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!







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