matharium

Terence Tao says that the AI proof on the Navier–Stokes problem is correct, yet mathematicians learned almost nothing from it. The proof is 100 pages, AI-generated and full of issues in its text, but a formal certificate in the language Lean confirms it.

Navier–Stokes is a Millennium Prize problem that asks whether the equations for fluids like water can start smooth and then blow up, with velocity or pressure becoming infinite. Human mathematicians came very close, and OpenAI ran its biggest model, with millions of dollars of compute, to get there first.

A human proof normally comes with talks on what was tried, what worked and what it connects to, like picking fruit and planting the seeds. Tao calls this one strip mining: the fruit taken, the landscape left barren. One colleague, using lots of AI, could explain only half of it after days of work.

A handful of symbols carry most statements in logic and set theory. ∀ means for all, ∃ means there exists, and ∃! means there exists exactly one.

The bar | reads such that, ∈ means is an element of, ∧ is and, ∨ is or, ⊆ is subset of and ∪ is union. Combined, they turn a sentence into a precise condition: {x ∈ ℝ | x > 0} is the set of positive real numbers, and ∀ε > 0 ∃δ > 0 begins the definition of a limit.

The notation grew from the work of Peano, Frege and Russell around 1900. The same operators appear in database queries, where ∧ and ∨ become AND and OR, and in formal verification of software.

Integration adapts to every dimension and setting while keeping one idea: add up a quantity over a set. The set can be an interval, a region, a solid, a curve, a surface or a general measure space.

A single integral gives area under a curve, a double integral gives volume over a region, and a triple integral totals a quantity inside a solid. Line integrals ∫_C F · dr give work along a path, surface integrals ∬_S F · dS give flux through a surface, and the Lebesgue integral ∫ f dμ integrates with respect to a measure.

The Lebesgue integral, introduced by Henri Lebesgue in 1902, is the foundation of modern probability, where an expected value is an integral against a probability measure.

Chaos is deterministic behaviour that cannot be predicted in practice. The Lorenz system, ẋ = σ(y − x), ẏ = x(ρ − z) − y, ż = xy − βz, has no randomness in it, yet with σ = 10, ρ = 28 and β = 8/3 its trajectories never settle and never repeat.

The reason is sensitivity to initial conditions. Two starting points a tiny distance d₀ apart separate roughly as d(t) ≈ d₀e^(λt), where λ > 0 is the Lyapunov exponent. On a log scale the gap grows as a straight line until it reaches the size of the attractor. Any measurement error, however small, eventually swamps the prediction.

Edward Lorenz found this in 1963 in a simplified model of convection, which is why weather forecasts lose skill after about two weeks. The Rössler attractor, Chua's circuit and the logistic map at r = 3.9 show the same behaviour.

Many flowers with six petals or tepals follow the geometry of a regular hexagon. Mark six points 60° apart on a circle, join every second one, and two equilateral triangles appear, forming a six-pointed star inside the hexagon.

The numbers follow from the circle. The central angle is 360°/6 = 60°. Each side of the hexagon spans a 60° arc, so its length is 2R sin 30° = R, equal to the radius. Each interior angle is (6 − 2) × 180°/6 = 120°, which is why hexagons tile the plane with three meeting at every corner.

Lilies, squills and the Star of Bethlehem show this pattern. In many monocots the flower is built from two whorls of three parts, offset by 60°, which produces the two-triangle arrangement.