5 ms·

Speaking about the "invention of zero" is not a good choice of words. All languages have had words for "zero", for at least a few thousand years. For instance

Speaking about the "invention of zero" is not a good choice of words. 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.

mensetmanusman 2y ago
Null is closer to ‘empty set’ than to zero, and the two concepts are similar but different enough to be distinct.
- dayjaby 2y ago
  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.
  - deleted 2y ago
    [deleted]
  - rdtsc 2y ago
    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.
- kreetx 2y ago
  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.
  - adrian_b 2y ago
    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.
    - kreetx 2y ago
      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.
- adrian_b 2y ago
  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.
  - badkitty99 2y ago
    [dead]
- kazinator 2y ago
  The word "null" in "null set" refers its Lebesgue measure being zero.
  - badkitty99 2y ago
    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
    - deleted 2y ago
      [deleted]
    - kazinator 2y ago
      > has to do with Lebesgue measure As in, the only thing Lebesgue that my comment mentions.
Someone 2y ago
> 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.
- adrian_b 2y ago
  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.
  - Someone 2y ago
    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 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)
rokisen157 2y ago
Null mean nothing but not zero. In Tamil language it is called as Paazh. Zero is called as Suzhiyam.
vishnugupta 2y ago
> 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.
- Chinjut 2y ago
  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.
  - vishnugupta 2y ago
    Agreed. I forgot to mention that positional numbering’s space complexity is log.
- kragen 2y ago
  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
kragen 2y ago
> 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
- srean 2y ago
  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.
  - kragen 2y ago
    i agree and have elaborated a bit more on the theme in https://news.ycombinator.com/item?id=40918326 https://news.ycombinator.com/item?id=40918326
- jacobolus 2y ago
  > 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.
  - kragen 2y ago

mensetmanusman 2y ago

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

dayjaby 2y ago
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.
- deleted 2y ago
  [deleted]
- rdtsc 2y ago
  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.
kreetx 2y ago
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.
- adrian_b 2y ago
  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.
  - kreetx 2y ago
    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.
adrian_b 2y ago
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.
- badkitty99 2y ago
  [dead]
kazinator 2y ago
The word "null" in "null set" refers its Lebesgue measure being zero.
- badkitty99 2y ago
  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
  - deleted 2y ago
    [deleted]
  - kazinator 2y ago
    > has to do with Lebesgue measure As in, the only thing Lebesgue that my comment mentions.

Someone 2y ago

> 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.

adrian_b 2y ago
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.
- Someone 2y ago
  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 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)

rokisen157 2y ago

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

vishnugupta 2y ago

> 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.

Chinjut 2y ago
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.
- vishnugupta 2y ago
  Agreed. I forgot to mention that positional numbering’s space complexity is log.
kragen 2y ago
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

kragen 2y ago

> 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

srean 2y ago
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.
- kragen 2y ago
  i agree and have elaborated a bit more on the theme in https://news.ycombinator.com/item?id=40918326 https://news.ycombinator.com/item?id=40918326
jacobolus 2y ago
> 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.
- kragen 2y ago