Re: goldbach's conjecture

Fernando wrote:

I would forget the reference to God, although I
understand that it isn't a bad literary reference.
All
what I can say if I understand something of this,
is
that natural numbers are an extreme way of freezing
time
and in that sense non- prime natural numbers are
exactly those that present discontinuities on
some
structures isomorphic to N and never the primes.
Curiously according to this, are just the primes,
"the
atomic entities", those whose behaviour is regular,
not
backwards. Well, this is not my own opinion, I
don't
say this, Mathematics, formulas, say this. Vortex
points, an "irregularity", characterize non-primes.
I
prefer now not to talk about Goldbach's Conjecture,
formulas and not me say something more about
addition
and multiplication in that context.

Once, an excellent mathematician wrote to me
something :

Hagman replied:

Please revisit the label "excellent mathematician"
after
reading my remarks below.


"...one may write an equation -- for example, 7 + 5
=
12 -- that satisfies all the characteristics of an
equation (equality, reflexivity, transitivity),
yet
does not satisfy a third characteristic necessary
to a
proof: that the numerical properties of the
equation
hold for all time. In other words, we cannot
capture
all of the information needed to answer the
question
("Can every even integer greater than 2 be
expressed as
the sum of two primes?") within the axioms and
rules of
classical arithmetic. There aren't enough
dimensions."

Ehm, how can a *single* equation satisfy e.g.
transitivity?
And I'd interprete tha statment that "the numerical
properties ...
hold for all time" more in the direction that 7+5=12
will
still be true in 100 years.
Whether or not Goldbach's conjecture is true or not,
however, has nothing to do with such a stability over
time,
hence the "In other words" is quite absurd.
And how does he come to the conclusion that
Goldbach's
conjecture is not provable/disprovable "within the
axioms
and rules of classical arithmetic" (at least that's
how I
see myself forced to interprete the last 5 lines)?
Wouldn't that handwaving "too many dimensions"
argument
imply as well that the question "Can every even
integer be written
as the sum of at most 6 primes?" cannot be answered
either?





I claim no excellence, nor even mediocrity.

I will own up to the words Fernando quotes, however,
which were written in private correspondence.

The statements are neither nonstandard nor controversial.
Transitivity is a property of the equals relation,
and thus a property of every equation; it contributes
the very meaning of a mathematical operation requiring
"a move in time," to borrow Brouwer's words.

The "...at most six primes ..." (a result on an upper
bound of the weak GC, due to Ramare using a Vinogradov
result) does not apply here. A proof of the strong
Goldbach Conjecture would in fact imply the weak version.

That 7 + 5 = 12 holds for "all time" is not at issue.
The "all time" result is for arbitrarily chosen primes
summing to some even integer. Simply calculating and
verifying any particular result does not count as
proof. Karl Popper (Realism and the Aim of Science,
Routledge 1983) demonstrated the difference between
verifiability and falsifiability:

Popper called the Goldbach Conjecture true if, G: For
every natural number x > 2, there exists at least one
natural number y such that x+y and (2+x)- y are both
prime. Popper called the Twin Primes Conjecture true if
H: For every natural number x > 2, there exists at
least one natural number y such that x+y and (2+x)+y
are both prime.

G is demonstrable by iterated arithmetic calculation.
A program to test the conjecture potentially halts
when it comes on a counterexample. H is not (in
Popper's context of computational falsifiability)
testable at all.

Neither conjecture is verifiable, but of these
propositions--which differ only by one sign change--
only GC is falsifiable. Significance?--the weak GC
belongs to the same class of non-falsifiable problems
(using Popper's context) as the twin primes conjecture.
One would think that a natively falsifiable conjecture
gives us a better chance at a computationally tractable
(therefore convincing)proof. The line about dimensions
refers simply to complex analysis, which is a two-
dimensional tool. Vinogradov's attack on the weak GC
(and therefore Ramare's result as well) removed the
necessity to assume the truth of the Riemann Hypothesis
(which of course lives in the complex plane) in order to
attack Goldbach. I still think RH and GC are
linked. I may be wrong, but I won't be alone.

Tom
.

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