Hi,

I think I have already mentionned it (four years ago?) but I have 
another Theory of Everything sleeping somewhere in my brain and books. 
I made an allusion about it yesterday in my post to Peter D Jones, and 
I take this as an opportunity to say some more words on it. Hope it is 
not too technical.
That theory is hardly original because it simply corresponds to the 
Polya-Hilbert conjecture in number theory. To sum it up, Euler 
discovered a rather cute relationship between the infinite sum of the 
inverse of natural numbers, with some exponents, and some infinite 
products related to the prime numbers. Precisely:

Sum (n = 1 to infinity) 1/n^s = Product (on all primes) 1/(1-1/p^s). 
This is the well known zeta Riemann function.

If you can read html:



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Or look at http://primes.utm.edu/notes/rh.html
or here: http://www.maths.ex.ac.uk/~mwatkins/zeta/tutorial.htm

The equivalence of that sum and product is actually equivalent with the 
fundamental theorem of arithmetic (which says that all numbers admit a 
unique decomposition in a product of primes numbers (not taking into 
account the factor order). Thanks to the work of Cauchy (complex 
analysis) Riemann realized that his zeta function zeta(s) defined a 
unique analytical function from C to C (C = the complex numbers), that 
is taking s as a variable on the C complex plane.
Zeta(1), which is the harmonic series, still diverge but all other 
value of s, crazily enough, makes zeta converging. In particular 
zeta(-1) = -1/12. And this is a little bit crazy because it means that 
on the complex plane there is a sense to say that the infinite serie 1 
+ 2 + 3 + 4 + 5 + ...  converge to minus 1/12. (When Ramanujan applied 
for a mathematician job in England, he just annonce that formula to 
show that he has some ability in computation stuff, and then he has 
been lucky enough that Hardy did understood it at once, knowing Riemann 
work).
Now some computation can show that the behavior of the prime numbers, I 
mean the way they are distributed among natural numbers, is controlled 
by a set of zero of the Riemann zeta function.
Actually that control is very fine once we assume those critical zero 
are all arranged on a line, and this is the famous Riemann hypothesis 
which fascinates so much mathematicians. The best introduction to that 
field (for those who remembers their high school math) is the dover 
book:
H. M. Edwards, Riemann's Zeta Function, Dover Publications, 2001. It 
contains the original paper by Riemann.

Now, all those zero could be justified being on a line in the case 
those zero would describe a spectrum of some quantum observable, and so 
this suggests a way to tackle that conjecture, and that way is 
attributed to Polya and Hilbert---from oral talks, see:
http://www.dtc.umn.edu/~odlyzko/polya/index.html

Now a logician can show that if you take the arithmetical formula 
having only the addition symbol (together with  all other logical 
symbols except the multiplication) or, contrarily, only the 
multiplication symbol (and the logical one but no addition) well, in 
each case you can find a corresponding decidable and complete theory 
(with respect to truth for those set of formula).
Put in another way, it is the mixing of addition and multiplication 
which is at the original of the richness of the relation between 
numbers. I have always suspected that the distribution of the prime 
numbers could encoded the logical mess coming from that mixing. In that 
case some "total" knowledge of the distribution of the primes would 
encode a necessarily non trivial part of the (incomplete, 
unaxiomatizable) arithmetical knowledge.

The fact is that, thanks to an algorithm due mainly to Turing and 
refined by Odlyzko, there is a sort of experimental verification of the 
Polya-Hilbert conjecture. Indeed it looks like if the zeros of the zeta 
function describe a spectrum already introduced by quantum physicists 
when they have been confronted to very complex quantum system like big 
atom's nucleus: actually they introduce for such system just very big 
random matrices up  the necessary condition of being unitary. It can be 
shown that such operator (interpreted in some way) describes a form of 
"quantum chaos". Such an "experimental" verification is of course very 
frustrating from the point of view of a mathematician who want to 
*prove* the Riemann hypothesis, but then it is a sort of delight for a 
pythagorean philosopher who want to extract physics from number theory. 
Actually, assuming comp and the 1-3 distinction,  it would be enough to 
show that this "quantum chaos" (or some semi-classical regime which can 
be associated with it) is rich enough to simulate (even in a weak local 
sense) a universal dovetailer, for getting freely a universal quantum 
dovetailer from the prime numbers.
I believe that an extraction of a universal dovetailer from the primes 
would also make the Riemann hypothesis undecidable in a very weak 
arithmetical theory (like Peano Arithmetic, or the weaker Robinson 
one), and that would be enough to prove the Riemann hypothesis (giving 
that if the Riemann Hypo is false then it is refutable in Peano 
Arithmetic (because PA can provably simulate the Turing-Odlyzko 
algorithm which search for zero out of the critical line).

Such a theory is obviously compatible with what we could find by the 
"pure" interview of the lobian machine, and if it is true, should be 
identical.

Such a Pythagorean TOE would give a more direct explanation of physics 
from numbers, as compared to the lobian interview which needs that 
godel encoding of propositions and machines, in term of numbers (many 
mathematicians hates such (intensional) use of numbers), but then such 
a theory would hide the 1-3 distinction in some "Everett like 
interpretation of prime number generated quantum chaos).

Unfortunately I don't know well about quantum chaos. If someone knows 
about a proof of non-universality of quantum chaos .... Actually I got 
a proof of turing universality of classical chaos, and I got evidence 
of many universal facets of the zeta riemann conjecture, thanks to a 
work by Voronin:
http://www.maths.ex.ac.uk/~mwatkins/zeta/voronin.htm
The rest are conjectures; even more difficult than the lobian one ...

Still I hope this post can help you to understand in a different way 
than my usual UDA/Lobian one, (or Hal Finney's universal absolute 
distribution, and other OM based toes) how realities could emerge from 
number theoretical relations. The advantage is that if it works, it 
would provide directly a universal *quantum* dovetailer. The lobian one 
leads only indirectly to it (if it leads to it at all).

Bruno


http://iridia.ulb.ac.be/~marchal/

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