History of Computer Part 1: The story of 0 and 1.
When did human first learn to count?
Once you learn how to add two numbers, having to explain why two plus two equals four but not five or ten becomes quite a task. When I was in college, I worked in the school’s Math lab as a tutor. There were quite a few times when it got emotional trying to calm down a college student who was struggling her way to solve an algebraic expression. I felt bad that I couldn’t explain something that seems so natural to me that I have been taking it for granted. Just like numbers, once you learn it, it’s hard to unwrap your brain from their embrace. However, as natural and innate as they may seem, “ mathematical concepts are not wired into the human condition” [1]. It’s not something we were born with. According to Caleb Everette, an anthropological and cognitive linguist at the University of Miami, “mathematical concepts are learned and acquired through cultural and linguistic transmission [1]”.
The origin of the concepts, algorithms, and the development in computation dates back to the very early cultures. To write about the origin of numbers and counting would require a lot of research. This topic likely deserves its own series of blogs. In order to not veer too far off the main topic of this blog series, we’re going to touch on a very small aspect of it just so that we can all agree on our starting point in this journey back in time.
Professor Everette elaborates on the idea that quantities exist in the physical world, but numbers neither exist in a material world, nor are they part of our intellect. His argument was based on research conducted on the indigenous Amazonian people known as the Pirahã (pronounced [piɾaˈhã]). This research was performed by professor Everette and other anthropologists, and the Pirahã tribe was selected for this research because their counting system only consists of words which means, approximately, “one”, “two”, and “many”. In this study, a row of batteries was lined up on one side of the table, and a group of Pirahã adults participated in the research was asked to place the same number of batteries on the other side. When one, two, or three batteries were presented, the task was completed with no difficulty. However, as soon as the number of batteries increased, the Pirahã began to make mistakes. This observation led the researchers to an extraordinary realization: “the Pirahã’s lack of numbers meant they couldn’t distinguish exactly between quantities above three” [2].
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 Caleb Everette — Numbers and the Making of Us: Counting and the Course of Human Cultures [1]
So, how did people count before the number and writing systems as we know today existed?
With the complex numbering system that we have today, it can be hard to grasp that people were using small stones or other objects as numerical devices from time immemorial. As an interesting fact, the word “calculate” itself comes from the Latin “calculus”, which means small stone. These calculation methods introduced some elementary kind of abstraction (the use of an object to represent quantity), which is a significant milestone as it laid the foundation for the advancement of computational concepts going forward. Although no one knows the exact origins of math, it’s no doubt that it has some connection to agriculture and trade in early human evolution. In an interview with Smithsonian Magazine in 2017, professor Everette suggested that it’s likely that “the invention of numbers is a result of a co-evolution with the invention of agriculture and trade” [2]. In other words, numbers were developed to allow trading in more precise ways.
According to professor Everette, numbers play a significant role in the development of our culture and societies. Although we don’t know for sure, the fact that early forms of writing found in Central America, Mesopotamia, and China are highly numeric-centered suggests that there is a possibility that numbers led to the development of the writing systems we know today.
With the domestication of plants and animals and the development of trade networks during the beginning of the Neolithic period, people started recording information about their agricultural goods in the form of small clay tokens. Mesopotamian clay tokens were not the first accounting method developed by humans. By 20,000 years ago, Upper Paleolithic people were leaving tally marks on cave walls and cutting hash marks onto portable sticks. Clay tokens, however, contained additional information including what commodity was being counted, a significant step forward in communication storage and retrieval.
The Prehistoric Period, i.e., when there was human life before records documented human activity, roughly dates from 2.5 million years ago to 1,200 BC. It is generally categorized in three archaeological periods: the Stone Age, Bronze Age, and Iron Age.
The Stone Age is divided into three periods: Paleolithic (Old Stone Age), Mesolithic (Middle Stone Age), and Neolithic (New Stone Age) marked by the use of tools by our early human ancestors (who evolved around 300,000 BC) and the eventual transformation from a culture of hunting and gathering to farming and food production.
The Paleolithic period, a.k.a. Old Stone Age, was dated from 2.5 million years ago to 10,000 BC. In this period, early humans lived in caves or simple huts and were hunters and gatherers. They used basic stone and bone tools, as well as crude stone axes, for hunting birds and wild animals.
The Neolithic period, a.k.a. New Stone Age from 10,000 BC to 3000 BC, was characterized by stone tools shaped by polishing or grinding, dependence on domesticated plants or animals, settlement in permanent villages, and the appearance of such crafts as pottery and weaving. [2]
Clay tokens with different shapes were used to express numerical quantities of goods being counted. These tokens also allowed for agriculturalists to keep track of animals and food that had been traded, stored, and/or sold. Tokens were publicly used to pool together community surplus for the preparation of the religious festivals that constituted the backbone of the redistribution economy, which was now administered at the temple by priestly rulers [4]. The tokens helped leaders keep track of the types and quantity of goods collected and redistributed as offerings to the gods and various community needs.
These tokens made of clay modeled into many shapes such as miniature cones, spheres, cylinders, disks, and tetrahedrons were counters. These “tools of the mind” gave us some insights into human cognition.
The tokens used a way of counting fundamentally different from ours. Our abstract number system, e.g., “one”, “two”, “three”, are independent of the type of items counted and, therefore, universally applicable. That was not the case at the time when tokens were used. Between 7500 BC and 3100 BC, each category of item was counted and with its own counter, reflecting the fact that counting was “concrete”, meaning that each category of item was counted with a special numeration or special number words specific to that particular item. For instance, small and large units of grain were counted with cones and spheres, oil with ovoids, animals with cylinders, and the units of labor with tetrahedrons[4]. This is referred to as one-to-one correspondence. Two jars of oil were shown by two ovoid tokens. These so-called “complex” tokens sometimes assumed the shape of the items they symbolized, such as garments, tools, furniture, etc.
During the Uruk period in Mesopotamia (4000–3000 BC), urban cities blossomed, and administrative needs for accounting expanded, and so did the token types. During the Late Uruk period (3500–3100 BC), tokens began to be kept in sealed globular clay envelopes called “bullae”. Bullae are hollow clay balls about 2–4 inches in diameter. The tokens were placed inside the bullae, and the opening pinched shut. The exterior of the bullae was stamped with a unique seal to identify the owner.
The Uruk period (4000–3000 BC) of Mesopotamia is known as the Sumerian state. It was the time of the first great blossoming of civilization in the Fertile Crescent of modern-day Iraq and Syria. The Early Uruk period (4000–3500 BC) is signaled by an abrupt change in settlement pattern from the preceding Ubaid period. During the Ubaid period, people lived primarily in small hamlets or one or two largish towns, across an enormous chunk of Western Asia, but at the end of it, a handful of communities began to enlarge. [5]
So, what was the purpose of “bullae”?
Bullae was more than an accounting tool; they were actual contracts. Everything we think of as a financial instrument today is, in fact, a contract. A government bond, for instance, is a contract between the government and the bond-holder to guarantee a series of payments in the future. A share of stock is a contract between the shareholder and the corporation that guarantees participation in the profits of the firm, and a right to vote on management. Although contracts existed before the invention of writing, and even the invention of bullae, the hollow clay balls, and their tokens appear to be the earliest archeological evidence of contracts.
The communal offerings to the temple continued, but the types of goods, their amounts, and the frequency of delivery to the temple became regulated, and non-compliance was penalized. The response to the new challenge was the invention of envelopes where tokens representing a delinquent account could be kept safely until the debt was paid.[4]The seals impressed on the exterior of the bullae are the Mesopotamian equivalents of signatures. They represent a personal mark indicating the promise of the owning party. These seals served a function comparable to that of our notary public today. They were a third party attesting to the validity of the tokens inside, which were kept by the lender as evidence of the agreement. These tokens tangibly symbolized the obligation. In order to be able to verify the content o the envelope without breaking it, the tokens were impressed on the surface before enclosing them. These impressed tokens on the exterior of the bullae act as a label representing both the type and quantity of goods being contracted.
The bullae do not specify the time or interest rates. Instead, they were contracts that bridged a period of time — from the moment when one party entered into an obligation, to the moment when an obligation was discharged. We do not know whether the obligation is a return of a loan, or simply a tax or tribute to the temple. All we know is that someone made a promise to give some commodities represented by the tokens (jars of honey, sheep, cattle, etc.) to the person possessing the bullae, the lender. The fact that the bullae were pinched shut seems to suggest that the owning party was concerned that the bullae holder might insert additional tokens without their consent.
How bullae led to the invention of Cuneiform, the ancient Sumerians of Mesopotamia’s first system of writing?
The use of clay tokens and bullae was simple and sufficiently solved the need for bookkeeping. However, as time went by, the Sumerians realized that the use of the clay tokens was redundant and cumbersome, especially when dealing with a large quantity of goods, since the impressed tokens on the exterior of the bullae already served the purpose of conveying both the type and quantity of trade goods. The cones and spheres symbolized the measures of grain became wedge-shaped and circular impressed signs. Reducing the 3-dimensional tokens to 2-dimensional signs proved revolutionary as it was a precursor to the invention of writing. The tokens enclosed in hollow envelopes were soon replaced by their marking impressed on the clay tablets, the early form of Cuneiform.
The unprecedented volume of goods to be administered, as the redistribution economy reached a regional scale, challenged writing to evolve in both form and content.
Form:
About 3100 BC, the form of the signs changed when a pointed stylus was used to sketch more accurately the shape of the most intricate tokens and their particular marking [4]. It seems someone realized it was quicker and easier to produce representations of such things as animals, rather than naturalistic impressions of them. The sign for oil, for example, reproduced the avoid token with its circular line at the maximum diameter.
Content:
Plurality was no longer indicated by one-to-one correspondence. The number of jars of oil was no longer shown by repeating the sign for “jar of oil” as many times as the number of units being recorded. People soon realized that the existing method of calculation did not go far enough to satisfy their increasing needs. For example, to count 1000 jars of oil, they would have had to impress the corresponding tokens used for “jar of oil” on the tablet 1000 times.
Surprisingly, instead of creating new signs to symbolize the numerals, the impressed signs for grain took on numerical values. The wedge that formerly represented a measure of grain came to mean , and the circular signs, formerly representing a measure of grain, meant . “ For the first time, signs expressing numerals abstracted the concept of number from that of the item counted”. While tokens were used in one-to-one correspondence counting, writing introduced the development in abstract numerals. For example, the figure below illustrates an account of 33 jars of oil indicated by three circular impressed signs (10 + 10 + 10) and three impressed wedges (1 + 1+ 1) followed on the right by the incised sign for “jar of oil”. The invention of abstract numerals made obsolete the use of different counters and numerations to count different products. With the abstraction of numbers, counting had no limit. [4]
So, how did we arrive at the numbering system as we know today?
As time went by, different civilizations came up with different ways of recording higher numbers. Many of these systems, Greek, Hebrew, and Egyptian numerals, were just extensions of the abstract numerals concept as described above with new symbols added to represent a larger magnitude of value. Each symbol was repeated as many times as necessary, and all were added together. Roman numerals added another twist. If a numeral appeared before one with a higher value, it would be subtracted rather than added. For example, the letter “V” represents the numerical value of “5” in our number system, and the letter “I” represents the numerical value of “1”. So, “IV” would correspond to the numerical value of “4” (5–1 = 4). This system eliminates the need to come up with a numeral symbol for the number “4” or having to use repeated numerals as in the case of an addition, e.g., four of the letter “I”. However, even with this innovation, it was still a cumbersome method for writing large numbers.
The way to a more useful and elegant system lay in something call positional notation. With the previous number systems, one needed to draw many symbols repeatedly and either added and subtracted the numerals accordingly to make up the final value. In addition, new symbols needed to be invented for each larger magnitude. The positional system solves those problems by reusing the same symbols and assigning them different values based on their position in the sequence.
“Several civilizations developed positional notation independently, including the Babylonians, the Ancient Chinese, and the Aztecs. By the 8th century, Indian mathematicians had perfected such a system, and over the next several centuries, Arab merchants, scholars, and conquerors began to spread it into Europe”. [6]
This was a decimal (base 10) system, which could represent any number using only ten unique glyphs. The position of these symbols indicates different power of ten, starting on the right and increasing as we move left. For example,
316 = 6 x 100 + 1 x 101 + 3 x 102
A key breakthrough of this system, which was also independently developed by the Mayans, was the number “0”. Older positional systems that lacked this symbol would leave a blank in its place, making it hard to distinguish between “63” and “603”. The understanding of zero as both a value and a place holder make for reliable and consistent notation. Of course, it’s possible to use any ten symbols to represent the numerals “0” through “9”. For a long time, the glyphs varied regionally. Most scholars agree that our current digits evolved from those used in the North African Maghreb region of the Arab Empire. And by the 15th century, what we know as the Hindu — Arabic numeral system had replaced the Roman numerals in everyday life to become the most commonly used number system in the world. [6]
And, of course, we can’t end this post without mentioning the base “2”, or binary system, used in all of our digital devices. Programmers also use base “8” and base “16” for more compact notation.
References:
- Everette, Caleb. Numbers and the Making of Us: Counting and the Course of Human Cultures. Harvard University Press, 2017.
- How Humans Invented Numbers-And How Numbers Reshaped Our World
https://www.smithsonianmag.com/innovation/how-humans-invented-numbersand-how-numbers-reshaped-our-world-180962485/ - “Wangfujing Paleolithic Site Mural” by Gary Lee Todd, Ph.D. is marked with CC0 1.0
- Tokens and Writing: the Cognitive Development | Denise Schmandt-Besserat
https://sites.utexas.edu/dsb/tokens/tokens-and-writing-the-cognitive-development/ - Uruk Period Mesopotamia: The Rise of Sumer | Kris Hirst
https://www.thoughtco.com/uruk-period-mesopotamia-rise-of-sumer-171676 - A Brief History of Numerical Systems | Kid PID
https://www.kidpid.com/a-brief-history-of-numerical-systems/
Originally published at https://www.spiritualitydecoded.com on October 4, 2020.
