Proportionality and Linearity I have a very basic question that a friend and I are disagreeing on. Does (direct) proportionality between two terms a and b imply linearity between the two terms?

Yes, $a$ is directly proportional to $b$ means that $a = k b$ for some $k$. This is a linear relationship. Imagine something like force and acceleration. If I double the force on an object, I double the acceleration. If I have two different forces acting on an object, the effect is the same as one bigger force corresponding to the addition.