Mathematics on NonMathematics
—– A Combinatorial Contribution
Linfan MAO
(Chinese Academy of Mathematics and System Science, Beijing 100190, P.R.China)Email: maolinfan@163.com
Abstract
:
A classical system of mathematics is homogenous without contradictions. But it is a little ambiguous for modern mathematics, for instance,the Smarandache geometry. Let
F
be a family of things such as those of particles or organizations. Then,
how to hold its global behaviors or true face
?Generally,
F
is not a mathematical system in usual unless a set, i.e., a systemwith contradictions. There are no mathematical subﬁelds applicable. Indeed,the trend of mathematical developing in 20th century shows that a mathematical system is more concise, its conclusion is more extended, but farther tothe true face for its abandoned more characters of things. This eﬀect impliesan important step should be taken for mathematical development, i.e., turnthe way to extending nonmathematics in classical to mathematics, which alsobe provided with the philosophy. All of us know
there always exist universal connection between things in
F
. Thus there is an underlying structure, i.e.,a vertexedge labeled graph
G
for things in
F
. Such a labeled graph
G
isinvariant accompanied with
F
. The main purpose of this paper is to showhow to extend classical mathematical nonsystems, such as those of algebraicsystems with contradictions, algebraic or diﬀerential equations with contradictions, geometries with contradictions, and generally, classical mathematicssystems with contradictions to mathematics by the underlying structure
G
.All of these discussions show that a nonmathematics in classical is in fact amathematics underlying a topological structure
G
, i.e., mathematical combinatorics, and contribute more to physics and other sciences.
Key Words
:
Nonmathematics, topological graph, Smarandache system,nonsolvable equation, CC conjecture, mathematical combinatorics.
AMS(2010)
:
03A10,05C15,20A05, 34A26,35A01,51A05,51D20,53A35
1
§
1
.
Introduction
A thing is complex, and hybrid with other things sometimes. That is why diﬃcult toknow the true face of all things included in “Name named is not the eternal Name;the unnamable is the eternally real and naming the srcin of all things”, the ﬁrstchapter of
TAO TEH KING
[9], a wellknown Chinese book written by an ideologist,
Lao Zi
of China. In fact, all of things with universal laws acknowledged come fromthe six organs of mankind. Thus, the words “
existence
” and “nonexistence” areknowledged by human, which maybe not implies the true existence or not in theuniverse. Thus the existence or not for a thing is
invariant
, independent on humanknowledge.The boundedness of human beings brings about a unilateral knowledge forthings in the world. Such as those shown in a famous proverb “the blind men withan elephant”. In this proverb, there are six blind men were asked to determine whatan elephant looked like by feeling diﬀerent parts of the elephant’s body. The mantouched the elephant’s leg, tail, trunk, ear, belly or tusk respectively claims it’s likea pillar, a rope, a tree branch, a hand fan, a wall or a solid pipe, such as those shownin Fig.1 following. Each of them insisted on his own and not accepted others. Theythen entered into an endless argument.
Fig.
1
All of you are right
! A wise man explains to them:
why are you telling it diﬀerently is because each one of you touched the diﬀerent part of the elephant. So, actually the elephant has all those features what you all said
. Thus, the best result on an2
elephant for these blind men isAn elephant =
{
4 pillars
}
{
1 rope
}
{
1 tree branch
}
{
2 hand fans
}
{
1 wall
}
{
1 solid pipe
}
What is the meaning of this proverb for understanding things in the world
? Itlies in that the situation of human beings knowing things in the world is analogousto these blind men. Usually, a thing
T
is identiﬁed with its known characters ( orname ) at one time, and this process is advanced gradually by ours. For example,let
µ
1
,µ
2
,
···
,µ
n
be its known and
ν
i
,i
≥
1 unknown characters at time
t
. Then,the thing
T
is understood by
T
=
n
i
=1
{
µ
i
}
k
≥
1
{
ν
k
}
(1
.
1)in logic and with an approximation
T
◦
=
n
i
=1
{
µ
i
}
for
T
at time
t
. This also answeredwhy diﬃcult for human beings knowing a thing really.Generally, let Σ be a ﬁnite or inﬁnite set. A
rule
or a
law
on a set Σ is amapping Σ
×
Σ
···×
Σ
n
→
Σ for some integers
n
. Then, a
mathematical system
is a pair (Σ;
R
), where
R
consists those of rules on Σ by logic providing all theseresultants are still in Σ.
Deﬁnition
1
.
1([28][30])
Let
(Σ
1
;
R
1
)
,
(Σ
2
;
R
2
)
,
···
,
(Σ
m
;
R
m
)
be
m
mathematical system, diﬀerent two by two. A Smarandache multisystem
Σ
is a union
m
i
=1
Σ
i
with rules
R
=
m
i
=1
R
i
on
Σ
, denoted by
Σ;
R
.
Consequently, the thing
T
is nothing else but a Smarandache multisystem(1
.
1). However, these characters
ν
k
,k
≥
1 are unknown for one at time
t
. Thus,
T
≈
T
◦
is only an approximation for its true face and it will never be ended in thisway for knowing
T
, i.e., “Name named is not the eternal Name”, as Lao Zi said.But one’s life is limited by its nature. It is nearly impossible to ﬁnd all characters
ν
k
, k
≥
1 identifying with thing
T
. Thus one can only understands a thing
T
relatively, namely ﬁnd invariant characters
I
on
ν
k
, k
≥
1 independent on artiﬁcialframe of references. In fact, this notion is consistent with
Erlangen Programme
ondeveloping geometry by Klein [10]:
given a manifold and a group of transformations
3
of the same, to investigate the conﬁgurations belonging to the manifold with regard to such properties as are not altered by the transformations of the group
, also thefountainhead of
General Relativity
of Einstein [2]:
any equation describing the law of physics should have the same form in all reference frame
, which means that a universal law does not moves with the volition of human beings. Thus, an applicablemathematical theory for a thing
T
should be an
invariant theory
acting on by allautomorphisms of the artiﬁcial frame of reference for thing
T
.All of us have known that things are inherently related, not isolated in philosophy, which implies that these is an underlying structure in characters
µ
i
,
1
≤
i
≤
n
for a thing
T
, namely, an inherited topological graph
G
. Such a graph
G
should beindependent on the volition of human beings. Generally, a labeled graph
G
for aSmarandache multispace is introduced following.
Deﬁnition
1
.
2([21])
For any integer
m
≥
1
, let
Σ;
R
be a Smarandache multisystem consisting of
m
mathematical systems
(Σ
1
;
R
1
)
,
(Σ
2
;
R
2
)
,
···
,
(Σ
m
;
R
m
)
. An inherited topological structure
G
[
S
]
of
Σ;
R
is a topological vertexedge labeled graph deﬁned following:
V
(
G
[
S
]) =
{
Σ
1
,
Σ
2
,
···
,
Σ
m
}
,
E
(
G
[
S
]) =
{
(Σ
i
,
Σ
j
)

Σ
i
Σ
j
=
∅
,
1
≤
i
=
j
≤
m
}
with labeling
L
: Σ
i
→
L
(Σ
i
) = Σ
i
and
L
: (Σ
i
,
Σ
j
)
→
L
(Σ
i
,
Σ
j
) = Σ
i
Σ
j
for integers
1
≤
i
=
j
≤
m
.
However, classical combinatorics paid attentions mainly on techniques for catering the need of other sciences, particularly, the computer science and children gamesby artiﬁcially giving up individual characters on each system (Σ
,
R
). For applyingmore it to other branch sciences initiatively, a good idea is pullback these individualcharacters on combinatorial objects again, ignored by the classical combinatorics,and back to the true face of things, i.e., an interesting conjecture on mathematicsfollowing:
Conjecture
1
.
3(CC Conjecture, [15],[19])
A mathematics can be reconstructed from or turned into combinatorization.
Certainly, this conjecture is true in philosophy. So it is in fact a combinatorial4
notion on developing mathematical sciences. Thus:(1)
One can combine diﬀerent branches into a new theory and this process ended until it has been done for all mathematical sciences, for instance, topological groups and Lie groups.
(2)
One can selects ﬁnite combinatorial rulers and axioms to reconstruct or make generalizations for classical mathematics, for instance, complexes and surfaces.
From its formulated, the CC conjecture brings about a new way for developingmathematics , and it has aﬀected on mathematics more and more. For example, itcontributed to groups, rings and modules ([11][14]), topology ([23][24]), geometry([16]) and theoretical physics ([17][18]), particularly, these 3 monographs [19][21]motivated by this notion.A
mathematical nonsystem
is such a system with contradictions. Formally, let
R
be mathematical rules on a set Σ. A pair (Σ;
R
) is nonmathematics if there isat least one ruler
R
∈
R
validated and invalided on Σ simultaneously. Notice thata multisystem deﬁned in Deﬁnition 1
.
1 is in fact a system with contradictions inthe classical view, but it is cooperated with logic by Deﬁnition 1
.
2. Thus, it lightsup the hope of transferring a system with contradictions to mathematics, consistentwith logic by combinatorial notion.The main purpose of this paper is to show how to transfer a mathematicalnonsystem, such as those of nonalgebra, nongroup, nonring, nonsolvable algebraic equations, nonsolvable ordinary diﬀerential equations, nonsolvable partialdiﬀerential equations and nonEuclidean geometry, mixed geometry, diﬀerential nonEuclidean geometry,
···
, etc. classical mathematics systems with contradictions tomathematics underlying a topological structure
G
, i.e., mathematical combinatorics.All of these discussions show that
a mathematical nonsystem is a mathematical system inherited a nontrivial topological graph, respect to that of the classical underlying a trivial
K
1
or
K
2
. Applications of these nonmathematic systems to theoreticalphysics, such as those of gravitational ﬁeld, infectious disease control, circulatingeconomical ﬁeld can be also found in this paper.All terminologies and notations in this paper are standard. For those not mentioned here, we follow [1] and [19] for algebraic systems, [5] and [6] for algebraicinvariant theory, [3] and [32] for diﬀerential equations, [4], [8] and [21] for topologyand topological graphs and [20], [28][31] for Smarandache systems.5