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From: Archimedes Plutonium on 2 Jul 2010 20:41


Archimedes Plutonium wrote:
(big snip)
>
> I want to say something further that I noticed and is probably a Lemma
> disease
> of reducto ad absurdum lemmas.
>
> Notice that Euclid's translated proof appears to have a lemma of
> contradiction,
> and that Ore seems to have retained that lemma of contradiction.
>
> But I said that the Euclid IP direct method needs no lemma of
> contradiction at all
> if you plugg in the Unique Prime Factorization Theorem UPFAT.
>
> So what I am wondering is if the world of math has a proliferation or
> reproduction of
> lemmas of contradiction by all those who forget that there is some
> theorem they should
> be applying and not be applying a argument of contradiction. So that
> if Ore had
> realized he was using UPFAT, only he did not use UPFAT, and instead
> argued there is
> a prime factor with a lemma contradiction.
>
> So that if Ore had realized or recognized he was using UPFAT, just
> needed to state that
> there exists a prime factor, not because P then P+1 has 1 divisible,
> not because of that,
> but because UPFAT was invoked and that Ore had not realized he was
> using UPFAT. Not
> realizing that UPFAT was used, then Ore launched a lemma of
> contradiction.
>
> So I am wondering whether a huge number of lemmas by contradictions in
> other proofs
> are used because the author invoked another theorem but did not
> realize it and then launched
> a needless lemma. I know alot of math proofs that seem to have strings
> and strings of
> lemmas. And for every lemma by contradiction, I would hazard to guess
> the author invoked an
> already established theorem, and did not realize he was using the
> theorem and thus created excess baggage of a lemma.
>
> So lemmas by contradiction are needless and heedless contraptions for
> which the author should have listed the theorem invoked and kept the
> proof as streamlined direct method.
>
> Now if memory serves me, there are some mathematicians who when faced
> with a lemma by
> contradiction, will stop at that point in the proof and search around
> and if they do not find
> an existing theorem, will pause in the proof and actually state a
> theorem and prove it there,
> then picking up the original proof to continue. They do this because
> they abhor most proofs, even lemmas
> by contradiction.

Well the story above gets very complicated. I was going under the
impression of
Wikipedia about the Unique Prime Factorization Theorem (UPFAT) which
Wikipedia calls
the Fundamental Theorem of Arithmetic.

--- quoting Wikipedia on UPFAT ---
The theorem was practically proved by Euclid (in book 7 of Euclid's
elements, propositions 30 and 32), but the first full and correct
proof is found in the Disquisitiones Arithmeticae by Carl Friedrich
Gauss.
--- end quoting ---

But I was struck by this quote of Weil's book "Number theory", 1984,
page 5: "Even in Euclid,
we fail to find a general statement about the uniqueness of the
factorization of an integer into primes; surely he may have been aware
of it, but all he has is a statement (Eucl.IX.14) about the l.c.m. of
any number of given primes. Finally, the proof for the existence of
infinitely many
primes (Eucl.IX.20).. "

Which brings up an interesting question. That if Euclid really
understood the UPFAT which was
referred to in IX.14, but which may not have been realized its
"uniqueness property", that Euclid could have avoided the lemma of
reductio ad absurdum in his proof of IX.20.

Euclid and Ore, do not need a lemma of contradiction if they simply
said that either P+1
is prime or has a prime factor not on the list and simply justified
that with UPFAT.

So was Euclid fully cognizant of UPFAT? Apparently not, and that
Weil's evaluation seems
to be accurate in that Euclid did not have UPFAT.

So here is probably another squabble about the history of mathematics.
Whether UPFAT
was not in existence until Gauss fully proved it, and only a notion
before Gauss?

If Euclid had been fully aware of Unique Prime Factorization, he would
not have needed that
lemma in Infinitude of Primes proof.

And my other point is also relevant, that it seems as though a direct
proof, if it has
any lemmas of reductio ad absurdum contained within that Direct proof,
is a sign of
weakness of the proof, in that it should be all direct method
throughout. That if there
appears a lemma of contradiction method, means that the author of the
proof is unaware
of a existing theorem that covers the issue at hand. Or in the case of
Euclid, to set
aside the proof and prove the Uniqueness of the Prime Factors.

So it is likely that a lemma of contradiction is most often a sign of
weakness in a proof or a mask for a theorem already in existence or a
theorem that needs to be proven and thus eliminate that lemma.

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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