The Nothing that Is: A Natural History of Zero (1999)

This interesting little book caught my eye in the stacks of the local public library when I was searching for something else. It recounts the history of the concept of zero and the struggle over thousands of years to understand how to use it in calculations. The problem for the ancient Sumerians was this: zero is certainly nothing, yet is must also be something since it is a number or a symbol. This presents the perfect validation of the basic tenet of semantics "the word is not the thing that it stands for" as presented by Alfred Korzybski in his book on general semantics. Korzybski also discusses the tricky word 'is' which is also an important part of that seeming contradiction, as well as part of Bill Clinton's defense when he responded to an interrogation by asking what did the questioner mean by 'is'. The author continues to tell the story of how various cultures Hindu, Greek, etc. responded in the face of the paradox of being nothing and something at the same time.

In view of the current mania concerning Dec. 21, 2012, the most interesting part of this book for the typical reader will be the chapter on the Maya concept of zero and how they used it in their calendars. The Mayans were clever at math and used zero as a place holder in their computing before the Europeans. The author tells us that the Mayans had a superstition that the gods might chose to end the universe at the end of a calendar. To prevent this, the Mayans used six calendars with incommensurable periods so that they would never all end on the same day. The longest of these stretched over 68,000,000 years. They also started the first day of a new calendar unit with human sacrifice to appease the gods and they numbered this day with zero. The author states that they sometimes used the symbol of the god of death for zero although apparently a shell like symbol was more common. (In an example of Mayan denumeration shown in an article by E.E. Krupp in the November 2009 issue of "Sky and Telescope", there were several different symbols for zero, possibily indicative of what kind of thing there was none of. Thus they may not have had the abstract idea of the number zero, but they did have the idea of a place holder, or operator concept of zero. The zero day may have been merely an interlude between the old and new calendars in which the gods could be influenced to continue the universe.) It seems that the primitive Mayan superstition concerning the end of calendars has been transplanted into modern American culture; we hope without the human sacrifice part.

The story continues with the history of zero in the European area. There it was associated with evil or the devil, ideas about as crazy as the Mayans, but at least without the sacrificial aspects. It seems that it was the commercial value of using zero in bookeeping that finally turned the tide and zero was finally accepted as a number, but not untill the renaissance.

In the last half of the book the author discusses the role of zero in the present day. He gives a method of factorization using zero which generalizes completion of the square. A set of postulates for an integral domain are introduced to discuss the problem that zero is still a chimera for many people since it is the only number which cannot be a divisor. Von Neumann enters the picture to show how to identify zero with the empty set and thus create all numbers out of absolutely nothing (at least no material thing) in a completely logical way. In a sense, this resolves the conflict between zero as nothing and as a number. The basic rules of differential calculus are given to discuss the problem which some people still have with the limit dx -> 0 when it is necessary to keep dividing by dx all the way. Real variable theory now provides a logical resolution to this problem. With real numbers defined by Cauchy series and Dedekind cuts, taking limits appears completely natural and is completely logical.

However in spite of the valiant efforts of Von Neumann, Cauchy, and Dedekind the struggle continues to the present day, but now the arena is more in physics than mathematics. The author (a mathematcian) touches only briefly on this aspect. The present situation can be understood by reference toFrank Wilczek's book "The Lightness of Being", p. 84, where Wilczek tells us that in a private conversation, Richard Feynman admitted that in his youth, his view of empty space was "there's nothing there". Later in the conversation he says that he was deeply disappointed that quantum electrodynamics (QED) could not be developed without the concept of the field; but the mathematics needed fields. Indeed, in his book "QED, the Strange Theory of Light and Matter", he discribes the fundamental principles of that subject without using the word 'wave' or the word 'field'. Since it is now an accepted experimenatal fact the particles can appear out of 'empty' space, one may partly agree with Feynman that there was NO THING there, but one must think that there was something there (possibly intangible or immaterial but not nothing), or equate that creation of particles to shear magic. This would mean that our logic and our sciences were all really worthless. The scientific answer to what 'something' was there is 'non material fields' of various types. Of course there are many people besides Feynman that are emotionally disturbed by the suggestion that the intangible or immaterial could be real, maybe because the word 'spiritual' could be used to describe those conditions. Feynman's dilemma continues for many people up to the present. So this confronts most of the current world with the same semantic riddle which confronted the Sumerians 5000 years ago: 'nothing' which is also 'something'. The more things change the more they are the same.

For a number nut (like me) this book is marvelous recreational reading, but it could be educational reading for a person who did not understand or believe that mathematics can have an important impact on our culture and intellectual life. This book could also be an easy introduction to abstract mathematical thinking for a mature person. ( )

Zero

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