Appeared-to-Blogly

July 21, 2010

Performing Inverse Operations on an Actual Infinite “In the Real World”

Filed under: Metaphysics — camcintosh @ 11:25 am

Here is an objection to Craig-style arguments against the existence of an actual infinite “in the real world”. It is often said that inverse operations on transfinite numbers are prohibited. But this is not entirely true. As Nowacki explains, “inverse operations are prohibited for transfinite numbers in cases where performing the operation would yield a result that is not actually infinite.” (Kalam Cosmological Argument for God, p. 281, n.12. My emphasis).  So where m = the number of books in an infinite library, n = the number of odd-numbered books, and o the number of books numbered 4 or higher, the following is permitted:

(i)            (mn) = 0

while the following is not:

(ii)           (mo) = 4

because (ii) yields a finite number. If inverse operations yielding finite numbers were permitted, then it would follow that we get inconsistent results by subtracting the same number from m since n = o (both n and o are infinite), and this of course is why operations like (ii) are prohibited. But this leads me to think the following.

Craig’s point is that while inverse operations yielding finite numbers may be prohibited on paper, nothing would prevent us from performing them in the real world (say, by checking out a book from the library). But is this right? What would performing such an operation “in the real world” really amount to? What (ii) makes clear is that one would have to check out an infinite number of books to get a finite difference, not just any finite number of books. But then does Craig’s point follow that “nothing would prevent us from performing them in the real world”? Here’s something that might prevent us: having to perform an infinite task (check out an infinite number of books).

I’m not convinced that this objection works, but I don’t know the answer to it.

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3 Comments »

  1. Hi Chad,

    I’m not sure this is much of an objection:

    But then does Craig’s point follow that “nothing would prevent us from performing them in the real world”? Here’s something that might prevent us: having to perform and infinite task (check out an infinite number of books).

    But that is a problem for one who believes in the existence of actually infinite sets of things, not a problem for Craig. That is, were we to grant the notion that such sets could exist, then it would be possible to perform such an operation in the real world. But, then it would be possible to achieve absurd results.

    Comment by Reidish — July 25, 2010 @ 1:04 pm | Reply

  2. Thanks for your comment, Reidish.

    I’m not convinced my “objection” is a good one, but I’m not sure your reply quite does it justice.

    My point is that if we grant for the sake of a reductio that such sets could exist, then it would not necessarily be possible to perform such operations in the real world, as Craig assumes.

    As I see it, Craig identifies two absurdities that would result should an actual infinite exist in reality. The first absurdity is that of adding or subtracting any number to or from an infinite set and ending up with no more or less members than what the set originally had. This corresponds to operation (i) in the post. Call this the “same results” absurdity. The second absurdity is that of subtracting identical quantities from identical quantities and ending up with contradictory results. The contradiction is between operations (i) and (ii) in the post: same quantities, different results. Call this the “contradiction” absurdity. Craig emphasizes the contradiction absurdity as the real problem, as is evident in the following remarks:

    But in reality, one cannot stop people from checking out of a hotel if they so desire! In this case, one does wind up with logically impossible situations, such as subtracting identical quantities from identical quantities and finding nonidentical differences…It is, of course, true that every time one subtracts all the even numbers from all the natural numbers, one gets all the odd numbers, which are infinite in quantity. But that is not where the contradiction is alleged to lie. Rather the contradiction lies in the fact that one can subtract equal quantities from equal quantities and arrive at different answers. For example, if we subtract all the even numbers from all the natural numbers, we get an infinity of numbers, and if we subtract all the numbers greater than three from all the natural numbers, we get only four numbers. Yet in both cases we subtracted the identical number of numbers from the identical number of numbers and yet did not arrive at an identical result. In fact, one can subtract equal quantities from equal quantities and get any quantity between zero and infinity as the remainder. For this reason, subtraction and division of infinite quantities are simply prohibited in transfinite arithmetic—a mere stipulation which has no force in the nonmathematical realm. (Craig and Sinclair, “The KCA” in Blackwell Companion, pp. 111-112)

    But my point is that one only gets the contradiction absurdity by assuming one actually can subtract an infinite number in the real world (i.e., that one can actually remove an infinite number of books). But why think someone can actually do that? Well, suppose we modify the example to include an infinite number of people, each of whom check out one book starting with 4 and above. But now the problem is that if we had an infinite number of people, there should have been a one-to-one correspondence between persons and books so that no remainder would be possible upon removal; hence, no contraction.

    What I think this illustrates is that there is something even deeper awry with instantiated infinite sets than possibly to perform inverse operations with them.

    Comment by Chad McIntosh — July 28, 2010 @ 2:16 pm | Reply

  3. I take it back! You would get a contraction in the modified case! Supposing we had an infinite number of people each of whom correspond to a book in an infinite library, we could have one case where every odd-numbered person checked out their corresponding book and another case where every person checked out their corresponding book. Again, where m = the number of books in an infinite library, n = the number of odd-numbered books, the contraction absurdity follows from these two cases:

    (i) (mn) = 0

    (ii*) (mm) = 0

    because n and m are the same number.

    Comment by Chad McIntosh — July 28, 2010 @ 2:43 pm | Reply


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