Hi all,
About the question by Stathis: "does any piece of matter implement all
computations", after reflection I would say that both comp and the
quantum should answer in the positive.
First in reasonable quantum field theories, the vacuum is already
turing universal (and 1-person relatively unstable with Everett). Also
to compute the probability that the piece of vacuum O go from state O
to state O you have to compute the (amplitude of) probabilty that O go
to C and that C go to 0, and this for *any* C, so that strictly
speaking you have to take into account the infinity of white rabbit
histories where for going from 0 to 0 the vaccum go toward the ten
thousands big bangs and crunches and other brane collisions (modulo
time question).
That is why both with comp and with the quantum we met infinities and
we take a look for renormalization strategies.
For example in string theory when you describe the "vacuum" state Zero
(of the relativistic open quantum string) you have to sum on a
reasonable combination of creation and annihilation operators (as
usual) but you have to add, actually, some shift which you hope to be
equal to minus one if you want the string spectrum to include the
massless photon, but instead of -1, string theory gives 1/2(D - 2)(1 +
2 + 3 + 4 + 5 + 6 + 7 + ...), where D is the dimension of the brane
(26). If that does not look like an infinity.
But here the String theorist are lucky because number theorist knew
already that *in the complex plane* the sum of the gaussian integers 1
+ 2 + ... = (1 + 0i) + (2 + 0i) + ... is equal to the value of Riemann
Zeta function on -1 zeta(-1), and this can be computed (by analytical
extension) and it gives -1/12 (fractional and negative!). But so there
are massless photon in the open string spectrum! (cf: zeta(s), s
complex number, is equal to (the provably unique analytical extension)
of the sum of the inverse of the natural numbers n up to s = Sum 1/n^s.
Euler showed us that this sum is deeply related to the prime numbers.
Hope you all recall that a^(-1) = 1/a.
Why do I talk about string theory now?
Remember the two "ontic" theories. I have already described two of
them: Robinson Arithmetic and the COMBINATORS. Their are ontologically
equivalent, and there are many others such theories. But the choice of
representation is important once we want to extract efficiently
information on the possible person views (plotinus hypostases).
- Robinson Arithmetic is important because it makes the Universal
Dovetailer "discourse" part of the much richer lobian discourse, and
that makes it possible to keep track of the difference between *true*
and *provable* (arithmetical) propositions. In Plotinus term it keep
tracks of the difference between earth and the divine.
- COMBINATORS is important because it gives a very fine grained on the
computations making it possible to sum up classical physics and quantum
physics in a very rough but illuminating way: classical physics = no
kestrel, quantum physics = no starlings: i.e. no loss and no creation
of information. This leads to a BCI combinator algebra with genuine
linear epistemic extension (and modelizable by symmetrical monoidal
categories).
- Now, for the measure corresponding to the 3-physical point of view
(the arithmetical fourth hypostase, "intelligible matter", the first
person plural) I think, since I have finished Matiyasevich's book(*),
that the diophantine representation could be used to provide a shortcut
to the 3-person physics, actually (like I said once) by the study of
(irreducible presentations of) the groups of permutations (on some
fields) keeping the roots of an universal (in turing sense, and more
general with CT) diophantine polynomial. See the summary of the ontic
theory (of everything) representation below.
Now each time I think (this includes reading books, goggeling ...)
about those diophantine equations I am driven toward those modular
functions and modular forms, which are basic tools in "advanced number
theory" (like the one used to settle Fermat for example, a very famous
diophantine problem. I guess the mathematically inclined everythinger
has heard about those modular forms.
But did you heard about Monstrous Moonshine? This is incredible and
nobody told me!
It appears that the coefficient of the most "basic" modular form (big
integer, like 196883) are related to the dimension of the irreducible
representation (complex matrices) of the
element of the Monster (the bigger simple finite "sporadic" group,
where sporadic means that it belongs to the 26 weird finite simple
groups which cannot be put in any reasonable classification (a simple
group is to a group what a prime number is to a number).
Like in the Polya Hilbert story that relation has been the object of a
conjecture. Conway conjectured it in a mathematically precise manner
by conjecturing the existence of some "graded algebra" relating the
Monster and the modular form, but unlike the Polya Hilbert conjecture
about Riemann, this one has been solved, by its student Borcheds (he
got the medal field for that).
And it appears that the genuine graded algebra has been found through a
direct inspiration of string theory. Indeed, apparently (I am still
discovering this) the algebra is at least a precise algebraical "toy"
string theory.
This gives two good news for the string theorists:
1) if their loose their job in physics, they will be welcomed in Number
Theory!
2) if comp is true, and if string theory is true, string should
(re)emerge in the first person plural (fourth) hypostases, and this can
be related to the irreducible representation of permutation group of
universal polynomial's roots, probably on all number theoretical rings
and fields. Somehow string theory suggests that the "many world" idea
extends itself to the many number systems, many topologies, ...
Greg Egan was correct with his permutation idea, but Adams is false, 42
is not the solution of the riddle of the universe, it is definitely 24
which plays some weird but big role here. I am looking to see if that
24 is related to the 24 from Ramanujan partition formula. I think so.
Bruno
APPENDICE:
The ontic theory: three equivalent representations (sketched):
1) RA (Robinson Arithmetic)
Classical logic + the successor axioms + the recursive definition of
addition + the recursive definition of multiplication (formalisable in
first order predicate logic, see Podnieks page). I have already give
you the formal presentation.
Advantage: RA is turing equivalent and at the same time a subtheory of
all correct lobian machine discourse, like Peano Arithmetic PA which is
RA + the induction axioms.
In this representation universal computability is a weak subcase of
provability.
The "B" in the description of the hypostases is for Godel's translation
of provability *in* PA, i.e. in term of addition and multiplication of
numbers. By Solovay theorem we inheritate the two modal logics G and G*
formalising the propositionnal level of self-correctness. It is here
that we get the 8 hypostases:
ONE: p (does not split)
INTELLECT: Bp (split by G* minus G)
SOUL: Bp & p (does not split)
INTELLIGIBLE MATTER: Bp & Dp (split by G* minus G)
SENSIBLE MATTER: Bp & Dp & p (split by G* minus G)
You can call them Truth, Reason, Knowledge, Observation and Feeling if
you prefer, but please keep the G/G* splitting in mind.
2) COMBINATORS:
They give a quasi-algebraical ontic toe with interesting categories as
models. From some related point of view, their little cousin LAMBDA
EXPRESSION are better at the job.
Suitable for the structure of computations (as opposed to
computability). Two important version emerge, the SK and the BCI, but
the second is not turing universal and is supposed to describe the
"uncrashable" (and thus non universal) rock-bottom physical reality.
a) With kestrel and starlings: (Eyes closed, SK quasi-algebra)
Axioms:
Kxy = x
Sxyz = xy(xz)
Rules: (no need of classical logic!). Just the rules:
- you can infer x = x
- from x = y and y = z, you can infer x = z
- from x = y you can infer y = x
- from x = y you can infer xz = yz
- from x = y you can infer zx = zy
b) Without kestrel and starling (Eyes open, BCI algebra)
Axioms:
Bxyz = x(yz)
Cxyz = xzy
Ix = x
Same rules of inference.
(careful BCI-algebra are not turing universal, unlike SK or BCK), the
epistemic extension is needed, it gives a sort of "dual point of view".
Explanations:
See my old post on this, + my last (Elsevier) paper. Note that this is
post thesis material. Like what follows:
3) Universal diophantine equation. See Matiyasevich's book.
The diophantine set are exactly the Wi of the recursion scientist, by
Matiyasevic result(*).
Like there is a universal Wu, there is a universal diophantine
polynomial Pu. Some of its parameter can be used for coding any turing
set or function in term of its roots or through its positive values.
For example for precise integer values on some of its parameters such a
polynomial has all and only all prime numbers as value.
This is really the golden bridge between number theory and recursion
theory (alias computer science).
And Monstrous Moonshine (see above, see more on this with Google) seems
to be a promising bridge between number theory, algebra, topology and
... physics.
(*) See Matiyasevich's book "Hilbert's Tenth Problem". The MIT Press,
1993. Third printing 1996. That book is a total chef-d'oeuvre. It is,
given on a plate, the bridge between number theory (Diophantine
equation) and recursion theory.
PS: part of this post has been written some time ago. Since then I have
read (quickly, not doing all exercices!) the marvellous book by Terry
Gannon "Moonshine beyond the Monster: The Brigde Connecting Algebra,
Modular Forms and Physics (Cambridge Monograph on Mathematical
Physics). Superb, very well written, but expensive (130 $).
Unexpectedly perhaps, Gannon is a Many-Worlder!
For String Theory, Barton Zwiebach's book "A first course in String
Theory" remains the more readable.
Of course, to read such books you have to love the numbers a bit ...
Louis Kauffman's book "Knots and Physics" remains quite genuine in this
context (cf my older post on knot theory, Yetter, etc.).
http://iridia.ulb.ac.be/~marchal/
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Received on Thu Sep 21 2006 - 11:21:38 PDT