| Topic: |
Science > Physics |
| User: |
"Deeps" |
| Date: |
11 Sep 2005 11:00:26 AM |
| Object: |
"The Limitations of Mathematics in Physics" |
"The Limitations of Mathematics in Physics"
The present philosophy in theoretical physics is to search for
mathematical relationships that predict experimental results that are then
used
to verify the validity of the mathematics in defining our physical Universe,
from the microcosmic to the macrocosmic. The philosophy was expressed
succinctly
by Dr. Hawking in Scientific American when he stated that he didn't care
about
"reality" because he didn't know what "reality" was, all that he cared about
was
whether the mathematics correctly predicted the results of experiment.
Unfortunately, requiring that the results of observation and of mathematics
be
in agreement is NECESSARY BUT NOT SUFFICIENT in our study of the Universe
around
us.
As a research tool, mathematics suffers from two basic
limitations:
A:- It is a linear logic process rather than an area logic
process. Any error along the line of logic being used can easily propagate
throughout the remaining logic chain without giving warning of its presence.
(A
solution using an area logic process, on the other hand, is similar to the
solution of a crossword or jigsaw puzzle. An error made anywhere in such a
process is revealed by an incongruity somewhere in the solution. As such,
such a
logic process is self-correcting because it forces one to go back and
rethink
the problem.)
B:- The logic associated with the mathematical treatment of a
process can lead to results in regions of the problem domain where another
constraint is imposed by Nature. To a mathematically trained physicist such
a
constraint may well go unrecognized because it is not inherent in the
mathematics. The existence of such a constraint may render the mathematical
conclusions involved useless.
The limitation described in paragraph A is illustrated by the
by the derivation of the Lorentz Transformation for Transverse Force. Unlike
the
derivation of the Lorentz Transformation for Parallel Force that provides
the
correct result, the derivation of the Lorentz Transformation for Transverse
Force is incorrect. (The derivation of this transformation is
straightforward
and is easily accomplished using the conventional Lorentz Transformations
for
Mass, Length and Time. The infamous "Right Angle Lever Paradox", see diagram
http://einstenhoax.com/rf511.gif , provided a "common sense"
warning that an error had been made in that mathematical derivation. In this
diagram, if forces are applied at the ends of the right angle lever and the
lever is observed not to rotate in both the "moving" and "stationary"
reference
frames, then the moments applied to each arm must be equal are equal in
magnitude and opposite in direction in each of the reference frames.
With the conventionally accepted values, the product of the
Lorentz Transformation for Parallel Length and the alleged Transformation
for
Parallel Force is not equal to the product of the Lorentz Transformation for
Transverse Length and the alleged Transformation for Parallel Length. This
inequality implies that, if the torques applied to the arms are equal in one
of
the reference frames ("moving" or "stationary") they cannot be equal in the
other reference frame ("stationary " or "moving") and the lever would then
rotate in one of the reference frames and not the other. Since the lever
does
not seem to know of this requirement, it remains stationary, and it was
necessary to reconcile the dilemma that resulted.
One would think that rational men would accept that an error
had been made in the derivation of the force transformations and search for
that
error. Instead, it was assumed that the derivation of the Lorentz
Transformations for the Forces had to be correct, after all they were
derived
mathematically. To account for the difficulty, a convoluted explanation was
devised which asserted that the rate of increase of energy in the transverse
lever produced by the parallel force was balanced by the rate of increase of
the
angular momentum supplied by the torque difference between the two
directions.
The explanation conveniently ignored the fact that for moments to be applied
to
the lever by the forces applied to its ends, equal and opposite forces would
result at the hinge pin. As a result, any energy that is added to the
transverse
lever arm by the applied parallel force at its end will be removed at the
hinge
pin, and the rate of change of energy in the lever is exactly zero.
What was also ignored was the fact that the angular momentum
of the lever is the product of its moment of inertia and its angular
velocity.
Since the lever was observed not to rotate in either reference frames, the
rate
of change of the lever's angular momentum is also zero. Properly done, the
mathematical explanation amounts to the statement that 0 = 0. This is
certainly
true but is also meaningless. It is frightening that the advanced academic
community could overlook such an error and appear in a large number of
postgraduate level texts. It is even more frightening that individuals who
would
embrace such an explanation are in a position to teach our best and
brightest.
For a derivation of the correct Lorentz Transformations for Forces see
http://einsteinhoax.com/relcor.htm ).
The type of error illustrated by paragraph B occurs in regard
to the idea of "action at a distance" being produced by the exchange of
"virtual
particles". Mathematically, such particles are capable of providing both
attractive and repulsive forces. In reality, in the absence of an overlying
substrate for space (e.g.- the classical Aether or Dirac's "Sea of Negative
Energy") with which the "virtual photons" could interchange momentum, the
production by "virtual photons, of attractive forces acting at a distance
would
seem to be impossible. But then, if such a substrate exists, why do we need
the
concept of "virtual photons"?
Perhaps it would be constructive to illustrate this type of
limitation on the reliability of a purely mathematical treatment by
considering
the analysis of a conventional FET transistor circuit, the cascode. This
circuit
consists of two FET transistors with the drain of the first one driving the
source of the second one and with the gate of the second biased at a
constant
positive voltage. For low frequency signals, the operation of such an
amplifier
is completely defined mathematically by a property known as
"transconductance".
The electron flow in the collector of the second FET is the product of the
signal voltage applied to the gate of the first FET times the
transconductance
of the first FET. The mathematics tells us that a positive signal will
produce a
positive flow of electrons at the output and a negative signal will produce
a
negative flow of electrons at the output.
What the mathematics doesn't tell us is that the process will
not work unless there was a bias flow of electrons in the amplifier to which
the
output signal was added. With the bias current, the amplifier can produce
both
positive and negative electron flow signal outputs by adding to or
subtracting
from the bias current. Without the bias current, the amplifier can only
produce
positive electron flow outputs. The mathematics is NECESSARY BUT NOT
SUFFICIENT
to define the process. Similarly, the writer asserts that the mathematics
associated with "virtual photons" is also NECESSARY BUT NOT SUFFICIENT. A
substrate (e.g.- the classical Aether or Dirac's "Sea of Negative Energy" or
one
of the several "background energy of space" concepts) would seem to be
necessary
for the idea of "virtual photons" as a force carrier to work, but, if the
substrate exists, why is the concept of "virtual photons" as force carriers
required?
Whether "virtual photons" exist can be evaluated by a rather
simple experiment as shown in the figure at
http://einsteinhoax.com/cf43.gif. There is an even simpler
means of evaluating the concept. A recent news item revealed that a loose
oxygen
cylinder was captured by the magnetic field of an MIR machine and flew
across
the room to crush the skull of a child being examined. If such a strong
magnetic
field existed in the room and that field involved "virtual photons", those
photons would be sufficiently numerous to produce an amount of
electromagnetic
noise that would be easily measured. I doubt very seriously if such noise
would
be found if s search were made.
Remember that Dr. Einstein warned "we have not proven that the
Aether doesn't exist, we have merely proven we do not need it (for
calculations). Also remember that it took 25 years of peer pressure to cause
Dr.
Einstein to relinquish the idea of "absolute time" (equivalent to believing
the
Aether) and accepting the unproven (and rather ludicrous) consensus
viewpoint of
space and time as a single entity.
Theoretical physics has taken the easy position that only
mathematics and experiment are required to deal with reality. Unlike every
other
branch of science, it is asserted that "mechanism" is irrelevant in physical
theory. Since we live in the "mechanism" represented by our Universe, the
prime
goal of any responsible theoretician should be an understanding of the
workings
of that "mechanism" with the mathematical and experimental results used to
iteratively improve our understanding of that "mechanization". What we have
today, with the idea of "mechanism" eliminated, is more akin to group
quasi-religious charlatanism (how many Angels can dance on the head of pin?)
practiced by a controlling cadre of self promoting "scientists" than it is
to an
organized science.
The source material for this posting may be found in
http://einsteinhoax/hoax.htm ("The Einstein Hoax" {1997});
http://einsteinhoax/gravity.htm; ("Gravity" {1987}); and
http://einsteinhoax/relcor.htm ("Corrections to Special Relativity" {1997}).
EVERYTHING WHICH WE ACCEPT AS TRUE MUST BE CONSISTENT WITH EVERYTHING ELSE
WE HAVE ACCEPTED AS TRUE, IT MUST BE CONSISTENT WITH ALL OBSERVATIONS, AND
IT MUST BE MATHEMATICALLY VIABLE. PRESENT TEACHINGS DO NOT ALWAYS MEET THIS
REQUIREMENT. THE WORLD IS ENTITLED TO A HIGHER STANDARD OF WORKMANSHIP FROM
THOSE IT HAS GRANTED WORLD CLASS STATUS.
All of the Newsposts made by this site may be viewed at
http://einsteinhoax.com/postinglog.htm.
Please make any response via E-mail as Newsgroups are not monitored on
a regular basis. Objective responses will be treated with the same courtesy
as they are presented. To prevent the wastage of time on both of our parts,
please do not raise objections that are not related to material that you
have read at the Website. This posting is merely a summary.
E-mail:-
The material at the Website has been posted continuously for over 5
years. In that time THERE HAVE BEEN NO OBJECTIVE REBUTTALS OF ANY OF THE
MATERIAL PRESENTED. There have only been hand waving arguments by
individuals who have mindlessly accepted the prevailing wisdom without
questioning it. If anyone provides a significant rebuttal that cannot be
objectively answered, the material at the Website will be withdrawn.
Challenges to date have revealed only the responder's inadequacy with one
exception for which a correction was provided.
.
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| User: "Uncle Al" |
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| Title: Re: "The Limitations of Mathematics in Physics" |
11 Sep 2005 06:12:11 PM |
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Deeps wrote:
"The Limitations of Mathematics in Physics"
[snip 240 lines of ignorant crap]
http://www.freefarts.com/farts.html
Move cursor over blinkers to hear Retic's lecture.
Psychotic ineducable boring spammer retic (Ernest Wittke),
You see yourself this way,
http://www.mazepath.com/uncleal/effete6.jpg
The entire remainder of the planet sees you this way,
http://www.mazepath.com/uncleal/effete3.png
http://www.edu-observatory.org/cranks.html
http://www.pagetutor.com/idiot/idiot.html
http://www.mazepath.com/uncleal/sunshine.jpg
<http://www.albinoblacksheep.com/flash/youare.swf>
http://www.fuckinggoogleit.com/
http://www.meninhats.com/d/20040430.html
http://www.you-moron.com/
http://groups.google.com/groups?q=group%3Asci.physics+author%3Awittke
http://www.apa.org/journals/psp/psp7761121.html
http://insti.physics.sunysb.edu/~siegel/quack.html
<http://www.firehead.org/~jessh/film/kubrick/Kubrick-Psycho.html>
<http://www.naturalchild.com/elliott_barker/prisons.html>
The source material for this posting may be found in "Gravity" (1987),
"The Einstein Hoax" (1997), and "Corrections to Residual Errors in Special
Relativity (1999)
[snip]
Hey, stooopid spammer Ernest Wittke - Do you want EVIDENCE? Each of
the 24 GPS satellites carries either four cesium atomic clocks or
three rubidum atomic clocks in orbit, with full relativistic
corrections being applied. NAVSTAR Block II GPS satellites (currently
being launched as replacements) have two rubidium and two cesium
atomic clocks.
Internal inconsistencies in SR (meaning inconsistencies of a purely
mathematical logical nature) automatically lead to contradictions in
number theory, itself, and arithmetic, since the mathematics of
Minkowski geometry is equiconsistent with the theory of real numbers
and with arithmetic.
<http://optoelectronics.perkinelmer.com/content/Datasheets/rfs2f.pdf>
<http://math.ucr.edu/home/baez/RelWWW/tests.html>
Mathematics of gravitation
<http://wugrav.wustl.edu/people/CMW/update98.pdf>
<http://www.astro.northwestern.edu/AspenW04/Papers/lorimer1.pdf>
<http://www.vallis.org/publications/tesidott.pdf>
Equivalence Principle testing
http://arXiv.org/abs/hep-th/0111236
Geometric structure of reality
http://arxiv.org/abs/gr-qc/0103044
http://arXiv.org/abs/hep-th/0307140
GR structure, especially Part 4/p. 7
<http://relativity.livingreviews.org/Articles/lrr-2001-4/index.html>
http://arXiv.org/abs/gr-qc/0311039
<http://www.physics.adelaide.edu.au/~dkoks/Faq/Relativity/SR/experiments.html>
Experimental constraints on General Relativity
<http://tycho.usno.navy.mil/ptti/ptti2002/paper20.pdf>
Nature 425 374 (2003)
http://www.eftaylor.com/pub/projecta.pdf
<http://www.public.asu.edu/~rjjacob/Lecture16.pdf>
<http://relativity.livingreviews.org/Articles/lrr-2003-1/index.html>
Relativity in the GPS system
http://arXiv.org/abs/gr-qc/9909014
Amer. J. Phys. 71 770 (2003)
Phys. Rev. Lett. 92 121101 (2004)
falling light
<http://hyperphysics.phy-astr.gsu.edu/hbase/relativ/airtim.html>
<http://metrologyforum.tm.agilent.com/pdf/flying_clock_math.pdf>
http://metrologyforum.tm.agilent.com/cesium.shtml
http://arxiv.org/abs/physics/0008012
Hafele-Keating Experiment
http://www.hawaii.edu/suremath/SRtwinParadox.html
<http://physics.syr.edu/courses/modules/LIGHTCONE/twins.html>
Twin Paradox
Science 303(5661) 1143;1153 (2004)
http://arXiv.org/abs/astro-ph/0401086
http://arxiv.org/abs/astro-ph/0312071
<http://relativity.livingreviews.org/Articles/lrr-2003-5/index.html>
<http://skyandtelescope.com/news/article_1473_1.asp>
Deeply relativistic neutron star binaries
http://arxiv.org/abs/hep-th/0405160
Black hole evaporation
Physics Today 57(7) 40 (2004)
http://physicstoday.org/vol-57/iss-7/p40.shtml
No aether
http://fsweb.berry.edu/academic/mans/clane/
http://physicsweb.org/articles/world/17/3/7
No Lorentz violation
http://arXiv.org/abs/gr-qc/0409089
Spin-2 gravitons have problems
<http://groups-beta.google.com/group/sci.physics.strings/msg/ba31a00f5f26277a>
(so does the proposal)
http://arXiv.org/abs/gr-qc/0411113
<http://www.npl.washington.edu/eotwash/pdf/prl83-3585.pdf>
http://arXiv.org/abs/gr-qc/0301024
Phys. Rev. Lett. 93 261101 (2004)
Nordtvedt Effect
http://map.gsfc.nasa.gov/
http://arxiv.org/abs/astro-ph/0403292
http://arXiv.org/abs/astro-ph/0310723
WMAP + Sloane Digital Sky Survey
http://arxiv.org/abs/hep-ph/0404175
Dark matter candidates
<http://nedwww.ipac.caltech.edu/level5/March01/Carroll/frames.html>
Carroll on what it all means.
Special Relativity is physics on a topologically trivial Lorentzian
manifold with a metric whose curvature tensor is zero. This is a
perfectly diffeomorphism-invariant condition and does not require
any particular coordinate choice. It is invariant under
the full group of diffeomorphisms. The Poincare group is
the group of *isometries* of the metric in special relativity.
The Special Relativity metric is *non-dynamical* (unlike GR). It
defines the coupling *constants* of your theory. If you change the
metric in any nontrivial way you are changing your theory. An
operation can only be called a "symmetry" of a special-relativistic
(non-gravitational) theory if it preserves the metric, and therefore
the symmetry of special-relativistic theories is the Poincare group
only. General Relativity (gravitation) has a dynamic metric.
NIM A 355 537 (1995)
Physics Letters B 328 103 (1994)
Physical Review Letters 64 1697 (1990)
Physical Review Letters 39 1051 (1977)
Physical Review 135 B1071 (1964)
Physics Letters 12 260 (1964)
Europhysics Letters 56(2) 170-174 (2001)
General Relativity and Gravitation 34(9) 1371 (2002)
http://fourmilab.to/etexts/einstein/specrel/specrel.pdf
<http://www.geocities.com/physics_world/sr/ae_1905_error.htm>
<http://www.physics.gatech.edu/people/faculty/finkelstein/relativity.pdf>
Longitudinal and transverse mass
Physics Today 58(3) 34 (2005)
Time passage, equator vs. poles
http://arxiv.org/abs/gr-qc/0306076
<http://www.metaresearch.org/solar%20system/gps/absolute-gps-1meter-3.ASP>
http://www.navcen.uscg.gov/pubs/gps/gpsuser/gpsuser.pdf
http://www.navcen.uscg.gov/pubs/gps/sigspec/default.htm
http://www.navcen.uscg.gov/pubs/gps/icd200/default.htm
http://www.trimble.com/gps/index.html
http://sirius.chinalake.navy.mil/satpred/
http://www.phys.lsu.edu/mog/mog9/node9.html
http://egtphysics.net/GPS/RelGPS.htm
http://www.schriever.af.mil/gps/Current/current.oa1
http://edu-observatory.org/gps/gps_books.html
<http://www-astronomy.mps.ohio-state.edu/~pogge/Ast162/Unit5/gps.html>
If anyone
provides a significant rebuttal that cannot be objectively answered, the
material at the Website will be withdrawn.
Right, like your head has ever been withdrawn from your ***** - even
when you *****.
--
Uncle Al
http://www.mazepath.com/uncleal/
(Toxic URL! Unsafe for children and most mammals)
http://www.mazepath.com/uncleal/qz.pdf
.
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| User: "Sam Wormley" |
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| Title: Re: "The Limitations of Mathematics in Physics" |
11 Sep 2005 11:22:49 AM |
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Deeps wrote:
"The Limitations of Mathematics in Physics"
Thanks for registering "The Einstein Hoax" at crank dot net.
http://www.google.com/search?q=einstein+hoax+site%3Awww.crank.net
.
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| User: "Orion" |
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| Title: Re: "The Limitations of Mathematics in Physics" |
11 Sep 2005 04:00:46 PM |
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The limitations are in number theory. Numbers are all identical.
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| User: "Robert J. Kolker" |
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| Title: Re: "The Limitations of Mathematics in Physics" |
11 Sep 2005 04:59:53 PM |
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Orion wrote:
The limitations are in number theory. Numbers are all identical.
are you saying 1 = 2?
Bob Kolker
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| User: "Orion" |
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| Title: Re: "The Limitations of Mathematics in Physics" |
11 Sep 2005 05:31:41 PM |
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All numbers are composed of 1s.
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| User: "Sam Wormley" |
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| Title: Re: "The Limitations of Mathematics in Physics" |
11 Sep 2005 05:47:48 PM |
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Orion wrote:
All numbers are composed of 1s.
Are you saying the number pi is composed of 1s?
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| User: "Orion" |
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| Title: Re: "The Limitations of Mathematics in Physics" |
11 Sep 2005 05:57:09 PM |
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I was talking about whole numbers.
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| User: "Sam Wormley" |
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| Title: Re: "The Limitations of Mathematics in Physics" |
11 Sep 2005 06:15:03 PM |
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Orion wrote:
All numbers are composed of 1s.
Orion wrote:
I was talking about whole numbers.
Actually you said "all numbers"... now you are saying that
All [whole] numbers are composed of 1s--Is that right?
.
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| User: "Al Zenner" |
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| Title: Re: "The Limitations of Mathematics in Physics" |
11 Sep 2005 06:36:12 PM |
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Sam Wormley <swormley1@mchsi.com> wrote in news:ou2Ve.323568$x96.300369
@attbi_s72:
Orion wrote:
All numbers are composed of 1s.
Are you saying the number pi is composed of 1s?
That's funny. I never think of pi as a number.
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| User: "Sam Wormley" |
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| Title: Re: "The Limitations of Mathematics in Physics" |
11 Sep 2005 06:49:33 PM |
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Al Zenner wrote:
That's funny. I never think of pi as a number.
Pi is an excellent example of an Irrational Number
http://mathworld.wolfram.com/IrrationalNumber.html
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| User: "Orion" |
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| Title: Re: "The Limitations of Mathematics in Physics" |
11 Sep 2005 07:04:38 PM |
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His point is that math has limitations but that's because we have
limitations.
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| User: "Sam Wormley" |
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| Title: Re: "The Limitations of Mathematics in Physics" |
11 Sep 2005 07:08:54 PM |
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Orion wrote:
His point is that math has limitations but that's because we have
limitations.
Orion wrote:
All numbers are composed of 1s.
Orion wrote:
I was talking about whole numbers.
Actually you said "all numbers"... now you are saying that
All [whole] numbers are composed of 1s--Is that right?
.
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| User: "Orion" |
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| Title: Re: "The Limitations of Mathematics in Physics" |
11 Sep 2005 07:17:47 PM |
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Sam Wormley what's your point?
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| User: "Al Zenner" |
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| Title: Re: "The Limitations of Mathematics in Physics" |
11 Sep 2005 07:39:10 PM |
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Sam Wormley <swormley1@mchsi.com> wrote in news:ho3Ve.323635$x96.289085
@attbi_s72:
Al Zenner wrote:
That's funny. I never think of pi as a number.
Pi is an excellent example of an Irrational Number
http://mathworld.wolfram.com/IrrationalNumber.html
Of course the numeric representation of pi is irrational.
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| User: "The Ghost In The Machine" |
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| Title: Re: "The Limitations of Mathematics in Physics" |
12 Sep 2005 07:00:06 AM |
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In sci.physics, Al Zenner
<azen@zenner.com>
wrote
on Sun, 11 Sep 2005 20:39:10 -0400
<Xns96CEC7C24E8ADzsp@63.223.5.248>:
Sam Wormley <swormley1@mchsi.com> wrote in news:ho3Ve.323635$x96.289085
@attbi_s72:
Al Zenner wrote:
That's funny. I never think of pi as a number.
Pi is an excellent example of an Irrational Number
http://mathworld.wolfram.com/IrrationalNumber.html
Of course the numeric representation of pi is irrational.
OK, dumb question: what other representations of pi are there?
- Number: pi
- Decimal representation: 3.141592653589... well never mind, one
will never finish it
- Number definition: ratio of circumference to diameter of a circle
in a flat plane
- Number construction: there are many ways to "construct" pi,
or at least its decimal representation; some of them are detailed
at http://mathworld.wolfram.com/PiFormulas.html
- Number characteristics: irrational, transcendental,
normality unknown, computable
- Number meaning: That's well outside physics; might as well try
to ascribe meaning to the number of stars in the Universe, or
the number of mosquitoes chasing ladders.
--
#191,
It's still legal to go .sigless.
.
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| User: "" |
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| Title: Re: "The Limitations of Mathematics in Physics" |
12 Sep 2005 07:11:44 AM |
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In article <k54gv2-mc7.ln1@sirius.tg00suus7038.net>,
The Ghost In The Machine <ewill@sirius.tg00suus7038.net> wrote:
In sci.physics, Al Zenner
<azen@zenner.com>
wrote
on Sun, 11 Sep 2005 20:39:10 -0400
<Xns96CEC7C24E8ADzsp@63.223.5.248>:
Sam Wormley <swormley1@mchsi.com> wrote in news:ho3Ve.323635$x96.289085
@attbi_s72:
Al Zenner wrote:
That's funny. I never think of pi as a number.
Pi is an excellent example of an Irrational Number
http://mathworld.wolfram.com/IrrationalNumber.html
Of course the numeric representation of pi is irrational.
OK, dumb question: what other representations of pi are there?
- Number: pi
- Decimal representation: 3.141592653589... well never mind, one
will never finish it
- Number definition: ratio of circumference to diameter of a circle
in a flat plane
- Number construction: there are many ways to "construct" pi,
or at least its decimal representation; some of them are detailed
at http://mathworld.wolfram.com/PiFormulas.html
- Number characteristics: irrational, transcendental,
normality unknown, computable
What do you mean by computable?
<snip>
/BAH
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| User: "The Ghost In The Machine" |
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| Title: Re: "The Limitations of Mathematics in Physics" |
13 Sep 2005 12:00:07 AM |
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In sci.physics,
<>
wrote
on Mon, 12 Sep 05 12:11:44 GMT
<dg3ra0$8qk_002@s960.apx1.sbo.ma.dialup.rcn.com>:
In article <k54gv2-mc7.ln1@sirius.tg00suus7038.net>,
The Ghost In The Machine <ewill@sirius.tg00suus7038.net> wrote:
In sci.physics, Al Zenner
<azen@zenner.com>
wrote
on Sun, 11 Sep 2005 20:39:10 -0400
<Xns96CEC7C24E8ADzsp@63.223.5.248>:
Sam Wormley <swormley1@mchsi.com> wrote in news:ho3Ve.323635$x96.289085
@attbi_s72:
Al Zenner wrote:
That's funny. I never think of pi as a number.
Pi is an excellent example of an Irrational Number
http://mathworld.wolfram.com/IrrationalNumber.html
Of course the numeric representation of pi is irrational.
OK, dumb question: what other representations of pi are there?
- Number: pi
- Decimal representation: 3.141592653589... well never mind, one
will never finish it
- Number definition: ratio of circumference to diameter of a circle
in a flat plane
- Number construction: there are many ways to "construct" pi,
or at least its decimal representation; some of them are detailed
at http://mathworld.wolfram.com/PiFormulas.html
- Number characteristics: irrational, transcendental,
normality unknown, computable
What do you mean by computable?
Good question. Best answer I can give is that, given sufficient time
and memory, one can approximate pi to any given degree, using
an algorithm (pick one; the page above gives some formulae as
starting points).
AIUI, a new processor is routinely tested by computing pi to a
billion or so places, just to see if it works properly.
<snip>
/BAH
--
#191,
It's still legal to go .sigless.
.
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| User: "Androcles Androcles@ MyPlace.org" |
|
| Title: Re: "The Limitations of Mathematics in Physics" |
13 Sep 2005 02:51:16 PM |
|
|
"The Ghost In The Machine" <ewill@sirius.tg00suus7038.net> wrote in
message news:tashv2-6gc.ln1@sirius.tg00suus7038.net...
| In sci.physics,
| <>
| wrote
| on Mon, 12 Sep 05 12:11:44 GMT
| <dg3ra0$8qk_002@s960.apx1.sbo.ma.dialup.rcn.com>:
| > In article <k54gv2-mc7.ln1@sirius.tg00suus7038.net>,
| > The Ghost In The Machine <ewill@sirius.tg00suus7038.net> wrote:
| >>In sci.physics, Al Zenner
| >><azen@zenner.com>
| >> wrote
| >>on Sun, 11 Sep 2005 20:39:10 -0400
| >><Xns96CEC7C24E8ADzsp@63.223.5.248>:
| >>> Sam Wormley <swormley1@mchsi.com> wrote in
news:ho3Ve.323635$x96.289085
| >>> @attbi_s72:
| >>>
| >>>> Al Zenner wrote:
| >>>
| >>>>> That's funny. I never think of pi as a number.
| >>>
| >>>> Pi is an excellent example of an Irrational Number
| >>>> http://mathworld.wolfram.com/IrrationalNumber.html
| >>>
| >>> Of course the numeric representation of pi is irrational.
| >>>
| >>
| >>OK, dumb question: what other representations of pi are there?
| >>
| >>- Number: pi
| >>- Decimal representation: 3.141592653589... well never mind, one
| >> will never finish it
| >>- Number definition: ratio of circumference to diameter of a circle
| >> in a flat plane
| >>- Number construction: there are many ways to "construct" pi,
| >> or at least its decimal representation; some of them are detailed
| >> at http://mathworld.wolfram.com/PiFormulas.html
| >>- Number characteristics: irrational, transcendental,
| >> normality unknown, computable
| >
| > What do you mean by computable?
|
| Good question. Best answer I can give is that, given sufficient time
| and memory, one can approximate pi to any given degree, using
| an algorithm (pick one; the page above gives some formulae as
| starting points).
| AIUI, a new processor is routinely tested by computing pi to a
| billion or so places, just to see if it works properly.
Roll a beer can across my desk, the distance moved in exactly one
revolution divided by the diameter of the can is EXACTLY pi to
whatever accuracy you can measure how far it rolled.
After I drink the beer and I'll roll the can back for you to check.
Androcles
.
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|
| User: "The Ghost In The Machine" |
|
| Title: Re: "The Limitations of Mathematics in Physics" |
13 Sep 2005 11:00:04 PM |
|
|
In sci.physics, Androcles
<Androcles@MyPlace.org>
wrote
on Tue, 13 Sep 2005 19:51:16 GMT
<U4GVe.25826$k22.11490@fe2.news.blueyonder.co.uk>:
"The Ghost In The Machine" <ewill@sirius.tg00suus7038.net> wrote in
message news:tashv2-6gc.ln1@sirius.tg00suus7038.net...
| In sci.physics,
| <>
| wrote
| on Mon, 12 Sep 05 12:11:44 GMT
| <dg3ra0$8qk_002@s960.apx1.sbo.ma.dialup.rcn.com>:
| > In article <k54gv2-mc7.ln1@sirius.tg00suus7038.net>,
| > The Ghost In The Machine <ewill@sirius.tg00suus7038.net> wrote:
| >>In sci.physics, Al Zenner
| >><azen@zenner.com>
| >> wrote
| >>on Sun, 11 Sep 2005 20:39:10 -0400
| >><Xns96CEC7C24E8ADzsp@63.223.5.248>:
| >>> Sam Wormley <swormley1@mchsi.com> wrote in
news:ho3Ve.323635$x96.289085
| >>> @attbi_s72:
| >>>
| >>>> Al Zenner wrote:
| >>>
| >>>>> That's funny. I never think of pi as a number.
| >>>
| >>>> Pi is an excellent example of an Irrational Number
| >>>> http://mathworld.wolfram.com/IrrationalNumber.html
| >>>
| >>> Of course the numeric representation of pi is irrational.
| >>>
| >>
| >>OK, dumb question: what other representations of pi are there?
| >>
| >>- Number: pi
| >>- Decimal representation: 3.141592653589... well never mind, one
| >> will never finish it
| >>- Number definition: ratio of circumference to diameter of a circle
| >> in a flat plane
| >>- Number construction: there are many ways to "construct" pi,
| >> or at least its decimal representation; some of them are detailed
| >> at http://mathworld.wolfram.com/PiFormulas.html
| >>- Number characteristics: irrational, transcendental,
| >> normality unknown, computable
| >
| > What do you mean by computable?
|
| Good question. Best answer I can give is that, given sufficient time
| and memory, one can approximate pi to any given degree, using
| an algorithm (pick one; the page above gives some formulae as
| starting points).
| AIUI, a new processor is routinely tested by computing pi to a
| billion or so places, just to see if it works properly.
Roll a beer can across my desk, the distance moved in exactly one
revolution divided by the diameter of the can is EXACTLY pi to
whatever accuracy you can measure how far it rolled.
After I drink the beer and I'll roll the can back for you to check.
Androcles
Define "exactly". I don't do beer but I do have some empty aluminum
cans lying about; I measure approximately 64 mm. Assuming 64 mm
exactly and with no regard at all for precision, that translates
into a diameter of 223541739434.1599 atoms -- OK, call it
223541739434; we'll just shave off about 4/25ths of an atomic shell.
Around the circumference we might get 702277086377 atoms plus
maybe 1/25th extra, give or take a quark or two.
So are you willing to claim that 702277086377.04 / 223541739434.16
is exactly equal to pi?
For all intents and purposes the precision of this approximation is
such that the error is at most a part in a trillion; in a roadway
that has a diameter of 1000 km that will translate into an error of
about two wavelengths of visible light. In an orbit of 1 AU
(1.5 *10^11 m) that translates into 15 cm. Close enough for most work.
But it's not exact, and never will be, as pi is transcendental.
--
#191,
It's still legal to go .sigless.
.
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|
| User: "" |
|
| Title: Re: "The Limitations of Mathematics in Physics" |
13 Sep 2005 04:24:37 AM |
|
|
In article <tashv2-6gc.ln1@sirius.tg00suus7038.net>,
The Ghost In The Machine <ewill@sirius.tg00suus7038.net> wrote:
In sci.physics,
<>
wrote
on Mon, 12 Sep 05 12:11:44 GMT
<dg3ra0$8qk_002@s960.apx1.sbo.ma.dialup.rcn.com>:
In article <k54gv2-mc7.ln1@sirius.tg00suus7038.net>,
The Ghost In The Machine <ewill@sirius.tg00suus7038.net> wrote:
In sci.physics, Al Zenner
<azen@zenner.com>
wrote
on Sun, 11 Sep 2005 20:39:10 -0400
<Xns96CEC7C24E8ADzsp@63.223.5.248>:
Sam Wormley <swormley1@mchsi.com> wrote in news:ho3Ve.323635$x96.289085
@attbi_s72:
Al Zenner wrote:
That's funny. I never think of pi as a number.
Pi is an excellent example of an Irrational Number
http://mathworld.wolfram.com/IrrationalNumber.html
Of course the numeric representation of pi is irrational.
OK, dumb question: what other representations of pi are there?
- Number: pi
- Decimal representation: 3.141592653589... well never mind, one
will never finish it
- Number definition: ratio of circumference to diameter of a circle
in a flat plane
- Number construction: there are many ways to "construct" pi,
or at least its decimal representation; some of them are detailed
at http://mathworld.wolfram.com/PiFormulas.html
- Number characteristics: irrational, transcendental,
normality unknown, computable
What do you mean by computable?
Good question. Best answer I can give is that, given sufficient time
and memory, one can approximate pi to any given degree, using
an algorithm (pick one; the page above gives some formulae as
starting points).
But that doesn't retain the value in memory so that it can
be used in calculations.
AIUI, a new processor is routinely tested by computing pi to a
billion or so places, just to see if it works properly.
It's a calisthenics style of code, common and reproducible.
There are a whole suite of these for any sanity check.
I was just curious about the use of the word, computable.
/BAH
.
|
|
|
| User: "Al Zenner" |
|
| Title: Re: "The Limitations of Mathematics in Physics" |
18 Sep 2005 10:40:11 AM |
|
|
wrote in
news:dg65sl$8ss_002@s768.apx1.sbo.ma.dialup.rcn.com:
In article <tashv2-6gc.ln1@sirius.tg00suus7038.net>,
The Ghost In The Machine <ewill@sirius.tg00suus7038.net> wrote:
In sci.physics,
<>
wrote
on Mon, 12 Sep 05 12:11:44 GMT
<dg3ra0$8qk_002@s960.apx1.sbo.ma.dialup.rcn.com>:
In article <k54gv2-mc7.ln1@sirius.tg00suus7038.net>,
The Ghost In The Machine <ewill@sirius.tg00suus7038.net> wrote:
In sci.physics, Al Zenner
<azen@zenner.com>
wrote
on Sun, 11 Sep 2005 20:39:10 -0400
<Xns96CEC7C24E8ADzsp@63.223.5.248>:
Sam Wormley <swormley1@mchsi.com> wrote in
news:ho3Ve.323635$x96.289085 @attbi_s72:
Al Zenner wrote:
That's funny. I never think of pi as a number.
Pi is an excellent example of an Irrational Number
http://mathworld.wolfram.com/IrrationalNumber.html
Of course the numeric representation of pi is irrational.
OK, dumb question: what other representations of pi are there?
- Number: pi
- Decimal representation: 3.141592653589... well never mind, one
will never finish it
- Number definition: ratio of circumference to diameter of a circle
in a flat plane
- Number construction: there are many ways to "construct" pi,
or at least its decimal representation; some of them are detailed
at http://mathworld.wolfram.com/PiFormulas.html
- Number characteristics: irrational, transcendental,
normality unknown, computable
What do you mean by computable?
Good question. Best answer I can give is that, given sufficient time
and memory, one can approximate pi to any given degree, using
an algorithm (pick one; the page above gives some formulae as
starting points).
But that doesn't retain the value in memory so that it can
be used in calculations.
AIUI, a new processor is routinely tested by computing pi to a
billion or so places, just to see if it works properly.
It's a calisthenics style of code, common and reproducible.
There are a whole suite of these for any sanity check.
I was just curious about the use of the word, computable.
You homed in on the central issue. Pi is not a number, it is a natural
ratio which can be represented as a more or less inaccurate (because
it is an irrational) number. Ghost had the *ratio* definition included
in his discussion but skirted by it because he called it a number
definition.
Numbers seem to occasionally capture our attention to the exclusion of
their underpinnings. Pi is notorious for that. You asked, IMO, an
excellent question. Growing out of this is an even more fascinating
discussion. How many real physical circles have we discovered in nature?
Anticipating that answer, what makes pi so important in the first place?
.
|
|
|
|
| User: "The Ghost In The Machine" |
|
| Title: Re: "The Limitations of Mathematics in Physics" |
13 Sep 2005 11:00:04 PM |
|
|
In sci.physics,
<>
wrote
on Tue, 13 Sep 05 09:24:37 GMT
<dg65sl$8ss_002@s768.apx1.sbo.ma.dialup.rcn.com>:
In article <tashv2-6gc.ln1@sirius.tg00suus7038.net>,
The Ghost In The Machine <ewill@sirius.tg00suus7038.net> wrote:
In sci.physics,
<>
wrote
on Mon, 12 Sep 05 12:11:44 GMT
<dg3ra0$8qk_002@s960.apx1.sbo.ma.dialup.rcn.com>:
In article <k54gv2-mc7.ln1@sirius.tg00suus7038.net>,
The Ghost In The Machine <ewill@sirius.tg00suus7038.net> wrote:
In sci.physics, Al Zenner
<azen@zenner.com>
wrote
on Sun, 11 Sep 2005 20:39:10 -0400
<Xns96CEC7C24E8ADzsp@63.223.5.248>:
Sam Wormley <swormley1@mchsi.com> wrote in news:ho3Ve.323635$x96.289085
@attbi_s72:
Al Zenner wrote:
That's funny. I never think of pi as a number.
Pi is an excellent example of an Irrational Number
http://mathworld.wolfram.com/IrrationalNumber.html
Of course the numeric representation of pi is irrational.
OK, dumb question: what other representations of pi are there?
- Number: pi
- Decimal representation: 3.141592653589... well never mind, one
will never finish it
- Number definition: ratio of circumference to diameter of a circle
in a flat plane
- Number construction: there are many ways to "construct" pi,
or at least its decimal representation; some of them are detailed
at http://mathworld.wolfram.com/PiFormulas.html
- Number characteristics: irrational, transcendental,
normality unknown, computable
What do you mean by computable?
Good question. Best answer I can give is that, given sufficient time
and memory, one can approximate pi to any given degree, using
an algorithm (pick one; the page above gives some formulae as
starting points).
But that doesn't retain the value in memory so that it can
be used in calculations.
I'd have to look, and in any event my brain got pi stuck in it to
about 20 decimal places long ago: 3.1415926535897932 just off
the tips of my fingers -- or some such. But 20 decimal places
would work out the Earth's orbit to a few atomic diameters and is
overkill for everything else. :-)
AIUI, a new processor is routinely tested by computing pi to a
billion or so places, just to see if it works properly.
It's a calisthenics style of code, common and reproducible.
There are a whole suite of these for any sanity check.
I was just curious about the use of the word, computable.
I'm wondering about that myself. My thinking is as follows:
- finite decimal expansion: 12.3456
- exactly representable: 22/7
- indefinitely approximatable: pi, e, various other constants
Of course one can cheat here a bit; 22/7 = 3.1 base 7.
and then one gets into territory inhabited by things such as
"the mots boring number" and Chaitin's Omega.
/BAH
--
#191,
It's still legal to go .sigless.
.
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