Marco Cimarosti <marco dot cimarosti at essetre dot it>

> And, IMHO, the concept of zero already existed even at that time: I
> would not be surprised if the concept of "number" was invented eactly
> as a cover term to include zero and the positive units used for
> counting in the same category.

The term "number" originally referred to what we call "natural numbers,"
or counting numbers, those beginning with 1.

Georges Ifrah in "Universal History of Numbers" observes that counting
with pebbles led to counting with different objects for different
magnitudes, followed by different written symbols to represent those
objects, eventually leading to the numerical use of letters of the
alphabet (thus bringing this thread back on-topic, if only briefly).

The concept of zero, as both a placeholder and a "null" quantity, did
not fully catch on until the fifth century or so, in India. The advent
of pure-positional number systems required a zero in ways that, say,
Roman numerals did not.

> IMHO, a mathematics which does not have zero is a mathematics which
> does not have subtraction. I.e., hardly a "mathematics" at all.

Certainly a very difficult type of mathematics. In that era, the
mathematical concept of subtracting 2 from 2 was about as abstract as
the modern notion of taking the square root of negative 1, or dividing 1
by 0 -- that is, not unthinkable, but not in the realm of "normal"
arithmetic used by ordinary people.

>> Like "discovery of America," the phrase "invention of zero"
>> describes an undeniably important event of history in very
>> misleading words.
>
> You are comparing a well-documented historical event and an
> undocumented hypothesis.
>
> America was "discovered" on the 12th of October of AD 1492, when a
> trans-oceanic expedition led by Cristobal Colón (or Cristoforo
> Colombo, Christopher Columbus), a Spanish explorer of Italian origin,
> arrived in the American islands now called Bahamas.
>
> Now, can you provide similar details about when, where and by whom
> zero was "invented"?

You missed my point. The word "discovery" is inappropriate to describe
the arrival of Europeans in a land filled with Native American cultures
dating back thousands of years.

Likewise, the word "invention" is inappropriate to describe the gradual
development of the concept of zero, from being outside the realm of
counting numerals to being a .placeholder, and finally a number in its
own right.

> Or, as a minimum, can you cite any evidence of an early *documented*
> mathematics which lacks the concept of zero and a way to express it (a
> symbol, a blank cell, the phrase "Sorry, no more of it")?

The Babylonians, for one, had no zero even though they had a positional
system. Ifrah writes, "For more than fifteen centuries, Babylonian
mathematicians and astronomers worked without a concept of or sign for
zero, and that must have hampered them a great deal." Eventually they
did leave blank spaces, but these were easily confused and often
omitted, and more importantly a blank space did not represent the
quantity "2 minus 2" the way a zero would today.

-Doug Ewell
Fullerton, California
http://users.adelphia.net/~dewell/