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