Very interesting Max. Filled with erudite ideas and a depth of knowledge far greater than mine!
Can't comment much on the technical stuff, but can talk about the ontological assumptions. The trouble with Platonism is that it's far too simplistic. *all* mathematical concepts are lumped into the same category, which are then defined to exist objectively. Of course, those who think mathematics is only a human invention (nominalists) make exactly the same mistake as the Platonists. They lump *all* mathematical concepts together, then define the lump to be a social construct. But before one starts talking about mathematical concepts, one must be careful to distinguish between *kinds* of mathematical concepts. I think that *some* mathematical concepts exist objectively, some don't. I think we need to be careful to distinguish between *Cognitive Models* (which make references to *mathematical objects* which really do objectively) and *Cognitive Tools* (which include *mathematical procedures* for reasoning about reality). So the distinction here is between *mathematical objects* and *mathematical cognitive tools*. We need to remember that if all the universe is math, intelligent observers have to *use* math to learn about math. The way observers use math *internally* to reason about mathematical things *externally* leads to the division between *mathematical cognitive tools* (subjective) and *mathematical objects* (objective). Thus if all the universe is math then I think we need to give up the idea of complete objectivity. The *mathematical objects* are objectively real, the *mathematical cognitive tools* aren't. However there is close relationship between the mathematical objects and the *mathematical cognitive tools* Let me give you an example of what I mean, because I think you were definitely on the right track when you were talking about the relationship between formal systems and computational models. Using my terminology, I think the formal systems are the objectively existing *mathematical objects*, the computational models are subjective *mathematical cognitive tools*. The computational model is not a mathematical *thing* , it's a mathematical *procedure* that observers use internally. Therefore, the computational model is not something objectively real. However, there is a close mapping between computational models and formal systems. This is hard to explain, but let me say that I think that the *formal system* is the more general concept. The computational model is a sort of *a subjective snap- shot* of the formal system. An apt analogy here might be the taking of photos - you can photograph a physical object from many different angles. In my analogy, the formal system is the externally real object being photographed, the computational model is the subjective 'photo' of the formal system. Another example might be the relationship between Algebra and Category Theory. Here I think standard Algebra is a tool-kit of (non- objective) mathematical *procedures* and therefore not objectively real. However, there is a mapping between standard Algebra and a more general theory: Category Theory. The concepts in Category Theory *are* I think objectively real (they are mathematical *objects*). For instance, the algebraic *operation* '2+2' does not correspond to anything objectively real. However the *category* - the number 4 - *is* objectively real - because it's not a procedure, it's a mathematical object. You see what I'm getting at? In general, each *mathematical object* maps to a corresponding *mathematical procedure*. The mathematical objects are objectively real general concepts, the mathematical procedures are the subjective internal snap-shots. What we need to remember, according to my suppositions, is that mathematical concepts which represent *objects* are objectively real, but mathematical concepts which represent *procedures* aren't. What all this is leading to is this punch-line: I f we believe your 2nd postulate that all the universe is math(which I do) I think we need to give up your first postulate. I do not see what is so bad about giving up the idea ofa completely objective description of reality. I do not believe that giving up the objectivity postulate would spell the end of the quest for a TOE. It would just mean that a TOE would have to *include* direct conscious experience (subjective elements) in order to be fully comprehended. This sounds highly strange, but it's not impossible. Onward! Cheers! --~--~---------~--~----~------------~-------~--~----~ You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to [EMAIL PROTECTED] To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~----------~----~----~----~------~----~------~--~---
