Magnus Anderson wrote:An infinite number of inches is equal to a smaller infinite number of 90 inch long segments and an even smaller infinite number of 93.75 inch long segments. There is no contradiction. You're just doing to the math the wrong way.

So you agree that the length of the lines is the same even though there are fewer 90-inch segments than 1-inch segments. But that would imply that if you counted the segments in each, you would finish counting the 90-inch segments before the 1-inch segments (that's what "fewer" means). That in turn implies that at least the number of 90-inch segments is finite.

Magnus Anderson wrote:The only reason you think that length implies a beginning and an end is because you're used to working with line segments.

That's sorta kinda how we derive the definition. We work with line segments and say: "Hey, why don't we refer to the distance between one end and the other as the line's 'length'". If you want to expand the definition to mean something different with infinite lines, you're on your own.

Magnus Anderson wrote:The other problem is that you're confusing conceptual matters with empirical ones. The same mistake that Silhouette is making.

Yeah, 'cause heaven forbid we confuse conceptual matters with empirical ones. You'd get

science!!! Magnus Anderson wrote:It does not.

Does so!

Magnus Anderson wrote:gib wrote:I think it went over your head

Very convenient.

I'm sure it

was convenient for you.

Magnus Anderson wrote:Why is infinity not a number?

Really???

Magnus Anderson wrote:It's not an unknown. And the word "infinity" does not mean "the highest number possible".

Exactly! You're arguing my point.

Magnus Anderson wrote:Meters, centimeters, inches, feet, etc are not undetermined things. \(1m = 100cm\) is not the same as \(1 \times x = 100 \times y\) where \(x\) and \(y\) are unknowns. Units \(\neq\) unknowns.

You're right about that, but variables (unknowns) can be treated like units, and visa-versa. 3X means 3 times the quantity X, but X here can be thought of as a unit, as in 3 times some object we call X (where this object is a set of X things). It works the other way around too: units can be treated like variables: 3cm = 3 x (100mm). You're right that we know how much a centimeter is and how much a millimeter is, but if I tell you 1cm = n gwackometers, it suddenly becomes unknown (you don't know how much a gwackometer is). So 1cm = an unknown number of gwackometers. The fact that you can divide a centimeter into any number of arbitrarily long segments means that it is an unknown in the sense that we don't know how many of these arbitrary segment there are.

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 \(\infty\) 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.

Magnus Anderson wrote:That's what it means with respect to line segments. And if we assume that the concept of length applies only to line segments, it does not follow that rays don't have something similar. Either way, you've got nothing.

Whatever it is you think this "similar" length is with respect to rays, you are once again on your own (and strictly speaking, we're talking about terms like "shorter" and "longer").

Magnus Anderson wrote:The symbol \(\infty\) represents an infinite quantity; believe it or not, even when I'm doing arithmetic with it. It certainly does not play the role of an unknown because the quantity is known -- it's not unknown. And yes, you can use any symbol you want (e.g. you can use \(\omega\) if you're into hyperreal numbers or \(\text{infA}\) if you prefer James's notation) but that does not mean we've somehow changed what we're representing. We're representing the same thing: infinite quantity.

"Infinite quantity" isn't really a quantity. That's just a phrase we use to talk about infinity. If you want to say you

know what \(\infty\) stands for in the equation (namely infinity), then you're simply doing something you can't do. It doesn't stop it from being a symbol which you can plug into the equation and manipulate according to the rules of algebra. I can do the same with the symbol for male:

\(\frac{{6}\times{♂}}{3}\) = \({2}\times{♂}\)

Does this prove anything about males? Do you think it proves males are quantities?

Magnus Anderson wrote:Saying "1 of something symbolized by ∞" is exactly the same as saying "an infinite number" (such as an infinite number of train carts) because ∞ means "infinite number".

Then you're saying \(\infty\) is both the unit

and the quantity at the same time. I take back what I said. I

don't agree that you can do this. You can still manipulate the symbols according to the rule of algebra, but you're doing nothing but shuffling symbols around. You're not actually deriving any insight into the nature of infinity.

The only time I've seen \(\infty\) used in an equation is:

\(\frac{1}{0}\) = \(\infty\)

...and \(\infty\) in this case explicitly means "undefined"--as in "you can't do that!" You're grade school teacher taught you not that dividing by zero gives you a really, really, really big number, but that you

can't divide by zero.

Magnus Anderson wrote:That's exactly what ∞ means. It means "an infinite number of things". The fact that I'm treating it like a unit DOES NOT change that fact.

Ok, Magnus, you're insisting that we treat \(\infty\) as an infinite quantity. Fine. I actually agree, \(\infty\) does mean "an infinite number of things". But then I deny your right to use it in math. You're trying to sneak it into math by, simultaneously, treat it as a unit--which, as I said, you can do but not while also treating it as a quantity. What number does 3cm equal? That is, how do you get rid of the units so that you're left with

only a number? 3cm = X. What is X? <-- That's what you're trying to do with \(\infty\). You're trying to say 3\(\infty\) = X, and since \(\infty\) is an infinitely large number, you can plug that number into \(\infty\), multiply it by 3, and solve for X.

Magnus Anderson wrote:When I use the symbol m to represent one hundred centimeters, does it suddently stop representing one hundred centimeters? Of course not. THAT'S EXACTLY THE PURPOSE OF SYMBOLS.

What about the formula E = mc^2? We know what c is (speed of light), but what is m? It's a symbol for mass, but do you know what the mass is?