1, 2, 3, 4, 5, 6, 7, 8, 9... and 0. With just these ten symbols, we can write any rational number imaginable. But why these particular symbols? Why ten of them? And why do we arrange them the way we do?
And on a related note, an article from: Smithsonia.com titled How Humans Invented Numbers—And How Numbers Reshaped Our World featuring Anthropologist Caleb Everett explores the subject in his new book, Numbers and the Making Of Us. Peersonally, I find the Pirahã study fascinating.
"Once you learn numbers, it’s hard to unwrap your brain from their embrace. They seem natural, innate, something all humans are born with. But when University of Miami associate professor Caleb Everett and other anthropologists worked with the indigenous Amazonian people known as the Pirahã, they realized the members of the tribe had no word used consistently to identify any quantity, not even one.
Intrigued, the researchers developed further tests for the Pirahã adults, who were all mentally and biologically healthy. The anthropologists lined up a row of batteries on a table and asked the Pirahã participants to place the same number in a parallel row on the other side. When one, two or three batteries were presented, the task was accomplished without any difficulty. But as soon as the initial line included four or more batteries, the Pirahã began to make mistakes. As the number of batteries in the line increased, so did their errors."
The researchers realized something extraordinary: the Pirahã’s lack of numbers meant they couldn’t distinguish exactly between quantities above three. As Everett writes in his new book, Numbers and the Making of Us, “Mathematical concepts are not wired into the human condition. They are learned, acquired through cultural and linguistic transmission. And if they are learned rather than inherited genetically, then it follows that they are not a component of the human mental hardware but are very much a part of our mental software—the feature of an app we ourselves have developed.”
How did you become interested in the invention of numbers?
It comes indirectly from my work on languages in the Amazon. Confronting languages that don’t have numbers or many numbers leads you inevitably down this track of questioning what your world would be like without numbers, and appreciating that numbers are a human invention and they’re not something we get automatically from nature.
In the book, you talk at length about how our fascination with our hands—and five fingers on each—probably helped us invent numbers and from there we could use numbers to make other discoveries. So what came first—the numbers or the math?
I think it’s a cause for some confusion when I talk about the invention of numbers. There are obviously patterns in nature. Once we invent numbers, they allow us access to these patterns in nature that we wouldn’t have otherwise. We can see that the circumference and diameter of a circle have a consistent ratio across circles, but it’s next to impossible to realize that without numbers. There are lots of patterns in nature, like pi, that are actually there. These things are there regardless of whether or not we can consistently discriminate them. When we have numbers we can consistently discriminate them, and that allows us to find fascinating and useful patterns of nature that we would never be able to pick up on otherwise, without precision.
Numbers are this really simple invention. These words that reify concepts are a cognitive tool. But it’s so amazing to think about what they enable as a species. Without them we seem to struggle differentiating seven from eight consistently; with them we can send someone to the moon. All that can be traced back to someone, somewhere saying, “Hey, I have a hand of things here.” Without that first step, or without similar first steps made to invent numbers, you don’t get to those other steps. A lot of people think because math is so elaborate, and there are numbers that exist, they think these things are something you come to recognize. I don’t care how smart you are, if you don’t have numbers you’re not going to make that realization. In most cases the invention probably started with this ephemeral realization [that you have five fingers on one hand], but if they don’t ascribe a word to it, that realization just passes very quickly and dies with them. It doesn’t get passed on to the next generation.
Another interesting parallel is the connection between numbers and agriculture and trade. What came first there?
I think the most likely scenario is one of coevolution. You develop numbers that allow you to trade in more precise ways. As that facilitates things like trade and agriculture, that puts pressure to invent more numbers. In turn those refined number systems are going to enable new kinds of trade and more precise maps, so it all feeds back on each other. It seems like a chicken and egg situation, maybe the numbers came first but they didn’t have to be there in a very robust form to enable certain kinds of behaviors. It seems like in a lot of cultures once people get the number five, it kickstarts them. Once they realize they can build on things, like five, they can ratchet up their numerical awareness over time. This pivotal awareness of “a hand is five things,” in many cultures is a cognitive accelerant.
How big a role did numbers play in the development of our culture and societies?
We know that they must play some huge role. They enable all kinds of material technologies. Just apart from how they help us think about quantities and change our mental lives, they allow us to do things to create agriculture. The Pirahã have slash and burn techniques, but if you’re going to have systematic agriculture, they need more. If you look at the Maya and the Inca, they were clearly really reliant on numbers and mathematics. Numbers seem to be a gateway that are crucial and necessary for these other kinds of lifestyles and material cultures that we all share now but that at some point humans didn’t have. At some point over 10,000 years ago, all humans lived in relatively small bands before we started developing chiefdoms. Chiefdoms come directly or indirectly from agriculture. Numbers are crucial for about everything that you see around you because of all the technology and medicine. All this comes from behaviors that are due directly or indirectly to numbers, including writing systems. We don’t develop writing without first developing numbers.
How did numbers lead to writing?
Writing has only been invented in a few cases. Central America, Mesopotamia, China, then lots of writing systems evolved out of those systems. I think it’s interesting that numbers were sort of the first symbols. Those writings are highly numeric centered. We have 5,000-year-old writing tokens from Mesopotamia, and they’re centered around quantities. I have to be honest, because writing has only been invented in a few cases, [the link to numbers] could be coincidental. That’s a more contentious case. I think there are good reasons to think numbers led to writing, but I suspect some scholars would say it’s possible but we don’t know that for sure.
Something else you touch on is whether numbers are innately human, or if other animals could share this ability. Could birds or primates create numbers, too?
It doesn’t seem like on their own they can do it. We don’t know for sure, but we don’t have any concrete evidence they can do it on their own. If you look at Alex the African grey parrot [and subject of a 30-year study by animal psychologist Irene Pepperberg], what he was capable of doing was pretty remarkable, counting consistently and adding, but he only developed that ability when it was taught over and over, those number words. In some ways this is transferrable to other species—some chimps seem able to learn some basic numbers and basic arithmetic, but they don’t do it on their own. They’re like us in that they seem capable of it if given number words. It’s an open question of how easy it is. It seems easy to us because we’ve had it from such an early age, but if you look at kids it doesn’t come really naturally.
What further research would you like to see done on this subject?
When you look at populations that are the basis for what we know about the brain, it’s a narrow range of human cultures: a lot of Americanundergrads, European undergrads, some Japanese. People from a certain society and culture are well represented. It would be nice to have Amazonian and indigenous people be subject to fMRI studies to get an idea of how much this varies across cultures. Given how plastic the cortex is, culture plays a role in the development of the brain.
What do you hope people will get out of this book?
I hope people get a fascinating read from it, and I hope they appreciate to a greater extent how much of their lives that they think is basic is actually the result of particular cultural lineages. We’ve been inheriting for thousands of years things from particular cultures: the Indo-Europeans whose number system we still have, base ten. I hope people will see that and realize this isn’t something that just happens. People over thousands of years had to refine and develop the system. We’re the benefactors of that.
I think one of the underlying things in the book is we tend to think of ourselves as a special species, and we are, but we think that we have really big brains. While there’s some truth to that, there’s a lot of truth to the idea that we’re not so special in terms of what we bring to the table genetically; culture and language are what enable us to be special. The struggles that some of those groups have with quantities is not because there’s anything genetically barren about them. That’s how we all are as people. We just have numbers.