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Notes
Simon Van Den Bergh, Notes to Tahafut al-Tahafut [The Incoherence of the Incoherence], 2 vols., by Averroës (London: Luzac & Co., 1954), 2: 8.
Aristotle Physica 3. 6. 206a15–20. For a good discussion, consult David Bostock, ‘Aristotle, Zeno, and the Potential Infinite’, Proceedings of the Aristotelian Society 73 (1972–3): 37–57.
To argue that there can be a potential infinite but no actual infinite does not commit one to the self-contradictory position that there are possibilities that cannot be actualised. (W. D. Hart, ‘The Potential Infinite’, Proceedings of the Aristotelian Society 76 [1976]: 247–64.)
Abraham Robinson, ‘The Metaphysics of the Calculus’, in The Philosophy of Mathematics, ed. Jaakko Hintikka, Oxford Readings in Philosophy (London: Oxford University Press, 1969), pp. 156, 159.
On some interesting medieval precedents for Cantor’s work, see E. J. Ashworth, ‘An Early Fifteenth-Century Discussion of Infinite Sets’, Notre Dame Journal of Formal Logic 18 (1977): 232–4.
On Leibniz’s view of infinitesimal terms as useful fictions, see John Earman, ‘Infinities, Infinitesimals, and Indivisibles: The Leibnizian Labyrinth’, Studia Leibnitiana 7 (1975): 236–51.
Karl Friedrich Gauss and Heinrich Christian Schumacher, Briefwechsel, 6 vols., ed. C. A. F. Peters (Altona: Esch, 1860–65), 2: 269.
Bernard Bolzano, Paradoxes of the Infinite, trans. Fr. Prihonsky with an Introduction by Donald A. Steele (London: Routledge & Kegan Paul, 1950), pp. 81–4.
Richard Dedekind, ‘The Nature and Meaning of Numbers’, in Richard Dedekind, Essays on the Theory of Numbers, trans. Wooster Woodruff Beman (New York: Dover Publications, 1963), p. 63.
For an exposition and defence of Cantor’s system, see Robert James Bunn, ‘Infinite Sets and Numbers’ (Ph.D. dissertation, University of British Columbia, 1975).
Georg Cantor, Contributions to the Founding of the Theory of Transfinite Numbers, trans. with an Introduction by Philip E. B. Jourdain (New York: Dover Publications, 1915), pp. 55–6.
David Hilbert, ‘On the Infinite’, in Philosophy of Mathematics ed. with an Introduction by Paul Benacerraf and Hilary Putnam (Englewood Cliffs, N.J.: Prentice-Hall, 1964), pp. 139, 141.
For example, R. L. Sturch dismisses the kalâm argumentation against the existence of an actual infinite with one sentence: the result of applying Cantorian theory to [these] paradoxes is to resolve them....’ (R. L. Sturch, ‘The Cosmological Argument’ [Ph.D. thesis, Oxford University, 1970], p. 79.)
W. I. Matson similarly asserts that since there is no logical inconsistency in an infinite series of numbers, there is no logical inconsistency in an infinite series of events, and therefore the first cause argument is incurably fallacious. (Wallace I Matson, The Existence of God [Ithaca, N.Y.: Cornell University Press, 1965], pp. 58–60.) Matson fails to understand that the kaldm argument holds that the existence of an actual infinite is really, not logically, impossible. That there is a difference can be seen in the fact that God’s non-existence, if He exists, is logically, but not really, possible; if He does not exist, His existence is then logically, but not really, possible. Analogously, the existence of an actual infinite is really impossible, even if it may not involve logical contradiction.
Bolzano, Paradoxes p. 101. Cantor also believed that his discoveries concerning the nature of infinity might be of great service to religion in understanding the infinity of God and even carried on a fascinating correspondence with Pope Leo XIII to this effect. (See Joseph W. Dauben, ‘Georg Cantor and Pope Leo XIII: Mathematics, Theology, and the Infinite’, Journal of the History of Ideas 38 [1977]: 85–108.)
B. Rotman and G. T. Kneebone, The Theory of Sets and Transfinite Numbers (London: Oldbourne, 1966), p. 61. Thus, when one selects from an infinite set an infinite subset, the actual possibility of such an operation is not implied. ‘The conception of an infinite sequence of choices (or of any other acts)... is a mathematical fiction—an idealization of what is imaginable only in finite cases.’ (Ibid., p. 60.)
Alexander Abian, The Theory of Sets and Transfinite Arithmetic (Philadelphia and London: W. B. Saunders Co., 1965), p. 68.
Pamela M. Huby, ‘Kant or Cantor? That the Universe, if Real, Must Be Finite in Both Space and Time’, Philosophy 46 (1971): 130.
Consult Joseph Breuer, Introduction to the Theory of Sets trans. Howard F. Fehr (Englewood Cliffs, N.J.: Prentice-Hall, 1958), pp. 35–6.
This story is recorded in an entertaining work by George Gamow, One, Two, Three,... Infinity (London: Macmillan & Co., 1946), p. 17.
For a good discussion of this issue, consult Abraham A. Fraenkel, Yehoshua Bar-Hillel, and Azriel Levy, Foundations of Set Theory, 2d rev. ed. (Amsterdam and London: North-Holland Publishing Co., 1973), pp. 331–45.
Stephen F. Barker, Philosophy of Mathematics, Foundations of Philosophy Series (Englewood Cliffs, N.J.: Prentice-Hall, 1964), pp. 69–91.
It might be protested that we need not bring in the notion of actual infinity when speaking of past events; we may simply say that the series of temporal events is beginningless. This does not appear to be counter-intuitive and avoids the problems of the actual infinite. But such an escape route is easily barred: for it is in analysing what a beginningless series of events involves that the absurdities become manifest. As G. E. Moore indicates, if we grant that events really occur in time, then only two alternatives are possible: either there was a first event or there has been an actually infinite series of events prior to the present one. For if there was no first event, then there must have been an event prior to any given event; since this one also could not be first, there must be an event prior to it, and so on ad infinitum. (George Edward Moore, Some Main Problems of Philosophy Muirhead Library of Philosophy [London: George Allen & Unwin, 1953; New York: Macmillan Co., 1953], pp. 174–5.) Therefore, a beginningless series involves the existence of an actual infinite.
F. Van Steenberghen, ‘Le “processus in infinitum” dans les trois premières “voies” de saint Thomas’, Revista Portuguesa de Filosofia 30 (1974): 128. Another Thomist of the same judgement is Lucien Roy, ‘Note philosophique sur l’idée de commencement dans la création’, Sciences Ecclesiastiques s (1949): 223.
P. J. Zwart, About Time (Amsterdam and Oxford: North Holland Publishing Co., 1976), p. 179.
Bertrand russell, The Principles of Mathematics, 2d ed. (London: George Allen & Unwin, 1937), pp. 358–9. Remarkably, even Fraenkel appears to agree with Russell on this score, though a mathematician of his status ought to be acquainted with the difference between a potential and an actual infinite. (Fraenkel, Theory, p. 30.)
Russell’s fallacy is also discerned by G.J. Whitrow, The Natural Philosophy of Time (London and Edinburgh: Thomas Nelson & Sons, 1961), p. 149. Whitrow argues that Russell presupposes the incompletable series of events in question may be regarded as a whole, when in fact it is not legitimate to consider the events of Tristram Shandy’s life as a completed infinite set, since the author could never catch up with himself.
See a similar argument in David A. Conway, ‘Possibility and Infinite Time: A Logical Paradox in St. Thomas’ Third Way’, International Philosophical Quarterly, 14 (1974): 201–8.
For example, Russell declares, ‘... when Kant says that an infinite series can “never” be completed by successive synthesis, all that he has even conceivably a right to say is that it cannot be completed in a finite time.’ (Bertrand Russell, Our Knowledge of the External World 2d ed. [New York: W. W. Norton & Co., 1929], p. 171.) Cf. Matson: ‘This... begs the question, since it is only impossible to run through an infinite series in a finite time.’ (Matson, God p. 60.)
John Hospers, An Introduction to Philosophical Analysis 2d ed. (London: Routledge & Kegan Paul, 1967), p. 434 Hosper’s own statement of the argument is defective, for he argues that it would take infinite time to get through an infinite series, and this is the same as never getting through. It is not the same, of course, but the argument has nothing to do with the amount of time allowed: it is inherently impossible to form an actual infinite by successive addition.
William L. Rowe, The Cosmological Argument (Princeton, N.J.: Princeton University Press, 1975), p. 122.
R. G. Swinburne, Space and Time (London: Macmillan, 1968), pp. 207–8.
J. R. Lucas, A Treatise on Time and Space (London: Methuen & Co., 1973), pp. 10–11
Lawrence Sklar, Space, Time, and Spacetime (Berkeley and Los Angeles: University of California Press, 1974), p. 274.
Ian Hinckfuss, The Existence of Space and Time (Oxford: Clarendon Press, 1975), pp. 72–3.
Stuart C. Hackett, The Resurrection of Theism (Chicago: Moody Press, 1957), p. 263.
Brian Ellis, ‘Has the Universe a Beginning in Time?’ Australasian Journal of Philosophy 33 (1955): 33.
G. J. Whitrow, What is Time? (London: Thames & Hudson, 1972) pp. 146–7.
For an introduction to scientific cosmology, see P. J. E. Peebles, Physical Cosmology, Princeton Series in Physics (Princeton, N.J.: Princeton University Press, 1971).
D. W. Sciama, Modern Cosmology (Cambridge: Cambridge University Press, 1971).
J. D. North, Measure of the Universe (Oxford: Clarendon Press, 1965).
and S. Weinberg, Gravitation and Cosmology (New York: Wiley, 1972).
Edwin Hubble, ‘A Relation Between Distance and Radial Velocity Among Extra-galactic Nebulae’, Proceedings of the National Academy of Sciences 15 (1929): 168–73.
H. Bondi and T. Gold, ‘The Steady-State Theory of the Expanding Universe’, Monthly Notices of the Royal Astronomical Society 108 (1948): 252–75.
F. Hoyle, ‘A New Model for the Expanding Universe’, Monthly Notices of the Royal Astronomical Society 108 (1948): 372–82.
A. A. Penzias and R. W. Wilson, ‘A Measurement of Excess Antenna Temperature at 4080 Mc/s’, Astrophysical journal 142 (1965): 419–21.
see also R. H. Dicke, P. J. E. Peebles, P. G. Roll, and D. T. Wilkinson, ‘Cosmic Black-Body Radiation’, Astrophysical journal 142 (1965): 414–19.
For a good synopsis of phases of the early universe, see E. R. Harrison, ‘Standard Model of the Early Universe’, Annual Review of Astronomy and Astrophysics 11 (1973): 155–86.
Allan R. Sandage, ‘Cosmology: A Search for Two Numbers’, Physics Today, February 1970, p. 34.
J. Richard Gott III, James E. Gunn, David N. Schramm, and Beatrice M. Tinsley, ‘Will the Universe Expand Forever?’ Scientific American, March 1976, p. 65.
Fred Hoyle, From Stonehenge to Modern Cosmology (San Francisco: W. H. Freeman & Co., 1972), p. 36.
Fred Hoyle, Astronomy and Cosmology: A Modern Course (San Francisco: W. H. Freeman & Co., 1975), p. 658.
Fred Hoyle, Astronomy Today (London: Heinemann, 1975), p. 165.
Fred Hoyle, ‘The Origin of the Universe’, Quarterly Journal of the Royal Astronomical Society 14 (1973): 278.
Adrian Webster, ‘The Cosmic Background Radiation,’ Scientific American, August 1974, pp. 26–8.
M. S. Longair, ‘Radioastronomy and Cosmology’, Nature 263 (1976): 372–4.
Ibid., pp. 686–9o; Fred Hoyle, ‘On the Origin of the Micro-wave Background’, Astrophysical Journal 196 (1975): 661–70.
Hoyle, Stonehenge p. 2. Hugo Meynell reports, ‘In his contribution to a symposium, [Religion and the Scientists, ed. Mervyn Stockwood, p. 56.] Hoyle says that he believes in a God in the sense of an intelligence at work in nature, but then he qualifies this by saying that he means by ‘God’ the universe itself rather than some being who transcends the universe.’ (Hugo Meynell, God and the World: the Coherence of Christian Theism [London: SPCK, 1971], pp. 18–19.)
Noting that the steady state theory failed to secure ‘a single piece of experimental verification’, Jaki points out that the theory’s exponents often had ‘overtly anti-theological, or rather anti-Christian motivations’ for propounding this cosmological model. (Stanley L. Jaki, Science and Creation [Edinburgh and London: Scottish Academic Press, 1974], p. 347.)
Ivan R. King, The Universe Unfolding (San Francisco: W. H. Freeman, 1976), p. 462.
John Gribbin, ‘Oscillating Universe Bounces Back’, Nature 259 (1976): 15.
J. Richard Gott III, James E. Gunn, David N. Schramm, and Beatrice M. Tinsley, ‘An Unbound Universe?’ Astrophysical Journal 194 (1974): 543–53. The previously cited article by the same authors in the Scientific American, is a re-write and revision of this article.
See Icko Iben, ‘Post Main Sequence Evolution of Single Stars’, Annual Review of Astronomy and Astrophysics 12 (1974): 215–56.
See also Icko Iben, Jr., ‘Globular Cluster Stars’, Scientific American, July 1970, pp. 26–39.
See David N. Schramm, ‘Nucleo-Cosmochronology’, Annual Review of Astronomy and Astrophysics 12 (1974): 383–406.
See also David N. Schramm, ‘The Age of the Elements’, Scientific American, January 1974, pp. 69–77.
For a good general review of the deuterium question, see Jay M. Pasachoff and William A. Fowler, ‘Deuterium in the Universe’, Scientific American, May 1974, pp. 108–18.
Fred Hoyle and William A. Fowler, ‘On the Origin of Deuterium’, Nature 241 (1973): 384–6.
Gott, Gunn, Schramm, and Tinsley, ‘Unbound Universe?’ p. 548. If deuterium were produced in supernovae, then we would have too much 7Li, 9Be, and 11B, according to Richard I. Epstein, W. David Arnett, and David N. Schramm, ‘Can Supernovae Produce Deuterium?’ Astrophysical Journal 190 (1974): L13–16.
Hubert Reeves, Jean Audouze, William A. Fowler, and David N. Schramm, ‘On the Origin of Light Elements’, Astrophysical Journal 179 (1973): 909–30.
Jean Audouze and Beatrice M. Tinsley, ‘Galactic Evolution and the Formation of the Light Elements’, Astrophysical Journal 192 (1974): 487–500.
Sandage and Tammann, ‘Steps Toward the Hubble Constant. VI’, p. 276. Davis and May have reported that they have determined as a result of observations of the 839.4 MHz absorption line in the spectrum of the quasar 3C 286 a redshift accuracy of one part in loe; observations over several decades could place ‘useful limits on q0.’ (Michael M. Davis and Linda S. May, ‘New Observations of the Radio Absorption Line in 3C 286, with Potential Application to the Direct Measurement of Cosmological Deceleration’, Astrophysical Journal 219 [1978]: 3.) Thus, indirect estimates of q0 may not be the exclusive possibility in the future.
Allan Sandage, ‘The Redshift-Distance Relation. VIII. Magnitudes and Redshifts of Southern Galaxies in Groups: A Further Mapping of the Local Velocity Field and an Estimate of q0’, Astrophysical Journal 202 (1975): 563–82.
J. Richard Gott III and Martin J. Rees, ‘A Theory of Galaxy Formation and Clustering’, Astronomy and Astrophysics 45 (1975): 365–76.
Edwin L. Turner and J. Richard Gott III, ‘Evidence for a Spatially Homogeneous Component of the Universe: Single Galaxies’, Astrophysical Journal 197 (1975): L.89–93.
J. Richard Gott III and Edwin L. Turner, ‘The Mean Luminosity and Mass Densities in the Universe’, Astrophysical Journal 209 (1976): 1–5.
Ibid., pp. 4, 5. By weighting each galaxy by its luminosity rather than weighting giant and dwarf galaxies equally, Turner and Ostriker determine a mass to light ratio that also yields ri o o8. (Edwin L. Turner and Jeremiah P. Ostriker, ‘The Mass to Light Ratio of Late-Type Binary Galaxies: Luminosity-versus Number-Weighted Averages’, Astrophysical Journal 217 [1977]: 24–36.)
S. Michael Fall, ‘The Scale of Galaxy Clustering and the Mean Matter Density of the Universe’, Monthly Notices of the Royal Astronomical Society 172 (1975): 23p–26p.
David Eichler and Alan Solinger, ‘The Electromagnetic Background: Limitations on Models of Unseen Matter’, Astrophysical Journal 203 (1976): 1–5.
Field and Perrenod argue further that both observational and theoretical constraints (including limitations of the energy source needed to heat the gas, the observed deuterium abundance, and the lack of evidence for any clumping of gas clouds in intergalactic space, without which they could not persist) combine to make a cosmologically significant amount of hot intergalactic gas uncertain. (George B. Field and Stephen C. Perrenod, ‘Constraints on a Dense Hot Intergalactic Medium’, Astrophysical Journal 215 [1977]: 717–22.)
Richard I. Epstein and Vahe Petrosian, ‘Effects of Primordial Fluctuations on the Abundances of Light Elements’, Astrophysical Journal 197 (1975) 281–4.
See also Donald G. York and John B. Rogerson, Jr., ‘The Abundance of Deuterium Relative to Hydrogen in Interstellar Space’, Astrophysical Journal 203 (1976): 378–85.
Toulmin objects that it does not make sense to ask if the universe is an isolated system because since it is all there is, it cannot be said to be isolated from anything. (Stephen Toulmin, ‘Contemporary Scientific Mythology’, in Metaphysical Beliefs, ed. Alasdair Maclntyre [London: SCM Press, 1957], p. 36.) It is hard to believe Toulmin intends this argument to be taken seriously. For as Sturch points out, when we say a system is closed or isolated, we mean that there is no other system with which the system in question has an interchange of energy; this is obviously the case with the universe. (Sturch, ‘Argument’, p. 230.) For a refutation of Toulmin’s other objections, see Ibid., pp. 229–34.
See R. E. D. Clark, The Universe: Plan or Accident? 3d ed. (London: Paternoster Press, 1961), p. 19. Clark supports Newton’s argument as reasonable.
P. C. W. Davies, The Physics of Time Asymmetry (London: Surrey University Press, 1974), p. 109.
Beatrice M. Tinsley, ‘From Big Bang to Eternity?’ Natural History Magazine, October 1975, p. 103.
Richard C. Tolman, Relativity, Thermodynamics, and Cosmology (Oxford: Clarendon Press, 1934), p. 444.
Davies, Physics p. 188. Novikov and Zel’dovich are less generous, urging technical considerations again the probability of a ‘bounce’ in a Friedmann model. (I. D. Novikov and Ya. B. Zel’dovich, ‘Physical Processes near Cosmological Singularities’, Annual Review of Astronomy and Astrophysics 11 [1973]: 401.)
Novikov and Zel’dovich, ‘Processes’, pp. 401–2. (My italics.) Russian-speaking readers are also referred to Ya. B. Zel’dovich and I. D. Novikov, Relativistic Astrophysics (Moscow: Nauka, 1967).
and Ya. B. Zel’dovich and I. D. Novikov, Structure and Evolution of the Universe (Moscow: Nauka, 1973).
Davies also concludes that the cycles of an oscillating universe will gradually increase, and furnishes a diagram like that of Novikov and Zel’dovich. (Davies, Physics pp. 190–1.) These findings are also confirmed by P. T. Landsberg and D. Park, ‘Entropy in an oscillating universe’, Proceedings of the Royal Society of London A 346 (1975): 485–95. See also Gribbin, ‘Oscillating Universe’, pp. 15–16.
Adolf Grünbaum, Philosophical Problems of Space and Time, 2d ed., Boston Studies in the Philosophy of Science, vol. 12 (Dordrecht, Holland and Boston: D. Reidel Publishing Co., 1973), p. 262.
Anthony Kenny, The Five Ways: St. Thomas Aquinas’ Proofs of God’s Existence ( New York: Schocken Books, 1969 ), p. 66.
G. E. M. Anscombe, ‘“Whatever Has a Beginning of Existence Must Have a Cause”: Hume’s Argument Exposed’, Analysis 34 (1974): 150.
Ibid. See also Kenny, Five Ways pp. 66–8; Aziz Ahmad, ‘Causality’, Pakistan Philosophical journal 12 (1973): 17–24.
Frederick C. Copleston, A History of Philosophy, vol. 5: Hobbes to Hume (London: Burns, Oates & Washbourne, 1959), p. 287.
Bella Milmed, Kant and Current Philosophical Issues (New York: New York University Press, 1961), p. 45.
See Gottfried Martin, Kant’s Metaphysics and Theory of Science, trans. P. G. Lucas (Manchester: Manchester University Press, 1955
reprint ed., Westport, Conn.: Greenwood Press, 1974), p. 48. According to Martin, ‘In this empty time before the beginning of the world there was the passage of time, but no events.’ (Ibid.) Accordingly, the antithesis asks why the world began at a certain point in time, when all moments are alike. (Sadik J. al-Azm, The Origins of Kant’s Arguments in the Antinomies [Oxford: Clarendon Press, 1972], pp. 44–5.)
Whitrow, Time, p. 566; Whitrow, Philosophy, p. 32; G. J. Whitrow, ‘The Age of the Universe,’ British Journal for the Philosophy of Science 5 [1954]: 217.
Hackett, Theism pp. 286–7. I think that it is within the context of Trinitarian theology that the personhood and timelessness of God may be most satisfactorily understood. For in the eternal and changeless love relationship between the persons of the Trinity, we see how a truly personal God could exist timelessly, entirely sufficient within Himself. Most writers who object to a timeless, personal God consider God only subsequent to creation as He is related to human persons, but fail to consider God prior to creation. (Eg., Nelson Pike, God and Timelessness [London: Routledge & Kegan Paul, 1970], pp. 121–9.) The former would appear to involve God in time, but the latter would not, for if God is tri-personal He has no need of temporal persons with whom to relate in order to enjoy personal relationships—the three persons of the Godhead would experience perfect and eternal communion and love with no necessity to create other persons. Thus, the answer to the question, ‘What was God doing prior to creation?’ is not the old gibe noted by Augustine: ‘He was preparing hell for those who pry into mysteries’; but rather, ‘He was enjoying the fullness of divine personal relationships, with an eternal determination for the temporal creation and salvation of human persons.’ Why did God so determine? Perhaps to share the joy and love of divine fellowship with persons outside Himself and so glorify Himself; on the other hand, perhaps we lack sufficient information to answer this question. Once these temporal persons were created, God would then begin to experience temporal personal relationships with them.
Thomas Aquinas Summa theologiae 1a. 13. 7. See also John Donnelly, ‘Creatio ex nihilo’, in Logical Analysis and Contemporary Theism, ed. John Donnelly (New York: Fordham University Press, 1972), pp. 210–11.
R. G. Swinburne, ‘The Timelessness of God’, Church Qyarterly Review 166 (1965): 331.
This serves to effectively rebutt the objection of Julian Wolfe to the kalām cosmological argument. (Julian Wolfe, ‘Infinite Regress and the Cosmological Argument’, International journal for Philosophy of Religion 2 [1971]: 246–9.
The crucial premiss is, in Wolfe’s opinion, that an infinite time cannot elapse. He argues that this is incorrect because prior to causing the first effect, the uncaused cause existed for infinite time. Since the first event did occur, then an infinite time must have elapsed. But in the first place, Wolfe’s formulation of the argument is defective, for the contention is that an infinite number of events cannot elapse, not that an infinite time cannot elapse. The Newtonian could hold that if God is changeless prior to creation, then an undifferentiated, measureless, infinite time could elapse before the first event, but that an infinite temporal series of definite and distinct events could not elapse. Because the argument concerns events, not time, Wolfe’s analysis is inapplicable, since prior to creation there were no events at all. Second, if the relationist is correct, then an infinite time does not elapse prior to creation because time begins at creation. God is simply timeless before the first event. As Harris urges, ‘A persistent state of affairs which is not contrasted with any series of changes, either internal or external to it..., could not be conceived as enduring..., because there would be nothing (no lapse) through which it could be thought to endure.’ (Errol E. Harris, ‘Time and Eternity’, Review of Metaphysics 29 [1976]: 467.)
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Craig, W.L. (1979). Notes. In: The Kalām Cosmological Argument. Library of Philosophy and Religion. Palgrave Macmillan, London. https://doi.org/10.1007/978-1-349-04154-1_10
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