Chaos
The Lorenz attractor is the wheels of fortune. The system spirals outwards: will it miss the hyperbolic fixed point and continue another turn, or will it shift to the other side? It is entirely deterministic, but each circulation makes the future less certain. Laplace's demon sits in a corner, cursing lack of precision.
The future cannot be predicted exactly except in the boring special cases. In all other situationssmall differences in initial conditions grow and grow until they make the outcome completely unexpected. We can make forecasts a bit into the future, and we can know the general shape of the attractors and universality classes we belong to, but we cannot know where we will end up.
Every grain of sand was placed there by interactions with the surrounding sand, the wind or the steps of an animal. Each grain has a complex history, despite having no memory ofwhere it came from. And each grain will influence anything interacting with it, sending itto an entirely new path.
Is the approaching sandstorm the work of a butterfly somewhere? Yes. But it is the work of every butterfly, every bird, every sneeze and every past hurricane. There are no single causes.
Chaos
The Lorenz attractor is the wheels of fortune. The system spirals outwards: will it miss the hyperbolic fixed point and continue another turn, or will it shift to the other side? It is entirely deterministic, but each circulation makes the future less certain. Laplace's demon sits in a corner, cursing lack of precision.
The future cannot be predicted exactly except in the boring special cases. In all other situationssmall differences in initial conditions grow and grow until they make the outcome completely unexpected. We can make forecasts a bit into the future, and we can know the general shape of the attractors and universality classes we belong to, but we cannot know where we will end up.
Every grain of sand was placed there by interactions with the surrounding sand, the wind or the steps of an animal. Each grain has a complex history, despite having no memory ofwhere it came from. And each grain will influence anything interacting with it, sending itto an entirely new path.
Is the approaching sandstorm the work of a butterfly somewhere? Yes. But it is the work of every butterfly, every bird, every sneeze and every past hurricane. There are no single causes.