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Alec McKenzie
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| Posted: Sun Dec 05, 2004 12:02 pm
Post subject: Re: Brand new colors |
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"Richard Maurer" <rcpb1_maurer@yahoo.com> wrote:
| Quote: | Alec McKenzie wrote:
I repeat, Euclid's proof that there is no largest prime is not a proof
by contradiction. It goes along these lines:
For any prime p (not the *largest* prime, note, but *any* prime) there
must exist a prime larger than p (because p! + 1 is either prime itself
or has a prime factor larger than p).
So, for every prime there exists a larger prime. So there is no largest
prime. QED
No contradiction involved anywhere.
You will have a pretty weak set of proved theorems if you do not
allow contradiction to be involved. Hint: look for words like
'obviously', 'must', ..., or missing parts like 'if it did not',
'if there were a largest prime'.
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Perhaps you should read what I said more carefully.
I have never suggested not allowing contradictions to be involved. I was
pointing out the fact that Euclid's proof was not a proof by
contradiction. He proved that for any prime there existed a larger
prime, without using a proof by contradiction.
--
Alec McKenzie
mckenzie@despammed.com |
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Torkel Franzen
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| Posted: Sun Dec 05, 2004 12:02 pm
Post subject: Re: Brand new colors |
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Alec McKenzie <mckenzie@despammed.com> writes:
| Quote: | I have never suggested not allowing contradictions to be involved. I was
pointing out the fact that Euclid's proof was not a proof by
contradiction. He proved that for any prime there existed a larger
prime, without using a proof by contradiction.
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Indeed. You often see Euclid's proof formulated as "Assume p1,..pn
are all the primes..." and a contradiction derived from this
assumption. Such formulations are utterly pointless, since the
assumption that p1,..pn are all the primes isn't even used in the
argument. It's a fine example of the kind of pointless recasting of
a proof as a proof by contradiction that students often produce.
Euclid simply noted that if A,B,C are primes, ABC+1 is either a prime
different from A,B,C, or is divisible by a prime different from A,B,C. |
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Donna Richoux
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| Posted: Sun Dec 05, 2004 12:02 pm
Post subject: Re: Brand new colors |
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Torkel Franzen <torkel@sm.luth.se> wrote:
| Quote: | Alec McKenzie <mckenzie@despammed.com> writes:
I have never suggested not allowing contradictions to be involved. I was
pointing out the fact that Euclid's proof was not a proof by
contradiction. He proved that for any prime there existed a larger
prime, without using a proof by contradiction.
Indeed. You often see Euclid's proof formulated as "Assume p1,..pn
are all the primes..." and a contradiction derived from this
assumption. Such formulations are utterly pointless, since the
assumption that p1,..pn are all the primes isn't even used in the
argument. It's a fine example of the kind of pointless recasting of
a proof as a proof by contradiction that students often produce.
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Maybe students do that, but this doesn't seem to be mere student
amateurism. The first discussion on this in Google,
http://www.utm.edu/research/primes/notes/proofs/infinite/euclids.html
gives "Ribenboim's statement of Euclid's proof" which starts right off
with
Suppose that p1=2 < p2 = 3 < ... < pr are all of the primes
and it does use that formulation to proceed with the proof.
| Quote: | Euclid simply noted that if A,B,C are primes, ABC+1 is either a prime
different from A,B,C, or is divisible by a prime different from A,B,C.
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A translation of what Euclid *actually* said is here:
http://aleph0.clarku.edu/%7Edjoyce/java/elements/bookIX/propIX20.html
and it's not exactly like any of the modern English versions. (Measure?
Multitude?). There are some good comments on that at the first URL.
Euclid didn't have modern concepts (like "infinity") to work with.
--
Best -- Donna Richoux |
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Torkel Franzen
Guest
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| Posted: Sun Dec 05, 2004 12:02 pm
Post subject: Re: Brand new colors |
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trio@euronet.nl (Donna Richoux) writes:
| Quote: | http://www.utm.edu/research/primes/notes/proofs/infinite/euclids.html
gives "Ribenboim's statement of Euclid's proof" which starts right off
with
Suppose that p1=2 < p2 = 3 < ... < pr are all of the primes
and it does use that formulation to proceed with the proof.
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A fine example! No use whatever is made in the argument of the
assumption that p1,..,pr are all the primes. Euclid, of course, did
not introduce this pointless assumption. He simply noted that if A,B,C
are primes, ABC+1 is either a prime different from A,B,C, or is
divisible by a prime different from A,B,C. |
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Bob G
Guest
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| Posted: Sun Dec 05, 2004 6:02 pm
Post subject: Re: Brand new colors |
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Enough of this!
I have found the last prime number and I'm noting it down on the margin of one
of my books.
Bob G |
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R H Draney
Guest
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| Posted: Sun Dec 05, 2004 6:04 pm
Post subject: Re: Brand new colors |
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Bob G filted:
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Enough of this!
I have found the last prime number and I'm noting it down on the margin of one
of my books.
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ObMusicalPun: An allusion to a single note held for over 300 years....r |
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Jordan Abel
Guest
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| Posted: Mon Dec 06, 2004 12:05 am
Post subject: Re: Brand new colors |
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On 2004-12-05, Torkel Franzen <torkel@sm.luth.se> wrote:
| Quote: | Alec McKenzie <mckenzie@despammed.com> writes:
Indeed. You often see Euclid's proof formulated as "Assume p1,..pn
are all the primes..." and a contradiction derived from this
assumption. Such formulations are utterly pointless, since the
assumption that p1,..pn are all the primes isn't even used in the
argument. It's a fine example of the kind of pointless recasting of
a proof as a proof by contradiction that students often produce.
Euclid simply noted that if A,B,C are primes, ABC+1 is either a prime
different from A,B,C, or is divisible by a prime different from A,B,C.
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This fact, however, doesn't prove that there are an infinite number of
primes unless you take the extra step [which requires a proof by
contradiction] - if Euclid didn't do that, fine, but it's not "utterly
pointless" to use his argument to prove that there _are_ an infinite
number [which, as you say, he did not himself prove] |
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Jordan Abel
Guest
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| Posted: Mon Dec 06, 2004 12:05 am
Post subject: Re: Brand new colors |
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On 2004-12-05, Torkel Franzen <torkel@sm.luth.se> wrote:
It's not pointless because...
| Quote: | He simply noted that if A,B,C
are primes, ABC+1 is either a prime different from A,B,C, or is
divisible by a prime different from A,B,C.
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....if that's all he noted, he proves nothing about primes being a finite
or infinite set. You may think it follows from it, and it does, but only
if you use it as a base for a proof by contradiction [you can't say
1+1=2 implies 2+2=4 without actually doing the math, even if it does] |
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don groves
Guest
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| Posted: Mon Dec 06, 2004 12:06 am
Post subject: Re: Brand new colors |
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In article <covcik014g@drn.newsguy.com>, R H Draney at
dadoctah@spamcop.net exposited:
| Quote: | Bob G filted:
Enough of this!
I have found the last prime number and I'm noting it down on the margin of one
of my books.
ObMusicalPun: An allusion to a single note held for over 300 years....r
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Eat your heart out, Kenny G.
--
dg (domain=ccwebster) |
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Donna Richoux
Guest
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| Posted: Mon Dec 06, 2004 12:06 am
Post subject: Re: Brand new colors |
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Jordan Abel <jmabel@purdue.edu> wrote:
| Quote: | On 2004-12-05, Torkel Franzen <torkel@sm.luth.se> wrote:
trio@euronet.nl (Donna Richoux) writes:
http://www.utm.edu/research/primes/notes/proofs/infinite/euclids.html
A fine example! No use whatever is made in the argument of the
assumption that p1,..,pr are all the primes. Euclid, of course, did
not introduce this pointless assumption.
It's not pointless because...
He simply noted that if A,B,C
are primes, ABC+1 is either a prime different from A,B,C, or is
divisible by a prime different from A,B,C.
...if that's all he noted, he proves nothing about primes being a finite
or infinite set. You may think it follows from it, and it does, but only
if you use it as a base for a proof by contradiction [you can't say
1+1=2 implies 2+2=4 without actually doing the math, even if it does]
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Did you look at those pages?
The ancient Greeks also did not have our modern
notion of infinity. School children now easily
understand lines as infinite, but the ancients were
again more concrete. For example, they viewed lines
as something that could be extended indefinitely
(not something infinite that we view just part of).
For this reason Euclid could not have written "there
are infinitely many primes," rather he wrote "prime
numbers are more than any assigned multitude of
prime numbers."
--
Best -- Donna Richoux |
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Jordan Abel
Guest
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| Posted: Mon Dec 06, 2004 12:07 am
Post subject: Re: Brand new colors |
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On 2004-12-05, Donna Richoux <trio@euronet.nl> wrote:
| Quote: | The ancient Greeks also did not have our modern notion of
infinity. School children now easily understand lines as
infinite, but the ancients were again more concrete. For
example, they viewed lines as something that could be extended
indefinitely (not something infinite that we view just part of).
For this reason Euclid could not have written "there are
infinitely many primes," rather he wrote "prime numbers are more
than any assigned multitude of prime numbers."
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He didn't make that leap if he didn't do the proof-by-contradiction
thing. |
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Peter Moylan
Guest
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| Posted: Mon Dec 06, 2004 6:07 am
Post subject: Re: Brand new colors |
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Torkel Franzen infrared:
| Quote: | Euclid simply noted that if A,B,C are primes, ABC+1 is either a prime
different from A,B,C, or is divisible by a prime different from A,B,C.
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Euclid wasn't working to modern standards of mathematical rigour.
Not that we can blame him for that. It took many centuries before
the discovery of the meta-theorem that says "any statement whose
proof includes words like 'obviously' is probably false".
--
Peter Moylan peter at ee dot newcastle dot edu dot au
http://eepjm.newcastle.edu.au (OS/2 and eCS information and software) |
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Bob G
Guest
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| Posted: Mon Dec 06, 2004 6:08 am
Post subject: Re: Brand new colors |
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| Quote: | The ancient Greeks also did not have our modern notion of
infinity. School children now easily understand lines as
infinite, but the ancients were again more concrete. For
example, they viewed lines as something that could be extended
indefinitely (not something infinite that we view just part of).
For this reason Euclid could not have written "there are
infinitely many primes," rather he wrote "prime numbers are more
than any assigned multitude of prime numbers."
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What is the difference between "there are infinitley many primes" and "prime
numbers are more than any assigned multitude of prime numbers"? I don't think
Euclid came to a mental block after imagining a multitude of, say, a thousand
prime numbers.
If there's any difference in the two statements it would be but a short step,
one that any ancient Greek, and not just a genius such as Euclid, could easily
take, to bridge the distance from the one to the other and arrive at our
"modern" notion.
Bob G |
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Bob G
Guest
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| Posted: Mon Dec 06, 2004 6:08 am
Post subject: Re: Brand new colors |
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| Quote: | So, for every prime there exists a larger prime. So there is no largest
prime. QED
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It's the second sentence that leads many to assume it's a proof by
contradiciton. That sentence is superfluous. Delete it and you have the essence
of Euclid's proof.
Bob G |
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don groves
Guest
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| Posted: Mon Dec 06, 2004 6:09 am
Post subject: Re: Brand new colors |
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In article <20041205204619.06675.00001332@mb-m15.aol.com>, Bob G
at bobjames27@aol.com exposited:
| Quote: | So, for every prime there exists a larger prime. So there is no largest
prime. QED
It's the second sentence that leads many to assume it's a proof by
contradiciton. That sentence is superfluous. Delete it and you have the essence
of Euclid's proof.
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Not exactly. The first sentence implies the second and the second
is the customary restatement of the theorem, followed by QED.
--
dg (domain=ccwebster) |
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