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From: Archimedes Plutonium on 28 Jun 2010 03:09


Archimedes Plutonium wrote:
> > [0] Michael *Hardy* and Catherine Woodgold,
> > "*Prime* *Simplicity*",  *Mathematical
> > Intelligencer<https://mail.google.com/wiki/
> Mathematical_Intelligencer>
>

There is a paragraph in that article that riles me. And shows how out
of place are the
authors of the article. There basic overall premiss is true that
Euclid wrote a Constructive
proof, what I called direct method. And an author trying to convince
the world of mathematics
that Euclid wrote a constructive proof, should not be running to Ore
for his book example, but
should be writing out a valid direct method and valid indirect method
and then compare and
eliminate the obviously wrong method.

For this is an exercise in logic, afterall, so the writeup, unlike
what Hardy/Woodgold/MI did
is not a logical writeup, rather a scatterbrained writeup.

The constructive proof in modern terminology which Euclid did not have
was that of set theory
increasing set cardinality by one more prime given any finite set of
primes. So someone with
set theory proof of Euclid IP, not Ore, should have been chosen, but
then again, neither Hardy,
Woodgold, or the MI editors are logicians. When Euclid did his proof,
he was talking about
"more than any assigned multitude" (if I remember the wording
correctly). Anyway, Euclid's
wording is what we recognize as modern day set theory of increasing
set cardinality.

But anyway, if the authors had been capable of writing their own two
proofs of Euclid IP, one
in construction and the other in contradiction. And if they were valid
proofs, they would notice
that the Euclid Number in contradiction method must be necessarily
prime in order for the proof to be valid. Now, nowhere in Euclid's
writing and no translator of ancient Greek has ever
found where Euclid says the P+1 number is necessarily prime, but
rather where Euclid does a
prime factor search. A valid Infinitude of Primes proof requires a
prime factor search in the Constructive method, never in the
Contradiction method because P+1 is automatically prime due to the
structure of the reductio ad absurdum with the definition of prime in
step one and the assumptive step in step two.

So, if Mathematical Intelligencer, had a superb write up of Euclid's
IP and why it is constructive proof, would have had the first
paragraph of a valid direct method proof and the
second paragraph of a valid indirect method proof. And the third
paragraph would simply say
that Euclid wrote a direct method because he was increasing set
cardinality, and impossible
for Euclid to have written a valid proof as reductio ad absurdum since
he never noticed
that P+1 was necessarily prime in that method (provided, of course we
insist that Euclid
gave a valid proof).

So what I have asked all interested people, whether mathematicians or
general public, is
to give both methods, and see for yourself whether you have the
logical mind needed to
deliver two valid proofs.

But what riles me about that article, other than no attribution to all
my work in sci.math
on this topic, is this paragraph by the authors:

--- quoting from Mathematical Intelligencer ---
Any proof that is not by contradiction can be rewritten
as a proof by contradiction in a way that superficially seems
trivial, but that can have quite unexpected consequences:
just prepend to the proof the assumption that the theorem
is false, then. . .

--- end quoting ---

Now I know that Michael Hardy is a statistician, not a logician, and
that Catherine
Woodgold is an electrical engineer and not a logician. But what riles
me is why
the editors of Mathematical Intelligencer did not contact a logician
such as Thomason
of Yale whose book I learned Symbolic Logic whilst in College. My
guess that a
statistician and electrical engineer was given the opportunity to fill
a number of
pages in MI, is because of the sensitive nature of this article. Keep
in mind that
many mathematicians, even living ones, are named as committing an
error of
logic by thinking the Euclid IP was indirect method. So any author is
going to
upset alot of living mathematicians and tarnishing their reputation as
not knowing
better that Euclid's IP was direct and not indirect. So not too many
takers for an
offer to fill pages, lambasting many a mathematician who erred in
Euclid's IP proof.
In that circumstance, one can thus see why a statistician and
electrical engineer
were allowed to write the article.

But let me address that riling paragraph about prepending and any
proof turned into a
proof by contradiction.

Proof by contradiction is vastly more complicated than merely
prepending.

And I know of several proofs in mathematics that have only a direct
method
and never a indirect was found. And I know of several math proofs that
have
only a indirect method but never a direct method found.

The trouble here and what riles me is that Hardy and Woodgold are way
out of their
realm of expertise. By gosh, they could not even do this article with
their own
direct and indirect Euclid IP. And for them to pontificate on "any
direct proof can
be prepended and turned into a indirect proof" is so misleading and so
far out in
left field, that the editors of Mathematical Intelligencer should be
ashamed of themselves.

The Indirect method is very complex and complicated procedure and is
not a simple
prepending. What the authors have in mind is that you can negate a
proof. We establish
as true A, then we negate it as not A, and then the proof ends up
again as A. But that is
not a proof by contradiction. See Thomason, Symbolic Logic, for a
proof by contradiction
is far more involved than any simple prepend.

And it is obvious that in mathematics and physics, that you can never
have both a constructive and contradiction method of proof for every
statement. Proof: the majority
of geometry statements have only a direct method proof. You cannot
prepend in geometry.

So I think that one paragraph is horribly misleading to many young
readers, who will
go away with the idea that every proof in mathematics has both a
contradiction and constructive proof. This subject of whether a proof
has both methods is an open area
of research in mathematics, and is not for some ribald authors in a
ribald article in
a ribald magazine to pontificate over.

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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