Dear Bruno:
je suis emu aux larmes. You really took your time and made it enjoyable.
The first that came to my dirty mind was that males are heterosexual
additions of any couple, females of any homosexual combination. Products of
heterosexuals are female, of homosexuals the same gender as multiplied.
I have to re-read your (confessedly long) essay on numbers again, because at
Jacobi I started to sweep through paragraphs and at Einstein my eyes got
glassy.
I have to find the punctum saliens where you jumped from number-topic into
the non-number world.
I learned about Lagrange, Euler, in 1940-42, then again for Ph.D. in 1947
then forgot them and the others before many of the esteemed list-members
were born.
So allow me to reflect later and thanks again
John
On 4/25/07, Bruno Marchal <marchal.domain.name.hidden> wrote:
>
> 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/ <http://iridia.ulb.ac.be/%7Emarchal/>
>
> >
>
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Received on Thu Apr 26 2007 - 07:09:39 PDT