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J. Howard Sobel devotes seventy pages of his wide-ranging analysis of theistic arguments to a critique of the cosmological argument.1 The focus of that critique falls on the argument a contingentia mundi; but he also offers in passing some criticisms of the argument ab initio mundi, or the kalam cosmological argument.
Sobel provides the following statement of the argument:
- Everything that begins to exist has a cause of its existence.
- The universe began to exist.
- Therefore, the universe has a cause of its existence [that did not begin to exist].2 [End Page 565]
Sobel will accept the causal premiss (1) only if 'begins to exist' means 'has a first instant of its existence,' and he disputes the arguments and evidence for (2).
Traditional proponents of the kalam argument sought to justify (2) by means of philosophical arguments against the infinity of the past, while contemporary interest in the argument arises from the empirical evidence of physical cosmology for the truth of (2). Both of these considerations arise in the course of Sobel's discussion of the possibility of an infinite temporal regress of events, as he weighs Thomas Aquinas' dictum that 'in efficient causes it is not possible to go to infinity.'3 Now, as Sobel recognizes, Aquinas applied that dictum only to simultaneous, essentially ordered causes, not to temporally successive, accidentally ordered causes. Nonetheless, Sobel considers whether Aquinas should have rejected the possibility of an infinite past on the basis of Thomas's own arguments against an infinite causal regress.
I Mathematical Finitism
Consider, first, Aquinas' rejection of the possibility of an infinite multitude. If an actually infinite number of things cannot exist, then it would seem that there cannot have been an actually infinite number of past causes.4 Aquinas rejected the possibility of an infinite multitude because multitudes are differentiated by numbers, and there are no infinite numbers. He argues, 'Now the species of multitude are to be reckoned by the species of numbers. But no species of number is infinite; for every number is multitude measured by one. Hence, it is impossible for there to be an actually infinite multitude....'5 Here Aquinas seems to take an intuitionist line: the reason there are no infinite multitudes in actuality is because there are no infinite numbers. Aquinas appears to deny even mathematical existence to the infinite, countenancing only the natural numbers constructed by mathematical induction.
Sobel's response to this argument is curious. He waves aside Rudy Rucker's Cantorian response that there are, in fact, infinite numbers by means of which various multitudes can be distinguished (perhaps realizing, pace Rucker, that Aquinas' constructivist challenge cannot be met [End Page 566] by merely 'exhibiting a theory of infinite numbers'6 ). Instead, Sobel attacks the claim that associated with every multitude is a number. He wants to say that there can be real multitudes of things with which no number is associated. He writes,
it is not after all obvious that there needs to be for every multitude a "number." "What is in the name 'number'?" Suppose ... we were ... [to] take back the word to its "home" in the finite, the numbers measured by one, the numbers 1, 2 = 1 + 1, 3 = (1 + 1) + 1, and so on. "Would not an erstwhile infinite multitude by any other name ... remain as multitudinous?" "Would not an erstwhile infinite multitude that is not finite remain a multitude?!" The species of multitudes are not necessarily reckoned by the species of number. There is prima facie the possibility of "multitudes beyond number." Whether it is realized depends on what multitudes there are and what numbers there are. Cantor himself believed in the existence of absolutely infinite and unlimited multitudes that are not subject to further increase. It is a good guess that he would have said, or proposed, that at least these multitudes...