| Author |
Message |
Torkel Franzen
Guest
|
| Posted: Mon Dec 06, 2004 12:07 pm
Post subject: Re: Brand new colors |
|
|
peter@seagoon.newcastle.edu.au (Peter Moylan) writes:
| Quote: | Euclid wasn't working to modern standards of mathematical rigour.
|
His proof is perfectly rigorous. It was clear to his readers that
A,B,C stood for any finite number of primes. |
|
| Back to top |
|
 |
Torkel Franzen
Guest
|
| Posted: Mon Dec 06, 2004 12:07 pm
Post subject: Re: Brand new colors |
|
|
bobjames27@aol.com (Bob G) writes:
| Quote: | 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.
|
Right. Euclid's proof is often needlessly recast as a proof of a
negative statement (and hence, of course, a proof by contradiction). |
|
| Back to top |
|
|
Jordan Abel
Guest
|
| Posted: Tue Dec 07, 2004 12:08 am
Post subject: Re: Brand new colors |
|
|
On 2004-12-06, Torkel Franzen <torkel@sm.luth.se> wrote:
| Quote: | peter@seagoon.newcastle.edu.au (Peter Moylan) writes:
Euclid wasn't working to modern standards of mathematical rigour.
His proof is perfectly rigorous. It was clear to his readers that
A,B,C stood for any finite number of primes.
|
But your claim that the proof-by-contradiction extension of his proof is
"pointless" shows ignorance about just what was proven - his proof, as
it is, isn't enough in modern mathematics to prove an unlimited/infinite
amount of primes. |
|
| Back to top |
|
|
Torkel Franzen
Guest
|
| Posted: Tue Dec 07, 2004 12:08 am
Post subject: Re: Brand new colors |
|
|
Jordan Abel <jmabel@purdue.edu> writes:
| Quote: | But your claim that the proof-by-contradiction extension of his proof is
"pointless" shows ignorance about just what was proven - his proof, as
it is, isn't enough in modern mathematics to prove an unlimited/infinite
amount of primes.
|
What is pointless is to introduce, in Euclid's argument, the
assumption that k1,..,kn are all the primes. From the conclusion of
that argument, it is a trivial logical step to the further conclusion
that it is not the case that there are finitely many primes. On the
other hand, the step from "it is not the case that there are finitely
many primes" to "for every k, there is a prime greater than k" is
trivial only if we use classical logic, whereas Euclid's argument is
constructive. Euclid's presentation is superior mathematical style -
in "modern mathematics". |
|
| Back to top |
|
|
Jordan Abel
Guest
|
| Posted: Tue Dec 07, 2004 12:10 am
Post subject: Re: Brand new colors |
|
|
On 2004-12-06, Torkel Franzen <torkel@sm.luth.se> wrote:
| Quote: | Jordan Abel <jmabel@purdue.edu> writes:
But your claim that the proof-by-contradiction extension of his proof is
"pointless" shows ignorance about just what was proven - his proof, as
it is, isn't enough in modern mathematics to prove an unlimited/infinite
amount of primes.
What is pointless is to introduce, in Euclid's argument, the
assumption that k1,..,kn are all the primes. From the conclusion of
that argument, it is a trivial logical step to the further conclusion
that it is not the case that there are finitely many primes.
|
No, it's not. You can't _formally_ make that step without the
proof-by-contradiction, however "obvious" you may think that is. |
|
| Back to top |
|
|
Torkel Franzen
Guest
|
| Posted: Tue Dec 07, 2004 12:10 am
Post subject: Re: Brand new colors |
|
|
Jordan Abel <jmabel@purdue.edu> writes:
| Quote: | No, it's not. You can't _formally_ make that step without the
proof-by-contradiction, however "obvious" you may think that is.
|
I said it is a trivial logical step, not that it does not involve
the use of (constructive) reductio. |
|
| Back to top |
|
|
Jordan Abel
Guest
|
| Posted: Tue Dec 07, 2004 12:10 am
Post subject: Re: Brand new colors |
|
|
On 2004-12-06, Torkel Franzen <torkel@sm.luth.se> wrote:
| Quote: | Jordan Abel <jmabel@purdue.edu> writes:
No, it's not. You can't _formally_ make that step without the
proof-by-contradiction, however "obvious" you may think that is.
I said it is a trivial logical step, not that it does not involve
the use of (constructive) reductio.
|
But you said it was pointless [the same logical step you make implicitly
is the one you ridicule the explicit statement of] |
|
| Back to top |
|
|
Torkel Franzen
Guest
|
| Posted: Tue Dec 07, 2004 12:10 am
Post subject: Re: Brand new colors |
|
|
Jordan Abel <jmabel@purdue.edu> writes:
| Quote: |
But you said it was pointless [the same logical step you make implicitly
is the one you ridicule the explicit statement of]
|
What is pointless is to introduce, in Euclid's argument, the
assumption that k1,..,kn are all the primes. From the conclusion of
that argument, it is a trivial logical step to the further conclusion
that it is not the case that there are finitely many primes. On the
other hand, the step from "it is not the case that there are finitely
many primes" to "for every k, there is a prime greater than k" is
trivial only if we use classical logic, whereas Euclid's argument is
constructive. Euclid's presentation is superior mathematical style -
in "modern mathematics". |
|
| Back to top |
|
|
| |
FAQ
Memberlist
Register
Profile
Log in to check your private messages
Log in