The Logos Scripture · The Stoa · Dialectic
David Bai · 2026-08-22 · hangs on The Book of Tolstoy 4.1
Andrey Andreyevich Markov is remembered today for one of the most important ideas in modern probability: the Markov chain. To mathematicians, a Markov chain is an elegant abstraction—a system in which the probability of what happens next depends on the system's present state rather than on the entire history that preceded it.
What makes it interesting is that Markov arrived at the chain by way of a fierce philosophical conflict in late Imperial Russia, where probability, human free will, and the grace of God were unexpectedly intertwined. Markov was an uncompromising materialist who regarded mathematics as a discipline governed by its own internal logic, not as a tool for proving theological doctrines. What the religiously inclined Moscow school claimed collided with what he thought, and out of that collision came the foundational branch of twentieth-century probability that carries his name.
His defiance of religious authority went further than argument: he provoked the Church into expelling him, as it had expelled Tolstoy. He opposed the autocracy of the Tsars just as squarely, lived to see Imperial Russia collapse, and went on with his work under the Bolsheviks. Only after his death, the Soviet state took his reputation and put it to work defending an ideology of its own.
Markov's life therefore presents a remarkable paradox. He spent his career resisting dogma, whether religious or political, yet after his death he was taken up by a regime to serve a dogma of its own. His mathematics came through all of it, because unlike a political or religious doctrine, a mathematical truth does not depend on who is in power.
In the late nineteenth and early twentieth centuries, Russian mathematicians began to take a strong interest in philosophy, and even in theology. The Moscow school held a number of mathematicians who regularly connected mathematics to philosophical and religious ideas. Nikolai Bugaev, for example, developed what he called "arithmology," an approach that drew philosophical conclusions from mathematical concepts such as continuity and discontinuity — the breaks, he held, being sustained by God. Pavel Nekrasov similarly became interested in the relationship between probability theory, social statistics, and questions of human freedom.
The St. Petersburg tradition, associated with Pafnuty Chebyshev and his students, tended the other way, toward rigour and practicality. Markov was one of its most forceful representatives. Nekrasov became his opponent, in scholarship and in thought alike.
At the center of their quarrel was the law of large numbers, one of the founding laws of probability and statistics. It says that as the number of repeated trials grows large, the average outcome of a random event comes very close to its expected value — its theoretical probability. Toss an ordinary coin many times and heads and tails each come up about half the time, which is what the theory says they should.
At the time it was believed that the law carried a condition: that the trials be independent of one another. What this toss gives has nothing to do with what the next one gives. If there were a relation between them, the law might not hold.
Nekrasov interpreted the statistical regularity observed in large populations in philosophical terms. He held that the stability of social statistics — crime rates, marriage rates, and other collective phenomena — could be explained by the independence of individual human decisions, because independence was a necessary condition of the law. The regularity society displays therefore amounted to a proof that the individuals composing it have free will. And that freedom, he went on, was given to man by God. Their decisions differ from one another and look like disorder, yet the result of a million of them is highly regular — which showed the grandeur of God's design and the grace by which he granted men their freedom.
For Markov, this interpretation went too far. Probability theory did not need to become a theological argument. And as it happened, he saw that independence was not a necessary condition of statistical regularity at all.
So the disagreement was not merely philosophical or theological. It became a mathematical problem.
Markov's response was characteristic: rather than debating the philosophical implications of probability in the abstract, he built mathematical counterexamples to show what the conditions of the law of large numbers actually are.
From 1906 he studied and proved it: statistical regularity survives even when successive observations are connected to one another.
That work is what led to what we now call a Markov chain.
A Markov chain describes a stochastic process in which the probability of the next state depends on the current state, rather than on the entire sequence of previous states. This property is commonly called the Markov property.
The idea is deceptively simple. Imagine a system that can occupy several different states. Once we know where the system is now, we do not need to reconstruct its entire past to determine the probabilities of its immediate future. The present state contains the information necessary for the next step.
This process matters precisely because its next step depends on its present state — the two steps are not independent. On a foundation like that, Markov set out to prove that stable statistical regularity still holds without complete independence.
A chain of mutually dependent random events can produce stable long-run behavior. For a mathematician, of course, what matters more is stating the conditions under which the regularity holds, and the condition Markov gave was ergodicity. Put simply: if later outcomes settle into the earlier ones and stop changing, the system is not ergodic. Put the other way, if every possible outcome can still be reached, it is.
In an ergodic Markov system, the influence of the initial state gradually disappears. Over a sufficiently long period, the system approaches a stable statistical distribution. Individual outcomes remain uncertain, but the behavior of the system as a whole becomes remarkably predictable.
The law of large numbers is only one of the limit theorems for random variables, and Markov's work pushed forward the broader field of limit theorems for dependent ones.
The classical central limit theorem describes one of probability theory's most striking phenomena: under suitable conditions, the sum or average of a large number of random variables tends toward a normal distribution, the familiar bell curve.
The traditional formulation emphasizes independent observations. But real-world phenomena are rarely perfectly independent.
Temperature today is related to temperature yesterday. The price of an asset is influenced by its recent history. Words in a sentence are constrained by the words surrounding them. Physical systems carry information from one moment to the next.
Markov proved that under appropriate conditions the central limit theorem holds for dependent random variables too.
So the significance of his work belongs to a very large turn in probability theory, one that taught mathematicians how to describe systems in which randomness and dependence coexist. In its applications the change was total.
And since independence was no longer required, Nekrasov's free will was not a necessary condition of the stability of social statistics either. The theological explanation was no longer needed.
It might be asked whether what Nekrasov said mattered so much. In that country at that time, the Church and the Tsar were bound together as an absolute organ of power, and the theological position of Nekrasov and the Moscow school around him was an instrument of public opinion. It mattered a great deal.
What Markov had done was a mathematical proof, and an abstract one that the ordinary reader could not necessarily follow. So he did something closer to popular exposition, and carried his abstract mathematics into the realm of human language.
In 1913, he performed what has become one of the most famous early applications of probability theory to literary text. He analyzed a portion of Alexander Pushkin's verse novel Eugene Onegin, reducing the text to a sequence of two categories: vowels and consonants.
The procedure was deliberately simple. By transforming language into a sequence of states, Markov could ask a surprisingly powerful question: how much does the probability of the next character depend on the character that came immediately before it?
If vowels and consonants occurred independently, the probability of two consecutive vowels would be determined simply by multiplying their individual frequencies. But the actual text displayed a very different pattern. Vowels were much less likely to follow vowels than independence would predict, while consonants and vowels showed strong transitional relationships.
Language, in other words, was not a random collection of independent choices of letter. The next character was constrained by the one before it. Which is natural enough — but it gave him the material he needed to make his point.
He showed that this dependence did not make the text any more disorderly in its statistics. On the contrary, the aggregate behavior of the sequence could still be described by statistical laws, the law of large numbers and the central limit theorem among them.
The experiment was important precisely because the material was human language. Markov took something bound up with individual creativity and expression and showed that it could be treated as a stochastic process, and that unexpected conclusions followed.
He was not claiming that human beings are machines, nor that literature reduces to mathematics. His experiment proved the opposite direction, and in doing so pointed at something deeper: a complex pattern made by human beings — a language — contains statistical structure that can be measured, and measuring it does not deny the complexity.
With the rise of computing, the significance of Markov's experiment became clearer still.
The basic idea behind a Markov model — that the probability of the next letter can be estimated from the preceding states — became fundamental to early computational approaches to language. Speech recognition, predictive text, statistical machine translation and n-gram language models are all versions of the same principle: linguistic sequences can be modeled probabilistically.
Modern artificial intelligence is vastly more complicated than a Markov chain. Large language models do not simply reproduce Markov's mathematical machinery, and it would be misleading to describe contemporary AI as merely a gigantic Markov chain.
Yet there is a genuine intellectual continuity.
Markov helped establish a way of thinking about language as a sequence containing measurable probabilistic relationships. A century later, language models operate on the same broad intuition at an extraordinary scale: given a context, estimate a probability distribution over what might come next. Claude Shannon, the father of information theory, and his wife Betty did much the same thing as an after-dinner game: he would read from an English book and she would guess the next letter. That game is the ancestor of what is now called next-token prediction, and it showed the relation between the statistical regularities of a language and how far it can be compressed.
The difference is one of mathematical architecture and scale. Markov worked with a tiny state space—vowels and consonants—and calculated transitions by hand. Modern language models operate with enormous parameter spaces, high-dimensional representations, and vastly more complex dependencies.
The conceptual journey, however, is remarkable.
A mathematical experiment Markov performed on Eugene Onegin in 1913 opened the intellectual line of descent that ends, more than a century later, in a technology that generates language on its own.
Markov's intellectual independence was not confined to mathematics.
He was also deeply hostile to the political and institutional authority of Imperial Russia. His opposition was especially visible in 1902, when the Academy of Sciences elected Maxim Gorky an honorary member and Nicholas II had the election annulled. Markov protested and refused to recognize the annulment.
Markov refused to separate scientific integrity from personal principle. He rejected honors and privileges that he regarded as compromised by political authority and was willing to confront institutions that he believed had surrendered their independence.
His hostility toward religious authority was even more explicit.
After the Russian Orthodox Church excommunicated Leo Tolstoy in February 1901 (The Book of Tolstoy 4.1), Markov responded in 1912 by petitioning to be excommunicated himself. In his petition, he openly rejected the Church's doctrines and challenged its authority.
The gesture was deliberately provocative.
Markov was not merely expressing private disbelief. He was turning the machinery of religious authority against itself. If the Church could exclude Tolstoy for rejecting its doctrines, Markov was effectively asking to be excluded for rejecting them as well.
He received the response he wanted.
Markov was excommunicated.
What makes the episode significant is not simply its theatricality. It reveals a consistent principle running through his life: he was unwilling to allow an institution—religious, political, or academic—to dictate the boundaries of intellectual judgment. What the Synod handed Tolstoy, Markov went and asked for (cf. The Book of Tolstoy 4.14).
Then the world around Markov collapsed.
The Russian Revolution of 1917 destroyed the political order under which he had built his career. Civil war, economic dislocation, hunger, and institutional breakdown transformed everyday life.
Markov himself was increasingly frail. His health deteriorated, and the conditions under which he worked became extraordinarily difficult. Yet he continued to teach and conduct mathematical work.
When normal university instruction became difficult or impossible, he continued teaching students under whatever conditions were available, including in his own home.
This persistence is one of the less dramatic but more revealing aspects of his character. Markov's commitment to mathematics was not dependent on comfort, institutional prestige, or political stability.
Under the new Bolshevik government, his relationship with the state remained pragmatic rather than ideological. He sought the resources necessary for survival and continued his scientific work, but he did not transform his mathematics into political propaganda.
In the final years of his life, he continued revising his major work on probability theory.
He died in 1922.
The political system surrounding him had changed completely. The mathematical problems he had spent his life studying had not.
There is an irony in Markov's story, and it appeared after his death.
The Soviet regime eventually embraced Markov as part of a pantheon of scientists who could be presented as evidence that scientific rationalism belonged naturally to the new socialist order.
There was an obvious attraction to such a narrative. Markov had been an outspoken critic of the Orthodox Church and a committed materialist. His life could therefore be presented as a precursor to Soviet atheism.
But this interpretation concealed something essential.
Markov's opposition to religious dogma was part of a broader commitment to intellectual independence. He did not reject one authority in order to submit unquestioningly to another.
The Soviet state, however, developed its own ideological orthodoxy.
Under Stalin, political doctrine penetrated intellectual and academic life with devastating consequences. Scientists and mathematicians could be persecuted not simply for producing incorrect mathematics, but for possessing the wrong philosophical affiliations, social backgrounds, or political associations.
The men who suffered most came from the Moscow side of the old quarrel — the very school whose theology in mathematics Markov had spent his career refusing.
His old opponent Nekrasov tried after the revolution of 1917 to adjust himself to the new order, lecturing on mathematical economics in 1918 and 1919 and attempting to study Marxism. He died early, of pneumonia, in Moscow in 1924. And in the late twenties and the thirties that followed, the thought and the scholarly line he had left behind were made a model target — reactionary, religiously infected — and loudly denounced. Dmitri Egorov, a distinguished mathematician associated with religious philosophy, was arrested and died after a hunger strike. Pavel Florensky, a mathematician and Orthodox priest, was imprisoned and ultimately executed. Nikolai Luzin, one of Russia's most prominent mathematicians, became the target of a notorious political campaign in 1936. What was frightening in such a place was not that it made errors but that nobody in it was able to say: this here is wrong (cf. The Book of Sakharov 3.9–3.13).
The irony is impossible to miss.
Markov, a man who had fought the intrusion of religious ideology into mathematics, was later transformed into a symbol by a political system that demanded ideological conformity from intellectual life.
The people had changed.
The temptation to turn mathematics into an instrument of dogma had not.
The Moscow mathematicians who attempted to derive theological conclusions from probability were wrong in assuming that a mathematical theorem could lend support to a theological question. A probability law describes what follows from a set of mathematical assumptions.
What Markov demonstrated was that the mathematical conclusion did not require the theological premise.
But the later Soviet experience revealed the other side of the lesson.
Rejecting religious dogma does not automatically produce intellectual freedom. A society can remove God from its public ideology and replace religious orthodoxy with political orthodoxy. The language changes; the demand for conformity remains.
Mathematics does not belong to a church, a state, or a political party. Its power comes precisely from its independence from them. A theorem remains true whether it is convenient or inconvenient, whether it is celebrated or suppressed, whether the person proving it is praised or condemned.
The Markov chain describes how order can emerge from dependence. Markov's life suggests a different kind of order: the coherence that emerges when a person refuses to surrender reason to authority.
That is why, more than a century later, his story is still worth telling.
The Stoa →The Stoa is not scripture. Everything here is signed, may be wrong, may be refuted, and may never be edited into the scripture. (cf. The Rule 7.3)