Re: goldbach's conjecture

thr:

The statements are neither nonstandard nor
controversial.
Transitivity is a property of the equals relation,
and thus a property of every equation;

hagman:

There is a difference between properties of the
equals relation
and properties of equations.
A single equation can have attributes like
"unsolvable over the
rational"
(e.g. "x^2 = 2") or "true" (e.g. "5 + 7 = 12") and
others,
but a single equation does not have the property of
being e.g.
transitive.
Nails I put into a glass jar are still made of iron.


The equivalence relation is transitive. To use an
analogy from language, like a transitive verb, "="
always has a direct object. That is, to say "Seven
plus five equals twelve" is not the same as saying
(as in the linguistic intransitive construction)
"Seven plus five has resulted in twelve." The former
assumes construction of the object in a transitive act.
We see this implied in the Theorem, line 72 of
Dedekind's "The Nature & Meaning of Numbers." "In every
infinite system S a simply infinite system N is
contained as a part." Just previous, Dedekind has
defined a simply infinite system, the conditions of
which "...are always the same in all ordered simply
infinite systems...(and)...form the first object of the
science of numbers or arithmetic." In the same
paragraph, Dedekind had said "With reference to this
freeing the elements from every other content
(abstraction) we are justified in calling numbers a free
creation of the human mind." An equation, therefore,
as an order-setting transformation (also Dedekind's
words) implies a direct object.

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.

These are just the *straightforward* tests for the
truth of the
corresponding conjectures.
How can we prove that 29 is prime?
29 is composite if there exist natural numbers x>1,
y>1
such that x*y=29.
We may try different combinations of x,y for ages
and will never find a counterexample (trust me), but
we can never be sure that there is no counterexample
just around the corner.
OTOH, with a bit of cleverness we may find out that
it
suffices to check only the finitely many x,y below
29.


Call me lazy. I prefer straightforward over clever
any day of the week. Clever is too much work.

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.

But isn't "... at most six primes..." in the same
"verifiability
class" as GC? Using the immediate formulation, we
can only search for counterexamples one by one.
And yet it has been proven...

As I (and Popper) said, neither conjecture is verifiable.
By throwing out verifiability as a criterion, and
concentrating on computational falsifiability (even
though such computation is obviously currently out of
reach)we possibly open up new lines of attack tractable
to algorithmic compression. If that is what you mean
by "clever," then fine. Testing specific cases, however,
no matter how numerous, does not constitute proof.
Appel and Haken could only be convincing with 4CT
because their algorithm reduced the checking cases to a
finite group. Their critical proof criterion was
falsifiability.

Tom


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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