The Book of Nothing

by John D. Barrow

  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.

  3
2. 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. Tor
ricelli 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).