The Elusive Origin of Zero
scientificamerican.com
scientificamerican.com
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.
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.
How intuitive is counting starting from no foundational knowledge?
I mean, every day you knew if you had 1 or 2 pieces of food to eat right?
It seems realizing when you had zero pieces of food would be top of mind.
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.
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.
But, no decimal placeholder system, like described in the article.
http://www.math.buffalo.edu/mad/Ancient-Africa/mad_ancient_e...
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.
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.
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".
"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."
Maybe you’re right and those mentions of the concept of nothing were misleading?
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.
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.
[0] https://en.wikipedia.org/wiki/Isha_Upanishad [1] https://en.wikipedia.org/wiki/%C5%9A%C5%ABnyat%C4%81 [2] https://www.myindiamyglory.com/2017/09/11/nalanda-9-million-... [3] https://en.wikipedia.org/wiki/Hindu_units_of_time
1305
+ 0935
----
2240
((5-5)+2)*0 = (0+2)*0 = 2*0 = 0Shukla 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.
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.
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.
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)
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.
Go on. Argue that please.
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.
[1] https://en.wikipedia.org/wiki/Algebraically_closed_field
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?
> 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.
*
* *
* * *
* * * *
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
Some of the non-Tamil seafaring polities: the Pallava, the Gupta (central-northern India), the Pala (Bengal coast and northward). https://en.wikipedia.org/wiki/Pallava_dynasty https://en.wikipedia.org/wiki/Gupta_Empire https://en.wikipedia.org/wiki/Pala_Empire
Any of these could have been the Indic culture at which "zero" moved eastward with merchants or religious record keeping.
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.
Enclosure empty = O
Yeah, Zero is a wonderful thing
In fact, Zero is my hero!
How can Zero be a hero?
Well, there are all kinds of heroes, you know
A man can get to be a hero
For a famous battle he fought
Or by studying very hard
And becoming a weightless astronaut
And then there are heroes of other sorts
Like the heroes, you know, from watching sports
But a hero doesn't have to be a grown up person, you know
A hero can be a very big dog
Who comes to your rescue
Or a very little boy who's smart enough to know what to do
But let me tell you about my favorite hero
My hero, Zero, such a funny little hero
But 'til you came along
We counted on our fingers and toes
Now you're here to stay
And nobody really knows
How wonderful you are
Why we could never reach a star
Without you, Zero, my hero
How wonderful you are
What's so wonderful about a zero?
It's nothing, isn't it?
Sure, it represents nothing alone
But place a zero after 1
And you've got yourself a 10
See how important that is?
When you run out of digits
You can start all over again
See how convenient that is?
That's why with only ten digits including zero
You can count as high as you could ever go
Forever, towards infinity
No one ever gets there, but you could try
10 billion zeros
From the cavemen to the weirdos
Who invented you
They counted on their fingers and toes
Or maybe some sticks and stones, rocks and bones
Their neighbors' toes
Yeah yeah
And nobody really knows
How wonderful you are
Why you could never reach a star
Without you, Zero, my hero
Zero, how wonderful you are
Place one zero after any number
And you've multiplied that number by 10
See how easy that is
Place two zeros after any number
And you've multiplied that number by 100
See how simple that is
Place three zeros after any number
And you've multiplied that number by 1, 000
Et cetera, et cetera, ad infinitum, ad astra, forever and ever
With zero, my hero, how wonderful you are
This concept seems intuitive at first, but when you think about it, it’s such an genius idea.
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.