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!