Errors of Bill Dubuque's perception of a valid math proof of IP; #123; 2nd ed; Euclid's IP
Archimedes Plutonium
Bill Dubuque wrote on Sep 17, 2009 12:39 PM
> juandiego <sttsc...@tesco.net> wrote:
> >
> > depending on how you formulate the argument
> > you can derive different contradictions ?
> >
> > 1) A natural >1 is prime iff it has no proper
> divisors
> > 2) Every n > 1 has a prime divisor
> > 3) assume a <b <c are the only primes
> > 4) z = abc+1 > c => z composite
> > 5) z has no prime divisors
> > 6) z is composite contra z has no prime
> divisors
> > or 6') z = 1 contra z > 1
>
> I've emphasized this and many other variations since
> at
> least 2003, e.g. see [1], [2], [3]. This has not
> convinced
> AP yet. So why do you think that merely repeating
> exactly
> the same old arguments will have any success? There
> is
> nothing in this thread that hasn't already been said
> many
> times before. You are severely beating a dead horse.
> Please TTTTT (terminate tickling the troll's tail)
> http://google.com/search?q=tickling+the+dragon%27s+tai
> l
>
> --BD
>
I should pause here and dwell on the above misconception of Bill
Dubuque about mathematical proofs and the Indirect method especially.
For it is
BD's misconception that fans others like Iain Davidson
in never a understanding.
The most worrisome misconceptions of BD is his statements such as "any
contradiction works" and
Euclid's IP proof has numerous valid variations. Another one of his
misconceptions is that W+1 as
necessarily prime is only one of many organized
variations.
So let me dive into this BD list of misconceptions with
some detail.
First of all, it is a poor mathematical judgement call on
the part of BD to think there are a numerous valid variations of a
mathematical proof of a statement. If one reflects for a moment on how
math is constructed,
its foundation and structure, that there should be only
one valid proof of a statement given a fixed set of
elements of that statement. And that variations are not
independent valid proofs but merely tacked on irrelevancies.
So for example, BD believes that Euclid IP with its fixed element of
construct W+1 has a dozen or more
valid Indirect proofs. AP believes that this Euclid IP
has only one valid proof and that any variation is just
added on irrelevant nonsense which when trimmed away, leaves only the
one valid proof.
So that when Iain Davidson comes running in with his
prime divisor theorem, he fails to discharge the reductio
ad absurdum step and he adds on the irrelevancies of
that theorem. So that BD would say that Davidson had
a independent and valid proof of IP indirect, whereas
AP would say that Davidson had a invalid proof because he failed to
properly do the logic to discharge the assumption.
So it all boils down to that BD thinks Euclid's IP indirect has a
dozen or thousands of independent and
valid proofs. AP would counter, and say that there exists only one
valid proof, and if someone varies from
that valid proof by tacking on irrelevancies, it makes the proof
longer, but if they fail to put together the logical steps, then their
attempt is invalid.
So AP says that there is only one valid proof without
irrelevancies. There can be other valid proofs but they are longer
since they have irrelevancies. But that most
proof attempts such as Davidson or Dubuque's fail to
have the correct steps of logic and thus are invalid.
Now why BD believes there are numerous valid and independent proofs of
Euclid IP indirect is beyond my reasoning. My reasoning follows from
my familarity of
mathematics in whole and such as the Hardy statement in his book A
Mathematician's Apology. Hardy remarked with words to the effect that
Reductio
ad Absurdum is the ultimate gambit where you offer up
the total subject of mathematics to squeeze out a contradiction.
Now if anyone pauses and reflects on that Hardy evaluation of
mathematics and has spent any time
in math making Indirect proofs, would instantly recognize that
Euclid's IP should not have a dozen or
more independent valid indirect proofs. That Euclid's
IP indirect should have only one narrowly constrained and confined
valid proof given W+1. That math is not
built so wide open and sloppy as to offer itself up in a
gambit where a dozen or more independent and valid IP
indirect exist.
As I said, there can be alot of offerings where irrelevant
garbage is packed into the offering, and although excessively long
when it can be shortened does not invalidate it. But as the case of
most of these irrelevant
packed offerings, it is the lack of logical inferences and
discharging of previous steps that makes their garbage
filled attempt invalid.
Neither Davidson nor Dubuque, as far as I have seen
on sci.math, deliver a LongForm of Euclid's IP direct
and indirect. Theirs has always been abbreviated steps
and steps in midair. Without the LongForm, noone can
assess that they ever reached a valid proof, but only a
invalid proof.
In the world of mathematics, it is wrapped around so tightly of its
structure, that there is only one valid proof
of Euclid IP. There can be tacked on irrelevancies that
makes the valid proof longer, but still valid. But there are no other
independent and valid proofs of Euclid IP
indirect. The offerings given by Dubuque and Davidson
have not been valid because of lack of proper logical
inferences-- they fail to discharge the assumption step.
Their attempts are not invalid because they tack on superfluous
irrelevancies, but invalid because they never have the proper sequence
of steps to prove the
problem.
Maybe BD is confused about mathematics proof because he sees a
topological Euclid Infinitude of Primes proof or some other area of
math that has
a proof of Euclid IP. If anyone stopped to think about
it, there are no dozen or so independent topological
proofs of Euclid IP, there is just one, and someone
can add irrelevancies to the topology IP that is still valid, but if
one were to omit key steps, then the topology proof is also invalid.
So it is about time that BD stops fanning his misconceptions for it
only further buries Iain Davidson
with his misconceptions.
Archimedes Plutonium
www.iw.net/~a_plutonium
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies
Owen Jacobson
<plutonium....@gmail.com> said:
[snipped]
Essentially, your point is given a finite set of primes p, then ∏p + 1
is also prime, yes? And that any proof that talks about cases where ∏p
+ 1 is not prime are "unnecessarily" adding "irrelevant" details?
Consider the case where the set of primes is p = {2, 3, 5, 7, 11, 13}.
Therefore,
∏p + 1
= (2 * 3 * 5 * 7 * 11 * 13) + 1
= 30030 + 1
= 30031
Note also that 59 * 509 = 30031, so 30031 is /not prime[0]/.
I know this has been raised to you before; if you addressed it, I
must've missed it and I'd appreciate a message-id so I can look it up.
-o
[0] given that a number is prime if and only if its only divisors are
itself and 1.
plutonium....@gmail.com
> On 2009-09-17 17:16:26 -0400, Archimedes Plutonium
> <plutonium....@gmail.com> said:
>
> [snipped]
>
> Essentially, your point is given a finite set of
> primes p, then ∏p + 1
> is also prime, yes? And that any proof that talks
In the Direct, no; in the Indirect, yes.
> about cases where ∏p
> + 1 is not prime are "unnecessarily" adding
> "irrelevant" details?
>
In the Indirect, yes.
plutonium....@gmail.com
> On 2009-09-17 17:16:26 -0400, Archimedes Plutonium
> <plutonium....@gmail.com> said:
>
> [snipped]
>
> Essentially, your point is given a finite set of
> primes p, then ∏p + 1
> is also prime, yes? And that any proof that talks
In the Direct, no; in the Indirect, yes.
> about cases where ∏p
> + 1 is not prime are "unnecessarily" adding
> "irrelevant" details?
>
In the Indirect, yes.
> Consider the case where the set of primes is p = {2,
> 3, 5, 7, 11, 13}.
> Therefore,
> ∏p + 1
> = (2 * 3 * 5 * 7 * 11 * 13) + 1
> = 30030 + 1
> = 30031
>
> Note also that 59 * 509 = 30031, so 30031 is /not
> prime[0]/.
>
In the Direct, a prime factor search is always conducted
in the Indirect, 30031 is necessarily prime itself since
the hypothetical assumption are all the primes that exist.
I know it is confusing for you, since about 90% of even
math professors are confused about it in their book writings.
> I know this has been raised to you before; if you
> addressed it, I
> must've missed it and I'd appreciate a message-id so
> I can look it up.
>
> -o
>
> [0] given that a number is prime if and only if its
> only divisors are
> itself and 1.
>
Sorry about the previous post for the keyboard hit the wrong button
before I had a chance to finish this reply.
AP
plutonium....@gmail.com
> Posted: Aug 25, 2009 8:43 PM
>
>Direct
>Th. 1 Every integer >1 is divisible by a prime
>If you can find n primes, then you can find n+1 primes.
>Add 1 to the product of the n primes you have found.
>The resulting number (>1) is not divisble by any of the
>primes that have been found, but by Th.1 this number must have
>another prime divisor.
>Therefore there is an n+1th prime and this implies the number
>of primes is infinite as the process could be repeated
>ad infinitum.
>
>Indirect
>Th. 1 Every integer >1 is divisible by a prime
>Assume that there are only a finite number of primes
>Add 1 to the product of all the primes assumed to exist.
>The resulting number (>1) is not divisble by any of the primes
-------------------------------------------------------------------
>assumed to exist, but by Th.1 this number must have a prime divisor.
-------------------------------------------------------------------
>Contradiction. Therfore, the number of primes is not finite.
---------------------
I remember when in High School some 45 years ago when in
Geometry class doing proofs. And where in one column we write
the statement of each step and in another column the reason that
warrants that statement. So that a proof is a long chain of ordered
reasoning, where each step is justified from previous steps. And
I hope this is how Geometry is still taught. And when in college
confronting Symbolic Logic, I was reintroduced to a "proof argument".
But it seems that proof argument on the Internet is a rare commodity.
That is unfortunate, for if both Davidson above and Dubuque below
could detail those steps underlined would find that their attempt
fails to deduce their conclusion.
Part of the problem is that Dubuque fans myths such as 1) any
contradiction will work 2) Euclid's IP has many independent
valid proofs. A contradiction does work if it follows from previous
steps, not pulled out of thin air. And Euclid's IP is too constrained
and narrow with its element of construct W+1 that it has only one
valid proof schemata. You can tack on irrelevant garbage to this
one schemata and still leave it valid. But as in the case of most
attempts such as Davidson's above, the garbage got in his
way so that he could not achieve a contradiction and thus invalid.
.......................................................................
Bill Dubuque calls this one as Direct Method
>
> Newsgroups: sci.math, sci.logic
> From: Bill Dubuque <w...@nestle.csail.mit.edu>
> Date: 06 Apr 2005 08:56:38 -0400
> Local: Wed, Apr 6 2005 7:56 am
> Subject: Re: G.H.Hardy gave an invalid proof of
> Infinitude of Primes
> in "AMathematician's Apology"
>
> My prior post gave a self-contained 'descent' proof
> of Euclid's
> theorem
> that there are infinitely many primes. Here is the
> proof in ascent
> form.
>
> THEOREM There are infinitely many prime integers.
>
> PROOF Given any finite set S of primes, we prove
> there exists a prime
> not in S. Let M = product of all the elements of S (
> M = 1 if S =
> {} )
> Let P be the least factor>1 of M+1 > 1. P is prime,
> else P would have
> a
> smaller factor>1, contra to our choice of P as the
> least factor>1 of M
> +1.
> P is not in S, since primes in S leave remainder 1
> when divided into M
> +1.
> Hence P is a prime not in S. QED
>
> Notice that this form of proof is not as subtle as
> those that proceed
> by way of contradiction, so it should prove more
> easily comprehensible
> to neophytes. Furthermore, it has a vivid
> constructive interpretation,
>
> --Bill Dubuque
Bill Dubuque calls this one as Indirect method:
What I have done is underline the part of Bill's offering
where the inferences do not follow from previous steps
and thus the attempt is invalid. Unless Mr. Dubuque
can fill in the details as to the reasons and missing steps
then we have to dismiss this attempt as invalid.
Bill throws around "contradictions" as a magician pulls a rabbit
out of a hat. Below I give my Direct and Indirect in longform, ie,
details. This is what is lacking in Bill's attempt--details to show
the steps follow from prior steps. And that the contradiction is
not pulled out of thin air.
> Newsgroups: sci.math, sci.logic
> From: Bill Dubuque <w...@nestle.csail.mit.edu>
> Date: 05 Apr 2005 19:59:20 -0400
> Local: Tues, Apr 5 2005 6:59 pm
> Subject: Re: G.H.Hardy gave an invalid proof of
> Infinitude of Primes
> in "AMathematician's Apology"
>
(snipped)
>
> SURPRISE One can bypass the lemma, yielding a
> self-contained
> 'elementary'
> proof, by 'inlining' the lemma into the theorem (in
> the same way that
> the
> classic proofs of unique factorization by Klappauf,
> Lindemann, Zermelo
> inline Euclid's Lemma on the Prime Divisor Property:
> P|AB => P|A or P|
> B).
> To do this smoothly we employ an alternative
> definition of 'prime'
> that
> is trivially seen to be equivalent to the standard
> definition, namely
>
> DEFINITION P is prime if P is the least factor>1 of
> an integer>1
>
> THEOREM There are infinite number of primes.
>
> PROOF Suppose not, i.e. suppose there are only a
> finite number of
> primes.
> We proceed by descent: given any integer N>1 we
> deduce there is a
> prime < N.
> Let S be the product of all primes >= N (convention:
> S = 1 if no
> primes >= N)
> Let P be the least factor>1 of S+1. By definition P
> is prime. But P is
> not
> one of the primes >= N since these leave remainder 1
> when divided into
> S+1.
> So P < N. This implies that among the primes there is
-----------------------------------------------------------------
> an infinite descending
------------------------------
> sequence of positive integers, a contradiction. QED
-------------------------------------------------------------
>
> NOTE that I have presented the proof in the pure form
> of infinite
> descent
> discovered by Fermat to emphasize that this proof
> could easily have
> been
> one of Fermat's (obviously one could shorten the
> proof by deviating
> from
> this older format). Although I have never seen this
> particular variant
> of
> the proof presented anywhere, it is so obvious it
> must surely be well-
> known
> (I found it and variants as a teenager while playing
> with inlined
> elementary
> proofs of unique factorization).
>
> --Bill Dubuque
>
For guidance to both Davidson and Dubuque, here is a valid direct and indirect
written in Longform and showing how the Contradiction follows from prior steps,
not like Davidson and Dubuque -- pulled out of thin air.
This is the trouble when you just sloppily write a attempt in shortform skipping about
5 or 6 steps. In that you lose contact with reasoning as to whether a step is warranted.
--- quoting my proofs of Euclid Infinitude of Primes---
Euclid IP Proofs, both Direct and Indirect methods
DIRECT Method, long-form; Infinitude of Primes Proof
(1) Definition of prime as a positive integer divisible
only by itself and 1.
(2) Statement: Given any finite collection of primes
2,3,5,7,11, ..,p_n possessing a cardinality n Reason: given
(3) Statement: we find another prime by considering W+1 =(2x3x...xpn)
+1 Reason: can always operate on given numbers
(4) Statement: Either W+1 itself is a prime Reason: numbers are either
unit, composite or prime
(5) Statement: Or else it has a prime factor not equal to any of the
2,3,...,pn
Reason: numbers are either unit, composite or prime
(6) Statement: If W+1 is not prime, we find that prime factor Reason:
We take the square root of W+1 and
we do a prime search through all the primes from 2 to
square-root of W+1 until we find that prime factor which
evenly divides W+1
(7) Statement: Thus the cardinality of every finite set can be
increased. Reason: from steps (3) through (6)
(8) Statement: Since all/any finite cardinality set can be increased
by one more prime, therefore the set of primes is an infinite set.
Reason:
going from the existential logical quantifier to the universal
quantification
INDIRECT Method, Long-form; Infinitude of Primes Proof
(1) Definition of prime as a positive integer divisible
only by itself and 1.
(2) The prime numbers are the numbers 2,3,5,7,11, ..,pn,... of set S
Reason: definition of primes
(3.0) Suppose finite, then 2,3,5, ..,p_n is the complete series set
with p_n the largest prime Reason: this is the supposition step
(3.1) Set S are the only primes that exist Reason: from step (3.0)
(3.2) Form W+1 = (2x3x5x, ..,xpn) + 1. Reason: can always operate and
form a new number
(3.3) Divide W+1 successively by each prime of
2,3,5,7,11,..pn and they all leave a remainder of 1.
Reason: can always operate
(3.4) W+1 is necessarily prime. Reason: definition of prime, step (1).
(3.5) Contradiction Reason: pn was supposed the largest prime yet we
constructed a new prime, W+1, larger than pn
(3.6) Reverse supposition step. Reason (3.5) coupled with (3.0)
(4) Set of primes are infinite Reason: steps (1) through (3.6)
--- the above are the Direct and Indirect, longforms of
Euclid Infinitude of Primes proof ---
juandiego
> On 2009-09-17 17:16:26 -0400, Archimedes Plutonium
This, of course means.
that 1 is prime (iself (1) divides 1 and 1 divides 1)
z =(1 * 2 * 3 * 5 * 7 * 11 * 13) + 1
1 divides z so the claim that none of the primes that
are assumed to exist divide z is false. z is necessarily
composite.
Of course. if you are AP you can accommodate the
idea that 1 is both prime ans not-prime in the
AP-exrectolocutionary domain of copristic numbers.