Euclid never gave a indirect proof of Infinitude of Primes; his proof was direct

From: Archimedes Plutonium (a_plutonium_at_iw.net)
Date: 03/29/05 

Date: Tue, 29 Mar 2005 07:31:40 -0600

The below is an old 1990s post of mine archived in File 106 of my
website
www.iw.net/~a_plutonium

  This exposition is not only a correction of one of mathematic's most
famous
proofs, Euclid's Infinitude of Primes (IP) but also a correction of the
history of mathematics as pertains to that proof. And as what Hardy
writes
in his eloquent book A MATHEMATICIAN's APOLOGY "no wrinkles",
the wrinkles are three major large gashes. As an outline of this
exposition,
it contains three main ideas and presented herein in that order as three
parts.

Part One: The Natural Numbers are not what they presently appear to be
of Finite Integers but are rather instead Infinite Integers.

Part Two: Most professors of mathematics were inept in seeing the
difference
between a Direct and Indirect proof of IP. This ineptitude caused
numerous
professors of mathematics to publish their version of IP, and 99% wrong
and
invalid proofs of IP. Most were wrong about the history of this proof,
calling it a reductio ad absurdum but never really looking into the
matter.

Part Three: Euclid's proof of the Infinitude of Primes was a Direct
Proof of that
of increasing set cardinality, and not as the often reported news that
his proof was Indirect. That is a falsehood, and the historians of
Mathematics
must correct their error.

----------------------------------------------------
From: Archimedes.Plutonium@dartmouth.edu (Archimedes Plutonium)
Newsgroups: sci.logic,sci.math
Subject: Euclid's IP proof was direct, and not a indirect proof
Date: 13 Jan 1998 07:32:01 GMT
Organization: PLutonium College
Lines: 103
Distribution: world
Message-ID: (69f59h$ms9$1@dartvax.dartmouth.edu>

--- quoting
http://aleph0.clarku.edu/~djoyce/java/elements/bookIX/propIX20.html 

---
            EUCLID's ELEMENTS
               BOOK IX
Proposition 20
    Prime numbers are more than any assigned multitude of prime
numbers.
  Let A, B, and C be the assigned prime numbers.
  I say that there are more prime numbers than A, B, and C.
  Take the least number DE
  measured by A, B, and C. Add the
  unit DF to DE.
  Then EF is either prime or not.
  First, let it be prime. Then the
  prime numbers A, B, C, and EF
  have been found which are more
  than A, B, and C.
  Next, let EF not be prime. Therefore it is measured by some prime
number. Let
  it be measured by the prime number G.
     VII.31
  I say that G is not the same with any of the numbers A, B, and C.
  If possible, let it be so.
  Now A, B, and C measure DE, therefore G also measures DE. But it also
  measures EF. Therefore G, being a number, measures the remainder, the
unit
  DF, which is absurd.
  Therefore G is not the same with any one of the numbers A, B, and C.
And by
  hypothesis it is prime. Therefore the prime numbers A, B, C, and G
have been
  found which are more than the assigned multitude of A, B, and C.
  Therefore, prime numbers are more than any assigned multitude of
prime numbers.
    Q.E.D.
--- end quoting
http://aleph0.clarku.edu/~djoyce/java/elements/bookIX/propIX20.html ---
  The language should have helped those in the future to ascribe
Euclid's IP as a direct proof, and not the mistaken indirect proof
method. By language I mean "assigned multitude" should have warned
those judging the proof of Euclid that Euclid gave a direct proof.
Assigned Multitude is set theory. And when you want to increase the
cardinality of any or every finite set of primes, that is the direct
proof of IP.
  The word "absurd" was a Roman word, not Greek and I suspect that later
writers such as Tartaglia, Cardan and others added the word "absurd".
                Word: ab*surd
        Etymology: French absurde , Latin absurdus.
 The proof method of Reductio Ad Absurdum is much more than simply
adding the word absurd somewhere in the text of the proof. Thus, I
suspect that the method was not known by Euclid and not by Archimedes
either. I suspect the method was not known by humanity until the first
official writing elucidating the Method of Reductio Ad Absurdum. Who was
that? Perhaps some logician.
And a nice correlating
fact is that the ancient Greeks did not have modern set theory, yet
they used set theory anyway. And the ancient Greek mathematicians such
as Euclid and Archimedes, I contend did not have the method of
Reductio Ad Absurdum, and that modern day historians of mathematics
falsely ascribe them with this method of proof. What I believe they
were doing was proving by geometrical construction, and if the word
or idea of "absurdity" cropped up, it was not that they were using
the indirect method but merely using the word. And I have traced back
the word "absurd" and it is a Roman word not a Greek word.
  Where in mathematics history does the full method of Reductio Ad
Absurdum appear? I would guess some logician detailed the method.
This is where in mathematics history the method of Reductio Ad
Absurdum started. Not with Euclid or Archimedes but much more recent.
 If we are to give Euclid the credit for Reduce to Absurdity method,
then, give the ancient Greeks also the credit for Set theory. But
both simultaneously would be a falsehood of math history.
  I do not know where in the history of mathematics that the entire
math community went astray and not until the 1990s is this able to be
corrected.
  Perhaps it was a desire of mathematics historians to look for the
first recorded indirect proof method, and their desire was more for an
earlier and earlier date. Thus, to them, whereever they could get away
with it they wanted to ascribe the earliest proof possible for the
indirect method. Even though the proof is clearly not the indirect
method. And probably none of the Ancient Greeks were consciously aware
of the logical mechanisms of the indirect method. I have seen it
written some place where Archimedes is ascribed with double reductio ad
absurdum methods in a proof. I suspect that is a falsehood and that the
authors are seeing and judging those proofs incorrectly. Those ancient
proofs of Archimedes were geometrical constructs in the first place.
And geometrical construction proofs are direct.
  The concept of "assigned multitude" is set theory and thus Euclid's
IP was direct proof method.
  I doubt that the wording of Euclid's original proof ever  had the
Reductio Ad Absurdum logic outline of this:
           Suppose a contrary statement,...
     steps of proof
     premissa step
     reach contradiction
     reversal of the supposition statement
  Euclid may have had a word like "absurd" or this
word may have been added in translated editions of Euclid,
and later readers judged
from that one word that Euclid gave a indirect proof when in fact he
never did. And I argue that the Ancient Greeks, none of them knew or
did any mathematics with the Reductio Ad Absurdum. I suspect this
method of proof was no older than several centuries when
logic was formalized. You need the logical formalization
to see how the method of proof delivers a true conclusion.
Perhaps the first person who discusses the
indirect proof method to any depth was the actual
first discoverer of the indirect proof method.
Archimedes Plutonium
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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