The Invention of Zero
themarginalian.org
themarginalian.org
Much as I have a rather low-level atavistic desire to credit India with the zero/nil, as there was so much exchange between mesopotamia and the early indus regions (just look at the idea of the alphabet going one way and then digits going the other), the sumerian "origins" are quite likely. More importantly, the vedic tradition didn't give rise to the formalisms developed by the later Greeks and, centuries later, their Islamic students. Thus for a long time, scholarly dissertations from the subcontinent on mathematics, philosophy etc tended to essays and explorations of conjectures, which makes pinning responsibility hard to do, the way you can, say, "Wiles did prove Fermat's Last Theorem".
Personally I find invention requires so much history and intertwined communication that the idea of "inventor" is kind of bogus anyway.
BTW in case this sounds like I'm dissing ancient indian scholars: you see this in the early days of any scientific field: early neuroscience in the early 20th century, the same with cognitive science in the mid 50s-70s (at least) etc. In fact most of contemporary ML just has a light layer of formalism painted on too. It feels like fields need names, but only really get them when they have attained some early level of abstraction and emerging rigor.
Sorry, that moved on beyond zero!
Trying to argue about whether one tribe's or another tribe's 50-generations-ago ancestor was the first one to do this or that thing seems to me like completely missing the point, when all of these steps were part of a long and gradual historical process, building ideas and tools up over centuries. (Similarly, it's annoying how many debates center on various ancient figures' ethnicity or religious affiliation, usually without much evidence.)
To anyone who tries researching ancient (or more recent) mathematics, it's clear that there usually isn't a single aha moment changing everything, but a broader culture that gradually evolves. We can see different flavors/aspects of a concept like "zero" which were developed different times and places (China, Mesopotamia, India, Greece, North America), none of which really draws any obvious line in the sand.
With regard to Indian innovations, however, it seems pretty clear that written arithmetic per se (performed on a "sand board") was developed there, as credited by all of the oldest extant texts on the subject from writers in Arabic (which call it something like "Indian arithmetic" or "Indian numbers"). Written arithmetic was then substantially elaborated in the Islamic world with a switch to using pen and paper, before making its way to Europe where it eventually kicked off the development of modern mathematical notation. The earlier Mesopotamian/Egyptian/Greek/European tradition, as well as the Chinese tradition, were generally based on using finger counting or some form of counting board, with written numerals used as a serialization format rather than a calculation tool. Arguably the invention and spread of physical materials like cheap good quality paper, writing implements, ink, and eventually printing presses were as important as the theoretical developments.
The intellectually honest account is that great people came at random from accommodating societies and cultures with the requisite technology and opportunity for those geniuses. There are countless stories of kings beheading inventors because their inventions risk offending the social order, or the gods, their pride, or some such. Just as there were in all likelihood millions of geniuses all over the world who were determined at birth to be subsistence farmers or slaves.
In any case, this must have been an independent invention of zero.
(now where did i put my aluminum foil)
All languages have had words for "zero", for at least a few thousand years.
For instance English has inherited "null" from Latin, whose literal meaning was initially "not even a small one", but whose meaning has become "zero". Latin also used very frequently the word "non-null", with the meaning "one or more". Latin had a few more other words that could be used to express the quantity "zero". The same was true for Ancient Greek and for the languages for which even older records exist.
What the parent article intends to discuss is the invention of a purely positional system for writing numbers. Before the invention of such a system for writing numbers, the words meaning "zero" were used in all languages only for the unique quantity "zero". They were not used as components of the numerals used to name bigger numbers.
The necessity to write very big numbers for accounting or computational purposes has made desirable the invention of a system that would be less cumbersome for such big numbers than the system used for the spoken numerals.
That was the positional number system, where a small set of symbols is sufficient to write even very big numbers. Any purely positional number system needs some kind of symbol for zero, which must be usable in any position.
When the sexagesimal Mesopotamian number system began to be used positionally, a symbol for "zero" had to be added. So nobody thought directly about a "zero" symbol. They just wanted to reuse the same symbols in all positions, and sooner or later someone understood that adding a symbol for "zero" is the solution to this problem.
The cardinal numbers are equivalence classes of the sets. For any other cardinal number except "zero", the equivalence class of that number contains a huge number of sets, potentially infinite.
For "zero", the equivalence class contains only a unique set, the empty set. Because of this one-to-one correspondence between the empty set and "zero", they may be interchanged in many contexts without causing any ambiguities.
It was used in precisely the same word contexts as the words for "one", "two", "three" etc. and in those contexts you could substitute any cardinal numeral. Like any cardinal numeral, it could be used to answer questions about how many things are in a certain place.
So it was really the number "zero".
For the empty set, the most appropriate Latin word was the noun "nihil" ("nothing"), sometimes contracted to "nil" (with long "i") hence the NIL of LISP for the empty list.
So the concepts of "zero" and "empty set" were distinguished in Latin and also in the other known ancient languages.
Some modern programming languages use "null" in a wrong way, when they should have stuck to the NIL of LISP. A null integer or floating-point number denotes the quantity "zero", but a null pointer is not a quantity. A null pointer points to nothing, so it denotes the object NIL.
As in, the only thing Lebesgue that my comment mentions.
> What the parent article intends to discuss is the invention of a purely positional system for writing numbers
I don’t see how positional is required there. Even if you just tally things, computations may produce zero, and you’ll want to write something to indicate that you made the calculation, and didn’t abandon it halfway though.
So it is not a new invention.
When the positional system of writing numbers was invented, the symbol for "zero" had to be used in a new way.
In the spoken language, one says "one hundred and five", without using any "zero" word.
In a non-positional writing system, one would mimic the spoken language, writing e.g. "CV". Symbols for "0" are not used inside any other number, but only for the unique quantity "zero".
In a positional system, one writes "one hundred and five" as "1" "0" "5", using a "0" in the appropriate positions, which allows the reuse of the symbol "1", instead of having to invent special symbols for hundreds.
For very big numbers, the economy of symbols brought by the positional system is great, so its invention has been very important.
The positional system did not need to invent a new word or symbol for "zero", but it had to invent a new way to use "zero", inside the strings used for writing big numbers.
And yes, to introduce a positional number system, you need some way to indicate “there is a position here, but there’s nothing there”, but you don’t need a positional number system to extend the integers by adding the novel concept of zero (and then, you need a, preferably compact, way to write them)
You have linked to a discussion of the word "integer", which has never been used by the Romans. "Integer" started to be used for numbers only in the late Medieval Latin.
As I have explained, the word "null" (i.e. nullus/nulla/nullum) had a grammatical distribution identical to that of any other number, with no difference from 1, 2, 3 etc.
Following the Greek tradition, the Roman grammarians did not classify their words based on their meanings or actual grammatical roles, but based on their kinds of morphological flexion (i.e. declension or conjugation).
For historical reasons, the Indo-European numerals from one to ten had a special declension, which was different from the declension of any other words.
Because of that, the Greeks and the Romans classified the numerals from one to ten and any other words derived from them into a special subclass of the nouns, the numerals.
The words used for "zero", both in Greek and in Latin, are much more recent than the words for "1" to "10", so when they have been coined they have received the regular adjectival declension, instead of the archaic numeral declension. This difference in declension prevented the classification of "null" as a numeral, even if it belonged to a group of words about which various ancient grammarians have expressed doubts about how they should be properly classified.
The ideas of the ancient grammarians about the relevance of declension for word classification do not matter for the classification of "null" from the point of view of modern grammar and mathematics. Even in antiquity, there were people like Aristotle, who have set the bases of a classification of the words based on meaning and grammatical roles, not on flexion (in the so-called "Categories"), even if, at least in the surviving works, this has not been applied to a detailed analysis of a language like the Ancient Greek.
As it is said in the well known quotation about ducks, if one reads the surviving Latin texts, there is no difference in usage between "nullus/nulla/nullum" and any other cardinal number, i.e. all the cardinal numbers including "nullus/nulla/nullum" appear in the same word contexts, where they are interchangeable, therefore "nullus/nulla/nullum" is a cardinal number, based on how the Romans were using it, regardless whether Priscianus would have agreed to this.
In fact, they didn't even consider 1 to be a number, let alone zero -- at least aristotle didn't, and his definition was the accepted one in Europe for many centuries.
Though I think you need to sort of separate out what philosophers and mathematicians thought about numbers from what regular people did. I think people obviously had an intuitive understanding of zero and one as quantities, even if it wasn't formally defined that way by philosophers for thousands of years.
It may produce zero but not necessarily.
Tallying things is more likely to produce a notion of "alright, no one owes anyone anything", and there's no need to track this "nothing". Only when things don't tally is it required to keep track.
Because there's no use persisting this "nothing"ness over space and time it's unlikely to have been abstracted further to zero.
Just the other day I was thinking about it and realised that positional number system enables us to represent infinite numbers with a finite set of symbols. Whoever (person/community) invented is a genius, as positional numbering unlocks so many further inventions down the line.
The advantage of positional notation is greater efficiency at representing large numbers.
I forgot to mention that positional numbering’s space complexity is log.
the previous sentence contains 125 words, and it should be evident that it could be extended indefinitely in any known human language (except possibly pirahã) without doing any violence to the rules of grammar, though perhaps great violence to the canons of courtesy to readers. if we use shannon's early estimate of 11.82 bits of entropy per english word, in english there are about 2¹⁴⁷⁸ ≈ 10⁴⁴⁵ perfectly unremarkable sentences of that precise length, a number which (it should be evident) grows exponentially with the sentence length
so, while i agree this concept is genius, it is part of the invention of language as we know it, and no isolated human tribe without language has ever been discovered. it probably dates to so-called behavioral modernity, at least 50000 years ago—probably longer than that
It allows us to represent numbers using log symbols. Symbols required = O(log10(N)). Or put another way, the max number we can represent with S symbols is exponential in S. N = O(exp(S)).
the sexagesimal mesopotamian number system was used positionally from the beginning (that's what was sexagesimal about it) and, as the article explains, did not have a symbol for zero in common use, or evidently at all for millennia. the early zero symbol mentioned in the article is from a tablet from 0700 bce or later, which is only 2700 years ago. at that point positional sexagesimal mesopotamian numerals were already about 1000–2000 years old. what popova doesn't explain is that they often used an empty space for zero instead
From what I understand, there were a variety of non-standardized symbols or place holders used by the Mesopotamians (space being one). Zero was not treated as a number, but as a special case in their evaluator.
This is oversimplified. The sexagesimal place-value system developed gradually over centuries+ from various inconsistent unit systems for length, weight, fluid volume, counting, ..., which were eventually (partially) standardized. This long development took place in tandem with the development of cuneiform writing, both ultimately originating in a record-keeping system involving clay tokens sealed in clay envelopes. For a detailed version, Eleanor Robson's book Mathematics in Ancient Iraq is excellent.
the book recommendation is greatly appreciated
The very earliest literate accounts – from late fourth-millenium Uruk – used commodity-specific metrologies with a variety of different numerical relationships between the units. Those original metrologies continued in use throughout the third milennium and beyond, whether essentially unchanged (for instance areas), or undergoing periodical reform (for instance capacities). They continued to be written with compound signs, which bundled both quantity and unit into a single grapheme, just as the preliterate accounting tokens must have done. Gradually, over the course of the later third millennium, scribes began to write those compound metrological numerals with a cuneiform stylus, rather than impressing a round stylus into the clay in imitation of accounting tokens. Throughout the Sargonic period, and even into the early Ur I period, impressed and incised number notations apear side by side on the same tablets – a phenomenon that has not yet been systematically documented or explained.
[...] But not every new metrological unit was sexagesimally structured. The smaller length measures, first attested in the Early Dynastic IIIb period, divide the rod into 2 reeds or 12 cubits, and the cubit into 30 fingers. None of these newly invented units of measure was recorded with compound metrological numerals, but always written as numbers recorded according to the discrete notation system followed by a separate sign for the metrological unit. This has implications for our understanding of the material culture of early Mesopotamian calculation, as well as for the shifting conceptualisation of number.
[...] At some point early in the Ur III period, the generalised sexagesimal fraction and the generalised unit fraction were productively combined to create a new cognitive tool: the sexagesimal place value system. This calculating device took quantities expressed in traditional metrologies and reconfigured them as sexagesimal multiples or fractions of a base unit, often at convenient meeting point between metrological systems.
[...] The SPVS temporarily changed the status of numbers from properties of real-world objects to independent entities that could be manipulated without regard to absolute value or metrological system. Calculations could thus transform numbers from lengths into areas, or from capacity units of grain into discretely counted recipients of rations, without concern for the objects to which they pertained. Once the calculation was done, the result was expressed in the most appropriate metrological units and thus re-entered the natural world as a concrete quantity.
The use of sexagesimal calculation probably had something to do with the use of a type of counting board about which we unfortunately know few details (it was called the "hand" and IIRC could generally support values with 5 sexagesimal places, but anything we know about it comes from scattered textual references and some inferences drawn from calculation mistakes; Robson's book doesn't talk about this subject much but there are some nice papers about it elsewhere).
The language we speak and the concepts we are taught literally form into structures in our brain, and we can't understand what it would be like not to have them, because those structures are part of the organ of understanding
For example, people normally count things by starting from one, so, at least following this usual procedure, counting to zero is technically speaking impossible. Also, we can't distinguish between zero apples and zero oranges, but we can tell two apples and two oranges apart.
In fact, even with the perspective of modern algebra zero remains special. For example, it is the only element of a field without a multiplicative inverse.
I'd be surprised if zero didn't take any extra effort to discover. It's clearly different than other integers.
But with zero, this idea converges on the same thing. No matter what things you were counting, if you have zero of them, you have the same idea. And so you take a step towards the idea of a number being a concept in its own right, rather than existing purely for the purpose of counting or measurement.
It is the same sort of conceptual freedom that allows you to do things like add a number to a square. To deal with an equation like x + x ^ 2 = 0. If you're stuck with numbers "meaning" something beyond themselves, then you'll never add x to x^2. One is a length, the other an area. They are different objects.
This intellectual leap is one that must be made by all students of mathematics - and many young people do not.
I’m not sure what this even means, but Sumerians had abstract mathematics, in addition to art and literature which are abstract by their nature. They were using numbers in the abstract sense before the number zero was named, and so while it seems like a logical and tempting narrative that naming zero is what abstracted numbers, history doesn’t seem to support this particular post-facto rationalization. Naming zero is very important in the history of math, it just isn’t the first abstraction.
And I think you know exactly what I meant, because you immediately countered with some historical evidence around those first abstractions.
So, thank you, and I will read much more about Sumerian mathematics with great interest.
And thus was born the everlasting confusion between cardinal and ordinal numbers.
> The nine Indian figures are: 9 8 7 6 5 4 3 2 1. With these nine figures, and with the sign 0 (...) any number may be written.
i'm not sure when it became conventional to consider it a number rather than the absence of one; it might not have been until the early modern era
https://news.ycombinator.com/item?id=40917674 suggests that european mathematicians still hadn't agreed that 1 was a number until the early modern era
______
† it doesn't
1) deductive geometry of Thales, where we could now prove things independent of the physical world (abstracting away from the physical)
2) Plato’s remarks on incommensurability (proto irrational numbers) being something real but not physical, because no physical process could prove to the mathematician that two lengths really have no common unit measure. Here the abstraction is again away from physical means.
3) infinity of numbers. Abstracting away from large but finite collections. We can only ever survey finite collections physically, again is an abstraction.
Aren't numbers themselves an abstraction?
They are not helped by educators insisting on presenting "word problems" when teaching maths. To get to the next level, you need to break the connection between numbers and the "real world". I've always felt that the number zero represented the first step that humanity took in this journey - and it's a step that every human also needs to individually take if they want to learn maths.
Putting the concept at some specific geographic location seems very strange to me. To me there is no doubt that in each mathematical culture there was some notion of it, just weaved into that conceptions of that particular culture.
Of course then there is zero as a symbol and positional number systems. The first one seems very uninteresting, the later one definitely more so, but the question of how to express numbers is definitely more interesting and broad than the history of just one component of it.
Mathematics to the Greeks was "mathematics" in the sense we understand it today. From a system of axioms they derived a complex system of theorems, which allowed for an abstract description of reality. It was mathematics as a system of truth, which then could be used for other purposes. E.g. with Archimedes who discovered integration, but also was a prolific engineer.
The Greeks were so far ahead of anything the Mesopotamians did, that even the comparison is unfair.
The comparison does make sense if you interpret the parent’s post simply to state that Babylonian mathematics (in Mesopotamia) were developed (and seemingly stagnated) before Greek mathematics began in earnest. Which seems to be pretty uncontroversial. There are extant clay tablets from 1800 to 1600 BC that would indeed predate the Greek Geometers by a millennia — and that’s if you’re counting Thales as the beginning.
Ie. your parent post is using “ahead of” == “temporally before” not as in “more advanced”.
I think it is fair to say that my statement is too strong to say it is wrong. My assertion would be that it is more complicated and almost certainly there is a lot lost in translation along the years.
The Greeks were ideologically opposed to the number zero. Aristotle outright refuses to acknowledge the existence of zero and of infinity. The Greeks were aware of the idea of zero, and they even used zero when they calculated using the Babylonian number system (which used zero as a placeholder number,) but they always converted the numbers back into their own system, and stubbornly refused to acknowledge its existence.
The fact that the Greeks saw geometry and math as interchangeable was their weakness here. There’s no way to represent the number 0 geometrically, but the Greeks weren’t gonna give up their belief in Geometry because it provided them with social and political power.
Pythagoras and Aristotle believed in a religious philosophy with logos at the center. Logos can be translated as “thought” or “word” (as it is in the Bible) or it can be translated as “ratio.” This is because they saw all these things as one (the Latin translation of the Greek word “logos” is “ratio.”) The ratio of numbers was thought the be the underlying mechanism that proved the order of the universe (which naturally saw the nobility as orderly and the peasantry as chaotic). This was a profoundly powerful sociopolitical tool that ended up spreading all across the world because Aristotles student just happened to be the greatest conqueror of the era: Alexander The Great.
Anything that threatened the philosophy of logos was suppressed, violently. Hippassus and Zeno were both murdered for the crime of talking about irrational numbers and infinity. Zero was one of these threats. 1:0 = infinity, 10:0 = infinity, anything:0 = infinity. This was not logos and therefore it was suppressed.
This philosophy extended beyond mathematics into the realm of astronomy and, weirdly enough, music (at the time, Pythagoras was actually most famous for his discovery of the golden ratio using an instrument called the monochord, which is a legend that seems to be false but nonetheless made him very famous.) This astronomical belief system was then later attributed to Ptolemy. This philosophy then was transplanted into Christian theology, and it took centuries for the monks to accept the existence of zero and infinity as a result. We even have cases of religious figures persecuting mathematicians about zero and infinity as late as the 1800s.
I don't think this claim is supportable, certainly not as such a broad generalization. Do you have a specific statement clearly attributable to an ancient Greek author to that effect?
> Greeks saw geometry and math as interchangeable
You're going to have to define these terms more explicitly. I don't think this statement is right. Ancient Greeks spent quite a lot of effort studying areas of what we now call mathematics but which were not geometrical per se.
> Greeks weren’t gonna give up their belief in Geometry because it provided them with social and political power
Any claim that geometers in general had social or political power owing to their mathematical work seems exaggerated. While a couple of geometers happened to incidentally also be local political leaders (e.g. Eudoxus), geometers explicitly lamented how little their contemporaries cared about their work and how few colleagues they could find to share it with.
> Aristotle outright refuses to acknowledge the existence of zero and of infinity.
This seems like a reductive summary. Aristotle had a pretty sophisticated idea about this which finds echoes in modern mathematics: here is Aristotle:
> Now there is no ratio in which the void is exceeded by body, as there is no ratio of 0 (οὐδέν) to a number. For if 4 exceeds 3 by 1, and 2 by more than 1, and 1 by still more than it exceeds 2, still there is no ratio by which it exceeds 0; for that which exceeds must be divisible into the excess + that which is exceeded, so that 4 will be what it exceeds 0 by + 0. For this reason, too, a line does not exceed a point-unless it is composed of points!
Alternate translation:
> But the nonexistent substantiality of vacuity cannot bear any ratio whatever to the substantiality of any material substance, any more than zero can bear a ratio to a number. For if we divide a constant quantity c (that which exceeds) into two variable parts, a (the excess) and b (the exceeded), then, as a increases, b will decrease and the ratio a :b will increase; but when the whole of c is in section a there will be none of c for section b; and it is absurd to speak of 'none of c' as 'a part of c.' So the ratio a: b will cease to exist, because b has ceased to exist and only a is left, and there is no proportion between something and nothing. (And in the same way there is no such thing as the proportion between a line and a point, because, since a point is no part of a line, taking a point is not taking any of the line.)"
Later, about motion in a vacuum:
> But if a thing moves through the thickest medium such and such a distance in such and such a time, it moves through the void with a speed beyond any ratio.
(The physical premise here is wrong, but the concept of division by zero is clear.)
See https://www.jstor.org/stable/pdf/2304187.pdf
Aside: anyone telling you what Pythagoras thought about numbers is pulling your leg. We have no idea about this whatsoever, only what various people claimed like 5+ centuries later, most of which is nonsense.
Even though others like the Arabs built on this idea later, India got the whole zero ball rolling.
1. I'm a huge fan and long-time supporter of Maria Popova; IMHO themarginalian.org is one of the finest websites ever, with a breadth and depth of consistently worthwhile content. Check it out!
2. This post reminds me of a book I really enjoyed maybe 10 years ago -- "Zero: the History of a Dangerous Idea". Recommended.
They cover how we went from objects to abstract concepts, and how 1 and 0 were "invented".
https://i.pinimg.com/originals/cd/55/b5/cd55b56cd40e1fa16e0b...
Is infinity the mirror image of nothingness? I cannot conceive infinity let alone its mirror image. Same with nothingness.
One of the best books I’ve read in a while. Really gives a great story about the evolving history of thought.