Magnus Anderson wrote:You need to address this argument:Max wrote:Suppose there is an infinite line of people somewhere in the universe and that YOU are one of the people waiting in it.

Suppose now that I take you by your hand, remove you from the line and place you somewhere outside of it.

The line is the same as before except that you're no longer part of it. Noone joined the line, noone left the line -- except for you.

If you say that the number of people waiting in that line is the same as before, it either means that I didn't really took you out of that line (that you're still there) or that I did but that someone else joined it. Both are contradictions.

You never addressed this argument.

What will be accomplished if I address this? What are you looking for? Do you want me to show how it's not really a contradiction? And if I do, what are you expecting to change in your thinking or the discussion?

Anyway, I'll do my best.

First of all, it's a loaded question. You're already presupposing that the notion that the line remains the same means one or more of your premises has been negated. You're boxing me in. But you know very well what my view is: the line remains the same despite that you really did take me out of it and no one else joined. The question you should be asking is: how is that possible? <-- I'll address this question.

The truth is, I'm not sure what to say about the number of people in the line. I understand that it seems intuitive to say: if you add someone to an infinite line of people, it has one more person. And if you take a person away, it has one less person. And if the notion of adding or subtracting from infinity wasn't so problematic, I would have no problem with this. But it is problematic. It's problematic because if you accept the notion that adding or subtracting a person changes the number of people in the line, it may square well with your intuition (thereby satisfying that aspect of the problem intellectually), but then you have to address other problems that crop up. For me, it's terribly problematic to do arithmetic with infinity. As soon as I start thinking about doing arithmetic on infinity, I can't help but to think of it as a quantity. Then I ask: well, what is that quantity? Is it bigger than a thousand? Less? Is it bigger than a million? Less? I don't know how to understand quantities except as existing one the number line, as existing between greater quantities and lesser quantities. But that, to me, defies the definition of infinity. Infinity means endlessness. It means: as soon as you've got a quantity, think bigger. Think: it's greater than this quantity. So if a line has an infinite number of people, and you add one more person to it, and you say that gives you \(\infty\) + 1, these are the thoughts that are going to come to my mind. They are problematic. I can't accept the notion that adding/subtracting a person from the line changes the number of people in the line without being bothered by these other considerations.

In other words, even if I were to agree that thinking of infinity as something to which you can add or subtract solves the problem of how counterintuitive it seems to say nothing in the line changes, I see this more as a trade of one problem for another, and in the larger picture, you still have a problem. You don't seem to be bothered by this. You're either dismissing the other problems I bring up as if they aren't there, or you have a different understanding of these problems such that they aren't really problems. So far, I've gathered that you think of infinity as a quantity that finds a place on the number line but is an infinite distance away from the finite numbers. That certainly solves the problem of how infinity can have a place on the number line (so we don't have to ask: is it greater than a thousand? Less? etc.), but for me it raises yet another problem: it seems to suggest that infinity has an end, or an "after". You find \(\infty\) + 1 after \(\infty\). But that, for me, flies in the face of the very definition of infinity: endlessness... infinity means no end. So how can there be an "after"?

Maybe you have a solution to this problem as well, maybe not. But as you can see, the problem to me is like a balloon. You try to solve little bits of it by squeezing on those bits. But this only results in other bits inflating, making other sides of the problem more emphatic. This is what I've been seeing throughout this whole discussion. I've been watching you squeeze certain parts of the balloon and seeing other parts inflating. And I hear from you: what inflation? There is no inflation!

Sorry that's such a long winded answer to your question. The short answer is: I don't know what to say about what happens to the number of people in the line when you add or subtract from it. If I agree with you, I see other problems crop up. If I don't, I have to deal with this counterintuitive notion that adding or subtracting doesn't change the quantity. At least I can say that intuition isn't the same as logic, so if it seems counterintuitive that adding or subtracting from the line doesn't change the line, maybe it's still logical. Therefore, I lean towards saying it doesn't change the line.

I think a huge part of the problem has to do with something you said: most of infinity lies outside what we can see. When we visualize the infinite line of people, we can't help but to visualize something that looks finite. We keep in mind that it's supposed to be infinite, but this is more how we conceptualize it, not how we visualize it. Because the visualization of the line is inevitably finite, it gives rise to the intuition that adding and subtracting from the line must change the number of people in the line (because that's how it works with finite things). I think if we had the capacity to visualize the entirety of an infinity of things, we would see how it really works, and the notion that adding or subtracting from an infinite set doesn't change the quantity of the set wouldn't seem nearly as problematic.