The Geometry of Illusion — Part II

The first essay ended with a question, not a ship: where can we place the fixed point from which a distortion is measured? The answer cannot be “inside the observer,” because the observer is precisely the instrument under examination. But “outside the observer” is not yet an answer either. Outside where? In another person’s judgment? In a laboratory result? In a theological claim? Each candidate carries its own rules of evidence.

Before deciding where the point belongs, we should ask what it is supposed to fix.

In geometry, a displacement is meaningful only relative to a coordinate system. Give me a position x and an apparent position x̂, and I can write an error: e = x̂ − x. Take away the independently established x, and I have two appearances, not an error. I can subtract them, of course. Mathematics allows subtraction with very little moral supervision. The result simply does not tell me which appearance is true.

That distinction explains why a Fata Morgana can be modeled but not decoded from its image alone. A ray-tracing model uses information about the atmospheric structure and the geometry of observation. The published optics literature shows how changing a refractive profile can change an image’s position and shape. It follows that a photograph, by itself, need not uniquely reveal the conditions that produced it. The “fixed point” in the physical case is not a metaphysical certainty. It is a collection of independently measured constraints. [1]

Now take the problem from the horizon to a conversation. Someone says, “You did well,” and I hear condescension. What is x here? The words are observable. The speaker’s intention is not directly visible. My interpretation is observable to me, but its truth is still at issue. There is no single scalar called what really happened from which I can subtract my feeling and obtain an angle of pride.

The initial essay wrote P = T_S(R): perception or interpretation P, reality R, and the observer’s condition S. The formula was explicitly philosophical. To develop it responsibly, we must expose an ambiguity it hides. P may mean a visual experience, a judgment about another person’s intention, or a ranking of what matters. These are different outputs. R may be a physical position, a recorded utterance, or a moral fact. They are different targets. If we let the letters slide between meanings, the equation becomes a little machine for manufacturing profundity while measuring nothing.

So let us narrow the claim. Suppose we want to know whether a person’s fear changes how often he judges an ambiguous remark to be hostile. We could present comparable remarks under defined conditions, record judgments before outcomes are revealed, and compare the rate of hostile interpretations. The measurable quantity would be a difference in classifications under those conditions. It would not be “the magnitude of fear in the soul.” It would not prove that the person is spiritually disordered. A psychological effect and a theological diagnosis do not share a unit.

This narrower approach has a philosophical consequence. The fixed point is often not one point. It may be a set of independent constraints: the exact words, the prior context, later conduct, another witness, and the speaker’s own account. None is infallible. Together they make some interpretations harder to maintain. Correction becomes possible without pretending to possess a view from nowhere.

There is a still deeper problem. Suppose I choose only the references that confirm my first interpretation. Then the coordinate system itself has been bent. Ziva Kunda’s account of motivated reasoning describes how desired conclusions can influence which beliefs and strategies are brought to bear on a question. [2] The danger is not merely that I calculate incorrectly. I may select the observations by which the calculation will later be judged.

The crooked ruler, then, is not just the mind that makes an error. It is the mind that quietly decides what counts as a ruler.

We do not solve that problem by declaring every judgment suspect. We solve a small, tractable part of it each time we specify the target, name the independent checks, and state what result would make us revise our view. The question from Part I remains open in theology and philosophy. In an empirical inquiry, however, it has at least this disciplined answer: place the reference where the hypothesis cannot move it after the measurement begins.

Notes on evidence

The equations in this essay define a possible measurement only when their terms are operationally specified; they are not laws of moral perception. The analogy between optical inversion and human interpretation is philosophical. The empirical claims concern ray-tracing models and research on reasoning, within the limits of their cited sources.

1. Siebren van der Werf, “Novaya Zemlya effect and Fata Morgana: Raytracing in a spherically non-symmetric atmosphere,” Comptes Rendus Physique 23 (2022), 365–389.
2. Ziva Kunda, “The Case for Motivated Reasoning,” Psychological Bulletin 108 (1990), 480–498.

Previously in The Geometry of Illusion: Part I.