Marilyn vos Savant and FLT
Joel Bradford Klammer
ludicrous IQ) attempted to refute the proof of Fermat's Last
Theorem and failed miserably, and that, furthermore, there was
discussion of this in sci.math. Is there a sci.math archive
somewhere where I can read about this? Or can anyone just fill
me in? Much thanks.
Joel
Timothy Y Chow
David T. Ose <o...@symcom.math.uiuc.edu> wrote:
>There is a review of her book in the most recent copy of
>The American Mathematical Monthly.
Yes, that should give you a representative view.
There is one thing that I would like to add, though. Vos Savant's most
crucial argument is that "if we reject a hyperbolic method of squaring
the circle, then we should also reject a hyperbolic method of proving
Fermat's Last Theorem." Most mathematicians are content to dismiss this
as nonsense; I prefer to regard it as an argument that *would* be valid
except that it's founded on a fundamental misconception.
More precisely, vos Savant appears to regard Euclidean geometry and
hyperbolic geometry as ALTERNATIVE AXIOMATIC SYSTEMS FOR *ALL* OF
MATHEMATICS. If we accept this (false) premise, her argument runs
approximately as follows. The problem of squaring the circle calls
for a proof of a certain statement from the axioms of Euclidean
geometry. Bolyai found a proof from the axioms of hyperbolic geometry,
but this doesn't count. What Wiles has done is to find a proof of
Fermat's Last Theorem from the axioms of hyperbolic geometry, but this
doesn't count, because the original problem calls for a proof from the
axioms of Euclidean geometry.
This argument is shot through with misconceptions, but it does have a
certain logic to it. I've thrashed this out on sci.math before, and
not everyone agrees with my interpretation of vos Savant, but I'm
convinced that the above is what she was thinking, and it's helped me
understand where she's coming from.
Peter Flor
>From: kla...@acs3.acs.ucalgary.ca (Joel Bradford Klammer)
>Subject: Marilyn vos Savant and FLT
>Date: 11 May 1995 21:30:53 GMT
>Joel
M. vos Savant´s article appeared in PARADE MAGAZINE, on November 21, 1993. It
contains the following argument: "...what would we think if it were discovered
that Janos Bolyai... managed to 'square the circle' - but only by using his
own hyperbolic geometry? Well, that´s exactly what happened. And Bolyai
himself said that his hyperbolic proof would not work in Euclidean geometry.
So one of the founders of hyperbolic geometry (the geometry used in the
current proof of Fermat´s last theorem) managed to square the circle?! Then
why is it known as such a famous impossibility? Has the circle been squared,
or has it not? Has Fermat´s last theorem been proved, or has it not? I would
say it has not; if we reject a hyerbolic method of squaring the circle, we
should also reject a hyperbolic proof of Fermat´s last theorem. ..."
And so it goes on. There was a discussion of this article last year in the
AMS "Notices", unless I am mistaken. (I have no time right now to check this.)
David T. Ose
>I heard from a friend that Marilyn vos Savant (chick with
>ludicrous IQ) attempted to refute the proof of Fermat's Last
>Theorem and failed miserably, and that, furthermore, there was
>discussion of this in sci.math. Is there a sci.math archive
>somewhere where I can read about this? Or can anyone just fill
>me in? Much thanks.
>Joel
Savant wrote a book in the summer of 1993, about Fermat's last
theorem and Wiles work (I have forgotten the exact title, sorry).
There is a review of her book in the most recent copy of
The American Mathematical Monthly. Interesting and entertaining
reading. The reviewers say that, while she does a very credible
job explaining some of the ideas of modern mathematics, most
of her book's criticism of Wiles and of modern mathematics result
from her own lack of understanding of both mathematics and mathematicians.
And her deceitful use of several prominent mathematicians' names
to give credibility to her book was highly unethical at best,
say the reviewers. But check out the review.
David Ose
David T. Ose
>In article <3ovqba$3...@vixen.cso.uiuc.edu>,
>David T. Ose <o...@symcom.math.uiuc.edu> wrote:
>>There is a review of her book in the most recent copy of
>>The American Mathematical Monthly.
>Yes, that should give you a representative view.
>There is one thing that I would like to add, though. Vos Savant's most
>crucial argument is that "if we reject a hyperbolic method of squaring
>the circle, then we should also reject a hyperbolic method of proving
>Fermat's Last Theorem." Most mathematicians are content to dismiss this
>as nonsense; I prefer to regard it as an argument that *would* be valid
>except that it's founded on a fundamental misconception.
>More precisely, vos Savant appears to regard Euclidean geometry and
>hyperbolic geometry as ALTERNATIVE AXIOMATIC SYSTEMS FOR *ALL* OF
>MATHEMATICS. If we accept this (false) premise, her argument runs
>approximately as follows. The problem of squaring the circle calls
>for a proof of a certain statement from the axioms of Euclidean
>geometry. Bolyai found a proof from the axioms of hyperbolic geometry,
>but this doesn't count. What Wiles has done is to find a proof of
>Fermat's Last Theorem from the axioms of hyperbolic geometry, but this
>doesn't count, because the original problem calls for a proof from the
>axioms of Euclidean geometry.
>This argument is shot through with misconceptions, but it does have a
>certain logic to it. I've thrashed this out on sci.math before, and
>not everyone agrees with my interpretation of vos Savant, but I'm
>convinced that the above is what she was thinking, and it's helped me
>understand where she's coming from.
I would contest your statement that Wiles found a "proof of Fermat's
Last Theorem from the axioms of hyperbolic geometry." The Shimura-
Tanyama-Weil conjectures have little to do with hyperbolic geometry,
although they can be written in the language of hyperbolic geometry,
they are not dependent upon them. This was Vos Savant's biggest
misunderstanding.
David Ose
Kevin D. Quitt
>> Apparently a lot of mathematicians said she was utterly ignorant about
>> this stuff (she did have a pretty lousy wording of the problem), etc., etc.
>>
>This happened. However when the mistake was pointed out to them, they
>saw it and she got a large number of apologies from them. In addition
>the journals were on her side.
Actually, her wording was so poor that her answer was wrong. She failed
to specify a necessary condition (that Monty did not show a door at random
and would never reveal the prize) that determines that strategy. The
mistake the mathematicians made was in not realizing the ambiguity, and
(some) in being pretty damn rude.
--
#include <standard.disclaimer>
_
Kevin D Quitt USA 91351-4454 96.37% of all statistics are made up
Kevin D. Quitt
>She seems to have a good grounding in probability theory, giving
>nice explanations for the Monty Hall paradox for instance (and defending
>herself against academics whose intuition is leading them astray).
After she misstated the problem, you mean? Her description of the
problem was ambiguous; if you didn't make the same assumption she did,
you got the same asnwer as all the mathematicians who "got it wrong".
Had she stated it correctly, there would have been no argument. On the
other hand, it seems typical of her style to leave off some of the
information needed. Maybe she's read too much Agatha Christie.
RKinnamon
proof will hold up. I have vos Savant's book and all she says is that
because of the methods Wiles used there may be errors and it may take a
while to verify. And remember, she wrote the book BEFORE Wiles admitted
his error a year ago December. Also, certainly Fermat, if he did indeed
have proof, used a more simplistic, "marvelleous" method.
Dik T. Winter
> What Wiles has done is to find a proof of
> Fermat's Last Theorem from the axioms of hyperbolic geometry, but this
> doesn't count, because the original problem calls for a proof from the
> axioms of Euclidean geometry.
>
> This argument is shot through with misconceptions, but it does have a
> certain logic to it.
What I understand of Wiles' proof is that it is even not geometric. (I do
not have a copy, but there was somebody who gave an introduction about it,
so I have a minimal understanding.)
Perhaps she was misled by the use of the term elliptic curves (that has
nothing to do with elliptic geometry)?
--
dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924098
home: bovenover 215, 1025 jn amsterdam, nederland; e-mail: d...@cwi.nl
Aaron Bergman
: I doubt it if Marilyn was "misled" . . . It is still uncertain that Wiles
: proof will hold up.
Most people are pretty sure now, IIRC.
: I have vos Savant's book and all she says is that
: because of the methods Wiles used there may be errors and it may take a
: while to verify.
She said that it was wrong because he used some hyperbolic geometry
which shows a complete non-understanding of how the proof works or
even how axiomatic systems are used.
: And remember, she wrote the book BEFORE Wiles admitted
: his error a year ago December.
And that error was completely unrealted to anything she said.
: Also, certainly Fermat, if he did indeed
: have proof, used a more simplistic, "marvelleous" method.
If he had one. What does that have to do with it, though?
Aaron
Benjamin J. Tilly
rkin...@aol.com (RKinnamon) writes:
> I doubt it if Marilyn was "misled" . . . It is still uncertain that Wiles
> proof will hold up. I have vos Savant's book and all she says is that
> because of the methods Wiles used there may be errors and it may take a
> while to verify. And remember, she wrote the book BEFORE Wiles admitted
> his error a year ago December. Also, certainly Fermat, if he did indeed
> have proof, used a more simplistic, "marvelleous" method.
I have read the book. That book is a joke. If you check, you will find
that she argues that since:
i) You cannot square the circle, and
ii) In hyperbolic geometry you can square the circle,
that
iii) hyperbolic geometry cannot be valid.
Thereby "overturning" over a century of well-established mathematics.
(The error is that the theorem involved in (i) only related to
EUCLIDEAN geometry, and says nothing about what you can or cannot do
with NON-Euclidean geometry. The situation is like my telling you
something about my dog Bob at one time, then something about my friend
Bob at another, and then having you conclude that I have lied to you
since the two comments do not jive with each other.)
In fact she goes on to argue that there is a possibility of disproving
Einstein's theory of general relativity!
All of which is complete hogwash.
Oh, and she has yet to admit any mistake, although a variety of people
have written to her, the mathematicians that she referred to having
disputed her comments, and a number of mathematical publications (the
latest being Math Monthly) having published highly critical articles.
Ben Tilly
keith green (simctr)
: I have read the book. That book is a joke. If you check, you will find
: that she argues that since:
: i) You cannot square the circle, and
: ii) In hyperbolic geometry you can square the circle,
It's pretty clear that Marylin is wrong. Check out recent comments in
sci.math, for a good discussion on the subject. They have a reference
to a review of Marylin's book in AMM, or something like that.
(You can also check out their faq at
http://daisy.uwaterloo.ca/~alopez-o/math-faq/math-faq.html for general
info on FLT. It looks pretty and is very well organized, but I wish it
had more details.)
Marylin is not necessarily wrong to hold her ground against a horde of
mathematicians. I remember reading several years ago a few comments
from a guy named Posner (or Poser or something like that) in The
Skeptical Inquirer about her column on the 3 door Monty problem.
Apparently a lot of mathematicians said she was utterly ignorant about
this stuff (she did have a pretty lousy wording of the problem), etc., etc.
Anyways the guy was using this as evidence that even college profs
(including some mathematicians) are innumerate. Really obnoxious. I
kept thinking this guy is an idiot. He's a mensan, so that just fit
in perfectly with my stereotype. HOWEVER, when I went on to read the
comments the mathematicians had sent her -- man, some of them were really
stupid and inappropriate. She was pretty polite, I think, not to get
too bent outta shape over this whole mess.
Well, she's out of her league on this. I think she may eventually
figure that out. (Kinda hard to do that after you've written a book
on something.) However, it is pretty silly that she would go out
on a limb like that in the first place. Probably thought she had
something to prove to herself. It happens.
I don't understand it all, but I'll go out on a limb and guess that
even if she turns out to be right, that her reasoning is incorrect.
I think she oughtta recognize that no matter how smart she is, she
just doesn't have to be the best at everything. She doesn't always
have to be the smartest or the rightest.
--
My employers disagree with everything I say, keith green, nan
write, think, believe, feel, do, or plan to do. <kgr...@ida.org>
They are not responsible for me, nor I for them.
Kevin Brown
DO> There is a review of her book in the most recent copy of
DO> The American Mathematical Monthly.
TC = Tim Chow
TC> There is one thing that I would like to add, though. Vos Savant's
TC> most crucial argument is that "if we reject a hyperbolic method of
TC> squaring the circle, then we should also reject a hyperbolic method
TC> of proving Fermat's Last Theorem"...
Every review and discussion of this book that I've seen has focused on
the "hyperbolic/Euclidean circle-squaring" argument, which was clearly
the weakest point of the book. However, the book also raised another
objection to Wiles' proof. As I recall, Ms Vos Savant observed that
mathematicians believe it is impossible to formally prove the consistency
of arithmetic. In other words, we have no rigorous proof that the
basic axioms of arithmetic do not lead to a contradiction at some point.
Therefore (she argues) if we assume a counter-example to FLT and
then, via some long and complicated chain of reasoning, produce a
contradiction, how do we know that the assumed counter-example is the
real cause of the contradiction? Could it not be that we have just
succeeded in producing the long-dreaded contradiction inherent in
arithmetic itself?
Of course, this kind of objection could be answered by just prefacing
every theorem with the words "If arithmetic is consistent, then...",
but this would get rather tedious. Nevertheless, the basic
observation has some merit. With short simple proofs we can usually
feel confident that the contradiction really is directly related to
our special assumption, but with very long proofs extending over
multiple papers by multiple authors, and involving the interaction
of many different branches and facets of mathematics, how would we
really distinguish between a subtle contradiction caused by one
specific false assumption vs a subtle contradiction inherent in the
fabric of arithmetic itself?
Matthew P Wiener
>I doubt it if Marilyn was "misled" . . .
Correct. She made her boners up all on her own.
> It is still uncertain that Wiles
>proof will hold up.
Uh no.
> I have vos Savant's book and all she says is that
>because of the methods Wiles used there may be errors and it may take a
>while to verify.
This is nonsense. She says there is something inherently improper about
the methods Wiles used. This is rank gibberish on her part.
> And remember, she wrote the book BEFORE Wiles admitted
>his error a year ago December.
So what?
> Also, certainly Fermat, if he did indeed
>have proof, used a more simplistic, "marvelleous" method.
He didn't have a proof.
--
-Matthew P Wiener (wee...@sagi.wistar.upenn.edu)
Benjamin J. Tilly
kgr...@dmsoproto.ida.org (keith green (simctr)) writes:
> Benjamin J. Tilly (Benjamin...@dartmouth.edu) wrote:
> : I have read the book. That book is a joke. If you check, you will find
> : that she argues that since:
> : i) You cannot square the circle, and
> : ii) In hyperbolic geometry you can square the circle,
> : iii) hyperbolic geometry cannot be valid.
>
> It's pretty clear that Marylin is wrong. Check out recent comments in
> sci.math, for a good discussion on the subject. They have a reference
> to a review of Marylin's book in AMM, or something like that.
>
sci.math is where I am posting from. :-)
> (You can also check out their faq at
> http://daisy.uwaterloo.ca/~alopez-o/math-faq/math-faq.html for general
> info on FLT. It looks pretty and is very well organized, but I wish it
> had more details.)
>
We wish that there were more details available at the moment...
> Marylin is not necessarily wrong to hold her ground against a horde of
> mathematicians. I remember reading several years ago a few comments
> from a guy named Posner (or Poser or something like that) in The
> Skeptical Inquirer about her column on the 3 door Monty problem.
> Apparently a lot of mathematicians said she was utterly ignorant about
> this stuff (she did have a pretty lousy wording of the problem), etc., etc.
>
This happened. However when the mistake was pointed out to them, they
saw it and she got a large number of apologies from them. In addition
the journals were on her side.
But when the shoe is on her foot? All that she has done so far is
produce a form letter. I cannot remember exactly what it says, but it
essentially it says that she knows that she is right and everybody else
is wrong...
> Anyways the guy was using this as evidence that even college profs
> (including some mathematicians) are innumerate. Really obnoxious. I
> kept thinking this guy is an idiot. He's a mensan, so that just fit
> in perfectly with my stereotype. HOWEVER, when I went on to read the
> comments the mathematicians had sent her -- man, some of them were really
> stupid and inappropriate. She was pretty polite, I think, not to get
> too bent outta shape over this whole mess.
>
I agree that the worst letters were pretty bad. But the overall tone of
the result was nowhere near the worst of the letters.
> Well, she's out of her league on this. I think she may eventually
> figure that out. (Kinda hard to do that after you've written a book
> on something.) However, it is pretty silly that she would go out
> on a limb like that in the first place. Probably thought she had
> something to prove to herself. It happens.
>
We will see. Probably even if she does figure it out she will never
admit it.
> I don't understand it all, but I'll go out on a limb and guess that
> even if she turns out to be right, that her reasoning is incorrect.
> I think she oughtta recognize that no matter how smart she is, she
> just doesn't have to be the best at everything. She doesn't always
> have to be the smartest or the rightest.
No guesses needed. Hyperbolic geometry is on as good a footing as
arithmetic. (Literally.) It has applications in practical areas such as
engineering, and it even has shown up in art. (Escher's circle-limit
drawings are actually representations of pictures on the hyperbolic
plane!)
In fact here is the representation that Escher used. Draw a circle. The
hyperbolic plane is the interior of that circle. However everything is
distorted so straight lines in the hyperbolic plane show up as lines
through the center of the circle, or as circles which intersect the
original circle at right angles.
If you run through it you can show that the first 4 axioms of Euclid
apply (digging those axioms up may take more work than the
demonstration, I recommend finding the Elements, a good encyclopedia
article, or _A History of Mathematics_ by Boyer), but the fifth one
fails.
Ben Tilly
Ben Tilly
Gary W. Takahashi
Theorem, but who is not heavily into number theory, but has a
bachelor's degree level understanding, the article "Fermat's Last
theorem and Modern Arithmetic" by Kenneth Ribet and Brian Hayes
in American Scientist, March-April 1994.
He summarizes FLT and reviews early efforts at solving it.
Then he talks about elliptic equations and what it means to be modular.
Then he presents that Taniyama-Shimura conjecture, and Frey's equation,
Galois groups, the Langlands-Tunnell theorem, then shows you where
the progress stops.
There is absolutely no reliance on hyperbolic geometry that vos-Savant
hinges her argument on.
The article, while recognizing that the intended reader does not have
a working familiarity with the field, doesn't dumb-down the discussion.
Four stars from this reviewer!
--Gary
y
--
Gary Takahashi, M.D. | "There is nothing that God hath established in a
Oregon Hematology/Oncology | a constant course of nature, which is done every
Portland, OR | day, but would seem a Miracle, and exercise our
| admiration, if it were done but once."
Kevin Brown
Benjamin P. Carter
> ... (she argues) if we assume a counter-example to FLT and
>then, via some long and complicated chain of reasoning, produce a
>contradiction, how do we know that the assumed counter-example is the
>real cause of the contradiction?
What could she possibly mean by "cause of the contradiction"? Events
and phenomena, not contradictions, have causes.
>Could it not be that we have just
>succeeded in producing the long-dreaded contradiction inherent in
>arithmetic itself?
That would follow if there were both a proof of and a counterexample
to FLT (or any other statement). Then we would know that arithmetic
is self-contradictory.
If MVS objects to the proof of FLT on this basis,
then she objects to almost all other proofs as well.
Very few axiomatic systems are known to be consistent.
--
Ben Carter internet address: b...@netcom.com
keith green (simctr)
: > It's pretty clear that Marylin is wrong. Check out recent comments in
: > sci.math, for a good discussion on the subject. They have a reference
: > to a review of Marylin's book in AMM, or something like that.
: >
: sci.math is where I am posting from. :-)
Ouch. I need to start paying closer attention to those header lines.
I thought your name looked familiar. But I didn't remember it from
there. Of course, I usually brush through pretty haphazardly. A
man with a mission.
: This happened. However when the mistake was pointed out to them, they
: saw it and she got a large number of apologies from them. In addition
: the journals were on her side.
Well. I would imagine the journals would be with her. I'm glad to hear
she got apologies. But the only people I would have expected to really
side with her and understand things well enough to agree with her would
be the statisticians. If math guys don't follow it, I don't view that
as evidence of innumeracy.
: But when the shoe is on her foot? All that she has done so far is
: produce a form letter. I cannot remember exactly what it says, but it
: essentially it says that she knows that she is right and everybody else
: is wrong...
Heya, if you guys could put this in the faq, it would be a nice.
(not necessary, but nice.)
: I agree that the worst letters were pretty bad. But the overall tone of
: the result was nowhere near the worst of the letters.
Okay. Well, I was reading a biased report. The guy in SI really sounded
like an ass to me.
: No guesses needed. Hyperbolic geometry is on as good a footing as
: arithmetic. (Literally.) It has applications in practical areas such as
I'll have to take your word for it (which I don't have a problem with).
: engineering, and it even has shown up in art. (Escher's circle-limit
: drawings are actually representations of pictures on the hyperbolic
: plane!)
: In fact here is the representation that Escher used. Draw a circle. The
: hyperbolic plane is the interior of that circle. However everything is
: distorted so straight lines in the hyperbolic plane show up as lines
: through the center of the circle, or as circles which intersect the
: original circle at right angles.
man, at times like this I really wish we could pass postscript.
: If you run through it you can show that the first 4 axioms of Euclid
: apply (digging those axioms up may take more work than the
: demonstration, I recommend finding the Elements, a good encyclopedia
: article, or _A History of Mathematics_ by Boyer), but the fifth one
: fails.
I'll probably check out Elements again in the future. To be honest,
when I first started reading it, I was enormously excited. But after
a few weeks, it really got dry. I did read Charles Boyer's _A History
of the Calculus and its Conceptual Development_ when I was in high
school, but was not aware of this other work. I'll definitely check it
out. I'm vaguely familiar with this stuff - VERY vaguely. I know one
or two of the buzzwords. I understand the 'thats' - well, some of them,
anyway. What I don't get are the whys and hows.
thanks for the reference,
k
William H. Jefferys
She came up with another doozy this week. Her claim is that
it is more dangerous to fly in a twin-engined plane than
a single-engined plane, on the grounds that one is more likely
to have an engine failure. It is, of course, true that if
you have two engines, and the probability of failure is
independent between them, one is more likely to experience a
failure than if one only has one. But what she fails to understand
is that twin-engined planes are designed to be able to fly
and to land even if one engine fails. This is just sound
engineering practice, eliminating a single-point failure.
So, it is safer to fly a twin-engined plane than a single
engined plane.
Bill
--
Bill Jefferys/Department of Astronomy/University of Texas/Austin, TX 78712
E-mail: bi...@clyde.as.utexas.edu | URL: http://quasar.as.utexas.edu
Finger for PGP Key: F7 11 FB 82 C6 21 D8 95 2E BD F7 6E 99 89 E1 82
--------------------------------------------------------------------------
Hugh LaMaster
> I'll probably check out Elements again in the future. To be honest,
> when I first started reading it, I was enormously excited. But after
> a few weeks, it really got dry. I did read Charles Boyer's _A History
> of the Calculus and its Conceptual Development_ when I was in high
> school, but was not aware of this other work. I'll definitely check it
> out.
For anyone with a moderate mathematical background who wants to
explore further the relationship of Euclidean to non-Euclidean
geometry, I recommend the following book:
Greenberg, Marvin J. (Jay), Euclidean and non-Euclidean
geometries: development and history, p. 183, W. H. Freeman,
New York, 1993, 1979, 1974. (LC: QA445 .G84 1993 DD: 516
dc20 ISBN: 0716724464 LCCN: 93-7207)
You don't need a PhD in Algebraic Geometry to understand this
book, and I think reading it will clarify why the geometry
arguments in M. V-S's book are so easily dismissed.
[But, Greenberg's book won't help you with Wiles' proof of FLT.]
--
Hugh LaMaster, M/S 233-9, UUCP: ames!lamaster
NASA Ames Research Center Internet: lama...@ames.arc.nasa.gov
Moffett Field, CA 94035-1000 Or: lama...@george.arc.nasa.gov
Phone: 415/604-1056 #include <std_disclaimer.h>
Randa...@jhuapl.edu
In article <3otvmd$1h...@ds2.acs.ucalgary.ca>, <kla...@acs3.acs.ucalgary.ca> writes:
> I heard from a friend that Marilyn vos Savant (chick with
> discussion of this in sci.math. Is there a sci.math archive
> somewhere where I can read about this? Or can anyone just fill
> me in? Much thanks.
Off the point, but I think the discrepancy between what people expect of
a "chick with a ludicrous IQ" and what vos Savant actually thinks and
writes, is a good argument against taking IQ numbers too seriously as
a good measure of brilliance. I was amazed to see how many questions
in her first columns were of the type "so, O wise guru, how many angels
CAN dance on the head of a pin?" She seems to spend most of the column
now on brain-teasers of the type that I (and I assume many of the people
here in the technical newsgroups) used to do for fun when I was around
10-13. She seems to have a good grounding in probability theory, giving
nice explanations for the Monty Hall paradox for instance (and defending
herself against academics whose intuition is leading them astray).
So she would probably be at least a competent grad student. But it's a
long leap from there to "the next Einstein", and if she truly has any
talent and brilliance in mathematics or anything else, it isn't going
to be measured by an IQ test. She could very well be brilliant, but
there just isn't any evidence from her newspaper column (or her test
score) to say one way or the other.
For those who don't follow the column, a typical puzzle was one sent
in by Martin Gardner (the king of Mathematical Recreation), which was
a coded message. The code was the simple A=1, B=2, C=3, etc... The
only thing that made it mildly interesting was that there were no
breaks between letter and word digits, so there was occasionally a
slight ambiguity.
Flapjack
than writing for Parade Magazine?
flapjack-who wonders how these posts wind up on alt.stupidity
--
The Former Official Pet Newbie of alt.stupidity, candidate for a Master
of Stupidity Degree and inventor of "and bacon."
"I bought a Super Nintendo. Kinda tasted like chicken."-Stick Figure
Man, "Western Massachusetts Avengers" #4
Here's that lame homepage you've been hearing about:
http://www-bprc.mps.ohio-state.edu/cgi-bin/hpp?flapjack.html
Matthew P Wiener
>Every review and discussion of this book that I've seen has focused on
>the "hyperbolic/Euclidean circle-squaring" argument, which was clearly
>the weakest point of the book. However, the book also raised another
>objection to Wiles' proof. As I recall, Ms Vos Savant observed that
>mathematicians believe it is impossible to formally prove the consistency
>of arithmetic. In other words, we have no rigorous proof that the
>basic axioms of arithmetic do not lead to a contradiction at some point.
It's very easy to prove the consistency of arithmetic. Just work in ZFC.
M. J. Saltzman
bi...@clyde.as.utexas.edu (William H. Jefferys) writes:
>She came up with another doozy this week. Her claim is that
>it is more dangerous to fly in a twin-engined plane than
>a single-engined plane, on the grounds that one is more likely
>to have an engine failure. It is, of course, true that if
>you have two engines, and the probability of failure is
>independent between them, one is more likely to experience a
>failure than if one only has one. But what she fails to understand
>is that twin-engined planes are designed to be able to fly
>and to land even if one engine fails. This is just sound
>engineering practice, eliminating a single-point failure.
>So, it is safer to fly a twin-engined plane than a single
>engined plane.
I never saw the original column that provoked the letter-writer in
this case, but a careful reading of the letter indicates that the
discussion was about comparing the safety of a single-engine plane to
that of a plane that *requires* two engines to remain airborne. Her
response is actually a passable explanation of the fact that if the
failure probability p for a single engine is small, then 2p is an
adequate approximation to the failure probability for such a
twin-engine plane, but as p gets larger, the approximation gets worse.
That's the thing about these little word puzzles--they are designed to
be puzzling, not to be a reasonable reflection of reality or "sound
engineering practice". Your complaint is the same as the complaints
about the Monty Hall problem being unrealistic. OK, but that doesn't
invalidate the mathematical ideas being illustrated.
--
Matthew Saltzman
Clemson University Math Sciences
m...@clemson.edu
William H. Jefferys
M. J. Saltzman <m...@hubcap.clemson.edu> wrote:
#[Alt.stupidity removed from the newsgroups line.]
#
#bi...@clyde.as.utexas.edu (William H. Jefferys) writes:
#
#>She came up with another doozy this week. Her claim is that
#>it is more dangerous to fly in a twin-engined plane than
#>a single-engined plane, on the grounds that one is more likely
#>to have an engine failure.
#
#I never saw the original column that provoked the letter-writer in
#this case, but a careful reading of the letter indicates that the
#discussion was about comparing the safety of a single-engine plane to
#that of a plane that *requires* two engines to remain airborne.
Hmm. You are right. I did not read her letter correctly. It
begins
"You recently stated that it is more dangerous to
fly in a plane that _requires_ two engines for
flight than in a plane requiring only one..."
So she is exonerated this time. My apologies to Ms. vos Savant,
and my thanks to you for pointing out my error.
Jeffery D. Sykes
William H. Jefferys <bi...@clyde.as.utexas.edu> wrote:
>
>She came up with another doozy this week. Her claim is that
>is that twin-engined planes are designed to be able to fly
>and to land even if one engine fails.
When this problem first appeared (this was a followup article),
she made the assumption that failure of one engine would result
in a crash. If I remember correctly, that assumption was part
of the original "questioner's" problem.
Jeff
--
Jeffery D. Sykes | "The cross that cost my Lord his life has given
sy...@ms.uky.edu | me mine!" -- Point of Grace, "The Great Divide"
University of Kentucky |
Department of Mathematics | Ask me about THE KRYPTONIAN CYBERNET!
Ross Garmil
: rkin...@aol.com (RKinnamon) writes:
: Thereby "overturning" over a century of well-established mathematics.
: (The error is that the theorem involved in (i) only related to
: EUCLIDEAN geometry, and says nothing about what you can or cannot do
: with NON-Euclidean geometry. The situation is like my telling you
: something about my dog Bob at one time, then something about my friend
: Bob at another, and then having you conclude that I have lied to you
: since the two comments do not jive with each other.)
See, now here's where I get confused. Do we live in a world dominated by
Euclidian geometry, and if so, then where does non-Euclidian come into play.
I'm not a matehmatician, but it seems to me that if humans cannot experience a
plane of existence (bad pun, sorry), where parallel lines indeed do intersect,
then why the fuck study it? Or is it just a different way of thinking about
and perceiving our universe? Or am I just being too philosophical for this
froup? And why do we call it a froup and not a group? I think it was Renes
Decartes who first said, "a froup is a group, unless it's a froup." And he
was a mathemetician, even though people laughed at him for having the name
Rene.
: In fact she goes on to argue that there is a possibility of disproving
: Einstein's theory of general relativity!
: All of which is complete hogwash.
Well, um, if there wasn't a possibility of disproving Einstein's theory, then
it would be a law, correct? Don't forget to get behind the pig's ears.
Ross--who'll be hanging with his dog after the war.
Bob Silverman
:Benjamin J. Tilly (Benjamin...@dartmouth.edu) wrote:
:: In article <3p2j10$i...@newsbf02.news.aol.com>
:: rkin...@aol.com (RKinnamon) writes:
:See, now here's where I get confused. Do we live in a world dominated by
:Euclidian geometry, and if so, then where does non-Euclidian come into play.
:I'm not a matehmatician, but it seems to me that if humans cannot experience a
But you DO experience non-Euclidean geometry every day!
You live on the surface of a (almost) sphere: the Earth.
A 'line' is determined by 2 points. It is the great circle containing those
two points. Note now that 2 lines are NEVER parallel..... Two great circles
must ALWAYS intersect.... You thus have a geometry violating Euclid's 5th
postulate which says you can draw exactly ONE line containing some given point
that is parallel to some fixed line. In spherical geometry there is NO such
line.
Try doing any sort of global navigation. You will experience non-Euclidean
geometry in a hurry.....
:plane of existence (bad pun, sorry), where parallel lines indeed do intersect,
:then why the fuck study it? Or is it just a different way of thinking about
Let me also add that on the scale of the entire universe we do not yet know
whether our geometry is Euclidean or non-Euclidean.
--
Bob Silverman
The MathWorks Inc.
24 Prime Park Way
Natick, MA
Ross Garmil
: In article <3paod7$l...@news.bu.edu>, Ross Garmil <lim...@bu.edu> wrote:
: :Benjamin J. Tilly (Benjamin...@dartmouth.edu) wrote:
: :: In article <3p2j10$i...@newsbf02.news.aol.com>
: :: rkin...@aol.com (RKinnamon) writes:
:
: stuff deleted....
: :See, now here's where I get confused. Do we live in a world dominated by
: :Euclidian geometry, and if so, then where does non-Euclidian come into play.
: :I'm not a matehmatician, but it seems to me that if humans cannot experience a
:
: But you DO experience non-Euclidean geometry every day!
: You live on the surface of a (almost) sphere: the Earth.
: A 'line' is determined by 2 points. It is the great circle containing those
: two points. Note now that 2 lines are NEVER parallel..... Two great circles
: must ALWAYS intersect.... You thus have a geometry violating Euclid's 5th
: postulate which says you can draw exactly ONE line containing some given
: point
: that is parallel to some fixed line. In spherical geometry there is NO such
: line.
: Try doing any sort of global navigation. You will experience non-Euclidean
: geometry in a hurry.....
: Let me also add that on the scale of the entire universe we do not yet know
: whether our geometry is Euclidean or non-Euclidean.
Oh. No really, I never studied much math (a little calculus, but let's just
say that me and Mr. Newton had a disagreement--he thought it made sense and I
thought it was....well, calculus). So although I had heard the term
"Non-Euclidian geometry" I didn't understand--but now it seems to make sense
that it's based on looking at things sperically. It appears to me now (give
that all the knowledge I have is what you just bestowed upon me) that
non-Euclidian deals with geometry based on a sperical, rather than a flat,
plane, or to be more precise on a sphere itself and therefore not on a plane
at all. I think I understand that, if I'm wrong then I don't understand that,
but I think I do (hey everyone, I'm learning something here!).
So....why can't a line just be called a curve and keep the definition of line
as straight line----no wait, because on a sphere as the surface a straight
line can't exist, I understand that, but how about as a tangent? Is that the
word? You know, touching one point of the sphere and then taking off outside,
or perhaps intersecting the sphere, so that there could exist what we perceive
(traditionally) as "straight" and these straight things could be called
"lines" and therefore two parallel "lines" wouldn't intersect.
I also don't understand how lines cannot necessarily intersect, such as lined
of longitude (to use your Earth example), or am I just fixating too much on
the word "line"?
But hey, I'm from alt.stupidity, so that's supposed to be an excuse, right?
Ross
Timothy Y Chow
Kevin D. Quitt <k...@emoryi.jpl.nasa.gov> wrote:
>Actually, her wording was so poor that her answer was wrong. She failed
>to specify a necessary condition (that Monty did not show a door at random
>and would never reveal the prize) that determines that strategy. The
>mistake the mathematicians made was in not realizing the ambiguity, and
I don't buy this for a minute. I have *never* seen convincing evidence/
argument that the mathematicians interpreted the problem differently
from the way vos Savant did and correctly solved the problem from their
own point of view, unaware of other possible interpretations. All the
evidence points to the mathematicians' making a blatant error. Can you
give any evidence for your point of view?
Tim.
Terry Moore
k...@emoryi.jpl.nasa.gov (Kevin D. Quitt) wrote:
>
> Randa...@jhuapl.edu wrote:
> >She seems to have a good grounding in probability theory, giving
> >nice explanations for the Monty Hall paradox for instance (and defending
> >herself against academics whose intuition is leading them astray).
>
> problem was ambiguous; if you didn't make the same assumption she did,
Welcome to the real world. Mathematicians are (sometimes/often)
careful to phrase a problem unambiguously. In the real world you
have to choose a reasonable model. Most non-mathematicians would
come up with Marilyn's model, and still get the answer wrong.
Terry Moore, Statistics Department, Massey University, New Zealand.
Moore's law: almost all real problems are ill-posed.
David Ullrich
I hope somebody points out that this is a joke...
Dave Ullrich
Sinan Karasu
>So....why can't a line just be called a curve and keep the definition of line
Staright line.
1) Is it the path light traverses? (The obvious problem is that how do you know
it eventually does not fold back unto itself...)
2) Is it the path you take by "not turning" ? Think of a creature who
is bound to surface of earth without a sense of "up direction"
3) Is it the shortest distance between 2 points? See number 2.
A Good example is in Weyl's book about a universe ( I am taking liberties here)
which is finite spherical, it goes from 100 deg. K at the center to 0 deg. K
(that is Kelvin) at the surface. Assuming matter shrinks to 0 size at 0K,
imagine the orbits of various "particles" in that universe.
Sinan
Terry Moore
So although I had heard the term
> "Non-Euclidian geometry" I didn't understand--but now it seems to make sense
> that it's based on looking at things sperically. It appears to me now (give
> that all the knowledge I have is what you just bestowed upon me) that
> non-Euclidian deals with geometry based on a sperical, rather than a flat,
> plane, or to be more precise on a sphere itself and therefore not on a plane
> at all.
Right idea, but the details are a bit fuzzy. Elliptic geometry is something
like the sphere, hyperbolic geometry is, in some sense, opposite.
In elliptic geometry, parallel lines don't exist, in hyperbolic geometry
there are several parallel lines to a given line through the same point.
> So....why can't a line just be called a curve and keep the definition of line
> as straight line----no wait, because on a sphere as the surface a straight
> line can't exist, I understand that, but how about as a tangent?
Good question.
Suppose the space we live in is curved. Just where is this "outside"
where you are going to draw the tangent? You see, the sphere
embedded in 3D space is just an example. It helps with visualisation,
but introduces extra confusion too.
Nevertheless, curved spaces do have things called "tangent spaces"
but they are purely formal mathematical constructs designed to
achieve aims similar to yours.
The nearest you can get to a straight line in a curved space is called a
geodesic. In navigation these are important. International airline
flights follow geodesics (also called great circles) to save fuel
(except during the Gulf War and some other conflicts).
Terry Moore, Statistics Department, Massey University, New Zealand.
Imagine a person with a gift of ridicule [He might say] First that a
negative quantity has no logarithm; secondly that a negative quantity has
no square root; thirdly that the first non-existent is to the second as the
circumference of a circle is to the diameter. Augustus de Morgan
Bryan
: I never saw the original column that provoked the letter-writer in
: this case, but a careful reading of the letter indicates that the
: discussion was about comparing the safety of a single-engine plane to
: that of a plane that *requires* two engines to remain airborne. Her
: response is actually a passable explanation of the fact that if the
: failure probability p for a single engine is small, then 2p is an
: adequate approximation to the failure probability for such a
: twin-engine plane, but as p gets larger, the approximation gets worse.
What is the actual probability? Cursory inspection suggests (2p-p^2).
Why would whe have to approximate? It doesn't seem like that hard of
a concept.
Regards,
Bryan
Bryan
: apply (digging those axioms up may take more work than the
: demonstration, I recommend finding the Elements, a good encyclopedia
: article, or _A History of Mathematics_ by Boyer), but the fifth one
: fails.
Errr...liked your post, but I think you mean the 5th *postulate*, don't
you? It would be big news if Euclid's axioms had been overthrown.
Regards,
Bryan
--
Bryan
: Let me also add that on the scale of the entire universe we do not yet know
: whether our geometry is Euclidean or non-Euclidean.
Isn't the question re lambda not Euc vs Non-Euc, but rather hyperbolic vs
elliptic?
regards,
bryan
--
Kevin Brown
KB> impossible to formally prove the consistency of arithmetic. In
KB> other words, we have no rigorous proof that the basic axioms of
KB> arithmetic do not lead to a contradiction at some point.
MW = Matthew Wiener
MW> It's very easy to prove the consistency of arithmetic. Just work
MW> in ZFC.
Let us say a proof of the consistency of system X is "incomplete" if
it's carried out within a system Y whose consistency has not been
completely proven.
Theorem:
Every proof of the consistency of arithmetic is incomplete.
In view of this, it isn't clear how "working in ZFC" resolves the
issue. There is no complete and absolute proof of the consistency
of arithmetic, so every arithmetical proof is subject to doubt. (By
focusing on arithmetic I don't mean to imply that other branches of
mathematics are exempt from doubt. Hermann Weyl, commenting on
Godel's work, said that "God exists because mathematics is undoubtedly
consistent, and the devil exists because we cannot prove the
consistency".)
Here is a quote from Morris Kline's book "Mathematics and the Loss of
Certainty":
"Godel's result on consistency says that we cannot
prove consistency in any approach to mathematics
by safe logical principles...
Here is a quote from John Stillwell's book "Mathematics and its
History":
"If S is any system that includes PA, then Con(S) [the
consistency of S] cannot be proved in S, if S is
consistent." [Godel's 2nd theorem]
I've received email suggesting that the contemplation of inconsistency
in the system of arithmetic is tantamount to a renounciation of reason
itself, i.e., if our concept of natural numbers is inconsistent then we
must be incapable of rational thought, and any further consideration is
pointless. ("We recoil in horror", as Poincare said regarding Cantor's
set theory.)
However, my view of formal systems is somewhat different. I think
they should be seen not as unordered ("random access") sets of
syllogisms, but as structured spaces, each layer of implicated
objects representing a region, and the implications representing
connections between different regions. The space may even possess
a kind of metric, although "distances" are not necessarily commutative.
For example, the implicative distance from an integer to its prime
factors is greater than the implicative distance from those primes
to their product.
According to this view a formal system does not degenerate into
complete nonsense simply because at some point it contains a
contradiction. A system may be "locally" consistent even if it
is not globally consistent. To give a crude example, suppose we
augment our normal axioms and definitions of arithmetic with the
statement that a positive integer n is prime if and only if
2^n - 2 is divisible by n. This axiom conflicts with our existing
definition of a prime, but the first occurrence of a conflict
is 341. Thus, over a limited range of natural numbers my crude
axiom system is "locally consistent".
Suppose I then substitute a stronger axiom by saying n is a prime iff
f(u^n) = 0 (mod n) where u is any root of f(x) = x^5 - x^3 - 2x^2 + 1.
With this system I might go quite some time without encountering a
contradiction. When I finally do bump into a contradiction (e.g.,
2258745004684033) then I could simply substitute an even stronger
axiom. In fact, I can easily specify an axiom of this kind for which
the smallest actual exception is far beyond anyone's (present) ability
to find, and for which we have no theoretical proof that any exception
even exists. Thus, there is no direct proof of inconsistency. I might
then, with enough imagination, develop a plausible (e.g., as plausible
as Banach-Tarski) non-finitistic system within which I can actually
prove that my arithmetic is consistent. In fact, it might actually BE
consistent. But I would have no more justification to claim absolute
certainty than with our present arithmetic.
fc...@lehigh.edu
writes:
>[Alt.stupidity removed from the newsgroups line.]
>
>bi...@clyde.as.utexas.edu (William H. Jefferys) writes:
>
>>She came up with another doozy this week. Her claim is that
>>it is more dangerous to fly in a twin-engined plane than
>>a single-engined plane, on the grounds that one is more likely
>>to have an engine failure. It is, of course, true that if
>>you have two engines, and the probability of failure is
>>independent between them, one is more likely to experience a
>>failure than if one only has one. But what she fails to understand
>>is that twin-engined planes are designed to be able to fly
>>and to land even if one engine fails. This is just sound
>>engineering practice, eliminating a single-point failure.
>>So, it is safer to fly a twin-engined plane than a single
>>engined plane.
>
>I never saw the original column that provoked the letter-writer in
>this case, but a careful reading of the letter indicates that the
>discussion was about comparing the safety of a single-engine plane to
>that of a plane that *requires* two engines to remain airborne. Her
>response is actually a passable explanation of the fact that if the
>failure probability p for a single engine is small, then 2p is an
>adequate approximation to the failure probability for such a
>twin-engine plane, but as p gets larger, the approximation gets worse.
I'm feeling silly, so try this on for size... Clearly a plane that
*requires* 3 engines is less safe yet, with a probability of 3p of failure.
And if you have more than 1/p number of engines, all of which are
*required* to fly, then, by gosh, you're better off walking... at least you
might get there, eventually... :-)
>That's the thing about these little word puzzles--they are designed to
>be puzzling, not to be a reasonable reflection of reality or "sound
>engineering practice". Your complaint is the same as the complaints
>about the Monty Hall problem being unrealistic. OK, but that doesn't
>invalidate the mathematical ideas being illustrated.
While your point is reasonable, it sounds like this particular puzzle was
somewhat contrived and artificial. What is the point of constructing a
2-engine plane if *both* engines are *required* to fly it, when one could
construct a 2-engine plane designed to work with only *one* engine?
Fred Chapman
o -------------------------------------------------------------------------- o
| Frederick W. Chapman, Mathematical Software Specialist, Lehigh University |
| Office Phone: (610) 758-3218 Internet E-mail: fc...@Lehigh.Edu |
o -------------------------------------------------------------------------- o
| "The ink of the scholar is more sacred than the blood of the martyr." |
o -------------------------------------------------------------------------- o
David C. Jones
>In article <mjs.800637351@hubcap>, m...@hubcap.clemson.edu (M. J. Saltzman)
>writes:
>>That's the thing about these little word puzzles--they are designed to
>>be puzzling, not to be a reasonable reflection of reality or "sound
>>engineering practice". Your complaint is the same as the complaints
>>about the Monty Hall problem being unrealistic. OK, but that doesn't
>>invalidate the mathematical ideas being illustrated.
>While your point is reasonable, it sounds like this particular puzzle was
>somewhat contrived and artificial. What is the point of constructing a
>2-engine plane if *both* engines are *required* to fly it, when one could
>construct a 2-engine plane designed to work with only *one* engine?
1) Heavy cargos may require more power for lift
2) Large wingspan and/or long compartment size may require an engine
on each side for aerodynamic stability
3) If you don't beleive me, my father has slews and slews and slews and
slews and slews and slews and slews of four engine bombers that require
at least one engine on each side to be going.
David
Colin Rosenthal
: Bob Silverman (bo...@mathworks.com) wrote:
Well Euc is the intermediate case. Also we know that spacetime is locally
curved in the solar system from several measurements
1) the precession of Mercury
2) the bending of light around the Sun
3) the change of clock rate for a clock at high altitude.
--
--Colin Rosenthal | ``Don't smell the flowers -
--rose...@obs.aau.dk | They're an evil drug -
--http://bigcat.obs.aau.dk/~rosentha | To make you lose your mind''-
--Aarhus University, Denmark | Ronnie James Dio, 1983 -
Colin Mclarty
In a previous article, dor...@asimov.nrl.navy.mil (Bryan) says:
>M. J. Saltzman (m...@hubcap.clemson.edu) wrote:
>
>: if the
>: failure probability p for a single engine is small, then 2p is an
>: adequate approximation to the failure probability for such a
>: twin-engine plane, but as p gets larger, the approximation gets worse.
>
>What is the actual probability? Cursory inspection suggests (2p-p^2).
>
>Why would whe have to approximate? It doesn't seem like that hard of
>a concept.
>
We have to approximate because we rarely know an actual number
for the probability, and the only useful thing to know about "small
squared" is that it is approximately zero.
It is useful to know the chance of one engine failing
out of two is about twice the chance of one out of one--whereas
the chance of one coin toss out of two being heads is not about twice
the chance of one out of one.
Hugh LaMaster
|> #[Alt.stupidity removed from the newsgroups line.]
|> #
|> #bi...@clyde.as.utexas.edu (William H. Jefferys) writes:
|> #
|> #>it is more dangerous to fly in a twin-engined plane than
|> #>a single-engined plane, on the grounds that one is more likely
|> #>to have an engine failure.
|> #
|> #I never saw the original column that provoked the letter-writer in
|> #this case, but a careful reading of the letter indicates that the
|> #discussion was about comparing the safety of a single-engine plane to
|> #that of a plane that *requires* two engines to remain airborne.
|> Hmm. You are right. I did not read her letter correctly. It
|> begins
|>
|> "You recently stated that it is more dangerous to
|> fly in a plane that _requires_ two engines for
|> flight than in a plane requiring only one..."
|>
|> So she is exonerated this time. My apologies to Ms. vos Savant,
|> and my thanks to you for pointing out my error.
I didn't see the original-original either. But, this is
yet another example of the importance of getting all the
assumptions clearly stated. I don't know about small
private twin-engined planes, but large commercial transports
are required to be able to sustain a single engine
failure at any point during the flight, and it is clear
that this is entirely different from the case of a plane
requiring both engines.
Ross Garmil
: Benjamin J. Tilly (Benjamin...@dartmouth.edu) wrote:
: : In article <3p2j10$i...@newsbf02.news.aol.com>
: : rkin...@aol.com (RKinnamon) writes:
[snipped everything hahahahahahhaah]
I'm replying to myself so that no one who responded will feel left out. Thank
you all for your explanations--I think I have a handle on this. Hopefully
they won't kick me out of alt.stupidity for using the group to <shudder> learn
something, but I think they'll be ok with it. They're very nice people.
See, it's weird that I though before that non-Euclidian was more theoretical,
but apparently I've really wrong. It's designed to actually figure out what
reality is more like since we live in a world/space/universe/thingy more
complex than the tradition concept of three dimensions, and what we learn in
school is less applicable. But my assumption is that it's still taught since
it's simpler to comprehend as a first step towards the more practical.
Again, thank you all, and I certainly wouldn't say "no" to a few more
explanations.
Ross--once an English major, always an English major (funny how that doesn't
hold true for Chemistry).
Jason Nafziger
>Ross Garmil (lim...@bu.edu) wrote:
>: Benjamin J. Tilly (Benjamin...@dartmouth.edu) wrote:
>: : In article <3p2j10$i...@newsbf02.news.aol.com>
>: : rkin...@aol.com (RKinnamon) writes:
>
>[snipped everything hahahahahahhaah]
>
>I'm replying to myself so that no one who responded will feel left out.
Thank
>you all for your explanations--I think I have a handle on this.
Hopefully
>they won't kick me out of alt.stupidity for using the group to <shudder>
learn
>something, but I think they'll be ok with it. They're very nice people.
Less talk, more pack, brain-boy...
>
>See, it's weird that I though before that non-Euclidian was more
theoretical,
>but apparently I've really wrong.
^^^^^^^^^^^^^^^^^
Okay, this clause proves you're still stoopid, you can stay...
Jason -- who's *watching* you, so... uh... do more... uh... dumb...
thingies...
____________________________________________
This IS the Crappy Homepage: http://metro.turnpike.net/C/crapco/index.htm
"Well at least I didn't tell you about the time I grew a penis and sired
those oxen...oops..." -- Gwyneth Kozbial
Karl Robert Loeffler
Benjamin J. Tilly <Benjamin...@dartmouth.edu> wrote:
>In article <3pb2q2$1...@news.bu.edu>
>lim...@bu.edu (Ross Garmil) writes:
>> But hey, I'm from alt.stupidity, so that's supposed to be an excuse, right?
>Ummm...if you keep on asking good questions then you might have to
>change groups... :-)
Actually, could you guys over on the math group consider not crossposting
the really intelligent answers to alt.stoopididty? It's hurting our
brains. Of course, if you have any dumb answers, post away.
Mad Czech-and if you know any palindromes, well....
--
Do you enjoy entropy? Then join the Worldwide Organization of
Independent Anarchists, Survivalists, and Conspiracy Theorists.
We're like the Illuminati, only better.
Karl Loeffler, Founder and sole member, WOIASCT
Matthew P Wiener
>KB> ...Ms Vos Savant observed that mathematicians believe it is
>KB> impossible to formally prove the consistency of arithmetic. In
>KB> other words, we have no rigorous proof that the basic axioms of
>KB> arithmetic do not lead to a contradiction at some point.
>MW = Matthew Wiener
>MW> It's very easy to prove the consistency of arithmetic. Just work
>MW> in ZFC.
>Let us say a proof of the consistency of system X is "incomplete" if
>it's carried out within a system Y whose consistency has not been
>completely proven.
This isn't a definition. What is "completely" proven?
So far as I know, the consistency of ZFC was completely proven by Goedel.
> Theorem:
> Every proof of the consistency of arithmetic is incomplete.
Proof?
>In view of this, it isn't clear how "working in ZFC" resolves the
>issue.
It gives what Ms Vos Savant claimed was missing--a formal proof of
the consistency of PA.
> There is no complete and absolute proof of the consistency
>of arithmetic, so every arithmetical proof is subject to doubt.
Of what relevance is that? You can doubt whatever you like.
>Here is a quote from Morris Kline's book "Mathematics and the Loss of
>Certainty":
> "Godel's result on consistency says that we cannot
> prove consistency in any approach to mathematics
> by safe logical principles...
So what? Kline doesn't know what he is talking about. That is a very
very very very very very pathetic and ignorant book you are referring
to, filled with mistakes, howlers, gibberish and all round stupidity.
>Here is a quote from John Stillwell's book "Mathematics and its
>History":
> "If S is any system that includes PA, then Con(S) [the
> consistency of S] cannot be proved in S, if S is
> consistent." [Godel's 2nd theorem]
I am quite familiar with Goedel's theorems.
>I've received email suggesting that the contemplation of inconsistency
>in the system of arithmetic is tantamount to a renounciation of reason
>itself, i.e., if our concept of natural numbers is inconsistent then we
>must be incapable of rational thought, and any further consideration is
>pointless.
I couldn't care less about such musings.
> ("We recoil in horror", as Poincare said regarding Cantor's
>set theory.)
Which was concerning an entirely different matter. What's your point?
>[miscellaneous comments omitted]
Daniel P. Johnson
kevi...@delphi.com (Kevin Brown) wrote:
> KB> ...Ms Vos Savant observed that mathematicians believe it is
> KB> impossible to formally prove the consistency of arithmetic. In
> KB> other words, we have no rigorous proof that the basic axioms of
> KB> arithmetic do not lead to a contradiction at some point.
> ...There is no complete and absolute proof of the consistency
> of arithmetic, so every arithmetical proof is subject to doubt...
When you say "complete and absolute proof", you are accepting a particular
definition of what complete and absolute proof means. In particular, you
are buying into Hilbert's formal axiomatics, and Hilbert's program for
establishing the consistency of mathematics. That program was the
hypothesis that, properly formalized, mathematics itself could be proven
consistent by the same formal theory. The particular theory that they
adopted was first-order predicate logic with a finitary proof theory.
Russell's "Principia Mathematica" showed that was sufficient to reproduce
large quantities of classical mathematics.
> Here is a quote from Morris Kline's book "Mathematics and the Loss of
> Certainty":
> "Godel's result on consistency says that we cannot
> prove consistency in any approach to mathematics
> by safe logical principles...
By "safe logical principles", Kline is referring to first-order logic and
finitary proof theory, which was regarded as "safe" by the classical
formalists.
> Here is a quote from John Stillwell's book "Mathematics and its
> History":
>
> "If S is any system that includes PA, then Con(S) [the
> consistency of S] cannot be proved in S, if S is
> consistent." [Godel's 2nd theorem]
By "system", Stillwell is again assuming a particular class of formalization.
To provide my own quote, Kleene, "Introduction to Metamathematics", (1971
sixth reprint), Ch. XV, sec 79, pp478,479 says:
"Gentzen's discovery is that the Goedel obstacle to proving
the consistency of number theory can be overcome by using
transfinite induction up to a sufficiently great ordinal...
The original proposals of the formalists to make classical
mathematics secure by a consistency proof did not contemplate
that such a method as transfinite induction up to eps0 would
have to be used. To what extent the Gentzen proof can be
accepted as securing classical number theory in the sense of
that problem formulation is in the present state of affairs
a matter for individual judgment..."
The basic reason why we haven't simply declared Hilbert's program and
first-order predicate logic and finitary proofs an interesting failure and
moved on, is from a wonderful bit of serendipity: that through Goedel's
work on recursive functions, and the later work by Church, Turing, Kleene,
and a host of others, we found out that first-order predicate logic and
finitary proofs is exactly the right mathematical model for computability
and our modern computer systems. Goedel's results thus becomes a theorem
on computability (E.G. Turing's Halting Problem formulation).
Unfortunately, we have now confused several aspects of the theory. It
shows there are limits to A) computability of digital computers, and B)
the ability to prove consistency of mathematics when limited to a
particular toolbox of proof techniques. It says nothing about C) the
ability to prove the consistency of mathematics, for which we do have
proof techniques that work for number theory at least.
--
Daniel P. Johnson Honeywell Technology Center
email:dr...@src.honeywell.com phone:612-951-7427 fax:612-951-7438
mail: MN65-2700, 3660 Technology Drive, Minneapolis, MN 55418
Quote: Truth always finds a natural way of telling her story,
and a natural way is an effective way, simple or not. --Charles Ives
Bruce W Appleby
>In <3pbsth$2h...@ns4-1.CC.Lehigh.EDU> fc...@Lehigh.EDU writes:
>>In article <mjs.800637351@hubcap>, m...@hubcap.clemson.edu (M. J. Saltzman)
>>writes:
>>... What is the point of constructing a
>>2-engine plane if *both* engines are *required* to fly it, when one could
>>construct a 2-engine plane designed to work with only *one* engine?
>1) Heavy cargos may require more power for lift
>2) Large wingspan and/or long compartment size may require an engine
>on each side for aerodynamic stability
>3) If you don't beleive me, my father has slews and slews and slews and
>slews and slews and slews and slews of four engine bombers that require
>at least one engine on each side to be going.
OTOH in the real world where people like me need to fly as passengers on
airlines I assume (until the "contract with big business" people are
done) that we have an FAA to have rules. I also assume that there are
some requirements for a two engine plane to fly and land with a single
engine. Further I *think* that the original article related to just this
situation.
Math puzzles can be fun, but when they are distributed to the general
public it would be nice to have them be generally consistent with common
sense and as completely stated as is reasonable.
Bruce Appleby
bapp...@world.std.com
if I'm wrong about this, then *never mind*
Marcus HUM
[snip]
>I'm feeling silly, so try this on for size... Clearly a plane that
>*requires* 3 engines is less safe yet, with a probability of 3p of failure.
>And if you have more than 1/p number of engines, all of which are
>*required* to fly, then, by gosh, you're better off walking... at least you
>might get there, eventually... :-)
But wait, don't we need 2 feet to walk? Perhaps we'd be better off
hopping :>. Or maybe we ought to just stay home.
M.Hum --- also feeling silly.
Benjamin J. Tilly
bo...@mathworks.com (Bob Silverman) writes:
> In article <3paod7$l...@news.bu.edu>, Ross Garmil <lim...@bu.edu> wrote:
> :Benjamin J. Tilly (Benjamin...@dartmouth.edu) wrote:
> :: In article <3p2j10$i...@newsbf02.news.aol.com>
> :: rkin...@aol.com (RKinnamon) writes:
>
> stuff deleted....
>
> :See, now here's where I get confused. Do we live in a world dominated by
> :Euclidian geometry, and if so, then where does non-Euclidian come into play.
> :I'm not a matehmatician, but it seems to me that if humans cannot experience a
>
> But you DO experience non-Euclidean geometry every day!
>
> You live on the surface of a (almost) sphere[...]
>
> Let me also add that on the scale of the entire universe we do not yet know
> whether our geometry is Euclidean or non-Euclidean.
Ummm...actually we do. It is (even on the scale of the Solar System)
non-Euclidean thanks to Einstein's general theory of relativity.
Furthermore note that the non-Euclidean nature of space on this scale
has been verified by experiments.
Ben Tilly
Benjamin J. Tilly
lim...@bu.edu (Ross Garmil) writes:
> Benjamin J. Tilly (Benjamin...@dartmouth.edu) wrote:
> : In article <3p2j10$i...@newsbf02.news.aol.com>
> : rkin...@aol.com (RKinnamon) writes:
> : Thereby "overturning" over a century of well-established mathematics.
> : (The error is that the theorem involved in (i) only related to
> : EUCLIDEAN geometry, and says nothing about what you can or cannot do
> : with NON-Euclidean geometry. The situation is like my telling you
> : something about my dog Bob at one time, then something about my friend
> : Bob at another, and then having you conclude that I have lied to you
> : since the two comments do not jive with each other.)
>
> See, now here's where I get confused. Do we live in a world dominated by
> Euclidian geometry, and if so, then where does non-Euclidian come into play.
> I'm not a matehmatician, but it seems to me that if humans cannot experience a
> plane of existence (bad pun, sorry), where parallel lines indeed do intersect,
> then why the fuck study it?...
We live in a world that is almost, but not quite, Euclidean. However
there are many applications (for instance some involving radar) in
which non-Euclidean geometry provides a nice format for describing the
problem.
>
> : In fact she goes on to argue that there is a possibility of disproving
> : Einstein's theory of general relativity!
>
> : All of which is complete hogwash.
>
> Well, um, if there wasn't a possibility of disproving Einstein's theory, then
> it would be a law, correct? Don't forget to get behind the pig's ears.
Sorry, I should have made that more clear. There is no possibility of
disproving Einstein's theory by proving that non-Euclidean geometry is
false. There is indeed a real possibility of demonstrating that his
theory does not describe the real world though. She claimed that the
first was possible. The second is not an issue of contention by
anybody.
Ben Tilly
Benjamin J. Tilly
> Bob Silverman (bo...@mathworks.com) wrote:
> : In article <3paod7$l...@news.bu.edu>, Ross Garmil <lim...@bu.edu> wrote:
> : :Benjamin J. Tilly (Benjamin...@dartmouth.edu) wrote:
> : :: In article <3p2j10$i...@newsbf02.news.aol.com>
> : :: rkin...@aol.com (RKinnamon) writes:
> :
> : stuff deleted....
>
> : :See, now here's where I get confused. Do we live in a world dominated by
> : :Euclidian geometry, and if so, then where does non-Euclidian come into play.
> : :I'm not a matehmatician, but it seems to me that if humans cannot experience a
> :
> : But you DO experience non-Euclidean geometry every day!
>
> : You live on the surface of a (almost) sphere: the Earth.
> : A 'line' is determined by 2 points. It is the great circle containing those
> : two points. Note now that 2 lines are NEVER parallel..... Two great circles
> : must ALWAYS intersect.... You thus have a geometry violating Euclid's 5th
> : postulate which says you can draw exactly ONE line containing some given
> : point
> : that is parallel to some fixed line. In spherical geometry there is NO such
> : line.
>
> : Try doing any sort of global navigation. You will experience non-Euclidean
> : geometry in a hurry.....
>
> : Let me also add that on the scale of the entire universe we do not yet know
> : whether our geometry is Euclidean or non-Euclidean.
>
> Oh. No really, I never studied much math (a little calculus, but let's just
> say that me and Mr. Newton had a disagreement--he thought it made sense and I
> thought it was....well, calculus). So although I had heard the term
> "Non-Euclidian geometry" I didn't understand--but now it seems to make sense
> that it's based on looking at things sperically. It appears to me now (give
> that all the knowledge I have is what you just bestowed upon me) that
> non-Euclidian deals with geometry based on a sperical, rather than a flat,
> plane, or to be more precise on a sphere itself and therefore not on a plane
> at all. I think I understand that, if I'm wrong then I don't understand that,
> but I think I do (hey everyone, I'm learning something here!).
>
That is one way that you can get a non-Euclidean geometry. (And a
fairly easy one to visualize.)
> So....why can't a line just be called a curve and keep the definition of line
> as straight line----no wait, because on a sphere as the surface a straight
> line can't exist, I understand that, but how about as a tangent? Is that the
> word? You know, touching one point of the sphere and then taking off outside,
> or perhaps intersecting the sphere, so that there could exist what we perceive
> (traditionally) as "straight" and these straight things could be called
> "lines" and therefore two parallel "lines" wouldn't intersect.
>
We think of lines as something inside our geometry (so they will lie on
more than one point:-) so "tangent lines" are not lines *in our
geometry*.
In the sphere they turn out to be "great circles" like the equator that
cut the Earth exactly in 2. Notice that two lines intersect in exactly
2 places.
> I also don't understand how lines cannot necessarily intersect, such as lined
> of longitude (to use your Earth example), or am I just fixating too much on
> the word "line"?
>
In spherical geometry two lines have to intersect. There is no way
around it.
In some other geometries they do not have to. Describing the picture is
not that useful, what you have to do is dig up a good book of Escher's
and look at his circle limit drawings. Look at them, and try to imagine
a universe where each of the images is the exact same size. (eg you
might look at his angel/devil one.) That is hyperbolic space. Now
taking the "lines" of symmetry (they look like circles on the page) it
is possible for two of them to not intersect either.
> But hey, I'm from alt.stupidity, so that's supposed to be an excuse, right?
Ummm...if you keep on asking good questions then you might have to
change groups... :-)
Ben Tilly
Benjamin J. Tilly
dor...@asimov.nrl.navy.mil (Bryan) writes:
> Benjamin J. Tilly (Benjamin...@dartmouth.edu) wrote:
> : If you run through it you can show that the first 4 axioms of Euclid
> ^^^^^^
> : apply (digging those axioms up may take more work than the
> ^^^^^^
> : demonstration, I recommend finding the Elements, a good encyclopedia
> : article, or _A History of Mathematics_ by Boyer), but the fifth one
> : fails.
>
> Errr...liked your post, but I think you mean the 5th *postulate*, don't
> you? It would be big news if Euclid's axioms had been overthrown.
I think of them as axioms, but you are correct.
Ben Tilly
Boudewijn W. Ch. Visser
>>I never saw the original column that provoked the letter-writer in
>>this case, but a careful reading of the letter indicates that the
>>discussion was about comparing the safety of a single-engine plane to
>>that of a plane that *requires* two engines to remain airborne. Her
>>response is actually a passable explanation of the fact that if the
>>failure probability p for a single engine is small, then 2p is an
>>adequate approximation to the failure probability for such a
>>twin-engine plane, but as p gets larger, the approximation gets worse.
>I'm feeling silly, so try this on for size... Clearly a plane that
>*requires* 3 engines is less safe yet, with a probability of 3p of failure.
>And if you have more than 1/p number of engines, all of which are
>*required* to fly, then, by gosh, you're better off walking... at least you
>might get there, eventually... :-)
Not exactly. 1-(1-p)^3 if the probability of failure for a 3 engine
plane that needs them all.
Or (1-p)^3 for the probability of survival.
There are places where you can't get by walking. And there are machines
that require a large number of parts to work correctly to prevent the
thing from crashing/blowing up etc.
For instance,the Space Shuttle.
According to the Feynman book (what do you care what other people think),
a realistic estimate of the chance of a serious accident was something
like 1/100,or 1/200 or so.
>Fred Chapman
>o -------------------------------------------------------------------------- o
>| Frederick W. Chapman, Mathematical Software Specialist, Lehigh University |
>| Office Phone: (610) 758-3218 Internet E-mail: fc...@Lehigh.Edu |
>o -------------------------------------------------------------------------- o
>| "The ink of the scholar is more sacred than the blood of the martyr." |
>o -------------------------------------------------------------------------- o
--
+-------------------------------------------------------------------+
|Boudewijn Visser |E-mail:vis...@ph.tn.tudelft.nl |finger for |
|Dep. of Applied Physics,Delft University of Technology |PGP-key |
+-- my own opinions etc --------------------------------------------+
Boudewijn W. Ch. Visser
>In article <3otvmd$1h...@ds2.acs.ucalgary.ca>, <kla...@acs3.acs.ucalgary.ca> writes:
>
>> I heard from a friend that Marilyn vos Savant (chick with
>> ludicrous IQ) attempted to refute the proof of Fermat's Last
>> Theorem and failed miserably, and that, furthermore, there was
>> discussion of this in sci.math. Is there a sci.math archive
>> somewhere where I can read about this? Or can anyone just fill
>> me in? Much thanks.
>Off the point, but I think the discrepancy between what people expect of
>a "chick with a ludicrous IQ" and what vos Savant actually thinks and
>writes, is a good argument against taking IQ numbers too seriously as
>a good measure of brilliance. I was amazed to see how many questions
>in her first columns were of the type "so, O wise guru, how many angels
>CAN dance on the head of a pin?" She seems to spend most of the column
>now on brain-teasers of the type that I (and I assume many of the people
>here in the technical newsgroups) used to do for fun when I was around
>10-13. She seems to have a good grounding in probability theory, giving
>nice explanations for the Monty Hall paradox for instance (and defending
>herself against academics whose intuition is leading them astray).
>So she would probably be at least a competent grad student. But it's a
>long leap from there to "the next Einstein", and if she truly has any
>talent and brilliance in mathematics or anything else, it isn't going
>to be measured by an IQ test. She could very well be brilliant, but
>there just isn't any evidence from her newspaper column (or her test
>score) to say one way or the other.
Here are some posts that appeared on sci.math some time ago. The high IQ of MvS
seems a bit artificial.
Boudewijn
+-------------------------------------------------------------------+
|Boudewijn Visser |E-mail:vis...@ph.tn.tudelft.nl |finger for |
|Dep. of Applied Physics,Delft University of Technology |PGP-key |
+-- my own opinions etc --------------------------------------------+
----------------------------------------------------------------
From: pe...@bignode.equinox.gen.nz (Pete Moore)
Subject: Re: Vos Savant:0 Math PrSummary:
Newsgroups: sci.math
References: <1994Aug25.2...@ttinews.tti.com> <alanfCv...@netcom.com> <33v7pc$4...@gaia.cc.gatech.edu> <340o65$9...@cutter.clas.ufl.edu>
Reply-To: pe...@bignode.equinox.gen.nz
Message-ID: <pete...@bignode.equinox.gen.nz>
Date: 2 Sep 94 14:46:10 +1200
Organization: Equinox Networks
Lines: 92
Scott G. Chastain (s...@math.ufl.edu) wrote:
>Quit a few comments have been made about Vos Savant since this thread
>began. The one I find most interesting is the statement that she began
>her career as an IQ test writer. The implication then made was that she
>took IQ test repeatedly until she did exceptionally well. I was telling
>some friends about this and realized I had no reference to back up this
>'folklore'. Can anybody supply a reference for autobiagraphical info
>on Ms Vos Savant.
Recent editions of the Guinness Book of Records no longer include Marilyn,
as the publishers apparently feel that the evidence in support of her IQ
claims is inadequate. In fact, the GBR no longer lists IQ records at all.
This probably doesn't qualify as "supplying a reference", but it's stuff
that I saved on previous occasions on which this subject was raised:
----------------------
~From: ch...@questrel.com (Chris Cole)
~Date: Tue, 1 Mar 1994 06:46:48 GMT
Marilyn's IQ score of 230 was on the
Stanford Binet taken at the age of 10 (she received a perfect score,
which the test book equates to a mental age of 23, and 23/10 = 230
percent). This is the score that got her into the Guinness Book of
Records. She also scored a 46 out of 48 possible on the Mega Test, but
Ron Hoeflin, the author of the Mega Test, claims that a raw score of 46
equates to an IQ of something around 180, not 230. Since then, other
people have achieved this and higher scores on the Mega Test.
----------------------
~From: ele...@atropos.acm.rpi.edu (Kenneth Lareau)
~Date: 28 Nov 1993 07:32:19 GMT
I've been seeing some discussion about Mrs. Savant's IQ being talked
about, and since I'm quite peeved at the woman myself, I've decided to give
a bit more info on how she got her revered status in the Guiness Book of
World Records.
First off, if anyone has noticed lately, the last few years of the Gui-
ness Book do _not_ contain the Highest IQ category. Therefore, the little
paragraph that Parade puts at the end of her column is quite incorrect, and
somewhat annoys me as I know the reason why it was yanked. But to get back
to the test...
Ah yes, the IQ test. Well, I can definitely tell you which test put Mrs.
(or is it Ms.? I can never remember) Savant in the Guiness Book. It is called
the Mega Test, and I myself am in the process of taking it...no, you don't
have to be a genius to take it, just willing to spend hundreds of hours bang-
ing your head against a wall. :) The test is a collection of 48 questions,
the first 24 being analogies (day : night :: sun : ?). The last 24 are var-
ious math problems, which range from mildly difficult to almost impossible.
The rules of the test are simple. You may take all the time to complete
the test as you want, i.e. there are no time limits. You can also use any re-
ference material that you might need, such as dictionaries, math tables, ar-
ticles in journals, computers...only thing you _can't_ use is another person,
as that would invalidate your score. When you feel you've answered all the
questions, or you've just given up on the ones you couldn't do, you send the
answers on a piece of paper with $25, payable to "Ronald Hoeflin", to the
scoring committee. Once your test is scored, they will send you a list of
the high-IQ societies (if any) you are eligible of entering.
The reason I happen to know about all this is that I have a friend who
tied Mrs./Ms. Savant's score on the test...46 out of 48, to be specific. In
fact, had it not been for the removal of the category from the Guiness Book,
he would have replaced her in it. But unfortunately there was a controversy,
which I will not proceed to tell. I'll leave it up to you to decide on Mrs./
Ms. Savant's integrity...
One of the first things that bothered Guiness about this claim was the
fact that came to light that Marilyn had been a member of the scoring com-
mittee BEFORE she submitted her answers. I don't know about anyone else, but
that would seem to nullify her score in my eyes. But wait, there's more... :)
Another fact that came to light was that for all the people who scored
better than a given number (somewhere around 35, I don't quite remember) had
all gotten a specific problem correct _except_ for Mrs./Ms. Savant. This to
me would seem rather unusual as well, considering it was one of the only 2
answers she was incorrect on.
Now this information, at least to me, is several years old, but it seems
that it hasn't been heard in too many places, judging by the discussions here.
I figured a few of you folks might want to hear about it, as it might answer
some question, though I have a feeling it will probably create even more.
--
+------------------ pe...@bignode.equinox.gen.nz ------------------+
GM/O -d+ -p+ c++ l u+ e+++ m++ s+/ n+ h+ f+ !g w+ t r- y+(*)
The effort to understand the universe is one of the very few things
that lifts human life above the level of farce, and gives it some
of the grace of tragedy - Steven Weinberg
Kevin Brown
>> basic axioms of arithmetic do not lead to a contradiction at some point.
>>
>I used to argue this line until I was set straight by some logicians.
>(Is Torkel available?:-) There is no proof in *first-order* logic that
>arithmetic is consistent. But that has more to do with the limitations
>of first-order logic than anything else and there are other more
>general types of logic in which proofs of the consistency of arithmetic
>are available.
Ask those logicians if they can prove the consistency of those more general
types of logic. (They can't.) Goedel's results applies to any formal system
that can be modelled by arithmetic. Granted, if you postulate a system that
cannot be modelled by arithmetic then other things are possible, but the
consistency of such a system would be at least as doubtful as the consistency
of the system you are trying to prove. For example, Gentzen's proof of the
consistency of PA uses transfinite induction. It seems rather pointless to
try to resolve doubts about arithmetic by working with transfinite induction,
since the latter is even more dubious.
Matthew P Wiener
the pop-poop version, the kind that sells.
There is a consistency proof of PA, and it is quite elementary, using
standard mathematical techniques (ie, ZF). It consists of exhibiting
a model.
Matthew P Wiener
>Ask those logicians if they can prove the consistency of those more general
>types of logic. (They can't.)
Sure you can. Same proof for most of them.
Michael Weiss
Every review and discussion of this book that I've seen has focused on
the "hyperbolic/Euclidean circle-squaring" argument, which was clearly
the weakest point of the book. However, the book also raised another
objection to Wiles' proof. As I recall, Ms Vos Savant observed that
mathematicians believe it is impossible to formally prove the consistency
of arithmetic. In other words, we have no rigorous proof that the
basic axioms of arithmetic do not lead to a contradiction at some point.
Therefore (she argues) if we assume a counter-example to FLT and
then, via some long and complicated chain of reasoning, produce a
contradiction, how do we know that the assumed counter-example is the
real cause of the contradiction? Could it not be that we have just
succeeded in producing the long-dreaded contradiction inherent in
arithmetic itself?
Proof-by-contradiction is just another way of phrasing the equivalence
between P --> Q and (not Q) --> (not P) (at least in standard
classical logic--- vos Savant wasn't discussing intuitionism, was
she?) A contradiction in arithmetic, on the other hand, would be a
proof of P & (not P). This objection is without merit, and just
displays again vos Savant's appalling ignorance of the subject matter.
Timothy Y Chow
Kevin Brown <kevi...@delphi.com> wrote:
>Let us say a proof of the consistency of system X is "incomplete" if
>it's carried out within a system Y whose consistency has not been
>completely proven.
>
> Every proof of the consistency of arithmetic is incomplete.
Supposing we grant this "theorem." So what? This is only an issue if
you somehow think that the lack of a complete proof engenders uncertainty.
Why should it? Simply because the lack of a complete proof means that
"it might be wrong"? But even if there *were* a "complete proof" it
still "might be wrong." For your concept of a complete proof implies
that ordinary proof is, by itself, insufficient to ensure certainty, and
once you have raised this question mark, we can also raise the question
mark over the notion of "complete proof." Namely, I can argue that a
complete proof of consistency is "non-metacomplete" if it's carried out
within a system whose consistency has not been metacompletely proven.
And in any case, on at least one reading of your "theorem," it isn't
true. The consistency of ZFC is provable in ZFC+Con(ZFC), the consistency
of ZFC+Con(ZFC) is provable in ZFC+Con(ZFC)+Con(ZFC+Con(ZFC)), etc., so
the infinite hierarchy of such systems provides a complete proof of the
consistency of ZFC. Or, just to press the point, there is actually a
system which proves the consistency of *any* system: take any inconsistent
system. This system even proves its *own* consistency. What more could
you ask for?
The point is that you can raise doubts about *anything*; the appeal to
the lack of an "absolute proof" (whatever that means) is essentially
irrelevant, because even if there were an "absolute proof" you could
still doubt it. On the other hand, asking for an ordinary proof is
not unreasonable, not so much because ordinary proof confers absolute
certainty, but because ordinary proof is what mathematicians are
interested in. And there does exist an ordinary proof of the
consistency of PA.
Benjamin J. Tilly
kevi...@delphi.com (Kevin Brown) writes:
> DO> David T. Ose <o...@symcom.math.uiuc.edu>
> DO> There is a review of her book in the most recent copy of
> DO> The American Mathematical Monthly.
>
> TC = Tim Chow
> TC> There is one thing that I would like to add, though. Vos Savant's
> TC> most crucial argument is that "if we reject a hyperbolic method of
> TC> squaring the circle, then we should also reject a hyperbolic method
> TC> of proving Fermat's Last Theorem"...
>
> Every review and discussion of this book that I've seen has focused on
> the "hyperbolic/Euclidean circle-squaring" argument, which was clearly
> the weakest point of the book. However, the book also raised another
> objection to Wiles' proof. As I recall, Ms Vos Savant observed that
> mathematicians believe it is impossible to formally prove the consistency
> of arithmetic. In other words, we have no rigorous proof that the
> basic axioms of arithmetic do not lead to a contradiction at some point.
>
I used to argue this line until I was set straight by some logicians.
(Is Torkel available?:-) There is no proof in *first-order* logic that
arithmetic is consistent. But that has more to do with the limitations
of first-order logic than anything else and there are other more
general types of logic in which proofs of the consistency of arithmetic
are available.
> Therefore (she argues) if we assume a counter-example to FLT and
> then, via some long and complicated chain of reasoning, produce a
> contradiction, how do we know that the assumed counter-example is the
> real cause of the contradiction? Could it not be that we have just
> succeeded in producing the long-dreaded contradiction inherent in
> arithmetic itself?
>
Think about it. Do you REALLY believe that there is a contradiction in
the positive integers?
[...]
Ben Tilly
Kevin Brown
excluded by...Goedel's analysis. In point of fact, meta-mathematical
proofs of the consistency of arithmetic have been constructed, notably
by Gerhard Gentzen, a member of the Hilbert school, in 1936. But such
proofs are in a sense pointless if, as can be demonstrated, they employ
rules of inference whose OWN consistency is as much open to doubt as is
the formal consistency of arithmetic itself. Thus, Gentzen used the
so-called "principle of transfinite mathematical induction" in his
proof. But the principle in effect stipulates that a formula is
derivable from an INFINITE class of premises. Its use therefore
requires the employment of nonfinitistic meta-mathematical notions,
and so raises once more the question which Hilbert's original program
was intended to resolve." [Ernest Nagel and James Newman]
"Goedel showed that...if anyone finds a proof that arithmetic is
consistent, then it isn't!" [Ian Stewart]
"...Hence one cannot, using the usual methods, be certain that the
axioms of arithmetic will not lead to contradictions." [Carl Boyer]
"An absolute consistency proof is one that does not assume the
consistency of some other system...what Goedel did was show that
there must be "undecidable" statements within ANY [formal system]...
and that consistency is one of those undecidable propositions.
In other words, the consistency of an all-embracing formal system
can neither be proved nor disproved within the formal system."
[Edna Kramer]
"Do I contradict myself? Very well then, I contradict myself.
I am large, I contain multitudes." [Walt Whitman]
Benjamin J. Tilly
ksb...@ksbrown.seanet.com (Kevin Brown) writes:
> >> of arithmetic. In other words, we have no rigorous proof that the
> >> basic axioms of arithmetic do not lead to a contradiction at some point.
> >>
> >I used to argue this line until I was set straight by some logicians.
> >(Is Torkel available?:-) There is no proof in *first-order* logic that
> >arithmetic is consistent. But that has more to do with the limitations
> >of first-order logic than anything else and there are other more
> >general types of logic in which proofs of the consistency of arithmetic
> >are available.
>
>
> Ask those logicians if they can prove the consistency of those more general
> types of logic. (They can't.) Goedel's results applies to any formal system
> that can be modelled by arithmetic. Granted, if you postulate a system that
> cannot be modelled by arithmetic then other things are possible, but the
> consistency of such a system would be at least as doubtful as the consistency
> of the system you are trying to prove. For example, Gentzen's proof of the
> consistency of PA uses transfinite induction. It seems rather pointless to
> try to resolve doubts about arithmetic by working with transfinite induction,
> since the latter is even more dubious.
Ask those logicians, or anybody else, whether they can prove that
first-order logic is perfect. If you want to be a philosophical
pedantic about it, it turns out that even this is impossible. But
nobody argues that they think that first-order logic is fundamentally
messed up. (If you are an intuitionist, then please stay quiet. :-)
Now if you think about it for a bit, there is no reasonable doubt that
arithmetic is just fine. And if that does not satisfy you, then
remember that the validity of ANY sort of reasoning, (including first
order logic) depends on similar considerations. Nothing more, and
nothing less.
Ben Tilly
Robert J. Pease
: In article <3pe8r6$f...@dartvax.dartmouth.edu>,: >In article <3pb2q2$1...@news.bu.edu>
: >lim...@bu.edu (Ross Garmil) writes:
: >> But hey, I'm from alt.stupidity, so that's supposed to be an excuse, right?
: >Ummm...if you keep on asking good questions then you might have to
: >change groups... :-)
: Actually, could you guys over on the math group consider not crossposting
: the really intelligent answers to alt.stoopididty? It's hurting our
: brains. Of course, if you have any dumb answers, post away.
: Mad Czech-and if you know any palindromes, well....
: --
: Do you enjoy entropy? Then join the Worldwide Organization of
: Independent Anarchists, Survivalists, and Conspiracy Theorists.
: We're like the Illuminati, only better.
: Karl Loeffler, Founder and sole member, WOIASCT
*
I am sure Marylin has a very high I.Q.
I am not so sur that this qualifies her to come to
goofy statistical conclusions without Studying anything very carefully.
We used to call this "sophmorism"
*
Kevin Brown
MW> There is a consistency proof of PA, and it is quite elementary, using
MW> standard mathematical techniques (ie, ZF). It consists of exhibiting
MW> a model.
When you speak about "exhibiting a model" you are referring to a relative
consistency proof, not an absolute consistency proof. Examples of
relative consistency theorems are
IF Euclidean geometry is consistent
THEN non-Euclidean geometry is consistent.
IF ZF is consistent
THEN ZFC is consistent.
Relative consistency proofs assert nothing about the absolute consistency
of any system, they just relate the consistency of one system to that of
another. Here's what the Encyclopedic Dictionary of Mathematics (2nd Ed)
says on the subject:
"Hilbert proved the consistency of Euclidean geometry by assuming
the consistency of the theory of real numbers. This is an example
of a RELATIVE CONSISTENCY proof, which reduces the consistency proof
of one system to that of another. Such a proof can be meaningful
only when the latter system can somehow be regarded as based on
sounder ground than the former. To carry out the consistency of
logic proper and set theory, one must reduce it to that of another
system with sounder ground. For this purpose, Hilbert initiated
metamathematics and the finitary standpoint...Let S be any consistent
formal system containing the theory of natural numbers. Then it is
impossible to prove the consistency of S by utilizing only arguments
that can be formalized in S...."
"In [these] consistency proofs of pure number theory..., transfinite
induction up to the first e-number e_0 is used, but all the other
reasoning used in these proofs can be presented in pure number theory.
This shows that the legitimacy of transfinite induction up to e_0
cannot be proved in this latter theory."
This implies to me that all known consistency proofs of arithmetic rely
on something like transfinite induction (or possibly primitive recursive
functionals of finite type), the consistency of which is no more self-
evident than that of arithmetic itself.
Timothy Y Chow
Kevin Brown <kevi...@delphi.com> wrote:
>MW = Matthew Wiener
>MW> There is a consistency proof of PA, and it is quite elementary, using
>MW> standard mathematical techniques (ie, ZF). It consists of exhibiting
>MW> a model.
>
>When you speak about "exhibiting a model" you are referring to a relative
>consistency proof, not an absolute consistency proof. Examples of
>relative consistency theorems are
>
> IF Euclidean geometry is consistent
> THEN non-Euclidean geometry is consistent.
The proof of the consistency of PA does not *explicitly* assume the
consistency of ZF, and thus is *not* a relative consistency proof.
The proof can be *encoded* in ZF, but this is *not* the same as saying
that the proof *assumes* the consistency of ZF.
If ZF turns out to be inconsistent, then the ZF formalization of the
proof of the consistency of PA will not be of much interest, but this
won't necessarily undermine the PA consistency proof *itself*. Given
your beliefs about "local consistency," it should not be too much of a
stretch of the imagination to think that a suitable replacement for ZF
would be found in which the PA consistency proof could find a new home.
Ross Garmil
: In article <3pdltn$6...@news.bu.edu>, lim...@bu.edu (Ross Garmil) wrote:
: >Ross Garmil (lim...@bu.edu) wrote:
: >: : In article <3p2j10$i...@newsbf02.news.aol.com>
: >: : rkin...@aol.com (RKinnamon) writes:
: >
: >[snipped everything hahahahahahhaah]
: >
: >I'm replying to myself so that no one who responded will feel left out.
: Thank
: >you all for your explanations--I think I have a handle on this.
: Hopefully
: >they won't kick me out of alt.stupidity for using the group to <shudder>
: learn
: >something, but I think they'll be ok with it. They're very nice people.
: Less talk, more pack, brain-boy...
Duh, okay Jason, whatever you say.
: >
: >See, it's weird that I though before that non-Euclidian was more
: theoretical,
: >but apparently I've really wrong.
: ^^^^^^^^^^^^^^^^^
: Okay, this clause proves you're still stoopid, you can stay...
I can? Really? "You like me, you really like me"
(Sally Field)
: Jason -- who's *watching* you, so... uh... do more... uh... dumb...
: thingies...
Ross---ok, what if I put my fork in this toaster. Argh, it was a toaster oven
and didn't do anything. Boy, was that stoopid, or what?
Matthew P Wiener
>MW = Matthew Wiener
>MW> There is a consistency proof of PA, and it is quite elementary, using
>MW> standard mathematical techniques (ie, ZF). It consists of exhibiting
>MW> a model.
>When you speak about "exhibiting a model" you are referring to a relative
>consistency proof, not an absolute consistency proof.
So what?
>Relative consistency proofs assert nothing about the absolute consistency
>of any system, they just relate the consistency of one system to that of
>another.
So what?
David Seal
>Every review and discussion of this book that I've seen has focused on
>the "hyperbolic/Euclidean circle-squaring" argument, which was clearly
>the weakest point of the book. However, the book also raised another
>objection to Wiles' proof. As I recall, Ms Vos Savant observed that
>mathematicians believe it is impossible to formally prove the consistency
>of arithmetic. In other words, we have no rigorous proof that the
>basic axioms of arithmetic do not lead to a contradiction at some point.
>
>Therefore (she argues) if we assume a counter-example to FLT and
>then, via some long and complicated chain of reasoning, produce a
>contradiction, how do we know that the assumed counter-example is the
>real cause of the contradiction? Could it not be that we have just
>succeeded in producing the long-dreaded contradiction inherent in
>arithmetic itself?
It doesn't matter: the contradiction still provides the proof you want
of FLT within arithmetic. (And since FLT is a theorem about
arithmetic, any proof of it must either be within arithmetic or within
a system that includes arithmetic as a subsystem).
The trouble with a mathematical system being self-contradictory is not
that you can't prove things within it: it is that you can prove too
much within it - in particular, that you can prove any statement at
all within it. This makes it the question of whether something is a
theorem within the self-contradictory system rather uninteresting...
So basically, assuming all steps of the proof are correct, then you
have proved FLT as an interesting theorem about arithmetic if
arithmetic is not self-contradictory, and as an uninteresting theorem
about arithmetic if arithmetic is self-contradictory. The difference
lies in whether the theorem and its proof are interesting, not in
whether the proof exists.
Of course, if arithmetic *is* found to be self-contradictory, it's not
just FLT that suddenly becomes uninteresting: an awful lot of other
mathematics would suffer exactly the same fate...
David Seal
ds...@armltd.co.uk
Daniel Kornhauser
<In article <9505162221591....@delphi.com>, kevin2003@delphi (Kevin
<Brown) writes:
<KB> ...Ms Vos Savant observed that mathematicians believe it is
<KB> impossible to formally prove the consistency of arithmetic. In
<KB> other words, we have no rigorous proof that the basic axioms of
<KB> arithmetic do not lead to a contradiction at some point.
<
< It's very easy to prove the consistency of arithmetic. Just work
< in ZFC.
<
< So far as I know, the consistency of ZFC was completely proven by Goedel.
<
showed showed the consistency of ZFC relative to that of ZF. Of course, this
says nothing about whether ZF is consistent or not, nor does it answer the
question about whether ZFC is consistent.
<
<
<
<KB>Here is a quote from John Stillwell's book "Mathematics and its
<KB>History":
<
<KB> "If S is any system that includes PA, then Con(S) [the
<KB> consistency of S] cannot be proved in S, if S is
<KB> consistent." [Godel's 2nd theorem]
<
<I am quite familiar with Goedel's theorems.
<
if ZF is consistent, then this fact cannot be proved within ZF.
Similarly for "ZF" replaced everywhere by "ZFC".
Just thought this might clarify the situation a bit.
--Dan Kornhauser
Kevin Brown
BT> Think about it. Do you REALLY believe that there is a contradiction
BT> in the positive integers?
What I believe and what I can prove are two different things. Having
said that, my answer is yes. In fact, I would suggest that every
(sufficiently complex) formal system contains a contradiction at some
point.
I suggested this to an email correspondent and he expressed utter
incredulity, saying "You can't possibly believe that simple arithmetic
could contain an inconsistency! How would you balance your check book?"
This is an interesting question. I actually balance my checkbook
using a formal system called Quicken. Do I have a formal proof of
the absolute consistency and correctness of Quicken? No. Is it
conceivable that Quicken might contain an imperfection that could
lead, in some circumstances, to an inconsistency? Certainly.
But for many sci.mathematicians this situation must be a real
paradox.
Suppose I balance my checkbook with a program called KevCash (so as
not to sully the good name of Quicken), and suppose this program
implements arithmetic perfectly - with one exception. The result
of subtracting 5.555555 from 7.777777 is 2.222223. Now if B is my
true balance then I should have the formal theorems
B = B + q ( (7.777777 - 5.555555) - 2.222222 )
for EVERY value of q. Thus, in the formal system of KevCash I can
prove that 1 = 2 = 3 = 4.23 = 89.23 = anything. Clearly the KevCash
system has a consistency problem, because I can compute my balance
to be anything I want just by manipulating the error produced by
that one particular operation. ("So oft it chances in particular
men...") But here is the fact that must seem paradoxical to most
sci.math posters: thousands of people have used KevCash for years,
and not a single error has appeared in the results. How can this
be, given that KevCash is formally inconsistent?
The answer is that although KevCash is globally inconsistent, it
possesses a high degree of local consistency. Travelling from any
given premise (such as 5+7) DIRECTLY to the evaluation (e.g., 12),
we are very unlikely to encounter the inconsistency. Of course, in
a perfectly consistent system we could take ANY of the infinitely
many paths from 5+7 to the evaluation and we would always arrive
at the same result, which is clearly not true within KevCash (in
which we could, by round-about formal manipulations, evaluate 5+7
to be -3511.1093, or any other value we wanted). Nevertheless, for
ALMOST ALL paths leading from a given premise to its evaluation, the
result is the same.
Now consider our formal system of arithmetic. Many people seem agog
at the suggestion that our formalization of arithmetic might possibly
be inconsistent at some point. Clearly our arithmetic must possess
a very high degree of local consistency, because otherwise we would
have observed anomalies long before now. However, are we really
justified in asserting that EVERY one of the infinitely many paths
from every premise to its evaluation gives the same result? As with
the system KevCash, this question can't be answered simply by
observing that our checkbooks usually seem to balance.
Moreover, the question cannot even be answered within any formal
system that can be modelled by the natural numbers. It is evidently
necessary to assume the validity of something like transfinite induction
to prove the consistency of arithmetic. But how sure are we that a
formal system that includes transfinite induction is totally consistent?
(If, under the assumption of transfinite induction, we had found that
arithmetic was NOT consistent, would we have abandoned arithmetic...
or transfinite induction?) The only way we know how to prove this is
by assuming still less self-evidently consistent procedures, and so on.
The points I'm trying to make are
(1) We have no meaningful proof of the consistency of arithmetic.
(2) If arithmetic IS inconsistent, it does NOT follow that our
checkbooks must all be out of balance. It is entirely possible
that we could adjust our formalization of arithmetic to patch
up the inconsistency, and almost all elementary results would
remain unchanged.
(3) HOWEVER, highly elaborate and lengthy chains of deduction
in the far reaches of advanced number theory might need to be
re-evaluated in the light of our patched-up formalization.
I believe that (3) was the point Ms vos Savant was laboring to make
in her book on FLT.
Jan Willem Nienhuys
#BT = Ben Tilly
#BT> Think about it. Do you REALLY believe that there is a contradiction
#BT> in the positive integers?
#What I believe and what I can prove are two different things. Having
#said that, my answer is yes. In fact, I would suggest that every
#(sufficiently complex) formal system contains a contradiction at some
#point.
[skip some]
#(1) We have no meaningful proof of the consistency of arithmetic.
#(3) HOWEVER, highly elaborate and lengthy chains of deduction
# in the far reaches of advanced number theory might need to be
# re-evaluated in the light of our patched-up formalization.
May I make a suggestion? Arithmetic's supposed inconsistency only
can make sense in a completely formalized system. The problem may
reside in the particular way of formalizing, rather than in `far reaches
of number theory'. Formalization has many conventions in it regarding
the use of variables, formulas, definitions.
It is one thing to present a few typical formulas and do a little
handwaving, it quite another thing to build up a complete system
entirely from scratch in such a way that it presupposes no previous
knowledge on the part of the reader while at the same time it is
completely clear that it corresponds with what we call `arithmetic'.
I'm not even sure you can do that without already assuming vast amounts
of information about numbers to be `known'. Just look at naming
conventions for variables. You'll have to suppose that a convention
whereby variables are named x , x' , x'' , x''' etc. is meaningful.
Writing `etc.' won't do, because you must suppose that this is a
concept unknown to the reader.
Maybe these problems can be solved satisfactorily. But these solutions
can have in them the seeds of contradictions that sneak up on you.
It is also imaginable that later generations of mathematicians will
think that these solutions are of an intolerable vagueness.
What would happen when a contradiction was found? I guess that
the foundations-of-mathematics people would try to find a way
to work around the problem. I think that what Ben Tilly meant
was a contradiction in the positive integers that won't go away,
no matter what formalization you use.
I share his belief that no such contradiction can be found.
JWN
Kevin Brown
JN> May I make a suggestion? Arithmetic's supposed inconsistency only
JN> can make sense in a completely formalized system. The problem may
JN> reside in the particular way of formalizing...in such a way that it
JN> presupposes no previous knowledge on the part of the reader while
JN> at the same time it is completely clear that it corresponds with
JN> what we call `arithmetic'.... Maybe these problems can be solved
JN> satisfactorily. But these solutions can have in them the seeds of
JN> contradictions that sneak up on you.
Excellent suggestion. I agree completely.
JN> What would happen when a contradiction was found? I guess that
JN> the foundations-of-mathematics people would try to find a way
JN> to work around the problem.
Yes, I agree. They would say "Ooppss, our formalization didn't quite
correspond to TRUE arithmetic. Now, here's the final ultimate and
absolutely true formalization... (I know we said this last time, but
THIS time our formalization really IS perfect.)"
JN> I think that what Ben Tilly meant was a contradiction in the
JN> positive integers that won't go away, no matter what formalization
JN> you use.
Here you are positing the existence of a Platonic ideal "ARITHMETIC"
that is eternal, perfect, and true, while acknowledging that any given
formalization of this Platonic ideal may be flawed. The problem is
that our formal proofs are based on a specific formalization of
arithmetic, not on the ideal Platonic ARITHMETIC, so we are not
justified in transferring our sublime confidence in the Platonic
ideal onto our formal proofs.
In effect your position is equivalent to saying that you are always
right, because even if you are found to have said something wrong,
that's not what you meant.
Kevin Brown
DS> ...the contradiction still provides the proof you want of FLT
DS> within arithmetic. (And since FLT is a theorem about arithmetic,
DS> any proof of it must either be within arithmetic or within a
DS> system that includes arithmetic as a subsystem).
FLT is a theorem about the ideal of ARITHMETIC, prior to any particular
formalization. (This is true both historically and conceptually.)
The first step in attempting to prove it is to select a formal system
within which to work. Of course, it's trivial to devise a formal
system labeled "arithmetic" and then prove FLT within that system.
For example, take PA+FLT. But the question is whether that system
really represents ARITHMETIC, one requirement of which is consistency.
DS> Of course, if arithmetic *is* found to be self-contradictory, it's
DS> not just FLT that suddenly becomes uninteresting: an awful lot of
DS> other mathematics would suffer exactly the same fate...
We don't know what, if any, parts of our present mathematics would
be rendered uninteresting by the discovery of an inconsistency in our
present formalization of arithmetic, because it would depend on the
nature of the inconsistency and the steps taken to resolve it. Once
the patched-up formalization was in place, we would re-evaluate all
of our mathematics to see which, if any, proofs no longer work in the
new improved "arithmetic". One would expect that almost all present
theorems would survive. The theorems most likely to be in jeopardy
would be the most elaborate, far-reaching, and "deep" results, because
their proofs tax the resources of our present system the most.
Carl Karl Miller
I thought Marilyn vos Savant's book was altogether stupid. Probably the
worst parts were her arguments against proof by contradiction and Godel's
theorem.
"'Proof by double negation' is fraught with dangers. For example, let's
say the human animal can only see in black and white. (To make this
point, let's also assume that nothing is gray.) We want to prove that an
object is white, so we manage to prove that it isn't black. By double
negation, the object is proved to be white. But considering all the
colors of the rainbow, this shouldn't be a valid proof. Maybe the object
is red. But how would we know it?"
As clever as this argument may sound, it has no mathematical equivalent.
If a statements negation creates a contradiction in the system, then A)
the system is consistent, and the statement is true, or B) the system is
inconsistent, and the statement creates a contradiction as well. (Any
type of proof assumes the system's consistency.) My impression was that
she was very misconceived about what proof by contradiction is.
She also seemed to believe that "proof by induction" was just checking a
few random examples and drawing a general conclusion. That is scientific
induction, not the mathematical induction which was used in Wiles'
proof. So, her entire section arguing against induction, which got
very ponderous eventually, was completely invalid.
"One wonder's whether Goedel's logic would be applied to his own
argument, like a dog chasing its tail, thus 'proving' itself invalid."
In this case, she didn't even bother to present any support. It's a very
naive assumption, which is a mood which persists througout the book.
Matthew P Wiener
>But ZF is such a system S. Therefore by Goedel's 2nd Incompleteness
>Thm, if ZF is consistent, then this fact cannot be proved within ZF.
>Similarly for "ZF" replaced everywhere by "ZFC".
But the consistency of PA can be proven in ZFC, as I claimed. Quoting
Goedel over and over again isn't going to change that.
Matthew P Wiener
>The points I'm trying to make are
>(1) We have no meaningful proof of the consistency of arithmetic.
The usual ZFC proof is quite meaningful, so far as I can tell.
>(3) HOWEVER, highly elaborate and lengthy chains of deduction
> in the far reaches of advanced number theory might need to be
> re-evaluated in the light of our patched-up formalization.
>I believe that (3) was the point Ms vos Savant was laboring to make
>in her book on FLT.
Obviously she wasn't aware of ZFC.
Virginia Krenn
> : >I'm replying to myself so that no one who responded will feel left
> : >out. Thank you all for your explanations--I think I have a handle
> : >on this. Hopefully they won't kick me out of alt.stupidity for using
> : >the group to <shudder> learn something, but I think they'll be ok
> : >with it. They're very nice people.
> : Less talk, more pack, brain-boy...
> Duh, okay Jason, whatever you say.
> : >See, it's weird that I though before that non-Euclidian was more
> : >theoretical, but apparently I've really wrong.
> : Okay, this clause proves you're still stoopid, you can stay...
> I can? Really? "You like me, you really like me"
> : Jason -- who's *watching* you, so... uh... do more... uh... dumb...
> : thingies...
> Ross---ok, what if I put my fork in this toaster. Argh, it was a toaster oven
> and didn't do anything. Boy, was that stoopid, or what?
If you want to feel really stupid, get someone to explain
"Projective Geometry" to you.
Bill Taylor
|> : If you run through it you can show that the first 4 axioms of Euclid
|>
|> Errr...liked your post, but I think you mean the 5th *postulate*, don't
|> you? It would be big news if Euclid's axioms had been overthrown.
Actually, they *have* been and it *was*, well... kinda, long ago.
Euclid's 5th postulate was the parallel, which was "overthrown", or at least
semi-sidelined, about 1830. But Euclid's 5th AXIOM, or what he called "common
notion", was observed to be screwy long before! Euclid's common notions were
"if things be added to equals, the sums are eqwual", and so forth, which were
held in the Elements to apply to *all* topics, not just geometry.
Coincidentally, it was the *5th*, again, which was troublesome...
"The whole is greater than any of the parts."
Galileo observed that the set of natural number squares was in some sense
"equal to" all the naturals. (This had apparently also been observed by the
scholastics, and even by the classical Greeks in the context of real intervals.)
However Galilaeo gets the credit.
Since Cantor and cardinality, we are used to this; so it is no longer news.
Odd coincidence, though... the 5th.
-------------------------------------------------------------------------------
Bill Taylor w...@math.canterbury.ac.nz
-------------------------------------------------------------------------------
The beauty of the Pythagorean theorem is hopelessly tainted by its
relevance to the approximately Euclidean space we actually inhabit.
[John Baez (misquoted)]
-------------------------------------------------------------------------------
Jason Nafziger
w...@math.canterbury.ac.nz (Bill Taylor) wrote:
>dor...@asimov.nrl.navy.mil (Bryan) writes:
>
>|> : If you run through it you can show that the first 4 axioms of Euclid
>|>
>|> Errr...liked your post, but I think you mean the 5th *postulate*, don't
>|> you? It would be big news if Euclid's axioms had been overthrown.
>
>Actually, they *have* been and it *was*, well... kinda, long ago.
Actually, *they* were *not*, but *I* will *forgive* you...*...*
>
>
>Euclid's 5th postulate was the parallel, which was "overthrown", or at
least
>semi-sidelined, about 1830.
I heard it pulled a hamstring, but it should be recovered in time for the
playoffs. Whew!
>But Euclid's 5th AXIOM, or what he called "common
>notion",
Actually, I think he called it "honey lotion". Or is that what he called
Barbara Bush?
>was observed to be screwy long before! Euclid's common notions were
>"if things be added to equals, the sums are eqwual", and so forth, which
were
>held in the Elements to apply to *all* topics, not just geometry.
What about sex? Do they work with sex? Add me to the list as well, please.
>
>Coincidentally, it was the *5th*, again, which was troublesome...
Always mocking the other axioms, never letting them join in any of his
reindeer games...
>
>"The whole is greater than any of the parts."
No, it's "the whole is greater than any of the parts, except in the case of
Pamela Anderson."
>
>Galileo observed that the set of natural number squares was in some sense
>"equal to" all the naturals. (This had apparently also been observed by
the
>scholastics, and even by the classical Greeks in the context of real
intervals.)
>However Galilaeo gets the credit.
What a ball-hog.
>
>Since Cantor and cardinality,
^^^^^^^^^^^^^^
You mispelled "Siegel".
>we are used to this; so it is no longer news.
>
>Odd coincidence, though... the 5th.
Ah, yes, odd, 5, Hahahahahahahahahhaahahahahahahaahaaaaa, eh... screw it.
Jason -- who would rather discuss Marilyn vos Savant and BLTs...
____________________________________________
This IS the Crappy Homepage: http://metro.turnpike.net/C/crapco/index.htm
"Well at least I didn't tell you about the time I grew a penis and sired
those oxen...oops..." -- Gwyneth Kozbial
Kevin Brown
MW> The usual ZFC proof is quite meaningful, so far as I can tell.
Let me be more explicit about the meaning of "meaningful". Consider
the two well known theorems
(1) con(ZF) implies con(ZFC)
(2) con(ZF) implies con(PA)
From a foundational standpoint, these two theorems act in opposite
directions. In case (1), if the result had been {con(ZF) implies
NOTcon(ZFC)} then it would have undermined our confidence in the "C"
part of ZFC. However, if the result of case (2) had been {con(ZF)
implies NOTcon(PA)}, it would presumably would have undermined our
confidence in ZF, not in PA (because the principles of PA are considered
to be more self-evidently consistent that those of ZF).
The only kind of proof that would enhance our confidence in PA would
be of the form
con(X) implies con(PA)
where X is a system whose consistency is MORE self-evident than that
of PA. (This is the key point.) For example, Hilbert hoped that with
X = 1st Order Logic it would be possible to prove this theorem, thereby
enhancing our confidence in the consistency of PA. That would have been
a meaningful proof of the consistency of PA. However, it's now known
that such a proof is impossible (unless you believe in the existence
of a formal system that is more self-evidently consistent than PA
but that cannot be modelled within the system of natural numbers).
Torkel Franzen
>The only kind of proof that would enhance our confidence in PA would
>be of the form
> con(X) implies con(PA)
>where X is a system whose consistency is MORE self-evident than that
>of PA.
I take it that by "enhance our confidence in PA" you mean "enhance
our confidence that PA is consistent". Now, if you have doubts about
the consistency of PA, you may or may not find any consistency proof
that alleviates those doubts. It is not correct that "it's now known
that such a proof is impossible" - for example, why should it be
impossible to regard the theory of primitive recursive functionals
as more evidently consistent than PA? But doubts about the consistency
of PA, if at all intelligible, must entail doubts about very many
proofs in mathematics, and it seems a bit odd to harp on the
particular (trivial) theorem "PA is consistent".
>(This is the key point.) For example, Hilbert hoped that with
>X = 1st Order Logic it would be possible to prove this theorem, thereby
>enhancing our confidence in the consistency of PA.
Hilbert was not looking for a theorem of the form "if FOL is
consistent, PA is consistent" - after all, FOL is trivially consistent
from practically any point of view - but for a "finitary" consistency
proof for PA. His aim was not just to "increase our confidence" in the
consistency of PA, but to show that the use of non-finitary methods
is in principle eliminable in proofs of "real statements".
Matthew P Wiener
>> We have no meaningful proof of the consistency of arithmetic.
>MW> The usual ZFC proof is quite meaningful, so far as I can tell.
>Let me be more explicit about the meaning of "meaningful". Consider
>the two well known theorems
> (1) con(ZF) implies con(ZFC)
> (2) con(ZF) implies con(PA)
>From a foundational standpoint, these two theorems act in opposite
>directions. In case (1), if the result had been {con(ZF) implies
>NOTcon(ZFC)} then it would have undermined our confidence in the "C"
>part of ZFC.
Well, yes. It would be a proof that AC is false.
> However, if the result of case (2) had been {con(ZF)
>implies NOTcon(PA)}, it would presumably would have undermined our
>confidence in ZF, not in PA (because the principles of PA are considered
>to be more self-evidently consistent that those of ZF).
Fascinating.
But you're speaking about some other universe, so, like, wow and woo.
For starters, ZF|-Con(PA). No need to assume Con(ZF).
>The only kind of proof that would enhance our confidence in PA would
>be of the form
> con(X) implies con(PA)
>where X is a system whose consistency is MORE self-evident than that
>of PA. (This is the key point.)
Huh? Who said anything about "confidence"?
My original statement was that we *do* have a consistency proof of PA,
using perfectly ordinary mathematics. That is true. But I also believe
you have the confidence game completely backwards.
> For example, Hilbert hoped that with
>X = 1st Order Logic it would be possible to prove this theorem, thereby
>enhancing our confidence in the consistency of PA. That would have been
>a meaningful proof of the consistency of PA. However, it's now known
>that such a proof is impossible (unless you believe in the existence
>of a formal system that is more self-evidently consistent than PA
>but that cannot be modelled within the system of natural numbers).
So far as *I'm* concerned, ZF is bigger than PA, and much more mathematics
has been done with it, and has thus we have *more* experimental evidence
in favor of Con(ZF), much more so than the amount of evidence that exists
in favor of Con(PA). If the two were completely unconnected, one would
naturally rate Con(ZF) as greater than Con(PA).
Given this, ZF->Con(PA) *increases* one's confidence in PA.
Who said confidence had to be *proven*? Hilbert? He invented that game
as a brilliant, but hindsight-says-unnecessary, response to the paradoxes.
To me, mathematical confidence is an experimentally derived fact. Unless
you *prove* that it needs to be proven, you're just gibbering.
tv's Spatch
Jason Nafziger <nafzi...@osu.edu> wrote:
>
>I'd rather have someone explain these things on my shoes... they're really
>long and I keep tripping over them... WHAT THE FUCK????
That's toilet paper.
--
tv's Spatch, father of alt.stupidity and an accident just waiting to happen
"Yeah, but bacon tastes good." - Dick York, "I Dream of Jeannie"
"There are two things I can't stand, one of them is your mom." - The BOBS
"It's conclusive. You WERE dropped on your head." - Liza Daly
Benjamin J. Tilly
wsa...@urc.tue.nl (Jan Willem Nienhuys) writes:
> kevi...@delphi.com (Kevin Brown) writes:
>
> #BT = Ben Tilly
> #BT> Think about it. Do you REALLY believe that there is a contradiction
> #BT> in the positive integers?
>
> #What I believe and what I can prove are two different things. Having
> #said that, my answer is yes. In fact, I would suggest that every
> #(sufficiently complex) formal system contains a contradiction at some
> #point.
> [skip some]
> #(1) We have no meaningful proof of the consistency of arithmetic.
>
> #(3) HOWEVER, highly elaborate and lengthy chains of deduction
> # in the far reaches of advanced number theory might need to be
> # re-evaluated in the light of our patched-up formalization.
>
> May I make a suggestion? Arithmetic's supposed inconsistency only
> can make sense in a completely formalized system. The problem may
> reside in the particular way of formalizing, rather than in `far reaches
> of number theory'. Formalization has many conventions in it regarding
> the use of variables, formulas, definitions.
> It is one thing to present a few typical formulas and do a little
> handwaving, it quite another thing to build up a complete system
> entirely from scratch in such a way that it presupposes no previous
> knowledge on the part of the reader while at the same time it is
> completely clear that it corresponds with what we call `arithmetic'.
>
Take a look at Peano's axioms. If the arithmetic of the positive real
numbers were to be found to not be a model for them, then I would sit
there moaning "... but ... but ..." until I got stuck in an insane
asylum!
> I'm not even sure you can do that without already assuming vast amounts
> of information about numbers to be `known'. Just look at naming
> conventions for variables. You'll have to suppose that a convention
> whereby variables are named x , x' , x'' , x''' etc. is meaningful.
> Writing `etc.' won't do, because you must suppose that this is a
> concept unknown to the reader.
>
I must assume that it is unknown to the reader? I have yet to see this
cause any problems for anybody yet...
> Maybe these problems can be solved satisfactorily. But these solutions
> can have in them the seeds of contradictions that sneak up on you.
> It is also imaginable that later generations of mathematicians will
> think that these solutions are of an intolerable vagueness.
>
It is imaginable that future mathematicians will be more picky than we
are. But I strongly doubt that PA will EVER be found to have problems.
Take a look at it. Then if you still have a question about them, then
get back to me with the one that you think could not be true for the
positive integers.
> What would happen when a contradiction was found? I guess that
> the foundations-of-mathematics people would try to find a way
> to work around the problem. I think that what Ben Tilly meant
> was a contradiction in the positive integers that won't go away,
> no matter what formalization you use.
>
What I meant is what I said. I do not believe that any contradiction
exists in PA.
> I share his belief that no such contradiction can be found.
What about PA? A collection of simple statements, all of which are
manifestly true about the positive integers.
Ben Tilly
Timothy Y Chow
Kevin Brown <kevi...@delphi.com> wrote:
>Let me be more explicit about the meaning of "meaningful". Consider
>the two well known theorems
>
> (1) con(ZF) implies con(ZFC)
>
> (2) con(ZF) implies con(PA)
What theorem are you talking about in (2) here? The usual proof of the
consistency of PA does *not* assume con(ZF). This was pointed out by me
in an earlier article in this thread (which you may have missed since I
haven't seen a followup to it by you).
If you can't even get your facts straight, your philosophical arguments
lose what force they might otherwise have had.
Tim.
Ross Garmil
: Virginia Krenn <asd...@osuunx.ucc.okstate.edu> wrote:
: >On 19 May 1995, Ross Garmil wrote:
: >> Jason Nafziger (nafzi...@osu.edu) wrote:
: >> : >Ross Garmil (lim...@bu.edu) wrote:
: >> Ross---ok, what if I put my fork in this toaster. Argh, it was a
: toaster oven
: >> and didn't do anything. Boy, was that stoopid, or what?
: >
: >If you want to feel really stupid, get someone to explain
: >"Projective Geometry" to you.
: I'd rather have someone explain these things on my shoes... they're really
: long and I keep tripping over them... WHAT THE FUCK????
Jason, those are salamanders. You're supposed to set them free in the wild
before you wear the shoes. Now you're stuck with them forever--'cuase you
know the old saying "don't release the salamander's in the wild before wearing
your shoes, and you're stuck with them forever."
: Jason -- who will lather and rinse, sure, but REPEAT? Never...
Ross--who has used the exact same joke before, and isn't accusing Jason of
anything, but is rather freaked out at the moment.
: ___________________________________________
Jason Nafziger
>Jason Nafziger (nafzi...@osu.edu) wrote:
>: In article
>: <Pine.ULT.3.91.95052...@osuunx.ucc.okstate.edu>,
>: Virginia Krenn <asd...@osuunx.ucc.okstate.edu> wrote:
>: >On 19 May 1995, Ross Garmil wrote:
>: >> Jason Nafziger (nafzi...@osu.edu) wrote:
>: >> : >Ross Garmil (lim...@bu.edu) wrote:
>: >> : Jason -- who's *watching* you, so... uh... do more... uh... dumb...
>: >> : thingies...
>: >> Ross---ok, what if I put my fork in this toaster. Argh, it was a
>: toaster oven
>: >> and didn't do anything. Boy, was that stoopid, or what?
>: >
>: >If you want to feel really stupid, get someone to explain
>: >"Projective Geometry" to you.
>
>: I'd rather have someone explain these things on my shoes... they're
really
>: long and I keep tripping over them... WHAT THE FUCK????
>
>Jason, those are salamanders. You're supposed to set them free in the
wild
>before you wear the shoes. Now you're stuck with them forever--'cuase you
>know the old saying "don't release the salamander's in the wild before
wearing
>your shoes, and you're stuck with them forever."
So does this mean I have to buy new shoes? Couldn't I just kill the
salamanders? Uh-oh, these aren't those magic(k) pixie salamanders, are
they?
>
>: Jason -- who will lather and rinse, sure, but REPEAT? Never...
>
>Ross--who has used the exact same joke before, and isn't accusing Jason of
>anything, but is rather freaked out at the moment.
You've never used it in my presence, have you? If so, I apologize. If not,
I freak out as well...
Jason -- who says "don't release the salamanders in the wild before wearing
your shoes, and you're stuck with them forever."
____________________________________________
Gmark
: Jason Nafziger (nafzi...@osu.edu) wrote:
: : In article
: : <Pine.ULT.3.91.95052...@osuunx.ucc.okstate.edu>,
: : Virginia Krenn <asd...@osuunx.ucc.okstate.edu> wrote:
: : >On 19 May 1995, Ross Garmil wrote:
: : >> Jason Nafziger (nafzi...@osu.edu) wrote:
: : >> : >Ross Garmil (lim...@bu.edu) wrote:
: : >> : Jason -- who's *watching* you, so... uh... do more... uh... dumb...
: : >> : thingies...
: : >> Ross---ok, what if I put my fork in this toaster. Argh, it was a
: : toaster oven
: : >> and didn't do anything. Boy, was that stoopid, or what?
: : >
: : >If you want to feel really stupid, get someone to explain
: : >"Projective Geometry" to you.
: : I'd rather have someone explain these things on my shoes... they're really
: : long and I keep tripping over them... WHAT THE FUCK????
: Jason, those are salamanders. You're supposed to set them free in the wild
SALAMANDERS!?
No wonder my damn shoes keep loosening up!
GMS
Ross Garmil
: >know the old saying "don't release the salamander's in the wild before
: wearing
: >your shoes, and you're stuck with them forever."
: So does this mean I have to buy new shoes? Couldn't I just kill the
: salamanders? Uh-oh, these aren't those magic(k) pixie salamanders, are
: they?
Well, of course they're pixie salamanders. You didn't think that they would
put just plain old salamanders on your shoes, do you? How silly of you.
Unless of course they're Reeboks, in which case they're the kind of
salamanders that drive on the left side.
: >
: >: Jason -- who will lather and rinse, sure, but REPEAT? Never...
: >
: >Ross--who has used the exact same joke before, and isn't accusing Jason of
: >anything, but is rather freaked out at the moment.
: You've never used it in my presence, have you? If so, I apologize. If not,
: I freak out as well...
I don't believe I've ever been in your presence. Let the freak out commence.
: Jason -- who says "don't release the salamanders in the wild before wearing
: your shoes, and you're stuck with them forever."
Ross--who also saw a great version of the joke in a poem called "Infinity" (by
my friend Joshua Liebster, although it was a different joke)
anyways
Infinity
Lather
Rinse
Repeat
Ross Garmil
: : : In article
: : : <Pine.ULT.3.91.95052...@osuunx.ucc.okstate.edu>,
: : : Virginia Krenn <asd...@osuunx.ucc.okstate.edu> wrote:
: : : >On 19 May 1995, Ross Garmil wrote:
: : : >> Jason Nafziger (nafzi...@osu.edu) wrote:
: : : >> : >Ross Garmil (lim...@bu.edu) wrote:
: : : >> : Jason -- who's *watching* you, so... uh... do more... uh... dumb...
: : : >> : thingies...
: : : >> Ross---ok, what if I put my fork in this toaster. Argh, it was a
: : : toaster oven
: : : >> and didn't do anything. Boy, was that stoopid, or what?
: : : >
: : : >If you want to feel really stupid, get someone to explain
: : : >"Projective Geometry" to you.
: : : I'd rather have someone explain these things on my shoes... they're really
: : : long and I keep tripping over them... WHAT THE FUCK????
: : Jason, those are salamanders. You're supposed to set them free in the wild
: SALAMANDERS!?
: No wonder my damn shoes keep loosening up!
No, your shoes keep loosening up-stream.
: GMS
Ross--who thinks that everone would get along better with their shoes if only
we better understood the nature of salamanders.
Jason Nafziger
<Pine.ULT.3.91.95052...@osuunx.ucc.okstate.edu>,
Virginia Krenn <asd...@osuunx.ucc.okstate.edu> wrote:
>On 19 May 1995, Ross Garmil wrote:
>> Jason Nafziger (nafzi...@osu.edu) wrote:
>> : >Ross Garmil (lim...@bu.edu) wrote:
>> : Jason -- who's *watching* you, so... uh... do more... uh... dumb...
>> : thingies...
>> Ross---ok, what if I put my fork in this toaster. Argh, it was a
toaster oven
>> and didn't do anything. Boy, was that stoopid, or what?
>
>If you want to feel really stupid, get someone to explain
>"Projective Geometry" to you.
I'd rather have someone explain these things on my shoes... they're really
long and I keep tripping over them... WHAT THE FUCK????
Jason -- who will lather and rinse, sure, but REPEAT? Never...
____________________________________________
Gareth Rees
> However, are we really justified in asserting that EVERY one of the
> infinitely many paths from every premise to its evaluation gives the
> same result?
If you allow associativity of modus ponens, then there is essentially
only one proof of any classical formula.
--
Gareth Rees
Jan Willem Nienhuys
[me commenting on BJT:]
>> I share his belief that no such contradiction can be found.
>What about PA? A collection of simple statements, all of which are
>manifestly true about the positive integers.
>Ben Tilly
These statements look simple. But they can do things for you only
if you have a vast mathematical apparatus that includes:
1. conventions regarding names of variables;
2. conventions regarding scopes of thoses names;
3. rules about what kind of substitutions are permittable;
4. conventions regarding definitions;
5. conventions regarding what is considered: sets, functions, products.
6. rules of inference (i.e. logic).
I just mentioned in my previous post 1 (variable names).
If one proposes to name variables x, x', x'' and so on, one certainly
needs to know that all these names are different. Of course, in my
heart I know they are different. But if the formal system is to be absolutely
watertight, I must prove that they are different. In other words, I
must apply the Peano axioms to the language used to talk about them.
The fact that my experience with nonnegative integers makes me believe
that the Peano axioms are manifestly true is irrelevant. My experience
certainly doesn't include having often verified that adding 1
consecutively 10^80 times gives each time a different result.
The basic trouble with the Peano axioms is that they implicitly
assert that any set of positive integers has a smallest element.
If you think of simple Ramsey numbers, then you'll realise that
even in simple cases this is more a metaphysical assertion than a
matter of "manifestly true". How can I have confidence that any
set of positive integers, no matter how unwieldy its definition
has a smallest element?
The unwieldyness of definitions is made possible by having
an unlimited amount of positive integers (and variable names) at
your disposal.
The Peano axioms plus a formal system in which they are embedded
must be free of contradictions, i.e. for no statement A both
A and not-A may be provable. (Note that I apparently want to allow
variables like A to denote statements; the formal system should
have watertight descriptions - no handwaving or illustration by
example allowed - of how to go about making variables represent
statements.)
Only such a system as a whole can conceivably contain a contradiction.
Without the formal framework a contradiction doesn't make much sense.
JWN
Torkel Franzen
>Each represents a non-
>finitistic extension of formal principles, which is precisely
>the source of uncertainty in the consistency of PA.
This kind of statement doesn't have any obvious objective
significance. Whose uncertainty?
>As to whether
>such doubts are intelligible, I will only comment that much of the
>most interesting and profound mathematics of this century has been
>concerned with just such doubts.
There are many perfectly intelligible expressions of such doubts.
But there are also versions that make little sense.
>personally worry much about con(FOL), but to accommodate Tim Chow's
>concern I decided to include the clause "If con(FOL)".
This is an example of a peculiar foundational delicacy or concern
that doesn't on the face of it make any sense. There are any number of
"if"-sentences that might with equal justification, which is to say
equally pointlessly, be made part of any mathematical assertion. If
the consistency of FOL is somehow a hypothetical matter, why should I
assume that you or I make any sense whatever in our babbling?
Kevin Brown
> be of the form { con(X) implies con(PA) } where X is a system whose
> consistency is MORE self-evident than that of PA.
TF> I take it that by "enhance our confidence in PA" you mean "enhance
TF> our confidence that PA is consistent".
Yes indeed. Thanks for the clarification.
TF> Now, if you have doubts about the consistency of PA, you may or
TF> may not find any consistency proof that alleviates those doubts.
Well, that narrows down the possibilities.
TF> It is not correct that "it's now known that such a proof is
TF> impossible" - for example, why should it be impossible to regard
TF> the theory of primitive recursive functionals as more evidently
TF> consistent than PA?
Torkel, let me remind you of the entire sentence from which you are
quoting:
"However, it's now known that such a proof is impossible (unless
you believe in the existence of a formal system that is more
self-evidently consistent than PA but that cannot be modelled
within the system of natural numbers)."
Since it's well known that both transfinite induction and the theory
of primitive recursive functionals cannot be modelled within the
system of natural numbers, my original statement clearly anticipated
your comment above. I do not claim it would be impossible to regard
such principles as more evidently consistent than PA; I simply observe
that no one does, and for good reason. Each represents a non-
finitistic extension of formal principles, which is precisely
the source of uncertainty in the consistency of PA.
Here's a little thought experiment that sometimes helps people sort
out their own heirarchy of faith: If, assuming the consistency of
the theory of primitive recursive functionals, one could prove that
PA is NOT consistent, would you be more inclined to abandon PA or
the theory of primitive recursive functionals?
TF> But doubts about the consistency of PA, if at all intelligible,
TF> must entail doubts about very many proofs in mathematics, and it
TF> seems a bit odd to harp on the particular (trivial) theorem "PA
TF> is consistent".
When considering doubts about the consistency of PA, I see nothing
odd about focusing on the theorem "PA is consistent". As to whether
such doubts are intelligible, I will only comment that much of the
most interesting and profound mathematics of this century has been
concerned with just such doubts. As to the number of proofs that
are cast into doubt by the possibility of inconsistency in PA, I've
explained in other posts that the "perfect consistency or total
gibberish" approach to formal systems evidently favored by residents
of sci.math is not justified.
TF> Hilbert was not looking for a theorem of the form "if FOL is
TF> consistent, PA is consistent" - after all, FOL is trivially
TF> consistent from practically any point of view...
If you regard the consistency of FOL as absolute, then the antecedent
clause "If con(FOL)" may be omitted from my statements without
affecting their content. In fact, that is actually what I did in
some earlier posts in which I described the goal of an absolute
(as opposed to relative) finitistic proof of con(PA). However,
Tim Chow commented that such a proof was not really absolute because
it assumed the consistency of first-order logic. Like you, I don't
personally worry much about con(FOL), but to accommodate Tim Chow's
concern I decided to include the clause "If con(FOL)".
TF> - but for a "finitary" consistency proof for PA. His aim was not
TF> just to "increase our confidence" in the consistency of PA, but to
TF> show that the use of non-finitary methods is in principle eliminable
TF> in proofs of "real statements".
Yes. It just so happened that Russell and Whitehead's FOL was a
conveinent finitistic vehicle to use as an example. Of course,
people sometimes raise the possibility of a finitistic system that
cannot be modelled within the theory of natural numbers but, as
Ernst Nagel remarked, "no one today appears to have a clear idea
of what a finitistic proof would be like that is NOT capable of
formulation within arithmetic".
Bill Taylor
|> Now consider our formal system of arithmetic. Many people seem agog
|> at the suggestion that our formalization of arithmetic might possibly
|> be inconsistent at some point.
Gog, gog, gooo...
|> However, are we really
|> justified in asserting that EVERY one of the infinitely many paths
|> from every premise to its evaluation gives the same result?
Heavens! You're right! God may be using a Pentium chip....
-------------------------------------------------------------------------------
Bill Taylor w...@math.canterbury.ac.nz
-------------------------------------------------------------------------------
Luke: you don't know the power of the dark side of the farce.
-------------------------------------------------------------------------------
david petry
(Timothy Y Chow) writes:
>
>In article <9505220123592....@delphi.com>,
>Kevin Brown <kevi...@delphi.com> wrote:
>>Let me be more explicit about the meaning of "meaningful". Consider
>>the two well known theorems
>>
>> (1) con(ZF) implies con(ZFC)
>>
>> (2) con(ZF) implies con(PA)
>
>What theorem are you talking about in (2) here? The usual proof of
the
>consistency of PA does *not* assume con(ZF).
Here's a comment that may not be relevant.
If the word "proof" is to have the meaning "a convincing argument",
then
the consistency of the formal system in which the proof lives is always
an implicit assumption. That is, you would not believe the proof is
convincing if you did not at least believe that all of the results
derived from the formal system are consistent with one another.
To my mind, this fact gives all discussion of consistency proofs a
surreal quality.