*deep breath* wow. First of all, let me say how thrilled I am to have just read through all of that. I know this reply is going to take me waaaaay too long to actually finish, but I am hoping someone, at least one person, reads it.
A little disclaimer first: I'm a high school student. I am not in an AP math course so I have not yet taken Calculus ... or philosophy. however, I am incredibly interested in topics like these, so all information that I know and will regurgitate I have acquired myself. So I don't know as much as I want to know...and probably never will. but I love trying. So here we go.
first and most importantly, "the wise man doesn't know how much he knows until he knows how much he doesn't know."
That said, infinities were discussed in repetitition here. What hasn't been mentioned, and may (I hope) provide some clairty is that you
can compare them. Georg Cantor, considered by some to be the founder of the "arithmetics of infinity," started questioning whether or not the number of all numbers is larger or smaller than the number of all points on a line. Anyway, skipping the intro and history of the topic, you get to the theory of how we compare infinities. Try to follow me in what I am attempting to explain (not as well as George Gamow did... which is how I learned this one - "One Two Three... Infinity by George Gamow) If we can pair the objects, or numbers, of two infinite groups so that each object, or number, of one infinite collection pairs with each object of another infinite collection, and no objects in either group are left alone, then the two infinities are equal. But if such an arrangement is impossible and in one of the collections some unpaired objects are left, then it is said that the infinity of objects in this collection is larger, or stronger than the infinity of objects in the other collection. So that while something may go on forever, all infinites may
not be equal. This is the most reasonable, and as a matter of fact, the only possible way of comparing infinite quantities. But, in application of this theory, some surprises might occur.
To explain this, an example would be comparing the infinity of all even numbers to the infinity of all odd numbers. Of course you would feel that the infinity of all even numbers is equivalent to the infinity of all odd numbers, and since a one to one correspondence of these numbers can be arranged, your intuition is right. i.e.:
1 2, 3 4, 5 6, 6 7, 7 8, etc. etc.
There is an even number to correspond with each odd number and vice versa, hence both infinities are equal. Seems practical, right? Well, here's where things get a little confusing. Which do you think is larger: the number of ALL numbers both even and odd, or the number of all EVEN numbers only? Of course all numbers, right? wrong! There is, in fact, a one-to-one correspondence of all numbers on one side, and even numbers only on the other:
1 2, 2 4, 3 6, 4 8, 5 10 6 12 etc. etc.
According to the rule of comparing infinities, we have to say that these infinities are equal! This sounds paradoxical, of course, since even numbers represent only a part of all numbers, but we have to remember that we are operating with infinite numbers, we must be prepared to encounter different properties.
And here's where things get relevant!
According to the above, we can infer and prove that in a world of infinity
a part may be equal to the whole! So... *whew* in reference to what you, Caleb, were saying... we could all be living in a circle...which would make it infinite -- and it could be the size of what we know as a marble. It is all possible. And this theory, for me, at least, just confirmed how many more possibilities there are out there. Made me think more at least...
And, again, in theory, an infinity might not be infinite at all, it may just be beyond our comprehension. You know, there are theories today that say we live in an 11 dimensional world, but we are so incapable of thinking outside our 3 (or 4, rather)dimensional perceptive minds, that we will never be able to fathom the 11 dimensions. It's similar to how a cat only sees in 3 dimensions. When they look into the mirror they stare blankly, or don't even bother to stare because they see nothing. This is not because cat's can't recognize themselves, but instead this is because a mirror's reflections are only 2 dimensional. The cat cannot fathom his reflection. Anyway, that's my piece on that
About time...while some claim that time has been proven to be nonexistent, I agree with most of you and think time is only one's perception. Einstein had a theory, which I'm sure you all are familiar with, as I forget it completely... about space traveling and time. How, theoretically, it would be possible for one half of a pair of identical twins travel into space and return to earth exactly one year later having not aged at all and reuniting with his twin, finding that he is exactly one year older. It went something like that I think. Anyway, I think you get my point of view.
Caleb... you brought up an interesting point in having faith in the intelligence of the masses. I have to contribute, I'm sorry. I think it was said in Men and Black (trite, I know), but "a man is smart. people are stupid" something like that. Anyway, I agree. I believe that every single person on this planet (and then some?) has something to offer, something I can learn from. But when we are herded together, we're morons. I sometimes seriously lack faith in humanity because of this. My pessimistic side assures me that we will, soon enough, destroy ourselves. It sucks to think the glass is half empty....but anyway....
hmm.. god.. now here's a touchy subject. I know you already clarified, but I don't think
anything besides an actual confrontation with the supposed being could prove the existence of god. To me, I don't need to know how everything started... how that first quark was created... I don't need a reason to feel insignificant. I don't need a god to supply me with a universal role in life.. or give me a soul... because of this, I cannot agree with any "proof" of an omnipotent being. I don't need a god to make me feel less small.. and I don't need to know all the answers.. or create something to answer them. (I am not meaning to offend anyone) However, as long as I am rambling.. I see the need for a belief in god. Religion (should) provide a system of ethics and morals that helps one to lead life. My own personal code of ethics and morality are not established by the belief in something higher than myself. I made them myself. based on the faith in myself that deep inside me, I know what's right and wrong, that I can make these decisions for myself, and that I am ready to be accountable for the repercussions of all my actions.
About chaos... so what if the entire universe is chaotic? Do you guys remember American Beauty? Well, do you remember that one video clip .. with the floating plastic bag. That was, in it's entirety - wind - pushing around trash, in no organized manner, in fact, chaotic, with no patterns, constants, or plan... just wind pushing trash. And yet, it was so beautiful. So maybe I take comfort in the belief in a chaotic world.. it's hope there's beauty out there
Anyway, thanks guys for reading (or scrolling through) my novel there. I didn't mean to go on.. and on... and on
And on a side note: Everyone is intelligent. Everyone is unique. And I love this stuff... don't ever stop thinking... and when you guys are ever in Philly... stop by cause there's nothing like finding people you can actually converse with.
[This message has been edited by Ro L L er G i r L (edited 09 April 2000).]