Hi John,
The 24 Feb 2007, à 23:59, John Mikes wrote in parts, to Jason:
Don't tell me please such "Brunoistic" examples like 1+1 = 2, go out
into the 'life' of a universe (or of ourselves).
All right, let me try to give you a less 'Brunoistic" relation among
numbers, if you can imagine this possible!.
I am afraid this will be a long post, as I will point toward the
unravelling of the origin of the multiverses---and the mess within
:-)
And I will not be theological, nor interview any universal machine, but
this little trip in the less brunoistic (hopefully) part of platonia
could perhaps help for explaining or illustrating the interview later.
But then I have to ask you, also, to accept a deep, but not so well
known, except by the Greeks of course, truth about numbers.
(... making this post not for children, and probably neither
politically correct ...)
But numbers have gender.
You have male numbers and female numbers. The rule is simple: odd
numbers are male, and even numbers are female:
ODD = MALE; EVEN = FEMALE
[Aparte: I guess the Greeks were a bit macho by thinking that the very
*first*, its majesty the one, 1, was a male. Today we are a bit more
modern, I guess, and we know that the big 1 is really situated in
between the two most terrible female of Platonia: the number 0 (death)
and the number 2 (Life, the Pythagorean Indefinite Dyad, ...). ... and
then comes 3, oh my! it's a boy! ...]
TOC, TOC, TOC!
What's that now? Ooooh!!! ... a categorical daemon!!!:
"--- about life and death, 0 and 2, add the arrows! add the arrows!"
(Categorical daemons always ask you to add arrows). ...
"0 -> 2 = elementary creation operator, 2 -> 0 = elementary
annihilation operator, cup and cap in quantum topological Temperley
Lieb categories, ...".
(ok, ok, keep calm, I manage the categorical daemon..)
Let us go back to pure (without arrows) numbers ...
SURPRISE PARTY!
Now, if numbers have gender, it obviously follows, mainly by addition
and multiplication, that, contrary to a widespread rumor about Platonia
austerity, there are many surprise parties in Platonia. Number
theorists would talk about *surprising partitions* but let us not be
distracted by vocabulary question (see the biblio below).
And, of course, it follows from this that numbers have clothes: you
would'nt imagine a number going naked to a party, all right?
Now, giving that we dispose only of numbers, together with addition and
multiplication, a clothe can only be a sum of product or a product of
sums, of numbers, including, for more decoration purpose (by pure
Platonia's frivolity!), integers: ..., -3, -2, -1, 0, 1, 2, ...
THE FOUR SQUARE PARTY
A very famous and popular party has always been the *four square* party
where numbers are permitted to participate only if disguised as a sum
of four squared integers. For exemple the number eleven can put the
following clothe: (-1)^2 + (-3)^2 + 0^2 + 1^2 = 1 + 9 + 0 + 1 = 11. The
order in the sum distinguishes the clothes: 1^2 + 2^2 + 3^2 + 4^2 and
4^2 + 1^2 + 2^2 + 3^2 are two different clothes of the number 30, for
example.
Death has only one clothe, in its garde robe: 0 = 0^2 + 0^2 + 0^2 + 0^2
And 1 ? One has eight clothes!
1^2 + 0^2 + 0^2 + 0^2
0^2 + 1^2 + 0^2 + 0^2
0^2 + 0^2 + 1^2 + 0^2
0^2 + 0^2 + 0^2 + 1^2
and
(-1)^2 + 0^2 + 0^2 + 0^2
0^2 + (-1)^2 + 0^2 + 0^2
0^2 + 0^2 + (-1)^2 + 0^2
0^2 + 0^2 + 0^2 + (-1)^2
And Life? The number 2, how big is her garde-robe? I let you play with
it.
[ .......................... (= symbolic time you are playing with it),
of course take *your* time.]
Answer: 24.
Hmmmm.....
POPULARITY
Where did the popularity of the four square party come from? Because
all natural numbers have four square clothes, i.e. any natural number
can be put in the form a^2 + b^2 + c^2 +d^2 for some a, b, c, d,
integers. Why? Because
LAGRANGE THEOREM (say so).
And what is remarkable about that theorem?
?
Err ... Well, what is remarkable is that EULER did not find the proof
of it, according to bad tongues. (Worst tongues say that Euler did not
find any proof, which is a comble of unfairness given that it is Gauss
who will introduce later the rigorous (although informal) notion of
proof.
To evaluate the unfairness,, remind yourselve that Euler solved BASEL
problem, at the age of 15. Everybody knew, experimentally, that the
infinite sum of the inverse of the squared natural numbers: 1/1^2 +
1/2^2 + 1/3^2 + 1/4^2 + 1/5^2 + ... converges to some anonymous real
number: 1,6..., but Euler recognizes it: one sixth of the square of PI.
1/6 (PI)^2, = 1, 64493406..., relating square numbers and the circle!
That was a famous problem which has defeated the big one like Leibnitz,
Newton, and even the whole Bernouilli family.
So, later Lagrange found a proof that indeed all natural numbers have a
non empty garde-robe, and I have myself found a short proof in ... the
first appendix of the marvellous Matiyasevitch book!
... and I have found another short proof in a book by Ian Stewart and
David Tall, based on an important theorem in geometrical number theory
by Minkowski (yes, the physicist, the treator was doing number theory
part time! Note that if you meet the name of Bohr in number theory,
don't conclude that Niels Bohr the physicist is also a physicist
treator, Harald Bohr the number theorist was just Niels Bohr's brother
..)
... and what all this has to do with the physical multiverse ? (Keep
on, John, we are driven toward it).
THE GARDE ROBE SIZE PROBLEM
A question arises. 8, 24, is there a pattern? How many clothes have
each natural numbers? Is there a rule? Is there a simple formula
capable of giving the size of each number garde robe?
Yes, there is one. Or more simply, there are two related simple rules,
as it is simpler (as we could have guess) to separate the rule for the
garde-robe of the male numbers and the rule for the garde-robe of the
female numbers.
JACOBI THEOREM:
1) All odd numbers have a garde-robe containing 8 times the sum of
its divisors.
Female numbers are both more demanding and more choosy:
2) All even numbers have a garde-robe containing 24 times the sum
of its *odd* divisors.
...
And what is remarkable about JACOBI theorem---beyond that feeling of
déjà-vu?
?
What is remarkable is that such relations have ever been able to be
proved. Unlike Lagranges theorem (the democratic character of the four
square party), which admits a two pages proof, well, I have not found
any short proof of Jacobi theorem.
One proof relates Partition Theory (the "surprise party" branch of
number theory, usually called the additive number theory), that branch
includes big set of marvellous formula by Euler ---so I said, one of
the proof relates partition theory, modular form, Heat Theory (sic),
Jacobi theta series ... Yes, you have to wandle in the space of heat
equation solutions space!
Ah! But I found a more modern proof where it is a "simple" consequence
of the existence of a ... super boson fermion correspondence in some
(abstract!) stringy quantum field theories!
This illustrates a new phenomenon in the story of the relation between
math and physics. The physicist, like Einstein, were used to pick
already existing mathematical theories (tensor calculus, Hilbert space
theory, ...) to develop their physical theories. Today, mathematical
physics has superseded the mathematical sophisticateness of the
mathematicians, and it looks like the physical universe is just God's
hint for solving purely number theoretical problems. It shows also that
number theorists, would they have the right fundamental motivations,
could as well find the physical TOE we are searching ...
But I don't want really push all this forward. One reason is that such
a line could lead us in really hard mathematical stuff, and this would
make people flying out of this informal list conversation. But the main
reason is that I would not like to encourage the number theorists into
searching this physical TOE.
Why?
Hmmm... Because, it is said, in the "Chronicles of the Multiple Third
Millenia" that in those realities where number theorists find the
physical TOE, the third millenia are wittnessing the suffering of acute
arrogant *numberism* capable of perpetuating the two Millenia old
Aristotelian physicalist person elimination. On the contrary, in those
third millenia-realities where the *correct and unphysicalist* TOE are
discovered by the platonist or neoplotinist theologians, the person
notions and rights will be saved so that we will not have to live again
one more millenia of authoritative obscurantism in the human
sciences---and in that case I will not have to come back again, so that
I will be able to take some well deserve (in that case only) holidays
:-)
And that is why I will preferably keep on with my brunoistic simple
examples of number relations, like "1+1=2", or "17 is prime", instead
of Jacobi theorems or any other deeply fascinating purely third person
number theoretical truth.
Instead, I will insist more than ever on the advantage of extracting
physics *and* actually extracting the whole conceptual unification of
*all* fields through the interview of the lobian universal
self-observing machine, which guarantees, through Lobian
incompleteness, the non-normative acceptance of the reality of the
many-many-person-many-many-points-of-view.
Best regards,
Bruno
Bibliography:
A very nice introduction to the general number partition theory is
given by the adorable little book by George E. Andrews and Kimmo
Eriksson: INTEGER PARTITIONS, Cambridge University Press, 2004.
I have already given the reference of Yuri Matiyasevich chef d'oeuvre:
Hilbert's Tenth Problem, MIT Press, 1996 (third printing). Lagrange
theorem is proved in the appendix.
The proof of Lagrange theorem based on Minkowski's geometrical number
theory can be found in the book by Ian Steward and David Tall:
Algebraic Number Theory and Fermat's Last Theorem, A.K. Peters Ltd,
Natick MA, 2002.
The first proof of Jacobi Theorem, the one who makes you make a non
standard trip through heat theory, can be found in the book by Yves
Hellegouarch, Invitation to the Mathematics of Fermat-Wiles, Academic
Press, Elsevier, (translated from the french) 2002.
The second proof of Jacobi theorem, the one which makes it a
consequence of the existence of super fermions-bosons correspondence in
physics, can be found in the book by Victor Kac: Vertex Algebras for
Beginners, American Mathematical Society, University Lectures Series,
Vol. 10, Rhode Island, 1997. The label "beginners" is misleading: you
don't have to know anything, indeed, about Vertex Operator Algebras,
but you have to know almost everything about almost all other known
operator algebras! ... Not easy stuff, even for mathematicians.
The awakening of the "categorical daemon" points toward the recent
works by
1) Samson Abramski : Temperley-Lieb Algebra: From Knot Theory to Logic
and Computation via Quantum Mechanics,
and by
2) Louis Kauffman: q-deformed Spin Networks, Knot Polynomials and
Anyonic Topological Quantum Computation. You can find both of them
freely on the net. They both confirmed my feeling that knot theory
provides the reasonable combinatorial discrete structures which should
appear in between number and physics. Similar such structures
*should* (and in some sense already) appear naturally in the semantics
of the modal theories (S4Grz1, Z1*, X1*) generated by the arithmetical
lobian interpretation of the third, fourth and fifth Plotinus
"hypostases". If confirmed, that would justify why our physical
neighborhood seems to be run by a *quantum* machine.
http://iridia.ulb.ac.be/~marchal/
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Received on Wed Apr 25 2007 - 06:31:25 PDT