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The reason there is so much difficulty in having Alex acknowledge time as a dimension, is because of his position that although we can move 'forward AND backward' along spatial dimensions, we cannot accomplish "backward motion" through time (thereby disqualifying time as a candidate for being deemed, "dimension").
However, just because we cannot "time-travel" into the past... doesn't mean that we must immediately jump to the conclusion that time is not a dimension. In fact if it isn't a dimension, then what the heck is it? Alex, you never did seem to come up with a satisfactory alternative...
The way I see it is this-
In junior-high we all learned one of the most significant theorems in all of geometry...
... a2 + b2 = c2 ...
(Pythagorean Theorem)
In first year algebra/introduction to geometry (9th grade), we proceed to learn Cartesian coordinate representation of Pythagorean Theorem...
... (X2 - X1)2 + (Y2 - Y1)2 = d2 ...
("Distance Formula"); where Xi & Yi are respective coordinates of two different points, and 'd' is measure of distance or 'magnitude of separation' evaluated in arbitrary system of units (meter, foot, furlong, stadiums, etc.)
One of the first things a 14 year old child will notice is that 'distance' can NEVER be negative! Even if we travel away from a point, stop, and reverse course traveling "BACKWARDS" along spatial dimension... we still are never allowed to say that we traveled "negative three meters" (in the opposite direction).
So for Alex I ask...
If we can't travel -3 meters in space, then why should we have to be able to travel -3 seconds in time (in other words "backward" in time, or "into the past") in order for "time" to qualify as a "degree of freedom" (=dimension)?? Under your reasoning Alex, space can no more be represented by the term "dimension" than time can be.
(Distance formula does not allow for negative distance in either case).
Let's even consider a more advanced "distance formula"...
... ds2 = c2dt2 - dx2 - dy2 - dz2 ... (distance formula extended to 4 dimensions to encompass the whole of space-time)
The Minkowski Metric says that two points can have the same spatial coordinates and still be separated because of "distance in time" (4-dimensional distance between two different events). The rate of separation between two rest-frame events just so happens to be valued at c... so in other words, we move through the dimension of time with speed c! (when at rest). Of course the word "rest" is in this sense relative; and hence motion through time is not absolute, but rather relative. |
