Friday, January 29, 2021

James Fodor vs. William Lane Craig on Counting to Infinity and the possibility of forming an actual infinite

  

Counting to Infinity

 

The most fundamental argument Craig provides against the possibility of forming an actual infinite by successive addition is that, however, long one ‘counts’ or continues to accumulate, there will still always be an infinite amount to go. For example, the difference between one and infinity is the same as the difference between any finite number and infinity – namely infinite. As such, the amount remaining to count is never diminished, and no progress towards reaching infinity is ever made. Another way of putting this is that every finite number is succeeded by another finite number, so whenever we start from a finite beginning and add a finite increment, we must get a finite result. There is simply no way to turn a finite series into an infinite series by adding finite quantities to it.

 

The problem with this argument is that it only establishes that one cannot turn a finite set into an infinite set by successive addition, not that it is impossible for an infinite set to be formed through successive addition. In Craig’s examples it seems that he implicitly assumes the existence of some sort of starting point which is then removed to an infinite distance. Craig rejects the notion that he assumes an infinitely distant starting point, retorting that:

 

“It is surprising that a number of critics, such as Mackie and Sobel, have objected that the argument illicitly presupposes an infinitely distant starting point in the past and then pronounces it impossible to travel from that point to today . . . But, in fact, no proponent of the kalam argument of whom we are aware has assumed that there was an infinitely distant starting point in the past.”

 

Nevertheless, I think it is clear that Craig’s examples do in fact rely on such an assumption. First, he states that):

 

“Given any finite number n, n + 1 equals a finite number. Hence ℵ₀ has no immediate predecessor.”

 

By this Craig means that the infinite value ℵ₀ can never be ‘reached’ by adding to any finite number, since there is no number x such that x + 1 = ℵ₀. This is true, but only establishes the impossibility of attaining an infinite result by adding to a finite starting point. He then argues that the result still holds even if infinite time were somehow available to perform the addition, since:

 

“Regardless of the time available, a potential infinite cannot be turned into an actual infinite by any amount of successive addition since the result of every addition will always be finite. One sometimes, therefore, speaks of the impossibility of counting to infinity, for no matter how many numbers one counts, one can always count one more number before arriving at infinity. One sometimes speaks instead of the impossibility of traversing the infinite. The difficulty is the same: no matter how many steps one takes, the addition of one more step will not bring one to a point infinitely distant from.”

 

Here again, Craig speaks of adding and counting, operations which require starting values to act upon. Addition is only possible if we have a number that we are adding to, and that number in this instance must clearly be finite, as otherwise we would already have the actual infinite that Craig is saying cannot be formed. So once again, all that Craig’s example shows is that adding a finite number to another finite number will never yield an infinite number, which is not in dispute. What Craig needs to show is not that adding two finite numbers always results in a finite result, but that it is impossible for an actual infinite to form by successive addition.

 

Even when he explicitly turns his attention to the question of whether it might be possible to form n actual infinity “by never beginning but ending at a point, that is to say, ending at a point after having added one member after another from eternity,” Craig still falsely assumes the existence of an infinitely distant beginning. This is evident from his remark:

 

“If one cannot count to infinity, how can one count down from infinity? If one cannot traverse the infinite by moving in one direction, how can one traverse it by moving in the opposite direction?”

 

The problem with this analogy is that counting is an operation that requires a beginning. One cannot just ‘count’; one always has to count by starting somewhere. Hence Craig finds that in order to apply a counting analogy to the series of temporal events, he must use absurd phrases like ‘count down from infinity’, as if infinity were a number from which we could commence counting. As Craig says, infinity has no successor or predecessor, so it is simply impossible to count either up or down ‘from infinity’. The problem thus has nothing to do with what direction we imagine counting in. Rather, the problem lies in Craig’s insistence on using the language of counting when such terms are only applicable if we have something finite to start out count with. In the case of forming an actual infinite by successive addition, however, not only is there no finite starting point, but no starting point at all. Counting analogies, therefore, simply are not applicable.

 

Craig’s fundamental difficulty seems to be that he insists on describing or conceptualising the process of forming an actual infinite by successive addition as literally an additive or counting process, which must start somewhere and then count up. This is clearly not how infinite time could work, as Craig’s examples show. Rather, a better way of describing how infinite time would operate is that there exists a process of continuous and eternal temporal becoming. Each moment continues from the previous moment and inexorably gives way to the next moment in time, a state of affairs which has always been (i.e. has been at every past moment of time). For every past moment, there was a moment of time before that. Time has always been passing, each moment giving way to the next, for an infinite length of time that has no beginning. It is this conception of a beginningless past that Craig needs to refute, not the absurd notion that involves counting up from some starting point. (James Fodor, Unreasonable Faith: How William Lane Craig Overstates the Case for Christianity [Hypatia Press, 2018], 88-91)

 

Further Reading:

 

Blake Ostler, Kalam Infinity Arguments and the Infinite Past