<^> Each truth has its context : . . Truth is consistent & complete, upto a point . . . (~ K.Goedel)
. . . . . . . . . . " A theorem's value amounts to the theory it makes superfluous ". . . . (~ D.Hilbert)

<^> If Reality R has a next-state model,
. . . then Truth G is a Global image or independent component of it : . . . . . <---- R ---->
. . . G is a simpler con-sistent (static) and con-sequent (dynamic) projection : [G] --> [S]
. . . Component S is a local Subsystem depending on (via its coupling to) G,

Including "dissipative effects" at inter-galactic distance travelling
photons (re: Redshift) the more global view might have to adjust to
a more complex "non-uniform medium" (ether-drag) model for light
propagation, to arrive at a simpler and more encompassing galactic-scale
model. It is - again - all a matter of context (with new orbiting
telescopes observing deep space), is'nt it?

Subj:   Re: Q re Goedel's work re unprovability
Date:   15 Nov 2002 (sci.math)
From:   Nico Benschop 
 Org:  Digital Research : Finite Associative Networks
----------
Arturo Magidin wrote:
>
> In article ,
> Leonard Blackburn  wrote:
> > [...] There are philosophical reasons for choosing each axiom.
>
> Could you please state the "philosophical" reasons
> for the axioms of semigroup theory?     -- AM

Interesting case. I guess this is about why 'associative' is
taken as composition property (taking 'closure' for granted).
I think it is not so much a 'philosophical' but a 'practical'
choice: for instance the rather ubiquitous and simple concept
of "iteration" [a basic concept for arithmetic, re Peano]
requires associativity, since otherwise (aa)a =/= a(aa) is
possible, while [using multiplicative notation]:
in a semigroup S: (aa)a = a(aa) = a^3 , for each  a in S.
I suspect that axioms arise for such 'trivial' and common sense
practical reasons, not so much for 'philisofical' reasons. -- NB

Subject:  Re: On prime producing polynomials
   Date:  12 Nov 2002 (sci.math)
   From:  Nico Benschop 
    Org:  Digital Research : Finite Associative Networks
--------
Erick Bryce Wong wrote:
>
> Gerry Myerson   wrote:
> >erick@sfu.ca (Erick Bryce Wong) wrote:
> >=> >Is it known whether or not Euler's polynomial can be
> >=> >non-trivially improved upon with just one indeterminant?
> >=>
> >=> This isn't known.  I think most number theorists believe that
> >=> it can be improved upon, as a consequence of the prime k-tuples
> >=> conjecture, but as far as I know, an example has yet to be found.
> >
> > At http://www.primepuzzles.net/problems/prob_012.htm it says,
> >
> >      Ruby found that the absolute value of (36X^2-810X +2753)
> >      assumes 45 distinct prime values for the 45 integers X,
> >      0<=X<=44.
> [snip]
> >     2.Ruby and Fung have the second record polynomial,
> >       47X^2-1701X+10181 which produces 43 distinct prime values
> >          for   0<=X<=42
>
> Oops, I should have looked more closely :}. These have in fact been
> known since around the late 80's.  There is also another polynomial
> that ties for second place, namely 103X^2 - 3945X + 34381.
>  -- Erick

How about generalizing a bit, to cover all primes by considering
      the prime values of a set QP of quadratic polynomials.
Must QP necessarily be infinite, or could a finite QP suffice?

"There's definitely something quadratic about the primes"...
-- NB:   all non-primes: ab = [ (a+b)^2 - (a-b)^2 ] /4

Subject:  Re: what is the logic behind zero
   Date:  16 Aug 2002 (sci.math)
   From:  Nico Benschop 
  Org'n:  Digital Research : Finite Associative Networks
-------
Teacherjh wrote:
>
> [...]
> Here's another mystery for you.
>
> 2+2 is the same as 2x2 which is the same as 2^2.      ..[*]
>   However, this is not true of any other number.
> Zero comes close, but 0^0 will start a thread of discussion here,
> whereas 2^2 will not.  So, what's the magic of 2? -- Jose

No 'magic', really.

NB: By definition (notational agreement):

      a + a = 2 x a,  and:  a x a = a^2

    Replace 'a' by '2' and you get the 'magic'...

The clue is that you combine just two numbers into one number:
a 'binary' operation (having two operands), the essence of the
main concept of 'closure' in math.

The complementary concept is 'generation' which is in extreme form:
anti-closure, as in x^p + y^p =/= z^p, the sum of two equal types
(p-th powers, p>2) is NOT of that type. If *that* holds, you have
an efficient set of generators (here: under addition), which usually
is very exeptional and not easy to find.

-- NB -  http://home.iae.nl/users/benschop/anti-cl.htm

Subject:  Re: A word on Galois               (sci.math  16 oct 2002)
 Author:  Nico Benschop 
    Org:  Digital Research : Finite Associative Networks
------
William Hale wrote:
>
> "Bob Pease"  wrote :
> > I was delighted to read the article as it gave me insight as to
> > Galois' involvement with the Avant Garde Mathematicians of his
> > day. I had thought he was a  just loner with uncommon insight.
>
> I believe Galois was well read in the research articles published
> by the leading mathematicians of the day. Galois was not working
> from scratch (or "alone").
>
> In particular, I believe Galois read LaGrange's article on why
> one can find radical solutions for equations of degree less than 5.
> I believe that article was some 90 pages long. LaGrange talks
> about the permutations of roots and the LaGrange resolvent used
> to solve solvable equations. I think LaGrange was so close to
> obtaining the results that Galois arrived at, but he couldn't make
> the insights that Galois made.
>
> I am not saying that Galois was not brilliant: he was. But, he
> achieve his brilliance by absorbing the ideas of the best minds
> of his time. I think this is an example of the usual (always?)
> breakthroughs of a creative mind: they know and have studied
> what others have achieved in the area that they are working in.

Precisely, good point. It is very much worth one's while to study
the original papers in a subject. For instance on Boolean Algebra:
read the original: George Boole "The Laws of Thought" - on a
reduced form of arithmetic ('idempotent': x^2 = x) introduced as
'calculus of properties' to be a model of Aristotelian logic,
                 (notice: not the set theory it later became;-)
which he applied to analyse the God-existence 'proofs' of Spinoza
and Clarke (resulting in his conclusion that with a math proof
one can never derive more than what is essentially already in the
used axioms: Spinoza's 'proof' is circular;-(

Or: Claude Shannon (1938, his MSc thesis at MIT) on the application
of Boolean Algebra as model for binary circuits (relay ccts in his
time, used to implement the complex logic of telephone exchanges).
That original paper is a beauty of clarity and of purpose, having
a separate section on the importance of symmetric BF's - until now
not really playing any important role at all, in logic synthesis. [1]

Or: B.D.Tellegen (Philips Research 1948), deriving the 'gyrator' as
fifth basic component in linear circuit theory. A very nice paper,
methodically analysing all cases of DE's involved in such networks.

> As an aside, there is a simliar case in physics around the same
> time: Sadi Carnot wrote his treatise on the "Motive Power of Fire"
> which introduced the three laws of thermodynamics, but his
> work was not accepted or appreciated until many years later
> (I belive after his death). -- Bill Hale

-- NB - http://home.iae.nl/users/benschop
        "Symmetric Logic Synthesis with Phase Assignment" [1]
22-nd Symp. on Inform'n & Comm'n Theory (p115-122, U-Twente may 2001)

Subj:   Re: A wonderful proof
News:   sci.math (31 Oct 2002)
From:   Nico Benschop 
 Org:   Digital Research : Finite Associative Networks
------
Virgil wrote:
>
> "H.G. Wells"  wrote:
>
> > I have stumbled upon an incredible proof that for two consequtive
> > primes a, b such that |a-b| = 2, there exists an infinite number
> > of such pairs.
>
> If the fall from your stumble did not cause you to lose your memory,
> please tell us where this wonderful proof is to be found.

NB:  I bet it did'nt fit in his (too small) margin... ;-)

Euler's method:
 If a problem is really too hard (inf. twin primes): generalize it.

 Say, look at differences 2^i (note: 30 = 2.3.5 and all primes
      5 < p < 30-5 are of form 30/2 +/- 2^k. Coincidence ?-)

-- NB - http://home.iae.nl/users/benschop/general.htm
        http://home.iae.nl/users/benschop/ng-abstr.htm