8.2 Aesthetics as Guide: Why Do We Find Certain Physical Equations “Beautiful”? Because Beauty Is the Perfect Balance Between Low Computational Complexity (Simplicity) and High Logical Depth (Profundity). Beauty Is the Heuristic Search Function of Cosmic Computation.

In the history of science, a mysterious phenomenon repeatedly appears: Great physical laws are often “beautiful.” Einstein once said: “If a theory is not beautiful, it cannot be correct.” Dirac even believed: “Beauty in equations is more important than agreement with experiment.”

Why? In a universe composed of cold QCA logic gates, why is there “beauty” as a subjective feeling? Why can this feeling guide us to discover truth?

This section will propose a physics aesthetics theory: Aesthetics is not an arbitrary cultural preference, but an evolved “Heuristic Search Algorithm.” Its function is to help limited observers quickly identify candidates closest to the universe’s underlying rules $\hat{U}$ in the vast space of theories.

8.2.1 Computational Definition of Beauty: Tension Between Simplicity and Profundity

In computation theory, we can quantify two core dimensions of “beauty”:

1. Simplicity:

   Corresponds to Kolmogorov Complexity $K(T)$.

   The simpler a theory $T$ (shorter formula, fewer free parameters), the smaller its $K(T)$.

   • Physical correspondence: Occam’s razor. If $F=ma$ can explain motion, we don’t use $F=ma+b{x}^3$.

   • QCA foundation: The universe’s underlying rules $\hat{U}$ should be extremely simple (just a few lines of code). Therefore, theories close to truth must be simple.

2. Profundity:

   Corresponds to Logical Depth $D(T)$.

   Although a theory’s form is simple, the phenomena it can derive must be extremely rich and non-trivial.

   • Physical correspondence: From simple Maxwell’s equations, we can derive infinite phenomena like light, electromagnetic waves, magnetic fields.

   • Counterexample: An all-zero sequence is simple ($K\approx 0$) but not profound ($D\approx 0$), because it’s dead. White noise has rich content but is random, also not profound.

Definition 8.2 (Physical Beauty):

Beauty is the ratio of low complexity to high logical depth.

$$\text{Beauty}\propto \frac{\text{Logical Depth}(Output)}{\text{Kolmogorov Complexity}(Rule)}$$

The most beautiful theory generates the most complex universe with the fewest bits.

8.2.2 Aesthetics as Cognitive Shortcut

In a computationally irreducible universe, we cannot exhaustively search all possible theories to verify which is correct. Our computational power is limited.

For survival, evolution endowed us with an intuition that instantly produces pleasure (dopamine reward) when we see certain patterns. This intuition is aesthetics.

• Symmetry: We find symmetry beautiful.

  • Physical reason: Symmetry means elimination of redundancy. If a system is rotationally symmetric, I only need to store $1/360$ of the information to reconstruct the whole. Symmetry = high compression ratio = low $K$.

• Fractal: We find fractals beautiful.

  • Physical reason: Fractals mean recursive generation. A simple recursive formula $z\to z^2+c$ can generate the infinitely complex Mandelbrot set. Fractal = extremely simple rule + infinite depth.

Conclusion:

When we find an equation “beautiful,” it’s because our brain (as an efficient compression algorithm) recognizes that this equation has extremely high “compression ratio” and “generative power.”

Our brain subconsciously computes: “This theory is very likely the universe’s source code.”

8.2.3 Feynman’s Path Integral and Beauty’s Guidance

This mechanism explains why pursuing beauty leads to truth.

• The universe itself operates according to optimized algorithms (principle of least action).

• Our aesthetic instinct is a pattern recognizer isomorphic to the universe’s underlying logic, filtered through billions of years of evolution.

When we are moved by a theory’s “elegance,” that’s not emotional impulse; it’s resonance between the miniature universe within us and the grand universe outside.

Corollary:

Physicists’ work is essentially art appreciation.

They are not piling up data; they are searching for that rhyming poem.

Because only rhyming poems (self-consistent and beautiful theories) could be written by God.

(Section 8.2 Complete)