In the post, someone has written WLC asking him to justify the following assertion he has made in another argument
A collection formed by successive addition cannot be actually infinite.The writer basically says that he intuitively feels that this assertion by WLC makes sense, and yet he was wondering if WLC can unpack the intuition and get into the the details. WLC says yes and starts his argument by thinking about adding finite things together. He says
In the case of beginning with some finite quantity and adding finite quantities to it we can pinpoint the problem clearly: since any finite quantity plus another finite quantity is always a finite quantity, we shall never arrive at infinity even if we keep on adding forever. Infinity in this case serves merely as a limit which we never attain.What he has written here is wrong, but the reason why it is wrong is a little bit subtle, and it depends on what he means by 'never' and 'forever'. To make his example a little bit more concrete, let's assume the things I am adding up are baseball cards. I have a pile of baseball cards in front of me and I keep adding more baseball cards to the pile, every time I add new cards to the pile, I am adding a finite number of them. You might ask at what point in the future will I have a pile with infinite cards, and the answer I would give is never. There is no point in time in the future at which I will have an infinite number of cards in front of me. This is the only way I can think to justify his statement that we will never arrive at infinity.
However, in this situation, I did not "keep on adding forever", I added for an arbitrarily large finite amount of time. These two things are very different. By talking about adding forever, you are talking about taking a limit (which he mentions). In the normal way we think about time, it is true that we can never arrive at that infinity, but we also can't keep adding forever. If we change the setting to where it is possible to add forever, then it is also possible to arrive at infinity. We are adding up all of the baseball cards that are now or ever will be in the pile. That collection is infinite.
There is something very important here that I want to highlight, at any given time on this timeline, how many baseball cards remain that have yet to be put onto the pile? Infinity! No matter how far into the future we go, there is always an infinite number of baseball cards that still need to be put onto the pile.
Back to WLC's argument, he now tackles an infinite past instead of an infinite future. Suppose we have been adding baseball cards to a pile with no beginning, in other words, we have been making this pile forever into the past, and we end at some time and wind up with an infinite pile of baseball cards. he says the following
Now notice that one still hasn’t explained how we are able to form our infinite collection of baseball cards by successive addition. For at any time in the past the collection is already infinite, and yet the total collection has not yet been formed. The total collection will not be formed until the last card is added. From any point in the past one need add only a finite number of cards to complete the collection. But that leaves unsolved the problem of how the entire infinite collection could have been formed by successive addition.Let's pick this apart a bit. He says "For at any time in the past the collection is already infinite", that's true, since we have an infinite past, we have already added up an infinite number of things, no matter how far back in time we go. He also says "From any point in the past one need add only a finite number of cards to complete the collection" which is also true, because every point in the past is a finite distance from the present. But then his last sentence is "But that leaves unsolved the problem of how the entire infinite collection could have been formed by successive addition." But there is actually no problem here. If we return to our other situation where we are counting into the future, at any point in time we saw that there is an infinite number of things remaining to be put onto the pile. Here everything is turned backwards, so at every point there is an infinite number of cards that have already been put on the pile. If at every second we have put a card on the pile, and time goes infinitely into the past, then we will have an infinite number of cards because an infinite number of seconds has past. There is no problem here.
He then writes the following paragraph, which I have seen in various forms at various places by him
in order for the collection to be completed, we must have already enumerated, one at a time, an infinite number of previous cards. But before the final card could be added, the card immediately prior to it would have to be added; and before that card could be added, the card immediately prior to it would have to be added; and so on ad infinitum. So one gets driven back and back into the infinite past, making it impossible for any card to be added to the collection.He seems to think he has a contradiction here, but I think he explains himself quite poorly. However, as far as I can tell, his final example is another version of this same problem, and it is much easier to understand what he is trying to say and identify the mistake he has made:
It gets even worse. Suppose we meet a man who claims to have been counting down from infinity and who is now finishing: . . ., -3, -2, -1, 0. We could ask, why didn’t he finish counting yesterday or the day before or the year before? By then an infinite time had already elapsed, so that he should already have finished. Thus, at no point in the infinite past could we ever find the man finishing his countdown, for by that point he should already be done! In fact, no matter how far back into the past we go, we can never find the man counting at all, for at any point we reach he will already have finished. But if at no point in the past do we find him counting, this contradicts the hypothesis that he has been counting from eternity. This shows again that the formation of an actual infinite by never beginning but reaching an end is as impossible as beginning at a point and trying to reach infinity.I think in this paragraph WLC once again, demonstrates his complete lack of understanding of what he is talking about. This guy has been counting up from negative infinity and he just now reached zero. WLC wants to say this is impossible, and to challenge it he is going to go back in time 10 seconds and confront the guy
WLC "what number are you on?"
WLC "how can you be on -10? you've already counted an infinite number of things!"
guy "be that as it may, I'm at -10 right now"
WLC "but if you've already counted an infinite number of seconds shouldn't you be done?"
guy "nope, you can go as far back in time as you want and talk to me, I will always have counted an infinity of numbers already, it won't mean I'm already at zero."
You see, the problem here is that WLC has conflated two different ideas. One is that the guy has counted down from infinity and finished. The other is that the guy has counted down from infinity, and when he reaches zero it is the first time he has reached infinity. This second idea makes absolutely no sense. You will never have a finite number, add 1 to it, and get infinity. But that is essentially what he is saying. "ah ha! you just got to zero which means you counted an infinite number of things, why didn't you reach infinity 1 number before that?" "well, I did".
The sad thing is that WLC is actually touching on a concept which is actually kind of neat, but of course since he is trying to say this stuff is impossible there is no way he will see how cool it is. Let's consider time with an infinite past as a number line. Every second is a point on that number line but let's not put any labels on the seconds at first. Now, let's say some guy is walking along our number line putting labels down and it just so happens that when he gets to us he puts down the zero. This is only 1 possible labeling of our timeline. We might ask why he didn't have the zero 10 seconds ago, and he will just say that it is not where it goes. He's been counting forever and this is how the labeling worked out. However, a different person, also counting down from infinity, might just have the zero in the same spot the first guy put -10. Absolutely any second on our timeline could potentially be the zero, but once it is set, all other numbers are determined.
Another of WLC's examples demonstrates his lack of understanding. He is searching for "absurdities" with infinities, and describes the following
Consider the scenario imagined by al-Ghazali of our solar system’s existing from eternity past, with the orbital periods of the planets being so co-ordinated that for every one orbit which Saturn completes Jupiter completes 2.5 times as many. If they have been orbiting from eternity, which planet has completed the most orbits? The correct mathematical answer is that they have completed precisely the same number of orbits. But this seems absurd. Think about it: the longer Jupiter and Saturn revolve, the greater becomes the disparity between them, so that they progressively approach a limit at which Jupiter has fallen infinitely far behind Saturn. Yet, being now actually infinite, the number of their respective completed orbits is somehow magically identical. Indeed, they will have “attained” infinity from eternity past: the number of completed orbits is always the same. So Jupiter and Saturn have each completed an infinite number of orbits, and that number has remained equal and unchanged from all eternity, despite their ongoing revolutions and the growing disparity between them over any finite interval of time. This strikes me as nuts.This paragraph demonstrates his complete lack of mathematical understanding. He is right though, it does seem absurd at first, and yet it's true. How can the 2 planets have taken the same number of orbits when one has taken 2.5 times as many orbits? Well let me ask you this? What is 2.5 times infinity? Still infinity, right?
Let's change this question slightly: Which set has more numbers in it, all integers or all even integers? Your first instinct is probably to say all integers, but that is wrong, the correct answer is they are the same size. This is hard to understand at first, but without too much trouble we can get most people to see why it is true. That is why it is a standard problem that most math majors will see in their "introduction to proof" class by the time they are half way through their sophomore year. If William Lane Craig was in that class while I was teaching it, he would have failed.