Ted Winslow wrote:
As I said, this conception of "preferences" assumes that, to the extent required by the argument, the identities of the related entities (the "variables") remain unchanged with changes in their relations. This is explained in the Whitehead passage. Where this isn't true, i.e. where relations are "internal" in a relevant way, the hypothesis "breaks down." This is true independent of how abstract you make the "variables."
With all due respect to Whitehead, if the "internal change" of variable X invalidates its use in a given reasoning, then how come the "internal change" of your term "internal change" doesn't invalidate your statements? I mean, your term "internal change" changed internally from the time you began to type it to the time you ended typing it. How seriously should I take your terms then? Don't you realize that your (or Whitehead's?) argument would make all discursive (not only all mathematical) logic impossible, since the content of *all* terms is subject to "internal change" as a result of their relations with other terms? Or are your terms only externally related to your overall argument? Shane Mage wrote:
But in this type of economic theory you do *not* have "ample room to determine... the form of u(.)..." The form of u(.) must be *linear* if it is to be in any sense workable.
Uh? I'm not sure what you mean, since a bunch of important economic results have been derived without having preferences posed as *functions*. They have been instead postulated as (mathematical) "relations." Gerard Debreu's proof of general equilibrium introduces, not a preference- or "utility" function, but a "preference relation" between sets. It's not even a function, let alone linear.
Now, mathematically "internal relations" means that each of the variables making up "u(.)" is a function of all the others, so that the "form of u(.)" has to be nonlinear, and therefore useless.
Uh? Are you serious? How would you respond if someone said to you that you can't use the term "internal relations" as the subject of a sentence, because "It can only be used in the predicate or else it's useless"? You'd probably say: "I'm the one who decides whether it's useful or useless as the subject in a sentence or playing any other syntactic role. I'll use it as and wherever I please, as long as it helps me express myself." Exactly!
