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.