The Core Insight
Consider √2.
It has infinite decimals. To multiply two numbers with infinite decimals should require infinite computation. Yet √2 × √2 = 2 comes for free. Why? Because the symbolic relationship—the Platonic structure—carries the answer. You never compute the decimals. The mathematical identity does the work.
Does nature exploit these shortcuts? Are there physical systems that "solve" hard mathematical problems not by computing, but by instantiating the structure?
The Answer: Yes, Everywhere
Soap bubbles find minimal surfaces—a brutal calculus of variations problem—in milliseconds. Mathematicians struggle with equations; the film just settles into the answer because surface tension is minimization.
Light finds the fastest path through different media, solving an optimization problem. But photons don't compute. Wave interference cancels all paths except the optimal one. The answer emerges from what waves are.
Slime mold builds networks approximating optimal transport solutions—problems that are NP-hard. It does this with simple local rules: reinforce what works, let the rest wither.
Proteins fold into precise shapes despite astronomically many possibilities. They don't search—they fall into the configuration where all forces balance.
Crystals solve packing problems that stump computers. Atoms just settle into what's stable.
The Unifying Principle: Feedback Loops
Every case has the same deep structure. Local elements interact through feedback. Some configurations reinforce themselves; others destabilize and dissolve. The system settles into a fixed point—and that fixed point is the solution.
The soap film's tension network, the slime mold's nutrient flows, the protein's atomic forces—all are parallel feedback systems searching for equilibrium. The "computation" is the dynamics playing out.
The Key Is Topology
Here's the crucial realization: which problem gets solved depends entirely on how the feedback loops are connected.
The soap film solves minimal surfaces because tension propagates equally in all directions across a 2D membrane. The slime mold solves network optimization because flow reinforcement creates competition between parallel paths. The crystal solves packing because neighboring atoms push back when they're too close.
Same principle—feedback seeking equilibrium—but different topologies yield different computations.
This means something profound: if you can design a system with the right feedback topology, you can get the solution to almost any problem for free.
This is exactly what analog computers did. Want to solve a differential equation? Wire up resistors, capacitors, and amplifiers so that voltage flows mirror the equation's structure. The circuit topology is the equation. Turn it on, and the system relaxes into the answer.
Want to find the shortest path through a network? Build a physical model where edges are resistors. Inject current at the start, drain it at the end. The current distribution is the solution—more current flows through shorter paths, automatically.
The free lunch is real, but it has a price: you must encode your problem into the topology of a physical feedback system. Get the topology right, and physics hands you the answer. Get it wrong, and the system solves a different problem—or no coherent problem at all.
The Deeper Implication
Computers do sequential symbol manipulation. Nature does something different: it builds systems whose stable states are the answers, then lets physics find them.
This suggests a different way to think about computation itself. Instead of asking "what steps solve this problem?" we can ask "what feedback topology has this problem's solution as its equilibrium?"
The first question leads to algorithms. The second leads to design—designing systems that fall into answers the way a ball falls into a valley.
Mathematics describes the structure. Physics is the structure.
Nature has been doing this for billions of years. We're just starting to learn the trick.