What do we mean by the dimensionality of behavior?

There is growing effort in the "physics of behavior" that aims at complete quantitative characterization of animal movements under more complex, naturalistic conditions. One reaction to the resulting explosion of data is the search for low dimensional structure. Here I try to define more clearly what we mean by the dimensionality of behavior, where observable behavior may consist either of continuous trajectories or sequences of discrete states. This discussion also serves to isolate situations in which the dimensionality of behavior is effectively infinite. I conclude with some more general perspectives about the importance of quantitative phenomenology.