Can Set Theory Teach Trinitarians?
Can Set Theory Teach Trinitarians a Thing or Three?
Mathematics is not my forte, not by a long shot. I think most people can relate to that sentiment. But we all know there’s knowledge and mystery there, and that’s why we’re riffing off maths tonight. Specifically, we’re exploring the philosophical dimensions of mathematics, which do—in fact—captivate my interest due to their potential to unfold truths and insights about theology, paving the way for theological contemplation, my primary field of study.
This piece endeavors to theologically interpret the insights that the philosophy of set theory might offer regarding orthodox Trinitarianism and reflections on the Trinity. A meticulous exploration of these parallels can significantly strengthen the discourse, revealing the intricate tapestry of relations within both domains.
From my perspective, set theory seems to be the most prominent of modern mathematical thought. Like the majority of mathematical frameworks, it is constructed axiomatically, with its foundational axioms resting on a seemingly simple concept: the set. However, the reality is more intricate. It resembles a triad of concepts that elevate set theory to its paramount status, as I perceive it. The triad consists of the set, the fuzzy set, and the class, forming the pillars of the axioms of set theory. Delving deeper into these foundational notions can illuminate the relational aspects inherent in them, opening up avenues for theological contemplation and understanding.
The set implies a distinct clear set, the fuzzy set accommodates varying degrees of elements within sets, and the class permits sets to be inclusive or to possess a common property within the domain of sets. Here are their definitions as per ChatGPT:
Class
In mathematical logic and set theory, a Class is a collection of sets or sometimes other mathematical objects that share a common property or attribute. It is a more generalized concept than a set. Not every class is considered a set, especially within the framework of set theory, due to restrictions imposed to avoid paradoxes, such as Russell’s paradox. Classes that are not sets are typically called proper classes.
Set
A Set is a foundational concept in mathematics, representing a well-defined collection of distinct objects, considered as an object in its own right. These objects can be anything, including numbers, alphabets, or even other sets, as long as they are clearly defined and unique within the set. Sets are fundamental to set theory, a branch of mathematical logic that studies collections of objects and has been widely used as a basis for understanding mathematics.
Fuzzy Set
A Fuzzy Set is a type of set in which the boundary is not sharply defined, allowing for elements to have degrees of membership between 0 and 1, unlike classical sets where membership is binary (either 0 or 1). Developed by Lotfi A. Zadeh in 1965, fuzzy sets are used in fuzzy logic and have applications in various fields such as artificial intelligence, control theory, and decision analysis, where ambiguity and uncertainty need to be accounted for. In a fuzzy set, an element can partially belong to the set, with the degree of membership representing the extent of its belongingness.
These concepts are inherently undefined. The definitions provided are more descriptive of what the concept represents rather than offering a higher-order definition or rationale. Defining them would constrain their applicability in the axioms that construct specific set theories. Hence, it is prudent to consider them as philosophical primitives. A thoughtful exploration of these undefined notions can provide a more coherent and persuasive understanding of their role and significance in both mathematical and theological realms.
This inherent primitiveness in the notions behind set theory is theologically reassuring, as it signifies their existence in our world; they are not merely entities of abstraction; they are like us, creatures. They are components of the world as they, like us, necessitate an observer to articulate the concept they make possible. By meticulously elucidating these primitive notions, we can enhance the coherence and persuasiveness of the argument, fortifying the connection between the theological and mathematical domains. Or, another way to put it, one doesn’t require a structured set of axioms to delve into the application of sets; the notions are sufficient, and the abstraction is relatively straightforward to realize from there. This indicates their resemblance to a power of the world, suggesting a discussion about our universe.
These notions provoke contemplation: Is the foundational concept of God in Christian theology a primitive? Or is it something distinct? An exploration of this question can open up a rich field of contemplation, allowing for a more nuanced understanding of the primitive and relational nature of God.
Initially, the answer seems to be perceived as a negative theological datum, specifically a revelation. However, Christian doctrine has historically refuted the concept of God being entirely a component of the world. Being a Catholic, I refer to the classical theistic perspective of God. Thus, the concept of the trinity as a notion is more than just a concept or notion. The key here is to engage in a thoughtful exploration of the trinity as a concept(s), exhaust it of its power as such, and from that make strides in connecting diverse ideas to arrive at the notion of relation.
So, I propose here that the Trinity condescends to the world and, in a manner, assumes it. However, it’s analogous to consuming a meal: it’s essential in some aspects to the agency or action of the consumer, but it doesn’t define the consumer. The Christian God is like a notion of the triunity, but it’s not fundamentally a notion.
The temptation here is to make the concept of God a primitive, a foundational entity that is inherently undefined and beyond human comprehension. And then to expound that out to be likened to the primitive notions in set theory, serving as the foundational entity from which other theological concepts and understandings are derived. And then that temptation would lead one to trade for 30 pieces of silver and say something like the Trinity, consisting of the Father, the Son, and the Holy Spirit, can be paralleled to the triad in set theory: the set, the fuzzy set, and the class. Each person of the Trinity is distinct yet fully God, similar to how each concept in the triad is distinct yet foundational to set theory. This is entirely incorrect and represents a flawed way of thinking about the concept. Rather, this bad illustration shows that the analogy does break down, when reflecting on set theory’s notions with the doctrine of contents on the surface. Here we have to go deeper and take the dogma, and reflect on what the eternal life of the Trinity may look like and be like.
Let’s take a step back. By extrapolating from the notions of set theory we see it’s helpful because what lies behind the mathematics is not merely more mathematics: its creatures expressing the world, and the mysteries of it. It’s theologically innovative as these concepts are also foundational, primitive, and interestingly are a triad. The trinity is a triadic concept but a singular truth. So, it challenges us to ask where does the actual connection between the two realms lie? A meticulous elucidation of these connections is getting us close to seeing some underlying relationality.
The point of divergence arises when the underlying notions of set theory are considered, which seem to grapple with the issue of constituent elements forming a cohesive whole. These notions possess something akin to an observer property; they are entities, components of the world, and their actions are manifested within the worldly realm, especially within the domain of mathematicians. Conversely, and strikingly different the truth of the Trinity is a revelation, illuminating a world profoundly intertwined with ours, but more radically, it represents another world that collides with ours, manifested through the Father, the Son, and the Holy Spirit.
There exists a significant Catholic prayer that articulates some of this here, “Glory be to the Father, the Son, and the Holy Spirit. As it was in the beginning, is now, and ever shall be. World without end. Amen.”
The Trinity, while being a revelation, is not inherently conceptual. However, it does encompass a logically inherent concept, namely, relation. This prayer bears open the insight revealing their triune nature, which is fundamentally based on the coequal concept (consubstantiality) that their world is relational. The existential knowledge imparted by the Trinity conveys that our world, inherently relational prays the truth of theirs. But most importantly the act expresses our world has always been and will always be capable of connecting with the revelation if for no other reason, then to pray. Which the implied capacity to pray and this relate to entities that would typically be deemed ineffable and unreachable, is no small feat. Lucky for us this revelation of an internal relation of this incomprehensible world without end chooses to descend to a world already capable of establishing a connection with it, in some manner. It is at this precise intersection that parallels with the notions of set theory can be drawn.
Lest one thinks I’m drawing an arbitrary concept into Trinitarianism and reflections on the Trinity, let me refer to the words of Joseph Aloisius Ratzinger (or Pope Benedict XVI) on this matter:
“Person is the pure relation of being related, nothing else. Relationship is not something extra added to the person, as it is with us; it only exists at all as relatedness.”
Here, he is asserting that the persons of the trinity are fundamentally relational, making the concept novel. To emphasize: “Relationship is not something extra added to the person, as it is with us; it only exists at all as relatedness.” A thoughtful exploration of these words can provide a deeper understanding of the relational nature of the Trinity, making the connection between the theological and the mathematical more profound and compelling.
And in another part of the same book, he states,
“Therein lies concealed a revolution in man’s view of the world: the sole dominion of thinking in terms of substance is ended; relation is discovered as an equally valid primordial mode of reality.”
This implies that relation is not a secondary or derivative aspect but is a fundamental and original element of reality, equivalent in primacy to substance. If it’s acceptable for a pope of exceptional intellect, which is rare, it’s acceptable for me, with my commonplace intelligence and my inconspicuous status. It truly is persons all the way down. And when we search for the foundation of the persons standing on each other’s shoulders, it’s nothing but relational entities. By considering the primacy of relation, we can arrive at a more compelling conclusion, making the notion of relation serving theology as set serves math more convincing.
We have ultimately arrived at the concept that serves theology as set serves mathematics: relation. But does this notion benefit or entail for a triadic expression to encompass the axioms that make Trinitarianism work?
The answer to that question would be much more complex than we can cover here. My guess is the development of the triad of notions in set theory was an organic and dynamic process, as most knowledge growing enterprises are. Adjudicating that process for the notions behind the Catholic view of the Trinity however has a much older and complex history. So, I’m going to avoid doing an analysis beyond my scope.
What one can do in a reflection like this is recognize that once one notices a legit notion in relation to the revelation of the Trinity, they probably can intuitively recognize others that have been important to or underlying theological discussions.
The creative offering of the reflection of this one, so far, is noticing an existential circumstance, where there is always and already a human inclination to a particular notion that is situated uniquely, to relate to the revelation of God in Christ by the Spirit. What other notions in their contingency are situated well to be related or logically entailed by this revelation of the triune God?
This piece would get way too long if we had to do the work required to connect the dots in a reasonable way. But two notions that come to mind are the notion of act and the notion of being. These seem to be general enough, and possibly primitive for the theological task. The ubiquity of the trinitarian commitment in theology has probably latched onto these notions in some interesting ways specifically to relate and cope with the revelation.
The point at this juncture is it is conceivable we could find three really important notions and emulate what set theorist have done, but for a theological task. Naturally, theologians have already well developed explications of these notions and have leverage them for centuries to do the theological task so, if I was to follow my line of reasoning, at this point of the piece, I would just be synthetically trying to contribute and emulate set theory as I’ve posited it, when that is not necessary.
What I will say before we end this piece is that notions like relation, and act, and being for that matter, are not abstract entities, which is an interesting reflection to consider, given we have been reflecting out from an enterprise that’s bread and butter is the abstraction. Perhaps, this is why the tradition has been able to so powerfully able leverage ideas that feel very concrete like person, particularities of history, and the hermeneutics of understanding them, and the ideas around the sacraments (to name some examples)? The notions behind these great reflections are themselves a World, already being assumes in that they can relate to Revelation. In a sense they are more concrete than these really tough subjects to understand.
The exploration of set theory has led us to a theological triad of notions: Relation, Being, and Act, each serving as a foundational pillar analogous to the set, the fuzzy set, and the class in set theory. These notions, deeply intertwined, could potentially form the essence of Trinitarian theology in a way similar to what set theory has done, by allowing the theologian to form some powerful metaphysical foundations that can extrapolate the contents of specific theological tasks. Just like mathematicians have done with their axioms to create powerful explications of sets, and use that to explain many fields of maths.
I think these three notions could be super helpful as the basis for a Trinitarian project towards a Catholic set of axioms for the broader Trinitarian task. The Father, Son, and Holy Spirit are One being in three Persons, subsisting in fully actualized relation, and their act is basic and intrinsically actual and One too. There’s something here to be learned from set theory, I think. However, the preliminary inquiry into the theology of “notions” as I have been is a pretty good start.
In conclusion, the exploration of the philosophical intricacies of set theory and a hypothetical analogous theological triad of Relation, Being, and Act allows us to delve deeper and to ponder what mysteries await to be explored of the Trinity?
