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Comment by drewcoo | Hacker News Reader

Zero was not a "counting number." It wasn't used in early calculations.

To an ancient Roman, tomorrow is 2 (ok, II) days from now. Today . . . one . . . tomorrow . . . two . . . two days!

> Because the first few numbers, especially 0, 1 and 2

Considering zero one of "the first few numbers," or even considering zero being a number, or for that matter considering zero at all are very recent concepts for humans.

A millennium ago, asking someone to count all the sheep in a field that had no sheep would be like today asking the sound of one hand clapping.

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A millennium ago or five millennia ago it would have been very normal to ask someone to count all the sheep in a field that had no sheep, because the one asking would not have known that there were none, hence the question. The asked would also have not been surprised by the question, because it would have been obvious that whoever had asked had not known the answer, and they would have provided the correct answer: none.

Already before the ancient Romans, Aristotle had made a classification of the parts of speech based on their meaning as exposed by the kind of questions to which they can provide answers (the so-called "Categories" of Aristotle). For some of his categories he has coined new names, including the name that has been later translated into Latin as "quantity", and Aristotle has divided the quantities into discrete quantities (providing answers to "How many?") and continuous quantities (providing answers to "How much?"). According to Aristotle's definitions, "zero" was certainly a quantity, together with the other numbers and with other words that can express quantities.

I do not think that there is any preserved text from the Classical period of Ancient Greece in which it is stated whether they thought that zero belongs to what they called numbers (arithmoi), or not. In any case whether they had chosen to call zero using the term "number" or not is less important than the fact that they understood perfectly the more general concept of "quantity", which included various subclasses, like numbers (what we call integer numbers) and measures (what we call real numbers), and to which zero also belonged.

About the ancient Romans, the most frequent way of saying in Latin "some", i.e. an unknown number, but at least 1, was "not zero". Therefore regardless whether they were using the word "number" for zero or not, they had a clear concept of a set consisting of 0, 1, 2, 3 and all the other cardinal numbers, so they identified the subset with 1, 2, 3 ... as the complement of the subset containing zero.

The ancient people were much less ignorant than some modern people believe.

Writing is expensive. I always wonder if this is why you don't see zero expressed as often. Why bother? Once writing is cheap, you can explicitly enumerate the zeroes, not because they convey useful information, but because they add a check that you can now afford. "I didn't forget to count sheep. I counted and there were actually none."

I have a hard time believing this though. If there are no sheep, then would it really not be "obvious" to them that there are no sheep, i.e. zero sheep, or none to be found? What would have been their answer if not something akin to this?

It's not that there's no concept of the absence of sheep, it's that there's no concept that that can be represented as some number of sheep.

This may sound clever, but it fails the experimental tests.

In most modern grammar theories and already in some of the works of the Ancient Greeks and Romans, a group of words is recognized as having the same nature if any of those words can be substituted for another of them in a given sentence.

Given the sentences (which have one-to-one correspondences in the ancient languages):

I see zero sheep. (In Latin, "null-" is the word stem for zero.) I see one sheep. I see two sheep. I see three sheep. ...

both a modern grammarian and also some of the ancient ones will conclude that zero, one, two, three ... have the same nature.

So zero was a concept that represented a certain quantity of sheep.

The Romans might have not used the word "number" in reference to zero, but there is no doubt that they understood the concept of a set whose members are 0, 1, 2, 3 and so on.

The proof is that the normal way of saying "some" in Latin, i.e. an unknown number that is at least 1, was to say "not zero", i.e. by identifying the subset 1, 2, 3, ... with the complement of the subset that contains 0 in the set of what now are called natural numbers.

While they did not have standardized words for concepts like set, subset, complement, membership or quantifiers, these concepts of the modern set theory were nevertheless well understood by the educated people in the Ancient World as they were studied within Logic a.k.a. Dialectic.

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