THE NOTHING THAT IS A natural History of Zero By Robert Kaplan Oxford University Press, 1999. 225 Pages |
Is 0 (zero) a number? Or is it a symbol representing "nothing"? The Romans had no symbol for zero among the numerous letters they used ("Roman numeral") to represent numbers. And whoever has tried to do arithmetic with Roman numerals knows the difficulty, for example, of multiplying MDCCCLIV by X. (The answer is 18,590--I don't have any idea how to write it in Roman numerals.) So it's not surprising that for millennia arithmetic was done by finger reckoning or "boards" of various types on which counters or beads were placed.^{1}
It may seem obvious to us today that zero is a number, but a little thought shows that its concept is a little hazy even today. For example, how many people celebrated the millennium on the wrong day, Jan. 1 2000 rather than Jan. 1, 2001? (There was no year 0.) Physicists routinely begin sequences with zero, while mathematicians prefer to start with one, arguing that the first term would be so labeled.^{2 }My (nine-year-old) granddaughter Bailey Zweifel, who is kibitzing me as I write this, points out that the question "Is zero a number?" is analogous to the physics question "Is black a color?"
Zero is important not only as a digit; the little Roman multiplication problem given above shows the importance of its use for positional notation in arithmetic. With such notation, a number is multiplied by 10 merely by adding zero to the end of it., and is divided by ten by putting a zero in front of it (but after the decimal point). So the construction of a symbol for zero was not enough to give us modern arithmetic; a positional form of writing numbers was still required. The actual symbol "0" was invented and reinvented many times. As the author says in his introduction, "You will see [zero] appear in Sumerian days almost as an afterthought, then in the coming centuries casually alter and almost as casually disappear, to rise again transformed…It will just tease and flit away from the Greeks, live at careless ease in India, suffer our Western crises of identity and emerge this side of Newton with all the subtlety and complexity of our times."
The author does a masterful job of tracing these peregrinations through the civilizations mentioned above and others.^{3} (It seems that positional notation came about some time after Arabic numerals were clearly adopted, but this is not made completely clear in the book.) And he then goes on to discuss more advanced topics in modern mathematics (but always at a level geared to a non-technical readership). These include the ideas of the "infinitesimal" as associated with the calculus of Newton and of Leibniz (Newton rejected them, Leibniz did not). He even ventures briefly (one paragraph) into nonstandard analysis. The author also describes how all the real numbers can be constructed, by Farey addition, from the two number 0 and 1, and uses this to argue that the rationals are countable.^{4} He carries this one step further by giving the von Neumann construction of the reals from zero alone. This leads to a philosophical discussion of the meaning of the empty set and to the structure of mathematical logic, using the truth-table. He even shows how to use the concept of zero to solve algebraic equations.
Well, there are a lot of other concepts and ideas presented and expounded
in this interesting little book. The author's liberal use of literary quotations,
attributed and not,^{5} serve to remind us of the cultural content
of mathematics, and of its beauty as the queen of sciences.^{6}
- A typical board, used in Norman England, looked like our present chess board, hence the term "exchequer." (The French word for "chess" is "échecs"; even our word "check" representing a bank draft comes from chess.) An abacus, a device on which beads slide, is a variant of the exchequer.
- Physicists must have constructed all of the buildings in Europe, since what Americans call the n^{th} floor is designated the (n-1)^{st} floor in Europe.
- The most interesting part of the book, by far, is Chapter 8, "A Mayan Interlude." It describes how the Mayans used a number of incommensurate calendars to keep track of time since they believed that their world would end when time "stopped." Thus when one of their linear calendars arrived at the end of a cycle, there was another to keep time going. But every 52 years two of their calendars would end together, and they then carried out hideous blood sacrifices to assuage the gods
- In more recent times, Mitchell Feignebaum has used Farey addition to construct models of chaotic systems. In Farey addition p/q + r/s = (p + q)/(r + s).
- Here are a few of the "nots;" Page 6: "…5000 years are like an evening gone." Page 237:
consisted of arithmetic, music, geometry and music, considered to be the four branches of
mathematics.