There are several different representation theorems, not just the one by VNM. They differ in what they take to be basic. See the table here in section 2.2.5. As the article emphasizes, nothing can be concluded from direction of representation about what is more fundamental:
Notice that the order of construction differs between theorems: Ramsey constructs a representation of probability using utility, while von Neumann and Morgenstern begin with probabilities and construct a representation of utility. Thus, although the arrows represent a mathematical relationship of representation, they cannot represent a metaphysical relationship of grounding. The Reality Condition needs to be justified independently of any representation theorem.
E.g. you could also trivially “represent” preferences in terms of utilities by defining
x≻y:=U(x)>U(y).
This case isn’t mentioned in the table because a representation proof based on it would be too trivial to label it a “theorem” (for example, preferences are automatically transitive because utilities are represented by real numbers and the “larger than” relation on the real numbers is transitive).
If we want to argue what is more fundamental, we need independent arguments; formal representation relations alone are too arbitrary.
There are indeed a few such arguments. For example, it makes both semantic and psychological sense to interpret “I prefer x to y” as “I want x more than I want x”, but it doesn’t seem possible to interpret (semantically and psychologically) plausible statements like “I want x much more than I want y” or “I want x about twice as much as I want y” in terms of preferences, or preferences and probabilities. The reason is that the latter force you to interpret utility functions as invariant under addition of arbitrary constants, which can make utility levels arbitrarily close to each other. So we can interpret preferences as being explained by relations between degrees of desire (strength of wanting), but we can’t interpret desires as being explained by preference relations, or both preferences and probabilities.
There are several different representation theorems, not just the one by VNM. They differ in what they take to be basic. See the table here in section 2.2.5. As the article emphasizes, nothing can be concluded from direction of representation about what is more fundamental:
E.g. you could also trivially “represent” preferences in terms of utilities by defining
x≻y:=U(x)>U(y).
This case isn’t mentioned in the table because a representation proof based on it would be too trivial to label it a “theorem” (for example, preferences are automatically transitive because utilities are represented by real numbers and the “larger than” relation on the real numbers is transitive).
If we want to argue what is more fundamental, we need independent arguments; formal representation relations alone are too arbitrary.
There are indeed a few such arguments. For example, it makes both semantic and psychological sense to interpret “I prefer x to y” as “I want x more than I want x”, but it doesn’t seem possible to interpret (semantically and psychologically) plausible statements like “I want x much more than I want y” or “I want x about twice as much as I want y” in terms of preferences, or preferences and probabilities. The reason is that the latter force you to interpret utility functions as invariant under addition of arbitrary constants, which can make utility levels arbitrarily close to each other. So we can interpret preferences as being explained by relations between degrees of desire (strength of wanting), but we can’t interpret desires as being explained by preference relations, or both preferences and probabilities.