Magnus Anderson wrote:We cannot put in anything for \(n\). There is an explicit condition: \(n\) must be greater than \(0\). It does not have to be something that satisfies Gib's narrow definition of the word "quantity", but at the same time, it cannot be something that cannot be said to be greater than \(0\). If we cannot say that \(cow > 0\) then we cannot say \(\sum_{i=1}^{cow} \frac{9}{10^i} < 1\). Fortunately for us, we all agree that \(\infty > 0\), so we can use it for \(n\).

You're not quite getting what \(\sum_{i=1}^{n} \frac{9}{10^i} < 1\) for n > 0 means. In the case of sums, valid values for n are typically integers greater or equal to whatever initial value is set for i. In other words, n must be an integer, and integers are only found on the number line.

You got this part right:

\(\infty\) > 0

But you're making an implicit assumption:

\(\infty\) has a specific place on the number line.

^ This you got wrong.

\(\infty\) is not among the values you have at your disposal to choose from for n. IOW, the condition that n > 0 does not

just mean "anything greater than 0", it also means values that fit with sums of the form \(\sum_{i=x}^{n} f(i)\) (that is, integers, those symbols above the tick marks on the number line).

\(\infty\) > 0 remains true because > and < don't signify specific points on the number line. They signify

directions on the number line. "Greater than" means: in the positive direction. "Less than" means: in the negative direction. \(\infty\) need not have a place on the number line for \(\infty\) > 0 to be true. It just has to be to the right of 0.

Magnus Anderson wrote:But you're missing a crucial step between 2) and 3): \(\infty\) is a valid value for n. I know you feel intuitively that \(\infty\) must be a valid value for n because \(\infty\) is a quantity, so no need to prove it. But when you're arguing with someone who disagrees with you on that, you do need to prove it. You can't just run on intuition.

And in order for me to do so, you have to tell me what you think is a necessary and sufficient condition for something to be considered a valid value for \(n\). And you may also need to explain

why.

Now we're cooking! You're finally asking the right questions.

I believe my explanation above should suffice. Valid values for n are any integer found on the number line greater that 0 (at least in the case where i = 1). Why? Because that's what the condition n > 0 is asking for. \(\infty\) is not an integer found on the number line. Why? Because it is the endlessness of the number line itself, at least in the positive direction. In other words, \(\sum_{i=1}^{\infty} \frac{9}{10^i}\) is not a special case of \(\sum_{i=1}^{n} \frac{9}{10^i} < 1\) for any n > 0. It's an entirely different statement. It says:

don't select a value for n. Just let the sum continue forever.

^ This may not satisfy your request for further clarification on what I need from you, but now your move is to probe deeper into my response. Is there something in the above you need even further clarification on? (wanna get into hyperreals?

)

Magnus Anderson wrote:I think that \(n > 0\) is a sufficient condition. You obviously don't. Why? You keep mentioning the word "quantity" without ever bothering to explain what you mean by it and why it's necessary for \(n\) to fit your definition of the word "quantity".

Again, I think the above addresses this. The important point is that \(\infty\) fails my definition of quantity. A quantity is what points on the number line represent (at least in terms of cardinality). Since \(\infty\) has no point on the number line, it doesn't represent a quantity.

Magnus Anderson wrote:If I argued that "cow" > 0, and therefore the formula applies to cows as well, you would insist I prove that "cow" is a valid value for n, wouldn't you?

No. I'd ask you what you mean by the word "cow" such that you can say \(cow > 0\).

You say tomato, I say tomato. (<-- That doesn't quite come through in text, does it?

).

Magnus Anderson wrote:Same onus falls on you to prove that \(\infty\) is a valid value for n.

It's not the same onus. We all know that \(\infty > 0\).

It doesn't matter what "we all know". I'm not a part of "we". You're dealing with me in particular. If you want to convince

me of your point, the onus falls on you to prove your point to

me.

Magnus Anderson wrote:But so far, all I've seen from you is re-assertion after re-assertion that \(\infty\) is a number--no proof--which tells me you believe it on intuition, not logic.

It should tell you that you're not doing a good job at explaining what needs to be done (and why) in order for you to accept the position that I'm putting forward.

You mean like, oh I don't know, asking you to prove that what applies to finite sets also applies to infinite sets? How 'bout proving that \(\infty\) is a valid value for n? Did I forget to ask for that?

It takes two to tango. I very well may not have been detailed enough in explaining to you what I need, but that's your cue to ask for further clarification. These things may go through a few rounds of back and forth before both parties are clear on what the other needs in order to be convinced. Nothing wrong with that. It's how it goes sometimes.

Magnus Anderson wrote:But what that means is: grab any number on the number line. \(\infty\) is not on the number line. It's a direction in which the number line extends.

That's an extremely narrow view. Basically, you're limiting yourself to what is widely-accepted (or at the very least, to what is familiar to you.)

What makes you think you cannot place \(\infty\) on the number line? The fact that it is not a widely accepted position?

There is a class of people known as

parrots. Parrots are people who adopt other people's conclusions without knowing how to arrive at them on their own. Since they are not independent thinkers, they can never accept a belief unless their trusted authority approves of it.

How do you convince a parrot? By convincing their trusted authority.

If this is the only way that I can convince you, then we have nothing to discuss.

You know, this is a desperate cry. You typically see it when your opponent has nothing left. It really means: I've got nothing but my going against the grain. I know that I'm taking an unpopular position. I know that all the professionals in the field, all the experts, all the really smart people disagree with me. I know the majority of people who follow this topic are not on my side. I know all my arguments are falling on deaf ears, that my attempts at convincing others seems to be an exercise in futility... but you know what, at least I'm not a sheep, at least I'm not mindlessly conforming to the masses. I'm thinking for myself, I'm exercising my independence of thought. And that's something to be proud of. I've exhausted everything else in the debate, might as well start using this.