And so beguile thy sorrow. - William Shakespeare, Titus Andronicus
The Infinite! No other question has ever moved so profoundly the spirit of man; no other idea has so fruitfully stimulated his intellect; yet no other concept stands in greater need of clarification than that of the infinite. - David Hilbert
The is a response to Wes Morriston's "Craig on the actual infinite" (2002).
1. Absurd Implications - Morriston charges Craig with arguing that 'there is no possible world' in which such absurd implications follow. I'm not sure that's the case. There may be a possible world in which there is a library with infinitely many books (infinitely many blue books and infinitely many red books). It just might not be feasible. Feasibility just is metaphysical possibility. Morriston is no doubt aware of these modal distinctions. I just feel the need to point that out. This distinction serves to distinguish feasibility from strict logical impossibility and nomological impossibility. The rest of Morriston's argument serves to motivate a possible world in which 'the way in which the number of elements in the set interacts with other features' of the examples Craig describes. But even if Morriston succeeds in his project, it won't serve to motivate feasibility. Many of the ways Morriston goes about motivating such a possible world involves the fixity of the past (I've addressed this worry in previous blogs in terms of counterfactual conditionals and the contingency with which such conditionals interact with the potential, mental annihilation of fixed elements, as well as how this is explored by supposing what would follow given the truth of an impossible antecedent in the aforementioned conditional).
Morriston explores what happens if there's an isomorphic correlation between all the natural numbers and such numbers printed on the spines of all the books in the infinite library. You walk into the library with another book! How could one do this? Renumbering all the books ad infinitum? You can't do that (argues Craig) because 'all' the numbers are already there: 'addition seems impossible'. Morriston makes a curious distinction between books being numbered (from 0 to ∞, indefinitely) 'before any new books' are added and the fixity of the numbers being on the spines. Morriston argues that Craig's worry (adding new books violates the 'initial conditions' of the example) is unfounded because 'the books were not initially assigned to the natural numbers, beginning with zero.' This is puzzling. Morriston himself paraphrased Craig by saying, "If an infinite library existed, its books could be numbered sequentially, beginning with 0, in such a way that every natural number is printed on the spine of exactly one book." This seems to me to imply (contrary to Morriston's argument undermining Craig's worry) that the books 'are' initially assigned the natural numbers, beginning with zero! And so Craig's complaint (that the initial conditions of the example are violated by adding new books) seems to me to be well-founded.
Morriston's objection to Craig's comparing the infinite library with the series of natural numbers (you can't add another book just as you can't add another integer) amounts to little more than underdeveloped (but interesting!) theses. He distinguishes between an infinite number of books, the set of all possible books, and the set of all actual books. The last descriptive just seems misplaced (I think Craig would admit that the set of all actual books won't be infinite). It's the second descriptive I find interesting. Are there more members in the set of all possible books than the members in the set of books in an infinite library? Morriston just seems to assume that the number of members in the two sets will at least be dissimilar. But why assume this? Suppose there are an infinite number of possibilities. If so, then each possibility could be put into a one-to-one correspondence with the series of natural numbers, right? If so, I'm confused as to why Morriston feels the need to underline the distinction between the infinite library and the set of all possible books. Morriston just seems to assert there's no reason to think that the correlation of the books with the natural numbers implies the former's immutability. Contrary to Morriston's protestations, even God couldn't add a book to the library. Morriston seems to think that 'we' couldn't add a book because we have limited power (we'd never complete the process). But to say that we could if we had the power (if we say that God could do it) is to just beg the question against the conclusion of Craig's argument.
The infinite-counter: Morriston addresses the alleged absurdity of 'counting down' from eternity-past to the present. He makes a distinction between counting all the numbers and counting infinitely many numbers. That's a valid distinction. It's just that the same collections exemplify both properties at one time and one property at another time. Each past day can serve as the counter's finally ending in 'zero' (all the numbers), and each past day can serve as the counter's counting an infinite number of numbers (an infinity of numbers). Once we get to tomorrow, tomorrow's 'today' will be the new 'zero'. Today's 'today' will involve a collection whose front edge will include all the negative numbers; tomorrow's 'today' will designate today's collection as not having all the negative numbers, if tomorrow's 'today' is 'zero' and today's 'today' is no longer 'zero', but '-1'. The infinite library has all the numbers up to 'zero', which corresponds to the numerals inscribed on the books' spines. So, Morriston's distinction doesn't seem to me to subvert Craig's point: bringing another book into an infinite library in which the books correlate with 'all' the natural numbers is absurd. This isn't because there are an infinite number of books; it's because there is an infinite number of books that correlate with all the natural numbers. As far as I can see, Morriston just assumes that 'the numbers on the spines of the books' 'can at least in principle be changed.' Further, his supposition that the series of past 'countings' differs from the presently inscribed numerals (since the countings are past and can't be changed) is misconceived. The inscribed numerals may be presently inscribed, but their present inscription was inscribed 'in the past'. One wonders why the same rules don't apply.
Next, Morriston addresses 'inverse mathematical operations', the aspect of an actual infinite (its metaphysical existence) that I find to be the toughest pill to swallow. Morriston alleges Craig's 'infinite library' (when people 'check books out') doesn't involve the subtraction of numbers. Of course! In the thought experiment, the numbers serve as rigid designators of the books (evidenced by the spinal numerals). Hence, with the operations, we're not subtracting numbers, we're subtracting what the numbers are rigidly designating. Morriston resides in the mathematical realm and thinks he's safe from Craig's absurdities. My issue with inverse operations with infinite quantities is that the operation itself is meaningless. Its being meaningless motivates Craig's claim that the operation is prohibited in transfinite arithmetic. Morriston is correct that once the variables are defined (e.g. Aleph-null), you can do to the quantities what you would instinctually try to do with finite quantities and get the quantitative results you're left with. The problem is that once you do this (assuming that subtraction is even meaningful), you can get an infinite number of 'differences' involving operations with identical quantities. Hence, Morriston's notion that you get 'exactly the same remainder' is bewildering to me. Morriston acknowledges that Cantor leaves subtraction undefined. The bigger issue though is that Morriston critiquing Craig's thought experiment under the assumption that numbers are the objects being subtracted. Morriston has to traverse the chasm that yawns between mathematical legitimacy and metaphysical possibility. That's the semantic function of the rigid designators.
Morriston thinks Craig thinks there is 'something wrong with infinite sets'. Not at all (for the reasons spelled out above). Craig subscribed to the mathematical legitimacy of infinite sets. Morriston does think that the idea of eliciting the absurdities by traversing the chasm noted above is 'badly confused'. Morriston distinguishes between 'checking out books' and 'subtracting books'. Presumably, the latter operation is reserved for numbers and the former act is reserved for metaphysically instantiated things. If inverse operations with infinite sets are prohibited (because undefined), checking books out isn't technically an inverse operation and so isn't undefined. Admittedly, this is an interesting distinction, but I see no reason to accept it. If we have a finite set (the set of 4 dogs, say), and I subtract two members from that set, then I'll have two members in the set left. I'm subtracted the quantity 'two' from the set of 4 dogs and was left with 2 dogs. The mathematical operation covaries with the metaphysical existents. I see no reason why the same wouldn't apply with the counterpossible supposition of actually infinite sets. Morriston's other distinction (adding/subtracting numbers from sets vs. constructing new sets) just reiterates the impotent point.
I, therefore, see no reason so far for thinking that the infinite library doesn't elicit the kinds of absurdities Craig thinks are lurking there.
2. A Deeper Analysis - Morriston thinks it difficult to interpret Craig when Craig argues that the principle of correspondence (PoC) and Euclid's Maxim (EM) apply to finite collections but not actually infinite collections, because, if they did (per impossible), absurdities would result. Morriston alleges that Craig argument that EM only applies to finite collections is guilty of circularity. I don't see it. It seems glaringly intuitive. Craig seems to use a reductio in restricting EM to finite collections. Assume EM applies to infinite sets. If it does, then absurdities follow. Therefore, EM can't apply to infinite sets. However, EM applies to metaphysically instantiated sets because such sets are finite. Such sets are finite because, if they were infinite, then absurdities follow. Therefore, EM applies to finite collections, including the finite collections that designate metaphysically instantiated sets. At this point, Morriston once again engages in motivating a thesis that serves to be his modus tollens against Craig's modus ponens. Morriston makes a distinction between one set S being larger than another set S*, not by virtue of having a larger cardinality, but in the sense of S having elements that S* doesn't have. Well, that 'latter sense' is exactly why Craig thinks that S would be larger than S*. Morriston just places a distinction between the two without warrant. That's precisely the reason why EM isn't applied to infinite sets: greater size wouldn't imply a difference of cardinality (because it doesn't, absurdities result if PoC relates the members of such sets to the realities of the thought experiments).
3. Existence in Reality vs. Existence in God's mind - This was back when certain things Craig was saying seemed to indicate that Craig was a divine conceptualist regarding abstract objects. I'll reserve comment here since it's now evident that Craig is an anti-realist regarding abstract objects. I'll just note one thing. Morriston expresses puzzlement regarding Craig not having a problem with the actual infinite in mathematics. If there are contradictions involving inverse operations, shouldn't mathematicians be concerned? This is easy to answer. Mathematicians don't divide or subtract in transfinite arithmetic! The contradictions result from the inverse operations. Such operations are prohibited.
Some summaries of thoughts:
I'll skip the section on Craig's former endorsement of conceptualism. I've already addressed Morriston's concerns regarding God's knowledging being non-propositional. Morriston mentions the Alston paper we saw in Alex's blog (Craig's List), and Alston's citation of Bradley is reiterated. I've also addressed the ostensible inconsistency between affirming propositional and non-propositional knowledge regarding God. Also addressed were Morriston's concerns regarding an infinite quantity of propositions or truths about the future. Morriston begins to develop his idea of the future not being potentially infinite because the future is endless. Also addressed in past blogs is Morriston's objections to Craig's appropriation of 'tense' to keep the endless future from being an actual infinite, and the idea that 'tense' indicates the possibility of 'adding' to infinite collections (God's omniscience - thought Plantinga has argued that God's omniscience shouldn't be understood in terms of sets and collections - see here) without absurdity, etc. Regarding space being continuous, if space is discrete, then space is susceptible to infinite divisibility. Morriston argues that even if space is continuous, space is susceptible to infinite divisibility. This relies on the same line of argumentation that motivates the endless future being an actual infinite. The divisions latent in continuous space are 'there', even if there won't be enough time for dividers to uncover them. But that misses the point. The divisions that won't ever be uncovered aren't 'there', and God's omniscience wouldn't encompass them (for reasons given in previous blogs): there aren't an infinite number of truths or propositions. Lastly, Euclid's notion of continuous extension could be construed by Craig as a potential infinite. Morriston's point that the infinitely extended line needs 'somewhere for it to go' is completely misconceived. It elicits an absurdity by imposing Euclid's mathematical modeling onto reality. A mathematical construct doesn't need anywhere to go. Further, if Euclidean space had no end, it doesn't follow that such a space is infinite (there are all kind of topologies that can illustrate this: a torus, for example). Morriston's citation of possible worlds in which 'straight lines never meet' is irrelevant - Craig is talking about that subset of possible world's that are metaphysically possible. Craig dismisses the possibility of Euclidean space and since Morriston's only objection to Craig dismissal is that Morriston has addressed Craig's arguments, it follows that if Morriston's objections to Craig can be answered, and so Craig's dismissal can be motivated.
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