Space, Time & Stephen Hawking Jive

[usmhot]

"I think I?ll skip it and go straight to String Theory.

I hear it has 20+ dimensions. That would make the BB&C (baffle befuddle & confuse) factor awesome."

My understanding was it was pretty much settled on 11 dimensions, though 27 (I think) was the other possibility.

Not at all confusing - you've seen one dimension, you've seen them all!!

"What do math mystics do when the limit is 0 (zero) and turns up in some bewildering complex of equations as a divisor? (n/0 = ?mathematical absurdity?)"

Its a very useful occurrence, in that it signifies the breakdown of your equation / theorem. What you do is go back to the beginning and find a proper solution. Mathematicians never sweep something under the carpet, and never invent something without rigorous justification.

Einstein was the first to admit that he was not a mathematician. And, indeed, he also admitted that adding a term to his equations to satisfy an intuitive contradiction was the biggest mistake of his life. See - there you go - one of the most brilliant men ever pointed out that giving into his intuitions at the expense of mathematical rigor was a mistake. See? [img]/images/graemlins/grin.gif[/img]

Actually, throughout history, the most brilliant men and women have learned, time and time again, that when it comes to a choice between 'intuition' and pure, hard, rigorous mathematics, its the latter that turns out to be correct. The whole reason that Relativity and Quantum Theory came about is because they followed where the maths and logic lead them.

[FNHinVA]

OK, I lied. Next last question.

"...you started by assuming _infinity does exist..."

I'm not sure what you mean by this.

I thought what I was saying was consistent with a rather "conventional" dictionary-like definition:

"Unbounded space, time, or quantity; an indefinitely large number or amount."

As opposed the a mathematical definition:

"The limit that a function f is said to approach at x = a
when f(x) is larger than any preassigned number for all x sufficiently near a."

... whatever that means.

Irrespective of whatever it means, the key difference ... at least to me ... is that the former cannot be quantified while the latter can, thus defining the key difference between "Real World" and "Math World."

Am I somehow being inconsistent?

[PairTheBoard]

FNMinVA --
"Am I somehow being inconsistent? "

Consistently so.

PairTheBoard

[FNHinVA]

usmhot is safe. But now I want to shoot you, PairTheBoard

[usmhot]

Its very difficult to answer that without appealing to mathematical rigor but I'll try ...

First the conventional dictionary-like definition that you quoted is not quite appropriate - or needs to be reworded to fit the problem you're addressing.

You start with two points - A and B - a finite distance apart. You assume that, as space is continuous, you can divide the distance by 2 repeatedly into an infinite number of points each of which must be passed. Now, that is a bounded infinity, as opposed to unbounded. It is clearly bounded as you started with the bounds - A and B. (Technically, its a 'closed bounded' infinity as you're including the two points - if you didn't include them it would be an 'open bounded' infinity.)

Anyway, the mathematical definition of a bounded infinity is that given the bound (B) and any arbitrary point (x where x is not B) there is at least one more point (y) such that the distance from x to B is greater than the distance from y to B.
Now, you need to convince yourself that this makes sense to you and that it says what you are getting at.
You are taking A and B and saying you'll start with the halfway point as that x point, but there's a point which is halfway from x to B. So, now you'll take that point as your point x and there's another point halfway , etc.
This is really the same as the mathematical definition that I gave you and its within this definition that "sum of series" maths is totally consistent.

In short, you're starting with the assumption that there are an infinite number of points within a finite distance / space. This puts you firmly within the system of mathematics that includes sum of series proofs.

[usmhot]

Incidentally, my deepest gratitude to you for this discussion. It has been many years since I've had to think this rigorously about mathematics, and its reawkening my appreciation for the beauty and awe of it.

[FNHinVA]

You are very welcome. You have been very generous with your time and I appreciate that. As I said before, I know stuff I didn't know a few days ago. Gotta be happy with that.

With respect to your last post/answer: I believe I essentially understand it but - as always - it leads to more questions.

But this has to end sometime. So I'll do some more reading & Googling on my own and let you get back to whatever it is you do when not tutoring math dolts.

Thanks again...

Later...

[spoohunter]

You guys are all really dumb, in a smart kind of way. I apologize if the answer has already been said.

You must move a minimum distance at one point. You are incapable of moving less than this distance. Problem solved.

[FNHinVA]

Apology accepted.

This thread is over.

Go to "Some More Infinite Series Jive" and tell them:

"You must move a minimum distance at one point. You are incapable of moving less than this distance. Problem solved."

Fun will ensue...

[K C]

Hehe. That's the paradox of this other problem though. Intuitively we would assume that.

KC