The Book of Nothing

by John D. Barrow

  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 Leibniz33

  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 Number38 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, naming39 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 few
er 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.”

  Edgworth44

  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 Lear where the Fool says to the King45

  “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 Pisa46 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 libra to 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 Dirac52

  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.’