Axiom of "Choice".
In a sense, the mathematical notion of "choice" in practice has a very particular feel to it in that choice is associated with arbitrariness.  When you can distinguish some object according to unique properties, and these properties are mathematically definable in the appropriate ways, you don't necessarily need "choice" to isolate the object.  But if you want to distinguish a point from other points that are not necessarily distinguishable from it, except that they are just not the same points, then the axiom tells you, "just pick one."   And on an intuitive level, why would that be a "wrong" mathematical move to make?  It's not in itself, if the mathematician is operating under a theory where mathematical choice is allowed, and a lot of fields like analysis assume it.  I myself have used the Axiom of Choice or some mathematical equivalent of it several times in my own work, often implicitly, but sometimes more explicitly.

But this notion of choice as unfettered arbitration is found in all sorts of fields, not just in math.  For instance, in neuroscience, there is the Libet experiment, which found that for a certain simple tasks like when to push one of two buttons, the time the subject made the decision to push the button was preceded by some milliseconds by neurons firing in certain regions of the brain before the subject was actually conscious of their decision. Such an experiment has been used by pop materialists and determinists to argue that free will does not exist, despite its flawed design and limitations of implication. Here, the experiment's structure itself (pressing one of two buttons without any prior prompting or reasoning) reveals an understanding of choice that, in its rawest and most rudimentary of forms, involves at some level mere arbitration. 

However, is this how all fields understand choice and free will? In the next article, let us see what a few theologians have to say about that!

 

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.

Reasoning mathematically About the Trinity, part I

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When it comes to doctrines that are hard to grasp, the Trinity takes 1st place for most people looking to study Christianity, and there are a few reasons why.

To boil it down, we use very strong metaphysical categories to describe the  three-ness of the Trinity.  In Western theology, The Trinity consists of three "personae", or "persons," or in the East, three "hypostases", often rendered as "substances" or "subsistences" in English. Either way, the concepts of personhood and substance are pretty metaphysically strong.  Hence we easily have to state that the Father is not the Son, the Son is not the Spirit, and the Father is not the Spirit. 

Fair enough, but then we use equally if not more strong language to convey unity of the Trinity!  We in the West speak of the three persons being of one "substantia", commonly rendered as "substance" (note the possible linguistic conflict with the Eastern terminology!), while in The East the unity is defined in terms of "ousia", which is rendered in English as "being" (but also "substance!"  The terms ousia and hypostasis at one point were synonymous in Greek.  It took Christian Trinitarian controversies to establish differences in meaning). 

The problem is compounded by the way we use language and reason.  Consider the image below, traditionally used to summarize the key point of orthodox Christian doctrine on the Trinity:

It's known as the"scrutum fidei", or "shield of the faith".  It encapsulates the minimum parts of what is considered Orthodox Trinitology:

• The Father is not the Son, nor the Son the Father
• The Father is not the Spirit, nor the Spirit the Father
• The Son is not The Spirit, nor the Spirit the Son
• The Father, Son, and Spirit are each fully God

So we look at the common verb "is", think that we are making identities between God and each of the persons, and then deduce naively that they must be each other, but we can't do this!  We also can't try to work our way around by claiming that each of the persons are a part of God, because they are each fully God.  So what gives?  Each tradition was trying to break down how God was both three and one.  However, He couldn't be both three and one in the exact same way.  There is, after all, a difference between something being a mystery and something being a contradiction.

In the end, some differences between the terminology arose.  I'm a Catholic formed mostly by Western theology, so I can at least talk about it in those terms.  Personhood basically answers who I am, and makes reference specifically to my rational nature.  We don't talk about inanimate objects, or even animals (except to personalize or anthropomorphize), as "Who's,"  but rather as what's.  Substance refers, however, to an instantiation of being or the particular nature of something.  In a sense, it answers what something is.  For us, we are of human substance or nature.  The persons of the Trinity all are of divine substance.  So if you were to ask me what I was, I would respond by mentioning my nature: "I'm a human."  For the persons of the Trinity, the response would be "I'm God." (though the Son would also say what we do!)

So we can use philosophical terminology to make some nuanced statements about the Trinity. At best though, this only eliminates error, but people still have trouble understanding the trinity in a way that is coherent, so analogies have been used to help them roughly understand how three-ness and one-ness can coincide.  But these have their weaknesses that also correspond to certain heresies:

  A prototypical example is the three-leaf clover.  A clover is technically one leaf-like structure, but it is composed of three "leaves" as well.  The vulnerability here is that people might be tempted to see the three persons as part of one God,  rather than fully God. 

Another more modern approach is thinking of a single person under different categories: for example, I am a mom, a sister, and daughter, but mom, daughter, and sister are not the same thing.  Another variation of this is looking at objects that assume different states, like how water can be ice, vapor, or its regular liquid form.  The problem is that this can lead people to blend the persons into one.  The differentiation are not fundamental enough.

So can math help in this situation? It can't fully lay bare the mystery, but it can expand our creativity in imagining ways of thinking analogically.  Mathematical objects and structures are not tied to physical limits.  We can extrapolate to infinity, and when we enter into the world of infinity, we come a bit closer to thinking about God, who is also not bound by physical limits.  For example, we can consider structures like the Sierpinski triangle.  Start with a triangle, break it into four as shown below, and break the three triangles adjacent to the corners each into four, and so on.

Now note that the triangle can be at best divided into three "parts" corresponding to the top, bottom left, and bottom right corners.  There is one Sierpinski triangle, each section is not the other, but each section is also a full Sierpinski triangle!  The analogy is still imperfect, but it better captures the fullness of divinity inherent in each person.

Now here is where the fun begins:  Most analogies capture only one aspect of the Trinity with failure arising from the inability to compare to it in more than one or two aspects.  How does the Sierpinski triangle fare? Stay tuned for part II!

 

Hello World! A new Math and Theology blog

So What's this all about?

As one in Catholic theological circles, I have often encountered three responses with regards to mathematics from my fellow theologians (and more generally humanities oriented people) when I tell them that I am a mathematician.  The first two are variations of

• So you study math? I hated math/ I was never good at math!
• So you study math?  Math is nice because things are always in black and white! You have only one right answer.

Maybe I'll talk about those two another day, but there is a third type of response that I think merits a bit more discussion.  It involves a well intentioned philosophizing about mathematics and how it connects to truth, or variations on that.  They might talk a bit about how mathematics traditionally was a precursor to philosophy or even make mention of the heading over Plato's Academy stating, "Let none ignorant of Geometry enter." These people do ascribe a high place to mathematics, but when they begin to talk about mathematics proper, their examples and actual use of mathematics to illustrate other truths stops at around a high school level.  I've seen college talks with reference to the Pythagorean theorem or trigonometry as illustrative of some deeper truth.  Surely the philosophical and theological contributions of mathematics don't end with arithmetic and rudimentary geometry!

Now those who do this aren't necessarily wrong, but they are severely limited by ignorance, even if it is an "invincible ignorance" of sorts.  Can you imagine, biologist readers, what a discussion on Catholic moral theology with respect to medical ethics would look like if you had no more than a high schooler's understanding of biology? Or can you imagine theologically oriented explorations of literature, but sticking to the content that a typical American high schooler reads (which, given the state of modern education, is quite lacking)? 

On the other hand, it might not be entirely our humanities oriented friends' faults.  We mathematicians tend to get so absorbed in our increasingly idiosyncratic problems that we turtle ourselves away from the broader human project of the quest for truth.  Sometimes, we even forget what the point of our work is.  The definition of mathematics itself has been and still is now a debate among its people.   The papers that we write only a portion of us can understand. 

 Now, I have also seen mathematicians, including non-religious ones, wonder if there is a spiritual side to the practice of mathematics.  They find themselves in an almost ecstatic state as they are enthralled by a problem or a theorem.  They see themselves as peering into the mind of God, or at least something like it. I myself have experienced such states when working on a problem or finally understanding the intuition behind theorems I have come to love.     These too I do not begrudge, and they are on to something, but in the midst of this I began to wonder if this too was not enough.  Most theorems that gave me that spiritual feeling had little actual spiritually formative content, and try as I might, I had trouble learning who God actually was in this supposed language of his! I wonder how much these mathematicians are familiar with real spiritual practice, which goes far beyond the experience of certain highs.


My project in this blog is to bridge, or at least begin to build a bridge, between my two intellectual loves at a higher level: Mathematics and Theology.   I will also write some posts that are specific to each field at different educational levels, so hopefully something should be there for people of all backgrounds.  Finally, some of these posts will also include issues more peripheral to either field, including topics like pedagogy (in both math and theology... pedagogy is a big deal in both of these!) The interaction with either field to real-world issues, math related or theology related topics in the news, etc.

What is this not about?

Numerology.... enough said.

A bit about me 

I recently graduated with a PhD in mathematics with my main research area in functional analysis, but I also like to dabble in mathematical logic.  Though now working in industry, I continue working on research projects on the side. I got my undergraduate degree in mathematics and theology.   If I were to have chosen theology as my graduate study, however, I would have focused on biblical theology (probably something in the wisdom literature).  My blog name is based on the fact that both the math world and the theology world contained very influential scholars surnamed Lagrange.