The Book of Nothing

by John D. Barrow

  However, as Frank Tipler and I also showed,47 if the vacuum energy exists then everything changes – for the worse. All evolution heads inevitably for a state of uniformity characterised by the accelerating universe of de Sitter. Information processing cannot continue for ever: it must die out. There will be less and less utilisable energy available as the material Universe is driven closer and closer to a state of uniformity. If the vacuum energy exists but there is insufficient matter in the Universe to reverse its expansion into contraction before the vacuum energy gets a grip on the expansion48 and begins to accelerate it, then the Universe seems destined for a lifeless far future. Eventually, the acceleration leads to the appearance of communication barriers. We will be unable to receive signals from sufficiently remote parts of the Universe. It will be as if we are living inside a black hole. The part of the Universe that can affect us (or our descendants) and with which they may be in contact will be finite. In order to escape this claustrophobic future we would need the ubiquitous vacuum energy to decay. We think it must stay constant for ever, but maybe it is slowly, imperceptibly eroding. Maybe one day it will decay suddenly into radiation and ordinary forms of matter and the Universe will be left to pick up the pieces, and slowly use gravity to aggregate matter and process information. But the decay may not be so benign. We have seen that it could herald a slump into an even lower energy state for the Universe with a sudden change in the nature of physics accompanying it. It is even possible for the vacuum to decay into a new type of matter that is even more gravitationally repulsive than the lambda force. If its pressure is even more negative then something very dramatic can lie in the future. The expansion can run into a singularity of infinite density at a finite time in the future.49

  There is one last line of speculation that must not be forgotten. In science we are used to neglecting things that have a very low probability of occurring even though they are possible in principle. For example, it is permitted by the laws of physics that my desk rise up and float in the air. All that is required is that all the molecules ‘happen’ to move upwards at the same moment in the course of their random movements. This is so unlikely to occur, even over the fifteen-billion-year history of the Universe, that we can forget about it for all practical purposes. However, when we have an infinite future to worry about all this, fantastically improbable physical occurrences will eventually have a significant chance of occurring. An energy field sitting at the bottom of its vacuum landscape will eventually take the fantastically unlikely step of jumping right back up to the top of the hill. An inflationary universe could begin all over again for us. Yet more improbably, our entire Universe will have some minutely small probability of undergoing a quantum-transition into another type of universe. Any inhabitants of universes undergoing such radical reform will not survive. Indeed, the probability of something dramatic of a quantum-transforming nature occurring to a system gets smaller as the system gets bigger. It is much more likely that objects within the Universe, like rocks, black holes or people, will undergo such a remake before it happens to the Universe as a whole. This possibility is important, not so much because we can say what might happen when there is an infinite time in which it can happen, but because we can’t. When there is an infinite time to wait then anything that can happen, eventually will happen. Worse (or better) than that, it will happen infinitely often.

  Globally, the Universe may be self-reproducing but that will merely provide other expanding regions with new beginnings. Perhaps some of their inhabitants will master the techniques needed to initiate these local inflations to order and engineer their outcomes in life-enhancing ways. For us, there is a strange symmetry to existence. The Universe may once have appeared out of the quantum vacuum, retaining a little memory of its energy. Then in the far future that vacuum energy will reassert its presence and accelerate the expansion again, this time perhaps for ever. Globally, the self-reproduction may inspire new beginnings, new physics, new dimensions, but, along our world line, in our part of the Universe, there will ultimately be sameness, starless and lifeless, for ever, it seems. Perhaps it’s good that we won’t be there after all.

  Notes

  “I must say that I find television very educational. Whenever somebody turns it on, I go to the library and read a book.”

  Groucho Marx

  “There are scholars who footnote compulsively, six to a page, writing what amounts to two books at once. There are scholars whose frigid texts need some of the warmth and jollity they reserve for their footnotes and other scholars who write stale, dull footnotes like the stories brought inevitably to the minds of after-dinner speakers. There are scholars who write weasel footnotes, footnotes that alter the assertions in their texts. There are scholars who write feckless, irrelevant footnotes that leave their readers dumb-struck with confusion.”

  M.-C. van Leunen

  A Handbook for Scholars

  chapter nought

  Nothingology – Flying to Nowhere

  1. First words spoken in a ‘talking film’, The Jazz Singer, 1927.

  2. P.L. Heath, in The Encyclopedia of Philosophy entry on Nothing, vols 5 & 6, Macmillan, NY (1967), ed. P. Edwards, p. 524.

  3. Indeed, who else?

  4. This word for nothing, still common in dialects in the north of England, has a Scandinavian origin in Old Norse. It had many other meanings – a bullock or an ox (the nowt-geld was a rent or tax in the north of England, payable in cattle), or a stupid or oafish person.

  5. Having no proper form.

  6. Worlds devoid of any unifying pattern, plan or purpose. This usage, as a contrast to the structure of a Universe, can be found in William James’ Mind, p. 192, where he writes that ‘The World…is pure incoherence, a chaos, a nulliverse, to whose haphazard sway … I will not truckle.’

  7. Who reject all religious beliefs and moral precepts; sometimes expressed as an extreme form of scepticism in which all existence is denied.

  8. Who maintained the heretical doctrine that in Christ’s nature there was no human element, only divine nature.

  9. Those who deal with things of no importance.

  10. Those who do nothing.

  11. Those who hold no religious or political beliefs.

  12. Those who have no faith in any religious belief.

  13. Those who believe that no spiritual beings exist.

  14. ‘Zeros’ are zero dividend preference shares. These are a relatively low-risk investment issued by split capital investment trusts: low risk because they are usually first in line to be paid out when the trust is wound up. They do not pay income (hence the ‘zero’) but are brought to mature in different years with the gains reinvested in further zeros.

  15. See, for example, the OED.

  16. Pogo, 20 March 1965, cited by Robert M. Adams, Nil: Episodes in the literary conquest of void during the nineteenth century, Oxford University Press, NY (1966).

  17. J.K. Galbraith, Money: Whence it Came, Where it Went, A. Deutsch, London (1975), p. 157.

  18. prope nihil. There has been a persistent notion that Nothing is a very small positive quantity – much less than one expected, but still a minimal thing. French rien (= nothing) is said to be derived from the Latin rem, the accusative singular form of the word for ‘thing’, and indeed Old French frequently uses rien in a positive sense: ‘Justice amez so tôte rien’, ‘Love justice above every other thing’.

  19. R.M. Adams, Nil: Episodes in the literary conquest of void during the nineteenth century, Oxford University Press, NY (1966), p. 3 and p. 34.

  20. R.F. Colin, ‘Fakes and Frauds in the Art World’, Art in America (April 1963).

  21. H. Kramer, The Nation (22 June 1963).

  22. B. Rose (ed.), Art as Art: The Selected Writings of Ad Reinhardt, Viking, NY (1975).

  23. J. Johns, The Number Zero (1959), private collection.

  24. −273.15 degrees centigrade.

  25. M. Gardner, Mathematical Magic Show, Penguin, London (1985), p. 24.

&
nbsp; 26. Quoted in N. Annan, The Dons, HarperCollins, London (1999), p. 264.

  27. E. Maor, To Infinity and Beyond: a cultural history of the infinite, Princeton University Press (1987).

  chapter one

  Zero – The Whole Story

  1. A. Renyi, Dialogues in Mathematics, Holden Day, San Francisco (1976). This quote is taken from part of an imaginary Socratic dialogue.

  2. J. Boswell, The Life of Johnson, vol. III.

  3. R.K. Logan, The Alphabet Effect, St Martin’s Press, NY (1986), p. 152, partly paraphrasing Constance Reid.

  4. A handy rule of thumb is Moore’s Law, named after Gordon Moore, the founder of Intel, who proposed in 1965 that each new chip contained roughly twice the capacity of its predecessor and was released within 18–24 months of it.

  5. The fact that you are reading this book shows that my computers survived. But I remain unconvinced that all was down to the prescience of computer scientists because it was disconcerting to discover that the computers that I was told needed adjustment, did not, while those that I was assured required none, did.

  6. J.D. Barrow, Pi in the Sky, Oxford University Press (1992).

  7. The word ‘score’ has an interesting number of meanings. It refers to counting, as in keeping the score; it also refers to making a mark; and it means twenty. The score was originally the mark for this quantity on the tally stick used by the treasury.

  8. The term hieroglyph used to describe the language and numbersigns of the ancient Egyptians was introduced by the Greeks. Because they could not read the symbols they found on Egyptian tombs and monuments, they believed them to be sacred signs and called them grammata hierogluphika, or ‘carved sacred signs’, hence our ‘hieroglyphs’.

  9. G. Ifrah, The Universal History of Numbers, Harvill Press, London (1998). This is an extension and retranslation from the French original of the author’s earlier volume From One to Infinity: a universal history of numbers, Penguin, NY (1987).

  10. The Book of Daniel 5, v. 5, 25–8. Daniel’s interpretation of the writing on Belshazzar’s wall.

  11. There have been many attempts to explain the existence of this sexagesimal base. It is clear that 60 is very useful for any commercial accounting involving weights, measures and fractions because it has so many factors. It may be that the arithmetic base derived from some pre-existing system of weights and measures adopted for these reasons. Another interesting possibility is that it emerged from the synthesis of two systems used by two earlier civilisations. A merger with a natural base-10 system seems unlikely since it needs a base-6 system to exist. None is known ever to have existed on Earth and there is no good reason why it should. More promising seems to be the idea that there was a merger of a base-12 and a base-5 system. Base-5 systems are natural consequences of finger-counting whilst base-12 systems are attractive for trading purposes because 2, 3, 4 and 6 are divisors of 12. We can still see the relics of its appeal in the imperial units (12 inches in a foot, 12 old pence in a shilling, buying eggs by the dozen) and the Sumerians used it extensively in their measurements of time, length, area and volume. The words for ‘one’, ‘two’ and ‘three’ just correspond to the concepts one, one plus one and many: a common traditional form. But the words for ‘six’, ‘seven’, ‘nine’, and so on have the form of words ‘five and one,’ ‘five and two’ and ‘five and four’. This is evidence of a base-5 system in the past. Of course, these attractive scenarios may be entirely post hoc. The choice of 60 as the base could have been made because some autocrat had a dream, an astronomical coincidence, or a lost mystical belief in the sacredness of the number itself. We know, for instance, that some of the Babylonian gods were represented by numerals. Anu, the god of the heavens, is given the principal number, 60, because it was seen as the number of perfection. The lesser gods have other, smaller, numbers attributed to them, each has some significance. It is hard to tell whether the theological significance preceded the numerical.

  12. By 2700 BC they had rotated the symbols by ninety degrees. This appears to have been because scribes had evolved from working with small hand-held tablets to large heavy slabs that could not be orientated easily in the hand, see C. Higounet, L’Écriture, Presses Universitaires de France, Paris (1969).

  13. A further economy measure appeared about 2500 BC when very large numbers were written using a shorthand multiplication principle. For example, a number like 4 × 3600 would be written by putting four ‘ten’ markers inside the wing symbol and placing it to the right of 3600.

  14. From the Latin word, cuneus, for ‘a wedge’.

  15. R.K. Guy, ‘The Strong Law of Small Numbers’, American Mathematical Monthly 95, pp. 687–712 (1988).

  16. In fact, the wedge symbols were also used to extend the Babylonian system down to fractions in a mirror image of large numbers so a vertical wedge not only stood for 1, 60, 3600, etc., but also for 1/60, 1/3600, etc. In practice, the whole numbers were distinguished from the fractions by writing the whole numbers from right to left in ascending size and the fractions from left to right in descending size.

  17. Non-astronomers appeared not to have done this and this misled some historians to conclude that the Babylonian zero was never used at the end of a symbol string (unlike our own zero). It was also used in the first position when employed for angular measure, so the [0;1] would denote zero degrees plus 1/60th of a degree, i.e. one minute of arc. For the most detailed analysis of the different accounting systems practised in different spheres of life in Babylonia see the detailed study by H.J. Nissen, P. Damerow, and R. Englund, Archaic Bookkeeping: writing and techniques of economic administration in the ancient near east, University of Chicago Press (1993).

  18. Some Babylonian texts contain subtle numerical puns, cryptograms and pieces of numerology.

  19. John Cage, ‘Lecture on Nothing’, Silence (1961).

  20. There were some differences in their system for counting time.

  21. The Mayan word for ‘day’ was kin.

  22. F. Peterson, Ancient Mexico, Capricorn, NY (1962).

  23. G. Ifrah, From One to Zero, Viking, NY (1985).

  24. B. Datta and A.N. Singh, History of Hindu Mathematics, Asia Publishing, Bombay (1983).

  25. See Datta and Singh, op. cit.

  26. Subandhu, quoted in G. Flegg, Numbers Through the Ages, Macmillan, London (1989), p. III.

  27. The Satsai collection; see Datta and Singh, op. cit., p. 220.

  28. Black for an unmarried woman but indelible red for a married woman. These marks symbolise the third eye of Shiva, that of knowledge.

  29. S.C. Kak, ‘The Sign for Zero’, Mankind Quarterly, 30, pp. 199–204 (1990).

  30. Ifrah, op. cit., p. 438. These synonyms are not confined to the number zero. The Sanskrit language is rich in synonyms and all the Indian numerals possess a collection of number words taking on different images. For example, the number 2 is described by words with meanings that span twins, couples, eyes, arms, ankles and wings.

  31. The use of zero as a number in India is displayed in both number words and number symbols. The number words were based on a decimal system and read like our own reference to 121 as ‘one-two-one’. If a zero is used in this scheme it is referred to as sunya, kha, akâsha, or by one of the other synonyms. See A.K. Bag, Mathematics in Ancient and Medieval India, Chaukhambha Orientalia, Delhi (1979) and ‘Symbol for Zero in Mathematical Notation in India’, Boletin de la Academia Nacional de Ciencias, 48, pp. 254–74 (1970).

  32. J.D. Barrow, Pi in the Sky, Oxford University Press (1992), pp. 73–78; N.J. Bolton and D.N. Macleod, ‘The Geometry of Sriyantra’, Religion 7, pp. 66 (1977); A.P. Kulaicher, ‘Sriyantra and its Mathematical Properties’, Indian Journal of History of Science, 19, p. 279 (1984).

  33. G. Leibniz, quoted in D. Guedj, Numbers: The Universal Language, Thames and Hudson, London (1998), p. 59. Leibniz is credited with inventing the representation of numbers in binary form, using 0s and 1s. He describes this discovery and presents a table of the repres
entation of powers of 2 from 2 to 214 in a letter written at the end of the seventeenth century, see L. Couturat, ed., Opuscules et fragments inédits de Leibniz, extraits des manuscripts de la Bibliothèque Royale de Hanovre par Louis Couturat, Alcan, Paris (1903), p. 284. There appears to have been an early Indian discovery of the binary representation, perhaps as early as the second or third century AD. It was used to classify metrical verses in Vedic poetry by Pingala, see B. van Nooten, ‘Binary Numbers in Indian Antiquity’, J. Indian Studies, 21, pp. 31–50 (1993).

  34. G. Ifrah, op. cit., pp. 508–9.

  35. Plato, The Sophist, Loeb Classical Library, ed. H. North Fowler, pp. 336–9.

  36. T. Dantzig, Number: The Language of Science, Macmillan, NY (1930), p. 26.

  37. It was my choice in an electronic poll of inventions of the millennium and also for La Repubblica’s choice of greatest inventions.

  38. The Hebrew title was Sefer ha Mispar; see M. Steinschneider, Die Mathematik bei den Juden, p. 68, Biblioteca Mathematika (1893); M. Silberberg, Das Buch der Zahl, ein hebräisch-arithmetisches Werk des Rabbi Abraham Ibn Ezra, Frankfurt am Main (1895), and D.E. Smith and J. Ginsburg, ‘Rabbi Ben Ezra and the Hindu-Arabic Problem’, American Mathematical Monthly 25, pp. 99–108 (1918).