I've heard RS2 uses 15 dimensions! What's up with that?

No, RS2 does NOT use 15 dimensions... it uses 3 dimensions with 15 "degrees of freedom". Since RS2 incorporated information determined from "virtual reality" models in computers, some of the techinques used in computer modeling were adopted for use with RS2:

Homogeneous Coordinates

Are the conventional X, Y, Z coordinates that includes a 4th coordinate that is a scale factor, w. Use of this form allows the plane at infinity to be modeled, thus eliminating the problem of division by zero or reaching infinity. The coordinates are represented in matrix form as: [ X/w Y/w Z/w w/1 ]. Note that in terms of "motion", it represents 3 SPEEDS: x/t, y/t, z/t, which is much closer to Larson's concept.

Quaternions

One of the big problems with rotation in computer models is the old "Gimbel Lock" problem, which happens when trigonometry is used solely for rotation. There are circumstances where the equations cancel each other out, causing a rotation to jump to an unexpected orientation. The use of quaternions, another 4-coordinate system that uses a scale factor and 3 "imaginary" operators to represent the rotation on X, Y and Z axes, eliminates the problem. Quaternions are also expressed in matrix form: [w/1 iX/w jY/w kZ/w]. In terms of motion, we are looking at the inverse of homogeneous coordinates: iX/s jY/s kZ/s.

Transformation Matrix

After a couple decades of thinking about "motion", Larson eventually concluded that he was dealing with nothing more than "abstract change in three dimensions" (from Video). "Abstract change" is represented in virtual models by a concept known as a "transformation matrix", a 4x4 matrix that is a composite of rotation, translation and scaling, with the "4th coordinate" being the scaling factor representing the point and plane at infinity. The requirements for a transformation require that ONE of the 16 cells be defined; so normally 1 cell is fixed at Unity, with the remaining 15 representing the degrees of freedom of motion. This is where the misconception of "15 dimensions" originates.

References

An excellent reference on Projective Geometry and how matrix mathematics are used in it can be found at: Marc Pollefeys Visual 3D Modeling from Images