I am interested in the difference between scales that intuitively have a direction of increase (such as height, which "tends" to taller) versus scales that do not (such as e.g. hair color, which does not "increase" in any direction). It would be nice to have clear, already existing terminology to discuss this difference and so I was wondering whether this distinction has been formalized. I am not completely sure what the difference is beyond the intuition, but I would be happy with anything that could be said to formalize that intuitive difference. I don't want to discuss the difference here, just to check whether someone has already worked on making that intuitive difference more precise.
From high school the closest thing I can think of is an ordered set, but that does not seem to model the distinction I have in mind (since it could model equally well a directed and an undirected scale, depending on how one interprets the ordering relation).
(I am unsure whether this question is too open ended. If so I apologize in advance)
Those two examples you gave sound like two of the four scales of measurement in statistics. To summarize some info from that link:
Nominal scale
Each value on the scale has a unique meaning, but no inherent numerical value. Gender, religion, and political affiliation are examples of variables typically measured on a nominal scale.
Ordinal scale
Values on the scale each have a unique meaning and have an ordered relationship with each other, i.e., one can be "larger" or "smaller" than others. For example, the order in which horses finish a horse race is measured on an ordinal scale.
Interval scale
Values on the scale each have a unique meaning, have an ordered relationship with each other, and there are units on the scale that are an equal distance from each other. The Fahrenheit scale for measuring temperature is an example of an interval scale. Each value is unique, there is a clearly ordered relationship among the values, and the scale units on the scale are equal distances from each other (could be 1 degree, or half a degree, etc., depending on the accuracy of the measuring device).
This may sound very similar to the ordinal scale but the big difference between the two is that the interval scale has the scale units, whereas the ordinal scale does not. With the horse race example in the ordinal scale, we know which horse came in first, second, third, etc., but we don't know by how much the first place horse won. An interval scale would be able to tell us how far ahead of the second place horse the first place one was. An ordinal scale can only tell us that the first place horse was actually first.
Ratio scale
Values on the scale each have a unique meaning, have an ordered relationship with each other, and there are units on the scale that are an equal distance from each other, and there is a zero value on the scale, below which there are no values. Age, height, and weight are examples of values measured on a ratio scale.