The problem of quantitative properties is better handled by the metric conception of similarity, which posits that there are certain dimensions of similarity concerning different respects, e.g. color, shape or weight, which constitute the axes of one unified metric space. This can be visualized in analogy to three-dimensional physical space, the axes of which are usually labeled with x, y and z. In both the qualitative and the physical metric space, the total distance is determined by the relative distances within each axis. The metric space thus constitutes a manner of aggregating various respective degrees of similarity into one overall degree of similarity. The corresponding function is sometimes referred to as a similarity measure. One problem with this outlook is that it is questionable whether the different respects are commensurable with each other in the sense that an increase in one type can make up for a lack in another type. Even if this should be allowed, there is still the question of how to determine the factor of correlation between degrees of different respects. Any such factor would seem to be artificial, as can be seen, for example, when considering possible responses to the following case: "[l]et one person resemble you more closely, overall, than someone else does. And let him become a bit less like you in respect of his weight by gaining a little. Now answer these questions: How much warmer or cooler should he become to restore the original overall comparison? How much more similar in respect of his height?" This problem does not arise for physical distance, which involves commensurable dimensions and which can be kept constant, for example, by moving the right amount north or south, after having moved a certain distance to the west. Another objection to the metric conception of similarity comes from empirical research suggesting that similarity judgments do not obey the axioms of metric space. For example, people are more likely to accept that "North Korea is similar to China" than that "China is similar to North Korea", thereby denying the axiom of symmetry.
