In this post I will consider two arguments which have at times been brought up in connection with the Kalam Cosmological Argument (KCA), which I will call the "subtraction argument" and the "argument from traversing an infinite," the former of which I have heard from Dr. William Lane Craig. The KCA goes as follows:
1. Whatever begins to exist has a cause.
2. The universe began to exist.
3. So the universe has a cause.
The arguments in question are designed to defend the second premise, which is presumably implied by the past's being finite. I should note that I think the second premise is true and there is strong evidence in favor of its truth. Alexander Pruss has given an excellent argument here, to which I have heard no compelling reply. I also think there is very strong scientific indication of the premise's truth, which Craig has adequately demonstrated. I just don't think these two arguments demonstrate its truth.
The "argument from traversing an infinite" goes something like this:
1*. If the past were infinite, one would have to cross an infinite temporal distance to get to the present moment.
2*. If one had to cross an infinite temporal distance to get to the present moment, then one could not get to the present moment.
3*. So if the past were infinite, then one could not get to the present moment.
4*. But we are at the present moment.
5*. So the past is not infinite.
The argument requires some unpacking. First of all, to say the past is finite is to say there was a beginning of time, and to say the past is infinite is to say there was no beginning of time. Second, 'temporal distance' means the length of time between one moment and another. There is a perfectly good way to define finite temporal distance. If we take our measure of time as a second, we can assign the current time the number 0, the time one second ago -1, the time two seconds ago -2, and so forth. To find the temporal distance from one time t1 to another t2, we take the number assigned to t1 and the number assigned to t2, and take the absolute value of the difference between the two. For instance, take the time 1000 seconds ago. To find the temporal distance from that time to the present time you take the absolute value of -1000 minus 0, which is of course 1000 seconds. Pretty simple.
However, problems begin to arise when we start to talk about an "infinite temporal distance." This phrase is ambiguous, and depending on which interpretation of this phrase we take it will either cause problems for premise 1* or for premise 2*. First, the phrase could mean something analogous to the way finite temporal distance has been defined above. However, infinity is not a real number, so you simply cannot define an infinite temporal distance the same way as above. There is no number "-infinity" from which you can subtract, say, -5. So if this is what is meant, then premise 1* appears to be false, since no real sense can be given to an infinite distance in this way.
On the other hand, crossing an infinite temporal distance could just mean that the set of all the numbers assigned to the seconds is infinitely large. This makes perfectly good sense of the phrase, but then in that case it is not clear why premise 2* is true. As Thomas Aquinas points out, there being an "infinite temporal distance" in this sense is perfectly consistent with all the temporal distances from the past to the present being finite, where "temporal distance" is defined as it was earlier:
"Passage is always understood as being from term to term. Whatever bygone day we choose, from it to the present day there is a finite number of days which can be passed through. The objection, however, is founded on the idea that, given two extremes, there is an infinite number of mean terms." [ST Ia q.46 a.2]
So for instance, the distance from the present to one second ago is 1 second, the distance from the present to two seconds ago is 2 seconds, etc. and so on forever and ever back into time. Hence, no matter how far you go back in time, the distance in the way I've defined above from any given past moment to the present will be finite, and thus you will only have to cross a finite number of seconds to get to the present moment. But of course any finite number of seconds can at least in principle be crossed; hence, premise 2* is false.
So much for the "traversing an infinite" argument then. The "subtraction argument" goes something like this:
1'. If the past were infinite, then an actual infinity would be possible.
2'. If an actual infinity were possible, then one could perform subtraction on infinities.
3'. But if one can perform subtraction on infinities, then one will get contradictory results.
4'. So if the past were infinite, then one would get contradictory results.
5'. So the past is not infinite.
This seems to be one of the arguments William Lane Craig gave in his debate against Peter Millican. Let me first point out an ambiguity in the phrase "actual infinity," after which I'll assess the argument under each interpretation. Here are two possible meanings of the phrase "actual infinity":
(ACT1) An actual infinity exists just in case for some time, at that time there exist distinct concrete objects such that the size of the set containing all and only them is infinite.
(ACT2) An actual infinity exists just in case there is some set containing only distinct concrete objects whose size is infinite.
Some elaboration is in order. First off, both definitions presume when speaking of actual infinities that we are dealing with concrete objects. While my understanding is that Craig does not believe there are infinitely many numbers (he is a nominalist), presumably his argument doesn't presuppose this view; Craig only wants to rule out the possibility of infinitely many concrete objects. Now as for the definitions themselves, the difference between (ACT1) and (ACT2) is that in (ACT1) you only have an actual infinity when all the concrete objects exist at the same time. In (ACT2) you could have finitely many objects at t2, finitely many at t1, and so forth, yet if you take a set containing concrete objects from different times, and the times go back to infinity, you will still have an actual infinity. So both of these definitions make fine sense. However, the assessment of the argument will depend on which interpretation we take.
Let's deal with the first definition, (ACT1). Given our definition of actual infinity in (ACT1), premise 1' does not appear to be true, or at least not obviously true. It is consistent with holding that the past is infinite that at each time there are only finitely many concrete objects. And if you believe only objects in the present moment exist, then 1' is definitely not true. The fact that there were objects at each time in the eternal past in no way implies an infinite collection of simultaneously-existing objects.
My main concern is with 2' and 3' though. Take 2', since it is also ambiguous to a certain extent. The problem is it is not immediately clear what is meant by "perform subtraction on infinities." Craig acknowledges that the ordinary operation of subtraction is not defined for "infinity". If he did mean this, i.e. the ordinary operation of subtraction, then clearly 3' would be true but 2' would have no support at all. So it cannot mean the ordinary operation of subtraction.
Now, Craig gives us an example to support 2' and 3', and this may help us understand what he means. Suppose we have an infinite number of coins. Then we can take away all the coins except three of them. And in this sense we can be said to perform subtraction on infinity, i.e. taking away some number of things from an infinitely large collection. This definition of performing subtraction on infinities makes sense. But then why is 3' true? Craig says, considering our infinite number of coins, that you can take away infinitely many coins and be left with 3 coins, and thus infinity minus infinity = 3; but you can also take away infinitely many coins and be left with 2 coins, and thus infinity minus infinity = 2; hence, 2 = 3, which is our contradiction.
The problem with this argument is that it runs on an equivocation: We agreed that we are not using "performing subtraction" or "minus" in the normal sense of the arithmetical operation, since this just makes no sense. So "infinity minus infinity = 3" must simply be shorthand for saying "taking away infinitely many objects from an infinite collection leaves us with 3", and similarly with "infinity minus infinity = 2." But then if "2 = 3" means that 2 is identical to 3, then it certainly does not follow that 2 = 3; all that follows is that you can take away an infinite number of things and be left with 3, and also take away an infinite number of things and be left with 2, and this is certainly not a contradiction! It only looks like a contradiction when we are illicitly inferring "2 = 3", as if the phrase "infinity minus infinity = 2" were using "minus" and "=" in the same way as "5 minus 3 = 2." It would be like if I had infinitely many pennies and dimes, and I said, "infinity minus infinity = a penny, infinity minus infinity = a dime, so a penny = a dime." Clearly I am making an illicit inference here, and for the same reason Craig's argument makes an illicit inference as well.
Now, this whole time I have been working under the assumption that throughout the argument "actual infinity" is meant in the sense of (ACT1). But under interpretation (ACT2) the situation is even worse, since it is not clear 2' is true. It seems that in order to "subtract" infinitely many coins in the sense defined above, all of them must exist at the same time. But if "actual infinity" is taken in the sense of (ACT2), then it is not required that all of the infinite number of coins exist at the same time, and thus 2' has no support. And of course, with the exception of premise 1', all the same criticisms I have just given apply equally well under (ACT2). So, interpreted charitably, the argument seems to be a failure, with the primary problem being in premise 3'.
I should note one more time that, in spite of all my criticisms of these two arguments, I think there are good reasons for thinking the KCA is sound. I just don't think these are among them.