Q:

How is traveling back in time different from traveling at a negative velocity?

A simplified explanation of why you can't travel through space faster than the speed of light is that you are already traveling at the speed of light, through spacetime. 
If you are stationary in space, you are traveling at c in time. If you speed up to c in space, time appears to be stopped and you are not traveling in time. 
To travel backward in time, you'd want to go faster than c in space, so that time will be forced to be negative. 
By my logic, that means that you have to also be able to do the opposite, travel back in space. Except, if you slow down to 0 and the keep accelerating, you'll just be traveling forward in the opposite direction. 
Is this line of reasoning correct? How would time travel be possible considering this? I'm not talking about alternate universes and similar things that are time travel in name only. 

A:

Is this line of reasoning correct?

First, according to SR, it is just the case that one cannot "speed up to $c$ in space".
Second, as has been stated here many times, there is no inertial frame of reference with relative speed $c$ so statements like "time appears to be stopped" are problematic (there are no clocks with relative speed $c$).
Third, despite some popular accounts, it seems that the notion of "traveling at the speed of light, through spacetime" is conceptually incorrect.  See, for example, this answer by Ben Crowell.

This idea seems to be something that the popularizer Brian Greene has
  perpetrated on the world. Objects don't move through spacetime.
  Objects move through space. If you depict an object in spacetime, you
  have a world-line. The world-line doesn't move through spacetime, it
  simply extends across spacetime.
Greene's portrayal of this seems to come from his feeling that because
  the magnitude of a massive particle's velocity four-vector is
  traditionally normalized to have magnitude c , it makes sense to
  describe the particle, to a nonmathematical audience, as "moving
  through spacetime" at c. This is simply inaccurate.

Fourth, and regardless of the above, speeds are positive.  That is, two velocity vectors $\mathbf{v} = v\hat{\mathbf{x}}$ and $\mathbf{v}' = -v\hat{\mathbf{x}}$ have the same speed $v$ so it isn't clear to me how your logic leads you to your conclusion.

