Stir a cup of coffee and let it settle, without spilling any and without tearing the liquid apart. Intuitively, after a thorough stir everything has moved somewhere else. A theorem in topology says otherwise, and the cup is only the illustration. What does it actually claim?
At least one point ends exactly where it began. The Brouwer theorem states that any continuous function from the unit ball into itself has a fixed point, a place the map sends to itself. It sounds like a trick, and it is a deep result about and shape, which also means it is a statement about functions rather than about fluids.
The claim sounds like a party trick. Stir your coffee, let it settle, and somewhere in the cup is a point of liquid that has ended exactly where it started. Stir it again, and the same thing is true again.
What is actually being asserted is the Brouwer fixed point theorem: any continuous function from the unit ball into itself has a fixed point. The liquid is standing in for a function that maps every point of the disc to its new position.
The conditions do all the work. Continuity forbids tearing and jumping, which stirring respects. The domain has to be , which a disc is and a ring is not, and a ring is exactly where the claim fails, since rotating one moves every point.
The deeper value is the genre of result it belongs to. It guarantees existence while giving no procedure for finding the point, That is precisely what economics, game theory and differential equations need when an equilibrium or a solution must be shown to exist. The coffee is the illustration. The theorem is about maps.
wat. Sometimes mathematics cannot tell you where the answer is, only that there is no way to escape its existence.
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Fixed point theorems underpin economics, differential equations and game theory, wherever somebody needs to prove a solution exists without being able to compute it.
How do we know?
The theorem was proved within topology and is stated for continuous functions from a compact convex set into itself. The coffee is an intuitive illustration and not part of the proof.
What is still unknown
The conditions are exact. If the map is not continuous, or the domain is the wrong shape, or the subject is real liquid with genuinely complicated mixing, the illustration stops being evidence.
Learn something?
Where to next
Did Alan Turing really prove the halting problem attributed to him?proof
How many colors does it take to color any map?proof
How do you prove the primes never run out?proof