Hi,
I have found an universal diophantine polynomial on the net (the one by
J.P. Jones). To be sure it is a *system* of equations, but by adding
the squares of all the polynomials (by writing them in the form P = 0)
in the system, you will get an equivalent unique polynomial which is
still (turing) universal. I let you do that very long but easy
computation :)
I cannot copy it without making disappear the exponent (for some
mysterious reason);
So, if you are interested, please look at:
http://www.math.ucalgary.ca/~jpjones/abst1982.htm
to see it, and how to use it to generate any recursively enumerable
set Wi.
It is also possible to reduce the degree of such polynomial to 4. It is
an open problem if 3 would be enough (2 can be shown not being enough).
If there is a universal diophantine polynomial of degree 3 (open
problem) this would considerably help to extract the "particle
structure" of the comp physics. I have evidence that if there is a
universal cubic (degree 3) polynomial, then we get "string theory like"
subparticle physics at least.
The role of the monster in string theory makes me think that such a
cubic universal polynomial should exist if both comp and string theory
is correct. In that case the communicable 3-comp-physics would be
related to a recursively presentable group, and the non communicable
3-comp-physics would be related to a non recursively representable
group. The 1-physics, i.e. the fifth hypostases, remains rather
mysterious, especially its non communicable part.
Well, I don't know if it is a good or a bad new, but Matiyasevich's
result makes it possible to use algebraic and analytical number theory
tools in recursion theory, and thus provides tools for studying the
needed models for the Z and X logics (4th and 5th hypostases).
Here a nice pdf by Yuri Matiyasevich on diophantine equation (256 kb):
http://logic.pdmi.ras.ru/~yumat/Journal/H10history/H10histe.pdf
Less technical, he wrote "My collaboration with Julia Robinson", a
paper which try to explain of Julia Robinson miss the discovery of the
Universal Polynomial (and thus the negative solution of Hilbert tenth
problem) although she paves the way, notably with Martin Davis (the
editor of the comp bible: "The undecidable"), and Hilary Putnam.
Sorry for those who dislike math, but there is no hope to prgress if we
don't take seriously our theory, and with comp, some computer science
and number theory is unavoidable. Here I just point on some of my
post-thesis technical progress. It looks I can no more avoid what I did
call a long time ago (in this lmist) the mermaid's song of mathematics
....
Bruno
Le 21-sept.-06, à 17:19, Bruno Marchal a écrit :
>
> 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/
>
>
> >
>
http://iridia.ulb.ac.be/~marchal/
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Received on Fri Sep 22 2006 - 09:59:03 PDT