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Originally posted by AThousandYoungI have answered that a number of times. The definition quite clearly states that it is the definition as used in mathematics. In mathematics 'a space' essentially means a type of set. I already pointed out to you several times that if they had meant 'space time' they would have said 'space' not 'a space'.
How do you reconcile that fact with the definition 4.a that you claim mass falls under?!
[4.a] The least number of independent coordinates required to specify uniquely the points in a space.
http://www.answers.com/topic/dimension
The the space in question is the set of all possible masses, then there is one dimension, as that is the least number of independent co-ordinates required to specify uniquely the points in the space.
If the space in question is the set of all possible pendulums, then we might say there are two dimensions - length and mass.
If you want to learn more see:
http://en.wikipedia.org/wiki/Space_%28mathematics%29
http://www.spiritus-temporis.com/space/mathematics-and-space.html
Originally posted by AThousandYoungI apologize if I seem patronizing.
You're extremely patronizing and refuse to see my point. I won't bother continuing arguing with you on this point. I believe you are wrong, and I think I've demonstrated that to any third parties.
I don't recall you demonstrating that mass does not fit the 4.a. definition. As far as I can tell, all you have done is claim that it doesn't fit, you haven't provided any argument as to why.
You claimed that mass was not a Euclidean space, yet 4.a. does not mention Euclidean space, but merely says 'a space' and I have provided references to show that 'a space' in mathematics is more general than Euclidean space.
Further, I have even given an example in which mass could be a dimension of a Euclidean plane, and it is trivial to extend that to a Euclidean space.
The fact that the same reference you give, gives mass as a concrete example of a dimension in 5. proves my case. Physics is, after all, applied mathematics.
Further to that, even if we totally ignore mass, there can be no doubt whatsoever that Southness is a dimension on the surface of a sphere and that any set of dimensions chosen on a sphere are finite. If you insist on Euclidean space then make it a four dimensional sphere.
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Originally posted by twhiteheadOK. That's not patronizing. Let's see if I can clarify my point.
I apologize if I seem patronizing.
I don't recall you demonstrating that mass does not fit the 4.a. definition. As far as I can tell, all you have done is claim that it doesn't fit, you haven't provided any argument as to why.
You claimed that mass was not a Euclidean space, yet 4.a. does not mention Euclidean space, but merely says 'a space' and I ha here are finite. If you insist on Euclidean space then make it a four dimensional sphere.
I will think more carefully about my position on the definition.
Southness represents distance around the Earth from the equator - until you reach the south pole.
Thus, it's equivalent to this:
[theta] - [pi/2] - [2 x pi x n]= S
Where:
theta = angle of inclination from the north pole
n = the number of times you have gone around the world and crossed the north pole again
minus pi/2 = because the zero point on latitude is the equator, not the north pole
minus 2 x pi x n because the range of southness has been defined out of convenience as being [-pi/2, pi/2].
The dimension here is theta - the angle of inclination. Southness is a function and a subset of the dimension theta.
Theta has a range of [-infinity, infinity].
Analogously, observable time, which began at the big bang, is a subset of the dimension of time, which did not ever begin as far as I can tell.
Originally posted by AThousandYoungEither you have just proved that the surface of the earth is infinite, and that that there is something further south than the south pole or you have made a mistake somewhere.
Southness represents distance around the Earth from the equator - until you reach the south pole.
....
Theta has a range of [-infinity, infinity].
Just because you can go around a circle forever does not give the circle an infinite circumference.
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Originally posted by AThousandYoungI think there is a problem with your formula for values of theta > pi.
OK. That's not patronizing. Let's see if I can clarify my point.
I will think more carefully about my position on the definition.
Southness represents distance around the Earth from the equator - until you reach the south pole.
Thus, it's equivalent to this:
[theta] - [pi/2] - [2 x pi x n]= S
Where:
theta = angle of inclination fro ...[text shortened]... ig bang, is a subset of the dimension of time, which did not ever begin as far as I can tell.
That aside, why couldn't somebody give a counter argument as follows, if southness is a function of theta, then just because theta (the domain) has a range of [-infinity, infinity] it doesn't follow that southness (the range) does.
Compare this with the function sin(theta) which has an infinite domain but whose range is [-1, +1].
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Originally posted by twhiteheadWhat I've proved is that if you continue going in the direction you were going when you were going south to the south pole, you will continue travelling until you reach the north pole, and then the south pole again, etc... i.e. there's something "south" of the south pole unless you want to argue semantics. If you do, look at my mathematical analysis above. The direction that was south before you reached the south pole will become north along the same Great Circle that you went south on but the opposite side of said Great Circle. Angular velocity will remain constant.
Either you have just proved that the surface of the earth is infinite, and that that there is something further south than the south pole or you have made a mistake somewhere.
Just because you can go around a circle forever does not give the circle an infinite circumference.
There is no limit to how far you can travel. However if you go far enough you will end up where you started. You will never reach the "end" and there is no "beginning" to the path.
If this is analogous to time, then the future would end up being identical to the past, with no beginning or end, just a rhythmic pattern.
Originally posted by Lord SharkWhat problem do you think exists?
I think there is a problem with your formula for values of theta > pi.
That aside, why couldn't somebody give a counter argument as follows, if southness is a function of theta, then just because theta (the domain) has a range of [-infinity, infinity] it doesn't follow that southness (the range) does.
Compare this with the function sin(theta) which has an infinite domain but whose range is [-1, +1].
Of course functions with infinite domains can have finite ranges. Southness is an example; you just gave another. So?
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Originally posted by AThousandYoungWhat problem do you think exists?
What problem do you think exists?
Of course functions with infinite domains can have finite ranges. Southness is an example; you just gave another. So?
The problem I have is that if you are going to apply the arbitrary convention that southness should be relative to the equator then for theta < pi/2 you get a negative value. That's ok, I don't have a problem with viewing northness from the equator as negative southness. But for theta > pi you have gone past the south pole and southness relative to the equator decreases again. But subtracting pi/2 doesn't give you the correct angle. You could argue, as you have above, that this just represents being further south-that's fine, but then wouldn't it have been simpler just to take the north pole as the starting point and define southness as theta? Hence if, for example, you started at the north pole and travelled south, by the time you'd gone around the world three times you degree of southness would be 6pi. Seems less fiddly to me.
Of course functions with infinite domains can have finite ranges. Southness is an example; you just gave another. So?
So you haven't shown whether time is analogous to the domain or the range, so you haven't made the case that time is infinite.
Originally posted by AThousandYoungMaybe I would argue semantics, because surely it is only semantics that makes one direction 'before' another.
i.e. there's something "south" of the south pole unless you want to argue semantics.
There is no limit to how far you can travel. However if you go far enough you will end up where you started. You will never reach the "end" and there is no "beginning" to the path.
But the key question is whether or not the surface is finite. That is all that matters. Whether you can run around on it forever is irrelevant.
If this is analogous to time, then the future would end up being identical to the past, with no beginning or end, just a rhythmic pattern.
And what is the problem with that? The fact remains that dimensions can be finite, and therefore any argument based solely on the properties of dimensions cannot prove that time is infinite.
OKay---wait. I'm late to this discussion, but have we decided yet whether or not time is infinite? Personally, I've always considered the answer to be yes, but if I understand what Hawking wrote in "Brief History of Time", physicists say no. Or that anything that happened prior to the Big Bang is unimportant (which is a cop-out, and doesn't answewr the question of when time began)...
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Originally posted by PinkFloydThe question is whether space-time is finite but unbounded. I think the jury's out.
OKay---wait. I'm late to this discussion, but have we decided yet whether or not time is infinite? Personally, I've always considered the answer to be yes, but if I understand what Hawking wrote in "Brief History of Time", physicists say no. Or that anything that happened prior to the Big Bang is unimportant (which is a cop-out, and doesn't answewr the question of when time began)...
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I'm not ignoring this thread. I'm letting it cook in the back of my mind.
So, twitehead, even if there is a beginning to time, you think there could very well be something before the beginning of time - that is, the end of time could be before the beginning of time right?
If you are comfortable with that when you refer to the "beginning of time" then it is indeed semantics we are arguing about. I'd never refer to the beginning or end of a cyclical pattern myself, but you can choose an arbritrary point on the loop and call it the beginning if you like.
Originally posted by PinkFloydNo, we haven't decided.
OKay---wait. I'm late to this discussion, but have we decided yet whether or not time is infinite? Personally, I've always considered the answer to be yes, but if I understand what Hawking wrote in "Brief History of Time", physicists say no. Or that anything that happened prior to the Big Bang is unimportant (which is a cop-out, and doesn't answewr the question of when time began)...
I feel nuggets of truth in both my position and what's he's saying. Not being able to reconcile it is really annoying. Cognitive dissonance is a pain in the butt for someone like me who likes to think he understands stuff.
Originally posted by twhiteheadYou say that we cannot say anything definitive about the Big Bang and yet when I ask "what came before the big bang" you seem quite definite that there was no before.
Science only describes the big bang from a certain point during its expansion. Anything before that is hypothesis. Too many people take it as fact that there was a singularity and that time started at the singularity. It is not something that is known scientifically.
The problem is that at times you come over all definite and catagorical but when I put you on a sticky wicket you get all floppy.