The current consensus among most researchers is that natural number is not built solely upon a foundation of mental magnitudes. On their way to the conclusion that magnitudes do not form any part of that foundation, Rips et al. pass rather quickly by theories suggesting that mental magnitudes might play some role. These theories deserve a closer look.

We agree that the approximate number system truly represents number. We endorse the authors' conclusions on the arguments from confounds, congruency, and imprecision, although we disagree with many claims along the way. Here, we discuss some complications with the meanings that undergird theories in numerical cognition, and with the language we use to communicate those theories.