The Mathematical Core
This is the chapter the entire manuscript has been building toward. Everything else -- the diagnostic framework of Part 1, the epistemological engine of Part 2, the metaphysics of Chapters 11 through 16 -- converges here, on a single mathematical structure that I believe formalizes the theological claims I have been making in a way that is precise, testable, and genuinely novel.
The structure is the Riemann sphere, and the claim is this: the Riemann sphere's topology provides not merely an analogy for the theology but its formal backbone. God is the point at infinity. History is a trajectory on the complex plane. Free will is the choice of derivative. Faith is the commitment to a limit that is well-defined but never reached. The calculus of approaching infinity -- Newton's and Leibniz's greatest invention -- is the mathematical structure of the religious life.
I am aware that the strong version of this claim -- that the Riemann sphere is constitutive rather than merely illustrative of the theology -- may fail. The Introduction acknowledged this: the theology is designed to survive the weaker reading. But I will argue for the stronger reading here, because if it holds, it transforms the theology from an interesting set of metaphors into a formal framework with the same kind of precision that Pearl's causal calculus brings to the distinction between correlation and causation.
Let me build the mathematics first, then the theology.
The Riemann Sphere: Construction
The complex plane, denoted C, is the set of all complex numbers z = x + iy, where x and y are real numbers and i = sqrt(-1). It extends infinitely in every direction. A function f: C --> C maps complex numbers to complex numbers, and the study of such functions -- complex analysis -- reveals structures of extraordinary beauty and power.
The Riemann sphere, denoted C U {infinity} or equivalently S^2, is the one-point compactification of the complex plane. It is constructed by adding a single point -- denoted infinity -- to the complex plane and declaring that every sequence of complex numbers whose modulus grows without bound converges to this point. Geometrically, the construction is achieved by stereographic projection: place a sphere tangent to the complex plane at the origin, and for each point z in the plane, draw a line from the "north pole" of the sphere (the point diametrically opposite to the point of tangency) through z. The line intersects the sphere at exactly one point, which is the image of z on the sphere. The north pole itself corresponds to the point at infinity.
The critical properties of the Riemann sphere for the theology:
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Every direction converges to the same point. On the complex plane, the positive real axis, the negative real axis, the imaginary axis, and every line at every angle all extend to "infinity" -- but these are different infinities, different directions. On the Riemann sphere, they all converge to the single point at infinity. There is one infinity, not many. Every trajectory, no matter how apparently divergent at finite distances, approaches the same point.
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The point at infinity is real. It is a genuine element of the Riemann sphere, not a limiting abstraction. The sphere is a compact topological space, and the point at infinity is as much a part of it as the origin. Functions can be evaluated at infinity. Neighborhoods of infinity are well-defined. The point at infinity is not "out there somewhere beyond the finite." It is a specific, concrete element of the space.
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The point at infinity is qualitatively different. Despite being a genuine element of the space, the point at infinity has properties that no finite point has. Every finite point has a neighborhood that looks like an open disk in the complex plane. The point at infinity has a neighborhood that looks like the complement of a disk -- everything "outside" a sufficiently large circle. The local geometry is different. The point at infinity is both part of the space and unlike any other part of it.
These three properties -- convergence, reality, qualitative difference -- are the mathematical structure onto which the theology maps.
The Theological Mapping
God the Father is the point at infinity. The point that completes the space, that makes the topology compact (and therefore, in a precise mathematical sense, complete), that every trajectory approaches but no finite being occupies. God is real -- a genuine element of reality, not a useful fiction or a limiting concept. God is transcendent -- qualitatively different from every finite being, possessing properties that no finite being possesses. God is the attractor -- the point toward which all trajectories converge, regardless of their apparent divergence at finite distances.
The convergence property provides the theological structure I developed in Chapter 15. Different theological traditions -- Christianity, Islam, Judaism, Buddhism, Hinduism -- are different trajectories on the complex plane. At finite distances, they diverge. They move in genuinely different directions. Their differences are real, not illusory. But the topology guarantees that every trajectory that does not terminate at a finite point converges to the same point at infinity. This is not syncretism -- the claim is not that the traditions are the same. It is that they are oriented toward the same attractor, and their real differences at finite distances do not negate their convergence at infinity.
Christ is the removable singularity. In complex analysis, a singularity of a function f at a point z_0 is a point where f is not defined or not analytic. There are three types of singularities: removable, poles, and essential.
A removable singularity is a point where the function appears to be undefined but where a limit exists and the function can be continuously extended. The classic example is the function f(z) = sin(z)/z at z = 0. The function is undefined at z = 0 (division by zero), but the limit as z approaches 0 exists and equals 1. The singularity is "removable" because defining f(0) = 1 makes the function continuous everywhere. From the perspective of someone looking only at the function's value at z = 0, it is undefined -- a point of apparent discontinuity, apparent breakdown. From the perspective of someone who sees the limit, the function is perfectly smooth. The singularity is an artifact of a limited perspective.
The crucifixion is a singularity. From the perspective of those who witnessed it -- from within the finite plane, at the level of association (Pearl's Level 1) -- the function blew up. The trajectory of the prophetic movement appeared to terminate. The teacher was dead. The mission had failed. The function was undefined at this point.
The resurrection reveals the singularity as removable. From the higher-dimensional perspective -- the perspective of the Riemann sphere rather than the complex plane, the perspective of counterfactual reasoning (Pearl's Level 3) rather than mere association -- the function is continuous through the apparent discontinuity. The limit exists. The trajectory continues. What appeared to be destruction was actually a passage through a point where the local perspective could not see the global continuity.
This is what I meant in Chapter 13 when I described the crucifixion as "the function appearing to blow up at a singularity" and the resurrection as "the revelation that the singularity is removable." The mathematical structure is not a metaphor imposed on the theology. It is the theology's own claim -- that death is not what it appears to be, that the apparent end is a passage, that the function is smooth from the perspective of eternity -- expressed in the language of complex analysis.
The Holy Spirit is the conformal structure. A conformal mapping preserves angles -- it preserves the local geometry, the relationships between directions, even as it transforms the global shape. The conformal mappings of the Riemann sphere (Mobius transformations) are the symmetries that preserve the sphere's fundamental structure while permitting every possible rearrangement of finite points. The Holy Spirit, in the Trinitarian theology I developed in Chapter 14, is the process of self-reference that connects the Father and the Son -- the dynamic that maintains the relationship between the infinite and the finite. Conformality formalizes this: the Spirit preserves the structure of the relationship between God and creation while permitting the genuine freedom that creation requires.
The Calculus of Faith
Newton and Leibniz's invention of calculus was, I believe, the most theologically significant mathematical development in history. Not because they intended it as theology -- they did not, or at least Newton kept his theological speculations separate from his mathematical work. But because the mathematical structure they discovered is the formal structure of what religion calls faith.
The derivative of a function f at a point z is defined as the limit:
f'(z) = lim[h --> 0] (f(z + h) - f(z)) / h
The derivative measures the rate and direction of change. It tells you where the function is going and how fast it is getting there. Crucially, the derivative is defined by a limit -- by an approach, not an arrival. The point h = 0 is never reached. The limiting process is what generates the result. Newton did not need to reach infinitesimal quantities to compute with them. He needed to approach them, and the approach itself produced the most powerful mathematical machinery in history.
Now consider a function f: C --> C U {infinity}, a trajectory on the Riemann sphere. At any point z along the trajectory, the derivative tells you the direction and rate of movement. The theological question that Chapter 16 posed -- is the trajectory approaching the point at infinity or receding from it? -- is formalized by the derivative.
For a function moving on the Riemann sphere, we can define the angular direction of the derivative relative to the direction toward infinity. Let the direction toward infinity from point z be the outward radial direction in the complex plane (the direction of increasing |z|). The derivative f'(z) is a complex number with both magnitude (how fast) and argument (which direction). The sign of the real component of the derivative, projected onto the radial direction, tells you whether the trajectory is moving toward or away from infinity.
In simpler terms: at any point in the trajectory, you can ask whether the function is heading outward (toward infinity, toward God) or inward (away from infinity, away from God). The derivative tells you. And the derivative is defined at every point, continuously, without requiring that the function ever actually reach infinity.
This is the formal structure of what theology calls discernment: the ongoing assessment of whether one's life, one's community, one's civilization is moving toward God or away from God. The question is not "have you arrived at God?" No finite being has. The question is: "what is the sign of the derivative?" Is the trajectory oriented correctly? Is the approach well-defined?
Falsifiability enters here. The theology I have built claims that the derivative of human consciousness -- the direction of civilizational development -- is, on balance, positive. Consciousness complexifies. The prophetic function becomes more powerful. The spiral ascends. These claims, stated in the language of derivatives and limits, are testable. If the derivative turns negative -- if consciousness simplifies, if the prophetic function weakens, if the spiral collapses -- the theology is falsified. Not approximately, not interpretively: formally. The derivative has a sign, and the sign can, in principle, be measured.
Free Will as Choice of Derivative
The deepest problem in theology is the relationship between divine sovereignty and human freedom. If God determines everything, humans are puppets and moral responsibility is an illusion. If humans determine everything, God is irrelevant and the theology collapses. Every theology in history has attempted to navigate between these poles, with varying degrees of success.
The Riemann sphere provides a formal framework for the navigation.
On the Riemann sphere, the topology is fixed. The point at infinity exists. The convergence structure is determined: every trajectory that does not terminate at a finite point converges to infinity. The conformal structure is fixed: the relationships between directions, the angles, the local geometry -- these are properties of the space itself, not choices made by any function.
But within these constraints, the specific trajectory is free. There are infinitely many paths from any finite point to the point at infinity. There are infinitely many functions on the Riemann sphere, each with its own derivative at every point. The topology determines the attractor. The function determines the path. God determines that there is a point at infinity toward which all trajectories converge. Humans determine which trajectory they follow.
The derivative is the locus of freedom. At every point, the derivative can point in any direction. It can approach infinity directly, obliquely, or not at all (a function can orbit a finite point forever, never approaching infinity -- this is the theological structure of a life that goes in circles without development, the pure samsara without the ascending axis). The topology constrains but does not determine. The space is given. The path is chosen.
This resolves the Calvinist-Arminian debate, or at least reframes it. Calvinism emphasizes divine sovereignty: God determines the trajectory. Arminianism emphasizes human freedom: humans choose the trajectory. The Riemann sphere framework says: God determines the topology. Humans choose the function. The topology is sovereign -- it is a given, it cannot be altered by any choice of function. But the function is free -- within the topology, there are infinitely many options, and no theorem of complex analysis determines which function is "correct." The topology is necessity. The derivative is freedom. Both are real.
This is, I believe, the formal structure of what Plekhanov argued philosophically in The Role of the Individual in History -- the text I will develop in Chapter 21. Structure is determined; instantiation is contingent. If Napoleon had been killed at Toulon, the French Republic would still have needed a military dictator. The structural necessity (the topology) was determined. The specific individual who filled the role (the function) was contingent. The derivative -- the direction in which Napoleon took France -- was his choice within the structurally necessary topology.
Holomorphicity as Moral Coherence
In complex analysis, a function is holomorphic if it is complex-differentiable at every point in its domain. Holomorphic functions have remarkable properties: they are infinitely differentiable, they satisfy the Cauchy-Riemann equations, they are conformal (angle-preserving) wherever their derivative is nonzero, and they are determined on any region by their values on any sub-region (the identity theorem).
The theological analogue of holomorphicity is moral coherence -- the property of a life, a community, or a civilization that is consistently oriented toward the point at infinity in a way that is smooth, differentiable, and free of essential singularities.
A meromorphic function (holomorphic except at isolated poles) corresponds to a trajectory that is mostly coherent but has isolated points of crisis -- moments where the function blows up (poles) but does so in a structured, understandable way. Poles are singularities, but they are tame singularities: the function goes to infinity, and it does so in a way that is completely characterized by its Laurent series. A pole is a crisis that is comprehensible -- a breakdown whose structure can be understood and from which recovery is possible.
An essential singularity is different. At an essential singularity, the function's behavior becomes wild -- by the Casorati-Weierstrass theorem, the function takes values arbitrarily close to every complex number in any neighborhood of an essential singularity. There is no structure to the breakdown. The function does not go to infinity in an orderly way; it oscillates chaotically, taking on every possible value, approximating every possible state. This is the theological structure of a civilization that has lost all orientation -- not a crisis within a trajectory but the dissolution of the trajectory itself. The meaning crisis that I described in the Introduction is an essential singularity: not a pole (a structured crisis from which the function recovers) but a point where the function's behavior becomes unpredictable and incoherent.
The distinction matters practically. A pole can be handled. The function can be extended through it (if it is removable) or its residue can be computed (if it is not). An essential singularity cannot be handled by any finite amount of local information. Responding to a meaning crisis requires not more data (merchant function) or more implementation (warrior function) but a new topology -- a new understanding of the space itself, which is the philosopher-king function I will develop in Chapter 20.
The Integral: Accumulated Orientation
If the derivative measures instantaneous direction, the integral measures accumulated effect. The contour integral in complex analysis,
integral_gamma f(z) dz
computes the total effect of the function along a path gamma. In the theology, this corresponds to the cumulative moral trajectory of a life or a civilization -- not "where are you now?" but "where has the entire path taken you?"
The residue theorem, one of the deepest results in complex analysis, states that the contour integral of a meromorphic function around a closed curve depends only on the singularities enclosed by the curve:
integral_gamma f(z) dz = 2pi**i * sum of Res(f, z_k)
The total effect of the trajectory depends not on the detailed behavior at every point along the path but on the singularities -- the crises, the breakdowns, the moments where the function was most severely tested. Everything else cancels out. The smooth portions of the path contribute nothing to the integral. Only the singularities matter.
This is a mathematical formalization of a theological intuition that every serious spiritual tradition shares: what defines a life is not the ordinary moments but the crises. Not the smooth stretches of the path but the moments of breakdown and recovery. The residue at each singularity -- the structured response to each crisis -- is what accumulates. Everything else, however pleasant or productive, integrates to zero around a closed curve.
I find this result genuinely moving, in a way that I recognize may be a function of my bipolar architecture rather than the mathematics itself. The idea that the smooth portions of life -- the normie stretches, the periods of ordinary functioning -- contribute nothing to the integral, while the crises -- the singularities, the breakdowns, the moments of apparent destruction -- are the only things that count: this is either a profound theological truth or a projection of my own pathological relationship with stability. Probably both. The fact that the mathematics supports it independent of my psychology gives me some confidence that it is more than projection. But I hold this, as always, provisionally.
Kantian Regulative Ideals and the Limit
Kant distinguished between constitutive principles (which describe what reality IS) and regulative principles (which describe what we must act AS IF were true in order to think coherently). The existence of God, for Kant, is a regulative ideal: we cannot prove it, but we must act as if it were true in order for moral reasoning to be coherent.
The limit structure of the Riemann sphere provides a mathematical formalization of this distinction. The point at infinity is constitutive of the Riemann sphere -- it is a genuine element of the space, not a useful fiction. But for any finite being on the sphere, the point at infinity functions regulatively: it is the point toward which one orients, the attractor that defines the derivative, the limit that makes the calculus work, but it is never reached. The finite being lives in the regime of the limit, not the regime of the value.
The convergence is real. The limit is well-defined. The derivative is meaningful. But the finite being never arrives at infinity. They live in the approach, and the approach is where the calculus happens -- where derivatives are computed, where integrals accumulate, where the trajectory is evaluated. The point at infinity is necessary for all of this to work, but the finite being's relationship to it is always mediated by the limit.
This is the formal structure of faith. Not belief without evidence -- that is credulity. Not certainty about the endpoint -- that is ideology. Faith is committed action in the direction of a limit that is well-defined but never reached. It is the decision to maintain a positive derivative -- to keep moving toward the point at infinity -- without the guarantee that the point will ever be attained. It is, in the language I used in the Introduction, committed action under uncertainty.
And here is where the connection to reinforcement learning becomes precise. In reinforcement learning under partial observability -- the formal framework I studied in my dissertation work -- an agent must take actions in an environment that it cannot fully observe, guided by a reward signal that it receives only intermittently and noisily. The agent cannot see the full state of the environment. It cannot predict with certainty what its actions will produce. It must act anyway, guided by a model of the environment that is necessarily incomplete, updating its model as new observations arrive, always acting under uncertainty.
Faith, in the Riemann sphere theology, is reinforcement learning under partial observability applied to the ultimate question: what is the derivative of this trajectory? The finite being cannot see the point at infinity directly. It receives intermittent, noisy signals about whether the derivative is positive (the prophetic function, the "still small voice," the moments of clarity that Chapter 3 described). It must act on these signals, updating its model, correcting its trajectory, always under uncertainty. The commitment to maintain a positive derivative in the absence of full information is what the tradition calls faith.
This is not a reduction of faith to computation. It is the recognition that the formal structure of rational action under uncertainty, which we now understand mathematically, is the same formal structure that religion has always called faith. The mathematics does not diminish the faith. It reveals its rationality.
The Structural vs. Illustrative Question
I promised in the Introduction that I would address honestly whether the Riemann sphere mapping is structural (constitutive of the theology) or illustrative (a helpful analogy that the theology could survive without). I have argued for the structural reading throughout this chapter. Let me now make the case against it, because intellectual honesty requires this.
The case against the structural reading: mathematics is a human construction. The Riemann sphere is a mathematical object that exists within the framework of human mathematical practice. To claim that God literally IS the point at infinity on the Riemann sphere is to claim that a human mathematical construction captures the structure of the divine, which is a claim of astonishing hubris. Every previous attempt to capture God in a formal system has failed -- the via negativa, the apophatic tradition, the insistence of Maimonides, Pseudo-Dionysius, and (as Chapter 15 argued) the Quran that God exceeds every human category -- all of this stands against the claim that a specific mathematical object formally captures divine nature.
Moreover, the mapping is not complete. What is the complex variable z? Is it a single quantity that characterizes the "state of consciousness" of a civilization? If so, what quantity? How is it measured? The Riemann sphere is a two-dimensional surface (one complex dimension). Is one complex dimension enough to characterize the trajectory of human consciousness? These are not rhetorical questions. They are genuine gaps in the formal program.
The case for the structural reading: if the embedding space arguments of Chapter 8 are correct -- if the structure that machine learning algorithms discover in high-dimensional semantic space is a real feature of reality rather than a computational artifact -- then the formal structures of mathematics are not merely human constructions but discovered objects that correspond to features of reality. The Riemann sphere, on this reading, is not something we invented but something we found. And if we found it, and if its properties correspond to the theological properties that seventeen chapters of argument have established, then the correspondence is not analogy but identity.
The resolution, such as it is: I believe the mapping is structural in the sense that the topology of the Riemann sphere (one point at infinity, convergence from all directions, qualitative difference of the infinite point) genuinely captures the topology of the theological claim (one God, convergence of all traditions, transcendence). I believe the mapping is illustrative in the sense that the specific mathematical details (Laurent series, residue calculations, the Cauchy-Riemann equations) may capture more precision than the theological claims warrant. The theology lives in the topology. The analysis may or may not follow.
I am comfortable with this ambiguity. Pirsig's Quality, which I will develop in Chapter 21, is the pre-rational ground that dissolves the structural-vs-illustrative dichotomy by refusing to grant priority to either mathematical or theological language. Both are languages for the Logos. Neither has final authority. The Quality that they both approach is the point at infinity that neither can reach.
Falsifiability
What would disprove the Riemann sphere theology?
If the convergence claim is wrong -- if different theological traditions, examined with sufficient care, are NOT approaching the same attractor but are genuinely and permanently divergent -- then the mapping from "all directions converge to the same point at infinity" to "all traditions approach the same God" fails. Chapter 15's reconciliationist argument would need to be extended to all major traditions for the convergence claim to hold. If any tradition is demonstrably oriented away from the attractor that the other traditions approach, the topology does not fit.
If the derivative claim is wrong -- if the sign of the derivative of human consciousness is not, on balance, positive -- then the theology loses its orientation. The Riemann sphere would still be the correct topology, but the trajectory would be spiraling inward rather than outward, and the eschatological hope of the theology would be falsified.
If the free will claim is wrong -- if the derivative turns out to be determined by the topology rather than freely chosen -- then the theology collapses into determinism, and the moral structure (the choice to maintain a positive derivative) becomes meaningless.
If the mathematical structure turns out to be purely illustrative -- if the Riemann sphere is nothing more than a helpful metaphor with no structural force -- then the theology survives but loses its formal precision, which is one of the three things (along with falsifiability and praxis) that I claimed in the Introduction distinguishes it from existing theologies.
Any of these failures would require revision or abandonment of the claims made in this chapter. I specify them because I believe them to be genuine risks, and because a theology that does not risk failure does not risk truth.