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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#Hi

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.
empath75 2y ago
> “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.

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.

empath75 2y ago

> “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.