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From: Archimedes Plutonium on 25 Jul 2010 03:25
Now I was asking Lwalk in a prior post, and hope he does not ignore
the question. The question was that since Euclid Infinitude of Primes
proof, both direct and indirect yield
the infinitude of Regular primes and since the twin primes are a
subset. One has to wonder,
logically, that logic demands the same apparatus of proof mechanism
"multiply the lot and add, subtract 1" that this same mechanism must
yield a proof of twin primes. Sounds logical,
sounds reasonable. And has math already proven in Algebra theory that
if you have a proof of the larger set, then the same mechanism must
prove the small set. Any thoughts on this LWalk? The Galois Group
theory should have some say on this.

Another thing I want to point out about the misleading or failing of
our education system on
the proof by Contradiction. I heard it in college in a Freshman Logic
class and several times in
math classes that in the contradiction method, any contradiction will
do. These are one of those statements that is half wrong and half
right, and so very misleading. Every math proof is a focused line of
arguement with many constraints. So the trouble with the statement
that
"any contradiction will do" is typified by Iain Davidson error filled
Euclid Indirect IP. He has the
erroneous assumption that just about any contradiction in Infinitude
of Primes will do. And so,
he, like thousands, millions of others who heard it say in classrooms
around the world "any
contradiction will do". And so these impressionable minds when
encountering a proof by
contradiction, have this false idea that like a menu list, they can
pick and choose from a wide
assortment of attacks and have a wide assortment of contradictions
waiting for each math proof. I think one of the reasons this
misleading or falsity of method is preached in college,
is because, the professor was preached that same idea, and like
parrots handing down to
the next generation of parrots. Or, the professor wanted something
more to say, rather than
a blank mind in class, harps on "any contradiction will do."

What Euclid IP shows us, is that the mechanism is "multiply the lot
and add 1." So now, let us examine the proof logically. Like, proving
the proof. Would anyone think that if you formed
"multiply the lot and add 1" that you had a menu of chooses for a
contradiction? Would it not seem highly logical that if the mechanism
is "multiply the lot and add 1" that the contradiction is narrowed
down to only one contradiction that gives one and only one valid
proof? Seems logical. And that the mechanism, "multiply the lot and
add 1" would only entail a contradiction of W+1 being a larger prime
than the supposed assumed finite list largest prime p_k. In other
words, when some bloke like Iain Davidson goes running around looking
for an alternative contradiction with his "prime divisors", does not
he miss the entire point of the proof, that if you construct W+1 the
contradiction is uniquely revolving around the fact that W+1 as prime
is larger than the p_k prime.

So I think our education system spreads alot of bogus ideas. It
spreads the bogus idea that
Euclid's proof was by contradiction when all along it was a Direct
method/ construction proof.
And the bogus idea that any contradiction will do. What was meant by
that is all contradictions in the reductio ad absurdum reverse the
assumption step. But when taught to
impressionable students, they pick that up as meaning there is a menu
of contradictions to choose from in a proof.

And even Hardy/Woodgold/and Davis in the Mathematical Intelligencer
are spreading more
bogus ideas when they spoke about every construction proof turned into
a contradiction proof
and vice versa, or words to that effect in their Fall 2009 magazine.
Here is where the authors and editor are spreading more bogus that
future impressionable minds will pick that up as meaning, the
Construction is the same as the Contradiction proof, or almost
identical after the first few steps. And nothing is further from the
truth, because as we can see, the Twin Primes
Infinitude comes easily out of the Contradiction method whereas
impossible in the construction method. And many proofs in mathematics
have only one method and never the other. So when Hardy/Woodgold ask
the question at the end of the article as to where the bogus error
that Euclid's proof was reductio ad absurdum got started, one can
easily ask how much bogus misleading ideas were in that article for
which Hardy/Woodgold are to be blame for a new generation of bogus
ideas.

Archimedes Plutonium
http://www.iw.net/~a_plutonium/
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies
From: sttscitrans on 25 Jul 2010 03:59
On 25 July, 08:25, Archimedes Plutonium
<plutonium.archime...(a)gmail.com> wrote:


If the theorem

"Every natural >1 has at least one prime divisor"

is false as you claim, you should be able
to state an n>1 that does not have a single
prime divisor.

(other mindless ramblings deleted)
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