The Book of Nothing: Vacuums, Voids, and the Latest Ideas about the Origins of the Universe - John D. Barrow (2002)
Chapter 1. Zero – The Whole Story
“Is it not mysterious that we can know more about things which do not exist than about things which do exist?”
Alfréd Renyi^{1}
“Round numbers are always false.”
Samuel Johnson^{2}
THE ORIGIN OF ZERO
“The great mystery of zero is that it escaped even the Greeks.”
Robert Logan^{3}
When we look back at the system of counting that we learned first at school it seems that the zero is the easiest bit. We used it to record what happens when nothing is left, as with a sum like 6 minus 6, and anything that gets multiplied by zero gets reduced to zero, as with 5 × 0 = 0. But we also used it when writing numbers to signal that there is an empty entry, as when we write one-hundred and one as 101.
These are such simple things – much simpler than long division, Pythagoras’ Theorem, or algebra – that it would be easy to assume that zero must have been one of the first pieces of arithmetic to be developed by everyone with a counting system, while the more difficult ideas like geometry and algebra were only hit upon by the most sophisticated cultures. But this would be quite wrong. The ancient Greeks, who developed the logic and geometry that form the basis for all of modern mathematics, never introduced the zero symbol. They were deeply suspicious of the whole idea. Only three civilisations used the zero, each of them far from the cultures that would evolve into the so-called Western world, and each viewed its role and meaning in very different ways. So why was it so difficult for the zero symbol to emerge in the West? And what did the difficulty have to do with Nothing?
As the end of the year 1999 approached, the newspapers devoted more and more copy to the impending doom that was to be wrought by the Millennium Bug. The reason for this collective loss of sleep, money and confidence was the symbol ‘zero’, or two of them to be more precise. When the computer programs that control our transport and banking systems were first written, computers were frugal with memory space – it was much more expensive than it is today.^{4} Anything that could save space was a money-saving bonus. So when it came to dating everything that the computer did, instead of storing, say, 1965, the computer would just store the last two digits, 65. Nobody thought as far ahead as the year 2000 when computers would be faced with making sense of the truncated ‘date’00. But if there is one thing that computers really don’t like, it’s ambiguity. What does 00 mean to the computer? To us it’s obviously short for year 2000. But the computer doesn’t know it isn’t short for 1900, or 1800 for that matter. Suddenly, you might be told that your credit card with its 00 expiry year is 99 years out of date. Born in 1905? Maybe the computer would soon be mailing out your new elementary-school application forms. Still, things didn’t turn out as badly as the pessimists predicted.^{5}
Counting is one of those arts, like reading, into which we are thrust during our first days at school. Humanity learned the same lessons, but took thousands of years to do it. Yet whereas human languages exist by the thousand, their distinctiveness often enthusiastically promoted as a vibrant symbol of national identity and influence, counting has come to be a true human universal. After the plethora of our languages and scripts for writing them down, a present-day tourist from a neighbouring star would probably be pleasantly surprised by the complete uniformity of our systems of reckoning. The number system looks the same everywhere: ten numerals −1, 2, 3, 4, 5, 6, 7, 8, 9 and 0 – and a simple system that allows you to represent any quantity you wish: a universal language of symbols. The words that describe them may differ from language to language but the symbols stay the same. Numbers are humanity’s greatest shared experience.
The most obvious defining feature of our system of counting is its use of a base of ten. We count in tens. Ten ones make ten; ten tens make one hundred; and so on. This choice of base was made by many cultures and its source is clearly to be found close at hand with our ten fingers, the first counters. Sometimes one finds this base is mixed in with uses of 20 as a base (fingers plus toes) in more advanced cultures, whilst less advanced counting systems might make use of a base of two or five.^{6} The exceptions are so rare as to be worth mentioning. In America one finds an Indian counting system based on a base of eight. At first this seems very odd, until you realise that they were also finger counters – it is just that they counted the eight gaps between the fingers instead of the ten fingers.
You don’t have to be a historian of mathematics to realise that there have been other systems of numbers in use at different times in the past. We can still detect traces of systems of counting that differ in some respects from the decimal pattern. We measure time in sets of 60, with 60 seconds in a minute, 60 minutes in an hour, and this convention is carried over to the measurement of angles, as on a protractor or a navigator’s compass. Else-where, there are relics of counting in twenties:^{7} ‘three-score years and ten’ is the expected human lifetime, whilst in French the number words for 80 and 90 are quatre-vingts and quatre-vingt-dix, that is four-twenties and four-twenties and ten. In the commercial world we often order by the gross or the dozen, witness to a system with a base of twelve somewhere in the past.
The ten numerals 0, 1, …, 9 are used everywhere, but one other system for writing numbers is still in evidence around us. Roman numerals are often to be found on occasions where we want to emphasise something dynastic, like Henry VIII, or traditional, like the numbers on the clock face in the town square. Yet Roman numerals are rather different from those we use for arithmetic. There is no zero sign. And the information stored in the symbols is different as well. Write 111 and we interpret it as one hundred plus one ten plus one: one-hundred and eleven. Yet to Julius Caesar the marks 111 would mean one and one and one: three. These two missing ingredients, 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.
EGYPT – IN NEED OF NOTHING
“Joseph gathered corn as the sand of the sea, very much, until he left numbering; for it was without number.”
Genesis 41
The oldest developed counting systems are those used in ancient Egypt and by the Sumerians in Southern Babylonia, in what is now Iraq, as early as 3000 BC. The earliest Egyptian hieroglyphic^{8} system used the repetition of a suite of symbols for one, ten, a hundred, a thousand, ten thousand, a hundred thousand and a million. The symbols are shown in Figure 1.1. The
Figure 1.1 Egyptian hieroglyphic numerals.
Egyptian symbols for the numerals one to nine are very simple and consist of the repetition of an appropriate number of marks of the vertical stroke, |, the symbol for one; so three is just |||. The symbols for the larger multiples of ten are more picturesque. Ten is denoted by an inverted u, a hundred by a coil, a thousand by a lotus flower, ten thousand by a bent finger, a hundred thousand by a frog or a tadpole with a tail, and a million by a man with his arms raised to the heavens. With the exception of the sign for one, they seem to have no obvious connection with the quantities they denote. Some connections are probably phonetic, deriving from the similar sounds for the things pictured and the original words used to describe the quantities. Only the bent finger marking ten thousand seems to hark back to a system of finger counting. We can only guess about the others. Perhaps the tadpoles were so numerous in the Nile when the frogs’ spawn hatched in the spring that they symbolised a huge number; maybe a million was just an awesomely large quantity, like the populations of stars in the heavens above.
The symbols were written differently if they were to be read from right to left or left to right in an inscription.
Hieroglyphs were generally written down from right to left so that our number 3,225,578 would appear as shown in Figure 1.2.
One of the oldest examples of these numerals appears on the handle of a club belonging to King Narmer, who lived in the period 3000–2900 BC, celebrating the fact that the loot seized in one of his military campaigns amounted to 400,000 bulls, 1,422,000 goats and 120,000 human prisoners. The symbols for these quantities beneath pictures of a bull, a goat and a seated figure can be seen on the bottom right of Figure 1.3. The order in which the symbols are written is not important because there are different symbols for one, ten and a hundred. The hieroglyph
would signal exactly the same quantity if written forwards or backwards. The symbols can be laid out in any way at all without changing the value of the number they are representing. However, Egyptian stonemasons were given strict rules of style for writing numbers: signs were to appear from right to left in descending order of size on a line underneath the symbol for the object that was being counted (as in Figure 1.3). However, there was a tendency to group similar symbols together over two or three lines to help the reader quickly read off the total, as shown in Figure 1.4.
Figure 1.2 The hieroglyph for our number three million, two hundred and twenty-five thousand, five hundred and seventy-eight.
Thus we see that the relative positions of the Egyptian counting symbols carry no numerical information and so there is no need for a symbol for zero. When the number symbols can sit in any location without altering the total quantity they are representing, there is no possibility of an empty ‘slot’ and no meaning to a signal of its presence. The need for a zero arises when you have nothing to count – but in that case you write no symbols at all. The Egyptian system is an early example of a decimal system (the collective unit is 10) with symbols for numbers which carry no positional information. In such a system there is no place for a zero symbol.
Figure 1.3 Hieroglyphics inscribed on the handle of King Narmer’s war club,^{9} 3000–2900 BC.
Figure 1.4 The grouping of number signs to help the reader.
BABYLON – THE WRITING IS ON THE WALL
“In the same hour came forth fingers of a man’s hand, and wrote over against the candlestick upon the plaster of the wall of the king’s palace … And this is the writing that was written, Mene, Mene, Tekel, Upharsin. This is the interpretation. Mene; God hath numbered thy kingdom and finished it. Tekel; Thou art weighed in the balances, and art found wanting. Peres; thy kingdom is divided.’
Daniel 5^{10}
The earliest Sumerian system, also in use around 3000 BC, was more complex than that employed by the Egyptians and seems to have developed independently. It was later adopted by the Babylonians and so the two civilisations are usually regarded as different parts of a single cultural development. The motivation for their systems of writing and counting was at first administrative and economic. They kept detailed records and accounts of exchanges, stores and wages. Often, a detailed list of items will be found on one side of a tablet, with the total inscribed on the reverse.
The counting system of early Sumer was not solely decimal. It made good use of the base ten to label quantities but it also introduced 60 as a second base number.^{11}
It is from this ancient system that we inherited our pattern of time-keeping with 60 seconds to the minute and 60 minutes to the hour. Expressing 10 hours 10 minutes and 10 seconds in seconds shows us how to unfold a base-60 counting system. We have a total of (10 × 60 × 60) + (10 × 60) + 10 = 36,000 + 600 + 10 = 36,610 seconds.
The Sumerians had number words for the quantities 1, 60, 60 × 60, 60 × 60 × 60, … and so on. They also had words for the numbers 2, 3, 4, 5, 6, 7, 8, 9 and 10, together with the multiples of ten below 60. A distinct word was used for 20 (unrelated to the words for 2 and 10) but ‘thirty’ was a compound word meaning ‘three tens’, ‘forty’ meant ‘two twenties’, and ‘fifty’ meant ‘forty and ten’. So there was a weaving of base 10 and base 20 elements to ease the jump up from one to sixty.
Whereas the Egyptians carved their signs in stone with hammers and chisels or painted them on to papyri with reeds, Sumerian records were kept by making marks in tablets of wet clay. Stone was not common in Sumer and other media like papyrus or wood would rapidly perish or rot, but clay was readily available. The inscriptions were made by impressing the wet clay with two types of reed or ivory stylus, shaped like pencils of differing widths. The round blunt end allowed notches or circular shapes to be impressed whilst the sharp end allowed lines to be drawn. The sharp end was used for writing whilst the blunt end was used for representing numbers. The original symbols are shown in Figure 1.5 and are called curvilinear signs. The number symbols^{12} usually appeared over an image of the thing being enumerated and reveal a new feature, not present in Egypt. The symbol for 600 combines the large notch, representing 60, with the small circle, representing 10. Likewise, the symbol for 36,000 combines the large circle, for 3,600, with the small circle, for 10. This economical scheme creates a multiplicative notation. There are fewer symbols to learn and the symbols for large numbers have an internal logic that enables larger numbers to be generated from smaller ones without inventing new symbols. However, notice that you have to do a little bit of mental arithmetic every time you want to read a large number! The system is additive and there is again no significance to the positions of the symbols when they are inscribed on the clay tablets. As in Egypt, similar symbols were grouped together for stylistic reasons and for ease of reckoning. The early style was to gather marks into pairs. For example, the decimal number 4980 is broken down as
and this would be written as shown in Figure 1.6 since tablets were read from right to left and from top to bottom.
Figure 1.5 The impressed shapes representing Sumerian numerals on clay tablets.
A tedious feature of this system is the huge number of marks that have to be made in order to represent large numbers that are not exact multiples of 60. To overcome this problem, scribes developed a shorthand subtraction notation, introducing a ‘wing’ sign that played the role of our minus sign so that they could write a number like 59 as 60 minus 1 by means of the three symbols (Figure 1.7) instead of the fourteen marks that would otherwise have been required.^{13}
By 2600 BC a significant change had occurred in the way that the Sumerian number characters were written. The reason: new technology – in the form of a change of writing implement. A wedge-shaped stylus was introduced which could produce sharper lines and wedge-shapes of different sizes. These became known as ‘cuneiform’^{14} signs and only two marks are used, a vertical wedge denoting ‘one’ and a chevron representing ‘ten’ (Figure 1.8). Again, the fusion of symbols can be used to build up large numbers from smaller ones. If the symbols for 60 and 10 were in contact they signified a multiplication of values (600) whereas if they were separated they signified an addition (70). However, some care was needed to make sure that juxtapositions of signs like these did not become confused. The Sumerian combinations of symbols avoided this problem because the individual marks were much more distinct.
Figure 1.6 The number 4980 in early Sumerian representation, before 2700 BC.
Another problem was the distinction of the signs for 1 and for 60. Their shapes are identical wedges and at first they were distinguished simply by making the 60 wedge bigger. Later, it was done by separating the wedge shape for 60 from those for numbers less than nine. The writing of the number 63 is shown in Figure 1.9.
Figure 1.7 The number 59 written as 60 minus 1.
Figure 1.8 The cuneiform impressions made by the two ends of the scribe’s stylus, denoting the numbers 1 and 10.
Many other systems of counting can be found around the ancient world which use the same general principles as these. The Aztecs (AD 1200) had an additive base-20 system with symbols for 1, 20, 400 = 20 × 20, and 8000 = 20 × 20 × 20. The Greeks (500 BC) used a base-10 system with different signs for 1, 10, 100, 1000, 10,000 but supplemented them with a further sign for 5 which they then added to the other signs to generate new symbols for 50, 500, 5000, and so on (see Figure 1.10).
All these systems of writing numbers are cumbersome and laborious to use if you want to do calculations that involve multiplication or division. The notation does not do any work for you, it is just like a shorthand for writing down the number words in full. The next step in sophistication, a step that was to culminate in the need to invent the zero symbol, was to introduce a positional or place value system in which the locations of symbols determined their values. This allows fewer symbols to be used because the same symbol can have different meanings in different locations or when used in different contexts.
Figure 1.9 Two ways of writing the number 63: (a) using a larger version of the 60 symbol to separate 63 as 60 and 3, or (b) by leaving a space between the symbol for 60 and those for 3.
Figure 1.10 Greek numerals, which first appeared around 500 BC, used combinations of symbols to generate higher numbers. As an example, we have written the number 6668.
A positional system appeared first in Babylonia around 2000 BC. It simply extended the cuneiform notation and the old additive base-60 system to include positional information. It was used by mathematicians and astronomers rather than for everyday accounting because the old system allowed the reader to see the relative sizes of numbers more easily. Many scribes must therefore have practised with both systems. However, it was used in the recording of royal decrees and so must have been understood by a broad cross-section of the Babylonian public. Thus, a number like 10,292 would be conceived in our notation as [2; 51; 32] = (2 × 60 × 60) + (51 × 60) + 32, and written in cuneiform as shown in Figure 1.11. This is just like our representation of a number like 123 as (1 × 10 × 10) + (2 × 10) + 3. Our notation just reads off the number that multiplies the number of contributions by each power of 10. We still retain the Babylonian system for time measures. Seven hours and five minutes and six seconds is just (7 × 60 × 60) + (5 × 60) + 6 = 25,506 seconds.
Figure 1.11 The number 10,292 in cuneiform.
The earliest positional decimal system like our own did not appear until about 200 BC when the Chinese introduced the place value system into their base-10 system of signs. Their rod number symbols, together with an example of their positional notation in action, are shown in Figure 1.12.
THE NO-ENTRY PROBLEM AND THE BABYLONIAN ZERO
“There aren’t enough small numbers to meet the many demands made of them.”
Richard K. Guy^{15}
These advances were not without their problems. The Babylonian system was really a hybrid of positional and additive systems because the marking of the number of each power of 60 was still denoted in an additive fashion. This could produce ambiguity if sufficient space was not left between one order of 60 and the next. For instance, the symbols for 610 = [10; 10] = (10 × 60) + 10 could easily be misread for 10 + 10 (see Figure 1.13). This was generally dealt with by separating the different orders of 60 clearly. Eventually, a separation marker was introduced to make the divisions unambiguous. It consisted of two wedge marks, one on top of the other, as shown in Figure 1.14.
Any difficulties of interpretation would be compounded further if there was no entry at all in one of the orders. The spacing would then be more tricky to interpret. Imagine that our system had no 0 symbol and relied on careful spacing to distinguish 72 (seventy-two) from 7 2 (seven hundred and two). With different writing styles to contend with there would be many problems which are exacerbated if one has to distinguish 7 2 (seven thousand and two) as well as 7 2 and 72. The more spaces that you need to leave, the harder it becomes to judge.^{16} This is why positional notation systems eventually need to invent a zero symbol to mark an empty slot in their positional representation of a number. The more sophisticated their commercial systems the greater is the pressure to do so. For nearly 1500 years the Babylonians worked without a symbol for ‘no entry’ in their register of different powers of ten or sixty; they merely left a space. Their success required a good feeling for the magnitudes of the astronomical and mathematical problems they were dealing with, so that large discrepancies from expected answers could be readily detected.
Figure 1.12 (a) Chinese rod numerals. They are pictures of bamboo or bone calculating rods. When these symbols were used in the tens or thousands position they were rotated, and written as in (b), so our number 6666 would have been expressed as shown in (c).
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 astronomers^{17} 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 interweaving^{18} 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 Cage^{19}
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 ornaments 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 Ifrah^{23}
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 like^{24} (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 how^{26}
“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 poetry^{27} by Bihârîlâl in which he expresses his admiration for a beautiful woman by referring to the dot (tilaka) on her forehead^{28} 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 circle 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.
Figure 1.26 The Sriyantra, a geometric construction used as a meditational guide in parts of the Tantric tradition. The earliest known examples date from the seventh century AD, but simpler patterns date back to the twelfth century BC. It consists of an intricate nested pattern of triangles, polygons, circles and lines, converging upon a central point, or bindu, which was either the end or the beginning of the meditational development as it moved inwards or outwards through the patterns. Of the nine central triangles, four point upwards marking ‘male’ cosmic energy, and five point downwards marking ‘female’ energy. Considerable geometric knowledge was required to construct these and other Vedic guides to worship.^{32}
INDIAN CONCEPTIONS OF NOTHINGNESS
“It is true that as the empty voids and the dismal wilderness belong to zero, so the spirit of God and His light belong to the all-powerful One.”
Gottfried Leibniz^{33}
The Indian introduction of the zero symbol owes much to their ready accommodation of a variety of concepts of nothingness and emptiness. The Indian culture already possessed a rich array of different concepts of ‘Nothing’ that were in widespread use. The creation of a numeral to de-note no quantity or an empty space in an accountant’s ledger was a step that could be taken without the need for realignment of parts of any larger philosophy of the world. By contrast, the Hebrew tradition regarded the void as the state from which the world was created by the movement and word of God. It possessed a host of undesirable connotations. It was a state from which to recoil. It spoke of poverty and a lack of fruitfulness: it meant separation from God and the removal of His favour. It was anathema. Similarly, for the Greeks it was a serious philosophical dilemma. Their respect for logic led them into a quandary over the treatment of Nothing as if it were something.
The Indian religious traditions were more at home with these mystical concepts. Their religions accepted the concept of non-being on an equal footing with that of being. Like many other Eastern religions, the Indian culture regarded Nothing as a state from which one might have come and to which one might return – indeed these transitions might occur many times, without beginning and without end. Where Western religious traditions sought to flee from nothingness, the use of the dot symbol for zero in meditational exercises showed how a state of non-being was something to be actively sought by Buddhists and Hindus in order to achieve Nirvana: oneness with the Cosmos.
The hierarchy of Indian concepts of ‘Nothing’ forms a coherent whole. It includes the zero symbol of the mathematicians in an integrated way. In Figure 1.27,^{34} the network of meanings gathered by Georges Ifrah is displayed. Notice how the network of meanings is linked to the ideas captured by the words for zero that we gave on pages 36–37. Amid this network of connected meanings, we begin to see some of the sources for our own multiple meanings for Nothing.
At the top level are words, including those which are associated with the sky and the great beyond. They are joined by bindu, reflecting its representation of the latent Universe. As we move down the tree we encounter a host of different terms for the absence of all sorts of properties: non-being, not formed, not produced, not created, together with another collection of terms that carry the meaning of being negligible, insignificant, or having no value.
These two separate threads of meaning merged in the abstract concept of zero so that, at least from the fifth century AD onwards, the concept of Nothing began to reflect all the facets of the early Indian nexus of Nothings, from the prosaic empty vessel to the mystics’ states of non-being.
The Greek tradition was a complete contrast to that of the Far East. Beginning with the school of Thales, the Greeks placed logic at the pinnacle of human thinking. Their sceptical attitude towards the wielding of ‘non-being’ as some sort of ‘something’ that could be subject to logical development was exemplified by Parmenides’ influential arguments against the concept of empty space. He maintained that all his predecessors, like Heraclitus, had been mistaken in adopting the view that all things (those of which we can say ‘it is’) were made of the same basic material, whilst at the same time speaking about empty space (that of which we can say ‘it is not’). He maintained that you can only speak about what is: what is not cannot be thought of, and what cannot be thought of cannot be.
From this statement of the ‘obvious’, Parmenides believed that many conclusions followed, among them the theorem that empty space could not exist. But more unexpected was the further conclusion that neither time, motion nor change could exist either. Parmenides simply believed that whenever you think or speak you must think or speak about something and so there must already exist real things to speak or think about. This implied that they must always have been there and can never change. Plato tells Theaetetus that
Figure 1.27 The array of interrelated meanings of the concepts associated with different aspects of nothingness in early Indian thought, culminating in the mathematical zero.
“the great Parmenides … constantly repeated in both prose and verse:
Never let this thought prevail, that not being is,
But keep your mind from this way of investigation.”^{35}
There are all sorts of problems with these ideas. How can Parmenides ever say that anything is not the case, or that something cannot be? Nevertheless, the legacy of his emphasis upon the need to be speaking about ‘something’ actual makes it very difficult to discuss concepts like the vacuum, Nothing, or even the zero of mathematics. From our vantage point this barrier seems strange. But whereas in India the zero could be introduced without straining any other philosophical position, in Greece it could not.
THE TRAVELLING ZEROS
“The ingenious method of expressing every possible number using a set of ten symbols (each symbol having a place value and an absolute value) emerged in India. The idea seems so simple nowadays that its significance and profound importance is no longer appreciated … The importance of this invention is more readily appreciated when one considers that it was beyond the two greatest men of antiquity, Archimedes and Apollonius.”
Pierre Simon de Laplace (1814)^{36}
The Indian system of counting is probably the most successful intellectual innovation ever devised by human beings.^{37} It has been universally adopted. It is found even in societies where the letters of the Phoenician alphabet are not used. It is the nearest thing we have to a universal language. Invariably, the result of trading contact between the Indian system of counting and any other system was for the former to be adopted by the latter or, at least, for its most powerful features to be imported into the local scheme. When the Chinese encountered the Indian system in the eighth century, they adopted the Indian circular zero symbol and a full place-value notation with nine numerals. The Indian system was introduced into Hebrew culture by the travelling scholar Ben Ezra (1092–1167), who journeyed widely in Asia and the Orient. He described the Indian system of counting in his influential Book of Number^{38} and used the first nine letters of the Hebrew alphabet to represent the Indian numerals from 1 to 9 with a place-value notation but retained the small Indian circle to symbolise zero, naming^{39} it after the Hebrew for ‘wheel’ (galgal). Remarkably, Ben Ezra single-handedly changed the old Hebrew number system into one with a place-value notation and a zero symbol, but there seemed to be no interest in his brilliant innovation and no one else took it up and developed it.
The Indian zero symbol found its way to Europe, primarily through Spain,^{40} via the channel of Arab culture. The Arabs had close trading links with India which exposed them to the efficiencies of Indian reckoning. Gradually, they incorporated the Indian zero into the notation of their own sophisticated system of mathematics and philosophy. Their great mathematician Al-Kharizmi (in whose honour we use the term algorithm) writes of the Indian calculating techniques that,^{41}
“When [after subtraction] nothing is left over, they write the little circle, so that the place does not remain empty. The little circle has to occupy the position, because otherwise there will be fewer places, so that the second might be mistaken for the first.”
The Arabs did not originate a system of numerals of their own. Even in works of mathematics, they wrote out numbers word by word and accompanied them with parallel calculations in other systems, for example in Greek.^{42}
Baghdad was a great cultural centre after its foundation in the eighth century and many mathematical works from India and Greece were translated there. In AD 773 the Caliph of Baghdad received a copy of a 150-year-old Indian astronomical manual, Brahmasphutasiddhanta (the ‘Improved Astronomical Textbook of Brahma’), which used Indian numerals and place-value notation with a zero. Al-Kharizmi wrote his classic work on arithmetic forty-seven years later, explaining the new notation and its expediency in calculation. He introduced the practice of grouping numerals in threes, separated by commas, when writing large numbers that we still use today – as in 1,456,386 – unless the numbers are dates – year 2000, not 2,000. His book was translated into Latin and widely known in Europe from the twelfth century onwards.
The use of words or Greek alphabetical forms for numbers persisted until the tenth century when we see two sets of numerals develop, the ‘East’ and ‘West’ Arab numerals. An interesting feature of both these systems was their adoption of the Indian symbols for the numerals 1,…,9 but not the zero. Instead, they developed a simple form of place-value notation that sidestepped it. If a numeral was denoting the number of tens then a dot was placed above it (e.g. 5 with one dot above meant 50), if it denoted the number of hundreds then two dots were placed above it, and so on. Thus the number three hundred and twenty-four, which we write as 324, would have appeared as
but 320 would have been written as and 302 as . Later, the East Arabs introduced the small circle for the zero system and aligned their notation fully with the Indian convention.
The introduction and spread of the Indo-Arab system of numbers into Europe is traditionally credited to the influence of a Frenchman, Gerbert of Aurillac (945–1003). He became acquainted with Arab science and mathematics during long periods spent living in Spain and was extremely influential in directing theological education in France, and later in other parts of Europe. He had humble beginnings but received a good education in a monastery, and went on to hold a succession of High Church offices, as Abbot of Ravenna and Archbishop of Rheims, before ultimately being elected Pope Sylvester II in 999. Gerbert was the first European to use the Indo-Arab system outside Spain and was one of the most important mathematicians of his time, writing on geometry, astronomy and methods of calculation: a unique mathematical pope.
Gradually, the advantages of the Indo-Arab system became compelling and by the thirteenth century it was quite widely used for trade and commerce. Yet, despite its efficiency, there was opposition. In 1299, a law was passed in Florence forbidding its use. The reason was fear of fraud. Its rival, the system of Roman numerals, is not a place-value system and contains no zero. In the days before the invention of printing, all financial records were handwritten and special measures had to be taken to prevent numbers being illicitly altered by unscrupulous traders. When a Roman number I appeared on the end of a number, for example in II, denoting ‘two’, it would be written IJ to signal that the right-hand symbol was the end of the number. This prevents it being turned into III (but, alas, not into XIII) and is akin to our practice of writing ‘only’ after the amount to pay on a personal cheque. Unfortunately, the Indo-Arab system appeared wide open to fraud of this sort. Unlike the Roman system the addition of a numeral on the end of any number creates another larger number (most such additions did not create a meaningful number in the Roman system). Worse still, the zero symbol lays itself open to artistic elaboration into a 6 or a 9. These problems played an important role in bolstering natural inertia and conservatism which held up the wholesale introduction of the Indo-Arab system amongst the majority of merchants in Northern Europe until well into the sixteenth century.^{43}
THE EVOLUTION OF WORDS FOR ZERO
“It was said that all Cambridge scholars call the cipher aught and all Oxford scholars call it nought.”
Edgworth^{44}
We have seen that our numerical zero derives originally from the Hindu sunya, meaning void or emptiness, deriving from the Sanskrit name for the mark denoting emptiness, or sunya-bindu, meaning an empty dot. These developed between the sixth and eighth centuries. By the ninth century, the assimilation of Indian mathematics by the Arab world led to the literal translation of sunya into Arabic as as-sifr, which also means ‘empty’ or the ‘absence of anything’. We still see a residue of this because it is the origin of the English word ‘cipher’. Originally, it meant ‘Nothing’, or if used insultingly of a person it would mean that they were a nonentity – a nobody – as in King Learwhere the Fool says to the King^{45}
“Now thou art an 0 without a figure. I am better than thou art now. I am a fool, thou art nothing.”
The path to this meaning is intriguing. The Arab word sifr was first transcribed into medieval Latin in the thirteenth century in the two forms cifra or zefirum, and into Greek as τσιφρα, which led to their use of the letter tau, τ, as an abbreviation for zero. But the two Latin words acquired quite different meanings. The word zefirum (or cefirum, as Leonardo of Pisa^{46} wrote it in the thirteenth century) kept its original meaning of zero. In fourteenth-century Italian, this second form changed to zefiro, zefro or zevero, which was eventually shortened in the Venetian dialect to zero which we still use in English and French. This same type of editing down was what reduced the currency from librato livra to lira.
By contrast, the word cifra acquired a more general meaning: it was used to denote any of the ten numerals 0, 1, 2, …, 9. From it comes the French chiffre and the English cipher. In French, the same ambiguities of meaning exist as in English. Originally, chiffre meant zero, but like cipher came to mean any of the numerals. The merger of the ideas for zero and Nothing gave rise to the name ‘null’ being used either to denote ‘Nothing’ or the circular symbol for zero. This meant a ‘figure of nothing’, or nulla figura in Latin. John of Hollywood (1256) writes in his Algorismus of the tenth digit that provides the zero symbol:
“The tenth is called theca or circulus or figura nihili, because it stands for ‘nothing’. Yet when placed in its proper position, it gives value to the others.”^{47}
A fifteenth-century French book of arithmetic for traders tells us:
“And of the ciphers [chiffres] there are but ten figures, of which nine are of value and the tenth is worth nothing [rien] but gives value to the others and is called zero [zero] or cipher [chiffre].”^{48}
It is interesting that both these commentators write of ten symbols, including the zero. We can conceive of how finger-counting culture might have devised a system in which the ten fingers were used to denote the quantities 0 to 9 rather than 1 to 10. Yet the conceptual leap needed to associate that first finger with nothing would have been vast. Needless to say, no finger counters did that, but we don’t know what use they made of a hand displaying no fingers to convey the intuitively simple piece of information that they had nothing left.^{49}
In German, the ambiguity between the word for numbers and for zero was broken, with numbers called Figuren while the words cifra or Ziffer were used for zero.^{50} The English word ‘figures’ was, as now, a synonym for numerals and ‘being good with figures’ became a familiar accolade for anyone possessing some ability as a computer.^{51}
The terms theca and circulus (‘little circle’) are sometimes encountered as synonyms for zero. Both refer to the circular form of the sign for zero. Theca was the circular brand burned into the forehead or the cheeks of criminals in the Middle Ages.
A FINAL ACCOUNTING
“A place is nothing: not even space, unless at its heart – a figure stands.”
Paul Dirac^{52}
So far, we have seen some of the history of how we inherited the mathematical zero sign that is now so familiar. It is part of the universal language of numbers. Obvious though it may now seem to us, very few ancient cultures appreciated its need and most of those that did needed a little prompting from its inventors. The system of attributing a different value to a numeral according to where it is located in a list was one of the greatest discoveries that humanity has ever made. Once made, it requires the invention of a symbol that signals that no value be attributed to an empty location in the list. The Babylonians and the Indian cultures first made these profound discoveries and their invention spread to Europe and beyond around the globe by the channel of the Arab culture and its sophisticated interest in mathematics, philosophy and science. Strangely, the ancient Greeks, despite their extraordinary intellectual achievements, failed to make these basic discoveries. Indeed, we have seen that their approach to the world and the use of logic to unravel its workings was a serious impediment to the genesis of the zero concept. They demanded a logical consistency of their concepts and could not countenance the idea of ‘Nothing’ as a something. They lacked the mystical thread that could interweave the zero concept into a practical accounting system. The fact that the Indian system worked, and transparently so, was sufficient to justify its spread. Their affinity to the philosophical concept of Nothing as a desirable thing in itself, not merely the absence of everything else, allowed the zero symbol to accrete a host of other meanings that persist today in our words for Nothing. Nothing started as a small step but it brought about a giant leap forward in the effectiveness of human thinking, recording and calculation. Its usefulness and effectiveness in commerce, navigation, engineering and science ensured that once grasped it was a symbol that would not be dropped. For, as Napoleon Bonaparte pointed out, ‘The advancement and perfection of mathematics are ultimately connected with the prosperity of the state.’
Zero was like a genie. Once released it could not be restrained, let alone removed. Once words existed for the concept that the zero symbol represented, it was free to take on a life of its own, unconstrained by the strictures of mathematics, and even those of logic. The mathematicians had played a vital role in making legitimate the concept of Nothing in a place where it was easiest to define and control. In the centuries to follow, it would emerge elsewhere in different guises, with even deeper consequences, and more puzzling forms.