The Book of Nothing

by John D. Barrow

  Figure 1.13 The Babylonian forms for 610 and 20 could easily be confused.

  Figure 1.14 The Babylonians first introduced a ‘separator’ symbol to mark empty spaces in the number expression. They were shaped like double chevrons and created by two overlapping impressions of the stylus wedge. This example was found on a tablet recording astronomical observations, dated between the late third and early second century BC.

  The Babylonian solution to the no-entry problem was to use a variant of the old separation marker sign to signal that there was no entry in a particular position. This appears in writing of the fourth century BC, but may have been in existence for a century earlier because of the paucity of earlier documents and the likelihood that some of those that do exist are copies of earlier originals. A typical example of the use of the Babylonian zero is shown in Figure 1.15, where the number 3612 = 1 × (60 × 60) + (0 × 60) + (1 × 10)+ 2 is written:

  Figure 1.15 An example of the Babylonian zero as used in the third or second century BC to write 3612 = (1 × 60 × 60) + (0 × 60) + (1 × 10) + 2.

  Babylonian astronomers17 also made extensive use of zero at the end of a character string and we find examples of 60 being distinguished from 1 by writing it as shown in Figure 1.16. So we begin to see how the Babylonian zero functioned in a similar way to our own. It began, like the positional notation, as shorthand used by Babylonian mathematicians. This ensured its extensive use by Babylonian astronomers, and it is because of the huge importance and persistence of Babylonian astronomy that their system of counting remained so influential as the centuries passed.

  Figure 1.16 The number sixty in an astronomical record was also used at the end of strings of numerals, as shown here.

  This is the culmination of the Babylonian development: the first symbolic representation of zero in human culture. In retrospect, it seems such a straightforward addition to their system that it is puzzling why it took more than fifteen centuries to pass from the key step of a positional notation to a system with an explicit zero symbol.

  Yet the Babylonian zero should not be identified totally with our own. For the scribes who etched the double chevron sign on their clay tablets, those symbols meant nothing more than an ‘empty space’ in the accounting register. There were no other shades of meaning to the Babylonian ‘nothing’. Their zero sign was never written as the answer to a sum like 6 −6. It was never used to express an endpoint of an operation where nothing remains. Such an endpoint was always explained in words. Nor did the Babylonian zero find itself entwined with metaphysical notions of nothingness. There is a total absence of any abstract interweaving18 of the numerical with the numinous. They were very good accountants.

  THE MAYAN ZERO

  “I have nothing to say

  and am saying it and that is

  poetry.”

  John Cage19

  The third invention of the positional system occurred in the remarkable Mayan culture that existed from about AD 500 until 925. Paradoxically, despite achieving great sophistication in architecture, sculpture, art, road building, writing, numerical calculation, calendar systems and predictive astronomy, the Mayans never invented the wheel, never discovered metal or glass, had no clocks which could measure time intervals of less than a day, and never made use of beasts of burden. Stone Age practices went hand in hand with extraordinary arithmetical sophistication. Why their culture ended so suddenly is still a mystery. All that remains are abandoned cities in the jungles and grasslands of present-day Mexico, Belize, Honduras and Guatemala. All manner of disasters have been suggested for the exodus of the population. Plague or civil war or earthquakes have all been blamed. A better bet is agricultural exhaustion of their soil through persistent intensive farming and overuse.

  The Mayan counting system was founded upon a base of 20 (see Figure 1.17) and the numbers were composed of combinations of dots (each denoting ‘one’) and rods (each denoting ‘five’). The first nineteen numbers were built up with dots and lines in a simple additive fashion, probably derived from an earlier finger-and-toe counting system.20 The dot (or sometimes a small circle) used as a symbol for ‘one’ is found throughout the Central American region at early times and was probably linked to the use of cocoa beans as a currency unit. As in the Babylonian culture, there was a distinction between everyday calculation and the higher computations of mathematicians and astronomers.

  When one needed to write numbers larger than 20 a tower of symbols was created, the bottom floor marking multiples of 1, the first floor multiples of 20. However, the second floor did not read multiples of 20 × 20. It carried multiples of 360! But the pattern then carried on unbroken. The next level up then carried multiples of 20 × 360 = 7200; then 20 × 7200 = 144,000 and all subsequent levels were each 20 times the level below. Numbers were read downwards. The number 4032 = (11 × 360) + (3 × 20) + 12 is shown in Figure 1.18.

  Figure 1.17 The numbers from 1 to 20 in the Mayan system used by priests and astronomers.

  Thus we see that the Mayans had a positional, or place-value, system and to this they added a symbol for zero, to denote no entry on one of the levels of the number tower. The symbol they used is very curious. It resembles a shell or even an eye, comes in a number of slightly different forms, and seems to have conveyed the idea of completion, reflecting its aesthetic role in representing the numbers which we will describe below. Some of the zero shapes are shown in Figure 1.19. Thus the number 400 = (1 × 360) + (2 × 20) + 0 would be written as shown in Figure 1.20. The Mayans used their zero symbol in both intermediate and final positions in their symbol strings, just as we do.

  The curious step in the Mayan system at level two, marked by 360 rather than 400 as would have been characteristic of a pure base-20 system, means that the zero symbol differs from our own in one very important respect. If we add a zero symbol to the right-hand end of any number then we multiply its value by 10, the value of our system’s base; thus 170 = 17 × 10. If a counting system of any base proceeds through levels which are each related to the previous one by a power of the base, whatever its value, then adding a zero to a symbol string will always have the effect of multiplying the number by the base value. The Mayan system lacked this nice property because of the uneven steps from level to level. It stopped the Mayans from exploiting their system to the full.

  Figure 1.18 The Mayan representation of the number 4032.

  Figure 1.19 Different symbolic forms for the Mayan zero (see note 9). They look like the shells of snails and sea creatures, or human eyes.

  The Mayans failed to introduce an even sequence of levels for a reason; they had other jobs for their counting system. It was designed to play a particular role keeping track of their elaborate cyclic calendar. They had three types of calendar. One was based upon a sacred cycle of 260 days, the tzolkin, which was split into 20 periods of 13 days. The second was a civil ‘year’ of 365 days, called the haab, which was divided into 18 periods of 20 days each plus a transition period of 5 days. The third calendar was based on a period of 360 days, called the tun, which was divided into eighteen periods of 20 days. Twenty tun equalled one katun (ka was the word for 20); twenty katuns was one baktun (bak was the word for 20 × 20); one uinal equalled 20 days.21 Special hieroglyphs were used to represent these periods. A complete picture denoting a period of time would then combine symbols for the time intervals with those signifying how many multiples of them were meant. The hieroglyph in Figure 1.21 should be read from left to right and from top to bottom and records the following times: 9 baktun, 14 katun, 12 tun, 4 uinal and 17 kin (days).

  Figure 1.20 The Mayan representation of 400.

  Figure 1.21 A Mayan hieroglyph denoting a length of time. For each of the units, baktun, katun, uinal and day, a special picture was used, usually of a head with other defining features or adornments. Alongside each picture was a numeral, composed of dots and bars, to indicate how many of those units should be taken. Sometimes small numbers, requiring only two dots or bars would have further orn
aments added to balance the space. Here, reading from left to right and top to bottom, we have a representation of 9 baktun and 14 katun and 12 tun and 4 uinal and 17 kin. This gives a total 3892 tun and 97 kin, or 1,401,217 kin (days).

  In these pictograms the zero was represented by a number of exotic glyphs,22 a few of which are shown in Figure 1.22.

  Figure 1.22 The various hieroglyphs for zero found on Mayan columns and statues.

  In this scheme the zero symbol is not essential for recording dates. What is novel about the Mayan zero is that it was introduced for aesthetic reasons. Without the zero picture, the pictogram for a date would have had a vacant patch and would look unbalanced. The elaborate zero glyphs filled the gap and created a dramatic rendering of a date which reinforced the religious significance of the numbers being represented.

  THE INDIAN ZERO

  “The Indian zero stood for emptiness or absence, but also space, the firmament, the celestial vault, the atmosphere and ether, as well as nothing, the quantity not to be taken into account, the insignificant element.”

  Georges Ifrah23

  The destruction of the Babylonian and Mayan civilisations prevented their independent inventions of the zero symbol from determining the future pattern of representation. That honour was to be given to the third inventor of the zero whose way of writing all numbers is still used universally today.

  The Hindus of the Indus valley region had a well-developed culture as early as 3000 BC. Extensive towns were established with water systems and ornaments. Seals, writing systems and evidence of calculation witness to a sophisticated society. Writing and calculation spread throughout the Indian sub-continent over the following millennia. A rich diversity of calligraphic styles and numeral systems can be found throughout Central India and in nearby regions of South-East Asia which made use of the Brahmi numerals. This notation appeared for the first time about 350 BC, although only examples of the numerals 1, 2, 4 and 6 still remain on stone monuments. Transcriptions in the first and second century BC show what they probably looked like24 (see Figure 1.23).

  The forms of the Brahmi numerals are still something of a mystery. The signs for the numerals from 4 to 9 do not have any obvious association with the quantities they denote, but they may derive from alphabets that have disappeared or be an evolutionary step from an earlier system of numerals with clear interpretations that no longer exist.

  Figure 1.23 The early Indian symbols for the numerals 1 to 9.

  The Brahmi system was transformed into a positional base-10, or decimal, notation in the sixth century AD. It exploited the existence of distinct numerals for the numbers 1 to 9 and a succinct notation for larger numbers and number words for the higher powers of ten. The earliest written example of its use goes back to AD 595 on a copperplate deed from Sankheda.25

  The inspiration for this brilliant system is likely to have been the use of counting boards for laying out numbers with stones or seeds. If you want to lay out a number like 102 using stones, then place one stone in the hundreds column followed by a space in the tens column and a two in the units. A further motivation for devising a clear logical notation for dealing with very large numbers is known to have come from the studies of Indian astronomers, who were influenced by earlier Babylonian astronomical records and notations. The commonest positional notation emerging from the Brahmi numerals was that using the Nâgarî script, shown in Figure 1.24.

  A unique feature of the Indian development of a positional system is the way in which it made use of the same numerals that were in existence long before. In other cultures the creation of a positional notation required a change of notation for the numerals themselves. The earliest known use of their place-value system is AD 594.

  Figure 1.24 The evolution of the Nâgarî numerals. Notice how similar many of them are to the numerals we use today.

  As we have learned from the Babylonians and the Mayans, once a positional system is introduced it is only a matter of time before a zero symbol follows. The earliest example of the use of the Indian zero is in AD 458, when it appeared in a surviving Jain work on cosmology, but indirect evidence indicates that it must have been in use as early as 200 BC. At first, it seems that it was denoted by a dot, rather than by a small circle. A sixthcentury poem, Vâsavadattâ, speaks of how26

  “the stars shone forth … like zero dots … scattered in the sky.”

  Later, the familiar circular symbol, 0, replaced the dot and its influence spread east to China. It was used to mark the absence of an entry in any position (hundreds, tens, units) of a decimal number and, because the Indian decimal system was a regular one, with each level ten times the previous one, zero also acted as an operator. Thus, adding a zero to the end of a number string effected multiplication by 10 just as it does for us. A wonderful application of this principle is to be found in a piece of Sanskrit poetry27 by Bihârîlâl in which he expresses his admiration for a beautiful woman by referring to the dot (tilaka) on her forehead28 in a mathematical way:

  “The dot on her forehead

  Increases her beauty tenfold,

  Just as a zero dot [sunya-bindu]

  Increases a number tenfold.”

  Although the Indian zero was first introduced to mark an absent numeral in the same way as for the Babylonians and the Mayans, it rapidly assumed the status of another numeral. Also, in contrast to the other inventors of zero, the Indian calculators readily defined it to be the result of subtracting any number from itself. In AD 628, the Indian astronomer Brahmagupta defined zero in this way and spelled out the algebraic rules for adding, subtracting, multiplying and, most strikingly of all, dividing with it. For example,

  “When sunya is added to a number or subtracted from a number, the number remains unchanged; and a number multiplied by sunya becomes sunya.”

  Remarkably, he also defines infinity as the number that results from dividing any other number by zero and sets up a general system of rules for multiplying and dividing positive and negative quantities.

  There have been some interesting speculations as to why the Indian zero sign assumed the circular shape.29 After all, we have seen it assume a very different form in the Mayan and Babylonian scripts. Subhash Kak has proposed that it developed from the Brahmi symbol for ten. This resembled a simple fish or a proportionality sign ∞. Later, in the first and second centuries, it looked like a circle with a 1 attached (see Figure 1.25). Hence, it is suggested that the symbol for ten may naturally have divided into the sign for 1, a single vertical stroke, and the remaining circle which had the zero value.

  Figure 1.25 A possible separation of the fishlike symbol for 10 into a circle and a line representing 1, leaving the circle for a zero sign.

  A fascinating feature of the zero symbol in India is the richness of the concept it represents. Whereas the Babylonian tradition had a one-dimensional approach to the zero symbol, seeing it simply as a sign for a vacant slot in an accountant’s register, the Indian mind saw it as part of a wider philosophical spectrum of meanings for nothingness and the void. Here are some of the Indian words for zero.30 Their number alone indicates the richness of the concept of nothing in Indian philosophy and the way in which different aspects of absence were seen to be something requiring a distinct label.31

  Word Sanskrit Meaning

  Abhra Atmosphere

  Akâsha Ether

  Ambara Atmosphere

  Ananta The immensity of space

  Antariksha Atmosphere

  Bindu A point

  Gagana The canopy of heaven

  Jaladharapatha Sea voyage

  Kha Space

  Nabha Sky, atmosphere

  Nabhas Sky, atmosphere

  Pûrna Complete

  Randhra Hole

  Shûnya/sunya Void

  Vindu Point

  Vishnupada Foot of Vishnu

  Vyant Sky

  Vyoman Sky or space

  Bindu is used to describe the most insignificant geometrical object, a single point or a circl
e shrunk down to its centre where it has no finite extent. Literally, it signifies just a ‘point’, but it symbolises the essence of the Universe before it materialised into the solid world of appearances that we experience. It represents the uncreated Universe from which all things can be created. This creative potential was revealed by means of a simple analogy. For, by its motion, a single dot can generate lines, by whose motion can be generated planes, by whose motion can be generated all of three-dimensional space around us. The bindu was the Nothing from which everything could flow.

  This conception of generation of something from Nothing led to the use of the bindu in a range of meditational diagrams. In the Tantric tradition the meditator must begin by contemplating the whole of space, before being led, shape by shape, towards a central convergence of lines at a focal point. The inverse meditational route can also be followed, beginning with the point and moving outwards to encompass everything, as in Figure 1.26, where the intricate geometrical constructions of the Sriyantra are created to focus the eye and the mind upon the convergent and divergent paths that link its central point to the great beyond.

  The revealing thing we learn from the Indian conception of zero is that the sunya included such a wealth of concepts. Its literal meaning was ‘empty’ or ‘void’ but it embraced the notions of space, vacuousness, insignificance and non-being as well as worthlessness and absence. It possesses a nexus of complexity from which unpredictable associations could emerge without having to be subjected to a searching logical analysis to ascertain their coherence within a formal logical structure. In this sense the Indian development looks almost modern in its liberal free associations. At its heart is a specific numerical and notational function which it performs without seeking to constrain the other ways in which the idea can be used and extended. This is what we would expect to find in modern art and literature. An image or an idea may exist with a well-defined form and meaning in a specific science, yet be continually elaborated or reinvented by artists working with different aims and visions.