Subject: Re: Can Science Say If God Exists? (long)
From: Henry Warwick
Date: Mon, 28 Oct 1996 13:53:08 +0000
Well, friends, I can for once and for all settle this
silly little argument. It just so happens that yours
truly has built a God Detector (tm), and the results
were truly astounding.
First, I had to figure out what would make "God" more
or less detectable. So what I did was I figured I'd hang
out with people who claim to be close to God, and might
know His habits. Well, wasn't THAT just a weird adventure!
The first place I go is to a local church. As it so
happens, I live 3 blocks from a very large Catholic
Church. I ask the priest, "Hey, Priest! I'm buildin'
me a God Detector (tm), and so I'm tryin' to find out
more about God, so I can detect him better. So can you,
like, tell me some of his habits, ya know, like where
he might like to hang out and stuff?"
So, anyway, the Priest, he just laughs and laughs, and
says that I'm a silly boy. I tell him I'm 38 and ain't
no kinda boy.And that I'm really serious about buildin'
my God Detector (tm). He said he wasn't too sure about
the viability of my God Dectector (tm) but that he could
give me some pointers about God. He talked to me about
silence and bein' good and stuff. I thanked him for
his time and snagged a Dixie Cup of Holy Water and
some bingo sheets on the way out.
Down the street was another church and so I bop in
there to see what da preecherman has to say. So I
goes up to da Preecherman and I tell him about my
God Detector (tm). He looks at me all angry like and
thne lets loose a whole lotta invective about blasphemy
and Jesus this and JeEEeeeEeZzus that, and I listen as
best I can, but his stuff is so fulla self servin'
crap that my Bullshit Detector (tm) damn near hadda
meltdown!
So, I asked him if God likes to Laugh much, seein'
as how he's all powerful and stuff, that he might find
his creation amusing. The Preecherman just reared up,
you know, & like the short hair on the back of his neck
came skippin' out of these little goosebumpy things,
and he just yelled "WHAT?!?!?!"
I just asked him if "Gowodd" hadda sense a humor, ya know,
like maybe dirty Limericks aren't his style, but maybe
knock knock jokes are OK?
He just yelled at me to get out. But on my merry way
I was able to snag some nasty little hate filled
and fear driven comic books on the way out. Filled
with ridiculous stories about people just leadin'
kinda sad mediocre lives bein' punished by some asshole
with a long white beard qua qua qua or whatever.
I always liked comic books, but these are kinda stupid.
So, I take a bus downtown and I go to the Rabbi at
a Jewish Temple, and I tell him about my God Detector (tm).
When I say that, he kinda freezes up for a second like
I'd spoken to him in Chinese or something, and then
goes on to say that they already invented one called
the Caalballa or somethin'and that I'm still too young
to be let in on it, as ya hafta be at least 40 before
they give ya any details. Then he offered me some flat
cruchy bread. And then he kicked me out.
But I grabbed a couple of little candles on the way out,
and stuck 'em in my pocket with the comic books.
So I go up the block to the Hindu Shrine and talk to
a fella there about my idea of a God Detector. He looks
at me and says, "Oh, but a God Dectector, yes- a very
fine idea, but for which God? We have so many! Here-
this is a book and it talks about many of our Gods.
I will lend it to you, but please- do not spill any of
the water from that Dixie Cup on it! If you can make
a God detector for even ONE of our Gods, I am sure
that I can find people interested in buying one!"
Anyway, his wife was there and she was wearing one of
those dot thingies in her forhead, but it blowed off
while we were talking.
When he kicked me out, I found the dot thingie on the
walk and took it with me.
I was kinda tired. I'd been hikin' all over town looking
for advice on how to make my God Detector (tm). I figured
it'd be great- ya turn it on, and if it says God ain't
around, then you can do the kinds of things that God
doesn't like people doin', ya know, lying, cheatin,
and killin' and stuff. And if it says God's watchin'
then ya just get smart straight up and whistle the
theme to Davy and Goliath or somethin'.
So, anyway, I go over to a Mormon Church and an Islamic
type of place, but the Mormons wouldn't let me in, and
the Islamic guy was going on and on about Allah and
Mohamed in such a way that I really didn't understand,
and though it was interesting but kinda creepy. Also,
they didn't have anything I could snag for my God
Detector (tm) research.
So, I wander over to a Zen Buddhist place, and it's
really clean, and everybody's sittin around examining
the wall in front of them really intent like, and keepin'
all motionless, like if they move, the wall might flinch
or run away like a scared deer or somethin'.
Anyway, i ask this fella in another part of the temple
about my idea for a God Detector (tm), and he sez he can't
reall help me, 'cuz Buddhism doesn't really care about
whether God is there or not or anything like that.
This had me kinda put out, 'cuz it was a long walk
over there. So I tell him that, and he says that there
is a famous Zen saying that I should pay attention to.
I asked "Whuzzat?" an he sez: "Life...is TOUGH!"
This seemed like a sensible response and I wanted
to pat him merrily on his bald little head, but I
decided not to, 'cuz he might get mad.
So, anyway, I leave there and grab some rocks from the
gravel garden (Man- these people rake rocks!) and
wandered over to the bus stop with my pockets full
of stuff I'd swiped from the religious folks. Some
gravel, a dot thingie, a book of Hindu Gods, three
candles, two nasty little comic books and some bingo
sheets. Oh. And a dixie cup fulla Holy Water.
So I took the candles and lit 'em up, and as they melted,
I used the wax on the comics and book of Hindu Gods.
for extra strength I ground up the Rabbi's flat bread
into powder and mixed it with the Holy Water to make a
flour paste. This I used to glue the bingo sheets onto
the box I had made. I glued the gravel in a circle
on the top and put the dot thingie right in the middle.
It was really hot that day, so it set up really pretty
when I let it sit on my windowsill. That was until the ants
discovered it and started chewing it down. And that's
what gave me the idea- the box IS the God Detector (tm)!
So, the way it works ya see, is ya get people to hold
the box and be real quiet and concentrate like if they
move the earth'll flinch and take off like a jack
rabbit. And while they're all concentrating ya talk to
'em about how inside this pasty brown nasty lookin' box
covered with bingo sheets and gravel is the answer to
their deepest existential and ontological questions
and how it's really important for them to keep
their eyes closed.
And so in their imaginations they are to focus on this
box, bringing all cathectic power to it, and then tell
them to open this box they see in their mind's eye
(which looks like the one ya glued out of ground up
Matzoh bread and Holy water) and tell ya what they see
in the box.
If they cry with joy and are overcome with happiness,
yellin' all kinds of stuff like "I see God!",
then ya know God is there, and ya better behave
yourself. If they slap you in the head, sayin "this is
the stupidest damn thing I've ever done in my life and
take your damn box back- it's leakin' ants," then God
probably ain't there, and you can carry on as usual.
My personal observation, judgin' from the way people
react, is that God's usually there but not paying real
close attention....
Mr Warwick
Subject: Re: "Essential" reality (was: When did Nietzsche wimp out?)
From: Steven Hines
Date: Mon, 28 Oct 1996 13:24:51 -0800
meron@cars3.uchicago.edu wrote:
>
> In article <3274E91F.5247@sdd.hp.com>, Steven Hines writes:
> >Russell Turpin wrote:
> >>[snip]
> >> There is no faith to which one
> >> must leap, no metaphysical tenets that one must accept.
> >
> >Don't you have to believe that every event has a cause? Doesn't
> >acceptance of this proposition constitute a "passage of faith"?
> >
> Assume, yes, believe, no. There is a difference. Science is self
> checking and self correcting. It makes assumptions, draws conclusions
> and checks the conclusions versus experimantal data. But the
> assumptions aren't sacred and if the data fails to support your
> conclusions you may be forced to modify your assumptions. It is the
> continuous cross checking that distinguishes assumptions from beliefs.
Okay.
I think I have a basic understanding of the model scientists
use in settling on hypotheses, theories, and laws. I accept that
each of these must be consistent with observations in order to hold
up and that theories that do not stand up to experimental rigor are
discarded in time.
I write the above paragraph in the hopes of establishing some common
ground here and so you do not think that I am attempting to discard
science and its accomplishments.
That being said, I have a hard time seeing how one can experimentally
determine that some events do not have causes and still be doing
science. That is, if a scientist observes an event, looks for a cause,
and finds none, what is the consensus among other scientists in the field?
Honestly... are they likely to say, "Ah, this event has no cause"
or instead will they say (perhaps to themselves) "This scientist has
not looked hard enough, or in the right places."
What I mean to say is that I can't see how science can proceed unless
is assumes beforehand that observed phenomena can be explained
(isn't that, after all, the job?). But what about this assumption?
Is it forced to stand up to the same rigor as the hypotheses, theories
and laws? That is, how can one know for certain that _all_ avenues
have been traveled in search of a cause, but to no avail, and that
therefore the phenomena has no cause? How many scientists would
accept this?
If the proposition "all events have causes" cannot be proven or
disproven by experimentation, then isn't it true that the proposition
is being accepted on faith?
-----------------
Steve Hines
shines@sdd.hp.com
Subject: Re: A photon - what is it really ?
From: platt@watson.ibm.com (Daniel E. Platt)
Date: 28 Oct 1996 22:04:01 GMT
In article <3272C9FD.6613@starbase.deepspace.nine.mil>, Quark writes:
|> magnus.lidgren wrote:
|> >
|> > Trying to educate myself within the subject - A photon - what is it really ? .
|> > Thanks to all those initiated, sharing their wisdom by responding to this issue !
|> >
|>
|> What you have observed is the classic conundrum involving the question
|> of the particle versus wave theory for light. Science will in general
|> develop models of the physical world that will attempt to describe it in
|> a fashion that will show precicely the way phenomenon will exist in
|> nature and be able to generate predictions with regard to how the
|> physical world will show itself to be under various circumstances.
Science tries to construct good descriptions of reality. It also tries
to explore connections and consequences of those descriptions. An
example is Newton's laws as a description of motion, with lots of
testable consequences.
|> In
|> general, in the late twentieth century there were two general theories
^^^^^^^^^ nineteenth?
|> that were used to describe light. Particle theory and wave theory.
|> Both theories were viable under different circumstances, and had
|> intrinsic merit when it came to predicting the properties of light in
|> general.
Light as a wave was supported by observation in the very end of the
1700's and early 1800's. Also around then, electricity and magnetism,
and induction and Ampere's laws were discovered. It wasn't until
Maxwell that the displacement current was posited by Maxwell, and he
realized electromagnetism could propagate as a wave. By the time
Maxwell actually admitted light was a form of electromagnetism, most
of the world had accepted the idea, and Hertz' experiments had verified
the propagation of electromagnetic energy via sparks.
|> However, when each of them were to be used, and not to be
|> used, and the circumstances under which they were valid or not valid
|> were not well defined, and thus did not provide a complete description
|> of electromagnetic radiation in general due to the inexact properties of
|> definition and a failure to define exactly under what conditions the
|> various descriptions would apply. Ther propagation of light through a
|> transparent media is a classic example of when the old 'basic photon
|> theory' of light would break down and be invalid.
Maxwell's equations did very well in describing electromagnetic fields.
Certainly, they did very well in relating mechanically measureable aspects
of the fields to matter (ie, through Newton's laws). They related well
to the more empirically based polar wave (in the sense of being polarizable,
which is a feature that was already experimentally established by the time
Maxwell wrote down his formulation) description of electromagnetism.
However, there were two problems which emerged with the realization that
electrodynamics was light. First is the thermodynamics of light -- what
happens when you heat up a cavity and bring the gas of electromagnetic
excitation into thermodynamic equilibrium with its environment? Stefan
had observed a T^4 law in the power spectrum, and the next year, Boltzmann
offered an analytical argument (based on the idea that the pressure of
electromagnetic radiation would be one third the energy density of the wave --
something that follows directly from Maxwell's formulation -- and this came
before Maxwell admitted the connection of light and electrodynamics).
Also, during Hertz' experiments, he noted that the spark of the receiver
seemed to perform better under violet light than under red light... strange
little side note to the overwhelming news that he'd experimentally verified
Maxwell's waves.
Boltzmann and others tried for a long time to connect microscopic mechanics
to the thermodynamic properties of light, in particular to try to predict
the observed spectrum. Off and on, the photoelectric effect had come up too.
Yet, no really good description came of either of these until Plank's
derivation of the black-body spectrum in 1900. Even then, he didn't offer
a strong connection to a ``lumpiness'' to light, rather focusing on the
radiation process itself. Einstein proposed an explanation of the
photoelectric effect in 1905 (for which he won a Nobel prize -- but that's
another story).
The exact description of that lumpiness is to say that only certain
discrete energy densities may be carried by light at any one particular
frequency. This is a very peculiar statement of the ``particulate'' nature
of light. It isn't at all like lumps of sand or marbles. Rather, it is
a statement that the amplitude of the waves of any one particular frequency
can only take on discrete values.
To some extent, even electrons don't act like marbles. You have electron
amplitudes in various modes, and you can only have amplitude values that
reflect the Pauli exclusion principle. All in all, the meaning of ``particle''
has come to be redefined in terms of the descritization of field amplitudes
rather than in terms of a simple marble counting mechanism.
|>
|> > A short summary of some of the questions and answers:
|> >
|> > (The summary below should of cource in no way be regarded as a judgement with respect
|> > to scientifical or pedagogical skill of above mentioned initiated, only, as I forget
|> > quickly, as an expression what it all has condenced into in terms of my personal
|> > understanding.)
|> >
|> > Q1. What is a photon really made of ? Does a photon constitute a collection of smaller
|> > identical elements, (some kind of basic photon parts), of which all different photons
|> > are made of, (but, with respect to different frequencies, with various amounts of
|> > these basic parts) ?
|> >
|> > Majority opinion was that the photon just is and does not consists of minor parts.
|> >
Minority parts? To some extent, this depends on how you define it.
The way minority parts are determined or recognized these days is to look
at whether there are scattering resonances that reflect very short-lived
excited states of a particle. For example, there are short-lived
excited states of protons that show up with scattering resonances
(surprisingly large scattering cross-sections that emerge as you crank
up the bombardment energy). This ends up looking like ``structure.''
Now, its possible to construct electron-positron pairs if you have an
energetic enough photon (you destroy the photon in the creation of the
pair), or you can anihilate the electron-positron pair to get two photons
(momentum conservation). Even if you don't construct a complete e-p
pair, they can partially participate in the scattering of two photons
from each other. Is that structure? Given you DO have photons at less
energy than an e-p pair, the answer is no. Does this mean electrons
are made of photons? Again, the characteristics of angular momentum and
other features of electrons and positrons would indicate that they aren't
photons or aggregates of photons. But the question is not really simple,
but rather follows from how electrons-positrons and photons are described.
|> There are of course, some types of particles or 'quanta' which are
|> divisible into smaller ones, however they might not be considered
|> 'elementary quanta' (like particles built up of several quarks). There
|> are, however, basic elementary quanta.
|>
|> > Q2. Do all different photons, if they are not absorbed or reflected, also
|> > travel with identical speed when traveling through a media, for example glass ?
|> >
|> > All pointed out that the photon travel with different speed in different media
|> > according to the index of refraction for the media.
|> >
|> I should remind you that basic ancient wave theory for light as derived
|> in the nineteenth century was able to predict even the speed of light as
|> it traveled through a non-vacuous media. It was related to the
|> permitivity and permeability of the media (ie, the general
|> susceptibility of the material to electricity and magnetism), and it
|> used the same general equations as those used to predict the speed of
|> light in a vacuum.
This is actually a little more complicated. Photons travel at the speed
of light. As photons get absorbed and re-radiated, and as the phases
of the photons get added back into the whole, the effect of the phase
shift is to retard the effective total wave. It is the nature of the
formulation that you really cannot tell whether or which photons are
destroyed. Rather, you count amplitudes and cross-sections for various
scattering events, and describe the macroscopic susceptability and
permeability in terms of those.
|>
|> > Q3. As I understand it, Quantum mechanics states that when a photon makes
|> > its way through a glass body, it is absorbed and (a new?) (re?)-emitted a number of
|> > times before it passes through. Every time "the photon" actually is a "real" photon it
|> > travels at c speed and during "the absorption period" "it" stands still ??
|> >
|> > Majority opinion was that the photon goes through repeated absorbtions
|> > and re-emissions (in the forward direction) thus delaying "its" passage
|> > through the glass in correlation with the index of refraction for glass.
You can make a Huygen's construction of photon propagation in vacuum, destroying
a photon at each point, and re-radiating a wave front to get a complete
construction of a free plane wave. Further, you cannot tell the difference
between the two results mathematically or by any constructable experiment.
The photons that were destroyed or created have no real identity within the
``collective'' (to borrow a borg-like idea). In this sense, its hard to
really answer how often and in what ways it is absorbed and reradiated by
a medium.
|> >
|> > Q4.If a photon is truly absorbed by the media glass, in what way does "it"
|> > know (as it then has ceased to exist ?) what direction to take when
|> > emitted again and how does it know what frequency to recover?? Is all
|> > information, needed to guide the "new" photon to the right path and frequency,
|> > delivered from the "old" photon to the glass atoms during absorption and
|> > present in the glass atoms while "the photon" is in "absorpted mode".??
|> >
|> > Majority opinion was that "true" absorption could not have happened, re-emission would
|> > then have random direction. Where the information about direction was situated when
Re-radiation is not independent of the incident photon's direction, wavelength,
and other characteristics. That is, the amplitude of scattered light depends
on the direction it is scattered in with respect to the incident source.
It is in the context that a photon gets scattered all over with different
amplitudes in different directions that the question of the probabilistic
nature of a photon as particle emerges. It was partly this element that
Einstein's derivation of the black-body spectrum from his photo-electric
hypothesis by means of stimulated emission caused Einstein so much vexation:
he was the first to recognize that this quantization stuff had to have
a probabilistic character, because there was the problem of getting
half-a-lump emitted -- you could only interpret this statistically or
probabilistically. Needless to say, the idea of stimulated emission
was important in developing the idea of lasers from the stimulated emission
of photons from mater where many of the constituents were in meta-stable
or states, a statistical ``inversion.''
|> > not carried by the photon was more an open question. Perhaps one photon on its way
|> > could guide another while this was emitted , (not quite clear, this issue). Recovering
|> > frequency, however, could be possible through the specific amount of energy delivered
|> > to electrons, going from one lower shell to a higher and then back again.
|> >
|>
|> As you can see, many of the humans of this time period could not figure
|> out whether the photon was absorbed or not. Basic quantum mechanics
|> would state that a particle would situate itself as a probability
|> distribution within the schrodinger wave, and would interact with the
|> glass at specified points and then be re-emitted at random directions.
|> This is precicely what occurs when you have non-transparent glass, but
|> that is not the question being considered. I should also remind you
|> that transparancy occurs when there are no energy levels being
|> interacted with, thus you do not have a sort of laser effect going on.
Not quite -- or rather sort of. What you have is elastic scattering,
with momentum transfer but no energy absorption. What you see is some
direction-dependent probability that photons will be scattered in various
directions that depends on the momentum transfer. The tolerance of
momentum transfer is related to the Heisenberg uncertainty principle
through the mechanism of trying to localize a wave (via the Schwartz
inequality applied to Fourier transforms). I've always felt this
made the meaning of the Heisenberg uncertainty princple more clear.
|>
|> For a good description of what goes on with respect to light in a
|> transparent medium and the question of particle and wave descriptors, it
|> is a good idea to remember what the mathematical expressions 'particle'
|> and 'wave' mean to begin with. When one has a phenomenon where you have
|> continuous free space, where each location in that space will affect the
|> surrounding medium, you have what in mathematics is called a 'continuous
|> function', and can build such properties as 'coherence' in a periodic
|> spatial function such as a wave. I should remind you that the spatial
|> location of 'potential quantum transitions' or 'photons' is confined
|> within the spatial characteristics of the schrodinger wave (or light
|> wave in general). In other words, if you have a node in a schrodinger
|> wave, at that location there is 0 probability, and there are still
|> probability constraints as defined by that wave. I should also remind
|> you what is meant by a particle. A particle in general can be said to
|> be a phenomenon that is defined as localized within a specific region of
|> space. In some ways like the 'point' on a Cartesian coordinate system,
|> though still capable of taking up space. Many relations of 'parts to
|> the whole' and 'points to the curve' were devised by ones like Newton,
|> who were even further back in human history. When you have the light
|> wave interacting with relatively localized phenomenon like specific
|> atoms and their orbitals, you have basic photon interaction. When you
|> have refraction and the slowing down of light in glass, the entire
|> schrodinger wave interacts with the glass to produce effects that reduce
|> down to those calculated in the nineteenth century for electromagnetism
|> and glass, because the entire schrodinger field will act much in the way
|> that an 'electromagnetic field' was calculated to then. In some ways
|> the phenomenon might be described as a 'virtual photon' interacting with
|> the glass and then being reabsorbed by the wavefront as it would pass
|> through the medium. I should remind you that it does not do so in the
|> 'random photon' fashion as described by basic particle mechanics, also,
|> each portion of schroding field interacts with the glass in an even
|> fashion. If it did not, the wave front would quickly get 'chopped up'
|> as it would move through the glass and we would not be able to see a
|> viable image as it would travel out the other side of the transparent
|> material. It is the whole schrodinger field that interacts. In theory
|> you could define it in terms of 'virtual photons' so long as you would
|> remember that they were behaving in coherence with each other, but then
|> they would not be engaging in the probabilistic fashion that we would
|> generally associate with photons. In general, when you consider light
|> as it moves through a transparent medium you are dealing with
|> interactions that are more macroscalar with respect to the light wave,
|> and thus deal with the whole schodinger wave, and thus use equations
|> that reduce to maxwell's equations, rather than localized phenomenon
|> with respect to the wave, in which case the phenomonon would be
|> photonic.
Maxwell's equations describe the transmission of light very well, which
includes the spatial and termporal characteristics of photons; what it
does poorly is describe the interaction of light with matter (paraphrasing
a quote from Born and Wolf's famous optics text). Whether you have to count
states in statistics or the absorption of photons in the photo-electric
effect, at some point you have to recognize the discrete character of the
electromagnetic wave's amplitude. This discrete character is the so-called
``particle'' character of light (far different from what Newton would have
had in mind). Further, this discrete character follows from specific
empirical observations that required DESCRIPTION; testable consequences
follow, whose testing has always come up supported. In other words, the
description relates disparate phenomena in a way that is internally consistent
-- as in the photoelectric effect and black-body radiation.
Dan
|>
|> For more information dealing with the ancient mysteries concerning light
|> I suggest you read 'Scientific American - The Ancient Quandries
|> Concerning Descriptors for Quantum Phenomenon', June, 2075.
|>
|> Maybe someday you humans might even be able to fix my replicators.
|>
|> Quark - for the finest in the Bajoran system, or anywhere.
|>
--
-------------------------------------------------------------------------------
Daniel E. Platt platt@watson.ibm.com
The views expressed here do not necessarily reflect those of my employer.
-------------------------------------------------------------------------------
Subject: Re: Using C for number-crunching (was: Numerical solution to Schrodinger's Eq)
From: shenkin@still3.chem.columbia.edu (Peter Shenkin)
Date: 28 Oct 1996 23:17:01 GMT
In article <551uiv$1hu@ys.ifremer.fr>,
Michel OLAGNON wrote:
>
>subroutine saxpy1 (n, alpha, x, y)
>integer :: n
>real :: alpha
>real, dimension (n) :: x, y
> do i = 1, n
> y (i) = y(i) + alpha * x (i)
> enddo
>end
>The Fortran compiler may unroll the loop and reschedule the instructions
>(and indeed, most of them do) because the standard prohibits that it is
> called as call saxpy1 (10, y(5), y(1:10), y(2:11))
>
>The C compiler may unroll the loop but not reschedule the instructions,
>because the above calling sequence equivalent is permitted.
However, if the C code were written as follows, the compiler would
know that x[] and y[] cannot overlap:
void sizpy1( const int n, const float alpha, const float x[], float y[] ){
int i;
for( i=0; iC
translators know how to add const qualification are separate questions.
-P.
--
****************** In Memoriam, Bill Monroe, 1911 - 1996 ******************
* Peter S. Shenkin, Chemistry, Columbia U., 3000 Broadway, Mail Code 3153,*
** NY, NY 10027; shenkin@columbia.edu; (212)854-5143; FAX: 678-9039 ***
MacroModel WWW page: http://www.cc.columbia.edu/cu/chemistry/mmod/mmod.html
Subject: Fermat's Last Theorem: A High School Algebra Problem
From: James Harris
Date: Mon, 28 Oct 1996 06:44:28 -0800
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I gave a dramatic title in the hopes of increasing the number of people
who will look this over because of curiosity; although, strangely
enough, I've found that statement to be basically true.
I've included the text version and a Write file version.
Some odds and ends.
I state without proof that if (a-b) is divisible by n, then a^n-b^n
must be divisible by n^2. This is obvious since a-b divisible by n
implies that a=jn+r and b=kn+r so you can do the substitution and figure
out the rest in your head.
I also state without proof that (x+y-z)^n is divisible by (z-x), (z-y)
and (x+y). Again, this is also obvious and can be done in your head by
expanding (x+y-z)^n.
James S. Harris, Georgia, USA
------------------------------------------
Introduction.
Fermat's Last Theorem has long been a magnet to the amateur and
professional mathematician alike because of its seeming simplicity; yet,
extraordinary difficulty. Although there is a proof by Andrew Wiles, I
think it is understandable that the problem still would incite
curiosity. I would also assume that a simpler solution would also be of
interest.
Note: The following proof makes extensive use of Fermat's Little
Theorem which isn't usually stated. I also make use of accepted results
which have came up in my previous posts on sci.math without going over
them in detail again.
1. Statement of the Problem: Fermat's Last Theorem
Given x,y,z, relatively prime, n odd prime
no solution exists for the equation x^n + y^n = z^n
2. Proof for Cases where x,y or z are divisible by n.
Let x=af, y=bg, z=ch which means that
x+y=h^n or n^{n-1}h^n, z-x=g^n or n^{n-1}g^n, and z-y=f^n or
n^{n-1}f^n.
For example,
x^n + y^n = (x+y)(x^{n-1}-x^{n-2}y+...+y^{n-1}) = z^n
Since x,y and z are relatively prime, (x+y) can only be divided out once
and the term it is multiplied times can have no factors of (x+y) except
for maybe one n factor.
And,
(x+y-z)^n = n(z-x)(z-y)(x+y)Q
where Q represents all those other terms that are hard to write out for
the general case. For
n=3 it is one. And for n=5
Q = z^2 - (x+y)z + x^2 + xy + y^2
.
Using the above the following is always true.
(x+y-z) = nfghq (Using Q=q^n)
I can then use my earlier relations to write my equation in terms of
f,g,h,q, n only. An intermediates step with z divisible by n is
x + y-z = af - f^n = bg - g^n = n^{n-1}h^n - nch = nfghq
From which I get
f^n + 2nfghq + g^n = n^{n-1}h^n and subtracting f^n + nfgp + g^n =
(f+g^)n
(p used for ease of writing the general case, for example, with n=3,
p=f+g)
gives
nfg(p-2hq) = (f+g)^n - n^{n-1}h^n
p is always divisible by (f+g) because nfgp=(f+g)^n - (f^n + g^n)
Since (f+g) must be divisible by n, because in this case z is divisible
by n, requires that f,g,h or q be divisible by n which creates an
infinite regression similar to the one in Fermat's proof for n=3.
The same comes up with x or y divisible by n since you get
g^n + 2nfghq - h^n = n^{n-1}f^n subtracting
g^n + nghp - h^n = (g-h)^n
gives
ngh(p-2fq) = (g-h)^n - n^{n-1}f^n
which requires that f,g,h or q be divisible by n because (g-h) is
divisible by n which is again a contradiction for the reason given
before.
3. Proof for Case x,y,z not divisible by n
Extension of Fermat's Little Theorem:
Given a-b divisible by n, a^n - b^n must be divisible by n^2
So Fermat's Last Theorem can be written as
Given x,y,z relative prime, none divisible by n; n odd prime
for a solution to exist (x+y)^n - (x^n + y^n) must be divisible by
n^2.
Now notice that x and y can be written in terms of n like
x=jn+r and y=kn+s with r,s < n
If (x+y)^n - (x^n + y^n)
were divisible by n^2 from a substitution it can be seen that this
would require that
(r+s)^n - (r^n + s^n) be divisible by n^2 but we can also write
u+r = c1n, v+s = c2n which still requires that (u+v)^n - (u^n +
v^n) be divisible by n^2 .
But by varying c1 and c2 one can create u's and v's with any desired
modulus with respect to n. Therefore, the above requires that for any
integers a,b
(a+b)^n - (a^n + b^n) must be divisible by n^2
But then I can use a=f, b=g which requires that (f+g)^n - (f^n + g^n)
must be divisible by n^2
(in working out the text version I've noticed that all of this isn't
required because f has the same modulus as x and g the same as y anyway)
Which from before equals nfgp and requires that p be divisible by n
But like before I can write
f^n + 2nfghq + g^n = h^n and again subtract f^n + nfgp + g^n
=(f+g)^n
which gives
nfg(p-2hq) = (f+g)^n - h^n
which conflicts with the requirement that x,y and z be relatively prime,
as before.
It is then seen that (x+y)^n - (x^n + y^n) is not divisible by n^2
which completes the proof of Fermat's Last Theorem.
Here is one example with an alternate proof of the above for one n.
Notice that for n=5, (x+y)^n - (x^n + y^n) equals 5xy(x+y)(x^2 + xy +
y^2)
which would mean that x^2 + xy + y^2 must be divisible by 5. It's easy
enough to see that it can't be just by trying different r's and s'
with (r+s)^n - (r^n + s^n).
James S. Harris
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Introduction.
Fermat's Last Theorem has long been a magnet to the amateur and professional mathematician alike because of its seeming simplicity; yet, extraordinary difficulty. Although there is a proof by Andrew Wiles, I think it is understandable that the problem still would incite curiosity. I would also assume that a simpler solution would also be of interest.
Note: The following proof makes extensive use of Fermat's Little Theorem which isn't usually stated. I also make use of accepted results which have came up in my previous posts on sci.math without going over them in detail again.
1. Statement of the Problem: Fermat's Last Theorem
Given x,y,z, relatively prime, n odd prime
no solution exists for the equation xn + yn = zn
2. Proof for Cases where x,y or z are divisible by n.
Let x=af, y=bg, z=ch means that
x+y=hn or nn-1hn, z-x=gn or
nn-1gn, and z-y=fn or nn-1fn.
For example,
xn + yn = (x+y)(xn-1-xn-2y+...+yn-1) = zn
Since x,y and z are relatively prime, (x+y) can only be divided out once and the term it is multiplied times can have no factors of (x+y) except for maybe one n factor.
And,
(x+y-z)n = n(z-x)(z-y)(x+y)Q
where Q
represents
all those other terms that are hard to write out for the general case. For
n=3 it is one. And for n=5
Q = z2 - (x+y)z + x2 + xy + y2
.
Using the above the following is always true.
(x+y-z) = nfghq (Using Q=qn or nn-1qn)
I can then use my earlier relations to write my equation in terms of f,g,h,q, n only. An intermediates step with z divisible by n is
x + y-z = af - fn = bg - gn = nn-1hn - nch = nfghq
From which I get
fn + 2nfghq + gn = nn-1hn and subtracting fn + nfgp + gn =
(f+g)n
(p used for ease of writing the general case, for example, with n=3, p=f+g)
gives
nfg(p-2hq) = (f+g)n - nn-1hn
p is always divisible by (f+g) because nfgp=(f+g)n - (fn + gn)
Since (f+g) must be divisible by n, because in this case z is divisible by n, requires that f,g,h or q be divisible by n which creates an infinite regression similar to the one in Fermat's proof for n=3.
The same comes up with x or y divisible by n since you get
gn + 2nfghq - hn = nn-1fn subtracting
gn + nghp - hn = (g-h)n
gives
ngh(p-2fq) = (g-h)n - nn-1fn
which requires that f,g,h or q be divisible by n because (g-h) is divisible by n which is again a contradiction for the reason given before.
3. Proof for Case x,y,z not divisible by n
Extension of Fermat's Little Theorem:
Given a-b divisible by n, an - bn must be divisible by n2
So Fermat's Last Theorem can be written as
Given x,y,z relative prime, none divisible by n; n odd prime
for a solution to exist (x+y)n - (xn + yn) must be divisible by n2.
Now notice that x and y can be written in terms of n like
x=jn+r and y=kn+s with r,s < n
If (x+y)n - (xn + yn)
were divisible by n2 from a substitution it can be seen that this would require that
(r+s)n - (rn + sn) be divisible by n2 but we can also write
u+r = c1n, v+s = c2n which still requires that (u+v)n - (un + vn) be divisible by n2 .
But by varying c1 and c2 one can create u's and v's with any desired modulus with respect to n. Therefore, the above requires that for any integers a,b
(a+b)n - (an + bn) must be divisible by n2
But then I can use a=f, b=g which requires that (f+g)n - (fn + gn) must be divisible by n2
Which from before equals nfgp and requires that p be divisible by n
But like before I can write
fn + 2nfghq + gn = hn and again subtract fn + nfgp + gn =
(f+g)n
which gives
nfg(p-2hq) = (f+g)n - hn
which conflicts with the requirement that x,y and z be relatively prime, as before.
It is then seen that (x+y)n - (xn + yn)
is not divisible by n2 which completes the proof of Fermat's Last Theorem.
For those who like the feel of playing with numbers, here is one example with an alternate proof of the above.
Notice that for n=5, (x+y)n - (xn + yn) equals 5xy(x+y)(x2 + xy + y2)
which would mean that x2 + xy + y2 must be divisible by 5. It's easy enough
to see that it can't be just by trying different r's and s' with (r+s)n - (rn + sn).
James S. Harris
--------------4C6E2D5E5392--
Subject: Re: Hermeneutics and the difficulty to count to three...
From: Anton Hutticher
Date: 29 Oct 1996 00:20:54 GMT
moggin@bessel.nando.net (moggin) wrote:
>
> meron@cars3.uchicago.edu (Mati):
>
> >> [...] So, you see, this concept of "wrong" as used by moggin is
> >> not only mot in use in science, it is also not accepted in general
> >> usage. So, where is it valid?
I do seem to have missed your answer to matis question. Where *is*
your concept of "wrong" valid.
> I answered all the questions you put to me, in some cases
> twice. I'm sorry if you don't consider the answers satisfactory.
> I don't think much of your comments, either. That's where things
> end up.
I did consider them evasions, but in light of Silkes difficulty
in understanding what generalization means to a scientist, maybe
they were bona fide. Well, lets try it again:
old post:
Anton
>You did not answer - as far as I know - the question posed earlier
>in the thread ( I think by Silke Weineck): In essence it was: If a
>conductor says:"The bus will arrive at 20h 45min" and the bus
>arrives at 20h 44min 59.9sec, was he right or wrong. (Numbers
>slightly improved).
moggin:
If the conductor is Newton, we need to add some details. Say
you're riding on his bus. As you drive along, he announces the time
that you'll arrive at each stop. The bus reaches the first one within
a second of the time he announced it would be there. It gets to the
second stop within a minute of the time he announced. When it reaches
the third stop, it's five minutes late. At the fourth stop, it's ten
minutes behind. Several stops after that, it's running an hour behind
the conductor's announcements. Then several hours. And so forth, as
it proceeds across country, until it's off by days, weeks, and months.
Is the conductor right or wrong? You could say he's right in
a "limited domain," or that he produces "useful approximations within
certain limits" (that is, the area of the first four or five stops).
But in general, his announcements can only be described as inaccurate.
They begin with a small inaccuracy, dismissable from a practical point
of view, which grows steadily as the bus travels along. And that says
something -- namely, that the theory he's using to produce the times
he announces is false.
Anton:
>I infer from your paragraphs above that you use "right" in the sense of
>"exactly right" and "wrong" as "wrong to the slightest degree" or
>"not exactly right". Only then it makes sense to say that " there's
>no domain where Newton is "right" -- just a range where the errors
>his theory generates are small enough to limit their practical
>consequences", because in science this is exactly the range where
>Newton is right.
moggin:
You misspelled "engineering."
end old post �^^^^^^^ ^^^^^^^^^
and you didn�t answer the question.
As to the above: Is the bus driver right or wrong when he says:
"The bus will arrive at 20h 45min" and the bus arrives at
20h 44min 59.9sec. Thats the question, not : "what shall we do, if
he deviates increasingly from the announced date".
> Obviously you missed my reply to Mati's question, but it
> was brief, so I'll repeat it here: I said, "A better question is,
> how is that an honest account of my position?"
So this is all your answer.
> I have the same question about your most recent reply to
> Silke, where you go on and on about my satanic qualities. I just
> don't recognize myself in your descriptions. The kindest thing I
> can say is that you're arguing with a strawman.
Not exactly: I am arguing with someone who said:
a) Newton was wrong. As wrong as Ptolemy
and
If it so happens that all theories are wrong, well,
they're wrong then
and, when asked what wrong meant in the specific example of a single
announcement of a bus driver proceeded to waffle on something else.
> >I planned to make several posts, but a tooth extraction which took
> >more than an hour made me decide otherwise for a few days.
>
> I'm genuinely sorry to hear that -- I know how painful it
> can be -- dentistry has never fully emerged from the Middle Ages.
> (Watch: somebody's gonna cross-post this to sci.dentistry, and a
> horde of angry dentists are going to come after me with drills.)
>
> -- moggin
>
Thanks, it got much better over the weekend. I had to ask them
explicitely for pain killers for after the extraction and they gave
me three tablets for the first day and night. Which, well, temporarily
inconvienenced me the days (and nights) after. But tomorrow the
sutures are removed and my sarcasm should be as biting as ever.
What I wanted to post in these days was this:
In a previous post I called you "not overly bright".
Did I, by chance, forget to mention that this phrase has a private
meaning, different from the usage of others, *exactly* as with your
usage of "wrong" etc.
You see, bright means: Having one nobel prize
very bright means: Having two nobel prizes
and overly bright means: Having three or more nobel prizes.
Since no person ever has gotten three nobels, you cannot be overly bright.
Did I really forget to mention that? REALLY??? ....----Oops.
You seem to have been less than pleased with a usage of "not overly
bright" which can be misunderstood by others, especially if no
explanation is given. Yet you persist in this with your usage of
"wrong", etc.
You wrote:
(moggin to Matt Silberstein):Yes, it's relevant whether you did or
did not agree with me previously. And in fact you did, stating both
that "Newton's physics was wrong" and "Newton was just incorrect
Obviously you think that he agrees with you when he writes "Newton
is wrong" and you write "Newton is wrong". Putting it to you that
you use wrong in a different way from almost everyone else, especially
those in the "science" camp, seems not to have influenced you in
any way.
Anton Hutticher
(Anton.Hutticher@sbg.ac.at)