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From: Archimedes Plutonium on 22 Jul 2010 11:56
Mathematical Intelligencer is a benchmark of the misunderstanding of
the correct valid
Indirect Method proof of Euclid's Infinitude of Primes proof. Euclid's
proof was a direct
method proof of increasing set cardinality. And the authors and editor
of Mathematical
Intelligencer did not and could not provide a valid Indirect Method
proof, and they could
not even provide a Direct Method without copying Ore's written text.

Such sad state of affairs that when writing about mistakes of direct
or indirect that Hardy/Woodgold/ Chandler Davis cannot even provide
the two methods together so that the reader
can compare the two.

The first valid Indirect Euclid Infinitude of Primes proof came circa
1991 and involves the
recognition that P+1 is necessarily a new prime, for which it is easy
to see, next, that P-1 along with P+1 are necessarily two new primes
for which Twin Primes infinitude is proven. This outcome is due to the
alliance in the proof of the definition of prime and the supposition
hypothetical step.

Trouble was in the history of mathematics, the two methods of direct
versus indirect became
tangled up together, where everyone was doing a prime factor search
near the end of the proof,
whether they were doing Direct or Indirect. One of the reasons is that
everyone who gets excited about math and wants to do a math proof,
usually does a Euclid IP proof but which is filled with error. So the
popularity of Euclid's IP proof may have contributed immensely to
never a "valid Indirect method." And thus, where the two methods
direct and indirect became
blurred into one big pile of mess, all of one method; hunting around
for prime factors.

Even those who never studied Symbolic Logic, knows that two different
methods are not going to be lock-step identical steps for their
respective proofs. In the Direct Method of set cardinality increase,
there is always a prime factor search
of Euclid's number. In the Indirect Method there can never be a prime
factor search in that hypothetical space, but rather, once Euclid's
number is formed in Indirect, it alone provides the extra needed new
prime number. Euclid's number is the only candidate to be a new prime
in Indirect and the definition of prime that is required for the first
step of the proof insures Euclid's number is prime.

The widespread sloppiness of mathematical proofs is remarkable. Rarely
if ever, in print or otherwise, does a person doing Euclid's
infinitude of primes proof, rarely do they begin the proof by step
one-- definition of a prime number.

Here is the Indirect Method in short form:

(1) definition of prime
(2) hypothetical suppose all primes are finite with 2,3,5,7,.., p_k
the complete list
of primes with p_k the last and largest prime number
(3) form Euclid's Number of multiply the lot and add 1 and call it W+1
(4) W+1 is necessarily a new prime by (1) with (2)
(5) contradiction to p_k being the last and largest prime since we
have W+1
(6) reverse the hypothetical supposition that primes are infinite.

It was not until the 1990s was the correct Indirect proof found and
which would
thus deliver the Infinitude of Twin Primes, because anyone can see
that there is
symmetry between W+1 and subtracting 1 from W in that W-1 and W+1 are
twin primes.

Anyone not in mathematics can understand that if you have Greek
mathematics
from Ancient Greek times, able to do a Direct method of Infinitude of
Regular primes. And
somewhere along the way of history, no mathematician is able to sort
out a correct Indirect
Method but that it gets all garbled up and messed up into one method,
that all those mistaken
and flawed Indirect proofs, would hide the Infinitude of Twin Primes
proof.

Anyone can understand that proof of Infinitude of Twin Primes should
be as simple and easy as proof of Regular Primes. And anyone can
easily see that if you never can straighten out the
mess of mixing Indirect with Direct, that you can never see the proof
of Twin Primes.

Neither Hardy/ Woodgold/ Davis at Mathematical Intelligencer can see a
valid Indirect method
of Euclid's Infinitude of Primes proof as evidenced by many of their
statements in that article
and as evidenced by Mr. Davis's emails to me. Their article, although
it points out that Euclid did a Direct Method, that is certain, but
still, by Fall of 2009 with that article in print, shows us the state
of misunderstanding by the mathematics community that they still do
not understand a valid Euclid Indirect.

The moment you have a valid Euclid Indirect, is the moment you have a
proof of the Infinitude of Twin Primes.

Once you recognize that W-1 and W+1 are necessarily new primes in the
Indirect Method, you instantly have a proof of Infinitude of Twin
Primes, and the reason it took over 2 thousand years to get Twin
Primes Infinitude is the scrambled up mess of not able to do a valid
Indirect method.

In the decade of the 1990s, there were only two persons with a strong
enough logical mind to do a valid Euclid Indirect method. It is
difficult because Euclid's proof was so misunderstood, and in fact, it
is difficult to see that variance of the direct versus indirect. When
you are sloppy, and most people, even mathematicians are sloppy, and
when you are sloppy by neglecting the definition in step one, it is
easy to fall into the trap of looking and hunting for a new prime in
the prime factors of Euclid's Number in Indirect.

So by Fall 2009, Mathematical Intelligencer article of Prime
Simplicity by Hardy/Woodgold and
editor Chandler Davis provides the history of mathematics a benchmark
that by 2009, still,
the mathematics community could not do a valid Euclid Infinitude of
Primes proof Indirect method. And Mr. Davis, via email, says this
field of study is closed and that I had better apply
my time elsewhere.

That by Fall 2009, the mathematics community as a whole is just
beginning to recognize that
Euclid did a Direct method proof. How long will it be before they
realize that only two persons
in the 1990s and up to 2009, only two persons in all of math history
knew of a valid Indirect method.

So math is a sad state of affairs, that such a simple and easy proof
of regular primes when
done in a valid Indirect method yields the Twin Primes proof, yet only
two persons have that
"good enough logical mind."

One would think, and I certainly think that if anyone with a degree in
math reads the above
six lines of proof can see and understand how Twin Primes infinity is
achieved. I would think
that Mr. Chandler Davis of Mathematical Intelligencer would understand
a correct valid Indirect
and see that Twin Primes is yielded. But it looks as though I expect
far too much of Mr. Davis.


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
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