One problem mathematics has in the public eye is its image as an esoteric pursuit that has no real-world, real-life applications. It’s just something that occupies some impossibly brainy professors in their ivory-tower institutes. Not for us plebeians and, in any case, we “can’t do numbers!", and “maths is scary!"—or so we like to claim. So, why study it?
Yet the truth is that mathematics is everywhere. It’s always been so and it’s getting more so every day. The sciences—chemistry, biology, physics—use mathematics all the time, but you knew that. But it’s also true that practitioners of the social sciences —what we might call the arts, or humanities—are using mathematical techniques in their work. Economics, sociology, history, even literature, all of them.
Case in point: Anthropology. Let me tell you about a certain tiny Pacific Ocean island, and about the people who lived there, probably starting in about 500 CE. What’s interesting about these gentle folk is that they carved several hundred statues (called “moai")—mostly just heads—then erected them along the coast of the island.
You probably know that I’m referring to Easter Island, or Rapa Nui, its native name. This 160 sq. km speck in the middle of the Pacific is often referred to as “the world’s most isolated piece of habitable land". Not a label you want to dispute, because the nearest land mass is South America, 3,200km away; the nearest island is Pitcairn, 2,200km away. How did Easter’s inhabitants get there all those centuries ago? If that’s a puzzle, consider too that this is an island without substantial timber or material to make ropes, without large domestic animals to help with dragging statues. How did those inhabitants drag 200 statues all over their island and erect them? It is a question worth asking, because the statues are enormous: The largest is over 30 feet tall and weighs over 80 tonnes. But there’s more. Another 700 statues were abandoned before being erected, and the largest of those is some 65 feet and 250 tonnes. There’s still more. The erected statues were installed on enormous stone platforms (“ahu")—200 of them, just as carefully crafted. And all this frenetic construction seems to have been done between 1250 and 1500 CE.
In addition, there’s evidence that the island was seriously deforested through the centuries these people lived there, which led to conflicts and a steep population decline. For all that, the author Jared Diamond suggests in his book Collapse that Rapa Nui might just stand for what’s in store for our entire planet.
Along with Diamond, others have theories that seek to explain all that happened on Easter Island. As you can imagine, they are vigorously contested. For now, I’ve been fascinated instead by some new research. It offers answers for more questions that Rapa Nui’s inhabitants raised for us when they carved and installed their moai on the ahu. The paper I’m referring to begins with this sentence: “Explaining the processes underlying the emergence of monument construction is a major theme in contemporary anthropological archaeology, and recent studies have employed spatially-explicit modelling to explain these patterns."
Right there is a reference—“spatially-explicit modelling"—to the mathematics these researchers used. But what were they trying to show with this mathematics? In a word, the “why". Why did the islanders build their huge ahu and then erect moai on them? Can we divine their purpose centuries after the construction and in the absence of records?
A team of anthropologists—Robert DiNapoli of the University of Oregon, Carl Lipo of Binghamton University and others—focused on where on the island the ahu are to be found (bit.ly/2Bdyo3X). They first noted that there did not seem to be any particular connection between these sites and the locations where the raw material for the construction came from. But what about the locations of what the team called “subsistence resources"? That is, “mulch gardens" where islanders grew food, the areas where they could gather fish from the sea, and the island’s few sources of freshwater. Since the island had limited food and water supplies, there must have been competition among its inhabitants for these resources. Did the ahu and moai play some role in that competition? The hypothesis in this paper is that “Rapa Nui’s monumental architecture served as territorial signals of control over subsistence resources". In other words, were the platforms and statues built to reinforce claims that one or the other island community made on particular resources? For, if so, you’d expect that the ahu are located close to those particular resources.
To answer this, DiNapoli, Lipo and their colleagues did a “spatial analysis" of the eastern part of the island, amounting to about one-third of its total area. The eastern sector of Rapa Nui is where the team had gathered the most data on the three subsistence resources as well as the ahu and moai themselves. Spatial analysis is still an evolving technique that tries to explain or predict human behaviour in a given space in mathematical and geometric terms.
Imagine looking at a map of the area you’re interested in—in this case, the east of Rapa Nui. It’s covered with dots that represent things of interest to you: Settlements, homes, wells, farms—or indeed, in this case, the ahu. Once you plot all the ahu on the map, there are two fundamental properties of the pattern of these dots that you might want to explore. The first is its intensity, meaning how many dots there are per square km, for example. If the dots are pretty uniformly distributed, that tells a story. If they are gathered densely in some places, but loosely elsewhere, that tells a story, too. The second property captures how the dots interact with each other. Do you find clusters, suggesting that the builders tended to build new ahus near existing ones? Or are they separated, suggesting an opposite impulse? Spatial analysis tries to understand and explain these phenomena.
On Rapa Nui, the authors pointed out that “ahu may appear clustered simply because they are predominantly coastal in their distribution". In other words, the fact that there are ahu clusters along the coast does not automatically imply that there is some link to the sea and the availability of fish. For example, their locations on the edge of a small bay might suggest a clustering that wasn’t intended at all. Any spatial analysis, then, must account for this aspect of ahu distribution. Once DiNapoli et al did this, they came to the meat of the process of checking their hypothesis: Tests that “explore potential relationships between ahu and distance from subsistence resource locations".
Various exotic techniques were useful here: Spatial Kolmogorov-Smirnov (SKS) tests, point-process models (PPMs), the Bayesian Information Criterion (BIC), the Akaike Information Criterion (AIC) and more. Now the nitty-gritty of these acronymed tests here is beyond the scope of a column like this one. But there’s an important detail about them that DiNapoli et al mentioned: They operated on a principle of “parsimony". You know that word to mean extreme thrift, no extravagance. In this specific mathematical and statistical context, the principle of parsimony seeks to explain the variability in the data it is applied to—on Rapa Nui, we’re talking about the clustering or otherwise of ahu—in the simplest possible way. No extravagance.
And what did all these techniques find, with the ahu on Rapa Nui and the island’s subsistence resources? First, that there is no real link to mulch gardens. Second, there is indeed a link to both marine resources and freshwater sources. But some more analysis showed that the relationship to marine resources is not particularly meaningful, and it shows up only “because marine resource and freshwater locations tend to occur in similar locations".
In short, the analysis showed that “ahu locations are most parsimoniously explained by distance from freshwater sources".
In conclusion, DiNapoli and his colleagues make a pointed reference to Diamond’s theory about Rapa Nui in his Collapse. “The long held orthodox view", they say, is of a complex society “that then ‘collapsed’ following a self-imposed ‘ecocide’. [N]early every aspect of this narrative has been shown to lack empirical sufficiency." Newer theories take into account the locations of the ahu. And those were probably determined “by the availability of the island’s limited freshwater". Which may not be particularly surprising, of course: We humans need water to live. But the ahu are distinct from where the islanders actually lived, where they needed water for daily consumption. That there is still a discernible link between ahu and freshwater sources, then, at least suggests that ahu were intended to demonstrate ownership or control of those sources.
On Rapa Nui, that’s what mathematics can show us.
Once a computer scientist, Dilip D’Souza now lives in Mumbai and writes for his dinners. His latest book is Jukebox Mathemagic: Always One More Dance. His Twitter handle is @DeathEndsFun