What is nilpotency and why is it the meaning of life?
Have you ever heard the word “nilpotency”? I hadn’t until recently, and once I learned about it, it sure didn’t seem particularly significant to me, at least initially. I’ll avoid technical definitions for now, but nilpotency is a mathematical property that essentially means a system squares to zero. Some mathematical “thing,” when multiplied by itself, disappears.
At this point, you might be wondering, “Isn’t 0 the only thing that equals zero when you square it (x2=0)?” and, “How could this idea of nilpotency possibly have anything to do with the meaning of life?” Well, hold on to your calculators, nerds, because we’re about to get real abstract and philosophical. That said, I think that if you stick with me for a few minutes, you will be blessed with a unique perspective on God, the universe, and your ultimate purpose in life, and yes, it does kind of involve nilpotency.
Math and the Trinity
Before explaining more about nilpotency and its significance, I think we should start with the question of why you should care about this at all, to which my main answer is: because it’s about God, the nature of the Trinity, and the key to a meaningful life. The Christian doctrine of the Trinity is fundamentally a mystery (and not one I’m attempting to fully explain here), but it is a glorious mystery to be explored and enjoyed. It is my conviction that diving into the deep mysteries of God and learning to see and experience more of His glory always bears good, practical fruit in our daily lives. If God is fundamentally a Tri-unity, that reality means something infinitely significant and ultimately relevant to us.
While this Christian idea that God is both one in essence and three in person is surely counterintuitive, does that mean it is a logical or mathematical contradiction? My Muslim friends like to point out that 1 + 1 + 1 = 3, not 1, so surely this Christian doctrine must be a mathematical impossibility. Is this an instance where we need to turn off our rationality and accept this doctrine on blind faith, or are there logically coherent ways of describing the Trinity?
Christians throughout history have argued that, while the Trinity may be beyond our abilities to fully comprehend, it is not fundamentally illogical. For example, we could point out to our Muslim friends that, while the addition of finite objects yields one result (1 + 1 + 1 = 3), multiplying an immaterial Unity by itself yields a different result (1 x 1 x 1 = 1). Or we could remember that God’s nature is unique, in that he is not composed of parts and he is absolutely infinite. So we shouldn’t use finite numbers in our calculations, but rather infinity, whether in terms of addition (∞ + ∞ + ∞ = ∞) or multiplication (∞ x ∞ x ∞ = ∞). But, most of all, we would be wise to remember that all of our ideas, analogies, and mental maps are going to be wrong in some capacity, because God is God and he will always transcend any attempts we make at describing Him, whether mathematically or otherwise.
One ancient description of God, intended to visually describe trinitarian relationships, is the Trinitarian Triangle, also called the Shield of the Trinity:
Obviously, no Christian believes that God is a triangle, and no one believes these models describe the fullness of who the trinitarian God is. This is simply a shorthand way of describing some historic, orthodox claims about the Trinity, and the geometry makes the model more visually intuitive.1
In a similar tradition, I have attempted to describe a “map” or “diagram” of the Trinity using a more sophisticated mathematical tool called a Hyperset (bearing in mind that a map is never the same as the territory and every map is always incomplete in some way).2
Set theory is a mathematical system that serves as the logical foundation for all other kinds of mathematics. The most standard version used, called ZFC Set Theory, includes an axiom that prevents a set from containing itself. While this is done in order to avoid certain paradoxes, the axioms of ZFC actually result in a ton of mathematical paradoxes, especially when describing infinities.3
An alternative to ZFC Set Theory is AFA Set Theory, which rejects the axiom that prevents sets from containing themselves. And why shouldn’t a set contain itself? A document can contain a link to the folder that contains that document, and you could follow that path in a recursive loop. If you and I are both looking at the same information, then I know that you know what I know, and you know that I know what you know in a recursive loop. Our knowledge contains the other’s knowledge of our own knowledge and vice versa. There is nothing illogical about mutual containment; in fact, the historic Christian claim about the Trinity is that it involves mutual containment.
Jesus told people to believe in him, “that you may know and understand that the Father is in me, and I in the Father.” (John 10:38) Later, Jesus sought to make sure his followers really understood this important truth:
"Do you not believe that I am in the Father and the Father is in me?” John 14:10
And then (I assume because he knew the concept would sound a bit counterintuitive), in the very next verse, he doubles down:
”Believe me when I say that I am in the Father and the Father is in me” John 14:11
Jesus is describing a kind of mutual containment that can't be expressed using the mathematical rules of normal set theory, but which is perfectly at home using the axioms of AFA Set Theory to define a hyperset.
Describing the Trinity with a Hyperset
The thing about hypersets is that they are not mere collections of elements that just sit in static piles. They describe dynamic systems that unfold infinitely. Doesn’t that sound like a more reasonable and logical description of ultimate reality? Hypersets are mathematical descriptions of recursive systems that must actively “oscillate,” or “revolve,” or “unfold” within themselves.
There’s an interesting scenario used to describe a famous mathematical paradox: Imagine a barber who shaves every man in his village who does not shave himself. Does the barber shave himself? This is paradoxical if we say that the barber has to remain in only one of the two groups: if he shaves himself, he shouldn’t shave himself, and if he doesn’t shave himself, then he should - thus the paradox. This scenario is related to Russell’s paradox, which is part of why ZFC Set Theory rejects the idea of sets that contain themselves.
But now imagine a simple tweak to the premise where we say, “This week, the barber shaves every man in the village who didn’t shave himself last week.” Well, now there is nothing paradoxical at all - the barber simply alternates which group he’s in, shaving himself one week and not the next. But dissolving this paradox requires that the elements of the set do more than just sit quietly in a boring, frozen, group. It requires that we use a mathematical system that is more reflective of the dynamic, unfolding universe we actually live in. Despite the claims of the most widely used logic underlying all of standard mathematics, the world is not best described as a collection of static sets that are prohibited from containing themselves. That’s not how the universe works, and that’s not what God is like.
It seems to me that, especially if you already believe in God, you might agree with me that ultimate reality should be described as an active, infinite, and self-contained relationship. God is alive and, because he is a Tri-unity, he can eternally love and act and relate to Himself in a way that would be impossible if he were a single unitary being. When Jesus prays in John 17, he says to the Father,
“Glorify your Son, that your Son may glorify you…Father, glorify me in your presence with the glory I had with you before the world began…All I have is yours, and all you have is mine…we are one…you, Father, are in me, and I in You.” (17:1,5,10,21)
The Father glorifies the Son, who glorifies the Father, who glorifies the Son, who glorifies the Father…And all this happens through the Spirit who is both in them and between them and around them…and on and on in an infinite, dynamic cycle.
In the Christian tradition, this has been described using the theological term perichoresis - sometimes translated as “mutual indwelling”. It is derived from peri - meaning “around” (with the connotation of surrounding or enclosing), and choreo - which means something like “to go and come” or “to make space for.” The image is of a perpetually revolving, surrounding, interpenetrating unity. It was translated into Latin as circuminsession, conveying the idea that members of the Trinity are simultaneously “around” and “within” each other. That’s an infinitely recursive hyperset of mutual containment!
So what?
If God were only a single solitary “person”, his eternal existence apart from creation would not include any speaking - because who would he talk to? He would not interact or be relational in any way - because who could he interact with and relate to? A unitary God could not love because there would be no one else to love. Or, if you want to say that he could only love himself, that would imply His love was fundamentally and ultimately self-centered.
I actually know many people, including some Christians, who imagine God as fundamentally and ultimately selfish. Selfishness, these Christians say, is wrong for us, but it’s right for God because he is the only one who is truly worthy of being at the center of all things. I’ve heard Bible school teachers insist that the motivation for all of God’s works, the one thing God is most concerned with, is His own glory. His existence is ultimately about receiving the most glory he can possibly receive. And there are some things, like His glory, that God just doesn’t share with anyone else…ever.
But there are some big problems with that picture of God. For one, the Bible very explicitly says that God absolutely does share his glory with humanity! It’s actually the first thing he does when he creates people - he makes us in his own image, sharing some of his authority to rule over creation (Genesis 1:26-28), and cronwing us with glory and honor (Psalm 8:5). Humanity has lost some of that glory and fallen short of the glory God intends for us (Romans 6:23), but when Christ appears, we ”will appear with him in glory” (Colossians 3:4), because his ultimate desire is that we “share in his glory” (Romans 8:17). And in case you are still thinking that surely God does not really intend on sharing his own glory with us (“I mean, it’s not like we’re going to share his divine nature or something”), you might want to read 2 Peter 1:4: “he has granted to us his precious and very great promises, so that through them you may become partakers of the divine nature.” I’m not saying we will ascend to become God or that he will not have certain attributes that remain wholly his own - but the language of 2 Peter certainly implies a much greater degree of generous self-giving on God’s part than the selfish view implies.
But doesn’t the Bible say something about God not sharing his glory? Yes, and in those cases, the context makes it clear that he’s talking about not giving his glory to worthless idols (Isaiah 42:8, 48:11). In contrast, God’s plan for humans includes:
• Adoption as his children, intimate relationship, and sharing his glory - the glory that will be revealed in us - the glory of the children of God who are justified and glorified by the God who will graciously give us all things. (Romans 8:15-29).
• Sitting on his throne and reigning with him (Revelation 3:21 & 2 Timothy 2:12)
• Knowing the Father and the Son as they know each other (John 17:2-3)
• Being given God’s own name - the same name the Father gave the Son (John 17:11, Revelation 3:12,14:1)
• Being given the same glory that the Father gives the Son (John 17:22)
• Being united to the Father the same way the Son is, and being loved with the same love. (John 17:22-23, 26)
In other words, those who imagine God as selfish have missed a huge part of God's purpose for his beloved creatures: He desires to generously share all of his good things with us! These verses also highlight the second big problem with the selfish God picture, which is that it leaves out all the most beautiful things about the Trinity.
The Trinity is Not Selfish
Jesus described himself as gentle and lowly (Mathew 11:29), and he modeled humble service (John 13:1-17) and self-sacrificial love even to the point of death (Philippians 2:3-8). But Jesus is the visible image of the invisible God (Colossians 1:15), who is the exact representation of his character (Hebrews 1:3) and who shows the world what God is like (John 1:14, 18, 1 John 1:1-3). This means that, as counter-intuitive and countercultural as it sounds, God is humble and gentle and lowly and self-sacrificial and a servant. Humility is not usually an attribute people associate with God, but it is possible because of the Trinity! The Father doesn’t seek his own glory, but glorifies the Son. The Son doesn’t seek his own glory, but the glory of the Father (John 8:50, 54). The Spirit doesn’t glorify himself or speak on his own authority, but glorifies Jesus by declaring what belongs to him. And everything that the Father has, he gives away to the Son (John 16:12-16). This is because God is not just loving; God is love (I John 4:7-12). God is love, and love is fundamentally a generous, other-focused, giving away of oneself.
The term kenosis is often used to describe the “self-emptying” done by the Son when he became flesh and lived as a humble human (Philippians 2:7). Theologians debate the degree to which the Son gave up his divine rights, privileges, and power in becoming human, but the idea I think should be much less controversial and much more emphasized is that, whatever the Divine Son did, it must have been totally consistent with the character of God. If the Son emptied himself, this means that self-emptying is something God does. The members of the Trinity have been eternally giving themselves away and pouring themselves out for the others in an infinite flow of other-oriented love.
This is God’s way of life. This is the heart and soul of God. This is the pattern of His existence - His “Spirit”. God is Spirit, and those who worship him must worship him in this Spirit and according to this Love-centered Truth. That’s what God wants. He is seeking people to join in this glorious flow of self-emptying, other-focused, mutually glorifying love.
The meaning of life is to enter into this infinitely beautiful, gloriously selfless, perpetually unfolding relationship; to pour ourselves out so that we can be refilled by the endless Source of all good things; to participate in the family of God as his children who live in him and act like him.
So what does nilpotency have to do with any of this?
I said in the introduction that nilpotency is the idea that a mathematical object squares to zero. This doesn’t work for regular numbers (except zero), but it does work for matrices. For example, the matrix A does not equal zero, but using the process of matrix multiplication, A2 = 0
Mathematical operations can also be nilpotent. Imagine a computer where pushing the delete key moves a file one step towards removal (e.g., from the desktop → to the trash → to total removal). This means that deleting any file requires two applications of the delete key operation to fully delete the file, and after that, repeated presses do nothing. The first operation isn’t nothing; it changes the file’s state, but it takes two applications of the operation to delete the file and make the operation equal zero.
Nilpotency actually includes more than just squaring to zero since an element or operation x, is nilpotent if xn = 0. So you could have an operation that deletes a file after running three times, or four times, or whatever. Likewise, you could have matrix B, where B4 = 0. Multiplying this matrix by itself multiple times eventually results in a matrix with all zeros (its index of nilpotency is 4).
Or you could have a bigger, more complex matrix that takes more steps to get to zero:
You could imagine these kinds of nilpotent matrices as shifting data along a track until it falls off a cliff into nothingness. The internet tells me that you can also have nilpotent linear operators in a vector space, nilpotent elements of dual numbers, nilpotent elements of a ring in abstract algebra, nilpotent Grassmann variables, nilpotent Lie algebras, and more…but I confess I have no idea what any of that means other than that in all those cases, the object multiplied by itself or applied to itself vanishes. It’s not that the thing is eliminated by an outside force or process (like being multiplied by zero); it’s that its nature is such that it will eliminate itself if applied to itself. Because of this aspect, I tend to imagine nilpotency as the idea that something “eats itself”.
Another example is from geometry and topology. A boundary operator (∂2=0) is a mathematical tool that takes an object and gives you its boundary. But a boundary operator is nilpotent because nilpotency is also the idea that you can't have a boundary of a boundary. A 2D flat disk has a boundary that is a circle…but a circle doesn't have a boundary. A 3D ball has a two-dimensional sphere as its boundary (you could imagine it as being like the outside “skin” of the ball), but a sphere also doesn't have a boundary. The boundary of a line segment is two points, but points don't have boundaries. Nilpotency says that you can't have the edge of an edge.
To get a bit more philosophical with the logic of nilpotency, we could say that there are things or processes in the universe whose fundamental nature is that they will consume themselves. Or you might imagine a nilpotent object as having a trajectory that curves around on itself such that it crashes or spirals into zero. Nilpotency reflects the deeper truth that contingent things can’t be self-grounding. Nilpotency is like the second law of thermodynamics because, without some outside intervention, a system tends towards disorder and eventual annihilation. You, me, and the created universe are all “nilpotent”.
The Mirrored Truth
If the Trinity involves the fundamental and universal principle that other-oriented, self-emptying love leads to life, joy, and an infinite unfolding of complex beauty…nilpotency is the mathematical dual or mirror image of that reality. Nilpotency is the fundamental and universal principle that selfishness consumes itself. If the hyperset of Trinitarian perichorises involves the glorious participation in the Jesus way of self-emptying service and love, nilpotency describes the tragic reality that attempting to fill and serve ourselves leaves us alone and empty. One path revolves around and dives into the infinite beauty of the Ultimate Source. The other revolves around itself, spiraling into death.
But wait, didn’t I say that nilpotency was the meaning of life? That sounds a lot more like the meaning of death. Actually, the principle of love and the nilpotency principle are both the same principle, just described in different ways. In the hyperset model of the Trinity, each element (Father, Son, and Spirit) must flow through the other. The principle of nilpotency says that they would go to zero if they ever turned in on themselves (which obviously can’t happen), and it is this very principle that keeps the unfolding hyperset from splintering into three separate gods. There are real distinctions between the persons of the Trinity, but nilpotency prevents the distinctions from growing into differences in essence. There is a first-order distinction between the members of the Trinity (a kind of “boundary”), but there is no second-order separation (no “hard edges” between them - no “boundary of the boundary”). Nilpotency says that the Father, Son, and Spirit can be distinct, but they can not be independent, self-contained subjects. There can be no “self-begetting” or “self-glorifying” among members of the Trinity because the divine life is always oriented towards the other. The Father must pour himself into the Son, who must pour himself back to the Father through the Spirit. If any person of the Trinity turned inward on himself, God would cease to be God. God can’t be selfish because God is perfect, self-giving, other-oriented love, and that kind of perfect love can never be selfish.4
Nilpotency is the meaning of life because it is the principle of self-giving love stated negatively. Nilpotency says that we, as created beings, cannot be our own source or center. We cannot sustain ourselves. We cannot relate exclusively to ourselves. We were made to pour ourselves out in sacrificial love and service for the One who is Himself Sacrificial Love and Service.
We are all nilpotent elements orbiting around something. If we try to make that something ourselves, we will spiral into ourselves on a trajectory approaching zero. If we seek to climb over others to lift ourselves up or take from others to fill ourselves up, we will be left empty and alone. But if we lower ourselves, seeking to raise and serve others (as Jesus did), and if we generously pour ourselves out to fill others up (as Jesus did), we will find ourselves lifted and filled. We will find ourselves being served by the One whose very existence is this Spirit of lifting up the other in love. We will orbit around and through and in a glorious community that orbits around and through and in us. The nilpotency principle says that denying yourself and following Jesus is the counterintuitive trajectory that leads to life. Nilpotency says that if you love your life, you will lose it, but if you give up your life, you will gain eternal joy.
And nilpotency says that there is no evil in the universe that can endure forever. Nothing that rotates against the perfect, unstoppable gravity of God’s love can last beyond a few feeble nilpotent cycles - everything contrary to the love of God must eventually burn itself into nothing.
So, this is all just an elaborate metaphor, right?
I suppose that’s probably true. I’m no mathematician, and I don’t really know what I’m talking about. It’s possible that all my talk about the profound implications and esoteric meaning of nilpotency would be comparable to someone claiming there is some mysterious and spiritual significance to “addition” or “multiplication” simply because those ideas show up in a lot of areas of math and science. This might all just be silly musings from someone too ignorant to know better.
But I suspect there might be more to it. For example, nilpotency is also used in Synthetic Differential Geometry to define infinitesimals - tiny micro-segments of practically zero length. These infinitesimals are so small that they square to zero (𝛜2 = 0), but they are not themselves zero (𝛜 ≠ 0). This provides a different way to do calculus that uses these practically zero infinitesimals rather than limits. I think reality is better described in terms of these “nilsquare infinitesimals" than by the standard idea of infinite divisibility.5 I find the standard concepts of infinitely divisible lines and zero-dimensional points to be pretty philosophically suspect. How can a point of zero width, zero length, and zero volume actually “exist” in any meaningful way?
I also have a hunch that hypersets, nilpotency, and Synthetic Differential Geometry might provide solutions to some of the perennial challenges in physics.6 Nilpotency is already a standard tool in several areas of quantum field theory7, and I think it’s possible that the nilpotency principle might be the mechanism by which God sets boundaries and limits on what can happen in the universe.
Of course, I could be wrong about all of that. Maybe none of those ideas has anything to do with the meaning of life. But, either way, I’m pretty confident in the earlier ideas that God is fundamentally unselfish, that his essence is other-oriented self-sacrificial love, and that we can follow his example as we center our lives on this Jesus Way of being.
Orient your existence around Him, not around yourself.
That’s the meaning of life.
1 This one diagram contains within it twelve theological statements: "The Father is God", "The Son is God", "The Holy Spirit is God", "God is the Father", "God is the Son", "God is the Holy Spirit", "The Father is not the Son", "The Father is not the Holy Spirit", "The Son is not the Father", "The Son is not the Holy Spirit", "The Holy Spirit is not the Father", "The Holy Spirit is not the Son." BACK^
2 A hyperset is also called a Non-well-founded set and was described by mathematician Peter Aczel in the late 1980s BACK^
3 I’ll try to cover this in more detail in a future post, but for now you can just trust me that current math involves some very counterintuitive claims. See for example, Hilbert’s Hotel (a fully occupied infinite hotel can take infinitely more guests), Cantor’s theorem on cardinalities (some sets contain subsets larger than themselves), Skolem’s Paradox (a mathematical universe containing uncountable sets can be described using countably many elements), the Banach-Tarski paradox (a 3D ball can be disassembled into pieces and then reassembled into 2 balls the same size as the original), the space between .99 and 1 contains all the encodable information that exists or can exist, other paradoxes of completed infinity (Thomson’s Lamp, the Diary of Tristam Shandy, Ross-Littlewood, and more). BACK^
4 I’m not saying the Trinity is nilpotent; it’s not. I’m saying the relational nature of the Trinity causes this nilpotent principle to be true of contingent realities. Math doesn’t limit God; God grounds math. BACK^
5 In the earlier analogy of deleting a file from the trash, it’s possible that the information could still exist somewhere on your hard drive even after being deleted. This “ghost” of your deleted file seems like a pretty good analogy for a nilsquare infinitesimal because it’s practically gone, even if it’s not technically gone, and it will certainly be totally gone with one more deletion operation. BACK^
6 The Trinitarian Hyperset might explain why the fundamental symmetries of physics can be described using only the three Reidemeister moves. Synthetic Differential Geometry might resolve the singularities that arise in standard mathematics by describing a discrete limit to space that prevents the infinities that arise at zero volume. It may be the solution to the tension between the smooth/continuous spacetime of General Relativity and the quantized/discrete nature of quantum mechanics, since SDG is inherently smooth. BACK^
7 For example, Quantum Field Theory uses a nilpotent operator to describe the Pauli-Exclusion Principle, which governs the behavior of all matter. It also employs a nilpotent operator (BRST) to isolate real-world physics from unphysical degrees of freedom (i.e. incoherent mathematical noise). BACK^