Colin Hales wrote:
> Hi Brian,
> I was wondering if you could connect (in the paper) the maths with our
> universe? As an example. What set operations or structures correspond to
> the standard particle model entities, what constitutes a chemical
> reaction or energy, what space is made of... that kind of thing. Maybe
> this is supposed to be obvious... if it is then sorry... but you've lost
> (as an audience) the entire world above mathematical
> physics...especially biofolks.
>
I think this is what usual theoretical physics is trying to do. As an
example, Torgny mentioned R^4 as being a relevant structure in General
Relativity. In string theory, the relevant structure is, as far as I've
read in the lay literature, some 11 (give or take) dimensional
manifold. As such, connecting the ultimate context structure I am
working towards to specific structures that represent things like
particle interaction would constitute a complete theory of physics and,
therefore, I myself am unable to see how this would be done. It perhaps
can be done but I lack the knowledge to do so.
Perhaps one thing to keep in mind is that this is a step towards a
mathematical representation of the so called -level 4- multiverse, by
which I'm referencing material here:
http://space.mit.edu/home/tegmark/multiverse.html
In Tegmark's "ultimate ensemble" paper, there is a diagram of physics
and maths structures, part of which is here:
http://space.mit.edu/home/tegmark/toe.gif
All structures, including those in the top row, which are the ones I
think you're asking about Colin, would have the property of being
elementarily embeddable within the ultimate structure I'm
investigating. (Keep in mind the deficency I mentioned in my previous
post.) Roughly speaking, to quote a wiki article, "In model theory
<http://en.wikipedia.org/wiki/Model_theory>, an *elementary embedding*
is a special case of an embedding
<http://en.wikipedia.org/wiki/Embedding#Model_theory> that preserves all
first-order formulas."
In short, the sub-structures, so to speak, of the ultimate structure I
am working towards that are relevant to Quantum Field Theory or General
Relativity (such as R^4) are covered in other texts. This paper I am
working on is to provide an answer to the question which is the subject
of this thread, raised my Tegmark. I'm afraid I don't know enough about
mathematical physics to be more explicit.
> I am a quintessentially visual/spatial thinker.. math does not speak
> very well to me unless I can 'see' the operations happening. in my mind.
> I don;t manipulate symbols. I manipulate 'stuff' and then retrofit symbols.
>
> I would also like to see how an observer with qualia might be
> constructed of it. In other words...how a universe thus constructed
> might create its own scientist describing it in the way you do....Having
> looked at the paper I hold some hope that it might contain a formalism I
> can use to construct the set theoretic basis of my own model... it might
> be within yours....maybe... not sure.
>
>
This is an excellent line of questioning and one I have high hopes to
one day seeing answered. In Tegmark's first of two papers along the
lines of a Mathematical Universe, he mentions what he calls Self Aware
Structures (SAS's). I have spent a lot of time wondering what type of
mathematical structures would have self-awareness. Two candidates that
might be just fumbling in the dark are these:
David Wolpert of NASA has written some interesting articles on what he
calls devices. These devices are mathematical models of scientists plus
investigative tools of scientists. In this mathematical device (not
completely unlike a Turing machine), it starts with a question and ends
with an answer; his papers form a theory of how devices operate. One of
his papers is entitled "the physical limits of inference." Anyway, he
talks at some point about self aware devices, and my understanding is
that these devices X are ones who correctly answer the question "is X a
device?" That is at least some form of self awareness. For a
pseudo-second example of mathematical self awareness, I was thinking of
self-referential first order logical formulas that, in essence, say "I
have property P," but let P be the property "X is a 1st order formula"
so these special self-referential 1st order formulas would essentially
be equivalent to "I am this 1st order formula." To simplify, that is
like the sentence "I am this sentence." The open question is what is
the nature of SAS's that corresponds to human self-awareness. I think
constructing an observer with qualia mathematically would be a most
excellent step and a necessary one to solve that open problem raised in
Tegmark's first MUH paper.
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