Zero: Nothingness Matters- Origins and Development of “A Dangerous Idea”
2 days ago, one of the better Twitter handles @Schandillia posted a very lucid thread- with credible enough sources, to put forth the idea that the credit of “inventing” zero can’t be unequivocally given to the Indians. He rightly cited instances of development of the concept of place value systems in Babylonian world, concept of Zero in Ancient Egypt and Mayans as well as the use of rounded form of the symbol in ancient China; ending with positing “No one civilization can claim monopoly over something as broad as zero. Different improvisations emerged in different parts of the world as and when the needs were felt. There’s little value, then, in romanticing a fictitious antiquity where one culture owns it all.”
This blog post (also posted as a twitter thread) is my take on the story; with some corrections to Amit’s post — where facts don’t match. A tale of nothingness, which may throw surprises at you, and while it follows pretty much the same course as posted in the original thread, I hope to conclusively prove different conclusion wherein Ancient Indians were the rightful “inventors” of Zero, more than Edison was of Electric Bulb, Galileo of Telescope or Newton of Calculus.
Before starting the history tour though, it’s important to discuss 2 basic concepts of arithmetic-
1. In arithmetic/ measurement, one has to distinguish between the notion of size, which leads to cardinal numbers, and the notion of position, which leads to the ordinal numbers. In other words- Cardinal numbers tell ‘how many’ of something, they show quantity. Ordinal numbers tell the order of how things are set, they show the position or “which one” of things.
2. The other key concept is the elementary notion of placevalue- the value of a digit based on where it is positioned in a number. Closely linked with the ordinal aspect of numbers; it is a very important notion, which took centuries to develop.
To elaborate- the place-value of the number 5 in 156 is different from that in 561; even as its representation and cardinality does not change. “How many” hundreds in 561, and “How many” tens in 156 will give you the same answer 5.
Notice that the ordinality in a place-value relates to its base. Thus in the decimal system (base 10); each place-value is 10 times its nearest one; in hexadecimal, the multiple is 16; and in Sexagesimal systems it would be 60. A number’s value comes from its ordinality — from its position compared with other numbers. Thus a number, has to have a value on its own AND is separate from the placevalue its numeral (the symbol for a number) may ordain ( that is to say, the numeral 6 in 562, 106 etc is different from number 6)
Even today, we sometimes treat zero as a non-number even though we know that zero has a value, using the digit 0 as a placeholder without connecting it to the number zero. Look at a telephone or the top of a keyboard. The 0 comes after 9, not before 1 where it belongs.
For telephones, it has to do with the fact that COUNTING precludes Zero, and the first (rotary) telephone systems were basically counting pulses sent over the lines. 0 was next to 9, corresponding to 10 pulses (except in NZ & Sweden, who had different pulse counting systems)
Moving on with the story of counting and my take on why Indians, are right in claiming “Zero diya mere bharat ne” (My India gave Zero)
The Story of Counting: A 40,000 year old saga
This is primarily a tale of man’s passion of counting & tabulating and problems lying therein. Starting from atleast 44,000 year ago with bones in South Africa used as tallying stick, through other bones found in Congo, Czechoslovakia, & pebbles, counting moved onto clay tokens around the same time as agriculture dawned.
By 8000 BC, complex tokens representing different quantities, & things were in vogue-mainly in mid-east & levant. And 2 distinct systems of token management — involving strings (called Bulla) and envelopes- had been developed by 4th Millenium BC. However, the concept of NUMBERS was limited to the “talent” of counting.
The use of tokens for tallying did mean development of “bases” by 3000BC- of 10 (in Egypt), 20 (Central Africa) & 60 (in Sumer/Babylon). Rather than count from 1–100 individually, it is quicker (and a sign of talent in early times) to count in 5s, 10s or 20s.
Counting, by definition excludes the concept of “nothing”. When answering “how many?”, one never needs to keep track of 0 sheep nor is it “natural” to start counting from 0. So there was no need for a zero as a number.
But that was about to change. ’cause between 3500 -3000 BC, for reasons still not well understood, the area of Southern Mesopotamia underwent a sudden growth and change, centred in the cities of Ur and Uruk. These folks-Sumerians-were the 1st people to introduce arithmetic & a mensuration system.
The main part of the third millennium, now called the Early Dynastic period, gave humanity its earliest “verifiably dated” literature in The Instructions Of Shuruppak (it outdates Vedas by 800–1000 years), the first poetess Enheduanna………& the first school mathematics.
By about 3000 BC, the Sumerians were already drawing images of the previous tokens on clay tablets, for ease of storage, with different types of goods represented by different symbols, & multiple quantities by repetition. So 3 units of grain were denoted by 3'grain-marks’, 5 sheep by 5'sheep-marks’ etc.
Advent of Numerals and number systems and the first “Zero”
But with increasing complexity of trade, & more importantly- with wars ending in increased territories and prisoners/ slaves- this was fast becoming an unmanageable system of too many symbols. The next great innovation in our story of numbers was not just the separation of quantity of the good from the symbol for the good (a NUMERAL) , but also the system of “measuring” non-integral things- like land area and volumes: introduction of reciprocals & fractions
By 2800 BC tablets like attached image were common. In this, The circular imprints stood for tens and the wedges for units. The incised figure to the right is a depiction of a jar of oil, and this tablet was a record of, in total, 33 jars of oil.
Unlike us, Sumerians used different systems for counting discrete objects, such as animals, vis-a-vis measuring areas or volumes. Each system had a collection of signs denoting various quantities. In the basic sexagesimal system used for counting discrete objects- sheep/slaves, a small cone =1.
10 cones = small circle, 6 small circles = big cone, 10 big cones = big cone around a circle, 6 of those = large circle ; 10 large circles = concentric large & small circle
The system for measuring halvable items-like bread -was different, so were those for land measures. Each system had a collection of signs denoting various quantities. Adding to the confusion was the fact that a single sign might be used in several systems, with diff values.
Gradually, over the course of the third millennium, these signs were replaced by cuneiform equivalents so that numbers could be written with the same stylus that was being used for the words in the text.
As trade grew, so did comms with other cultures (eg Indus Valley). The need of book-keeping system to track exchange of goods, & standardized notations in harmony with prevalent ones-i.e. bases 5,6,10,20 led to the entry around 2100 BC, of sexagesimal place value system.
While the story of sexagesimal system was progressing around Euphrates; Peruvian Andes had seen Norte Chico folks building cities without knowing sophisticated maths, Indus Valley bloomed in India, unfortunately with yet undeciphered evidence of their writings & maths.
This was also the beginning of Mayans in central America who would go on to create the complex long count Calendar and indeed independently invent zero, depicted by a shell/eye in their Vigesimal (base 20) system- but 15 centuries later.
In Central asia Sintashta culture was already established, which would give rise to India’s next civilization 1000 years later, just around when cultures would spring in China, North America and Europe
And offcourse- Pyramids had been built around Nile following a decimal system of computation, but Egyptians didn’t have a place value system- thereby 10,100,1000 each had a distinct symbol- much like the Greeks and then Romans who were influenced by them.. Egyptians had a calendar, were doing astronomical predictions, managing taxation and accounting systems -all without needing a zero- well, nearly!
For, they did use a symbol — nfr for depicting a base for directional change first in construction measurements for pyramids- x cubits “above/ below” nfr and then much later, in accounting tables.
Papyrus texts from much later period of around 1750 BC, written in Heiretic, have shown its usage in accounting texts to show “balancing” of accounts on income and expense side- a form of directional change.
It is at around this time that the Old Babylonians came to the forefront in place of Sumerians- under Hammurabi and Ammi-saduqa. They continued to use the sexagesimal place value system created around 2100 BC.
Instead, Babylonians used gaps between numerals, to handle the confusion caused by “absence” of place values as depicted in Plimpton 322, a tablet from 1800 BC.
This system continued through another 1500 years, while Kassites, Assyrians, Elamites, Neo-Babylonians, and First Persians came & went; then arose on the scene, the empire of Alexander. In this intervening period, cultural exchanges flourished from Greece to Egypt to India.
Beginnings of Arithmetic in Greece, South America & India
While there was this sort of stagnancy in middle-eastern maths; Indians maths had flourished, primarily in geometry and astronomy but also in trigonometry- first through the Vedanga Jyotish of Lagadha around 1200–800BC and later through Sulabha Sutras of 8th century BC.
By the time Alexander arrived on the Scene, Kharosthi numerals were already in use in parts of India for 200 years. More like Roman numerals, these are the earliest recorded “numbers” in Indian subcontinent; There is no zero and no separate signs for the digits 5–9
In parallel, China had seen the rise of its own mathematics,with Zhou Bi Suan Jing being written around 500–300 BC. They had developed their own numerals, first around 1400 BC (Shang numerals) and then a newer Rod numerals (475 BC).
In Europe, Greeks had inherited Aegean numerals from Linear B of Mycenaen and used them extensively till 5th century BC when the Miletus’ alphabet based system took firm root, to be copied by Romans later.
In South America, we had Mayans mathematics now beginning to flourish and the first recorded long count calendars start to appear between 5th-3rd century BC. The Incas continue with their Quipus, an abacus styled system of ropes with knots, which are not “written” numerals anyways.
Just to put things in perspective, its now ~300BC and we do not have a recorded use of ZERO, or an equivalent symbol- even as a placeholder- ANYWHERE in the world. The nearest version we have are GAPs in the sexagecimal system of Babylonians. The same gaps that appear in Incas’ Quipus, causing even more ambiguity: 210 and 2100 cannot be distinguished on a rope.
A Void: And then there was None
It is at this time of Chandragupta in India, Seleucids in Babylon, ROman republic in Europe; when we get our first of the many back to back Eureka moments in this 4000 yr long saga. Somewhere in the realms of early period of Seleucid empire (circa 300–280 BC), the Babylonian system introduced the slanted double wedge, as a placeholder, to replace that blank space. It was not the first symbol for this purpose, but one which would stick long enough to be popular.
However, It didn’t really have a numerical value of its own- it was a digit, not a number. How do we know that? It was used only in the medial positions, not on right-hand side of the number, as in numbers like 100,1000 etc. Nor did it appear anywhere in isolation, on its own.
Thus to remove any ambiguity, Babylonians DID NOT give the “number” zero to the world.
While Babylonians introduced a placeholder symbol for zero; within next 100–150 years, Pingala in India gave “nothingness” a notation symbol in his Chhanda-Sutra, a name- Sunya. Neither of these two still had a VALUE associated with the respective symbols.
More importantly, Indian numerals were still more like Greek ones, even during the reign of Ashoka- alphabet like representations for digits in Brahmi and NO place value. In 1st century CE pottery found in Tamilnadu, Brahmi inscriptions have been found with numerals in non-place value system.
And at just around the start of common era, while the rest of the world was grasping with problems of place-values, numerals in alphabets etc; Mayans introduced a ZERO- at around 37 BC, in form of a crescent moon glyph. It was an independent number in its own right, and used in their placevalue system as well. But their system was isolated from the rest of the world.
It used glyphs — written or carved symbolic signs — that were not suitable for economy of notation. Net result, Mayan system remained largely intrinsic to them. Mayans DID NOT “give” Zero to the world.
Now sometime around the 2nd-4th century, Indian mathematicians changed their style of numbering; & moved from Greek-type system to Babylonian-type place-value one. An important difference being that Indian numbers were decimal based. Why they did so, is anybody’s guess, but some attribute this to rise of Indo-Scythians, also associated with the rise of new, Shaka calendar.
It ought to be noted, that while Bakhshali manuscript (variously dated from 3rd-8th century AD), has the first recorded use of a dot like symbol for zero, in a place value form; the script used is Sharda; which has a very different numeral representation from what is the common “Indian-Arabic numerals” we identify with.
The same dot like symbol later appeared in Khmer temple inscriptions of Sambor, Cambodia in 683AD.
This was also the beginning of a period of mathematical golden age in India, along with literary references to numbers- be it counting till 10 to power 53 in Lalitavistara, or the Jain cosmological text Lokavibhaga- originally written in Prakrit, on a very specific date- corresponding to 25/08/458 CE. However, the actual surviving text is a Sanskrit translation, written in 6th century.
This text would have held the title to oldest known mention of zero, in its date, had it been not destroyed; and further re-dating of Bakhshali manuscript.
And then came the era of Aryabhata, Varahamihir, & most importantly Brahmagupta who enumerated the rules for operations involving Zero, in his seminal work of 629 CE- Brahmasphuṭasiddhānta- rules which stand correct; barring that of division by zero. THAT is when zero becomes a number in its own right; with a place of its own on the number line.
What’s interesting is that this text doesn’t itself use numerals and thus we can’t say whether the round circular symbolism for zero was around by then or not. However, texts/ inscriptions from lands in SE Asia, with Sanskrit influence began showing the eponymous 0 around the same time.
There are the Kedukan Bukit, Talang Tuo inscriptions in Pallava script from 682/83CE,in Malay Language, but borrowing heavily Sanskrit words- which show the numbers 604 and 608 with the round circular symbol representing zero.
These along with those found in Cambodia, become significant once you consider that these places were enroute to the cultural and trade route between India & China.Champa, Khmer, Java,Malay people borrowed not only words but also numerals from India, just as they did the Shaka Calendar. Given that the oldest Chinese record for round circle comes from 13th century onwards, its almost likely that they got their Zero symbol from Indian sources, through these routes. In any case, China DID NOT give zero to the world.
Note the circular symbol- Xing in Chinese characters was a temporary inclusion, in 689 AD by Empress Wu; and didn’t represent any numeral; but rather was a word for ‘star’. When the Wu dynasty ended in 705 AD, the original words were recovered as people eventually forgot how to write the introduced characters.
Appropriating credit where it is due
Aryabhata DID NOT “invent” zero- he used the term kha for a concept in decimal place-value system- which wasn’t numeric- but rather an alpha-syllabic numeral system which attributes a numerical value to each syllable in Sanskrit of the form consonant+vowel possible- from Ka=1 to Hau. That actually is quite odd, as the new Brahmi numerals had begun gaining favour over the old Greek-style alpha-numerals by this time in India. Some historians have thereby given credence to the theory that Aryabhata was from Kerela, where it took a couple of centuries for the new system to reach.
Thus the place-value status to “nothing” was given during Seleuids, the concept of an independent symbol to represent it was Pingala’s doing 500 yrs ago; and the actual association of it as an independent number with value, which could be operated upon was given later by Brahmagupta. Its round circular shape came around same time, in India, but from yet unknown source.
It would take another 200 odd years for this notation to gain sufficient hold on inscriptions and texts, 500 years for it to travel via the Arabs to reach Europe and even longer to get universally accepted as a number in the form we know it today.
If we are to construct an unbiased alternative to the Eurocentric (or even an Indo-centric) trajectory, our guiding principle should be to recognize that different cultures in different periods of history have contributed to the world’s stock of mathematical knowledge, but as far as current ZERO goes in all its relevant bits, definitely “Zero diya mere Bharat ne”!
