essay on evolution of number

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The Evolution and History of Numbers and Counting

History of numbers: essay introduction, the egyptians/babylonians number history, the hindu-arabic number history, the mayan number history, history of numbers: essay conclusion, works cited.

This paper explores the evolution of number system from ancient to modern. Here, you’ll find information on the development of number system of the Egyptians/Babylonians, Romans, Hindu-Arabics, and Mayans.

The evolution of numbers developed differently with disparate versions, which include the Egyptian, Babylonians, Hindu-Arabic, Mayans, Romans, and the modern American number systems. The developmental history of counting is based on mathematical evolution, which is believed to have existed before the counting systems of numbers started (Zavlatsky 124).

The history of mathematics in counting started with the ideas of the formulation of measurement methods, which the Babylonians and Egyptians used, the introduction of pattern recognition in number counting in pre-historical times, the organization concepts of different shapes, sizes, and numbers by the pre-historical people, and the natural phenomenon observance and universe behaviors. This paper will highlight the evolution history of counting by the Egyptians/Babylonians, Romans, Hindu-Arabic, and Mayans’ counting systems. Moreover, the paper will outline the reasons why Western counting systems are widely used contemporarily.

The need for counting arose from the fact that the ancient people recognized the measurements in terms of more or less. Even though the assumption of numbers based its arguments on archeological evidence about 50,000 years ago, the counting system developed its background from the ancient recognition of more and less during routine activities (Higgins 87). Moreover, ancient people’s need for simple counting in history developed odd or even, more or less, and other forms of number systems evolved into the current counting systems. The need for counting developed from the fact that people needed a way of counting groups of individuals through population increase by birth. In addition, Menninger asserts that the daily activities of the pre-historical people, like cattle keeping and barter trade led to the need for counting and value determination (105).

For instance, in order to count cows, prehistoric people used sticks. Collecting and allocating sticks to count the animals helped determine the total number of animals present. The mathematical history evolved from marking rows on bones, tallying, and pattern recognition, which led to the introduction of numbers. The bones and wood were marked, as shown below.

Moreover, the development of numbers evolved from spoken words by pre-historical people. However, the pattern of numbers from one to ten has been difficult to trace. Fortunately, any pattern of numbers past ten is recognizable and easily traceable. For instance, eleven evolved from ein lifon, which was used to mean ‘one left’ over by the prehistoric people. Twelve developed from the lif, which meant “two leftovers” (Higgins 143). In addition, thirteen was traced from three and four from fourteen, and the pattern continued to nineteen. One hundred is derived from the word “ten times” (Ifrah and Bello 147). Furthermore, the written words used by the ancient people, like notches on wood carvings, stone carvings, and knots for counting, gave a solid base for the evolution of counting.

The Incas widely used counting boards for record-keeping. The Incas used the “quip,” which helped the pre-historical people record the items in their daily lives. The counting boards were painted with three different color levels. These were the darkest parts, representing the highest numbers; the lighter parts, representing the second-highest levels; and the white parts, representing the stone compartments (Havil 127). In addition, the quip was used to do fast mathematical computations (Zavlatsky 154). Generally, the quip used knots on cords, which were arranged in a certain way to give certain numeral information. However, the quip systems of record keeping and information have been associated with several mysteries which have not yet been established. Examples of how the knots looked are shown below.

This form is the common system of counting and numbers used in the 21st Century. In India, Al-Brahmi introduced the numbers 1, 2, 3, 4, 5, 6, 7, 8, and 9 (Menninger 175). The Brahmi numerals kept changing with time. For instance, in the 4th to 6th Century, the numerals were as shown below.

Finally, the numerals were later developed to 1,2,3,4,5,6,7,8,9 with time. The earliest system of using zero was developed in Cambodia. The evolution of the decimal points emerged during the Saka era, whereby three digits and a dot in between were introduced (Hays and Schmandt-Besserat 198). The Babylonians introduced the positional system, whereby the place value of the numerical systems was established. Moreover, the positional system by the Babylonians developed the base systems to the numerical, and the Indians later developed it further. The Brahmi numerals took different incarnations to develop, which resulted in the current number system (Higgins 204).

The Gupta numerals were one of the processes passed by the Hindu-Arabic number system to become the commonly used American number version. Currently, theories about the formation and development of the Gupta numerals remain debatable by researchers.

In addition, the Europeans adopted the Hindu-Arabic system through trading, whereby the travelers used the Mediterranean Sea for trade interactions (Havil 190). The use of the abacus and the Pythagorean dominated the European number evolution. The Pythagorean used “sacred numbers” even though the two systems diminished after a short while. With time, the Europeans borrowed the Hindu-Arabic number system to establish their mathematical number systems (Ifrah and Bello, 207). However, the process through which the Europeans adopted the Hindu-Arabic system has not been proven fully. It is believed that the Europeans adopted the Hindu-Arabic number system by relying heavily on it to build their current strong numerals (Higgins 210). For instance, the scope of the positional base system is quite large, which involves the conversion of different bases using the numerical number 10.

The Mayan civilization of counting and number systems developed in Mexico through ritual systems. The rituals were calendar calculations involving two ritual systems, one for the priests and the other for the ordinary civilians (Higgins 217). For instance, priestly calendar counting used mixed base systems involving numerical number multiples. The Mayan number systems form the base of mathematical knowledge. Moreover, the Mayan system of numbers used the positioning of numbers to allocate the place value of the combined digits (Havil 223).

The Mayans used the place value of numerical numbers, which were tabled to add and multiply numbers. Ultimately, the Hindu-Arabic and the Mayan number systems contributed highly to the evolution of numbers as opposed to the Egyptian/Babylonian number systems (Menninger 199). Nevertheless, the Western number system of counting and mathematics incorporated the strong features of all the other evolutions to get a standard solid number system. For instance, the American system, commonly used in most countries, uses decimal points, place values, base values, and Roman numbers from 1 to 10 (Ifrah and Bello 225). The figure below represents a sketch of the tabled digits by the Mayans.

The American version of numbers and counting used all the development features of the Mayans, Babylonians, Incas, Egyptians, and Hindu-Arabic systems to develop a reliable and universally-accepted number system (Hays and Schmandt-Besserat 214). This aspect is outstanding as it makes the American system stand out of all the number systems and counting. Nevertheless, the commendable work of the Mayans, Babylonians, Egyptians, and Indians cannot be underrated, as the historical trace of counting and number systems would be impossible without them.

The historical trace of number systems and counting covers a wide scope of pre-historical archeological evidence. Tracing ancient times by researchers poses a significant challenge in establishing counting and number systems. The research on number systems and counting has not yet been settled on the actual source information for evidence. Ultimately, the most effective number systems that led to the current dominant Western number system are the Mayans, Hindu, and Babylonian systems relying on the Incas’ developments. The prehistoric remains left mathematical evidence as stones and wood carvings, which led to the evolution of counting. Hence mathematical methodologies evolved. The methodology of research and arguments varies on the evolution of numbers. Consequently, there are no universally-accepted research findings on the mathematical and number systems evolution.

Havil, Julian. The Irrationals : A Story of the Numbers You Cant Count on, Princeton: Princeton University Press, 2014. Print.

Hays, Michael, and Denise Schmandt-Besserat. The History of Counting , Broadway: HarperCollins, 1999. Print.

Higgins, Peter. Number Story: From Counting to Cryptography, Gottingen: Copernicus, 2008. Print.

Ifrah, Georges, and David Bello. The Universal History of Number: From Pre-history to the Invention of Computer , Hoboken: Wiley, 2000. Print.

Menninger, Karl. Number Words and Number Symbols; Cultural History of Numbers, Mineola: Dover Publications, 2011. Print.

Zavlatsky, Claudia. Africa Counts; Number and Pattern in Africa Cultures, Chicago: Chicago Review Press, 1999. Print.

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The Innovative Spirit fy17

A Smithsonian magazine special report

How Humans Invented Numbers—And How Numbers Reshaped Our World

Anthropologist Caleb Everett explores the subject in his new book, Numbers and the Making Of Us

Lorraine Boissoneault

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.”

To learn more about the invention of numbers and the enormous role they’ve played in human society, Smithsonian.com talked to Everett about his book.

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.

Numbers and the Making of Us: Counting and the Course of Human Cultures

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 American undergrads, 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.

Get the latest stories in your inbox every weekday.

Lorraine Boissoneault | | READ MORE

Lorraine Boissoneault is a contributing writer to SmithsonianMag.com covering history and archaeology. She has previously written for The Atlantic, Salon, Nautilus and others. She is also the author of The Last Voyageurs: Retracing La Salle's Journey Across America. Website: http://www.lboissoneault.com/

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• NATURE PODCAST
• 02 June 2021

On the origin of numbers

• Nick Petrić Howe &
• Benjamin Thompson

You can also search for this author in PubMed   Google Scholar

Listen to the latest from the world of science, with Nick Petrić Howe and Benjamin Thompson.

In this episode:

00:45 Number origins

Around the world, archaeologists, linguists and a host of other researchers are trying to answer some big questions – when, and how, did humans learn to count? We speak to some of the scientists at the forefront of this effort.

News Feature: How did Neanderthals and other ancient humans learn to count?

07:47 Research Highlights

How sea anemones influence clownfish stripes, and the benefits of skin-to-skin contact for high-risk newborns.

Research Highlight: How the clownfish gets its stripes

Research Highlight: Nestling skin-to-skin right after birth saves fragile babies’ lives

09:48 Briefing Chat

We discuss some highlights from the Nature Briefing . This time, an upper limit for human ageing, and could tardigrades survive a collision with the moon?

Scientific American: Humans Could Live up to 150 Years, New Research Suggests

Science: Hardy water bears survive bullet impacts—up to a point

Subscribe to Nature Briefing, an unmissable daily round-up of science news, opinion and analysis free in your inbox every weekday.

Never miss an episode: Subscribe to the Nature Podcast on Apple Podcasts , Google Podcasts , Spotify or your favourite podcast app. Head here for the Nature Podcast RSS feed.

doi: https://doi.org/10.1038/d41586-021-01491-0

Host: Nick Petrić Howe

Welcome back to the Nature Podcast . This week, the ancient history of counting…

Host: Benjamin Thompson

And the latest from the Nature Briefing . I’m Benjamin Thompson.

And I’m Nick Petrić Howe.

First up on the show, reporter Adam Levy has been pondering some important questions.

Interviewer: Adam Levy

Hello there. How old are you? How tall are you? How many siblings do you have, and how much money do you have in your wallet right now? Whatever your answers, they reveal a fundamental way in which we see the world – through numbers. But how did our ancient ancestors start to think in numbers, and how can researchers even begin to unearth the history of such an intangible idea? Well, in this week’s Nature , reporter Colin Barras has written a feature all about the efforts to uncover the human history of numbers.

Interviewee: Colin Barras

It’s something that seems so familiar and yet when you begin to look into the origins of something like numbers it sort of quickly becomes a little unfamiliar when you realise how strange it is that we do this.

So, what is it about numbers that is strange and unique to us humans? After all, other species can roughly assess quantities. But humans have formalised numbers into concrete ideas and symbols far beyond what any wild animal does. We can…

Understand quantity using these abstract ideas such as twoness or symbols such as the number ‘2’ written on a piece of paper or words like the number ‘two’ spoken out loud.

To answer this question of how we built up our understanding of numbers, researchers are using evidence from across the globe and spanning disciplines of science. Through such work, scientists hope they can begin to dig up the origins of humans’ relationship with numbers. Archaeologist Francesco d’Errico has found instances of bones marked by humans, hinting at the origins of counting, one of these dates back some 40,000 years, uncovered in a cave in South Africa as Francesco describes.

Interviewee: Francesco d’Errico

A fibular of a baboon with 29 notches deeply cut into it, which showed that they have not just decorated, so the best interpretation for that is that something was recording numerical quantities.

One theory is that humans first created notches on bones for other purposes, maybe through butchering meat or through decoration. This allowed them to make the mental leap that these symbols could be used for counting. But interpreting such evidence, let alone creating a complete theory from it, is incredibly challenging. Here’s Colin again.

It’s fascinating but very, very difficult to understand what an individual who was living 50,000-60,000 years ago was thinking, what they had in mind when they were doing particular activities. All we have is artefacts that they leave behind and then we have to try to interpret what those artefacts might mean.

For cognitive archaeologist Karenleigh Overmann, the artefacts that researchers uncover can only tell a part of the story. She argues that the first steps humans took could have been much the same as the first steps we all take today as little kids – representing numbers with fingers on the hand.

Interviewee: Karenleigh Overmann

Sooner or later, you need to use your hand for something other than representing quantity, so you might go to some kind of device that does what the hand does but can do it for longer and a typical device that you might use would be a tally or maybe a group of stones.

Of course, a group of stones, let alone finger counting, can’t be preserved easily in the archaeological record, so researchers are using evidence from fields as wide-ranging as psychology, anthropology, archaeology and evolutionary biology to shed light on the matter.

You have to look at a wide variety of different sources of evidence and then infer from them how it might have worked in the past.

For Karenleigh, contemporary cultures have a lot to teach us about the evolution of numbers in our ancestors. Researchers can, for example, look at certain hunter-gatherer societies today who only have words for the first few numbers – ‘one’, ‘two’, ‘three’, perhaps before jumping to ‘many’. The varying use of numbers across different groups suggests that their use is strongly connected to possessions and materials.

So, we do tend to count the things that are important to us. If you’re worried about survival then you might be counting how much food you need to get you through the long, cold winter into the spring when the food comes back.

Other researchers look at how our words for numbers vary across languages. Comparing the words for small numbers suggests that such words have changed very slowly over the time since those languages split.

So, when they project that rate of change, or I should say that rate of stability, backwards, it suggests that words for the numbers ‘one’ through ‘five’ in some language families could be 100,000 years or older.

The origin of numbers may date back many thousands of years, but the research that hopes to get at these origins is still relatively new. For Colin, speaking to different researchers and putting together his feature has revealed a blossoming field.

Yeah, it’s an exciting time. It feels like there’s going to be quite a lot of interesting work coming out over the next five or ten years or so as more people begin to think about this topic.

And different researchers come to different conclusions as they think more about this topic. Were fingers on our hands or notches on a bone the crucial ingredient to start representing numbers? Or perhaps our ancestors used something else entirely. It may be impossible to ever know for sure. But one thing we can count on is that researchers aren’t going to stop trying to dig up the answer. Here’s Francesco.

There is a strong disagreement, and I think that is the key for science. Disagreement is something we need to build on in order to test the theories.

That was Francesco d’Errico of the University of Bordeaux in France. You also heard from Karenleigh Overmann of the University of Colorado in the US and Colin Barras who’s now at New Scientist , all speaking to Adam Levy. To read more, there’ll be a link to Colin’s feature article in the show notes.

Coming up in the show, we’ll be hearing the latest estimates for humans’ maximum lifespan and the likelihood that tardigrades survived a crash landing on the Moon. Both of those stories coming up in the Briefing Chat. Before that, though, it’s time for the Research Highlights, read this week by Noah Baker.

Clownfish are some of the most recognisable creatures in the oceans. In part, that’s down to some successful cinematic outings, but it’s also because of their distinctive patterns and colours. Now, a team of researchers have observed that the pattern of white stripes on clownfish is dependent on the species of sea anemone in which the fish developed. The researchers analysed the concentration of a key growth hormone – thyroid hormone – in clownfish that lived in different species of anemone, and found that the levels of the hormone varied depending on the anemone they grew up in. Then they exposed clownfish larvae to varying concentrations of thyroid hormone and found that the clownfishes’ characteristic white stripes appeared sooner in fish that received the highest dose. The authors say this could explain why white stripes develop faster in fish that live in certain species of anemone. Find that research in full in the Proceedings of the National Academy of Sciences USA .

Nestling skin-to-skin immediately after birth has been found to sharply cut the risk of death for high-risk newborns. So-called ‘kangaroo mother care’ is known to cut the risk of death for babies born early or very small, but such care, which also entails feeding babies exclusively on breast milk, is usually started after babies show signs of a stable condition, even though most mortality occurs before these babies’ conditions stabilise. In the new trial, researchers started kangaroo mother care immediately after birth and found that it improved babies’ chance of survival by around 25% – a benefit so clear that the trial monitors stopped the trial early. Read that research in full in the New England Journal of Medicine .

Finally on the show this week, it’s time for the Briefing chat, where we highlight a couple of stories from the Nature Briefing . Nick, what have you got for us this time round?

Well, Ben, I’ve been looking into the latest estimate for how long humans can live for and this was an article in Scientific American .

Right, I mean, I guess humans are living maybe longer and longer as generations go on, but I suppose I still consider maybe 100 as being a really, really good innings. Is that there or thereabouts right?

Well, people have lived longer than that, and this is trying to work out what would be the maximum ever. So, you may remember – I’m going to mangle this name, sorry French listeners – Jeanne Calment, who lives until she was 122, and she was the oldest ever person. And this article discusses a study that has been trying to work out like if nothing really went wrong, there was no sort of disease and things like that that normally cause people to die, how long could you possibly live for? And their estimate is somewhere between 120 and 150 years old.

Goodness, so maybe in an idealised scenario – which of course very rarely exist – we might be able to push 150. I mean, that’s a lot longer than we’ve got now.

Yeah, almost certainly. I mean, you have to solve some big problems. For instance, cancer becomes a lot more common as you get older. But the important thing about this study, according to the article, is that it gives us an estimate of the pace of ageing, so how quickly things start to deteriorate as we get older. And so, they looked at things like blood cell counts and step counts, and as we get older, those things change, and in the case of step counts, they start to decline. And also, when things happen – so you have an illness or something like that – it takes longer to get these things back to normal as well. And the authors of the paper, they discuss how if we know the sort of pace of change, we can perhaps then work out what we can do as sort of interventions into that and potentially get people closer to this limit.

Well, my goodness, Nick. I mean, I suppose there aren’t many people then, as you say, that reach 120+. Where is the data coming from that these researchers have been using and have they just sort of extrapolated it out from there?

So, this data comes from three different cohorts in the US, UK and Russia. And there aren’t any people who are over 120 years old alive today and so they’re basically working out how these things start to degenerate over time as you age and then from that they extrapolate out to work out, well, this is where things would be at a point where you wouldn’t be able to continue to live. And so, that doesn’t necessarily mean that that’s the upper, upper limit but it’s just with what we currently have and how ageing progresses according to this study, between 120 and 150 years is about as long as we could live if we can solve everything else, which is not an easy task.

Yeah, I mean, that’s probably the kicker isn’t it, Nick. I mean, it’s very much an ‘if’. We could live to 150 if we can solve a multitude of problems. So, it may be that we’re not going to get there any time soon.

Well, I’m going to hope so for my lifespan but we’ll see. I’ve got another 100 years yet. But what’s your story for this week, Ben?

Well, Nick, I am fast becoming the Nature Podcast ’s unofficial tardigrade correspondent, and I’ve been reading a story from a couple of weeks back that was reported in Science and based upon a paper in Astrobiology and I really wanted to share it with you.

Okay, well you’ve piqued by interest. So, what is happening in the world of tardigrades?

Well, of course, tardigrades are these little animals and they are pretty amazing, right? Extreme temperature, freezing, radiation – they just shrug it off. It doesn’t really bother them that much. But there’s a question that’s been going around a little bit and that is could tardigrades have survived a crash landing on the Moon.

Wow, okay. The Moon is quite an inhospitable place from what I know about it but if anything could survive, I guess tardigrades could because they are very hardy. But I guess the question is could they?

Well, let’s find out, Nick. Let’s give a little bit of context here. So, back in 2019, you may remember the Israeli space mission Beresheet it was called and it crashed on the Moon, and in turned out that on board were some tardigrades. And this question, could they have survived, has inspired some research and some researchers wanted to find out, but it’s also thrown up some interesting insights which maybe we can talk about in a little bit. And this is kind of how they did it. So, they fed 20 tardigrades some moss and some mineral water and then they froze them for a couple of days so they entered this state of hibernation, and then they fired them out of a gun.

Laughs. Okay, I’m guessing, is that to simulate crashing on the Moon?

Bingo. And it’s not any sort of gun, Nick. This is a two-stage light gas gun, which has a higher velocity than a conventional gun. And so these deeply hibernating tardigrades were placed in a nylon bullet a few at a time and fired at increasing speeds into some sand, and it seems that the upper limit that they could survive was 900 metres per second, and this gives a momentary shock pressure of up to 1.14 gigapascals, which trust me is really, really high. And anything kind of above this was sadly curtains for the tardigrades, but below this some of them actually made it.

Well, that sounds like a heck of an impact, but is that similar to the impact that actually happened when Beresheet crashed onto the Moon?

So the top line is that the tardigrades that hit the Moon were unlikely to survive. The lander crashed at a slower speed than the bullet, but apparently the shock pressure caused by the frame of the spacecraft meant that this kind of shock would have been much higher and that really would have done for the tardigrades. But this research is interesting because it’s not necessarily just about that. It actually may give some information into this theory called panspermia, and I don’t know if you know what this is, Nick, but this is the theory that life could hitchhike from maybe a meteorite onto another planet.

Oh, okay, yeah, so I guess if they could survive landing on the Moon or something, potentially they could seed more life. But it doesn’t seem like they were able to this time, but would a meteor be a different thing and maybe they could survive that?

So, I think that this theory of panspermia is considered pretty unlikely and this isn’t going to change that view, but it’s not necessarily impossible. So, the speeds that meteorites hit the Earth or Mars are higher than the bullets but some of the sections within a meteorite might experience lower shock pressure. So, it’s a possibility, but maybe putting that to one side, this research and the calculations in it could offer some useful insights into other things as well. So, in this article I read, it could be used maybe to check for life on one of Saturn’s moon called Enceladus, which ejects plumes of water out into space. And I think what this research has shown is that if a probe could move slow enough maybe it could pass through these plumes and see if there’s any life there, for example. And that’s an interesting one to check, right, because if you detect something that’s dead, you don’t know whether it died because you hit it at hundreds of miles a second with a spacecraft or whether it’s been dead for a very, very long time. So, maybe finding that sweet spot where you can sort of gently cruise through without causing any serious damage could be a useful one to answering that question is there life out there in the Solar System, and it seems that tardigrades have played a little role in maybe helping calculate how it could be discovered.

Well, thanks for that insight, Ben. It’s always fascinating to hear more about these strange creatures, and it’s good that we’ve got our own tardigrade correspondent so we’ll hear more about them in the near future. And listeners, if you want more stories like these but delivered directly to your inbox, make sure you sign up to the Nature Briefing . Look out for a link of where to do so in the show notes, and you’ll also find links to today’s stories as well.

That’s all for this week, but of course you can drop us a line any time on email – [email protected] – or on Twitter – we’re @NaturePodcast. I’m Benjamin Thompson.

And I’m Nick Petrić Howe. Thanks for listening.

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The Evolution of Numbers

I want to take you on an adventure ...

... an adventure through the world of numbers.

Let us start at the beginning:

Q: What is the simplest idea of a number?

A: Something to count with!

The Counting Numbers

We can use numbers to count : 1, 2, 3, 4, etc

Humans have been using numbers to count with for thousands of years. It is a very natural thing to do.

• You can have " 3 friends",
• a field can have " 6 cows"

So we have:

Counting Numbers: {1, 2, 3, ...}

And the "Counting Numbers" satisfied people for a long time.

The idea of zero , though natural to us now, was not natural to early humans ... if there is nothing to count, how can we count it?

Example: we can count dogs, but we can't count an empty space:

An empty patch of grass is just an empty patch of grass!

Placeholder

But about 3,000 years ago people needed to tell the difference between numbers like 4 and 40. Without the zero they look the same!

So they used a "placeholder", a space or special symbol, to show "there are no digits here"

5  2

So "5  2" meant "502" (5 hundreds, nothing for the tens, and 2 units)

The idea of zero had begun, but it wasn't for another thousand years or so that people started thinking of it as an actual number .

But now we can think

"I had 3 oranges, then I ate the 3 oranges, now I have zero oranges...!"

The Whole Numbers

So, let us add zero to the counting numbers to make a new set of numbers.

But we need a new name, and that name is "Whole Numbers":

Whole Numbers : {0, 1, 2, 3, ...}

The Natural Numbers

You may also hear the term " Natural Numbers " ... which can mean:

• the "Counting Numbers": {1, 2, 3, ...}
• or the "Whole Numbers": {0, 1, 2, 3, ...}

depending on the subject. I guess they disagree on whether zero is "natural" or not.

Negative Numbers

But the history of mathematics is all about people asking questions, and seeking the answers!

One of the good questions to ask is

"if we can go one way, can we go the opposite way?"

We can count forwards: 1, 2, 3, 4, ...

The answer is: we get negative numbers:

Now we can go forwards and backwards as far as we want

But how can a number be "negative"?

By simply being less than zero.

Negative Cows?

And in theory we can have a negative cow!

Think about this ...If you had just sold two bulls , but can only find one to hand over to the new owner... you actually have minus one bull ... you are in debt one bull!

So negative numbers exist, and we're going to need a new set of numbers to include them ...

If we include the negative numbers with the whole numbers, we have a new set of numbers that are called integers

Integers: {..., −3, −2, −1, 0, 1, 2, 3, ...}

The Integers include zero, the counting numbers, and the negative of the counting numbers, to make a list of numbers that stretch in either direction indefinitely.

Try it yourself (click on the line):

If you have one orange and want to share it with someone, you need to cut it in half.

You have just invented a new type of number!

You took a number (1) and divided by another number (2) to come up with half (1/2)

The same thing happens when we have four biscuits (4) and want to share them among three people (3) ... they get (4/3) biscuits each.

A new type of number, and a new name:

Rational Numbers

Any number that can be written as a fraction is called a Rational Number.

So, if "p" and "q" are integers (remember we talked about integers), then p/q is a rational number.

Example: If p is 3 and q is 2, then:

p/q = 3/2 = 1.5 is a rational number

The only time this doesn't work is when q is zero, because dividing by zero is undefined.

Rational Numbers : {p/q : p and q are integers, q is not zero}

So half ( ½ ) is a rational number.

And 2 is a rational number also, because we could write it as 2/1

So, Rational Numbers include:

• all the integers
• and all fractions .

And also any number like 13.3168980325 is rational:

13.3168980325 = 133,168,980,325 10,000,000,000

That seems to include all possible numbers, right?

But There Is More

People didn't stop asking the questions ...and here is one that caused a lot of fuss during the time of Pythagoras:

When we draw a square (of size "1"), what is the distance across the diagonal?

The answer is the square root of 2 , which is 1.4142135623730950...(etc)

But it is not a number like 3, or five-thirds, or anything like that ...

... in fact we cannot answer that question using a ratio of two integers

square root of 2 ≠ p/q

... and so it is not a rational number (read more here )

Wow! There are numbers that are NOT rational numbers! What do we call them?

What is "Not Rational" ...? Irrational !

Irrational Numbers

So, the square root of 2 (√2) is an irrational number. It is called irrational because it is not rational (can't be made using a simple ratio of integers). It isn't crazy or anything, just not rational.

And we know there are many more irrational numbers. Pi ( π ) is a famous one.

So irrational numbers are useful. We need them to

• find the diagonal distance across some squares,
• to work out lots of calculations with circles (using π ),

So we really should include them.

And so, we introduce a new set of numbers ...

Real Numbers

That's right, another name!

Real Numbers include:

• the rational numbers, and
• the irrational numbers

Real Numbers: {x : x is a rational or an irrational number}

In fact a Real Number can be thought of as any point anywhere on the number line:

This only shows a few decimal places (it is just a simple computer) but Real Numbers can have lots more decimal places !

Any point Anywhere on the number line, that is surely enough numbers!

But there is one more number which has turned out to be very useful. And once again, it came from a question.

Imagine ...

The question is:

"is there a square root of minus one ?"

In other words, what can we multiply by itself to get −1 ?

Think about this: if we multiply any number by itself we can't get a negative result:

• and also (−1)×(−1) = 1 (because a negative times a negative gives a positive )

So what number, when multiplied by itself, results in −1 ?

This is normally not possible, but ...

"if you can imagine it, then you can play with it"

Imaginary Numbers

And we can use it to answer questions:

Example: what is the square root of −9 ?

Answer: √(−9) = √(9 × −1) = √(9) × √(−1) = 3 × √(−1) = 3 i

OK, the answer still involves i , but it gives a sensible and consistent answer.

And i has this interesting property that if we square it ( i × i ) we get −1 which is back to being a Real Number. In fact that is the correct definition:

Imaginary Number : A number whose square is a negative Real Number.

And i (the square root of −1) times any Real Number is an Imaginary Number. So these are all Imaginary Numbers:

There are also many applications for Imaginary Numbers, for example in the fields of electricity and electronics.

Real vs Imaginary Numbers

Imaginary Numbers were originally laughed at, and so got the name "imaginary". And Real Numbers got their name to distinguish them from the Imaginary Numbers.

So the names are just a historical thing. Real Numbers aren't "in the Real World" (in fact, try to find exactly half of something in the real world!) and Imaginary Numbers aren't "just in the Imagination" ... they are both valid and useful types of Numbers!

In fact they are often used together ...

"what if we put a Real Number and an Imaginary Number together?"

Complex Numbers

Yes, if we put a Real Number and an Imaginary Number together we get a new type of number called a Complex Number and here are some examples:

• 27.2 − 11.05 i

A Complex Number has a Real Part and an Imaginary Part, but either one could be zero

So a Real Number is also a Complex Number (with an imaginary part of 0):

• 4 is a Complex Number (because it is 4 + 0 i )

and likewise an Imaginary Number is also a Complex Number (with a real part of 0):

• 7 i is a Complex Number (because it is 0 + 7 i )

So the Complex Numbers include all Real Numbers and all Imaginary Numbers, and all combinations of them.

And that's it!

That's all of the most important number types in mathematics.

From the Counting Numbers through to the Complex Numbers.

There are other types of numbers, because mathematics is a broad subject, but that should do you for now.

Here they are again:

The history of mathematics is very broad, with different cultures (Greeks, Romans, Arabic, Chinese, Indians and European) following different paths, and many claims for "we thought of it first!" , but the general order of discovery I discussed here gives a good idea of it.

And isn't it amazing how many times that asking a question, like

• "what happens if we count backwards through zero" , or
• "what is the exact distance across the diagonal of the square"

first led to disagreement (and even ridicule!), but eventually to amazing breakthroughs in understanding.

I wonder what interesting questions are being asked now?

Over to You!

Here are two questions you can ask when you learn something new:

Can it go the other way?

• Positive numbers lead to negative numbers
• Squares lead to square roots

Can I use this with something else I know?

• If fractions are numbers, can they be added, subtracted, etc?
• Can I take the square root of a complex number? (can you?)

And one day your questions may lead to a new discovery!

Development in Brown 1933, by Wassily Kandinsky. Photo by Christophel Fine Art/Getty

How natural is numeracy?

Where does our number sense come from is it a neural capacity we are born with — or is it a product of our culture.

by Philip Ball   + BIO

Why can we count to 152? OK, most of us don’t need to stop there, but that’s my point. Counting to 152, and far beyond, comes to us so naturally that it’s hard not to regard our ability to navigate indefinitely up the number line as something innate, hard-wired into us.

Scientists have long claimed that our ability with numbers is indeed biologically evolved – that we can count because counting was a useful thing for our brains to be able to do. The hunter-gatherer who could tell which herd or flock of prey was the biggest, or which tree held the most fruit, had a survival advantage over the one who couldn’t. What’s more, other animals show a rudimentary capacity to distinguish differing small quantities of things: two bananas from three, say. Surely it stands to reason, then, that numeracy is adaptive.

But is it really? Being able to tell two things from three is useful, but being able to distinguish 152 from 153 must have been rather less urgent for our ancestors. More than about 100 sheep was too many for one shepherd to manage anyway in the ancient world, never mind millions or billions.

The cognitive scientist Rafael Núñez of the University of California at San Diego doesn’t buy the conventional wisdom that ‘number’ is a deep, evolved capacity. He thinks that it is a product of culture, like writing and architecture. ‘Some, perhaps most, scholars endorse a nativist view that numbers are biologically endowed,’ he said. ‘But I’d argue that, while there’s a biological grounding, language and cultural traits are necessary for the establishment of number itself.’

‘The idea of an inherited number sense as the unique building block of complex mathematical skill has had an unusual attraction,’ said the neuroscientist Wim Fias of the University of Gent in Belgium. ‘It fits the general enthusiasm and hope to expect solutions from biological explanations,’ in particular, by coupling ‘the mystery of human mind and behaviour with the promises offered by genetic research.’ But Fias agrees with Núñez that the available evidence – neuroscientific, cognitive, anthropological – just doesn’t support the idea.

If Núñez and Fias are right, though, where does our sense of number come from? If we aren’t born equipped with the neural capacity for counting, how do we learn to do it? Why do we have the concept of 152?

‘U nderstanding number as a quantity is the most essential, most basic part of mathematical knowledge,’ explained Fias. Yet numbers seem to be out there in the world, no less than atoms and galaxies; they seem to be pre-existing things just awaiting discovery. The great insights of mathematics, especially in number theory, are simply found to be true (or not). That 3 2 + 4 2 = 5 2 is a delightful property of numbers themselves, not an invention of Pythagoras.

Yet whether numbers really exist independently of humans ‘is not a scientific debate, but a philosophical, theological or ideological one’, said Núñez. ‘The claim that, say, five is a prime number independently of humans is not scientifically testable. Such facts are matters of beliefs or faith, and we can have conversations and debates about them but we cannot do science with them.’

Still, it seems puzzling that we can figure out these things at all. Geometry and basic arithmetic were handy tools for the ancient builders and lawmakers – ‘geometry’, after all, means ‘measuring the Earth’ – but it’s hard to see how they served any function as human cognition was evolving over the previous million or more years. There certainly was no biological need to be able to prove Fermat’s last theorem, or even to state it in the first place.

To explore such dizzying questions of number theory, even the most gifted mathematicians have to start in the same place as the rest of us: by learning to count to 10. To do that, we need to know what numbers are. Once we know that the abstract symbol ‘five’ equates with the number of fingers on our hand, and that this is one more than the ‘four’ that equates to the number of legs on a dog, we have the foundations of arithmetic.

The capacity to discriminate between different quantities happens extremely quickly in the development of a child – before we even have words to express it. A baby just three or four days old can show by its responses that it can discern the difference between two items and three, and by four months or so babies can grasp that the number of items you get by grouping one of them with another one is the same as two of them. They have a sense of the elementary operation that they will later learn to express as the arithmetic formula 1+1=2.

Our capacity for tennis doesn’t mean that we evolved to play it, or that we have a tennis module in our brain

Monkeys, chimps, dolphins and dogs can likewise tell which of two groups of food items has more, if the numbers are below 10. Even pigeons ‘can be trained to press a certain amount of times on a lever to obtain food’, said Fias.

Such observations gave rise to what has long been the predominant view that we humans are born with an innate sense of number, says the cognitive neuroscientist Daniel Ansari of the University of Western Ontario in London, Canada. The neuroscientific evidence seemed to offer strong support for that view. For example, Ansari said: ‘Studies with newborns and infants show that, if you show them eight dots repeatedly and then change it to 16 dots, areas in the right parietal cortex of the brain respond to a change in numerosity. This response is very similar in adults.’ Some researchers have concluded that we are born with a ‘number module’ in our brains: a neural substrate that supports later learning of our culture’s symbolic system of representing and manipulating numbers.

Not so fast, responds Núñez. Just because a behaviour seems to derive from an innate capacity, that doesn’t mean the behaviour is itself innate. Playing tennis makes exquisite use of our evolutionary endowment (present company excepted). We can coordinate our eyes and muscles not just to make contact between ball and racket but also to knock the ball into the opposite corner from our opponent. Most impressively, we can read the trajectory of a ball, sometimes at fantastic speed, so that our racket is precisely where the ball is going to be when it reaches us. But this capacity doesn’t mean that our early ancestors evolved to play tennis, or that we have some kind of tennis module in our brains. ‘The biologically evolved preconditions making some activity X possible, whether it’s numbers or snowboarding, are not necessarily the “rudiments of X”,’ explained Núñez.

Numerical ability is more than a matter of being able to distinguish two objects from three, even if it depends on that ability. No non-human animal has yet been found able to distinguish 152 items from 153. Chimps can’t do that, no matter how hard you train them, yet many children can tell you even by the age of five that the two numbers differ in the same way as do the equally abstract numbers 2 and 3: namely, by 1.

What seems innate and shared between humans and other animals is not this sense that the differences between 2 and 3 and between 152 and 153 are equivalent (a notion central to the concept of number) but, rather, a distinction based on relative difference, which relates to the ratio of the two quantities. It seems we never lose that instinctive basis of comparison. ‘Despite abundant experience with number through life, and formal training of number and mathematics at school, the ability to discriminate number remains ratio-dependent,’ said Fias.

What this means, according to Núñez, is that the brain’s natural capacity relates not to number but to the cruder concept of quantity . ‘A chick discriminating a visual stimulus that has what (some) humans designate as “one dot” from another one with “three dots” is a biologically endowed behaviour that involves quantity but not number,’ he said. ‘It does not need symbols, language and culture.’

‘Much of the “nativist” view that number is biologically endowed,’ Núñez added, ‘is based on the failure to distinguish at least these two types of phenomena relating to quantity.’ The perceptual rough discrimination of stimuli differing in ‘numerousness’ or quantity, seen in babies and in other animals, is what he calls quantical cognition. The ability to compare 152 and 153 items, in contrast, is numerical cognition. ‘Quantical cognition cannot scale up to numerical cognition via biological evolution alone,’ Núñez said.

A lthough researchers often assume that numerical cognition is inherent to humans, Núñez points out that not all cultures show it. Plenty of pre-literate cultures that have no tradition of writing or institutional education, including indigenous societies in Australia, South America and Africa, lack specific words for numbers larger than about five or six. Bigger numbers are instead referred to by generic words equivalent to ‘several’ or ‘many’. Such cultures ‘have the capacity to discriminate quantity, but it is rough and not exact, unlike numbers’, said Núñez.

That lack of specificity doesn’t mean that quantity is no longer meaningfully distinguished beyond the limit of specific number words, however. If two children have ‘many’ oranges but the girl evidently has lots more than the boy, the girl might be said to have, in effect, ‘many many’ or ‘really many’. In the language of the Munduruku people of the Amazon, for example, adesu indicates ‘several’ whereas ade implies ‘really lots’. These cultures live with what to us looks like imprecision: it really doesn’t matter if, when the oranges are divided up, one person gets 152 and the other 153. And frankly, if we aren’t so number-fixated, it really doesn’t matter. So why bother having words to distinguish them?

Some researchers have argued that the default way that humans quantify things is not arithmetically – one more, then another one – but logarithmically. The logarithmic scale is stretched out for small numbers and compressed for larger ones, so that the difference between two things and three can appear as significant as the difference between 200 and 300 of them.

The arithmetic and logarithmic scales for numbers up to 16. The higher you go on the arithmetic scale, the more the logarithmic scale gets compressed.

In 2008, the cognitive neuroscientist Stanislas Dehaene of the Collège de France in Paris and his coworkers reported evidence that the Munduruku system of accounting for quantities corresponds to a logarithmic division of the number line. In computerised tests, they presented a tribal group of 33 Munduruku adults and children with a diagram analogous to the number line commonly used to teach primary-school children, albeit without any actual number markings along it. The line had just one circle at one end and 10 circles at the other. The subjects were asked to indicate where on the line groupings of up to 10 circles should be placed.

Whereas Western adults and children will generally indicate evenly spaced (arithmetically distributed) numbers, the Munduruku people tended to choose a gradually decreasing spacing as the numbers of circles got larger, roughly consistent with that found for abstract numbers on a logarithmic scale. Dehaene and colleagues think that for children to learn to space numbers arithmetically, they have to overcome their innately logarithmic intuitions about quantity.

Maybe the industrialised cultures are odd, with their pedantic distinction between 1,000,002 and 1,000,003

Attributing more weight to the difference between small than between large numbers makes good sense in the real world, and fits with what Fias says about judging by difference ratios. A difference between families of two and three people is of comparable significance in a household as a difference between 200 and 300 people is in a tribe, while the distinction between tribes of 152 and 153 is negligible.

It’s easy to read this as a ‘primitive’ way of reasoning, but anthropology has long dispelled such patronising prejudice. After all, some cultures with few number words might make much more fine-grained linguistic distinctions than we do for, say, smells or family hierarchies. You develop words and concepts for what truly matters to your society. From a practical perspective, one could argue that it’s actually the somewhat homogeneous group of industrialised cultures that look odd, with their pedantic distinction between 1,000,002 and 1,000,003.

Whether the Munduruku really map quantities onto a quasi-logarithmic division of ‘number space’ is not clear, however. That’s a rather precise way of describing a broad tendency to make more of small-number distinctions than of large-number ones. Núñez is skeptical of Dehaene’s claim that all humans conceptualise an abstract number line at all. He says that the variability of where Munduruku people (especially the uneducated adults, who are the most relevant group for questions of innateness versus culture) placed small quantities on the number line was too great to support the conclusion about how they thought of number placement. Some test subjects didn’t even consistently rank the progressive order of the equivalents of 1, 2 and 3 on the lines they were given.

‘Some individuals tended to place the numbers at the extremes of the line segment, disregarding the distance between them,’ said Núñez. ‘This violates basic principles of how the mapping of the number line works at all, regardless of whether it is logarithmic or arithmetic.’

B uilding on the clues from anthropology, neuroscience can tell us additional details about the origin of quantity discrimination. Brain-imaging studies have revealed a region of the infant brain involved in this task – distinguishing two dots from three, say. This ability truly does seem to be innate, and researchers who argue for a biological basis of number have claimed that children recruit these neural resources when they start to learn their culture’s symbolic system of numbers. Even though no one can distinguish 152 from 153 randomly spaced dots visually (that is, without counting), the argument is that the basic cognitive apparatus for doing so is the same as that used to tell 2 from 3.

But that appealing story doesn’t accord with the latest evidence, according to Ansari. ‘Surprisingly, when you look deeply at the patterns of brain activation, we and others found quite a lot of evidence to suggest a large amount of dissimilarity between the way our brains process non-symbolic numbers, like arrays of dots, and symbolic numbers,’ he said. ‘They don’t seem to be correlated with one another. That challenges the notion that the brain mechanisms for processing culturally invented number symbols maps on to the non-symbolic number system. I think these systems are not as closely related as we thought.’

If anything, the evidence now seems to suggest that the cause-and-effect relationship works the other way: ‘When you learn symbols, you start to do these dot-discrimination tasks differently.’

This picture makes intuitive sense, Ansari argues, when you consider how hard kids have to work to grasp numbers as opposed to quantities. ‘One thing I’ve always struggled with is that on the one hand we have evidence that infants can discriminate quantity, but on the other hand it takes children between two to three years to learn the relationship between number words and quantities,’ he said. ‘If we thought there was a very strong innate basis on to which you just map the symbolic system, why should there be such a protracted developmental trajectory, and so much practice and explicit instruction necessary for that?’

But the apparent disconnect between the two types of symbolic thought raises a mystery of its own: how do we grasp number at all if we have only the cognitive machinery for the cruder notion of quantity? That conundrum is one reason why some researchers can’t accept Núñez’s claim that the concept of number is a cultural trait, even if it draws on innate dispositions. ‘The brain, a biological organ with a genetically defined wiring scheme, is predisposed to acquire a number system,’ said the neurobiologist Andreas Nieder of the University of Tübingen in Germany. ‘Culture can only shape our number faculty within the limits of the capacities of the brain. Without this predisposition, number symbols would lie [forever] beyond our grasp.’

If you’re inherently good at assessing numbers visually, you’ll be good at maths

‘This is for me the biggest challenge in the field: where do the meanings for number symbols come from?’ Ansari asks. ‘I really think that a fuzzy system for large quantities is not going to be the best hunting ground for a solution.’

Perhaps what we draw on, he thinks, is not a simple symbol-to-quantity mapping, but a sense of the relationships between numbers – in other words, a notion of arithmetical rules rather than just a sense of cardinal (number-line) ordering. ‘Even when children understand the cardinality principle – a mapping of number symbols to quantities – they don’t necessarily understand that if you add one more, you get to the next highest number,’ Ansari said. ‘Getting the idea of number off the ground turns out to be extremely complex, and we’re still scratching the surface in our understanding of how this works.’

The debate over the origin of our number sense might itself seem rather abstract, but it has tangible practical consequences. Most notably, beliefs about the relative roles of biology and culture can influence attitudes toward mathematical education.

The nativist view that number sense is biological seemed to be supported by a 2008 study by researchers at the Johns Hopkins University in Baltimore, which showed 14-year-old test subjects’ ability to discriminate at a glance between exact numerical quantities (such as the number of dots in an image) correlated with their mathematics test scores going back to kindergarten. In other words, if you’re inherently good at assessing numbers visually, you’ll be good at maths. The findings were used to develop educational tools such as Panamath to assess and improve mathematical ability.

But Fias says that such tests of supposedly innate discrimination between numbers of things aren’t as solid as they might seem. It’s impossible to separate out the effects of the number of dots from factors such as their density, areal coverage and brightness. Researchers have known since the studies of the child-development guru Jean Piaget in the 1960s that young children don’t instinctively judge number independently from conflicting visual features. For instance, they’ll say that a row of widely space marbles contains more than a densely spaced (and hence shorter) row with the same number. Furthermore, many studies show that arithmetic skill is more closely linked to learning and understanding of number symbols (1, 2, 3…) than to an ability to discriminate numbers of objects visually.

A s much as educators (and the researchers themselves) desire firm answers, the truth is that the debate about the origin of numerical cognition is still wide open. Nieder remains convinced that ‘our faculty for symbolic number, no matter how much more elaborate than the non-symbolic capacity of animals, is part of our biological heritage’. He feels that Núñez’s assertion that numbers themselves are cultural inventions ‘is beyond the reach of experimental investigation, and therefore irrelevant from a scientific point of view’. And he believes that a neurobiological foundation of numerosity is needed to understand why some people suffer from dyscalculia – an inability of the brain to deal with numbers. ‘Only with a neurobiological basis of the number faculty can we hope to find educational and medical therapies’ for such cases, he said.

But if Núñez is right that the concept of number is a cultural elaboration of a much cruder biological sense of quantity, that raises new and intriguing questions about mathematics in the brain. How and why did we decide to start counting? Did it begin when we could name numbers, perhaps? ‘Language in itself may be a necessary condition for number, but it is not sufficient for it,’ said Núñez. ‘All known human cultures have language, but by no means all have exact quantification in the form of number.’

‘How and when the transition from quantical to numerical thinking happened is hard to unravel,’ said Andrea Bender, a cognitive scientist at the University of Bergen in Norway, ‘especially if one assumes that language played a pivotal role in this process, because we don’t even know when language emerged. All research in developmental psychology seems to indicate that one needs to have culture before one can understand number concepts.’ Some archaeologists date numerical thinking back to the Palaeolithic era a few tens of thousands of years ago, Bender said, based on material remains such as notched bones or finger stencils – ‘but this is speculative to some extent’.

The digital age has made binary seem perfectly logical: what works best depends on what you want to do

Further complicating things, when different cultures developed the concept of number, they came up with varying solutions of how best to count. Although many Western languages count in base 10 – probably guided by the number of digits on our hands – they typically have a language rooted in a base-12 calendar, so that only at 13 (‘three-ten’) do the number words become composite. Chinese was more logical and consistent from the start, denoting 11 in characters as ‘ten-one’ and continuing that logical structure to higher orders, with 21 being ‘two-ten-one’ and so forth. Some researchers have claimed that this relative linguistic transparency accounts for China’s impressive numeracy (although the difficulty of the written script works in the other direction for literacy).

Or we could have adopted a different number system altogether. Take the people of the small island of Mangareva in French Polynesia. Bender and her coworkers found that the Mangarevans use a counting system that is a mixture of the familiar decimal system and another that is equivalent to binary. That might have seemed a peculiar choice before the digital age, which has made binary seem, well, perfectly logical. But which number system works best depends on what you want to do with numbers, Bender says.

For certain arithmetical operations involved in the distribution of food and provisions in Mangarevan society, binary can be simpler to use. In this setting, at least, it’s a good solution to a cultural problem. ‘Mangarevan and related Polynesian cultures seem to be great examples of inventing counting systems because they were more efficient for the tasks at hand,’ Bender said.

She feels that her findings support Núñez’s contention that, although humans have biological, evolved preconditions for numerical cognition, ‘the tools they need and invent are a product of culture, and hence are diverse’.

Núñez thinks that many of his colleagues might be too eager to attribute to biology and evolution certain capacities that also derive from culture, such as music. ‘Many animals have capacities for sound discrimination, for vocalising with various frequencies and intensities, and so on,’ he said. ‘That doesn’t necessarily mean that those are “rudiments of music”. Vocal tracts are needed for bel canto, but they did not evolve for bel canto.’

Perhaps at the root of the impassioned disputes over number sense is a desire to valorise certain traits and capacities – not just mathematics, but also art and music – by giving them a naturalistic imprimatur of biology, as if they would be somehow diminished otherwise. Certainly, the fierceness of arguments triggered by the cognitive scientist Steven Pinker’s proposal that music is parasitic on capacities evolved for other reasons – he called it ‘auditory cheesecake’ – betrayed a sense that the intrinsic worth of music itself was at stake.

Which is odd, when you come to think about it. The idea that a grand mental capacity comes from our culture – that we conjured up something beyond our immediate biological endowment, through the sheer power of thought – seems rather ennobling, not dismissive. Perhaps we should give ourselves more credit.

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The evolution of number in Otomi

Related Papers

Krasnoukhova, Olga. 2022. In Paolo Acquaviva & Michael Daniel (eds.) Number in the World’s Languages. A Comparative Handbook. [Comparative Handbooks of Linguistics, 5]. De Gruyter Mouton.

Olga Krasnoukhova

This chapter offers a typological overview of the number category in the indigenous languages of South America (SA). The focus is placed on number in independent personal pronouns and on nouns, as well as on verbal number. The discussion is centered around tendencies and patterns that SA languages show in number marking in the (pro)nominal and verbal domains. Whenever relevant, SA data are situated in a larger cross-linguistic context. The section on verbal number constitutes a first comparative account of this phenomenon in SA languages. It is shown that both types of verbal number (i.e. event number and participant number) are widespread in SA, occurring in most language families surveyed. Teasing apart verbal number (of the participant plurality type) and nominal number manifested on the verb (argument indexing) turns out to be an interesting challenge for SA, with many intermediate cases.

Manuel Delicado Cantero

David Felipe Guerrero-Beltran

Karijona is a Cariban language from Northwest Amazonia spoken by only 15 people. This paper describes the systems of grammatical number in the language. The data, collected in the Karijona settlements of Puerto Nare and La Pedrera in Colombia, were analyzed according to Basic Linguistic Theory. Karijona has a minimal-augmented number system, on which the first person inclusive have both minimal and augmented values. The number is expressed in pronouns, verbs, nouns, and postpositions.

This paper investigates the syntax and semantics of the NP in Karitiana, and assesses its implications for a semantic theory concerning the expression of the notion of number and of the mass/count distinction in natural languages. The paper also aims to assess the possibility of occurrence of arguments without the presence of functional material. We argue that the bare nominals in Karitiana have denotations of a cumulative nature. However, we also affirm that Karitiana makes a lexical distinction between mass and count nouns. In addition, we maintain that nominal arguments in Karitiana do not have functional constituents.

François, Alexandre. 2019. Verbal number in Lo-Toga and Hiw: The emergence of a lexical paradigm. In Frans Plank & Nigel Vincent (eds), The life-cycle of suppletion. Special issue of Transactions of the Philological Society. 117 (3). 338-371.

Alexandre François

Several languages around the world encode number through a regular alternation between verb roots, in a pattern sometimes called "verbal number suppletion" (Veselinova 2006). Lo-Toga and Hiw, two Oceanic languages of Vanuatu (Torres Islands), thus alternate certain verbs according to their absolutive argument's number — e.g. Hiw tō 'go:Sing' vs. vën 'go:Plural'. The pattern affects 17 verb pairs in Lo-Toga, 33 in Hiw. This rich system is a local innovation in the Torres Islands, not found elsewhere in Oceanic. This structure is here analysed for the first time. Verbal number is not just agreement: its principles and categories differ from nominal number. Despite its similarity with suppletion, the structure really involves separate words, organised into a "lexical paradigm" — a structured set of lexical pairs — contrasting individual vs. collective events. The comparative method helps reconstruct the system's development. A former circumfix encoding pluractionality was the source for the number alternation; yet most verbs encoded the contrast lexically, as near-synonyms were harnessed into the emergent paradigm. Crucially, even after it was recruited into the number paradigm, each verb remained an autonomous lexeme. While nominal number belongs to the morphology, the paradigm of verbal number in the Torres languages pertains entirely to the lexicon.

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Diachronica

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This paper advances the case that linguistics requires a unified theory of number, serviceable to both semantics and morphology, by proving that the morphological concept of augmentation and the semantic concept of cumulation are near logical equivalents. From this emerge an inventory of number features incorporating the categories ‘paucal’ and ‘unit augmented’, a typology of number systems crosslinguistically, and indication of other areas of likely convergence between semantic and morphological research.

Previously published in Portuguese as Diferenças entre termos numéricos em algumas línguas indígenas do Brasil. https://www.sil.org/resources/archives/2851

Diana Green

This article presents a brief description of numerical terms in more than forty indigenous languages of Brazil, bringing out the striking differences between numerical systems based linguistically on the concepts of 'one,' 'two,' `three,' 'five,' 'ten' and 'twenty.' Languages with systems based on one and two have limited numerical terminology, rarely more than terms for 'one' through 'six.' Those with systems based on ten and twenty sometimes have terms for numbers over one hundred. In these languages, the terms sometimes have an extremely complex system of affixes. Numerical terms in systems based on one or two may simply indicate a relational or global type of reasoning; the others demonstrate an analytical or synthetic approach. All the systems are quite logical and adequately serve the necessities of the peoples who use them.

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Understanding Evolution

Your one-stop source for information on evolution

The History of Evolutionary Thought

Natural selection: charles darwin & alfred russel wallace.

Pre-Darwinian ideas about evolution

It was Darwin’s genius both to show how all this evidence favored the evolution of species from a common ancestor and to offer a plausible mechanism by which life might evolve. Lamarck and others had promoted evolutionary theories, but in order to explain just how life changed, they depended on speculation. Typically, they claimed that evolution was guided by some long-term trend. Lamarck, for example, thought that life strove over time to rise from simple single-celled forms to complex ones. Many German biologists conceived of life evolving according to predetermined rules, in the same way an embryo develops in the womb. But in the mid-1800s, Darwin and the British biologist Alfred Russel Wallace independently conceived of a natural, even observable, way for life to change: a process Darwin called  natural selection.

The pressure of population growth

Interestingly, Darwin and Wallace found their inspiration in economics. An English parson named  Thomas Malthus  published a book in 1797 called  Essay on the Principle of Population  in which he warned his fellow Englishmen that most policies designed to help the poor were doomed because of the relentless pressure of population growth. A nation could easily double its population in a few decades, leading to famine and misery for all.

When Darwin and Wallace read Malthus, it occurred to both of them that animals and plants should also be experiencing the same population pressure. It should take very little time for the world to be knee-deep in beetles or earthworms. But the world is not overrun with them, or any other species, because they cannot reproduce to their full potential. Many die before they become adults. They are vulnerable to droughts and cold winters and other environmental assaults. And their food supply, like that of a nation, is not infinite. Individuals must compete, albeit unconsciously, for what little food there is.

Selection of traits

In this struggle for existence, survival and reproduction do not come down to pure chance. Darwin and Wallace both realized that if an animal has some trait that helps it to withstand the elements or to breed more successfully, it may leave more offspring behind than others. On average, the trait will become more common in the following generation, and the generation after that.

As Darwin wrestled with  natural selection  he spent a great deal of time with pigeon breeders, learning their methods. He found their work to be an analogy for evolution. A pigeon breeder selected individual birds to reproduce in order to produce a neck ruffle. Similarly, nature unconsciously “selects” individuals better suited to surviving their local conditions. Given enough time, Darwin and Wallace argued, natural selection might produce new types of body parts, from wings to eyes.

Darwin and Wallace develop similar theory

Darwin began formulating his theory of natural selection in the late 1830s but he went on working quietly on it for twenty years. He wanted to amass a wealth of evidence before publicly presenting his idea. During those years he corresponded briefly with Wallace (right), who was exploring the wildlife of South America and Asia. Wallace supplied Darwin with birds for his studies and decided to seek Darwin’s help in publishing his own ideas on evolution. He sent Darwin his theory in 1858, which, to Darwin’s shock, nearly replicated Darwin’s own.

Charles Lyell  and Joseph Dalton Hooker arranged for both Darwin’s and Wallace’s theories to be presented to a meeting of the Linnaean Society in 1858. Darwin had been working on a major book on evolution and used that to develop  On the Origins of Species , which was published in 1859. Wallace, on the other hand, continued his travels and focused his study on the importance of biogeography.

The book was not only a best seller but also one of the most influential scientific books of all time. Yet it took time for its full argument to take hold. Within a few decades, most scientists accepted that evolution and the descent of species from common ancestors were real. But natural selection had a harder time finding acceptance. In the late 1800s many scientists who called themselves Darwinists actually preferred a Lamarckian explanation for the way life changed over time. It would take the discovery of  genes  and  mutations  in the twentieth century to make natural selection not just attractive as an explanation, but unavoidable.

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• Go right to the source and read Darwin's  On the Origin of Species by Means of Natural Selection .
• Explore the  American Museum of Natural History's Darwin exhibit  to learn more about his life and how his ideas transformed our understanding of the living world.

Discrete Genes Are Inherited: Gregor Mendel

Early Evolution and Development: Ernst Haeckel

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