Shane Mage wrote:
mathematically "internal relations" means that each of the variables making up "u(.)" is a function of all the others
This isn't what's meant by "internal relations" in the sense I'm using it (which is why I usually put it in quotes). This sense makes the identities of the related entities dependent on their relations. When their relations change, so, to some degree, do their identities. This is relevant to any "logical reasoning which proceeds by the use of the variable." It's independent of whether a mathematical relation between a set of "variables" is linear or non- linear. As Whitehead puts it in the passage to which I pointed: "In logical reasoning, which proceeds by the use of the variable, there are always two tacit presuppositions - one is that the definite symbols of composition can retain the same meaning as the reasoning elaborates novel compositions. The other presupposition is that this self-identity can be preserved when the variable is replaced by some definite instance. Complete self-identity can never be preserved in any advance to novelty. The only question is, as to whether the loss is relevant to the purposes of the argument. The baby in the cradle, and the grown man in middle age, are in some senses identical and in other senses diverse. Is the train of argument in its conclusions substantiated by the identity of vitiated by the diversity?" He sometimes illustrates this with arithmetic: one thing plus another thing doesn't always make two things, e.g (one of his examples) "a spark plus gunpowder." Ted
