Appendix A: Aesthetic Computing — A Metric for Beauty based on Kolmogorov Complexity

In Chapters 4 and 8 of this book, we proposed that “aesthetics is a heuristic guide to truth.” This appendix aims to provide a quantitative mathematical framework explaining why certain structures (such as fractals, physical laws, artworks) are judged as “beautiful” by consciousness networks.

A.1 The Dilemma of Aesthetic Measurement

Traditional Shannon Entropy cannot measure beauty.

• Crystal (low entropy): $S\to 0$. Completely ordered, but dull.

• White noise (high entropy): $S\to max$. Completely random, but meaningless.

Beauty seems to exist in a critical region between “order” and “disorder.”

A.2 Birkhoff-Bennett Formula

George Birkhoff once proposed $M=O/C$ (aesthetic measure = order/complexity). In computation theory, we upgrade this to a formula based on Kolmogorov Complexity ($K$) and Logical Depth ($D$).

Let object $X$’s description be a binary string.

• $K(X)$: Length of the shortest program generating $X$ (compressed information content).

• $D(X)$: Number of logical steps required to run that shortest program to output $X$ (computation time).

Definition A.1 (Aesthetic Value Function $\mathcal{A}$):

$$\mathcal{A}(X)=\frac{\mathcal{D}(X)}{K(X)}\times \text{Resonance}(X,{\mathcal{M}}_{observer})$$

1. Simplicity Benefit ($1/K$):

   Our brains prefer high compression ratios. If a complex phenomenon $X$ can be explained by a short law (like $F=ma$), the brain saves enormous storage energy, producing a sense of “elegance.”

   • Example: Fractals are beautiful because the code generating them $z\to z^2+c$ is extremely short ($K$ small), but the generated images are infinitely rich.

2. Profundity Benefit ($D$):

   If an object has short code but an extremely trivial decompression process (like printing a million “A“s), it is boring.

   Only when the decompression process involves non-trivial computation (like life evolution, story development) does it have logical depth.

   • Example: A Taihu stone weathered over hundreds of millions of years, its form contains a long history of fluid dynamics computation ($D$ large).

3. Resonance Correction:

   The $\text{Resonance}$ term depends on the observer’s internal model $\mathcal{M}$. Aesthetics are activated only when object $X$’s topological structure undergoes Homology with the observer’s mental structure.

Conclusion:

Beauty = Encapsulating the longest history with the fewest bits.