The Limitations of Mathematics in Physics

"Great spirits have always encountered violent oppositions from mediocre
minds." - A. Einstein

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 ,
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
).

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
. 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 "Gravity" (1987),
"The Einstein Hoax" (1997), and "Corrections to Residual Errors in Special
Relativity (1999) located at
.
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.

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exception for which a correction was provided.

Oidua wrote:
The Limitations of Mathematics in Physics

"Great spirits have always encountered violent oppositions from mediocre
minds." - A. Einstein

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.

You must state the goals of your study before denying sufficiency.
It is *sufficient* for prediction i.e. extrapolation of past behavior
into the future. It may not be sufficient to satisfy a person's
asthetic sense, or emotional need to believe he "really knows what
is underneath it all". But this knowledge must come down to
faith as it by definition extends beyond what can be empirically
confirmed. It is metaphysics as in "beyond physics". Play that
game if you enjoy it but don't try to call it *physics*.



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.

But an error along the line of logic means you've stepped outside the
mathematics into random noise. It is not a flaw or limit to mathematic
rather a flaw in someone attempting mathematical deduction.

I would rather say that the limitation of mathematics is that it
can only rewrite your axioms and definitions in different form.
It cannot say anything about anything by itself.


(A solution using an area logic process, on the other hand, is similar to the solution
of a crossword or jigsaw puzzle.

You seem to confuse mathematics with the congnative process of
mathematicians. Just because one proof is sufficient doesn't many
many alternative proofs are not likewise possible in the same
standard mathematical system. Mathematicians have long found
that alternative chains of deduction for a given theorem is
a very good means to assure no mistake in reasoning has occured.
But mathematics is just rules to an abstract game.

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.

Yes but the whole point of translating scientific phenomena into
mathematical language is that one can re-use well tested chains
of reasoning. The mathematics is a logic template. Nature doesn't
impose constraints on mathematics.

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.

If the physicist is properly trained in mathematics he will distinguish
mathematical inferences from physical ones. It is the poorly trained
speculator in physics who thinks his 'reality model' is anything but
a mathematical construct.

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.

And heaven forbid anyone should pay attention to the predictions instead
of their own opinion about the asthetics of their method of
construction. Let's worry about whether the scaffolding used to
construct a building is as durable as the building needs to be.

Regards,
James Baugh

Well, let me rephrase this. "I found out that despite all my efforts I
coouldn't learn the mathematics necessary to do physics. Therefore, I
am going to declare all mathematics unnecessary."

This sort of attitude is absolutely standard on this board. Always has
been, at least ever since I rememeber (I started posting to the Usenet
in 1985 - at that time newsgroups were naturally idiot-free. The good
old days.)

Jan Bielawski

"Jan Bielawski" wrote in message
ups.com...
Well, let me rephrase this. "I found out that despite all my efforts I
coouldn't learn the mathematics necessary to do physics. Therefore, I
am going to declare all mathematics unnecessary."

The trouble is they do not seem to even be aware they do not understand the
math. A number claim they understand concepts like Hilbert spaces etc and
yet are unable to elucidate even the most basic of facts. The most you get
out of them is a cut and paste from a quick internet search - ask them
something that can not be easily looked up and they are flummoxed. The
usual response is to resort to the old crank standby of ignoring it,
claiming you have no idea what you are talking about or like this idiot
Heretic simply continue to post the same tripe.


This sort of attitude is absolutely standard on this board.


Too true.

Always has
been, at least ever since I rememeber (I started posting to the Usenet
in 1985 - at that time newsgroups were naturally idiot-free. The good
old days.)

I wish I was around then. But hey at least they make life interesting in a
weird sort of way. And like Baugh's response the different tacks people
take in refuting them makes interesting reading.

Thanks
Bill



Jan Bielawski