Mach's Principle and an Algorithmic Theory of Subjective Valence: A Relational Approach

Feelings are relative — just like inertia — and this paper builds the math to prove it.

The central idea is borrowed from Ernst Mach, who argued that a body's inertia isn't an intrinsic property but emerges from its relationship to all the mass in the universe. The authors apply the same logic to subjective experience: how good or bad you feel right now isn't determined by some absolute internal score, but by how that score compares to your recent history. They call the raw internal score "pre-valence" — the unprocessed output of whatever objective function an agent (biological or artificial) is running — and "valence" is what you actually experience after your brain has contextualized it.

The transformation from pre-valence to valence has three moving parts. First, a dynamic baseline B(t) tracks a fading memory of past pre-valence, with a nonlinear twist: large transient spikes — positive or negative — punch above their weight in shifting the baseline. Second, a local variability measure σ(t) is computed on the detrended signal (long-term drift removed), capturing how noisy the recent environment has been. Third, the gap between current pre-valence and the baseline is divided by σ(t), then squeezed through a sigmoid so the final valence lives in [-1, 1]. A suppression factor λ compresses the range further to match the limited dynamic bandwidth of real neural coding. The whole system is governed by a single differential equation and a handful of timescale parameters.

The clinical hook is depression. In the model, major depressive disorder (MDD) corresponds to sluggish plasticity — a high τ, meaning the baseline B(t) updates too slowly. When something good happens, the baseline doesn't rise fast enough to register it as a positive deviation, so valence stays flat or negative. The simulation in the paper shows this directly: a healthy agent with fast plasticity bounces back after both a positive and a negative event; an MDD agent with slow plasticity gets stuck. The authors suggest that psychedelics may work therapeutically by temporarily lowering τ — restoring the brain's ability to recalibrate its baseline and feel things again.

This paper sits inside the authors' broader "Algorithmic Psychodynamics" framework, which treats consciousness and emotion as computational processes grounded in Kolmogorov complexity theory. The suppression factor λ is acknowledged as somewhat ad hoc — the paper offers three possible justifications (empirical calibration, neural dynamic range, theoretical constraints) but does not derive it from first principles. The philosophical survey of Mach, Leibniz, and Greek relativists is genuinely motivating rather than decorative, grounding the math in a long tradition of arguing that only relational quantities carry meaning.