Kolmogorov Complexity
The complexity of a string can be described by the length of its shortest program that produces it.
The Mechanism: Kolmogorov complexity measures the length of the shortest program that outputs a given finite string, relative to a chosen universal description language. Different reasonable languages change the measure only by a bounded additive amount for long strings.
The Philosophical Pivot: Compression captures one sense of pattern: a highly compressible sequence has a shorter description than its raw form. This gives a precise lens on the relationship between regularity and information.
The Horizon: Exact Kolmogorov complexity is uncomputable in general, so practical compression algorithms provide upper bounds rather than definitive shortest descriptions. The concept is important in algorithmic information theory but should not be confused with everyday complexity.
A short description is not necessarily a meaningful explanation. The usefulness of a model depends on the question being asked and the context in which the data matter.
Randomness and complexity are also distinct: a sequence can be hard to compress without being profound or consequential.
The idea invites us to ask whether a theory compresses observations by revealing structure or merely hides complexity in its assumptions.
A pattern is one way of saying that the world can be described more briefly than it first appears.
When I call an explanation elegant, has it revealed a real pattern or merely moved the complexity somewhere I am not looking?