John H. Jenkins wrote:

> On Feb 8, 2005, at 2:25 AM, Marco Cimarosti wrote:
>
> > IMHO, a mathematics which does not have zero is a mathematics which
> > does not
> > have subtraction. I.e., hardly a "mathematics" at all.
>
> Remember, there are two things going on here. One is the concept of
> zero and the other is the ability to represent it in numerals. The
> concept is certainly very old and generally predates the ability of
> mathematicians to represent it as a quantity.

Actually, there are *three* things going on: the two you mentioned plus the
invention of *digit* zero.

IMHO, only the lattere is the only recognizable "invention of zero",
although it would be more precise to call it "invention of the positional
decimal notation"; the existance of a digit zero is a necessary corollary.

We don't know all the details about this invention, but we do know the big
picture: it was certainly invented in India, perhaps after Alexander reached
India and before the Muslims did. (And, BTW, we now know that digit zero was
also invented independently in Central America.)

In the positional notation, representing number zero is not the main task of
digit zero: normally it represent just an empty position in the string of
digits (each one of which is a multiplier for a specific power of ten).

However, a string composed of just one digit zero is also a convenient
symbol for *numer* zero, so convenient that it replaced all pre-existing
symbols for zero, hence the urban legend about the Indians who "invented"
number zero.

> [...]
> Zero as a *concept* is very old. A notation for zero is not
> and stems from the development of our modern numeral system
> in India.

Wrong, IMHO. Greek mathematicians had a symbol (more than one, actually) for
*number* zero, and so did, probably, any other civilization who had anything
ehich could be defined "mathematics".

Even the Indian symbol for zero (an empty circle) possibly predates the
positional numbering system and could have been originally a symbol for the
*number*, which was then used as a digit.

--
Marco