Mathematics on Non-Mathematics —– A Combinatorial Contribution

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  Mathematics on Non-Mathematics —– A Combinatorial Contribution  Linfan MAO (Chinese Academy of Mathematics and System Science, Beijing 100190, P.R.China)E-mail: Abstract :  A classical system of mathematics is homogenous without contra-dictions. 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 mathe-matical 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 non-mathematics 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 vertex-edge 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 non-systems, such as those of algebraicsystems with contradictions, algebraic or differential equations with contra-dictions, geometries with contradictions, and generally, classical mathematicssystems with contradictions to mathematics by the underlying structure  G .All of these discussions show that a non-mathematics in classical is in fact amathematics underlying a topological structure  G , i.e., mathematical combi-natorics, and contribute more to physics and other sciences. Key Words :  Non-mathematics, topological graph, Smarandache system,non-solvable 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 well-known 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 “non-existence” 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 multi-system    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 multi-system(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 charac-ters  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 uni-versal 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 philoso-phy, 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 multi-space is introduced following. Definition  1 . 2([21])  For any integer   m  ≥  1 , let   Σ;   R   be a Smarandache multi-system 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 vertex-edge 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 cater-ing 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 non-system   is such a system with contradictions. Formally, let  R   be mathematical rules on a set Σ. A pair (Σ;  R  ) is non-mathematics if there isat least one ruler  R  ∈  R   validated and invalided on Σ simultaneously. Notice thata multi-system 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 mathematicalnon-system, such as those of non-algebra, non-group, non-ring, non-solvable alge-braic equations, non-solvable ordinary differential equations, non-solvable partialdifferential equations and non-Euclidean geometry, mixed geometry, differential non-Euclidean 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 non-system is a mathematical sys-tem inherited a non-trivial topological graph, respect to that of the classical underly-ing a trivial   K  1  or   K  2 . Applications of these non-mathematic 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 men-tioned 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
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