Diversionary History¶
If there’s one thing everybody seems to agree about the history of zero, it’s that it was driven primarily by notational considerations. More specifically, zero was required to enable positional notation. The historical record reinforces this assumption, to such an extent that it becomes apparently obvious, and thus unproblematic.
For instance (grabbing what’s immediately to hand), John D Barrow’s The Book of Nothing organizes its discussion of ‘the Origin of Zero’ by relating how
… the zero sign and a positional significance when reading the value of a symbol, are features that lie at the heart of the development of efficient human counting systems.
Robert Kaplan, when discussing the retardation of Greek arithmetical notation, explains:
… the continuing lack of positional notation meant that [the Greeks] still had no symbol for zero.
As everyone ‘knows’, the Babylonians, and later the Indians, got it right: discovering or inventing a sign for zero to mark the empty place required for unambiguous positional-numerical values. Zero arose, and spread, because it allowed modular number systems to develop. Except that, conceptually, there is no basis to this story at all.
Counting is primarily practical, so that no argument counts for much besides a demonstration. In this case, demonstration is peculiarly simple, especially when it is noted that nobody seems to think it possible.
Modulus-2 is convenient, but there is nothing magical about it in this regard. A decimal demonstration, for instance, would be no more intellectually taxing, although it would be considerably more cumbersome. Any modulus works.
Start with the basics. The positions or places of a modular notational systems represent powers. If we count from zero, the number of each successive place (ascending to the left by our established convention) corresponds to the modular exponent. The zeroth power for a single digit number, the first and then zeroth power for two digits, the second, first and zeroth power for three digits, and so on.
As the accepted story goes, each place must be filled, if only by a marked nothing (zero), if the proper places, and their corresponding (modular exponential) values, are to be read. The places must indeed be filled. There is no need whatsoever for a zero sign to do this.
The demonstration, then. Our non-zero modulus-2 positional system has two signs, 1 and 2, each bearing its familiar values. The places also have their mod-2 values, counting in sixteens, eights, fours, twos, and units as they decline to the right. Here we go, counting from 1 to 31 (watch carefully for the point at which the supposedly indispensable zero sign is needed):
1, 2, 11, 12, 21, 22, 111, 112, 121, 122, 211, 212, 221, 222, 1111, 1112, 1121, 1122, 1211, 1212, 1221, 1222, 2111, 2112, 2121, 2122, 2211, 2212, 2221, 2222, 1111 …
Conclusion: the positional function of zero is wholly superfluous. The Greeks, or anybody else, could have instantiated a simple, fully-functional positional-numerical notation without any need to accommodate themselves to the trauma of zero. In regard to this matter, the history of numeracy is utterly diversionary (not just the historiography, but the substantial history — the facts).
Perhaps this won’t seem puzzling to people, but it puzzles the hell out of me.
ADDED: Mathematical lucidity on the topic from Alan Liddell. Part 2.