Wow. Things sure exploded. You'd think I was gone for a week.
It would be nice if you at least acknowledged that you have read my post before wanting to have a discussion, Steve. But sure, I'll bite. Just a fair warning: I may not be able to reply to you right away, and my answers may disappoint you. Also I'm not the kind to beat my head against the wall very much, so don't be surprised if I drop out this thread for good if it becomes clear that you expect me to listen to you without you doing the same for me. In the meantime, however, I'll gladly weigh in on your thoughts.
steve waterman wrote:1 Cartesian coordinate systems existed in the 1800's, thus existed before the Galilean transformation had been created/conceived.
This is probably true, but it is irrelevant.
2 A Cartesian coordinate system contains an infinite set of points, including the point (0,0,0) called the origin.
As you wrote it, no. This is because (0,0,0) is not a point; it is an ordered list of numbers (often called a [url="http://en.wikipedia.org/wiki/Tuple"tuple[/url]). To use the analogy I put forth, you must not confuse the measurement "zero inches" with a dot on the ground. There is a one-to-one correlation, sure, but that does not mean they are the same thing. In most cases this would be me just nitpicking, but it is absolutely crucial to understand.
But you are partially right. A three-dimensional Cartesian coordinate system implies that there are an infinite number of points, and the point corresponding to the tuple (0,0,0) is typically called the origin. That does not mean that the origin is special in any way, though.
3 Given coincident systems S and S',
mathematically allow a point P at (2,0,0) in/wrt S and also allow a point P' at (2,0,0) in/wrt S'.
From here on out, my statement that a tuple and point are two separate things still stands. It is just pointless for me to repeat myself. Back to the question though: If two coordinate systems are coincident, then the same coordinates in each coordinate system would both correspond to the same point. Calling one P and the other P' implies that they are not the same point. You can just call them both P because they are one in the same.
steve waterman wrote:cyanyoshi,
In an attempt for you and I to have a valid discussion, I am initially trying to establish a working scenario for the terms "point" and "coordinate system" that we can mutually agree upon.
That's fine by me. May I then suggest following accepted naming conventions? It's why they are there, after all: to let people communicate their ideas with each other.
4 Given a Cartesian coordinate system with a line drawn between point A(1,1) and point B(-1,-1).
Allowed? ( noting that there is only just the one system in this mathematical statement )
This is precisely why we need to get on the same page notation-wise. Are A and B functions? If so, then how are they defined? Are you simply naming the points that correspond to the tuples (1,1) and (-1,-1)? We are speaking two different languages.
You can certainly draw a line between any two distinct points in a Cartesian coordinate system, if that's what you are asking.
5 IFF number 4 is allowed, we could make a distinction between
the "coordinate point" (1,1) [ btw, I am quite aware that Relativity has no such term ]
and the "named point" A(1,1).
Do you allow these two differentiating terms, cyanyoshi, for the sake of our discussion?
Sorry, but unless you provide a rigorous definition of their meanings, then there is no chance of me using those words. I can't read your mind. It makes much more sense for us to use only words that we can look up the meaning of from an outside source (like Wikipedia, or a dictionary). How else can I we determine the truth of a statement? If however you want to try and give us definitions or suitable metaphors for those terms, then I'm all ears.
I will skip to the bottom line...
Given S(x,y,z)...x = distance from S(0,0,0) to S(x,0,0) and
Given S'(x',y',z')...x' = distance from S'(0,0,0) to S'(x',0,0) and
Given S and S' coincident with x = x'.
Allow S' or S to move by vt along the common x/x' axis, in which case, it is absolutely obvious that x = x'.
Therefore, one can deduce that x' = x-vt is mathematically incorrect unless vt = 0, since x = x'.
Just because this disagrees with the Galilean transformation equation results does not make my logic invalid. Just because point P in your manifold shares the same spatial location has zero impact upon the absolute fact that x = x'.
I do not comprehend why this simple logic is sooo difficult for everyone here at xkcd to grasp/agree with.
Prove to me that x does not equal x' after vt gets applied using the definitions above for x and x', or at least explain why you disagree with that definition for x and x'.
Okay, straight to the good stuff (for some definition of "good"). Read my original post. Seriously, read it and try to visualize what is going on with the dot and the rulers. Have you read it yet? No? What a shame.
I will still address your post, even though you might not like it. You start off by defining coincident coordinate systems where x=x'. Then you say that since x=x' that the Galilean transformation must be wrong? That's not how that works, no matter how much you want to believe it. More to the point, the Galilean transformation is applicable to certain pairs of coordinate systems that are not coincident. Your analysis is not. Have you seriously considered that people might have issue with your logic not because they are close minded, but rather because your conclusion does not logically follow from your premise?
Now your turn. What part of my
logic do you find fault with in my other post, the one with the colored rulers?
btw, combining these two threads on this topic, has now reached an unbelievable 6000 posts.
steve waterman wrote:
JudeMorrigan wrote:Groovy. Now why should the distance from S(0,0,0) to S(x,0,0) be the same as the distance from S'(0,0,0) to S'(x',0,0) if the axes are not coincident?
Simply because is was the Galilean given that at t = 0, x = x' so AFTER vt gets applied, x still equals x'.
You know what, I think we're done here. It is abundantly clear that you have no interest in understanding what the Galilean transformation is referring to, or basic logic for that matter. Dozens have come before me and dozens have failed, and the time has come for me to join their likes. Take care of yourself, Steve.