Magnus Anderson wrote:When we say "The number of people on planet A is less than the number of people on planet B" it does not mean that we can "finish counting" the number of people on either planet.

Well, geez, man, not sure how else to put it. "Fewer" implies finitude. It implies an actual quantity relative to a larger quantity. I shouldn't need proof of this, that's just the common definition. If you need proof, what you're really asking is: why should I accept your definition when I have my customized one that fits with my theories on infinity. And to that I say: fine, we can go with either definition, but then you gotta help me understand your definition.

Let's try this: imagine the scenario I described earlier, the one with two infinite parallel lines. For all intents and purposes, the same length. Now remove every odd point from one of the lines. Then move all remaining point into the spots left behind by the points you removed. According to you, the line with the points removed is now "shorter". But since we moved all points into the spots left empty from the points we removed, the lines are perfectly identical. It's as if we didn't remove any points at all. We're back to the initial state of the scenario. So here's your chance to shine. Help me understand how the lines are different now. Help me understand what it means that the line we removed points from is now "shorter".

Magnus Anderson wrote:By definition, if you have a bunch of things in one place (regardless of their number) and you subtract one thing, you get a smaller number of things. To say otherwise is to say that you didn't really perform the operation of subtraction, which is a logical contradiction.

You are, once again, dismissing counter-examples. Nothing wrong with following a string of logic to arrive at certain conclusions, but when someone shows you how counter-examples can be drawn in special case (like when the number is infinite), you can do two things: 1) dismiss it and just reiterate your original logic, or 2) consider the counter-example and rethink your original logic. You can then show how your original logic still holds and give an explanation for why, or you can concede that your original logic doesn't actually hold in the special case of the counter-example. Seems to me like you're choosing 1).

Magnus Anderson wrote:gib wrote:Yeah, 'cause heaven forbid we confuse conceptual matters with empirical ones. You'd get science!!!

You'd get confusion.

This thread is about concepts.

Unless we're talking about Anderson's magical fantasy world of shorter and longer infinities, this thread is about reality. The only way confusion would arise is if we insist on having contradictions between the concepts and the empirical evidence while still maintaining the concepts represent reality. If I say the concept of a flat world implies that you'd fall off the edge of the world if you walked far enough, and empirical evidence shows that you don't, you'd only be confused if you insisted your flat-earth concept is still reality. The alternative is to accept that the concept is wrong. You can still say that the concept of a flat-earth implies that you'd fall off the edge if you walked far enough, but what good does holding onto that concept do if the empirical evidence shows that it's wrong?

Magnus Anderson wrote:Why are infinite quantities not quantities?

*Ugh* Ok, let's try this. Infinity means no end. That means surpassing all numbers. As soon as you reach a particular number, you know you're not at infinity because otherwise you could say you've reached the end of all numbers, and you know that just doesn't make sense (right?). An infinite quantity just means beyond quantity. <-- That's why it's not a quantity. Give me any quantity, any quantity at all, and I will show you it's not infinity.

Magnus Anderson wrote:Your n is a symbol representing an unknown. 1cm is not representing an unknown. You're comparing a known value (1cm) to an unknown number (n) of gwackometers.

So then tell me, how many gwackometers is 1cm?

Magnus Anderson wrote:gib wrote:The central insight in algebra is that it doesn't matter what the variables (or units) stand for--they might as well be unknowns--its the rules for manipulating them that matter. And that was my point about plugging ∞ into algebraic equations--it doesn't matter what it stands for, it's just a symbol--manipulate to your hearts content. You don't end up prooving anything.

Of course I do.

This should be good.

Magnus Anderson wrote:Because Gib says so. Again, no arguments whatsoever. You are merely declaring that infinite quantity isn't really a quantity.

I am often amazed at what people need proof of.

Magnus Anderson wrote:I am certainly not saying that the reason ∞ represents quantity is because you can plug it into a mathematical equation.

That would be your misunderstanding.

Glad we got that cleared up.

Magnus Anderson wrote:gib wrote:Then you're saying ∞ is both the unit and the quantity at the same time.

I am saying that every unit can be represented as a number of other, smaller, units. For example, m can be represented as 100cm.

So \(\infty\) stands for: an infinite number of Xs (organisms, train carts, points in line, whatever) where X is the unit, correct? Then I forbid you to use it in arithmetic. If \(\infty\)

was the unit (as in 2\(\infty\) meaning 2 infinities), then I'd say run wild, have fun, but you're not saying that. You're saying you want to take the quantitative value of \(\infty\) and multiply it by 2, leaving whatever there is an infinity of as the unit, and to that I say: STOP IT! *slaps hand*

Magnus Anderson wrote:By following a rule that says that all numbers used in an expression represent a quantity of the same unit. So instead of writing something like 3cm + 10cm = 13cm you can simply write 3 + 10 = 13 because every number in the equation represents a number of units of the same kind (centimeters.)

No, no, no, you're just making the unit implicit. I mean, really get rid of it, as in you just have the number 13, not 13 centimeters, just 13. But this is a non-issue as I see above you don't mean to get rid of units entirely but simply use whatever object there is an infinity of as the unit. Again, I forbid you!!! *ominus voice echoing*

Magnus Anderson wrote:That's true. But how is that relevant?

You're insisting that because symbols serve to represent something, we know what those are. But I'm saying that the whole point of variables (which are symbols) in algebra is to have a way of doing arithmetic with unknowns. You

can know them, but you don't have to. I was saying that if you want to use \(\infty\) as a variable, it has to stand for an unknown because the actual value of infinity is off limits.