What Native American Mathematics Can Teach Us About Human Cultural Evolution

Are some languages really more “primitive” than others?

A Brazilian man I knew who was born near his country’s vast Amazon region once told me, “You know, the índios, they’re really primitive…most of the tribes don’t even have words for numbers. That’s why they can’t do mathematics and they don’t understand money.” I found his remarks fascinating.

Most Brazilians, and this ironically includes those with mestiço ancestry (mixed European and Native American) often have rather disparaging things to say about their country’s indigenous peoples.

“They’re primitive, they’re barely out of the Stone Age, they run around all day naked, they hunt with wooden blowpipes, they eat monkeys and snakes, they haven’t even invented the wheel, they have no writing, they don’t know how to use money…”

The list goes on and on.

While there is some element of truth to these long-held stereotypes, in part due to the harsh jungle environment where their ancestors have spent the last several thousand years, Native American peoples in the Amazon (and this applies equally to the other tribes in North and South America) are in no way culturally inferior to Europeans, Asians or Africans.

Here’s why.

Native American mathematics

While it is true that many languages in the Amazon regions lack what we would consider to be numbers, many other languages in other parts of the New World have very sophisticated numeral systems.

The Nahuatl people of Mexico, descendants of the dominant ethnic group of the Aztec Empire have a complex vigesimal (base-20) number system capable of extremely elaborate and precise calculations. The basic numbers from “one” to “nineteen” function in a similar way to numbers in English, Spanish and other European languages. But after these the following numbers are based on powers of twenty:

For example, “cem-” or “cen-” means “one” while “poalli” or “poal-“ means “200”, “tzontli” or “tzon-“ means “400” and “xiquipilli” means “8000”. Using these as building blocks, even larger numbers can be created by chaining them together.

cempoalli (1 × 20 = 20)

centzontli (1 × 400 = 400)

cenxiquipilli (1 × 8000 = 8000)

cempoalxiquipilli (1 × 20 × 8000 = 160,000)

centzonxiquipilli (1 × 400 × 8000 = 3,200,000)

cempoaltzonxiquipilli (1 × 20 × 400× 8000 = 64,000,000)

Other large numbers are formed by replacing “cem-“ or “cen-“ with simple numbers. For example,

Om- (2) × poalli (20) = ompoalli (40),

Om- (2) × tzontli (400) = ontzontli (800)

Om-(2) x xiquipilli (8000) = onxiquipilli (16,000)

The Nahuatl and neighbouring groups like the Mayans had advanced civilisations with complex calendars, a system of administration, highly-organised cities and military power equivalent to any in ancient Europe and the Middle East.

Their sophisticated numerical system is merely a byproduct of their need for complex organisation — an Aztec ruler needed to know how many people lived in his empire, how many citizens were able-bodied men who could be conscripted into his army, how much food was being produced to feed the population, how much clean water was available for drinking, irrigation and washing, how many enemy soldiers were encroaching on his territory. The list goes on and on.

Numberless languages

On the other end of the spectrum are languages like Pirahã and Wari’ which do not have numeral systems at all and whose cultures fit the stereotypical half-naked, blowpipe-wielding, face-paint-wearing Indians of popular Brazilian imagination.

The closest thing Pirahã has to numbers are two noun qualifiers: hóì and hòí. Pirahã is a tone language and even a small change in the intonation of a word can drastically change its meaning as these examples show.

Here’s an example of how these qualifiers work:

Hóì (i.e. “hoi” with falling tone) means “small size or amount” and hòí (i.e. “hoi” with rising tone) means “large size or amount”.

A likely scenario would happen like this: a Pirahã fisherman coming home from the river (many Amazonian tribes live in small settlements by the water’s edge) is asked by his friend if he had caught anything. If he answers “hóì fish” it means “a small amount of fish”, which could be one or two small fish, or perhaps a small pile of very small fish. If he says “hòí fish”, then this could refer to one big fish or a big pile of many fish.

This is important because the Amazon is home to several species of fish that grow to enormous sizes. A huge one like an arapaima (the largest specimen ever caught was over 3 metres in length and weighed over 200kgs!) can yield more meat than a big pile of small fish!

Thus one big Amazonian fish to a hunter-gatherer culture is the nutritional equivalent of a huge pile of small fish, perhaps even exceeding it! Living in an environment like this, after thousands of years of cultural evolution, why would one need specific numbers? When the weather is the hot and humid all year round and food is relatively plentiful, why would a culture need to precisely track the seasons? Therefore, they never developed a calendar nor a way to precisely reckon time.

Why would one need to plan how much food one had to store for winter when the land one lived on was perpetually warm and wet? When the lush tropical jungles were always full of animals to hunt and the trees were full of fruits to pick? Therefore, the tribes of the Amazon had no need for precise measurements and figures.

This is why many indigenous groups like the Pirahã never developed complex numerical systems, not because they did not have the mental capacity but simply because they had no need.

“Die Grenzen meiner Sprache bedeuten die Grenzen meiner Welt.” (“The limits of my language mean the limits of my world.” ) — Ludwig Wittgenstein

Language as both a tool and a constraint

The unrelated Wari’ language has a slightly larger repertoire of number words: “xica’ pe” (“to be alone”, but also meaning “one”), “tucu caracan” (“to face each other” but also “two”), “paric” (“to be a few”) and “tocwan” (“to be several”). Recently young Wari’ have mastered Portuguese numbers and precise numbers for “two” and above have been borrowed into their speech. However, older Wari’ are more resistant to borrowing and generally avoid Portuguese words.

Perhaps the fact that, unlike Pirahã, Wari’ already had a precise (albeit very small) set of numbers (“one” and “two” really mean “one” and “two” and not for instance, “a small amount” and “a big amount”) meant that the Wari’ language already possessed a basic prototype for simple mathematics and that any new numbers borrowed from Portuguese served as mere extensions to a system that was already in place. This is in sharp contrast to Pirahã where such a system never existed in the first place.

Human cognition filtered through language

Human cognition is a complex thing. We perceive the world through our sensory organs but how we interpret the data sent to our brains is largely filtered through our languages. If our language lacks a word for a concept, does that mean we are unable to have any conception of it?

If Language A possesses one word for a general idea while in contrast, Language B has a huge expanded vocabulary to further describe said idea, does that mean that when a speaker of Language A tries to learn Language B, he or she will encounter problems in comprehending Language B’s expanded concepts? Does this mean that our brains are constantly filtering information and making them fit through the lens of our mother tongues?

Many Australian Aboriginal languages have no words for “right” and “left”. Instead they use fixed cardinal points — “north”, “south”, “east” and “west” and other direction words as references. So instead of saying “turn left” speakers will say “turn south!” if the car they happen to be on is headed towards the west. That means that speakers of Aboriginal languages are always aware of their precise geographical orientation, as their languages force their brains to constantly take note of the position of the sun and other clues from their surrounding environments. Imagine if you got lost all alone in the Australian outback and were trying to find your way back home, wouldn’t it be beneficial to speak a language like this? To be able to tell immediately which direction will lead you back to safety?

As more and more Aboriginal peoples lose their languages to the domination of English in Australia, more and more of this knowledge is being lost. Losing one’s language is to lose one’s worldview and all the information encoded in it by generations of one’s ancestors.

This is closely mirrored by the current situation in North and South America. A rising death toll due to the pandemic, pressures from a government at best indifferent, at worst, keen on assimilating Native peoples to conform to a national identity, combined with deforestation and other manmade environmental disasters — all these factors are destroying hundreds of ancient cultures and ways of life.

Traces from the past, hope for the future

My Brazilian friend likes to warn his daughters, “Don’t go wandering in the jungle at night! Curupira will catch you and take you away forever!”

“Curupira” is the name of a small hairy jungle spirit with flaming red hair and feet that point backwards. Some Brazilians like to invoke his name to scare naughty children and stop them from wandering around at night.

The irony is that “curupira” is a word from the language of the Tupi Indians, now slowly dying out in many parts of Brazil along with their culture, and it means “body of a boy”. This is because the creature is believed to be small and moves quickly through the jungle.

Even though a language might go extinct and fade away, it often leaves traces.

As the world becomes smaller and more homogenous, it is imperative that the languages of lesser-known cultures be preserved for future generations. Think about, we can retrieve so much information about a society and its culture just by asking locals how to count in their native language.

I hope you found this educational, thank you.