1 Jordan Howard Sobel, Logic and Theism: Arguments for and against Beliefs in God (Cambridge: Cambridge University Press 2004), chaps, v-vi. For his critique of the kalam cosmological argument, see pp. 181-9,198-9, and relevant endnotes.

2 It is odd that Sobel adds the parenthetical phrase to his formulation of the argument, since it renders the argument plainly invalid. The practitioner of kalam will need to offer further argument to justify his belief that the cause of the universe is itself uncaused.

3 Summa theologica 1.2.3.

4 Sobel does not address Aquinas’ conviction that an infinite multitude of past causes would be a merely potential infinity, since all the causes do not exist in actuality.

5 Summa theologica 1.7A

6 Sobel, Logic and Theism, 185, citing Rucker.

7 Ibid.

8 Ibid.

9 On this notion see Georg Cantor, ‘Letter to Dedekind/ 28 July, 1899, in From Frege to Godel: A Sourcebook in Mathematical Logic, 1879-1931, ed. Jean van Heijenoort, Sourcebooks in the History of the Sciences (Cambridge, MA: Harvard University Press 1967), 114.

10 Sobel, Logic and Theism, 187.

11 Ibid., 29.

12 Ibid., 560.

13 Ibid., 202.

14 A rebutting defeater attempts to show that the disputed claim is false. An undercutting defeater aims to show that the disputed claim has not been shown to be true.

15 Sobel, Logic and Theism, 185

16 Ibid., 188-90. For another example, consider David Hume's comment on something's coming into being without a cause: ‘But allow me to tell you that I never asserted so absurd a Proposition as that anything might arise without a cause: I only maintain'd that our certainty of the Falshood of the Proposition proceeded neither from Intuition nor Demonstration; but from another source’ (David Hume to John Stewart, February 1754, in The Letters of David Hume, 2 vols., ed. J.T. Grieg [Oxford: Clarendon Press 1932], 1: 187).

17 Al-Ghazali, Tahafut al-Falasifah, trans. S.A. Kamali (Lahore: Pakistan Philosophical Congress 1963), 20

18 See Waclaw Sierpinski, Cardinal and Ordinal Numbers, Polska Akademia Nauk Monografie Matematyczne 34 (Warsaw: Panstwowe Wydawnictwo Naukowe, 1958), 146.

19 Sobel, Logic and Theism, 186-7

20 Sobel does claim that Aquinas could have been persuaded to give up (i) or (ii), since he believed in the reality of the natural numbers and so could have felt compelled to admit that there is an infinite multitude of them. But that fails to reckon with Thomas's view on the nature of abstract objects. Aquinas like other medievals embraced the Augustinian view that Platonic objects were, in fact, not mind-independent realities but rather the divine ideas. Moreover, since Aquinas held God's cognitive state to be absolutely simple, He does not really have a multiplicity of ideas. Rather the plurality of divine ideas is the representation we finite knowers are constrained to make of Gods’ simple intuition of all truth. The implication would seem to be that numbers have only a conceptual reality as they are cognized by finite knowers. Thus, we have in Aquinas a sort of intuitionistic constructivist view of mathematical objects.

21 Sobel, Logic and Theism, 187

22 It will not do, in order to avoid the contradiction, to assert with an anonymous referee that there is nothing in transfinite arithmetic that forbids using set difference to form sets. Indeed, the thought experiment assumes that we can do such a thing. Removing all the guests in the odd-numbered rooms always leaves an infinite number of guests remaining, and removing all the guests in rooms numbered greater than four always leaves four guests remaining. That does not change the fact that in such cases identical quantities minus identical quantities yields nonidentical quantities, a contradiction.

23 Summa theologica 1.46.2

24 Sobel, Logic and Theism, 182

25 In this and other ways the kalam cosmological argument is intimately bound up with a tensed theory of time. For discussion see my The Tensed Theory of Time: A Critical Examination, Synthese Library 293 (Dordrecht: Kluwer 2000); The Tenseless Theory of Time: A Critical Examination, Synthese Library 294 (Dordrecht: Kluwer 2000).

26 For a review see my ‘Naturalism and Cosmology,’ in Analytic Philosophy without Naturalism, ed. A. Corradini, S. Galvan, and J. Lowe (London: Routledge 2005) and my ‘Time, Eternity, and Eschatology,’ in Oxford Handbook on Eschatology, ed. J. Walls (Oxford: Oxford University Press, forthcoming).

27 Sobel, Logic and Theism, 198

28 Ibid.

29 Ibid.

30 Ibid., 198-9.

31 Stephen Hawking and Roger Penrose, The Nature of Space and Time, The Isaac Newton Institute Series of Lectures (Princeton, NJ: Princeton University Press 1996), 20

32 Paul Davies, About Time (New York: Simon & Schuster 1995), 132

33 I owe this suggestion to an anonymous referee. I encountered a similar suggestion from Quentin Smith on the occasion of the forum ‘Science and Religion/ University of California, Santa Barbara, January 30, 2004, featuring William Lane Craig, Richard Gale, Alvin Plantinga, and Quentin Smith, available in DVD format through www.veritas-ucsb.org. See also Adolf Griinbaum, ‘A New Critique of Theological Interpretations of Physical Cosmology/ British Journal for the Philosophy of Science 51 (2000) 16-17.

34 It will depend crucially on one's theory of quantum gravity. A typical approach to marrying quantum theory to General Relativity involves describing the evolution of spacetime as a path integral (a sum over all possible paths) in superspace, which is a space of points respresenting three-dimensional configurations of the universe. The points of this configuration space can be regarded as instantaneous states or even as instants, but the fact that in the quantum theory one has a path integral, rather than a single path, makes it impossible to ‘stack’ these instants into a unique history constituting a spacetime. So eventually the dividing process envisioned by Sobel becomes ill-defined in the theory. The fact that the first split-second of the universe's existence as measured in cosmic time is not resolvable into a unique sequence of ever briefer states is not inconsistent with there being a first second of its existence. I am grateful to Donald Page for an accurate articulation of this concern. 35 See Richard Sorabji, Time, Creation, and the Continuum (Ithaca, NY: Cornell University Press 1983), 403-21.

36 Sobel, Logic and Theism, 200

37 Ibid., 193-4

38 On the personhood and other attributes of the transcendent cause of the beginning of the universe see my and Paul Copan's Creation out of Nothing (Grand Rapids, MI: Baker 2004), 252-4.

39 My thanks to J. Howard Sobel for his kind responses to my interpretive questions concerning his text. Thanks are due as well to an anonymous referee for constructive criticisms.