Measurement, as a principle, is always a ratio of two quantities. In physics, you might measure speed as distance divided by time; in our framework, the Reality Equation measures surprise as the ratio of Actual to Expected outcomes:
Here, \(A\) is the Actual outcome, and \(E\) is the Expected outcome. This ratio tells us, in pure proportional terms, how reality compares to prediction.
Why the natural log belongs
Ratios compound multiplicatively over time. To make them additive — so that multiple experiences can be summed — we take the natural log. This follows the same logic Claude Shannon used in defining self-information in 1948:
Shannon’s formula for the information content of an event with probability \(p\) is:
By identifying the Reality ratio \(\frac{A}{E}\) with an odds-like factor, we use the log to translate multiplicative differences into an additive measure of surprise.
: Positive (pleasant) surprise
: No surprise
: Negative (unpleasant) surprise
Logs naturally match continuous growth/decay, entropy, and information gain — all of which align with our scale for experience.
“Self-information”
— C. E. Shannon, A Mathematical Theory of Communication (1948)
The Complex Form of Expectation
We model the Expected outcome \(E(t)\) as a complex signal with two components:
- \(\rho(t)\) — magnitude (prediction confidence)
- \(\theta(t)\) — phase (idea influence)
The Reality ratio in this form is:
Four Classroom-Useful Readouts
- Surprise:
- Phase (side):
- Intensity rate:
- Angular velocity:
Desire and the Idea’s Influence
An Idea has an Ideal outcome \(I\) — the imaginary component of the Expected outcome — which may differ from Actual \(A\). If Actual is not Ideal, the Idea works to adjust the Predicted outcome (real component of \(E\)) so that \(E\) is more like \(I\) than \(A\).
This substitution — replacing \(A\) with \(I\) in the numerator — creates the Desired Outcome:
The measurable influence of the Idea on its host is then:
When \(\Delta S\) is large, the Idea’s pull on the host is strong; when small, the Idea’s influence is weak.
Why This Matters
By combining the Reality ratio with the natural log, we can measure surprise in a way that is both additive and meaningful. Adding the dimension of Desire lets us quantify not just what happened versus what was expected, but how strongly an Idea tries to bend the host’s perception of reality toward its own ideal.