Theology and Indeterminate Infinity
Apologists often claim that actual infinites are logically impossible. One of the arguments which they utilize to support this claim deals with subtracting quantities from infinite quantities. One example of this comes from Blake Giunta’s Belief Map:
Infinity minus an infinity yields logically impossible scenarios. Notably, one can take away identical quantities from identical quantities and arrive at contradictory remainders.
On the face of it, this claim appeals to our intuitive understanding of subtraction. If I were to claim that there exists some Integer, , such that and , then we stumble upon the contradiction that . Subtracting identical quantities from identical quantities should yield identical results.
Belief Map further cites the work of William Lane Craig and James Sinclair for a particular example of how infinities can yield such contradictions.
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.
Unfortunately, this example is not quite so logical as its authors would like it to be. The reason for this is that they don’t seem to have a very good grasp of the mathematics which they are trying to discuss. The final statement of this paragraph makes a blatantly false claim: “For this reason, subtraction and division of infinite quantities are simply prohibited in transfinite arithmetic.” It is not the case that subtraction and division are “simply prohibited in transfinite arithmetic.” Rather, subtraction and division are not defined in these cases. This is a very important distinction.
Imagine, for a moment, that I have asked you to perform the calculation “seven vike four.” You would rightly ask me what ‘vike’ means? How does one ‘vike’ two numbers? Until there is a definition for viking numbers, it is not possible for anyone to vike those numbers. It is not that we are prohibited from viking numbers. It is simply that the phrase is undefined and, therefore, meaningless.
Similarly, let’s say that I asked you to perform the calculation “seven divided by zero.” Now, despite the fact that division is well-defined for almost all numbers, you may recall from your grammar school math classes that division is not defined when dividing by zero. In this case, once again, we are not prohibited from dividing by zero. Dividing by zero is simply not a meaningful concept.
Similarly, in the field of mathematics known as “transfinite arithmetic,” subtraction and division are not defined operations. Despite the similar name, “transfinite arithmetic” is not the same arithmetic which children learn in grammar school. It is a collection of rules for manipulating sets. It doesn’t matter that these operations are well-defined for other areas of mathematics– they are not defined for transfinite arithmetic. Again, these operations are not prohibited, as Dr. Craig claims; they are simply meaningless.
But doesn’t Dr. Craig’s example cite clear ways of subtracting from infinite sets? How can it be undefined if there exist ways to do it?
We are not saying that one cannot choose and eliminate any arbitrary number of elements from a given infinite set. Rather, what we are saying is that there is no general definition which can account for all cases of subtraction between two different sets in transfinite arithmetic. The manner in which one eliminates the elements from an infinite set can change the result, and therefore there is no single, correct way to do it. So, contrary to Dr. Craig’s claim, it is not the case that we are prohibited from removing elements of an infinite set, in mathematics; we simply do not have a general operation for subtracting any given set from any other given set, in transfinite arithmetic. Special cases can always be considered by themselves.
When different special cases of performing an operation can be done in order to produce different results, we say that the more general statement is indeterminate in that case. For example, as we mentioned earlier, division by zero is not defined in the arithmetic which we all learned in grammar school. However, when mathematicians see the statement they immediately recognize it as an indeterminate form. Depending on the special case, the mathematical phrase can be equal to any number.
Take, for instance, the phrases and . In both cases, these limits resolve to . However, a simple application of L’Hopital’s rule tells us that while . I can just as easily show cases where is equal to 5 or to 17 or to π or to any other number. This is not inconsistent or contradictory, because is not a number; it is a mathematical phrase.
Similarly, we can imagine two numbers and which are both infinite. The phrase is indeterminate. Depending on the particular case, could result in any number. This is not a contradiction. This is not an inconsistency. This is simply an indeterminancy.
The other examples which apologists usually bring out in support of their opposition to actual infinites tend to fall to similar problems. They are misunderstandings or misstatements of the underlying mathematics. The simple fact of the matter is that the idea that actual infinities are logically and metaphysically possible has not even been controversial in the Philosophy of Mathematics for a century. As such, when an apologist makes pronouncements upon the nature of mathematics, we should be more than a little skeptical.