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,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 iswe 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.

**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,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:making it impossible for any card to be added to the collection.

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?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 guyBy 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.

WLC "what number are you on?"

guy "-10"

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.This paragraph demonstrates his complete lack of mathematical understanding. He is right though, it doesBut 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.

**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.

Great post Haus! The thing with Craig is, I can't tell if he doesn't understand what you are saying or if he is purposefully dishonest about it, knowing that many people simply end up confused and he can lead them to the conclusion that he wants them to hear (and which they often want to hear).

ReplyDeleteSuch an awesome post Hausdorff! :) I do agree with ReasonBeing as well. He lets his audience hear what they want to hear. They've already come to their conclusion and its just a lot of confirmation bias. If they understand his argument(s), they don't critically think about them, and on the other side of the coin if they don't, they probably just assume he's right. I've seen WLC in debates hear some of his arguments torn apart, but then later he will still cling to the same arguments as though he doesn't realize that there are problems with them.

DeleteI definitely considered that. In fact, my original version of this post had some thoughts along those lines. WLC is an intelligent person, he has debated these topics a number of times with smart people who understand the mistakes he is making with infinities. I'm sure he's had it explained to him. So the question is, does he know his arguments are wrong and he still uses them anyway because he thinks it will still work on the average person, or is he just being willfully ignorant? I go back and forth on this, he's clearly very invested in what he is trying to promote, maybe he just always chalks his defeats to problems with debating rather than problems with the ideas, I don't know.

DeleteAnyway, the post was already long and that part wasn't coming out well, so I cut it. There was a little too much conjecture and personal attack and not enough evidence and fact. Although going down those lines might be good for a post of its own.

You could very well be right. Only WLC knows for sure. I think I'm skeptical of his motives since it is how he makes a living. A snake oil salesman will continue to sell what he's going to sell and pretend he believes in his product. However on the other side of the coin, he probably believes whole heartily the conclusion even when he has trouble justifying the premises or trying to prove this "truth" by using logic.

DeleteIt's hard to know what he does an doesn't believe, it is certainly possible that he think Christianity itself is bunk and he feels he is in deep enough that he can't turn back...but I doubt it. It's possible he knows that his arguments are faulty, but if the conclusion is correct and it will work to convince people, then the ends justify the means. I would put my money on that being his thinking.

DeleteI was trying to think of how I'd relate to the conclusion thing. I accept that one and one is two. I couldn't tell you why that's the case, I'd assume there is a proof out there somewhere or maybe its just a mathematical premise that's part of the foundation for math. I'm pretty ignorant when it comes to that sort of stuff. "Leave that to the mathematicians like hausdorff." But I guess the difference between math and theology is that math theories are falsifiable. Theology is pretty hard to do so. Math and science are always open to change in evidence and ideas, and welcomes it with open arms. Religion seems to be different.

ReplyDeleteAs to the 1+1=2 thing, what you are really doing here is defining two things, what numbers are, and what addition is. This might sound silly, but it is not necessarily what we would expect. What is 10+10 for example? Is it 20 or 8? If we are thinking in the normal sense, it is of course 20, but what if we are doing addition with respect to time? 10 o'clock + 10 hours is 8 o'clock.

DeleteIf we are to focus on the "normal" sense of addition on the integers, we can drop down into set theory and define these things so they work the way we expect. 1+1=2 because we defined it that way, and adding any other 2 numbers together works that way because we have defined them that way. The question that you need to ask yourself when you adding things up is whether or not this way of adding is appropriate to what you are doing. If you are adding up baseball cards, regular addition is appropriate. If you are adding up hours, regular addition may or not be appropriate. If I want to know how many hours I have spent playing a video game, normal addition is appropriate. On the other hand if I want to know what time it will be after I play a game for 8 hours, regular addition is useless to me, I need to use addition mod 12 (12=0)

Using time, you can see a situation where you have to be a little bit careful with what you mean by things (what does it mean to add things) in the finite realm. When you start to get into infinities and you are adding things up, you need to be really damn careful. Often times our intuition fails us, which isn't actually that much of a surprise, we live in a finite world, and infinities are hard.

I guess I stand corrected. I had never thought about it like that. It is rather situational. I guess 1 and 1 could be 10 as well if you were talking binary.

DeleteThat's a good point too. It would be reasonable to write 1+1=10 and you can also write 1+1=2. If the first is in binary and the second is in decimal, they mean the same thing. In fact, if I were to say either one verbally, they are both "one plus one equals two". In this case, we have two different ways to display the same information.

Delete"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."

DeleteIt seems to me that you are saying here that if we change the setting to where there is an infinite, why then there is an infinite. If "all the cards that ever will be" are infinite, why then they are. It strikes me as circular. It also strikes me that there is a difference between the act of adding up a series of finites, and the existence of the total objects that add up. We could, at least in theory, say that I could go through the action of adding up baseball cards without an end, but this is quite different than having a pile of cards in front of me.

Further, it would seem that the existence of an actual infinite of anything physical would be impossible for the simple fact that other things exist. An infinite amount of baseball cards is not possible because they would fill all space, which is impossible since other things occupy some of that space.

But I never claimed to be a mathematician, so I'm open to correction if I'm wrong. I asked some of these questions to a mathematician once, and he referred me to a book titled "The Mystery of the Aleph" by Amir Aczel. It explores these mathematical theories such as the possibility of infinites and the limitations of math. The book is in my reading pile, which will take me quite a long time to get through.

Excellent comments humblesmith. You touched on a number of really interesting points that I would like to respond to. Thanks for getting my brain going on this again, I love writing about infinity :). It got too long for a comment reply pretty quickly and I decided to just make it a new post. I'll post another reply here when I finish writing it, should be up in the next few days.

DeleteFYI, I finally had time to do a bit of writing, here is the follow up post

Deletehttp://hausdorffbb.blogspot.com/2013/11/is-actual-infinite-possible.html

Yeah, but what do you know? It's not like you're a math teacher or something.

ReplyDelete