It seems to me that you fail to understand the natural direction of interpretation. Given an "object" O, an "interpretation" of O in terms of I is a map M: I -> O that preserves some structure from I into O. Not the other way around.
Your physical scale (or your physical thermometer) is before all a physical object S. Calling such a physical object S a scale is asserting the existence of an "interpretation map" M: I -> S that preserves the structure of I into S, in a relevant and satisfying way. What is the domain I here? For an engineer, it's g... (read more)