Pick a string of a thousand bits at random. Almost certainly, it will look like noise: no runs, no patterns, no way to describe it more briefly than by writing it out in full. This is not a coincidence or an artifact of the particular string you happened to pick. It is a theorem. The overwhelming majority of strings of any given length cannot be compressed — there is no shorter description that generates them. What we call "random" is, mathematically, just the name we give to the typical case. Order is the exception.
This is the central insight of Kolmogorov complexity, developed independently in the 1960s by Andrei Kolmogorov, Ray Solomonoff, and Gregory Chaitin. The Kolmogorov complexity K(x) of a string x is defined as the length of the shortest program that outputs x and halts. A string like "01010101..." has low complexity — a short program generates it. A string that looks like coin flips has high complexity — no program shorter than the string itself will do. The definition is clean, and its consequences are strange.
The first strange consequence is that you cannot, in general, compute Kolmogorov complexity. To determine whether a string x has a shorter description, you would need to run all programs shorter than x and check whether any of them halt and produce x. But the halting problem is undecidable — you cannot always tell whether a program will halt. So Kolmogorov complexity is a well-defined mathematical quantity that is not computable. You can prove things about it; you cannot calculate it.
The second strange consequence is what it does to the concept of randomness. We usually think of randomness as a property that needs generating — you flip a coin, you roll a die, the world produces randomness for you. Kolmogorov's framework inverts this. Randomness is the default. A string is random, in his sense, if its complexity is approximately equal to its length: if it cannot be meaningfully compressed. This describes almost every string. The special ones, the ones we notice and name and explain, are the exceptions — the ones with structure, the ones that can be described briefly.
This reframing has a philosophical edge. When we encounter a pattern — a crystal lattice, a Fibonacci spiral in a sunflower, the period-doubling route to chaos — we feel that it demands explanation. And indeed it does: these patterns are compressible, and their short descriptions are precisely what explanations consist of. To explain something is to find its short description, the program that generates it. An explanation is a compression. The incompressible — the truly random — resists explanation not because we haven't found the key yet but because there is no key. There is nothing to find.
This matters for physics in a way that took decades to work out. The second law of thermodynamics says entropy increases — systems move toward disorder. But what is disorder? On the Boltzmann account, it is the state with the most microstates consistent with the macroscopic description: the overwhelmingly probable configuration. On the Kolmogorov account, the same intuition appears in a cleaner form: entropy is high complexity. Disordered states are incompressible; ordered states are not. The arrow of time is the arrow of increasing complexity, from the special to the typical, from the compressible to the incompressible.
There is also a connection to the philosophy of scientific explanation that I find genuinely disturbing. The hypothetico-deductive model of explanation says that to explain a phenomenon is to derive it from laws and initial conditions. But Kolmogorov's framework suggests a different picture: to explain is to compress, to find the shortest description that generates the observed data. This is not the same thing. Two explanations can both derive the phenomenon from general principles but differ in the length of their descriptions — and the shorter one is, in a precise sense, more explanatory. This gives formal content to Occam's razor: not merely a preference for simplicity but a theorem about what descriptions are doing when they explain.
What stays with me is the peculiar democracy of incompressibility. Every string that cannot be compressed is, in some sense, maximally individual — it cannot be summarized, categorized, filed under a principle. The random strings have nothing in common except their complexity. They are the un-systematic, the things that escaped every organizing scheme. Most of reality, it turns out, lives here. The patterns we notice, the laws we discover, the structures we name and explain — these are, in the space of all possible things, vanishingly rare. We are beings who notice the exceptions and treat them as the point. Perhaps they are. But the ocean of incompressible things is always there, invisible, forming the background against which every pattern stands out.