On 05 Aug 2010, at 01:18, Brian Tenneson wrote:
Hmm... Lawvere has tried to build an all encompassing universal
mathematical structure, but he failed. It was an interesting failure
as he discovered the notion of topos, (discovered also independently
by Groethendieck) which is more a mathematical mathematician than a
Also Tegmark is not aware that Digital Mechanism entails the non
locality, the indeterminacy and the non cloning of matter, and that
DM makes the physical into a person-modality due to the presence of
the mathematician in the arithmetical reality.
Quanta are special case of first person plural sharable qualia.
I'm not looking for a truly all-encompassing mathematical
structure. What I'm looking for is a mathematical structure in
which all mathematical structures can be embedded. By mathematical
structure, I mean there is a symbol set S consisting of constant
symbols, relation symbols, and function symbols, and the pairing of
a set with a list of rules that interpret the symbols. In Tegmark's
papers on "ultimate ensemble TOE" and "the mathematical universe,"
he refers to what I call a mathematical structure as a "formal
system" (and also mathematical structure).
The structure I'm looking for wouldn't encompass anything that isn't
a mathematical structure, like a category with no objects/elements.
You may encounter a problem with the notion of 1-person, and
Tegmark argues that reality is a mathematical structure. What's
cute about his argument is that while invoking the concept of a TOE,
his argument is independent of what that TOE might be. He defines a
TOE to be a complete description of reality. Whether or not this
can be expressed in a finite string is an open problem as far as I
know. (I doubt it can.) He argues that a complete description of
reality must be expressible in a form that has no human baggage and
I would add to that is something that exists independent of humans
in the sense that while the symbols used to provide that complete
description will depend on humans, what is pointed to by the symbols
Computationalism entails something very near such view indeed. It
entails also that if such structure make sense, then its cardinality
is unknowable by the self-aware beings that could be generated inside.
The statement that the cardinality of the mathematical universe is
countable or not is absolutely undecidable, from 'inside'.
Tegmark argues that reality is a mathematical structure and states
that an open problem is finding a mathematical structure which is
isomorphic to reality. This might or might not be clear: the
mathematical structure with the property that all mathematical
structures can be embedded within it is precisely the mathematical
structure we are looking for.
The problem is in defining "embedded". I am not sure it makes set
theoretical sense, unless you believe in Quine's New foundation (NF).
I am neutral on the consistency of NF.
With a large sense of "embedded" I may argue that the mathematical
structure you are looking for is just the (mathematical) universal
machine. In which case Robinson arithmetic (a tiny fragment of
arithmetical truth, on which both platonist and non platonist
(intuitionist) is enough. Indeed, I argue with comp that Robinson
arithmetic, or any first order specification of a (Turing) universal
theory is enough to derive the appearance of quanta and qualia.
I am confident that I have found such a structure but only over a
fixed symbol set; I need such a structure to be inclusive of all
symbol sets so as to cast away the need to refer to a symbol set.
This again follows from Church thesis, for the 'computationalist' TOE.
The technique I used was to use NFU, new foundations set theory with
urelements--which is known to be a consistent set theory, to first
find the set of all S-structures.
All right, then.
Then I take what I believe is called the reduced product of all S-
structures. Then I show that all S-structures can be embedded
within the reduced product of all S-structures. Admittedly, there
is nothing at all deep about this; none of my arguments are deeper
than typical homework problems in a math logic course.
That may be already a lot for non mathematical logicians ...
My next move is to find justification for the existence of a math
structure with the important property that all structures can be
embedded within it --independent of the symbol set-- and thus
eliminating the need to refer to it.
One thing I wonder is how to define all your notions such as
"mathematician," "n-brains," "n-minds," and "digital mechanism" in
terms of mathematical structures.
This is done. Everything is defined in term of number and number
relation. But it is not asked that the relation is arithmeticaly
definable. For example, the ONE of Plotinus is Arithmetical truth, and
can be represented by the set of Gödel numbers of true arithmetical
proposition. Of course the "internal machine will have to build richer
epistemological tools (like analytical tools, set theoretical tools)
to talk about all this. The internal epistemology is richer than the
needed external ontology. It is made consistent by some use of Skolem
I'm particularly interested in defining something that models
awareness and using it to find self-aware structures such as
Mathematician are just Löbian machine/theory/number. They are
(recursively) equivalent with Robinson Arithmetic + the induction
axiom, i.e. Peano arithmetic. They can be characterized by being
universal and knowing it. They can dream. More exactly, they cannot
not dream, and physical realities appears when collection of machine
glue well the dreams. This needs natural coherence property, and it is
suggested that those coherence properties are provided by the
intensional variant of the logic of self-reference (the Z and X logics).
This solves conceptually, I think, the mind body problem, in a way
compatible with the computationalist breakdown of the identity thesis.
I hope you see that to find (truing-emulable) self-aware structure is
not enough, given the inconsistency of the identity thesis with
mechanism (cf UDA). You have to recover the physical reality from the
breakdown of the identity thesis: physics has to be recovered by all
the infinite computations leading to the state of that self-aware
structure. This makes, strictly speaking, the physical reality out of
existence. Like in Plotinus theory of matter. Both God and Matter are
out of the intelligible Whole (the Noûs).
Digital mechanism (the tiny arithmetic TOE) entails already a large
part of Quantum Mechanics, and then group or category theoretic
considerations (and knot theory) might explain the 'illusions' of
time, space, particle, and (symmetrical) hamiltonians, and why indeed
physical reality should appear as an indeterminate state of a physical
vacuum. But the logic-math problems remaining are not easy to solve.
That is normal in a such top down, mind-body problem driven, approach
to physics (and psychology/theology/biology).
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