The Strange Math That Predicts (Almost) Anything

A century-old math feud in Russia didn’t just settle a philosophical argument about free will—it produced tools that later powered everything from nuclear-bomb simulations to Google search. The key pivot came when Andrej Markov showed that probability can still “settle down” even when events are dependent, breaking Pavel Nekrasov’s claim that convergence in real-world statistics proves independence (and therefore free will).

Nekrasov leaned on the law of large numbers, the familiar idea that repeated independent trials (like fair coin flips) produce averages that drift toward expected values. He went further: if marriage rates, crime rates, or birth rates appear to stabilize over time, then the underlying decisions must be independent—acts of free will that can be treated as measurable. Markov, an atheist who criticized what he saw as sloppy reasoning, attacked that leap. He argued that dependence doesn’t prevent probability from working; it only invalidates the inference from “stable averages” to “independent causes.”

To make the case, Markov turned to language. In Alexander Pushkin’s Eugene Onegin, he treated letters as a sequence where the next letter’s type (vowel or consonant) depends on the current one. Counting overlapping letter pairs showed strong dependence: vowel-vowel pairs occurred far less often than they would under independence. Then Markov built a predictive “machine” using states (vowel or consonant) and transition probabilities derived from the observed frequencies. Even with dependence, the long-run vowel/consonant proportions converged to the same stable split found in the original text. The takeaway was blunt: convergence in social data doesn’t prove independence, and independence isn’t required for probability theory to function.

Markov’s framework—later known as Markov chains—became a practical engine for simulation. During the Manhattan Project, Stanislaw Ulam used a card-game intuition to approximate hard problems by sampling many random outcomes. But neutrons inside a bomb don’t behave independently; each step depends on the neutron’s current state. Von Neumann recognized the need for a chain model, and together they used Markov chains on ENIAC to estimate the multiplication factor k and determine whether a chain reaction would die out, stabilize, or grow explosively. The method was christened “Monte Carlo,” and it spread quickly beyond weapons research.

That same probabilistic logic later shaped the internet. Sergey Brin and Larry Page modeled web surfing as a Markov chain, treating links as endorsements and using a damping factor to prevent getting trapped in loops. Their PageRank algorithm—built on long-run state probabilities—ranked pages by relative importance and helped Google overtake Yahoo. Meanwhile, Markov ideas also underlie text prediction: Claude Shannon’s experiments with predicting letters and words foreshadow modern token-based language modeling, where context matters and attention mechanisms refine predictions.

The through-line is memorylessness: for many complex systems, it’s enough to track the current state and ignore most of the past. That simplification is what makes Markov chains powerful—and what turned a personal intellectual feud into a mathematical backbone for 20th-century science and today’s information systems.