Can Math Explain History?



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One of the greatest puzzles of social science is how human societies evolved from small groups of relatives and friends to the huge, anonymous and complex societies of today.

demonstrationA peace demonstration. Today we live in huge societies of strangers, who nevertheless are capable of coming together for cooperative purposes

Ten thousand years ago everybody lived in a village. Strangers were rare, and most of them were enemies. Then the first hierarchically organized societies appeared – chiefdoms, simple and complex. Around 5,000 years ago the first states evolved and 2,500 years ago they transformed themselves into huge multiethnic empires, governing tens of millions of people.

How can we explain the rise of such large-scale societies? Archaeologists, sociologists, and political scientists proposed a multitude of theories. But the vast majority of them can be boiled down to just two general mechanisms.

Most anthropologists and archaeologists think that the driving force has been the invention of agriculture. It made possible high population densities as well as surpluses that could be appropriated by newly emerging ruling elites. In a way, this theory suggests that once agriculture created sufficient resources for the evolution of complex societies, such societies inevitably evolved.

A different theoretical perspective, one based on cultural evolution and multilevel selection theory, disagrees. Yes, intensive agriculture is a necessary condition for the evolution of complex societies. But it is not enough. Institutions of complex societies, such as bureaucracies, organized religion, and constraints on the ruling elites, which induce them to promote common good, are all costly. How can they evolve in spite of such costs? The theory of cultural multilevel selection says that this evolution is only possible when societies compete against each other, so that those that do not have the right institutions fail. Costly institutions of complex societies are spread because societies that have them destroy societies without them.

This may sound quite abstract, but it is actually possible to take this general theory and build a specific and detailed model that predicts where and when complex large-scale societies should arise, and how they spread during the Ancient and Medieval eras of human history. This is what we have done in a paper that was published today in the prestigious journal, Proceedings of the National Academy of Science.

The trick is to focus on factors that intensify intersocietal competition, which until very recently meant military competition.

In other words, warfare.


And between 1500 BC and 1500 AD the intensity of military competition in the Old World maps extremely well on the spread of military technologies based on warhorses. So we built a model around this factor, and it did an incredibly good job of predicting when and where large empires arose in Eurasia and Africa.

Here’s a press release that explains our model, written by Catherine Crawley of the National Institute for Mathematical and Biological Synthesis.

Math explains history: Simulation accurately captures the evolution of ancient complex societies

The question of how human societies evolve from small groups to the huge, anonymous and complex societies of today has been answered mathematically, accurately matching the historical record on the emergence of complex states in the ancient world.

Intense warfare is the evolutionary driver of large complex societies, according to new research from a trans-disciplinary team at the University of Connecticut, University of Exeter, and the National Institute for Mathematical and Biological Synthesis (NIMBioS) that appears this week as an open-access article in the journal Proceedings of the National Academy of Sciences.

The study’s cultural evolutionary model predicts where and when the largest-scale complex societies arose in human history.

Simulated within a realistic landscape of the Afro-Eurasian landmass during 1,500 BC to 1,500 AD, the mathematical model was tested against the historical record. During the time period, horse-related military innovations, such as chariots and cavalry, dominated warfare within Afro-Eurasia. Geography also mattered, as nomads living in the Eurasian Steppe influenced nearby agrarian societies, thereby spreading intense forms of offensive warfare out from the steppe belt. On the other hand, rugged terrain inhibited offensive warfare.

The study focuses on the interaction of ecology and geography as well as the spread of military innovations and predicts that selection for ultra-social institutions that allow for cooperation in huge groups of genetically unrelated individuals and prevent large-scale complex states from splitting apart, is greater where warfare is more intense.

While existing theories on why there is so much variation in the ability of different human populations to construct viable states are usually formulated verbally, by contrast, the authors’ work leads to sharply defined quantitative predictions, which can be tested empirically.

The model-predicted spread of large-scale societies was very similar to the observed one; the model was able to explain two-thirds of the variation in determining the rise of large-scale societies.

“What’s so exciting about this area of research is that instead of just telling stories or describing what occurred, we can now explain general historical patterns with quantitative accuracy. Explaining historical events helps us better understand the present, and ultimately may help us  predict the future,” said the study’s co-author Sergey Gavrilets, NIMBioS director for scientific activities.

Citation: Turchin P, Currie T, Turner E, Gavrilets S. 2013. War, space, and the evolution of Old World complex societies. PNAS.

Watch the movie by co-author Tom Currie showing model-predicted dynamics and actual historical data side by side.

Next blog: questions and answers about our model

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Peter Richerson

I think that the creative part of social evolution is as much in the peacemaking as in the warfare itself. It is the peacemakers who innovate the institutions, such the Axial Age religions, that enlarge the sphere of peace and build the means to deter predatory regimes.

Peter Turchin

Agreed. Figure 2 in our article actually shows the model-predicted spread of such ultrasocial institutions. For example, religious injunctions to the rulers to promote the public good. But such institutions have costs, and without competition from other societies they would collapse internally. The elites become selfish and despotic. So intersocietal competition is the key part to weed out societies with ‘lapsed’ ultrasocial institutions.


That is a pessimistic message for the future I fear. In a nuclear armed world, war could be an unprecedented disaster. It would be nice to think that asabiyya can be maintained by means short of war.

Peter Turchin

Who says that competition has to be violent? My take is that we need between group and between societies competition, otherwise all ultrasocial and prosocial norms will unravel. What we have to solve is preventing war, which I think is quite doable. But competition and even nonviolent conflict are good and necessary.


Yes. And I suppose that nuclear war and environmental change are global existential threats that might generate asabiyya the way existential wars did in the past.

Peter Turchin

Short of invading Martians, those are our best bets. I am optimist enough to think that the humanity has evolved to the point where such more abstract threats (compared to Chinggis’ hordes) would work the magic.


“the model was able to explain two-thirds of the variation in determining the rise of large-scale societies.”

An extension of the model which might increase that proportion might be the secondary effects of intense warfare i.e.
– the source regions for the neccessary raw materials of war at different times: horses, copper, tin, iron, wood (for ships and smelting)
– the source regions for the universal raw material of war, money: silver, gold, gems, silk, spices, sugar – and if sugar plantations then slaves.

You might expect certain regions to show up a little earlier in the historical record than in the simulation if they possessed those resources as a result of colonies, for example (just guessing here)
– Iberia (copper)
– SW Britain and Ireland (copper and tin)
– Switzerland/Bohemia (iron)
– Crimea (slaves and/or horses)
– Nubia/Axum (slaves and/or ?)

(cool painting)


also the gold producing region of west africa.

Peter Turchin

Right, we are moving in that direction, but the next iteration of the model will focus on getting the agricultural productivity right.

James Hedman

Describing the journal as “prestigious” really should have gone without saying. It makes you sound like an absolute twit.

Gene Anderson

I looked up the whole article…really neat. The few exceptions are interesting. Egypt was easy to conquer from north to south. But the really big problem with the theory is the New World civilizations. They not only rose without benefit of steppe nomads, they rose in very, very similar ways to the Old World ones; the parallels are absolutely astonishing. Warfare was, of course, a constant, and seems to have driven the process some, but it seems to me that more predictive of the exact locations of early civilizations are trade and fertility. The early-civilized places are trade nodes and are very fertile valleys, especially ones–as Carneiro points out–circumscribed by harsh deserts. (Not true in China, incidentally.) I certainly agree with the warfare and steppe nomad findings, but I think that those other factors were about equally important, not just as residual 34%.

Peter Turchin

Gene, this is the next step. We are busily gathering data to characterize agricultural productivity, so that we can go beyond crude measures that we used in the PNAS article. So we will be able to test the theory.

So it is an open question, what explains complex societies better – productive agriculture or intersocietal competition/warfare. We will obviously need different proxies for warfare intensity than pressure from nomadic pastoralists. Although I can’t resist pointing out that there was only one place in the Americas where there were pastoralist societies – and that’s the Andes. The significance of this observation remains to be assessed by future research…

T. Greer

What is up with Iran? Mountainous terrain throw off the formula there?

Peter Turchin

No, most of central Iran is deep desert. Our simulation excludes those squares from attacking and been attacked (no agriculture there). But historical atlases color that uninhabited territory as part of one Persian empire after another. So this discrepancy is an artifact of how historical atlases are constructed. This is the case where the model is actually closer to reality than data!


Looking at the movie, one mismatch between the simulation and the data appears to be Eastern Europe, where the model seems to say that complex polities should have appeared there earlier; the areas are red by about 500-600 AD in simulation, but red by only 1000 AD in data. Interesting. Parts of India get a bit weird too I think, no? A pretty impressive match overall though.

I can see the difficulties of extending this to the New World. Maybe look at the bow & arrow instead of cavalry … but none of the large New World empires really favored the bow.

Peter Turchin

There are lots of mismatches – but the model has only 4 tunable parameters. What’s remarkable is that it manages to capture data so well.

On North America: we are going to see whether the spread of bow and arrow MilTech helps to explain transitions to more complex societies (but of course nowhere near as large scale and complex as in the Old World, or Mesoamerica and Peru)


I’ve probably been staring at that movie too much by now. One thing that jumps out – and this isn’t meant as a criticism because the match between data and model is pretty amazing – is that in the model once places turn red they tend to stay red but in the data they can go back to being green. This is probably most of that 35% mismatch.

In terms of Europe, where I’ve got enough background knowledge to be able to think along with the animation (as in “oh yeah, that’s what happened then” as colors change) what the model seems to be missing is strong centrifugal forces which broke up complex large scale societies into still complex but much smaller ones. Like feudal fragmentation in the Holy Roman Empire, Kievan Rus, or Piast Poland in the high middle ages. These either didn’t have a high enough population before the fragmentation, so that a split even into a few smaller principalities leaves each descendant society with low population, or broke up into so many puzzle pieces (like HRE) that they all turn back green.

Theoretically, the costs of maintaining cohesive institutions probably rise with population size/density – which is something you (and others) address elsewhere. There may even be some discontinuities here where a state may grow too big to be sustainable given the existing level of “social technology” and then it takes only a small external shock for it to splinter.

Peter Turchin

Radek, two thoughts. First, remember that model predictions are an average of 20 realizations. In each realization you can have much more weirdness, and my guess would be that if you run the model enough times, you could get a much better match in one realization,

However, and second, I think your insight is very valid. In the real world, Europe developed what I called ‘asabiya black holes’. Central areas of an old empire that imploded tend to be very resistant to new state formation. The model doesn’t include this mechanism. But I think it’s quite real.


What is up with Iran? Mountainous terrain throw off the formula there?

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