"The tracing of zero’s heritage has been elusive. For a country to be able to claim the number’s origin would provide a sense of ownership and determine a source of great nationalistic pride."
As seen in some of the comments below. As if they personally made it possible. It's like your country winning a sports tournament and claiming that "WE" won. No, you didn't. You just watched it.
Yeah, Because Nationalism across the world is all time low!
Besides, The two main contenders for the title from Cambodia and Indonesia have stone inscriptions with 'Zero' for the 'Saka' calendar year; So even if the inscriptions are older, The origin of zero likely points to somewhere in Indian subcontinent of Saka Era.
The author seems to conflate between 'Origin of zero' and 'Latest evidence for oldest occurrence of zero'.
I don't think professors dealing with Epigraphy, Philology (or) Archeology can make such a mistake unless writing a click-bait article with misleading content was the goal.
Almost. Behind "WE/US" is the society which builds and runs sports facilities (operating quite often at loss) implements sport education programmes for children and youth etc. So, no it is us being represented by the few talented ones.
The difficulty, I think, is in establishing that "nothing" is a tangible object that can be manipulated as if it were a "regular" number.
This may be a little easier to grasp if you look at language. In many languages (most? nearly all?), you generally don't express lacking food as saying "I have 0 food" but as saying "I do not have food"--the concept of nothing is essentially reflected as turning the verb (or clause/sentence as a whole) into a negative mood, rather than indicating that the count of an object is 0.
I like to imagine the situation of a tax collector in ancient Mesopotamia, going around and asking the farmers how many animals they have.
If the farmer has 5 cows, they have a number of cows that you can put in a form on a clay tablet.
If the farmer has no goats, they don't have "a number of goats", they have no goats to count. Writing a number on the tax form doesn't make sense.
If numbers are only used to count actual things in the real world, zero is a very odd concept.
We realized later that having "a number" to represent "none" simplifies many things and rids us having to deal with special cases. And if we teach it to children early enough it isn't too confusing.
Yes, but when you have a set of goods to proportion out and you are complete, then there is a zero. So it may be quite common. The Egyptians indicated a concept like zero with the NFR sign, for perfect. That’s the one with a circle and a cross on top.
But, no decimal placeholder system, like described in the article.
> This may be a little easier to grasp if you look at language. In many languages (most? nearly all?), you generally don't express lacking food as saying "I have 0 food" but as saying "I do not have food"--the concept of nothing is essentially reflected as turning the verb (or clause/sentence as a whole) into a negative mood, rather than indicating that the count of an object is 0.
It is equally common to say: "I have no food" or "I have nothing".
I can go on: "I have no money". "I have no cattle".
So I don't know that your argument is a compelling one.
No is a boolean operator, not a number. "Do you have any bread?" ... "No. I have no bread." It's the opposite of yes, or some. No or none isn't countable, it's the opposite of any positive quantity, that's why the concept of zero was so counter-intuitive.
It's far more common to say "I haven't got any bread". There's no word or phase in there that's proxying for a number.
Your example ("No. I have no bread"), however, uses "no" in two different ways.
The first is negation and the second represent the null/empty set ("no bread"). And the empty set is not the same as zero.
You don't have to go back to the Middle Ages to find people having difficulty with the concept of zero. A couple of more recent examples from a paper by Eric Hehner (http://www.cs.toronto.edu/%7Ehehner/RPUTP.pdf):
"In the 1991 Toronto phone book, there is a page that helpfully gives the time difference to various places in the world; to the U.K. it says "+5", and to Costa Rica it says "-1". But to Cuba it says "NA", and the legenda explains "time difference not applicable". By 1996 they tried to correct it; for Cuba it says "=", with the same explanation. In 1997 they discovered the number 0 , but they felt the need then, and still do today, to explain that 0 means "no time difference"
"The Fortran language of 1955 had a loop construct, but its body had to be executed at least once; I suppose it seemed senseless to have a loop whose body might be executed 0 times. The error was corrected in Algol in 1958, and in PL/I, and in Pascal, in part: iteration might be 0 times, but the data structure over which one is iterating, the array, had to have at least one element. In Pascal that meant there was no null string. And that put the algebra of data structures back where the algebra of natural numbers was prior to 1930."
The article, and the overall discussion is really about having a symbol in your numeric alphabet for "none," not about having a conception of nothing. Even more particularly, it's about the use of that symbol in a positional symbolic system for representing natural numbers. That is the breakthrough generally accredited to India, not the idea that before you have 1 thing, you have no (zero) things.
I'd say it is about answering the question "How many?".
It becomes natural to give a distinct symbol for each answer smaller than a given limit say 10. If you have symbols 1, 2, 3, .... 9 then it would be pretty obvious you would also want a symbol for zero.
The positional way of expressing numbers is a separate more evolved development of course.
In Roman numbers you had I, II, and III, ... So why didn't they have say '-' to symbolize less than "I"? Maybe they just expressed it with the word "none".
> The stone bore a Khmer epigraphic inscription that included the date for the Khmer year 605, reckoned within the Hindu Saka system, a historical calendar based on the rule of the Indian emperor Shalivahana.
Seems there is a pretty good chance that the origin of zero is still Indian and was adopted by the Khmer just as they adopted the Indian calendar.
Isha Upnishad[0] was written in 1st half of 1st millennium BCE as per the conservative estimates. Its Shanti Mantra(wishing peace to all living beings) of says:
ॐ पूर्णमदः पूर्णमिदं पूर्णात्पूर्णमुदच्यते ।
पूर्णस्य पूर्णमादाय पूर्णमेवावशिष्यते ॥
ॐ शान्तिः शान्तिः शान्तिः ॥
Om Puurnnam-Adah Puurnnam-Idam Puurnnaat-Puurnnam-Udacyate |
Puurnnasya Puurnnam-Aadaaya Puurnnam-Eva-Avashissyate ||
Om Shaantih Shaantih Shaantih ||
Rough Translation:
That (god) is complete(puurnnam), this living being is complete. From completeness emerges completeness. When this completeness (living beings) emerges from that completeness (god), only completeness remains.
This verse describes God. But it can also be describing concept of zero and infinity. Buddhist philosophy emerging in 500 BCE is based in shoonyata[1] (nothingness) so concept of zero was definitely known for a long time.
As far as mathematical applications are concerned, we can't say for sure as a lot of knowledge was lost to invasions which specifically targeted universities like Nalanda[2].
Hindus imagined time scales of microseconds to trillions of years which in IMHO difficult to imagine without zero based mathematics[3].
Having said that there are a large number of manuscripts in the hands of small groups and families are yet to be discovered so these dates.
Thanks, but that's such a narrow definition of the concept.
Shukla Yajurveda[0] conservatively estimated to be written in 1200- 800 BCE period in 17.2.20 constructs powers of 10 upto a billion by names by adding shunya successively to 10.
This is the concept of zero. The title said "The Elusive Origin of Zero". Widening the definition to something else doesn't expand understanding but creates confusion.
As Swami Sarvapriyananda is fond of saying, this verse encapsulates the essence of Advaita Vedanta. This verse is in fact his favourite. He has given hours and hours of talks on it. There is a lot packed into it.
Maybe. It would be odd, though, not to find any Indian examples of zero from earlier than ~200 years after the oldest Khmer one if the invention came from India.
That area first came under Indian influence centuries before these dates. Some parts of the country, Chenla, were dependent on states in Java and Srivijaya (the one mentioned in the article) later on. The Indian cultural sphere as a whole covered a huge area and the Subcontinent itself certainly did not have a monopoly on culture there.
India as a whole didn't even exist back then. Why don't historians describe only in terms of ancient kingdoms or whatever geographical entities existed back then and then describe what current political boundaries contains that area today?
Given the only entity you think of India to be is a Nation State, it didn't. Which for most "west" educated people is the only type of state that can exist. But entities can exist beyond concept of a Nation State. Unfortunately, its not very easy to unlearn years of "one good way" to think.
Did Germany exist during Roman Empire or during Holy Roman Empire ? It did not as a nation state, but the people almost always felt connected.
India had much stronger bonds among people than say typical Germany, the religion/culture had elements from the entire nation, created by leaders from different parts of continent. Hinduism has verses conveying rivers and Gods from around the subcontinent as important aspects of life. Adi Shakaracharya, the religious teacher who is considered the person who made Hinduism the dominant form from among other existing interpretations of "Dharma" belongs to the southernmost tip of the region and created hindu institutions in all corners of subcontinent.
Bhakshali Manuscript is one of the oldest Indian documents with use of Zero in Maths written by Buddhist scholars in Sanskrit. https://en.wikipedia.org/wiki/Bakhshali_manuscript . This was discovered in modern day Pakistan [then British India]. You could argue that this is a part of Pakistan's history, but what is Pakistan's History then ?
You can somewhat imagine if a Mesopotamian state survived till today, parts of it unconverted to Islam in a sub area of fertile crescent. Would you call it the successor state to entire Mesopotamia or you would say Uruk history is not their history because the region is now part of nation state of Iraq. That is the case with India as well ! There is a reason we read Mesopotamian history as a composite and as a different topic than say Egyptian History, because Mesopotamia was a different and connected civilization. Institutions like that don't survive now and hence a modern western scholar has no ontology to relate it to, but India exists as an outlier. That is the reason what most orientalists write and what we observe on ground about India are super disconnected.
This is one of the points I always bring up with people discuss Islam, Kashmir or Pakistan. The current incarnation of India with the current borders was an invention of the British, even with Pakistan and Bangladesh included, and nothing about the country's borders are native. There is generally a "civilization" akin to Europe that starts somewhere in North-West India and south to the various islands off the southern coast of India, but was never and has never been united under the western concept of "nation state" and has only at the most been united under an imperial hegemony where one entity dominates the continent, and hence it is really silly to argue about the borders of India.
Vishnu Puran composed circa 400 BCE to 900 CE says.
उत्तरं यत् समुद्रस्य हिमाद्रेश्चैव दक्षिणम् | वर्षं तद् भारतं नाम भारती यत्र सन्ततिः ||
Land that lies north to the sea and south of Himalaya is called Bharat(India) and it's people Bharti (Indian)
Political system was diffrent from what we see today. But that doesn't mean land called Bharat didn't exist culturally and politically.
They were referring to a region, like europe! Even the name India indicates this: it's the greek pronunciation of the Iranian word Hindu, which is the Iranian pronunciation of the river Sindhu, which is the traditional western border between India and the Iranian lands.
I get a bit annoyed on these claims of "the first zero", since it never bothers to be very explicit about defining what first "zero" they're talking about.
Is it the first symbol for no quantity? If you're only working with positive numbers you're not missing much by just saying "nothing". Having a zero matters a lot in positional number systems, since it means something else there. The Babylonians still got by with just putting a space instead of a symbol. Lastly, when using negative numbers on top of this, being able to work with the number between -1 and 1 just as any other becomes more necessary.
All these examples of "first zero" originate from the need to simplify and strengthen an already innovative system, and each individual first zero needs to be tied to that concept of number.
'Zero' was most likely invented for the same reason Complex Numbers were invented: to make some rare cases of contradictory mathematical formulas work. Zero is more a mathematical device than a number.
On the contrary, complex numbers are one of the most "number" things that exist. It could be convincingly argued that either ℕ or ℂ are the most "real" of all sets of numbers; ℝ is a rather artificial restriction of the true richness of structure of its less popular superset.
FWIW I don’t think this is contrary to the comment you replied to, but I started reading your comment hoping you’d have the convincing argument. Are there some examples you’re thinking of? There’s no denying the mathematical utility of complex numbers, they are a great mathematical device. There’s also no question that natural and real numbers existed in math and language for thousands of years before complex numbers were invented. But why would complex be more “number” or more “real” than scalars, what does that mean? How are reals an artificial restriction of complex numbers? (Does it make real number sense to have 2+3i dollars or apples?) Surely the choice to include the real number line in the complex plane isn’t the only reason? Otherwise, I’d argue that vectors are the more natural choice for what you’re trying to say.
> There’s also no question that natural and real numbers existed in math and language for thousands of years before complex numbers were invented.
Real numbers very definitely were not known to humans until the 1700s, except in the vaguest sense that rational numbers seemed to have gaps in them that are the roots of certain polynomials. It was only Newton and Leibniz who really had to start thinking about what continuity and limits mean, and what they did manage was very hand-wavy, just enough to sort of convince themselves that analysis works.
Fully rigorous construction of the reals was not achieved until 1871, by Cantor, although Cauchy's work earlier in the 19th century, particularly the introduction of 𝛅–𝛆 reasoning, was certainly vital. It is important to note that by that time, complex numbers had already been introduced, and indeed were of great help in figuring out the reals, exactly because reals themselves were incomplete.
But really, I digress.
> Does it make real number sense to have 2+3i dollars or apples?
Does it make sense to have 𝛑 dollars or apples? Tangible quantities are integral or rational.
Every extension of number systems was inspired by the fact that some reasonable calculations did not seem to have an answer within the set of what was at the time understood as "numbers".
Zero was introduced to reason about the question of what remains if you have n sheep and give all of them away.
Negative numbers were introduced to reason about what happens if you want to buy a sheep that costs n coins and you only have n-1 coins, ie. debt.
Rational numbers were introduced to reason about dividing a deceased father's sheep and coins among several brothers, or a year or a day into sub-periods.
Real numbers were introduced to… Well, at this point it becomes more abstract, because the only justification for the reals was the philosophical desire to talk about the length of the hypothenuse of an ideal right isosceles triangle or the length of the circumference of an ideal circle.
Complex numbers were introduced because not all polynomials of degree n > 1 seemed to have real roots at all, and others only had some number < n. It was found that this, again, was only because reals were in some way incomplete, and the roots were all there if only we could have more numbers in some sense. Today we say that the field of complex numbers is algebraically closed [1] and reals are not. This is a tremendously important property and the very reason that ℂ is seen as mathematicians as the real deal and ℝ only interesting as a subfield of ℂ.
I missed the part where complex numbers are convincingly “number” or convincingly “real”. Maybe you’re making a different argument than I thought. Just because the full philosophical understanding and development of irrational numbers is recent doesn’t (to me) result in complex numbers being as natural as a scalar quantity. Vectors can be real numbers with a physical analog, and arguably a more natural algebra, and are more widely used than complex numbers. Does that have any bearing on what you were trying to express?
> Zero was introduced to reason about the question of what remains if you have n sheep and give all of them away.
Hmmm, why do you think this? I have a doubt. We already know that language had ways to express having none of something long before zero appeared in math.
> Does it make sense to have 𝛑 dollars or apples? Tangible quantities are integral or rational.
We aren’t talking about what’s tangible, that wasn’t your original claim. Yes absolutely it makes complete sense to have pi apples, it’s a completely natural concept one can easily imagine, whether it’s tangible or not. It does not make any sense to have 2+3i apples. Not only is that not tangible, it also can’t be approximated and has no physical analog. I can easily obtain 3.14 apples or dollars, a rational approximation to pi. As you know, the rationals are a subset of the reals, so for the purposes of this discussion I’m even fine with talking about only finite reals or rationals. If you’re trying to make a point about an infinite irrational number or that I can’t have exactly pi apples, then I suggest that’s perhaps a straw man argument irrelevant to the question of whether 2+3i apples makes “number” sense. Pi apples is easy to imagine, easy to approximate, and easy to draw a picture of, while 2+3i apples is not.
I see no contradiction to my statement at all: "Zero was introduced to reason about the question of what remains if you have n sheep and give all of them away".
People invented mathematics to be able to explain things happening in nature, and refined their mathematical devices even further to even explain the nature of mathematics itself. Numbers did not exist before people invented them as sets for different tools to calculate stuff, as you have so eloquently put. To argue that "Real numbers very definitely were not known to humans until the 1700s" means nothing else than these numbers were not invented yet. Um, what is your point exactly?
This is a Pythagorean Tetractys. It is a mystical symbol that shows how 1+2+3+4=10, which supposedly shows the presence of harmony in pure numbers. Harmony in numbers was believed to manifest in the physical cosmos (eg, Harmony of the Spheres). This Tetractys relates also to musical harmony (which was famously formalized by the Pythagoreans). The top dot shows that, when you press a lyre string in the middle, you produce an octave. The two dots show where you can press a string to produce a musical fifth (3:2), and so on. The presence of zero is represented esoterically in the open string, which has no dot. Whether or not this was actually discussed by the Pythagoreans cannot be known—and clearly the concept of zero was otherwise unknown to the Greeks. But, who knows what might have been: the Pythagoreans were nearly all massacred by their enemies in the 5th century BC.
"We are out of stuff" is not a concept that can be traced to recorded history, because it much predates humans. Literal single cell life forms behave differently under conditions of scarcity vs abundance.
Just use of Abacus would make one think that there must have been some thinking about concept of zero. Just using ten system and having 123 and removing 3 from that would mean that you have 0 left for 1s. What is this zero, clearly it is not none as there is 1 100 and 2 10s... Now how to notate this is different question.
IMHO the big picture is missing. Examples of Indic scripts are found inside of China, as far northeast as central Vietnam, at some (later) sites in Japan and Korea, and all over Southeast Asia. Most of the use of these scripts is related to religion and to a lesser extent record-keeping. Unfortunately, the majority of these scripts were written on palm leaves which do not keep very well, so we are only left with the relatively rare inscriptions on stone and inscriptions on metal.
The Indian seafaring polities which are associated with Southeast Asia (ie. Srivijaya et al) are from India's south and eastern coasts. The south of India is also the traditional home of the Tamil, a Dravidian (non-Aryan) people whose language and script predate and differ from the prakrit languages which dominate Aryan-influenced northern and western India. What is interesting about the Tamil is that they had a record keeping culture in which inscriptions were made in to metal plates which were then bound on giant keyrings like a giant keychain. There are a great deal of these preserved. One wonders if zero exists in this corpus, whether or not it goes back to a directly contemporary period. If not, then the origin would appear to be clearly non-Tamil and could be ascribed to the subset of enthusiastically seafaring empires of the southern and eastern Indian coasts. The apparent main Tamil/Dravidian seafaring polity was called the Chola. https://en.wikipedia.org/wiki/Chola_dynasty but there was also the Pandyas https://en.wikipedia.org/wiki/Pandya_dynasty and Cheras (west coast, modern Kerala) https://en.wikipedia.org/wiki/Chera_dynasty
Agree, I think it would have existed from the day a cat ate all chickens and one had to tell their neighbor they have no more chicken in house. Just that the neanderthal or Cro-Magnon did not need to write it down for the modern paleontologist to discover.
For me the importance of zero is not about assigning a symbol to nothing. It’s about to come up with the idea that the symbol for the number of things one more than 9 should be 10 instead of another arbitrary symbol.
This concept seems intuitive at first, but when you think about it, it’s such an genius idea.
> For me the importance of zero is not about assigning a symbol to nothing. It’s about to come up with the idea that the symbol for the number of things one more than 9 should be 10 instead of another arbitrary symbol.
By the middle of the 2nd millennium BC, the Babylonian mathematics had a sophisticated sexagesimal positional numeral system. The lack of a positional value (or zero) was indicated by a space between sexagesimal numerals. [...] The Babylonian placeholder was not a true zero because it was not used alone, nor was it used at the end of a number. Thus numbers like 2 and 120 (2×60), 3 and 180 (3×60), 4 and 240 (4×60) looked the same, because the larger numbers lacked a final sexagesimal placeholder. Only context could differentiate them.
LOL, how could two "emeritus professors" write an article like this without a single mention of the Mayan civilization, which is regarded by many as the earliest use of an explicit zero in a numerical system. Are they not aware of this?
The Mayans may or may not have been first, but due to their isolation their use of zero was unknown to the rest of the world. By the time Europeans and Mayans met, the Hindu-Arabic numeral system was well established and that was the system that became the world's.
As seen in some of the comments below. As if they personally made it possible. It's like your country winning a sports tournament and claiming that "WE" won. No, you didn't. You just watched it.