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> 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 position

> 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)
  - adrian_b 2y ago
    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.
vishnugupta 2y ago
> 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.

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)
- adrian_b 2y ago
  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.

vishnugupta 2y ago

> 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.