| Author |
Message |
Jerry Friedman
Guest
|
| Posted: Wed Dec 01, 2004 12:05 am
Post subject: Re: Brand new colors |
|
|
twiggy <twiggy@example.com.invalid> wrote in message news:<Xns95ABCB9F9122728762153tmichanet@ptn-nntp-reader02.plus.net>...
| Quote: | bobjames27@aol.com (Bob G) wrote in
news:20041124141754.06555.00000677@mb-m24.aol.com:
On a TV commercial someone announces the sweaters being peddled will
soon be available in brand new colors.
Anyone object to the word "brand" in that sentence? Obviously colors
have no brand, being naturally occurring phenomena.
Or is the meaning of "brand new" now completely divorced from any
connection with a particular brand, as such, or even from the idea of
a manufacturing process?
The 'brand' in 'brand-new' comes from 16th century 'brand'; ie it's like
bright new metal just taken out of the fire.
(Roughly what my Chambers says)
|
Thank you! I suspected that might be true, but it's good to know.
--
Jerry Friedman |
|
| Back to top |
|
 |
Guest
|
| Posted: Wed Dec 01, 2004 6:03 pm
Post subject: Re: Brand new colors |
|
|
In article <mckenzie-85A979.13263630112004@news.aaisp.net.uk>,
Alec McKenzie <mckenzie@despammed.com> wrote:
| Quote: | In article <31375iF33987dU7@uni-berlin.de>, blmblm@myrealbox.com wrote:
In article <20041128013920.06574.00001221@mb-m15.aol.com>,
Bob G <bobjames27@aol.com> wrote:
A possibly interesting exercise would be to consider the set of all
subsets of the set of integers - would that have ordinality Aleph
Null or Aleph One?
Definitely not Aleph Null, if I understand you right. Cantor also
proved (well, I think it was Cantor) that for any set A, A and its
"power set" (set of all subsets of A) do not have the same cardinality
("size", roughly speaking). The proof is short and similar in spirit
to the proof that the real numbers are uncountable -- you show that
any map from A to its power set "misses" something in the power set.
The details of the proof, though, are -- interesting? mind-twisting?
something. For the curious, this Wikipedia article seems pretty good:
http://en.wikipedia.org/wiki/Cantor's_theorem
To me, Cantor's proof is disturbingly reminiscent of Russell's paradox
concerning the set of all sets that do not include themselves, which is
thereby contradicted by its own definition.
I find the proof unconvincing, because the "missing something" in the
power set turns out to be precisely the subset that consists of those
integers whose mapping does not include that integer. This apparent
contradiction is then used to support the 'proof by contradiction'.
|
Hm. I can't quite get my head around why this would make the proof
unconvincing to someone, unless it's that the "missing something" set
is one that's hard to visualize in any sensible way? for me anyway.
But I've never been too bothered by that; I can follow along with the
rest of the argument in the spirit of "let's turn the algebra/logic
crank and find out if we can derive anything interesting/useful ...."
Which eventually happens, though the derived contradiction seems
peculiarly self-referential, or something. But again, that doesn't
make me think the whole proof's unconvincing, just -- strange, or
mind-boggling, or something.
"YMMV", maybe.
--
| B. L. Massingill
| ObDisclaimer: I don't speak for my employers; they return the favor. |
|
| Back to top |
|
|
jerry_friedman@yahoo.com
Guest
|
| Posted: Thu Dec 02, 2004 12:06 am
Post subject: Re: Brand new colors |
|
|
Alec McKenzie wrote:
| Quote: | In article <31375iF33987dU7@uni-berlin.de>, blmblm@myrealbox.com
wrote:
In article <20041128013920.06574.00001221@mb-m15.aol.com>,
Bob G <bobjames27@aol.com> wrote:
A possibly interesting exercise would be to consider the set of
all
subsets of the set of integers - would that have ordinality Aleph
Null or Aleph One?
Definitely not Aleph Null, if I understand you right. Cantor also
proved (well, I think it was Cantor) that for any set A, A and its
"power set" (set of all subsets of A) do not have the same
cardinality
("size", roughly speaking). The proof is short and similar in
spirit
to the proof that the real numbers are uncountable -- you show that
any map from A to its power set "misses" something in the power
set.
The details of the proof, though, are -- interesting?
mind-twisting?
something. For the curious, this Wikipedia article seems pretty
good:
http://en.wikipedia.org/wiki/Cantor's_theorem
To me, Cantor's proof is disturbingly reminiscent of Russell's
paradox
concerning the set of all sets that do not include themselves, which
is
thereby contradicted by its own definition.
I find the proof unconvincing, because the "missing something" in the
power set turns out to be precisely the subset that consists of those
integers whose mapping does not include that integer. This apparent
contradiction is then used to support the 'proof by contradiction'.
|
It seems to me like a standard proof by contradiction. For instance,
Euclid proved that there's no largest prime by saying, in effect,
"Assume there's a largest prime; call it p. Then p! - 1 would also be
a prime and it would be larger. That contradicts the hypothesis, so
there can be no largest prime."
Likewise Cantor said, in effect, "Assume there's a list that contains
all real numbers. Then by this diagonal trick, we can generate a real
number that's not on the list. This contradicts the hypothesis, so
there can be no such list."
Does it help to know, by the way, that given any list of real numbers,
you can come up with an *infinite* numnber of real numbers that aren't
on the list? In fact, there's a one-to-one correspondence between the
real numbers and the numbers that aren't on the given list. Hint
below.
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
Use base 10 and base 9.
--
Jerry Friedman is trying to remember or re-generate the correspondence
between the real numbers all subsets of the integers. |
|
| Back to top |
|
|
jerry_friedman@yahoo.com
Guest
|
| Posted: Thu Dec 02, 2004 12:06 am
Post subject: Re: Brand new colors |
|
|
the Omrud wrote:
| Quote: | Wood Avens typed thus:
Each time I see the title of this thread I find myself wondering
what
a genuinely, truly new color could possibly look like.
|
Stay tuned.
| Quote: | There is an SF story in which people are being transported to an
alternative universe, one of the characteristics of which is that
there is a new colour called (IIRC) "varm". However, I can't
remember its name or author, so that's not much help, really.
|
I haven't read that, but I have read H. P. Lovecraft's "The Colour out
of Space"
<http://www.dagonbytes.com/thelibrary/lovecraft/thecolouroutofspace.htm>.
Here's a sentence.
"Stubbornly refusing to grow cool, it soon had the college in a state
of real excitement; and when upon heating before the spectroscope it
displayed shining bands unlike any known colours of the normal spectrum
there was much breathless talk of new elements, bizarre optical
properties, and other things which puzzled men of science are wont to
say when faced by the unknown."
That won't satisfy any Lovecraft cravings anyone may have, so here's
another. However, not to spoil anything, I'm not quoting the vocabular
climax.
"It was a monstrous constellation of unnatural light, like a glutted
swarm of corpse-fed fireflies dancing hellish sarabands over an
accursed marsh, and its colour was that same nameless intrusion which
Ammi had come to recognize and dread."
--
Jerry Friedman |
|
| Back to top |
|
|
jerry_friedman@yahoo.com
Guest
|
| Posted: Thu Dec 02, 2004 12:06 am
Post subject: Re: Brand new colors |
|
|
jerry_friedman@yahoo.com wrote:
....
<http://www.bunnysneezes.net/lovecraft11.html> is much easier to read.
--
Jerry Friedman wonders about bunny sneezes. |
|
| Back to top |
|
|
Alec McKenzie
Guest
|
| Posted: Thu Dec 02, 2004 12:07 am
Post subject: Re: Brand new colors |
|
|
"jerry_friedman@yahoo.com" <jerry_friedman@yahoo.com> wrote:
| Quote: | It seems to me like a standard proof by contradiction. For instance,
Euclid proved that there's no largest prime by saying, in effect,
"Assume there's a largest prime; call it p. Then p! - 1 would also be
a prime and it would be larger. That contradicts the hypothesis, so
there can be no largest prime."
|
This is wrong on two counts:
Euclid's famous proof that there is no largest prime is *not* a proof by
contradiction.
The proof above fails anyway. If p is prime there is no reason why p! -
1 would also be prime. For example, 5 is prime, but 5! - 1 = 119, which
is not prime.
--
Alec McKenzie
mckenzie@despammed.com |
|
| Back to top |
|
|
R H Draney
Guest
|
| Posted: Thu Dec 02, 2004 12:07 am
Post subject: Re: Brand new colors |
|
|
jerry_friedman@yahoo.com filted:
| Quote: |
It seems to me like a standard proof by contradiction. For instance,
Euclid proved that there's no largest prime by saying, in effect,
"Assume there's a largest prime; call it p. Then p! - 1 would also be
a prime and it would be larger. That contradicts the hypothesis, so
there can be no largest prime."
|
Minor quibble the first: Euclid's proof referred to p!+1...p!-1 would not
necessarily be larger than p (it isn't for values p less than or equal to
two)....
Minor quibble the second: p!+1 is not necessarily prime...it must, however, have
a prime factor greater than p, which is sufficient to trigger the necessary
contradiction....r |
|
| Back to top |
|
|
jerry_friedman@yahoo.com
Guest
|
| Posted: Thu Dec 02, 2004 12:07 am
Post subject: Re: Brand new colors |
|
|
Alec McKenzie wrote:
| Quote: | "jerry_friedman@yahoo.com" <jerry_friedman@yahoo.com> wrote:
It seems to me like a standard proof by contradiction. For
instance,
Euclid proved that there's no largest prime by saying, in effect,
"Assume there's a largest prime; call it p. Then p! - 1 would also
be
a prime and it would be larger. That contradicts the hypothesis,
so
there can be no largest prime."
This is wrong on two counts:
Euclid's famous proof that there is no largest prime is *not* a proof
by
contradiction.
|
Sorry. More precisely, "the proof I remembered being told was due to
Euclid". According to
<http://www.math.unicaen.fr/~reyssat/largest.html#proof>, it's
"Kummer's restatement of Euclid's proof", though "restatement" seems a
little strong.
(In fact, Euclid's proof at
<http://aleph0.clarku.edu/%7Edjoyce/java/elements/bookIX/propIX20.html>
does contain a proof by contradiction of one point. However, it's not
what I meant above.)
| Quote: | The proof above fails anyway. If p is prime there is no reason why p!
-
1 would also be prime. For example, 5 is prime, but 5! - 1 = 119,
which
is not prime.
|
But if p is the *largest* prime, then there's no prime factor available
for p! - 1, so it would have to be a larger prime.
That's what makes this proof by contradiction so tidy. You don't need
a procedure to get a real larger prime (which as far as I know is
impossible). You just need a procedure to get a number that
contradicts the hypothesis by not being divisible by the hypothesized
largest prime or any smaller one.
--
Jerry Friedman |
|
| Back to top |
|
|
jerry_friedman@yahoo.com
Guest
|
| Posted: Thu Dec 02, 2004 12:07 am
Post subject: Re: Brand new colors |
|
|
jerry_friedman@yahoo.com wrote:
| Quote: | Alec McKenzie wrote:
"jerry_friedman@yahoo.com" <jerry_friedman@yahoo.com> wrote:
It seems to me like a standard proof by contradiction. For
instance,
Euclid proved that there's no largest prime by saying, in effect,
"Assume there's a largest prime; call it p. Then p! - 1 would
also
be
a prime and it would be larger. That contradicts the hypothesis,
so
there can be no largest prime."
.... |
| Quote: | That's what makes this proof by contradiction so tidy. You don't
need
a procedure to get a real larger prime (which as far as I know is
impossible).
.... |
Well, you could always use the sieve of Eratosthenes.
--
Jerry Friedman |
|
| Back to top |
|
|
jerry_friedman@yahoo.com
Guest
|
| Posted: Thu Dec 02, 2004 12:07 am
Post subject: Re: Brand new colors |
|
|
jerry_friedman@yahoo.com wrote:
| Quote: | Alec McKenzie wrote:
"jerry_friedman@yahoo.com" <jerry_friedman@yahoo.com> wrote:
It seems to me like a standard proof by contradiction. For
instance,
Euclid proved that there's no largest prime by saying, in effect,
"Assume there's a largest prime; call it p. Then p! - 1 would
also
be
a prime and it would be larger. That contradicts the hypothesis,
so
there can be no largest prime."
This is wrong on two counts:
Euclid's famous proof that there is no largest prime is *not* a
proof
by
contradiction.
Sorry. More precisely, "the proof I remembered being told was due to
Euclid". According to
http://www.math.unicaen.fr/~reyssat/largest.html#proof>, it's
"Kummer's restatement of Euclid's proof"
.... |
Second correction: it's not quite that either. It's my restatement
(and maybe my teacher's) in a form whose logic resembles Cantor's
diagonal proof.
--
Jerry Friedman |
|
| Back to top |
|
|
Alec McKenzie
Guest
|
| Posted: Thu Dec 02, 2004 12:07 am
Post subject: Re: Brand new colors |
|
|
"jerry_friedman@yahoo.com" <jerry_friedman@yahoo.com> wrote:
| Quote: | That's what makes this proof by contradiction so tidy. You don't need
a procedure to get a real larger prime (which as far as I know is
impossible). You just need a procedure to get a number that
contradicts the hypothesis by not being divisible by the hypothesized
largest prime or any smaller one.
|
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.
--
Alec McKenzie
mckenzie@despammed.com |
|
| Back to top |
|
|
Peter Moylan
Guest
|
| Posted: Thu Dec 02, 2004 6:00 am
Post subject: Re: Brand new colors |
|
|
Alec McKenzie infrared:
| Quote: | "jerry_friedman@yahoo.com" <jerry_friedman@yahoo.com> wrote:
It seems to me like a standard proof by contradiction. For instance,
Euclid proved that there's no largest prime by saying, in effect,
"Assume there's a largest prime; call it p. Then p! - 1 would also be
a prime and it would be larger. That contradicts the hypothesis, so
there can be no largest prime."
This is wrong on two counts:
Euclid's famous proof that there is no largest prime is *not* a proof by
contradiction.
The proof above fails anyway. If p is prime there is no reason why p! -
1 would also be prime. For example, 5 is prime, but 5! - 1 = 119, which
is not prime.
|
I suppose you're going to claim that 119 is divisible by 7. Sorry, but
7 doesn't qualify, since we already known that 5 is the largest prime.
(That's the hypothesis: not that 5 is prime, but that 5 is the
_largest_ prime.)
--
Peter Moylan peter at ee dot newcastle dot edu dot au
http://eepjm.newcastle.edu.au (OS/2 and eCS information and software) |
|
| Back to top |
|
|
jerry_friedman@yahoo.com
Guest
|
| Posted: Thu Dec 02, 2004 6:02 am
Post subject: Re: Brand new colors |
|
|
Alec McKenzie wrote:
| Quote: | "jerry_friedman@yahoo.com" <jerry_friedman@yahoo.com> wrote:
That's what makes this proof by contradiction so tidy. You don't
need
a procedure to get a real larger prime (which as far as I know is
impossible). You just need a procedure to get a number that
contradicts the hypothesis by not being divisible by the
hypothesized
largest prime or any smaller one.
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).
|
This is not quite what he proves. Instead, he proves that given any
finite set of primes, there must be a prime outside the set.
| Quote: | So, for every prime there exists a larger prime. So there is no
largest
prime. QED
No contradiction involved anywhere.
|
As it happens, there is. Euclid uses a contradiction to prove that if
DE, the product of the primes A, B, and C in his set, is not itself a
prime, it has a prime factor G that's not in the set. Here's
<http://aleph0.clarku.edu/%7Edjoyce/java/elements/bookIX/propIX20.html>:
"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 [divide] DE, therefore G also measures DE. But
it also measures EF [= ABC + 1]. Therefore G, being a number, measures
the remainder, the unit DF, which is absurd."
[brackets mine]
This fact is irrelevant to your point that the main idea of Euclid's
proof is not contradiction, and your point that the proof I outlined is
not Euclid's. (Neither is yours, though it's closer.) However, math
is a good subject for picking nits, in my opinion.
Obaue: "the same with"? Translatorese?
--
Jerry Friedman |
|
| Back to top |
|
|
Jordan Abel
Guest
|
| Posted: Sun Dec 05, 2004 1:38 am
Post subject: Re: Brand new colors |
|
|
On 2004-12-01, Alec McKenzie <mckenzie@despammed.com> wrote:
| Quote: | "jerry_friedman@yahoo.com" <jerry_friedman@yahoo.com> wrote:
It seems to me like a standard proof by contradiction. For instance,
Euclid proved that there's no largest prime by saying, in effect,
"Assume there's a largest prime; call it p. Then p! - 1 would also be
a prime and it would be larger. That contradicts the hypothesis, so
there can be no largest prime."
This is wrong on two counts:
Euclid's famous proof that there is no largest prime is *not* a proof by
contradiction.
The proof above fails anyway. If p is prime there is no reason why p! -
1 would also be prime. For example, 5 is prime, but 5! - 1 = 119, which
is not prime.
|
it's p!+1, and it's "is a prime or has a prime factor greater than p"
and it is a proof by contradiction - the hypothesis "p is the largest
prime" contradicts the conclusion "p!+1 does not have any number up
to and including p as a factor" |
|
| Back to top |
|
|
Richard Maurer
Guest
|
| Posted: Sun Dec 05, 2004 1:38 am
Post subject: Re: Brand new colors |
|
|
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'.
-- ---------------------------------------------
Richard Maurer To reply, remove half
Sunnyvale, California of a homonym of a synonym for also.
---------------------------------------------------------------------- |
|
| Back to top |
|
|
| |
FAQ
Memberlist
Register
Profile
Log in to check your private messages
Log in