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://members.aol.com/einsteinhoax/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://www.members.aol.com/einsteinhoax/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://www.members.aol.com/einsteinhoax/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 "Gravity" (1987),
"The Einstein Hoax" (1997), and "Corrections to Residual Errors in Special
Relativity (1999) located at http://www.members.aol.com/einsteinhoax/site.htm .
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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provides a significant rebuttal that cannot be objectively answered, the
material at the Website will be withdrawn.

Srettel15 resent the same drivel:
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.)


Linear logic? Area logic? Who does this idiot think he is kidding? To him
linear logic is obviously anything he does not like while area logic is
anything he does like. What a turkey. As any book on logic will tell you
logic is what necessarily follows for the premises by the rules of logic.
We all have a pretty good intuitive ideas what they are and they are not
divided into linear or area.

Sick of reading this crap.
Bill

Bill Hobba wrote:

Linear logic? Area logic? Who does this idiot think he is kidding? To
him
linear logic is obviously anything he does not like while area logic is
anything he does like. What a turkey. As any book on logic will tell
you
logic is what necessarily follows for the premises by the rules of
logic.
We all have a pretty good intuitive ideas what they are and they are not
divided into linear or area.


Robert J. Kolker correctly wrote:
For sure, the rules of logic that scientists and mathematicians use are
well known and quite rigorous in their application. But one thing logic
will NOT tell you, is what to assume in the first place! What are the
sources of fruitful scientific hypotheses? It is not combing over vast
heaps of facts to come up with an inductive hypothesis, although this
sometimes happens. Good theories and hypotheses do not leap out of piles
of act, as did Athena from the Brow of Zeus. The process by which
hypothesis are made was studied by the American logician, philosopher
and mathematician Charles S. Peirce. He called the process abduction
(nothing to do with kidnapping!) as opposed to induction. No, there are
no algorithmic or mathematical rules by which one can come up with the
best hypothesis proposing causes to a given set of effects. This, like
mathematical creativity is based on subconscious processes and
intuition. That is what makes theory formation an art as well as a
discipline.

Here is an example: the equivalence principle, which Einstein called the
happiest thought of his life. It is an abduction which characterizes
gravitation and leads to the hypothesis that gravitation is an effect of
spatial transformation by the presences of mass. A bend in the manifold,
as it were. The is no logical necessity for this hypothesis and others
have proposed covariant theories of gravitation using forces that do
almost as well as GTR. Einstein asked a key -question-. Is there a
difference between being in an elevator (or a uniformly accelerated
vessel in space) and being subject to a uniform gravitational field. The
answer, in the small, locally, as that one cannot determine the
difference. From this question follows the geometry of spacetime. Others
might have asked the question -- how do we modify Newton's law to
account for certain well known gravitational effects, such as the
precession of the perihelion of Mercury, for example.

So there is an area of -thought- that leads to winning hypothesis, that
is not entirely algorithmic or mechanical. But this area is -not-
deductive logic. Deductive logic is still as rigorous and rule based as
ever. There does not appear to be a surefire mechanical method that can
be programmed on a computer, that will produce the best hypotheses. Just
as there is no computerizable algorithmic methed for telling a
mathematician -what theorems- are important. From a purely formal and
logical point of view all theorems are equal, but we know from
experience that some theorems are more important to prove than others,
because in seeking the proof, other areas of mathematics are opened up.
The quest for Fermat's (so-called) Last Theorem (which was really a
conjecture on his part) is a good case in point. In the area of
mathematics, fruitful conjectures are the counterparts to fruitful
hypotheses in physics.


Deduction uses logic - induction uses creativity. Absolutely necessary but
it is not logic.

Thanks
Bill