The Book of Nothing

by John D. Barrow

  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 wh
ich 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).