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DOI: 10.14738/aivp.92.10067
Publication Date: 25
th April, 2021
URL: http://dx.doi.org/10.14738/aivp.92.10067
Research on Basic Scientific Characters of Complexity, and
Their Mathematics and Applications
Yi-Fang Chang
Department of Physics, Yunnan University, Kunming 650091, China
ABSTRACT
Based on the brief introduction of complexity, first we propose four basic scientific
characters of complexity: much elements, various interactions, hierarchies,
evolutions. Next, some corresponding mathematical methods of complexity are
researched. Third, we discuss complexity in physics, and entropy is an important
quantity. Fourth, nonlinearity of complexity is searched. Fifth, we research
complexity in biology and neural networks. Finally, we study general sciences of
complexity from chemistry, social sciences, to various applications. In a word,
complexity is a complex science. Its investigations, developments and applications
must be simplification and quantification.
Key words: complexity, elements, interaction, hierarchy, evolution, mathematics,
physics, nonlinearity, biology, social science.
INTRODUCTION
At present the study of complexity is very important, and is also very complex, and
the complexity may be shown in any field [1]. Even the hypercomplexity is
researched.
In 1973 E. Morin proposed the thinking and paradigm of complexity, which include
three rules: dialogic, cycle and hologram. Prigogine, et al., explored complexity
science [2,3]. Santa Fe Institute studies the science of complexity, and published a
series of lectures from 1989, and becomes the center of world complexity research.
Complexity can be defined in various ways [4] from information, computation [5,6],
network, statistics [7,8], thermodynamic depth [9], entropy [10,11] to effective
complexity [12,13] and philosophy [1], etc. Rescher proposed that the complexity of
systems has some problems [1]: First, it is related to the number of system
components and diversity. Second, the ontological complexity has three main
aspects: the compositional complexity, the structural complexity, the functional
complexity, and list many models of complexity. They are more complex complexity.
Only for the mechanism of complexity, Rescher discussed four theories: intelligent
theory, inherent teleology theory, chance-plus-self-perpetuation theory, and
automatic self-potentiation theory [1]. Mainzer research thinking in complexity, and
try to construct an interdisciplinary general methodology by the computational
dynamics of matter, mind and mankind [14].
We discussed generally the four variables and the eight aspects in social physics, and
searched social thermodynamics and the five fundamental laws of social complex
systems, and proposed the nonlinear whole sociology and its four basic laws [15].
Further, we researched the evolutional equation of system, the educational
equation, the nonlinear theory of economic growth and its three laws: Economic
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European Journal of Applied Sciences, Volume 9 No. 2, April 2021
Services for Science and Education, United Kingdom
takeoff-growth-stagnancy law, social conservation and economic decay law, and
economic growth mode transition and new developed period law. The social open- reform is a necessary and sufficient condition for further economic development
[15]. We proposed the multiply connected topological economics, and four
theorems on knowledge economic theory, etc [16]. In this paper we propose four
basic scientific characters of complexity and research corresponding mathematics
and applications.
BASIC SCIENTIFIC CHARACTERS OF COMPLEXITY
Any systems are determined by their elements and interactions each other.
Therefore, first the complexity has two basic characters:
I. Much elements. They may be the same or be different (diversity). Single
element is simple system theory.
II. Various interactions. This includes internal and external interactions.
Nonlinearity is an important mathematical method, which can also
produce chaos, fractal and so on. Synergetics describes various phase
transitions of nonlinear complex systems by the order parameters and
the slaving principle, etc [17-19].
III. Only I exist, and interaction may be neglected, it corresponds to
statistics. If interactions are simple, it may be crystals, rigid body, etc.
Different interactions may form solids, liquids, Brown movement and so
on.
IV. I and II are two necessary bases of complexity. Two correspond to
element and relation in mathematics. Further, we can derive two
important emerged results:
V. Hierarchies of complexity. A combination between I and II can produce
hierarchies and different structures. Such they will be more complex,
and to complicate (i.e. complexification) [20], and may show emergence,
hypercycle and so on.
VI. Simon defined complexity by the degree of hierarchy [21], whose
important character is emergence [22]. McShea discussed the
hierarchical structure of organisms as a scale and documentation of a
trend in the maximum [23]. In The Emergence of Complexity, Fromm
proposed some outstanding problems, self-organization, evolution,
cooperation, differentiation and so on [24]. Rescher’s first problem is
basis on element and interactions, and corresponds to two basic
characters. Second problem is some results [1].
VII. Complexity with time. If the system is dynamic, it will be from being to
becoming [25]. This not only produces difficulty of prediction, but also
more complex. Bonner searched the evolution of complexity [26].
Flood, et al., discussed complexity [27,28], in which people with power, and system
are all elements in I; nonlinear, non-symmetry, non-holistic constraints, etc., are all
interactions in II; and values, beliefs, and interests are the result of comparison.
This mode agrees with the Occam’s Razor. Four aspects can produce all kinds of
contingency, these are the results. The self-organization is not a general result of
complexity, and it is opposite with chaos.
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Chang, Y. (2021). Research on Basic Scientific Characters of Complexity, and Their Mathematics and Applications.
European Journal of Applied Sciences, 9(2). 302-318.
URL: http://dx.doi.org/10.14738/aivp.92.10067
MATHEMATICS OF COMPLEXITY
From I to IV we can define different complexities. Elements in I may have k types.
Interactions in II may have j types. Hierarchies in III may have i layers. Therefore,
we try to define a mathematical function of complexity
C (e , g , x ,t)
oml k j i
, in which
k j i e , g , x
represent different elements, interactions, and hierarchies of space,
respectively.
The four quantitative methods contribute to the computation of complexity.
Based on I we may use system theory, and set theory and union ∪, intersection ∩,
subser
A B
, difference
B − A
, etc. Discrete elements can be described by matrix
and matrix mechanics. If they are invariant, there is a corresponding conservation
law. General form of the generalized conservation law may be equation [29]:
+ [ , ] = 0
= G W
q
G
dq
dG
. (1)
It represents the quantity G is conservation for generalized coordinate q. These
generalized conservation laws may be represented by four dimensional divergence
forms:
i
i
x
S
=0. (2)
Based on II we may use graph theory, network [30], topology, etc.
Based on III we may use the fractal dimension D [10], which may be extended to
complex dimension in both aspects of mathematics and physics [31]:
Dz
= D + iT . (3)
When the complex dimension is combined with relativity, whose dimensions are
three real spaces and one imaginary time, it expresses a change of the fractal
dimension with time or energy, etc., and exists in the fractal’s description of
meteorology, seismology, medicine and the structure of particle, etc. [32-35].
Based on IV we may discuss dynamics [36], arrows of time and semi-group [37], etc.
Any life system must be related to evolution.
Symmetry principle and its violation are the first basic principles of physics [29].
Symmetries and groups exist probably for elements (I) and interactions (II), for
example, in four basic interactions in physics and their possible unification [38]. But,
they must be violations of symmetries [39,40] for hierarchies (III) and time,
evolution (IV). Symmetry is uniform and beautiful, but symmetry breaking can
produce complexity and evolution.
We may use various corresponding partial differentials:
k
oml
e
C
, j
oml
g
C
, i
oml
x
C
, t
Coml
, (4)