The Book of Nothing: Vacuums, Voids, and the Latest Ideas about the Origins of the Universe - John D. Barrow (2002)

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.

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 representation 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).

39. He also used the Arab word sifra, meaning ‘empty’.

40. The oldest known European manuscript containing the Arab numerals, the Codex Vigilanus, comes from Logroño in northern Spain and dates from AD 976. It contains a listing of the numerals 1 to 9 but not the zero. It is now in the Escorial.

41. B.L. van der Waerden, Science Awakening, Oxford University Press (1961), p. 58.

42. It is interesting that in the eighth century this use of Greek numerals was allowed even when use of the Greek language was banned.

43. The Indian zero spread East as well as West. In the eighth century, the Chinese left a gap in their representations of numbers, just like the Babylonians. Thus the number 303 is found in word form and also written in the simple ‘rod’ numerals as ||| |||. A circular symbol for zero did not appear until 1247 and then we find a representation of 147,000 as | ≡ Π000, see D. Smith, History of Mathematics, Dover, New York (1958), vol. 2, p. 42.

44. OED.

45. King Lear, Act 1, scene iv.

46. Better known to mathematicians as Fibonacci.

47. John of Hollywood (1256), Algorismus, cited in Numbers Through the Ages, ed. G. Flegg, Macmillan, London (1989), p. 127.

48. Cited in Numbers Through the Ages, op. cit., p. 127.

49. An interesting example of counting without counting is that of a farmer who wants to make sure he has not lost any of his sheep. If he puts down one stone in a pile when each sheep enters the field in the morning and then removes one from the pile when each sheep leaves the field at dusk then he only needs to check that there are no stones remaining when the last sheep has left. This example of how the farmer can use mathematics without understanding it in some respects is reminiscent of John Searle’s ‘Chinese Room’ argument against the possibility of artificial intelligence having semantic ability; see J.R. Searle, The Mystery of Consciousness, Granta, London (1997).

50. K. Menninger, Number Words and Number Symbols, MIT Press, MA (1969).

51. Until the development of the first calculating machines in England during the 1940s, the word ‘computer’ was solely a description of a person who performed calculations. It was then adopted to describe machines that can compute. Ironically, today it is used only to describe non-human calculation. During the interim period there have been many other mechanical ‘adding machines’ and non-programmable devices which became known as ‘calculators’. The word computer as a description of a human calculator derives from computare, the medieval Latin for ‘cut’, which echoes the cutting of notches on a tallying stick in the way that the English word ‘score’ means to keep count and make a mark (and the quantity 20). The medieval Latin for ‘calculate’ was calculare, and the Latin calculus ponere meant to move or place pebbles, an echo of the movement of stones as counters on a counting board. We recognise the source here for the naming of the differential and integral calculus, invented by Newton and Leibniz (and also for Hergé’s Professor Calculus).

52. P.A.M. Dirac, The Principles of Quantum Mechanics, Oxford University Press (1958).

chapter two

Much Ado About Nothing

1. Leonardo da Vinci, The Notebook, translated and edited by E. Macurdy, London (1954), p. 61.

2. U. Eco, The Name of the Rose, Secker & Warburg, London (1983).

3. Lyric from ‘Me and Bobby McGee’(1969).

4. This phrase has made several appearances in recent years in popular expositions of modern physics (see for example the books by Paul Davies and James Trefil, whose titles use it). It is used as a synonym for the laws of Nature or the underlying structural features of the Universe which may be partly (or wholly) independent of the laws, for example the existence and dimensionality of time and space.

5. The only slightly counter-intuitive property is the requirement that we define zero factorial, 0! = 1 = 1! since the factorial operation is defined recursively by (n + 1)! = (n + 1) × n!

6. See J.D. Barrow, Pi in the Sky, Oxford University Press (1992), pp. 205–216, for a more detailed account of these manipulations.

7. David Hilbert was one of the world’s foremost mathematicians in the first part of the twentieth century.

8. M. Friedman, ed., Martin Buber’s Life and Work: The Early Years 1878-1923, E.P. Dutton, NY (1981).

9. J.-P. Sartre, Being and Nothingness (transl. H. Barnes), Routledge, London (1998), p. 16.

10. Sartre, op. cit., p. 15.

11. B. Rotman, Signifying Nothing: The Semiotics of Zero, Stanford University Press (1993), p. 63.

12. The Odyssey, Book IX, lines 360–413, Great Books of the Western World, vol. 4, Encyclopaedia Britannica Inc., University of Chicago (1980).

13. In Greek ουύτις, meaning nobody.

14. G.S. Kirk and J.E. Raven, The Presocratic Philosophers: a critical history with a selection of texts, Cambridge University Press (1957).

15. Fragment quoted by S. Sambursky, The Physical World of the Greeks, Routledge, London (1987), pp. 19–20.

16. Sambursky, op. cit., p. 22.

17. Sambursky, op. cit., p. 108.

18. B. Inwood, ‘The origin of Epicurus’ concept of void’, Classical Philology 76, pp. 273–85 (1981); D. Sedley, ‘Two conceptions of vacuum’, Phronesis, 27, pp. 175–93 (1982).

19. Except for a few fragments, Leucippus’ and Democritus’ writings do not survive and his ideas have been partially reconstructed from the commentary of others, particularly Lucretius, Aristotle and the latter’s successor as head of the Academy, Theophrastus. When considering his views on whether atoms could be observed or not it is interesting to refer to some of the fragments of Democritus’ writings that do survive. He appears to have adopted a rather ‘modern’ (at least nineteenth-century) Kantian view that there is a distinction between what we can know about things and their real nature: ‘It will be obvious that it is impossible to understand how in reality each thing is,’ he writes, for ‘we know nothing accurately in reality, but as it changes according to the bodily conditions, and the constitution of things that flow upon the body and impinge upon it …for truth lies in an abyss.’ S. Sambursky, op. cit., p. 131.

20. Democritus endowed his atoms only with the properties of size and shape; Epicurus also allowed them to have weight in order to determine aspects of their motion under gravity.

21. Lucretius II, 308–322.

22. Aristotle quoted in J. Robinson, An Introduction to Greek Philosophy (1968), Boston, p. 75.

23. S. Sambursky, Physics of the Stoics, Routledge, London (1987); R.B. Todd, ‘Cleomedes and the Stoic conception of the void’, Apeiron, 16, pp. 129–36 (1982).

24. F. Solmsen, Aristotle’s System of the Physical World, Cornell University Press, Ithaca (1960); R. Sorabji, Matter, Space & Motion, Duckworth, London (1988); E. Grant, Much Ado About Nothing: Theories of Space and Vacuum from the Middle Ages to the Scientific Revolution, Cambridge University Press (1981).

25. Quoted by C. Pickover, The Loom of God, Plenum, NY (1997), p. 122.

26. In the nineteenth and twentieth centuries, mathematicians have come to appreciate the systematic way of constructing so-called ‘space-filling’ curves which will ultimately pass through every point in a specified region.

27. See for example The Complete Works of John Davies of Hereford, Edinburgh (1878) and for a review, V. Harris, All Coherence Gone, University of Chicago Press (1949).

28. The efficient cause in Aristotle’s sense.

29. For a detailed discussion of these arguments see W.L. Craig, The Cosmological Argument From Plato to Leibniz, Macmillan, London (1980).

30. R. Adams, Nil: Episodes in the literary conquest of void during the nineteenth century, Oxford University Press, NY (1966), p. 33.

31. For example, on the ground that if empty space were a body then when another body were placed in empty space there would be two bodies at the same place at the same time, and if two bodies could be coincident like this then why not all bodies, which he regarded as absurd.

32. It is intriguing to note that Aristotle formulates what we now call Newton’s first law of motion, that bodies acted on by no forces move at constant velocity, but rejects it as a reductio ad absurdum.

33. It was rediscovered at this time. Lucretius’ De Rerum Natura, where it appears in Bk I, p. 385, was unknown in Europe until the fifteenth century.

34. De Rerum Natura, Bk I, pp. 385–97.

35. For a classic study of these and a host of other medieval investigations into the nature of space, infinity and the void, see the beautiful book by Edward Grant, Much Ado About Nothing: Theories of Space and Vacuum from the Middle Ages to the Scientific Revolution, Cambridge University Press (1981), p. 83.

36. Grant, op. cit., p. 89.

37. This problem is discussed by Galileo in his Discourse Concerning Two New Sciences on the first day.

38. Here one is reminded of the modern idea, introduced by Roger Penrose, of cosmic censorship, the idea that Nature abhors the creation of singularities in space time which are visible from far away and which can causally influence events there. The ‘cosmic censor’ (not a person, just an internal property of Einstein’s equations, suspected to be necessary for the physical self-consistency of Einstein’s theory of general relativity) is hypothesised to cloak all singularities that could form with an event horizon. This horizon prevents information from the singularity, where the laws of physics break down, from passing out to affect events far from the singularity. The simplest example of this device is that of the black hole where, unless quantum gravitational effects always intervene to prevent an actual physical singularity of infinite density forming at the centre of the black hole, an event horizon always stops outside observers seeing it or being causally influenced by it.

39. Aristotle did not believe in the existence of this imaginary extra-cosmic void, of course, and commented that some people wrongly deduced its existence merely because they were unable to imagine an end to some things.

40. This quote is usually attributed to Nicholas of Cusa but was widely cited as early as the twelfth century. Grant, op. cit., pp. 346-7, gives Alan of Lille as the first known source; for a detailed study of the question see also D. Mahnker, Unendliche Sphäre und Allmittelpunkt, Hale/Salle: M. Niemeyer Verlag, 1037 (1937), pp. 171–6.

41. For a detailed account of these Design Arguments, see J.D. Barrow & F.J. Tipler, The Anthropic Cosmological Principle, Oxford University Press (1986).

42. I. Newton, Opticks, Book III Pt.1, Great Books of the Western World, vol. 34, W. Benton, Chicago (1980), pp. 542–3.

43. See E. Grant, op. cit., p. 245; A. Koyré, From the Closed World to the Infinite Universe, p. 297 note 2, Johns Hopkins Press, Baltimore (1957); and W.G. Hiscock, ed., David Gregory, Isaac Newton and their Circle: Extracts from David Gregory’s Memoranda, 1667–1708, Oxford (1937), printed for the editor.

44. Quoted by Robert Lindsay on Parkinson, BBC Television, 15 January 1999.

45. R.L. Colie, Paradoxia Epidemica: the Renaissance tradition of paradox, Princeton University Press, NJ (1966), pp. 223–4. See also A.E. Malloch, ‘The Techniques and Function of the Renaissance Paradox’, SP, 53, pp. 191–203 (1956).

46. Attributed to Edward Dyer by most commentators and to Edward Daunce by R.B. Sargent in The Authorship of The Prayse of Nothing, The Library, 4th series, 12, pp. 322–31 (1932); the passage quoted here is the second stanza. See also H.K. Miller, ‘The Paradoxical Encomium with Special Reference to its Vogue in England, 1660–1800’, MP, 53, p. 145 (1956).

47. Facetiae, chap. 2, pp. 389–92, London (1817), cited by Colie, op. cit., p. 226.

48. J. Passerat, Nihil, quoted by R. Colie, op. cit., p. 224.

49. Act I, scene 2, line 292.

50. The main sources for the story are a translation of a novella by the Italian short-story writer Matteo Bandello (1485–1561) and Ludovico Ariosto’s Orlando Furioso, an Italian epic poem (1532).

51. P.A. Jorgenson, ‘Much Ado about Nothing’, Shakespeare Quarterly,5, pp. 287–95 (1954).

52. Much Ado About Nothing, Act 4, scene 1, line 269.

53. Macbeth, I, iii, 141–2.

54. Macbeth, V, v, 16.

55. Colie, op. cit., p. 240.

56. Hamlet, III, ii, 119–28.

57. King Lear, I, i, 90.

58. R.F. Fleissner, ‘The “Nothing” Element in King Lear’, Shakespeare Quarterly, 13, pp. 62–71 (1962); H.S. Babb, ‘King Lear: the quality of nothing’, in Essays in Stylistic Analysis, Harcourt, Brace, Jovanovich, NY (1972).

59. It has also been suggested by David Willbern that in the theatre of Shakespeare’s day, Nothing would have sounded like ‘noting’ (thus, ‘Much Ado about Noting’) and this would have added a further contrasting meaning, the sense of ‘noting’ being our usual one together with observing, eavesdropping and overhearing, see R.G. White, The Works of William Shakespeare, Boston (1857), III, p. 226, but this seems to sell short the ingenuity of Shakespeare’s multiple meanings and creates a less enticing title. This idea does not seem to have been taken up by other commentators; see D. Willbern, ‘Shakespeare’s Nothing’, in Representing Shakespeare, eds M.M. Schwarz & C. Kahn, Johns Hopkins University Press, Baltimore (1980), pp. 244–63, and B. Munari, The Discovery of the Circle, transl. M. & E. Maestro, G. Witterborn, NY (1966) and H. Kökeritz, Shakespeare’s Pronunciation, Yale University Press, New Haven (1953). Willbern also pursues the psychoanalysis of Shakespeare to what many will consider to be an unconvincing extent. There are several other studies of this general sort which have looked at aspects of Shakespeare’s sense of Nothing, see D. Fraser, ‘Cordelia’s Nothing’, Cambridge Quarterly, 9, pp. 1–10 (1978), L. Shengold, ‘The Meaning of Nothing’, Psychoanalytic Quarterly, 43, pp. 115–19 (1974).

60. Sartre, op. cit., p. 23. Note that ‘nihilate’ (néantir) is defined by Sartre as ‘nihilation is that by which consciousness exists. To nihilate is to encase with a shell of non-being.’

61. The curtain was brought down by John Dunton’s huge compendium Athenian Sport: or, Two Thousand Paradoxes merrily argued to Amuse and Divert the Age (1701).

62. G. Galileo, Dialogue Concerning Two World Systems, transl. S. Drake, California University Press, Berkeley (1953).

63. Galileo, op. cit., pp. 103–4.

64. The medieval historian Edward Grant remarks that ‘…approximately two thousand Latin manuscripts of the work of Aristotle have been identified. If this number of manuscripts survived the rigors of the centuries, it is plausible to suppose that thousands more have perished. The extant manuscripts are a good measure of the pervasive hold that the works of Aristotle had on the intellectual life of the Middle Ages and Renaissance. With the possible exception of Galen …, no other Greek or Islamic scientist has left a comparable manuscript legacy’. E. Grant, The Foundations of Modern Science in the Middle Ages, Cambridge University Press, NY (1996), pp. 26–7.

chapter three

Constructing Nothing

1. Radio 3 broadcast Close Encounters with Kurt Gödel, reviewed by B. Martin, The Mathematical Gazette (1986), p. 53.

2. One of the curious facts about this view of the physical make-up of the world is that it arose amongst the Stoics as purely a religious belief at a time when there neither was, nor could be, any experimental evidence in its favour. However, it has turned out to be correct in its general conception of the hierarchical structure of matter.

3. G. Galileo, Dialogues Concerning Two New Sciences (1638), Britannica Great Books, University of Chicago (1980), p. 137. See also C. Webster, Arch. Hist. Exact. Sci., 2, p. 441 (1965).

4. This is what Aristotle would have called an ‘efficient cause’.

5. A British gold coin with a face value of one pound, minted first in 1663 for trade with Africa. After 1717 it became legal tender in Britain with a value fixed at twenty-one shillings or £1.05 in present UK currency. It is still used by auction houses and to fix the prize money of some horse races.

6. Letters announcing the discoveries of E. Torricelli are translated in V. Cioffair, The Physical Treatises of Pascal etc., Columbia University Press, NY (1937), p. 163. The originals are in E. Torricelli, Opera, Faenza, Montanari (1919), vol. 3, pp. 186–201.

7. If the column of mercury has height h, density d, the acceleration due to gravity is g and the tube has cross-sectional area A, then the downward force of the weight of mercury in the column is given by hAdg. When the mercury column comes into equilibrium, this down-ward force is balanced by the upward force due to the pressure exerted by the column, P, and this equals PA. Notice that both forces are proportional to A and so the height of the mercury in equilibrium is the same regardless of the value of the area A.

8. W.E.K. Middleton, The History of the Barometer, Johns Hopkins Press, Baltimore (1964).

9. S. Sambursky, Physical Thought from the Presocractics to Quantum Physics, Hutchinson, London (1974), p. 337.

10. S.G. Brush, ed., Kinetic Theory, vol. 1, Pergamon, Oxford (1965), contains extracts from Boyle’s original papers; see in particular ‘The Spring of the Air’ from his book New Experiments Physico-Mechanical, touching the spring of the air, and its effects. Boyle’s work on air pressure is discussed in M. Boas Hall, Robert Boyle on Natural Philosophy, Indiana University Press, Bloomington (1965); R.E.W. Maddison, The Life of the Honourable Robert Boyle, F.R.S., Taylor & Francis, London (1969); J.B. Conant, ed., Harvard Case Histories in Experimental Science, Harvard University Press, Cambridge Mass. (1950). For an excellent historical overview, see S.G. Brush, The Kind of Motion We Call Heat: a history of the kinetic theory of gases in the 19th century, vol. 1, New Holland, Amsterdam (1976).

11. M. Boas Hall, Robert Boyle on Natural Philosophy, Indiana University Press, Bloomington (1965).

12. This was based upon the deduction that the product of the pressure and volume occupied by the gas remains a constant when both change without altering the temperature. This result has become known as Boyle’s Law although he did not discover it himself (or claim to have). He merely confirmed the earlier experiments of Richard Townley. For the story, see C. Webster, Nature, 197, p. 226 (1963), and Arch. Hist. Exact. Sci., 2, p. 441 (1965).

13. A German translation of O. von Guericke, Experimenta nova (ut vocantur) Magdeburgica de vacuo spatio primum a R.P. Gaspare Schotto, Amsterdam (1672), was made by F. Danneman, Otto von Guericke’s neue ‘Magdeburgische’ Versuche über den leeren Raum, Leipzig (1894).

14. O. von Guericke, The New (so-called) Magdeburg Experiments of Otto von Guericke, translated by M.G.F. Adams, Kluwer, Dordrecht (1994), original publication 1672 by K. Schott, Würzburg, front plate.

15. O. von Guericke, op. cit., p. 162.

16. The second volume of his Treatise expounded his opinions about the nature and extent of void space. He believed in a universe of stars surrounded by an infinite void.

17. O. von Guericke, Experimenta nova, p. 63. The translation is from E. Grant, Much Ado About Nothing, p. 216.

18. A. Krailsheimer, Pascal, Oxford University Press (1980), p. 18.

19. B. Pascal, Pensées, trans. A. Krailsheimer, Penguin, London (1966).

20. Pascal planned a book on the vacuum entitled Traité du vide, but it was never completed. The Preface exists but the remaining parts have been lost. Two posthumous papers appeared in 1663, one on the subject of barometric pressure, L’Équilibre des liqueurs, the other about the hydraulic press, entitled La Pesanteur de la masse d’air.

21. Blaise Pascal, by Philippe de Champagne, engraved by H. Meyer; reproduced by permission of Mary Evans, Picture Library.

22. Spiers, I.H.B. & A.G.H. (transl.), The Physical Treatise of Pascal, Columbia University Press, NY (1937), p. 101.

23. Adapted from H. Genz, Nothingness, Perseus Books, Reading, MA (1999), p. 113.

24. Independent, 15 April 2000.

25. Second letter of Noël to Pascal, in B. Pascal, Oeuvres, eds 1. Brunschvicq and P. Boutroux, Paris (1908), 2, pp. 108–9, transl. R. Colie, Paradoxia Epidemica, Princeton University Press (1966), p. 256.

26. Oeuvres, 2, pp. 110–11.

27. This is because the Universe is expanding, hence its size is linked to its age. In order for nuclei of elements heavier than hydrogen and helium to have sufficient time to form in stars, billions of years are needed and so the Universe must be billions of light years in size, see J.D. Barrow & F.J. Tipler, The Anthropic Cosmological Principle, Oxford University Press (1986).

28. G. Stein, The Geographical History of America (1936).

29. F. Hoyle, Observer, 9 September 1979.

30. Defined by the distance that light has been able to travel during the age of the Universe, since its expansion began, about 13 billion years.

31. It is possible for the dark matter to be supplied by much lighter neutrinos which we already know to exist. We have only upper limits on their possible masses. These experimental limits are very weak. However, although these light neutrinos could supply the quantity of dark matter required in a natural way, they cause the luminous matter to cluster into patterns that do not look like those displayed by populations of real galaxies. Large computer simulations show that, in contrast, the much heavier neutrino-like particles (WIMPS = weakly interacting massive particles) seem to produce a close match to the observed clustering of luminous galaxies.

chapter four

The Drift Towards the Ether

1. D. Gjertsen, The Newton Handbook, Routledge & Kegan Paul, London (1986), p. 160.

2. Newton’s 1st law does not hold for observers who are in a state of accelerated motion relative to ‘absolute space’; for example, if you look out of the window of a spinning rocket you will see objects rotating about you, and hence apparently accelerating, even though they are acted upon by no forces. Thus Newton’s laws will be seen to be true only for a special class of cosmic observers, called ‘inertial’ observers, who are moving so that they are not accelerating relative to ‘absolute space’. One of the ways in which Einstein’s general theory of relativity supersedes Newton’s is that it provides laws of gravity and of motion which are true for all observers regardless of their motion: there are no observers for whom the laws of Nature are always simpler than they appear for others. See J.D. Barrow, The Universe that Discovered Itself, Oxford University Press (2000), pp. 108–24, for a fuller discussion of this development.

3. Experimental accuracy did not permit the detection of the very small change in the fall of a body in air compared to that in a vacuum.

4. Opticks (1979 edn), p. 349.

5. Bentley, a distinguished classical scholar, sought Newton’s advice when preparing his Boyle Lectures on natural theology. He was anxious to propose a new form of the argument from design, in which he would claim that it was the special mathematical forms of the laws of motion and gravity that were evidence for the existence of an intelligent Designer – a view with which Newton did not disagree. For a detailed discussion of this and other arguments of this sort, see J.D. Barrow & F.J. Tipler, The Anthropic Cosmological Principle, Oxford University Press (1986).

6. I.B. Cohen, Isaac Newton’s Papers and Letters on Natural Philosophy, Harvard University Press (1958), p. 279, letter of 25.2.1693.

7. R. Descartes, The World, or a Treatise on Light (1636).

8. Much has been made of the role that ‘beauty’ or some other human opinion of ‘elegance’ or ‘economy’ plays in the physicist’s conception of Nature (see, for example, S. Chandrasekhar, ‘Beauty and the Quest for Beauty in Science’, Physics Today, July 1979, pp. 25–30); however, this is often over-romanticised by physicists long after the creative process took place. Freeman Dyson has an interesting opinion of the work of Dirac and Einstein in this respect, arguing that their most important work was not guided by aesthetic considerations, but by experiment. Moreover, when they did become overtaken by the quest of beauty in their equations their useful scientific contributions ceased. Another interesting remark on the aesthetic appeal of Einstein’s theory was made by the experimental physicist, and operationalist philosopher, Percy Bridgman in his book Reflections of a Physicist, Philosophical Library Inc., New York (1950). He regarded the search for ‘beautiful’ equations to be a dangerous metaphysical diversion: ‘The metaphysical element I feel to be active in the attitude of many cosmologists to mathematics. By metaphysical I mean the assumption of the “existence” of validities for which there can be no operational control … At any rate, I should call metaphysical the conviction that the universe is run on exact mathematical principles, and its corollary that it is possible for human beings by a fortunate tour de force to formulate these principles. I believe that this attitude is back of the sentiment of many cosmologists towards Einstein’s differential equations of generalised relativity theory – when, for example, I ask an eminent cosmologist in conversation why he does not give up the Einstein equations if they make him so much trouble, and he replies that such a thing is unthinkable, that these are the only things that we are really sure of.’

9. Opticks, Query 18.

10. Op. cit., Query 21.

11. As he had first proposed to Boyle many years earlier.

12. Opticks, Query 21.

13. Remark to George FitzGerald of Trinity College Dublin, 1896.

14. We call this Olbers’ Paradox although Edmund Halley (famous for discovering the periodicity of the comet that now bears his name) appears to have been the first astronomer to highlight its significance, calling it a ‘metaphysical paradox’, in 1714. For an illuminating discussion of the dark sky paradox, see E.R. Harrison, Darkness at Night, Harvard University Press (1987). The account in S. Jaki, The Paradox of Olbers’ Paradox, Herder and Herder, NY (1969), is not recommended and resolution to the paradox suggested therein is incorrect, see Harrison, op.cit., p. 173.

15. E.R. Harrison, Darkness at Night, Harvard University Press (1987), p. 69.

16. J.E. Gore, Planetary and Stellar Studies, Roper and Drowley, London (1888).

17. J.E. Gore, op.cit., p. 233, cited by E.R. Harrison, Darkness at Night, pp. 167–8.

18. S. Newcomb, Popular Astronomy, Harper, NY (1878).

19. Figure adapted from E.R. Harrison, Darkness at Night, p. 169.

20. The Independent newspaper, Saturday magazine supplement, 17 January 1998, p. 10.

21. This version of the Design Argument made its first considered appearance in Bentley’s Boyle Lectures. These lectures were very significantly informed by the letters from Newton to Bentley. Newton was extremely sympathetic to his work being used for such religious apologetics even though he did not publish on this subject himself; for a detailed discussion see J.D. Barrow & F.J. Tipler, The Anthropic Cosmological Principle, Oxford University Press (1986).

22. J. Cook, Clavis naturae; or, the mystery of philosophy unvail’d, London (1733), pp. 284–6.

23. W. Whewell, Astronomy and General Physics considered with reference to natural theology, London (1833). The 3rd Bridgewater Treatise.

24. Whewell argued in some detail for the appeal of the ether hypothesis by pointing to the simplicity of the hypothesis of an ether governed by mechanical laws when compared with the complexity of the optical phenomena that it was able to explain. Elsewhere, Whewell supposes that there must exist several different ethereal fluids in order to explain the different propagation properties of sound, electricity, magnetism and chemical phenomena because their effects appear to be so qualitatively different.

25. B. Stewart and P.G. Tait, The Unseen universe; or, physical speculations on a future state, London (1875).

26. F. Kafka, Parables.

27. Fresnel’s ether was stationary. Stokes imagined that the Earth dragged the ether along as it rotated on its axis each day and orbited the Sun annually. Maxwell proposed an ether that was a magneto-electric medium consisting of a fluid filled with spinning vortex tubes as a model of the electromagnetic field.

28. B. Jaffe, Michelson and the Speed of Light, Doubleday, NY (1960); H.B. Lemon and A.A. Michelson, The American Physics Teacher, 4, pp. 1–11, Feb. (1936); R.A. Millikan and A.A. Michelson, The Scientific Monthly, 48, pp. 16–27, Jan. (1939).

29. J.R. Smithson, ‘Michelson at Annapolis’, American Journal of Physics 18, 425–8 (1950).

30. J.C. Maxwell, Encyclopaedia Britannica (9th edn), article on ‘The Ether’.

31. The interference of light was first demonstrated by Thomas Young in 1803.

32. We are assuming that one of the light paths is aligned with the direction of motion of the ether. In general it would not be, but this is easily incorporated into the calculation and does not alter the significance of a null result.

33. A. Michelson, ‘The Relative Motion of the Earth and the Luminiferous Ether’, American Journal of Science, series 3, 22, pp. 120–9 (1881).

34. R.S. Shankland, ‘Michelson at Case’, American Journal of Physics 17, pp. 487–90 (1950).

35. A. Michelson and E. Morley, American Journal of Science, series 3, 34, pp. 333–45 (1887).

36. Lorentz proposed that the values of mass and time are also changed. The transformation for mass, length and time are now generally known as the Lorentz transformations and form part of Einstein’s special theory of relativity.

37. Lorentz seems to have regarded the ether as inadequate as a representation of the vacuum. It had too many attributes; see A.J. Knox, ‘Hendrik Antoon Lorentz, the Ether, and the General Theory of Relativity’, Archive for History of Exact Sciences, 38, pp. 67–78 (1988).

38. Radio conversation released by UK Chief of Naval Operations, quoted in The Bilge Pump, the newsletter of the Sunshine Coast Squadron in British Columbia, October 1994.

39. A. Einstein, ‘Zur Elektrodynamick bewegter Körper’, Annalen der Physik, 17, pp. 891–921. Librarians often remove this volume from open library shelves because of the risk of theft.

40. M for matrix or mystery.

41. This discussion should be compared with that of ‘paradigms’ introduced by the late Thomas Kuhn, and which is popular in some circles. Kuhn made popular the idea of scientific ‘revolutions’ in which new paradigms periodically sweep away old ones. Kuhn’s thinking was strongly influenced by his historical studies of the Copernican ‘revolution’ which overthrew the Ptolemaic system of astronomy that preceded it. However, this example was special and not typical of the evolution of theories of physics which we see from Newton onwards. The evolution of those theories did not involve the overthrow of the old theory or paradigm. Rather the old theory was revealed to be a limiting case of the new, more general, more widely applicable, theory.

42. Cited in Jaffe, op.cit, p. 168.

43. A. Einstein, ‘Über die Untersuchung des Îtherzustandes im magnetischen Felde’, Physikalische Blätter, 27, pp. 390–1 (1971).

44. Einstein Archive FK 53, Letter to M. Maric, July 1899.

45. Einstein denied the existence of a physical ether consistently between 1905 and 1916 in scientific articles and in the popular press.

chapter five

Whatever Happened to Zero?

1. A. Marvell, The Poetical Works of Andrew Marvell, Alexander Murray, London (1870), ‘Definition of Love’, stanza VII.

2. The philosopher Immanuel Kant argued that Euclidean geometry was the only geometry that is humanly thinkable. It was forced upon us like a straitjacket by the way minds work. This was soon shown to be totally incorrect by the creation of new geometries. In fact, Kant should not have needed new mathematical developments to tell him this. By looking at any Euclidean geometrical example (for example a triangle on a flat surface) in a curved mirror it should have been clear that the laws of reflection guarantee that there must exist geometrical ‘laws’ on the curved surface which are reflections of those that exist on the flat surface.

3. Euclid, ElementsGreat Books of the Western World, Encyclopaedia Britannica Inc., Chicago (1980), vol. 11.

4. Euclid’s original axiom stated that ‘If a line A crossing two lines B and C makes the sum of the interior angles on one side of A less than two right angles, then B and C meet on that side.’ A simpler statement found in many geometry textbooks has the form ‘through any point not on a given line L there passes exactly one line parallel to L’. Euclid’s other four postulates were that: 1. It is possible to draw a straight line from any point; 2. It is possible to produce a finite straight line continuously in a straight line; 3. It is possible to describe a circle with any centre and radius; 4. All right angles are equal to one another.

5. B. de Spinoza, Ethics (1670), in Great Books of the Western World, vol. 31, Encyclopaedia Britannica Inc, Chicago (1980).

6. This can be done either by stating that through any point not on a given line L there must pass more than one line parallel to L, or no lines parallel to L.

7. If it is possible to deduce that 0 = 1, the system is inconsistent. Note that if any false statement of this sort is derivable then one can use it to deduce that any statement holds in the language of the system.

8. J. Richards, ‘The reception of a mathematical theory: non-Euclidean geometry in England 1868–1883’, in Natural Order: Historical Studies of Scientific Culture, eds B. Barnes and S. Shapin, Sage Publications, Beverly Hills (1979); E.A. Purcell, The Crisis of Democratic Theory, University of Kentucky Press, Lexington (1973); J.D. Barrow, Pi in the Sky, Oxford University Press (1992).

9. R.L. Graham, D.E. Knuth & O. Patashnik, Concrete Mathematics, Addison Wesley, Reading (1989), p. 56.

10. In fact, Euclid’s intuitively selected axioms were found to contain some strange omissions. For example, only in 1882 did Moritz Pasch notice that some things that seemed ‘obviously’ true could not be proved from Euclid’s classical axioms. One example is the following: if A, B, C and D are points arranged on a line so that B lies between A and C, and C lies between B and D, then show that B has to lie between A and D. This has to be added to Euclidean geometry as an additional axiom, if it is needed. Other observed facts which Euclid did not formulate as axioms, but which cannot be established from his chosen axioms, are that an unending straight line passing through the centre of a circle must intersect the perimeter of the circle and that a straight line that intersects one side of a triangle, but which does not intersect any of the triangle’s vertices, must intersect one of its other sides.

11. One of the strange things about the discovery of non-Euclidean geometries by mathematicians is that it took so long and proved so controversial. Artists and sculptors had discovered the rules that govern lines and angles on curved surfaces centuries earlier. In my book Pi in the Sky, there is a picture of an early Indian meditation symbol, a Sri Yantra, in which a nested pattern of triangles is arranged so that many lines intersect at a single point. Such objects were commonly drawn on flat surfaces but this one, made of rock salt, is unusual in that it was made on a curved spherical surface and must have required considerable appreciation of non-Euclidean geometry in order to be made. Another interesting factor that is hard to reconcile with the slowness of mathematicians to catch on to non-Euclidean geometries is the existence of curved mirrors, of glass or of polished metal. If you look in a curved mirror at a rightangled Euclidean triangle drawn on a flat surface then you are seeing a direct mapping of that triangle and the rules that govern its properties (like Pythagoras’ theorem) on to a curved surface. The rules themselves have direct reflected counterparts in the distorted triangle that is seen in the mirror. This tells you that there must be a set of rules governing the properties of the triangle in the mirror. This interrelationship was eventually captured more formally by Beltrami, Poincaré and Klein, who showed that Euclidean and non-Euclidean geometries are equiconsistent, that is, the logical self-consistency of one demands the self-consistency of the other.

12. It is easy to see that this element must be unique. For if there were two elements, I and J, with the identity property then I must equal I combined with J which must equal J, so I is the same as J.

13. If we included zero then we could form fractions like 2/0 which are not finite fractions and closure would be violated.

14. In fact, the German mathematician Felix Klein initiated a programme in 1872 (the so-called ‘Erlangen programme’) which aimed to unify the study of all geometries by defining them as mathematical structures with certain transformation properties. For example, one might define Euclidean geometry as the study whose properties remain the same under rotations, reflections, similarities and translations in space.

15. Unexpectedly, the Austrian mathematician Kurt Gödel showed that if a mathematical structure is rich enough to contain arithmetic then it is not possible that its defining axioms are inconsistent. If they are assumed to be consistent then the structure is necessarily incomplete in the sense that there must exist statements framed in the language of the structure which can neither be proved to be true nor false using the rules of reasoning of the system. Euclidean and non-Euclidean geometries are not rich enough to contain the structure of arithmetic and so this incompleteness theorem does not apply to them; see J.D. Barrow, Impossibility: the limits of science and the science of limits, Oxford University Press (1998), chapter 8, for more details.

16. This freedom to specify axioms allows a statement to be ‘true’ in one axiomatic system but ‘false’ in another.

17. It should be noted that, although the simple mathematical structure of a group that we have introduced requires the existence of an identity element which looks like the zero or arithmetic in some cases, not all mathematical structures have a zero element.

18. F. Harary & R. Read, Proc. Graphs and Combinatorics Conference, George Washington University, Springer, NY (1973).

19. M. Gardner, Mathematical Magic Show, Penguin, London (1977).

20. B. Reznick, ‘A Set is a Set’, Mathematics Magazine, 66, p. 95 (April issue 1993).

21. The full title was An Investigation of the Laws of Thought on Which are Founded the Mathematical Theories of Logic and Probabilities. He also developed some of these ideas in his earlier book The Mathematical Analysis of Logic.

22. This is obviously the case for finite sets and (not so obviously) is also the case for infinite sets as well, as proved by Georg Cantor. It means that there is a never-ending ascending staircase of infinities, each infinitely bigger (in the well-defined sense of there not being a one-to-one correspondence between the members) than the previous one. The set of all subsets of a given set is called its power set.

23. These diagrams are named after their inventor, John Venn (1834–83).

24. The basic idea of this construction was discovered by the German logician Gottlob Frege and then rediscovered by Bertrand Russell. The form presented here is simpler in its treatment and was introduced as a refinement of Frege’s scheme by John von Neumann.

25. R. Cleveland, ‘The Axioms of Set Theory’, Mathematics Magazine, 52, 4, pp. 256–7 (1979).

26. R. Rucker, Infinity and the Mind, Paladin, London (1982), p. 40.

27. D.E. Knuth, Surreal Numbers: How Two Ex-Students Turned On to Pure Mathematics and Found Total Happiness, Addison Wesley, NY (1974). In this quote you notice that Conway’s initials conveniently provide the Hebrew consonants for Jehovah = Yahweh.

28. J.H. Conway, On Numbers and Games, Academic, NY (1976).

29. D.E. Knuth, op.cit.

30. In the postscript to the book (p. 113) Knuth writes, ‘I decided that creativity can’t be taught using a textbook, but that an “anti-text” such as this novel might be useful. I therefore tried to write the exact opposite to Landau’s Grundlagen der Mathematik; my aim was to show how mathematics can be “taken out of the classroom and into life”, and to urge the reader to try his or her own hand at exploring abstract mathematical ideas.’ Knuth picks on Landau during the dialogues but his most general target is probably the Bourbaki approach to presenting mathematics.

31. Negative numbers are defined analogously −x = {−R|−L}.

32. If x and y are given byx={xL | xR} and y = {yL | yR} then the sum x + y = {xL + y, x + yL | xR + y, x + yR} and the product xy = {xLy + x yL − xL yL, xR y + x yR − xR yR | xL y + x yR − xL yR, xR y + x yL − xR yL.

33. J.H. Conway, ‘All Games Bright and Beautiful’, American Mathematics Monthly 84, pp. 417–34 (1977).

34. A. Huxley, Point Counter Point, Grafton, London (1928), p. 135.

35. J. Hick, Arguments for the Existence of God, Macmillan, London (1970).

36. Anselm, Proslogion 2.

37. C. Hartshorne, A Natural Theology for our Time, Open Court, La Salle (1967). A fuller discussion and bibliography can be found in J.D. Barrow & F.J. Tipler, The Anthropic Cosmological Principle, Oxford University Press (1986), pp. 105–9.

38. B. Russell, ‘Recent work on the principles of mathematics’, International Monthly, 4 (1901).

39. G. Cantor, Grundlagen einer allegemeinen Mannigfaltigkeitslehre, B.G. Treubner, Leipzig (1883), p. 182; English transl. as Foundations of the Theory of Manifolds, transl. U. Parpart, the Campaigner (The Theoretical Journal of the National Caucus of Labor Committees), 9, pp. 69–96 (1976). The translation here is from J. Dauben, Georg Cantor, Harvard University Press, Mass. (1979), p. 132.

chapter six

Empty Universes

1. P. Kerr, The Second Angel, Orion, London (1998), p. 201.

2. By ‘strong’ we mean that the gravitational force gradient can induce particles to move at speeds close to that of light.

3. Light moves more slowly through a medium than it does through a vacuum. It is possible for objects to travel through a medium at a speed which exceeds the speed of light in that medium. When this occurs then radiation, called Cerenkov radiation, is produced and is routinely observed. It is used by experimenters to detect high-speed particles from space.

4. We talk of mass and energy together because they are equivalent, related by Einstein’s famous formula E = mc2, where E is energy, m is mass and c is the velocity of light in a vacuum.

5. The ripples are called gravitational waves. They travel at the speed of light and can be viewed as the propagating influence of gravity fields. The long-range effect of rotation, called the dragging of inertial frames, pulls objects around in the same sense as that of the rotation possessed by a nearby source of gravity. Both of these phenomena are absent in Newton’s theory of gravity.

6. Curved space is easy to visualise but curved time sounds strange. In practice it amounts to a change in the rate of flow of time compared to the rate at a place, ideally infinitely far from all masses, where the space is flat. In general, clocks measure time to pass slower in strong gravitational fields than it passes in weak gravitational fields. This is also observed.

7. An interesting and controversial consequence of this picture is that it implies that the spacetime is the primary concept, rather than space or time separately or added together. The block of spacetime can be sliced up into a stack of curved sheets in an infinite number of different ways, all apparently as good as any other. This corresponds to a choice of time. Events on each slice are simultaneous but different moving observers create different slicings, different standards of time, and make different observations which they judge to be simultaneous. This block spacetime picture implies that the future is already ‘out there’. By contrast, in other sciences, the flow of time is associated with unfolding events, increase in information, entropy or complexity, and there is no suggestion that the future is out there waiting. For an interesting discussion of the theological and philosophical implications of the block spacetime picture, see C.J. Isham and J.C. Polkinghorne, ‘The Debate over the Block Universe’, in Quantum Cosmology and the Laws of Nature (2nd edn), eds R.J. Russell, N. Murphy and C.J. Isham, University of Notre Dame Press (1996), pp. 139–47.

8. All fundamental forces appear to possess ‘carrier’ (or ‘exchange’) particles which convey the interaction. The carrier for the electro-magnetic interaction between electrically changed particles is the photon which does not possess charge and so is not self-interacting. Gravity is carried by the graviton (which is the same as the gravitational waves discussed above) which possesses mass energy and so feels the force of gravity and is self-interacting. You can have a gravitating world that contains only gravitons but not an electromagnetic world that contains only photons.

9. Isaiah 34 v. 11–12.

10. J.D. Barrow, The Origin of the Universe, Orion, London (1994).

11. So that as the distribution of mass and energy changes from one slice to another there will be conservation of energy and electric charge and angular momentum.

12. M.J. Rees and M. Begelman, Gravity’s Fatal Attraction, Scientific American Library, New York (1996), p. 200.

13. C.S. Peirce, The Collected Papers of Charles Sanders Peirce (8 vols), ed. C. Hartshorne et al, Harvard University Press, Cambridge, Mass. (1931–50), vol. 4, section 237.

14. E. Mach, The Science of Mechanics, first published in 1883, reprinted by Open Court, La Salle (1911).

15. J.D. Barrow, R. Juszkiewicz and D. Sonoda, ‘Universal Rotation: How Large Can It Be?’, Mon. Not. Roy. Astr. Soc., 213, pp. 917–43 (1985).

16. The inflationary universe theory, which will be described in the next chapter, leads us to expect that the rotation of the Universe will be very small. Any rotation that existed before inflation (a period when the expansion of the Universe accelerates) occurs will be dramatically reduced during a period of inflation. Moreover, the matter fields expected to produce inflation cannot rotate and so inflation cannot create rotations in the way that it can produce variations in density and in gravitational waves. Indeed, the observation of large-scale rotation in the Universe would be fatal for the inflationary theory, see J.D. Barrow & A. Liddle, ‘Is inflation falsifiable?’ General Relativity & Gravitational Journal, 29, pp. 1501–8 (1997).

17. Of course, if you are interested in particular questions like how the small deviations from homogeneity and isotropy arose, and why they have the observed patterns, then you do not begin with such an assumption. Instead, you might assume that the irregularities are small (but non-zero) and that the Universe was just homogeneous and isotropic on the average.

18. A. Friedmann, Zeitschrift für Physik, 10, p. 377 (1922) and 21, p. 326 (1924). Translations appear in Cosmological Constants, eds J. Bernstein and G. Feinberg, Columbia University Press (1986). R.C. Tolman, Relativity, Thermodynamics and Cosmology, Oxford University Press (1934).

19. The alternative scenario of contraction is ruled out on the grounds that it would have resulted in a ‘crunch’ of high density already.

20. Friedmann was a daring balloonist in the cause of science and even held the world altitude record at one time. These flights appear reckless by modern standards, with the balloonists often undergoing calculated periods of unconsciousness in extreme weather conditions. For biographical details of these adventures, see E.A. Tropp, V. Ya. Frenkel and A.D. Chernin, Alexander A. Friedmann: The Man Who Made the Universe Expand, transl. A. Dron and M. Burov, Cambridge University Press (1993).

21. R. Rucker, The Fourth Dimension, Houghton Mifflin, Boston (1984), p. 91.

22. G. Lemaître, ‘Evolution of the expanding universe’, Proceedings of the National Academy of Sciences, Washington, 20, p. 12 (1934).

23. If a fluid has pressure p and energy density ρc2, where c is the speed of light, then the condition for its gravitational effect to be attractive (repulsive) is that ρc2 + 3p be positive (negative). In a homogeneous and isotropic universe the cosmological constant is equivalent to a ‘fluid’ with p = – ρc2 and hence it is gravitationally repulsive.

24. This description of expanding universes that begin at a past moment of high (infinite?) density was coined pejoratively by Fred Hoyle in a radio broadcast in 1950, to contrast it with the steady-state theory.

25. W.H. McCrea, Proc. Roy. Soc. A 206, p. 562 (1951). Lemaître’s early article (ref. 22) on the interpretation of the lambda term as a fluid with pressure and density in the context of general relativity was not known to McCrea.

26. A. Sandage, Astrophysical Journal Letters 152, L 149–154 (1968).

27. D. Sobel, Longitude, Fourth Estate, London (1995).

28. With a 95% statistical confidence level.

29. S. Perlmutter et al, ‘Measurements of Ω and Λ from 42 high-redshift supernovae’, Astrophysical Journal, 517, pp. 565–58 (1999) B.P. Schmidt et al, ‘The high-Z supernova search: measuring cosmic deceleration and global curvature of the Universe using type Ia supernovae’, Astrophysical Journal, 507, pp. 46–63 (1998). Updated information about the Supernova Cosmology Project can be obtained from the Project website at panisse.lbl.gov/public/papers.

chapter seven

The Box That Can Never Be Empty

1. B. Hoffman, The Strange Story of the Quantum, Penguin, London (1963), p. 37.

2. A. Einstein, letter to D. Lipkin, 5 July 1952, quoted in A. Fine, The Shaky Game, University of Chicago Press (1986), p. 1.

3. My own version, with many references to others, can be found in J.D. Barrow, The Universe that Discovered Itself, Oxford University Press (2000).

4. Quoted by N.C. Panda in Maya in Physics, Motilal Bonarsidass Publishers, Delhi (1991), p. 73.

5. A. Einstein, letter to Max Born, 4 June 1919, quoted by Max Born in The Born–Einstein Letters, Walker & Co., New York (1971), p. 11.

6. R. Feynman, The Character of Physical Law, MIT Press, Cambridge, Mass. (1967), p. 129.

7. W. Heisenberg, Physics and Beyond: Encounters and Conversations, Harper and Row, New York (1971), p. 210.

8. H.A. Kramers, quoted in L. Ponomarev, The Quantum Dice, IOP, Bristol (1993), p. 80.

9. Black bodies are perfect absorbers and emitters of light.

10. Zero degrees Centigrade equals 273·15 degrees Kelvin.

11. Its numerical value is measured to be h = 6.626 × 10−34 Joule-seconds.

12. It was predicted that the spectrum should have a Planckian shape over most of the wavelength range but there was great interest in how accurately it would follow the Planck curve in certain wavelength ranges. This interest arose because, if the history of the Universe had undergone violent episodes associated with the formation of galaxies, other sources of radiation with higher temperatures could have been added to the primeval radiation left over from the Big Bang. This can distort the spectrum slightly from the Planckian form. The observations showed no such distortions of the pure Planck spectrum to very high precision. This tells us important things about the history of the Universe.

13. J.C. Maxwell, Treatise on Electricity and Magnetism, Dover, NY (1965).

14. The zero-point energy idea was first introduced by Planck in 1911 in an attempt to understand how matter and radiation interact to create the black-body Planck spectrum. Planck first proposed that whilst the emission of radiation occurs in discrete quantum packets the absorption of radiation is continuously possible over all values. This hypothesis (which Planck abandoned three years later) led to the conclusion that the system would have energy hf/2 even at absolute zero of temperature. In 1913 Einstein and Otto Stern showed that the correct classical (non-quantum) limit for the energy is only obtained from the Planck black-body distribution if the zero-point energy is included (Annalen der Physik, 40, pp. 551–60 [1913]). For some further discussion see D.W. Sciama, in The Philosophy of Vacuum, Oxford University Press (1991), pp. 137–58.

15. Casimir, H.B.G., ‘On the attraction between two perfectly conducting plates’, Koninkl. Ned. Akad. Wetenschap. Proc., 51, pp. 793–5 (1948); Casimir’s first study of these effects dealt with the more specific situation of the attractive force between two polarisable atoms. An attractive force arises and led Casimir to replace the atoms by the simpler situation of parallel plates. H.B.G. Casimir and D. Polder, ‘The Influence of Retardation on the London-van der Waals Forces’, Phys. Rev. 73, pp. 360–72 (1948); G. Plumien, B. Muller and W. Greiner, ‘The Casimir Effect’, Phys. Rep., 134, pp. 87–193 (1986). A complete calculation of the effect requires several important details to be taken into account, for example the fact that the plates cannot be regarded as perfect conductors down on the scale of single atoms and smaller. The most complete book on the subject is P.W. Milonni, The Quantum Vacuum: an introduction to quantum electrodynamics, Academic, San Diego (1994). A simple account can be found in T. Boyer, ‘The classical vacuum’, Scientific American (Aug. 1985).

16. The formula gives 0.02 Newtons per square metre. These numbers are taken from the experimental investigation of Sparnaay.

17. J. Ambjorn and S. Wolfram, Ann. Phys., 147, p. 1 (1983); G. Barton, ‘Quantum electrodynamics of spinless particles between conducting plates’, Proc. Roy. Soc. A, 320, pp. 251–75 (1970).

18. M.J. Sparnaay, ‘Measurement of the attractive forces between flat plates’, Physica, 24, p. 751 (1958).

19. S.K. Lamoreaux, ‘Demonstration of the Casimir force in the 0.6 to 6µM range’, Phys. Rev. Lett. 78, pp. 5–8 (1997) and 81, pp. 5475–6 (1998).

20. Careful account must be taken of the fact that the experiment is not being performed at absolute zero of temperature and that the plates (made of coated quartz) are not perfect conductors as assumed in the simple calculation we have described.

21. C.I. Sukenik, M.G. Boshier, D. Cho, V. Sandoghdar and E. Hinds, ‘Measurement of the Casimir–Polder force’, Phys. Rev. Lett., 70, pp. 560–3 (1993).

22. H.E. Puthoff, ‘Gravity as a zero-point fluctuation force’, Phys. Rev. A, 39, pp. 2333–42 (1989); R.L. ‘Forward, Extracting electrical energy from the vacuum by cohesion of charged foliated conductors’, Phys. Rev. B, 30, pp. 1700–2 (1984); D.C. Cole & H.E. Puthoff, ‘Extracting energy and heat from the vacuum’, Phys. Rev. E, 48, pp. 1562–5 (1993); I.Y. Sokolov, ‘The Casimir Effect as a possible source of cosmic energy’, Phys. Let. A, 223, pp. 163–6 (1996); P. Yam, ‘Exploiting zero-point energy’, Scientific American, 277, pp. 82–5 (Dec. 1997).

23. J. Schwinger, ‘Casimir light: field pressure’, Proc. Nat. Acad. Sci. USA, 91, pp. 6473–5 (1994); C. Eberlein, ‘Sonoluminescence as quantum vacuum radiation’, Phys. Rev. Lett., 76, pp. 3842–5 (1996).

24. K.A. Milton and Y.J. Ng, ‘Observability of the bulk Casimir effect: can the dynamical Casimir effect be relevant to sonoluminescence?’, Phys. Rev. E, 57, pp. 5504–10 (1998); V.V. Nesterenko and I.G. Pirozhenko, ‘Is the Casimir effect relevant to sonoluminescence?’, Sov. Physics JETP Lett., 67, pp. 420–4 (1998).

25. J. Masefield, Salt-water Ballads, ‘Sea Fever’ (1902).

26. P.C. Causeé, L’Album du Marin, Charpentier, Nantes (1836).

27. S.L. Boersma, ‘A maritime analogy of the Casimir effect’, American J. Physics, 64, p. 539 (1996). The author says that his attention was drawn to this problem in Causeé’s book by Hazelhoff Roelfzema of the Amsterdam Shipping Museum.

28. The force of attraction calculated by Boersma is equal to F = 2π2mηhA2/(QT2) where m is the mass of each ship (the two ships are assumed to be of equal mass), A is the angle in radians of their rolling in the swell, h is the metacentric height of the ship, T is the period of the oscillations of the ships, Q the quality factor of the oscillation, and η is the efficiency of the energy losses through friction. Substituting m = 700 tons, h = 1.5 metres, A = 8 degrees (= 0.14 radians), T = 8 seconds, Q = 2.5, η = 0.8 we find F = 2000 Newtons.

29. In J. Weintraub, Peel Me a Grape (1975), p. 47.

30. W. Lamb and R.C. Retherford, Phys. Rev., 72, p. 241 (1947). The theoretical interpretation was supplied by T.A. Welton, Phys. Rev., 74, p. 1157 (1948).

31. P. Kerr, The Second Angel, Orion, London (1998), p. 316.

32. To see what else is needed to understand the world, see J.D. Barrow, Theories of Everything, Vintage, London (1988).

33. Because the quantum wavelength of a particle is inversely proportional to its mass.

34. F. Close and C. Sutton, The Particle Connection, Oxford University Press (1987).

35. Despite its intrinsic weakness gravity wins out over electromagnetism in controlling the behaviour of matter in large aggregates because electric charges come in two varieties, positive and negative, and it is hard to assemble a large amount of matter that has a non-zero charge. Gravity acts on mass, and mass, by contrast, comes only in a positive variety and so its effect is cumulative when large aggregates of material are assembled.

36. Lao-tzu, Tao Te Ching, chap. 11.

37. This number, first defined by Arnold Sommerfeld in 1911, is called the fine structure constant and is given by 2πe2/hc. For further details of these developments see Chapter 4 of J.D. Barrow and F.J. Tipler, The Anthropic Cosmological Principle, Oxford University Press (1986).

38. C. Pickover, Computers and the Imagination, St Martin’s Press, NY (1991), p. 270.

39. The positron is the antiparticle of the electron. It has the same mass but opposite sign of its electric charge. When an electron encounters a positron they will annihilate to produce two photons of light. The electric charges cancel out to zero.

40. The term ‘black hole’ was invented by the American physicist John A. Wheeler in 1968 in an article entitled ‘Our Universe: the known and the unknown’, American Scholar, 37, p. 248 (1968), reprinted in American Scientist, 56 p. 1 (1968). Eleven years earlier he had coined the term ‘wormhole’.

41. The radius of a black hole, R, is proportional to its mass, M, therefore its density, proportional to M/R3, varies as 1/M2. Hence, the more massive the black hole, the lower its density.

42. The ‘killer’ feature of a black hole is the strong variation in gravitational pull that it exerts over an object of a finite (the ‘tidal force’). For a point particle of zero size no such variation exists and it would feel nothing as it fell freely under gravity into a black hole, big or small. Objects of finite size get stretched out because the part of them closest to the hole gets pulled in more strongly than the part furthest away. The density of the black hole (see the previous footnote) is a good measure of the strength of this tidal force. It becomes more significant for small black holes. Black holes smaller than about one hundred million times the mass of our Sun are able to tear stars apart when they fall through the horizon. Bigger black holes allow whole stars to fall through the horizon without breaking them up.

43. J.P. Luminet, Black Holes, Cambridge University Press (1992).

44. M. Begelman and M.J. Rees, Gravity’s Fatal Attraction, W.H. Freeman, San Francisco (1996).

45. S.W. Hawking, ‘The Quantum Mechanics of Black Holes’, Scientific American, January (1977).

46. The black hole does not gain mass as a result of capturing one of the pair of particles. The black hole loses mass as a result of this process after one takes into account the change in the potential energy of the particle-antiparticle pair. The pairs will preferentially appear at a separation that gives them zero total energy.

47. The temperature of the black hole is inversely proportional to its mass. The time required to radiate away all of its mass is proportional to the cube of its mass.

48. In Einstein’s theory of gravitation the local effects of gravity should be indistinguishable from those experienced by undergoing accelerated motion at a suitable rate. Thus, for very short intervals of time, we should not be able to distinguish being in a small lift that is freely falling under gravity from one that is being accelerated downwards. If we apply this to the situation of black holes in the vacuum, we should not be able to distinguish the situation that exists in the gravitational field of the black hole very close to the event horizon from that experienced by an observer moving with an acceleration equal to the acceleration due to gravity at the horizon. In fact, as first shown by Bill Unruh and Paul Davies, this is exactly what is predicted. If an observer is accelerated through the quantum vacuum at a uniform rate, A, then that observer should detect thermal radiation with a temperature given by T = hA/4πck, where c is the speed of light, k is Boltzmann’s constant and h is Planck’s constant.

chapter eight

How Many Vacuums Are There?

1. Quoted in the Observer newspaper, 4 July 1999.

2. Contrary to the situation that exists in many other subjects, research journals are now largely superfluous in subjects like physics and astronomy. All research articles are ‘published’ electronically and are revised in the light of the comments, requests for credit, or corrections, that come in by email to the authors.

3. Note that Newton’s famous laws of motion did not satisfy this dictum of Einstein’s. The first law, which was that ‘all bodies acted upon by no forces remain at rest or move at constant speed’, is not one that would be found true by all observers. Newton specified that it would be seen only by observers who were not accelerating or rotating with respect to absolute space. These are known as ‘inertial observers’. For example, if an astronaut in a rotating spaceship were to look out of the window he would see a neighbouring satellite accelerating past his window even if it were acted upon by no forces. The astronaut in his rotating spaceship is not an inertial observer.

4. Quoted in Observer, 12 December 1999, p. 30.

5. J.D. Barrow, Theories of Everything, Vintage, London (1991); B. Greene, The Elegant Universe, Vintage, London (2000).

6. A. Linde, ‘The Self-Reproducing Inflationary Universe’, Scientific American, no. 5, vol. 32 (1994).

7. A. Guth, The Inflationary Universe, Vintage, London (1998).

8. Fractal surfaces do not. They can have structure on all scales of magnification.

9. W. Allen, Getting Even, Random House, NY (1971), p. 33.

10. This hierarchy of clustering of clusters does not carry on indefinitely. The clustering of galaxy clusters into so-called ‘superclusters’ seems to be the end of the line.

11. A term coined by the sociologist Robert Merton to describe the way in which people who are awarded honours and prizes then seem to be awarded even more honours and prizes. It is taken from Christ’s words in the Gospel of St Matthew chap. 13 v. 12: ‘For whosoever hath, to him shall be given, and he shall have more abundance: but whosoever hath not, from him shall be taken away even that he hath.’

12. When it was about ten million years old.

13. P. de Bernadis et al, ‘A flat universe from high-resolution maps of the cosmic microwave background radiation’, Nature, 404, pp. 955–9 (2000). See also www.physics.ucsb.edu/~boomerang/for further pictures and information about the experiment and the significance of its results.

14. The Independent, quoted in third Leader article in Review section, p. 3, 13 November 1999.

15. Based on data presented by the Boomerang Collaboration on their website www.physics.ucsb.edu/~boomerang/.

16. J.D. Barrow, ‘Dimensionality’, Proc. Roy. Soc. A., 310, p. 337 (1983); J.D. Barrow & F.J. Tipler, The Anthropic Cosmological Principle, Oxford University Press (1986), chap. 6; M. Tegmark, ‘Is “the theory of everything” merely the ultimate ensemble theory?’, Annals of Physics (NY), 270, pp. 1–51 (1998).

17. There was once some interest in science-fiction stories about biochemists based upon silicon chemistry. These do not look promising (as explained in Barrow and Tipler, op. cit.), but, ironically, it is silicon physics that looks the most likely route to a form of artificial life brought into being by means of the catalytic help of (human) carbon-based life.

18. During inflation the pressure contributed by the scalar matter field responsible is negative and so a change in energy of the expanding material actually provides energy for work.

19. A few years ago Sidney Coleman proposed a partial solution of this sort. He suggested that if the topology of the Universe was sufficiently complicated, with many holes, handles and tubes (‘wormholes’), then the presence of any lambda term would tend to create an opposing stress that cancelled out the lambda term to very high precision. The most probable value of lambda that would be measured when the Universe expanded and became very large would be zero with great accuracy. Unfortunately, the appealing idea did not survive further scrutiny and there are no similar general arguments which have so far provided us with an understanding of lambda’s tiny value.

20. I Corinthians chap. 15 v. 51–2.

21. The later editions of inflation, like chaotic inflation, could make do with a single vacuum.

22. P. Hut and M.J. Rees, ‘How stable is our vacuum?’, Nature, 302 (1983), pp. 508–9; M.S. Turner and F. Wilczek, ‘Is our vacuum metastable?’, Nature 298 (1982), pp. 633–4. For a wider review of possible ‘sudden’ ends to the world see J. Leslie, The End of the World.

23. Recently, there seems to have been some public worry in the United States that a planned sequence of high-energy particle collisions at a national laboratory might induce just such a catastrophe.

24. N. Eldridge and S.J. Gould, ‘Punctuated equilibria: an alternative to phyletic gradualism’, in T.J.M. Schoof (ed.), Models in Paleobiology, W.H. Freeman, San Francisco (1972), pp. 82–115.

25. From P. Bak, How Nature Works, Oxford University Press (1997), p. 39.

26. A.S.J. Tessimond, Cats, p. 20 (1934).

27. T. Kibble, ‘Topology of Cosmic Domains and Strings’, Journal of Physics A, 9, pp. 1387–97 (1972).

28. These should not be confused with superstrings or superstring theories. Superstring theories may permit the existence of cosmic strings but not necessarily.

29. From P. Bak, op. cit.

30. This gravitational lensing phenomenon, predicted by Einstein, is now commonly observed but is believed to be created by objects other than cosmic strings in the cases where it is seen. In our own galaxy and nearby in the Large Magellanic Cloud (a neighbouring very small galaxy) it is created by non-luminous objects that have masses similar to those of stars.

chapter nine

The Beginning and the End of the Vacuum

1. M. Proust, Le Côté de Guermantes (1921), transl. as Guermantes’ Way, by C.K. Scott-Moncrieff (1925), vol. 2, p. 147.

2. G.K. Chesterton, The Napoleon of Notting Hill, first published in 1902, Penguin, London (1946), p. 9.

3. E.O. James, Creation and Cosmology, E.J. Brill, Leiden (1969); C. Long, Alpha: the Myths of Creation, G. Braziller, New York (1963); C. Blacker and M. Loewe (eds), Ancient Cosmologies, Allen & Unwin, London (1975); M. Leach, The Beginning: Creation Myths around the World, Funk and Wagnalls, New York (1956).

4. M. Eliade, The Myth of the Eternal Return, Pantheon, New York (1954); see also J.D. Barrow & F.J. Tipler, The Anthropic Cosmological Principle, Oxford University Press (1986).

5. T. Joseph, ‘Unified Field Theory’, New York Times, 6 April 1978.

6. An interesting collection of articles is to be found in R. Russell, N. Murphy & C. Isham, Quantum Cosmology and the Laws of Nature, 2nd edn, University of Notre Dame Press, Notre Dame (1996).

7. A. Ehrhardt, ‘Creatio ex Nihilo’, Studia Theologica (Lund), 4, p. 24 (1951), and The Beginning: A Study in the Greek Philosophical Approach to the Concept of Creation from Anaximander to St. John, including a memoir by J. Heywood Thomas, Manchester University Press (1968); D. O’Connor and F. Oakley (eds), Creation: the impact of an idea, Scribners, New York (1969).

8. With St Augustine, this idea was made more sophisticated by including the idea that time must have come into being along with the world, thus avoiding questions about what existed ‘before’ the world was.

9. S. Jaki, Science and Creation, Scottish Academic Press, Edinburgh (1974), and The Road of Science and the Ways to God, University of Chicago Press (1978).

10. G.F. Moore, Judaism in the First Centuries of the Christian Era I, Cambridge, Mass. (1966), p. 381.

11. Wisdom chap. 11 v. 17.

12. 7 v.28.

13. G. May, Creatio ex Nihilo, transl. A.S. Worrall, T & T Clark, Edinburgh (1994), p. 8.

14. H.A. Wolfson, ‘Negative Attributes in the Church fathers and the Gnostic Basilides’, Harvard Theol. Review, 50, pp. 145-56 (1957), J. Whittaker, ‘Basilides and the Ineffability of God’, Harvard Theol. Review, 62, pp. 367–71 (1969).

15. A beautiful expression of this is found in the Tripartite Tractate of Valentius in the Jung Codex, cited by May, op.cit., p. 75, translation by H.W. Attridge and E. Pagels: ‘No one else has been with him from the beginning; nor is there a place in which he is, or from which he has come forth, or into which he will go; nor is there a primordial form, which he uses as a model as he works; nor is there any difficulty which accompanies him in what he does; nor is there any material which is at his disposal, from which he creates what he creates; nor any substance within him from which he begets what he begets; nor a co-worker with him, working with him on the things at which he works. To say anything of this sort is ignorant.’

16. Four centuries later the argument will still be used by John Philoponus to counter the same suggestions. He defines the creativity of the artist as an activity which rearranges existing elements in a new way and the creativity of the natural world as the bringing of living beings out of non-living matter. Divine creation is superior to both because it can create the material out of nothing.

17. Nevertheless, other aspects of his view of the world were different to those that would be adopted by the central Christian tradition. Basilides appears to have been a Deist in believing that God played no further role in the unfolding of the Universe after laying down the starting conditions. God’s creative activity was limited to a single act.

18. F.M. Cornford, Microcosmographia Academica, Cambridge University Press (1908), p. 28.

19. See N. Rescher, The Riddle of Existence, University Press of America, Lanham (1984), p. 2.

20. J.D. Barrow, Impossibility, Oxford University Press (1998).

21. N. Malcolm, Ludwig Wittgenstein: A Memoir, Oxford University Press (1958), p. 20.

22. M. Heidegger, Introduction to Metaphysics, Yale University Press, New Haven (1959); L. Wittgenstein, Tractatus Logico-Philosophicus, London (1922), section 6.44.

23. H. Bergson, Creative Evolution, trans. A. Mitchell, Modern Library, NY (1941), p. 299. This type of set-theoretic basis for conjuring something out of nothing is also hinted at in the discussion to be found in the final chapter of Gravitation by C. Misner, K. Thorne & J.A. Wheeler, W.H. Freeman, San Francisco (1972); and in P. Atkins, The Creation, W.H. Freeman, San Francisco (1981).

24. For example, the three interior angles of a triangle sum to 180 degrees in a Euclidean geometry but not in a non-Euclidean geometry.

25. A. Hodges, The Enigma of Intelligence, Unwin, London (1985), p. 154.

26. J.D. Barrow, Impossibility, Oxford University Press (1998).

27. R. Penrose, The Emperor’s New Mind, Oxford University Press (1989).

28. The feature that there exist statements of arithmetic whose truth or falsity cannot be established using the rules and axioms of arithmetic; see J.D. Barrow, Impossibility, Oxford University Press (1998), for a fuller discussion.

29. N. Rescher, The Riddle of Existence, University Press of America, Lanham (1984), p. 3.

30. T. Joseph, ‘Unified Field Theory’, New York Times, 6 April 1978.

31. Einstein thought that infinities and singularities were unacceptable in physical theories. His assistant at Princeton, Peter Bergmann, writes: ‘It seems that Einstein always was of the opinion that singularities in classical field theory are intolerable. They are intolerable from the point of view of classical field theory because a singular region represents a breakdown of the postulated laws of nature. I think that one can turn this argument around and say that a theory that involves singularities and involves them unavoidably, moreover, carries within itself the seeds of its own destruction.’ Contribution in H. Woolf, Some Strangeness in the Proportion, Addison Wesley, MA (1980), p. 156.

32. For a recent overview of the mathematical ideas see the first chapter of S.W. Hawking & R. Penrose, The Nature of Space and Time, Princeton University Press (1996). For a descriptive account see J.D. Barrow & J. Silk, The Left Hand of Creation (2nd edn.), Penguin, London, and Oxford University Press, New York (1993).

33. The recently discovered microwave background radiation was sufficient to meet this requirement.

34. It is known that incomplete histories occur which are not accompanied by infinities of physical quantities, like density or temperature. However, whilst it is suspected that these examples are in some way atypical of solutions to Einstein’s equations, this has not been proven in general. The original theorem of Penrose was proved for the situation of a collapsing cloud of matter (like the expanding universe in reverse). Subsequently, Hawking and Penrose proved a version of the theorem which applies specifically to cosmologies. For a detailed survey, see S.W. Hawking & G.F.R. Ellis, The Large Scale Structure of Space-time, Cambridge University Press (1973).

35. J. Earman, Bangs, Crunches, Whimpers, and Shrieks: singularities and acausalities in relativistic spacetimes, Oxford University Press (1995).

36. If one interprets the lambda force as a vacuum energy in Einstein’s equations then it behaves like a form of matter that exhibits gravitational repulsion because its pressure p and density ρ satisfy the relationship p = −ρc2. Gravitational repulsion arises whenever matter satisfies the weaker condition 3p < −ρc2. The singularity theorems assume that 3p > –ρc2.

37. M. Eliade, The Myth of the Eternal Return; see also J.D. Barrow & F.J. Tipler, The Anthropic Cosmological Principle, Oxford University Press (1986).

38. It is very likely that these two singularities would be very different in structure. Irregularities tend to grow during the evolution of the Universe in its expanding phase. These irregularities will be amplified even further during the contraction phase and the final singularity should be extremely irregular.

39. The big assumption here is that nothing counter to the second law of thermodynamics occurs at the moments when the Universe bounces (or indeed, that any such ‘law’ is applicable).

40. This was first pointed out by R.C. Tolman in two articles, ‘On the problem of the entropy of the universe as a whole’, Physical Review, 37, pp. 1639–1771 (1931), and ‘On the theoretical requirements for a periodic behaviour of the universe’, Physical Review, 38, p. 1758 (1931). Recently, a detailed reanalysis was given by J.D. Barrow and M. Dbrowski, ‘Oscillating Universes’, Mon. Not. Roy. Astron. Soc., 275, pp. 850–62 (1995).

41. See, for example, E.R. Harrison, Cosmology: the science of the universe, Cambridge University Press (1981), pp. 299–300.

42. A. Swinburne, ‘The Garden of Proserpine’, Collected Poetical Works, p. 83.

43. F. Dyson, ‘Life in an open universe’, Reviews of Modern Physics, 51, p. 447 (1979).

44. J.D. Barrow & F.J. Tipler, The Anthropic Cosmological Principle, Oxford University Press (1986), chap. 10.

45. The absolute minimum amount of energy required to process a given amount of information is determined by the second law of thermodynamics. If ∆I is the number of bits of information processed, the second law requires ∆I ≤ ∆E/kTln2 = ∆E/T(ergs/K)(1.05×1016), where T is the temperature in degrees Kelvin, k is Boltzmann’s constant and ∆E is the amount of free energy expended. If the temperature operates at a temperature above absolute zero (T > 0, asrequired by the third law of thermodynamics), there is a minimum amount of energy that must be expended to process a single bit of information. This inequality is due to Brillouin.

46. See S.R.L. Clark, How to Live Forever, Routledge (1995).

47. See The Anthropic Cosmological Principle, op.cit., p. 668.

48. The current observations are indicating that this is not the case in our Universe. It appears to be destined to keep expanding for ever, locally, and if the eternal inflation scenario is true it will continue expanding globally as well. Recently, João Magueijo, Rachel Bean and I (‘Can the Universe escape eternal inflation?’, Mon. Not. Roy. Astron. Soc., 316, L41–44 [2000]) have found a way for the Universe to escape from accelerating expansion. If it contains a scalar field, which is falling in a potential energy landscape which descends steeply but has a small U-shaped crevice on it, with a local minimum, then the scalar field can pass through along this valley and produce a short period of inflation. It carries on up the slope and then continues to fall down the slope again. When this happens the expansion stops accelerating and reverts to the usual decelerated expansion that it experiences for most of its history. Potential landscapes with this shape have been identified in string theories at high energy. They were suggested for cosmological applications by A. Albrecht and C. Skordis, Phys. Rev. Lett., 84, pp. 2076–9 (2000), but they envisaged that they would lead to a state of never-ending inflation.

49. This must lie at least about thirty billion years in the future. It should be noted that it is possible for us to encounter a singularity in the future without this lambda energy decay, even if the expansion appears to be going to carry on for ever. There could be a gravitational shock-wave travelling towards us at the speed of light that hits us without warning.