Yi Lin. General Systems Theory: A Mathematical Approach
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Some Unsolved Problems in General Systems Theory
365
this kind of application of mathematics has been very successful. However, in
the past 40 years, theoretical physicists have often been faced with difficulties
in mathematical calculation. According to Wang (1991), such difficulties arise
mainly because the modern world of physics is multilevel in its character, while
applicable mathematics still remains single level where the only infinities are
denoted by –
∞ or +
∞
, and are considered in our macroscopic world as ideal
objects and are treated with a different set of rules, compared with those of ordinary
real numbers. Now, in
GNS
the field of all real numbers is isomorphic to
(0)
=
{x ∈ GNS : x
(
t) = 0 if t ≠ 0}, which represents the quantities measurable within
the macroscopic world. Furthermore, all numbers in GNS, whether infinities or
infinitesimals, are treated with the same set of rules.
To give the reader a taste of how GNS can be applied to the study of physics,
we look at the calculation of the rest mass of a photon. According to Einstein, the
inertial mass m of a moving particle relates to its rest mass m
0
as follows:
(12.1)
where c stands for the speed of light and v is the particle speed.
Since the
photon’s speed is v = c, the rest mass m
0
should be m
0
= 0. Otherwise, one
would obtain from Eq. (12.1) that m
=
∞, contradicting observations. However,
even it is assumed that m
0
= 0, injustice involved in the calculation still remains.
This is because in classical mathematics on the real number line, m = 0
÷
0 is
undetermined.
This difficulty can be overcome by using generalized numbers. Suppose the
rest mass m
0
of a photon is
and that the inertial mass m in Eq. (12.1) is a finite real number; that is, m
∈
(0)
has a representation
Now the velocity of a photon is
where v
2
< 0, since c is assumed by Einstein to be the maximum speed in the
universe. So Eq. (12.1) can be written
366
Chapter 12
where
c
is understood in the same way as we did with
m
∈
(0)
. By substituting the
GNS
values of
m
, m
0
, 1
(0)
, v, and c into this equation and comparing the leading
nonzero terms, one obtains (–2
c
)
m
2
v
2
= m
2
1
. That gives the relation between v
2
and m
1
. For technical details, see (Lin and Wang, 1998).
Comparing the generalized number system with those contained in mathemati-
cal analysis, real, and complex analysis, we see that our understanding of the
GNS
is still superficial. More work needs to be carried out to gain deeper understanding
about nature and the world in which we live.
For instance, it is necessary to
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
(11)
Study the ordering relation. It is obvious that the GNS field is not complete
in the sense of Dedekind cuts.
Explore applications of the theory of GNS in other branches of human
knowledge. The future success of the theory of GNS may well depend on
various successful applications.
Develop workable and concrete formulas for the operations of
multiplication, division, and taking roots.
Redevelop quantum mechanics based on GNS so that, hopefully, more
puzzles in the current quantum mechanics can be explained more rationally.
Establish a law, similar to the transfer law in nonstandard analysis
(Davis, 1977), governing the relation between a set of properties of real
numbers and a set of properties of generalized numbers.
Develop the results of generalized numbers, corresponding to theorems in
mathematical analysis, such as the intermediate value theorem, maximum
and minimum theorem, etc.
Find detailed formulas to compute various derivatives, definite and
indefinite integrals of given functions from GNS into GNS.
Generalize the Fundamental Theorem of Calculus to the case of GNS,
linking GNS differential calculus to GNS integral calculus.
Develop an integral calculus of GNS based on the same mechanism of
Lebesgue integrals.
Use GNS representations of Schwartz’s distributions to explore new
explanations of equations, appearing in the theory of distributions, which
do not make sense in classical mathematical analysis.
Employ GNS representations of Schwartz’s distributions to solve open
problems in the theory of distributions, such as defining products of
distributions. Can the theory of GNS be used to explain why in general the
products of distributions might not exist?
Some Unsolved Problems in General Systems Theory
367
12.7. A Few Final Words
To face challenges, the study of general systems theory must further the re-
search of the concept of (general) systems and related topics so that the theory
possesses more systems, more large-scale combination, more pragmatism, more
accuracy, and more unification (blending philosophy, mathematics, physical sci-
ence, technology, etc., into one body of knowledge).
Based on the history of general systems theory and its current development,
among the keys for its future success are publicity and practicalities. Only with
these two tasks being accomplished, will the scientific community be truly moved
so that research of general systems theory and applications moves to a higher level.
Man is both a reflection reduced with the size of history and an enlarged image
of history. He is the creature of interactions between himself and the physical
world. Research and application of general systems theory are a natural extension
of his intelligent and comprehensive pursuits for the past several thousand years.
They are also creatures of interactions between them and the physical world and
reduced reflections and enlarged images of the history of the world of learning.
Any one with a prepared mind can achieve a great deal in the development and
application of such a theory.
The general systems enterprise has been, and will continue to be, developing
and self-completing. In the scientific garden with various theories which will
stand the test of time and history, general systems theory has been set off from the
grayer background, self-realizing, self-coquetting, self-dedicating, self-rebeling
and self-surpassing. Even though the general systems enterprise has not been, and
will not be, pursuing eternity, it will definitely become a powerful force sweeping
through the world of learning.
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