The Book of Nothing

by John D. Barrow

  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 Harr
ison, 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, Elements, Great 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.