The reason there is so much difficulty in having Alex acknowledge time as a dimension is because of his belief 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 answer...

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 points' coordinates and 'd' is measure of distance evaluated in arbitrary system of units (meter, foot, furlong, 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 a 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 "dimension" then time can (distance formula does not allow for negative distance).

Even consider the Minkowski Metric...

... ds2 = c2dt2 - dx2 - dy2 - dz2 ... (extended Space-Time distance formula; evaluated in 4 dimensions)

It 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.