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> When the sexagesimal Mesopotamian number system began to be used positionally, a symbol for "zero" had to be added

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

Having a symbol (for zero) is half the solution. It takes another leap to treat zero as number that participates in the usual arithmetic operations.

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.

i agree and have elaborated a bit more on the theme in https://news.ycombinator.com/item?id=40918326

> sexagesimal mesopotamian number system was used positionally from the beginning

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.

that's true, but to the extent that at any given time it was purely sexagesimal, it was also positional, wasn't it?

the book recommendation is greatly appreciated

I think it's fuzzier than that, both as a historical process (stretching across more than a millennium of development), and as a tool at any particular point in time. Here's some quotation from Robson:

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.

this makes it sound like robson agrees that there's no purely-sexagesimal non-positional transitional phase like you seemed to be positing?

As far as I understand there wasn't ever a "purely sexagesimal" phase. Some of the units were sexagesimal or sexagesimal-ish and others weren't, all the way along (but gradually trending toward more that were, as various unit systems were reformed). The sexagesimal system (per se) arose after literally centuries of prior experience with some of the unit systems being more or less sexagesimal, accompanying a transition in writing from using marks that were a concrete representation of clay tokens of various shapes toward using cuneiform. Even once there were people using sexagesimal calculations as an intermediate representation, they had to use tables for converting all of the non-sexagesimal units to sexagesimal quantities in terms of a specific unit of each type, and vice versa.

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).

aha, i see! thank you very much!

> That was the positional number system, where a small set of symbols is sufficient to write even very big numbers.

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.

You can already represent infinitely many numbers with a finite set of symbols without positional notation, in unary. You can write "1 + 1 + 1 + 1" or "This number of stars: ****" or whatever. This idea (make N marks to represent the natural number N) has surely been understood about as long as counting has been understood.

The advantage of positional notation is greater efficiency at representing large numbers.

Agreed.

I forgot to mention that positional numbering’s space complexity is log.

this is also true of clauses in grammar. the use of clauses makes it possible to write an infinite number of sentences using a finite number of words, each of which is drawn from a finite vocabulary, but whose assemblages are more infinite than the stars in the sky which so drew the attention of archimedes when he approximated the finite number of grains of sand that would be required to fill up the observable universe, a calculation which unfortunately he got low by several orders of magnitude due to an unfortunate lack of evidence for the hypothesis that the stars were other suns, not to mention the idea that some of the nebulae in the sky were entire galaxies of other stars, a fact which wasn't known until only a century ago

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

No, Log10(infinity) = infinity.

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)).

They are speaking of a finite set of symbols (the digits 0 through 9), not the number of glyphs in the representation of an arbitrary number.

You can represent any arbitrarily large number with a single symbol without positional notation, though. Let's use *. The number 3 is ***. The number 9 is *********. That interpretation of GP's comment (which, on re-read, is probably correct) makes it even more trivially false. Another comment by the same poster clarifies that I was actually responding to what he meant.

Null is closer to ‘empty set’ than to zero, and the two concepts are similar but different enough to be distinct.

In German, it's the same. I hope you are not talking about null in coding? Because I'm pretty sure human 2000 years ago did not know any programming code or even empty sets.

Empty set idea doesn’t seem so abstract. A bag of stuff can have nothing in it, so it’s empty. Combining and splitting up containers of items would have been an everyday occurrence.

I'm just nitpicking, but it's unlikely that the inventors of both had empty set and zero formalized to a degree to make them distinct.

While the empty set and the number "zero" are distinct concepts, the relationship between them is much closer than between any other sets and numbers.

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.

Right. What I was aiming at was that when someone invents one of these, they are probably thinking about the other, too, without the set theoretic rigor you recite.

In Latin, "null" was not a noun (i.e. substantive), but an adjective applied to nouns.

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.

The word "null" in "null set" refers its Lebesgue measure being zero.

Null sets in and of themselves have nothing to do with Lebesgue integrals. Null sets are definitely are not defined by Lebesgue integrals, you might be thinking of a set of measure zero which has to do with Lebesgue measure

> has to do with Lebesgue measure

As in, the only thing Lebesgue that my comment mentions.

> All languages have had words for "zero", for at least a few thousand years

> 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.

That matches in writing the meaning of the word "zero", "null" or whatever word is used in the spoken language.

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.

I disagree. The Romans had the word “nulla” to (loosely) mean “nothing”, but for them, the smallest whole number was 1 (https://en.wikipedia.org/wiki/Integer#History: “Historically the term was used for a number that was a multiple of 1, or to the whole part of a mixed number. Only positive integers were considered, making the term synonymous with the natural 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)

As I have said "null" does not mean "nothing". The latter is a substantive noun, while the former was an adjective in Latin, where such words were not interchangeable. "Nihil", i.e. "nothing", could be the answer to the question "What is in that box?". On the other hand, the question "How many eggs are in that box?" could be answered with "nullum ovum" or "nulla ova" ("no eggs"). In English, "nothing" and "no", when the latter means "zero", are also not interchangeable.

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.

> “Historically the term was used for a number that was a multiple of 1, or to the whole part of a mixed number. Only positive integers were considered, making the term synonymous with the natural numbers”

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.

> Even if you just tally things, computations may produce zero

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.

Null mean nothing but not zero. In Tamil language it is called as Paazh. Zero is called as Suzhiyam.