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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