This episode features mathematician Eva Miranda presenting new results connecting computability theory, fluid dynamics, and geometry, building on ideas from Alan Turing, Roger Penrose, and Terence Tao. The central theme is the discovery of “logical chaos” — a form of unpredictability rooted not in sensitivity to initial conditions (as in classical chaos), but in mathematical undecidability, where no algorithm can determine the outcome of a physical process even in principle.
From Hilbert’s Dream to Turing’s Limits
David Hilbert championed the idea that all mathematical questions should have definitive answers — “We must know, we will know.” This vision of formalization drove early 20th-century logic.
Alan Turing shattered this optimism in 1936 by proving the halting problem is undecidable: there is no general algorithm that can determine whether an arbitrary computer program will eventually stop or run forever.
This means the answer may exist in some abstract sense, but there is no logical or computational procedure to find it in finite time — a fundamental limit on knowledge itself.
In proving this, Turing invented the Turing machine, the theoretical foundation of all modern computers.
Gödel’s incompleteness theorems (1931) showed that any sufficiently powerful mathematical system contains true statements that cannot be proven within that system — another blow to Hilbert’s program.
These results reveal that mathematics and computation have inherent limits: some questions are not just practically hard, but logically impossible to decide.
Classical Chaos vs. Logical Chaos
Classical chaos (the “butterfly effect”) arises from extreme sensitivity to initial conditions: tiny changes in starting state lead to vastly different outcomes over time.
Discovered by Edward Lorenz in meteorology; popularized by his 1972 talk: “Does the Flap of a Butterfly’s Wings in Brazil Set Off a Tornado in Texas?”
This unpredictability is practical, not fundamental — with perfect measurements and infinite precision, the future would be determined.
Real-world examples include asteroid trajectory predictions (e.g., the 2024 YR4 scare) and weather forecasting limits (~7–10 days).
Logical chaos is fundamentally different: it arises when a physical system is Turing-complete, meaning its evolution can simulate any computation.
For such systems, predicting whether a trajectory enters a certain region becomes equivalent to solving the halting problem — which is undecidable.
This is not due to lack of information or measurement error, but a barrier inherent in logic itself.
The Rubber Duck Metaphor
In 1992, a cargo ship lost 29,000 rubber ducks in the Pacific Ocean. Oceanographers like Curtis Ebbesmeyer and James Ingraham used their dispersal to study ocean currents.
Despite sophisticated models, most ducks were never recovered — only about 2% of tracked bottles were found.
Miranda uses these ducks as a metaphor: finding them may be an undecidable problem if their motion encodes a Turing machine.
The rubber duck becomes the mascot of logical chaos, just as the butterfly represents classical chaos.
Can Fluids Compute? The Work of Chris Moore and Terence Tao
In the 1990s, mathematician Chris Moore asked: Can physical systems like fluids perform arbitrary computations?
He showed that a certain transformation of the square Cantor set (a fractal constructed by repeatedly removing middle thirds of intervals) can simulate any Turing machine.
Specifically, he encoded the state of a Turing machine’s tape (a sequence of 0s and 1s) as coordinates in the Cantor set using ternary expansions.
The dynamics of the Turing machine correspond to a discontinuous map on the Cantor set — a “puzzle” whose solvability is equivalent to computation.
Terence Tao later asked a sharper question (2019): Can we find solutions to the Euler equations (ideal fluid flow) that are Turing-complete?
His motivation was to explore whether such computational power could lead to finite-time blow-up — a singularity where fluid velocity becomes infinite — which would resolve one of the Clay Mathematics Institute’s Millennium Prize Problems on Navier-Stokes equations.
Tao had found blow-up in an averaged version of Navier-Stokes, but not in the true equations. He speculated that embedding a Turing machine in the initial conditions might force blow-up.
Constructing a “Fluid Computer” via Geometry
Miranda and collaborators (Daniel Peralta-Salas, Robert Cardona, Francisco Presas) answered Tao’s question yes in 2020: they constructed Turing-complete solutions to the 3D Euler equations.
Their method bridges discrete computation and continuous fluid flow using contact geometry — the odd-dimensional cousin of symplectic geometry.
Key idea: Use a Poincaré section — a 2D plane that a 3D trajectory intersects repeatedly. The sequence of intersection points encodes the Turing machine’s state.
They extended Moore’s Cantor set map to a smooth, area-preserving map on a disk, then embedded it as the return map of a stationary Euler flow (a Beltrami field).
A deep result (Grigori Perelman, John Etnyre, Robert Ghrist) provides a “magic mirror” between Beltrami fields (solutions to Euler) and Reeb vector fields (geometric objects in contact geometry), allowing them to translate the construction into fluid dynamics.
Corollary: There exist undecidable fluid paths — no algorithm can determine whether a particle in such a flow will ever enter a given region of space.
This is the first example of logical chaos in fluid dynamics.
Beyond Euler: Turing-Complete Navier-Stokes and Hybrid Computers
The original construction used Euler equations (zero viscosity). The Navier-Stokes equations (with viscosity) are more physically realistic and the subject of the Millennium Problem.
A result by Bournais, Graça, and Henry shows that adding viscosity destroys Turing-completeness under perturbation — suggesting Navier-Stokes might not support such computational universality.
However, Miranda announces a new result (presented here for the first time): using cosymplectic geometry, they have constructed Turing-complete solutions to the full Navier-Stokes equations.
This does not yet prove blow-up, but opens a new pathway toward that goal.
Even more recently (March 2025), Miranda, Ángel González-Prieto, and Peralta-Salas introduced a hybrid computer model combining:
Fluid computation (Euler/Navier-Stokes flows as “fluid qubits”)
Topological quantum field theory (TQFT) tools to assemble and entangle these components
This model is more powerful than either classical or quantum computing alone, and they are investigating whether it can achieve quantum supremacy — or surpass it.
Simulations are underway, with interest from Stephen Wolfram.
Open Questions and Future Directions
Can logical chaos explain undecidability in celestial mechanics? For example, is it undecidable whether an asteroid will hit Earth? Miranda speculates that combining classical chaos (butterfly) with logical chaos (rubber duck) in systems like the three-body problem could reveal deeper layers of unpredictability.
Entropy serves as a bridge between classical and logical chaos, measuring disorder in both contexts, but the full relationship remains unclear.
Do these models violate the Church-Turing thesis? No — the thesis concerns what is computable in principle, not efficiency. The hybrid computer aims for speed, not new computability.
The ultimate goal remains: resolving the Navier-Stokes blow-up problem, potentially winning the $1 million Clay Prize — though Miranda emphasizes the journey matters more than the reward.
