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Several epistemological problems related to the concept of systems
Math1Compuf.Modeliing,Vol.14,pp. 52-57, 1990
0895-7177/90 $3.00+0.00 Pergamon Press plc
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Methodology
Yi Lin', YonghaoMa* and Rick Port3 3
Departmentof Mathematics, Departmentof Psychology, lipperyRock University,SlipperyRock, PA 16057, USA ; Departmentof Mathematics,Universityof Utah, Salt Lake City, Utah 84112, USA. Abstract. A brief historicalbackgroundfor the general systemstheory to appear, and the followingproblemsare discussed.1. How do we know which definitionof general systemswill canfortus with the followingideal outccme: the general systemstheory,based on the definition,will offer a solid and unified theoreticalfoundationfor all systemsanalysismethods developed in differentfields ? 2. What is the differenceof the concept "set" in theories and in applications? 3. How can the general systemstheory be built on the axicaaticset theory ? 4. Why is mathematicsso unreasonablyeffectivein the study of natural science ? 5. What is the meaning of truth ? 6. IS the general systemstheory a languageof science and technology? 7. How can some results and conceptsin systemstheory be applied to unify differentconcepts in differentfields ? Keywords.General system, set, collection,nth-levelobject system,centralized system, a-type hierarchyof systems. systelns theory.
1RTR0DUc!r1oN
Cne of the basic problemson the study of general systemstheory is how to define the conceptof systemsrigorously,because the definitionwill affect the developmentof the general systrstheory greatly.Many effortshave been made ,,lthis problem.
Roughly speaking,the word "system"means a structure which consistsof a set of objects together with relationshipsbetween the objects and between their attributes1121.This idea of systems appeared in the early germinanttime of human society.For example,the ancient Chinese back many thousandsof years ago began to study huskanbody as an organicwhole, but not a pile of organs; and the Aristotle'sstatement"the whole is more than the sum of its parts" is a definitionof a basic systemsproblem . Along with the developmentof human kncNedge about the world and thatofmcdemtechnology, for details see [31, the concept of systemshas been describedby many greatestthinkersin the history in the languageof their times [41. However, the concept had never been introducedformally until the 1920s when L. von Bertalanffy noticed the importanceof the followingfact.The basic characterof the world is its organizational structure,the custcmaryinvestigationof single parts and processescannot provide a ccmplete explanationof natural phenomena.This investigationgives us no informationabout thecoordinationof parts and processes. Apollo dream, landinghuman foot on the moon, made the whole world realize that today'sscience and technologyhad been in such a historicalsituationthat we could design and producemiraculous things we never imaginedbefore if new systems of structurewere discovered. Since von Bertalanffyintroducedthe conceptof systems formallyin the 192Os,more and more systemsproblemshave been found in all scientific fields,and the word "system"has been given differentmeanings in differentdisciplines.Von Bertalanffyin 1968 [31 suggestedthe study of a general systemstheory to build a unified theoretical foundationfor all differentapproachesof
Historically,NorbertWiener in 1949 formulated a method, based on the work done by R.H.Camaron and W.T.Martin[71, for the analysisand synthesis of non-linearinput-outputfunctionalsystem 1351. 'Iheresearchalong this line was conducted by Y.W.Lee and the the researchgroup under his directionduring the period 1950-1965.Most of the resultsobtainedwere reportedin the R.I.E. QuarterlyProgressReports of that period. A.Tarski in 1954 [37] introduceda concept,termed as a system with relations,as a non-emptyset, called the danain of the system with relations, and a finite sequenceof familiesof relations defined on the set. A Ball and R.Fagen in 1956 [121 discussedthe definitionof systems in ordinary language (whichis given in the first paragraph above). J.Schmidtin 1961 [361 studieda concept in universalalgebra,called a general algebra,as follows:an ordered pair (A,f) = (A, {fi},sI) is said to be a general algebraof type A = iK+ 1 lc~ if and only if for any ie1, f. is a a fundamentaloperationfrcm AKi into A, where I is an index set, for any i E I, Ki is a set and AKi is the set of all functionsfrom Ki into A. M.D.Mesarovicin 1964 gave a definitionfor systems in the languageof set theory [331.His definitionwas that a general system was a proper subset of a finite Cartesianproduct: XC Xl x X2 X ... X Xnr where Xi, for i = 1, .... n, ware given sets. G. Gratzer in 1968 [ill introduceda mathematical definitionof systemsas follows:A relational system U is a pair (A;R),where A is a non-empty
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Proc. 7th Int. Conf. on Mathematical and Computer Modelling set and R is a family of (finitary) relations on A. In 1975, Masarovic and Y.Takahara [341 n-cdified Mesarovic's definition in 1964 as follows: A (general) system is a relation on non-empty sets: SC
II{Vi: i
E I},
II denotes the cartesian product and I is an index set. In 1979, M.Bunge [61 consideredthe inter-relationship between the systems under consideration and some environments of the systems. He gave the following definition: Let T be anonempty set. 'l&n the ordered triple a = (C,E,S) is a system over T if and only if C and E are mutually disjoint subsets of T and S is a non-empty set of relations on the union of C and E; and the sets C and E are called the composition and an environment of the system, respectively. G.J. Klir in 1985 [I71 surveyed the most important aspects of systems theory; and a philosophical definition of systems was given, which says that a system is what is distinguished as a system. In 1987, Lin and Ma 1291 applied the axicanatic set theory to the study of systems theory. Ccmbining the ideas listed above, their definition reads: S is said to be a system if and only if S is equal to an ordered pair (M,R) of sets, where R is a set of some relations defined on the set M. lhe elements in M are called the objects of the system S, and the sets M and R are called the object set and the relation set of the system S, respectively. Here, for any relation r in the relation set R, the relation r is defined as follows: there is an ordinal number n = n(r) such that r is a subset of the cartesian product Mn. Lin in [22] generalized the definition of relations in this concept of systems and introduced a concept, called L-fuzzy systems. In 1988, Lin and Ma gave a more general definition of systems as follows [301: Let I be an index set. S is said to be a general system of type K = {Ki}isI over I, where Ki is a partially ordered set, for each i E I, if S is an ordered triple (M,R,K) such that M is a set of all objects in S, for each i E I there is a subset ri of MKi as an element of R; Ki is called the type of the relation ri and K is called the type of the system S.
where
Fran the long list of definitions of systems, a question arises naturally: how do we know which definition will comfort us with the following ideal outcome: the general systems theory, based on the definition, will offer a solid and unified theoretical foundation for all systems analysis methods developed in different fields. The following discussion will show that each definition has its own advantages over all others. CNEERIMARYPROBLFMIN SYSTEMS THECRY General systems theory deriving frcm the work of von Bertalanffy [31 has had much influnce, but, has not been really concerned as a practical systems analysis method by scme systemstheorists 1391. Since the identification and propounding of the general systems theory many people have been trying to apply the theory to solving practical problems [161. This attempt has been notably unsuccessful [20, 11. The reason for the lack of success has been understood in many different ways. A.T.Wcod-Harper and G.Fitzgerald [391 believed that it was because the generality of the general systems theory made it difficult to use and to develop a methodological solution. Lin [21] believed that there were two sources
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which caused the unsuccessful attempt in the applications of general systems theory. A well-known fact is that the basic concepts of the general systems theory have been introduced with a primary notation "set". The concept of sets can be defined in two different ways: 1) A set is a collection of some objects; 2) A set is a mathematical object satisfying ZFC axian system (see details in [21]). The first definition seems very intuitive and trustworthy. But,unfortunetely, the following famous Russell's paradox shms that this definition of sets cancause some inconsistency in any theory develom on the definition. Russell's puadx 1191. Let X be the totalityof all sets. Then, fran the definition of sets,the totality X is a set. The set X can be written ‘2 a union of t+zo sets A and B, where A contains all sets such that none of the sets containsitself as an element; equivalently, a set xbelongs to A if and only if x does not belong to x; and B contains all the sets such that each set in B contains itself as an element; equivalently, a set x belongs to B if and only if x belongs box. Now, consider the set A. If A belongs to A, frcm the definition of A and B it follows that A does not belong to A, a contradiction. If A does not belong to A, it follows frcgnthe definition of the set A that A belongs to A, a contradiction. Therefore, A does not belong to the totality X of all sets, a contradiction. It is now natural for us to accept the seconddefinition of sets. As discussed in [211, this definition can also cause problems in applica?ions of the general systems theory, built on tl:xdefinition of sets. For example, in set theray we have the following result : lhearem 2.1. (2X). If A is a collection of objects and there is a l-lontomapping h: A + X, for scme set X, then A is a set. In applications, situations vary greatly franthe theorem; and we generally do not know how to prove whether or not a collection of objects can be studied as a set. For example, A. let N bathe collection of all integers and define the membership relation "E" as follows: for any two integers n and m in N, n E m if and only if n < m. In this case, the collection N and the membership relation "El'do not satisfy ZFC axiom system, whence, the collection N is not a set. B. Let M be 10 people standing in a circle. Define the membership relation “E” as follows: for any two people m and n in M, m E n if and only if the person m is standing on the left of the personn. Now, the membership relation "E" does not satisfy the ZFC axiom system, either. The fact is that in applications, it has never been checked whether or not the collection ofobjects and the membership relation between the objects satisfy all the ZFC axioms. Gnly when a membership relation IIs"makes all ZFC aXimS!Xcane true propositions, the systems theory can be applied to the situation under consideration. Question 2.1. Develop a method to check whether or not a collection of objects and a membership relation defined on the collection satisfy ZEC axiom system. SCMElHEDREl'ICAL RESULTS
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In this section, a few current results will begiven to show what a general systems theory can be develo& on the definition of systems given by Lin and Ma in [291 and how the axiomatic set theory plays an important role in the study of the structures of systems. All unexplained symbolsappearing in the following are from 1181.
'Ihecaen 3.3. ([241). If A is a system with anuncountable object set, each object in A is a system with a finite object set, and there is anobject belonging to at least uncountable many objects in A, then there is a partial system B of A with an uncountable object set, and B forms a centralized system.
S is said to be a system, in the rest of this section, if S is an ordered pair (M,R) of sets,where R is a set of some relations on the set M. Any element in M is called an object of the system S; and the sets M and R are called the object set and the relation set of S, respectively. See [29] for details.
Let Si = (MirRi), i = 1, 2, be systems. The SYStern S1 is said to be l-level similar to the system S a if all objects in the bth systems are still systems and there is a similar mapping h: S1 -t S2 such that for each object m E Ml, the systems m and h(m) are similar [251. A system S = (M,R) is said to be centralizable [241 if there is a centralized system C = (i$.,Rc)such that S and C are l-level similar. Any center of the system C is also called a center of the system S.
Ezmmple 3.1. A. Any system described by equations can be redescribed in the form above. B. Each topological space (X,T) is a system with the object set X and the relation set T. C. Each group (G,+) is a system with the object set G and the relation set {+I, where the relation + c G3 is defined by: (x,y,z) E G3belongs to + if and only if x + y = z. Suppose that Si= (MirRi), i = 1, 2, are two systems. The two systems are said to be identical, denoted by SI = S2, If Ml = Mz and RI = RP. The system S1 is similar to the system S2, if there exists a mapping h:Ml + M2, called a similar mapping from S1 into S2, such that h(Rl) = {h(r): r E RI} = R2, where for any relation r E RI, h(r)= I(h(x,),h(xl), ....h(x.), ...). (xO,xl,...,xCl,.. .) E r). It can be shown that if the system S1 is similar to the system Sp, the system S2 is also similar to the system Si. Thus, we can just say that they are similar systems. The system S1 is a partial system of S2 1231, if Ml c M2 and there is a subset R*c R2 such that RI = {r/Ml: r E R*j. The following result can be proved in a way similar to Russell's paradox.
l't~eorem3.4. ([241). Let K be an infinite cardinality and 0 > K be a regular cardinality and satisfy that for any c1 < 0, IUIb: h < Kl I < 0. Assume A is a system with an object set of cardinality 2 0 and any object in A is a system with an object set of cardinality < K. Then there is a partial system A* of A such that the object set of A* has cardinality _> 0 and A* forms a centralizable system. Let T be a partially ordered set with ordering5 and the order type of the ordered set (T,li be a. An a-type hierarchy S of systems 1271 over the ordered set (T,c) is a function defined on the set T such that for any t E T, St = S(t) = (Mt, Rt) is a system, which is called the state of the a-type hierarchy S of systems at the mcanent t. Without confusion, we omit the words "over a partially ordered set (T,z)". For any t, r ; T wiz r 2 t, let Qtr: Mr -fMt be a mappinq .~ch
Qts = QtrOQrs meOren 3.1. ([231). There is not such a system that the object set consists of all systems. A system S, = (M,,R,) is said to be an nth-level object system of a system So = (M,,Ro) [291, if there are systems Si = (Mi,Ri), for i = 1, .... n-l, such that Si E Mi_1, i = 1, . . . . n,where n is a natural number. -rem 3.2. ([291). Assume ZFC. For any system S, there does not exist any &h-level object system Sn of S such that S is a partial system ofSn, for any natural number n > 0. A sequence {Si: i E ni of systems, where n is an ordinal number, is said to be a chain of objcet systems of a system S [291, provided that there is a natural number i, such that S, is an i,thlevel object system of S, and for any i, j E n with i < j, there is a natural number ij such that the system Sj is an ijth-level objectsystem Of Sj_. 'IY'EO~ 3.2. ([291). Assume ZFC. For any system S, each chain of object systems of S must be finite. A system S = (M,R) is referred to as a centralized system I241 if each object in M is a system and there is a system C = (M,_,G) with M, z fl such that for any distinct elements x and y in M, say x = (Mx,Rx) and y = (My,Ry), then Mc = MxfiMy and RcC (RxlMcc)fi(Ry/Mc). The system C is called a center of the system S.
and
Qtt = idMt,
for any r, s, t E T with s > r > t, where idMt, is the identity mapping on the set Mt. The faml1Y 1&j: s, t E T with s 2 t] is termed as a family of linkage mappings of the a-type hierarchy S, and each Qts is termed as a linkage mapping from the state Sr into the state St. An a-type hierarchy S of systems is referred to as a linked &type hierarchy of systems, if a family IQts : t, s E T with s 2 t} of linkage mappings of S is given; now, the linked cl-typehierarchy S of systems is denoted by iS,Qts,T1 or {S(t), RtsrTj. 'Theorem 3.5. ([27]). For anyn-type hierarchy S of systems over T, if for each t E T, the object set M * 0, there is a family {Qts: s, t ET with s > t! of linkage mappings of the u-type hierarchy of systems. meOrem 3.6. ([27]). Suppose that K is a cardinality > 0, and S is ana-type hierarchy ofgrstems over a partially ordered set (T,c) such that i. for each t E T, St = (Mt,Rt) and /Mtl 2 K; ii. there is a cofinal and coinitial subset DC T such that(Qts: s, tE Dwiths>t}isafamily of linkage mppings for the hierarchy {St: t E D) such that for any s, t E D with s > t, lQt,(M,)j ( K; then there is a subset PC D such that ,r* is COfinal and coinitial in T, and a family {fits: s, t E T with s 2 t} of linkage rrappinqs for S such
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that for any s, t E Tk with s 2 t, &s= Qs.
(TUK)-truth ?
In the rest of the section,let the partially ordered set (T,z)be linearlyorderedby the ordering5 and there is an operation+ on the set T such that CT,+) is an abeliangroup satisfying that for any t, t', s and s' E T, t < s and t' < s' imply t + t' 5 s + s'. Let 0 be the identityin the group CT,+);and for any x E T, the symb01 1x1 indicatesthe absolutevalue of x defined by 1x1 = max{x, -x1, where t-x) is the additive inverseof x; and for any integern,
As a consequenceof the previousdiscussionin this section,the whole mathematicalstructure might be said to be true by virtue of mere definitions (namely,of the non-primitive mathematical terms) providedthat ZFC axioms are true. However,strictlyspeaking,we cannot,at this juncture,refer to ZFC axioms as propositions which are either true or false, for they contain free primitiveterms, "set" and the membership relation"E", which have not been assignedany specificmeanings.All we can assert so far is that any specificinterpretation of the primitives which satisfiesthe axioms -- i.e., turns them into true statements-- will also satisfy all the theoremsdeduced from them. For detailed discussions,see [18, 191. Examplescan be given to show that ZEC axiom systempermits of many differentinterpretations, in everydaylife as well as in the investigation of laws of nature. Frcaneach differentinterpretation we understand somethingm3re about nature.Becauseof this fact, we feel that mathematicsis so unreasonably effectivewhen appliedto the analysisof practicalproblems.
a. nx = 0, if n = 0; b.nx=x+x+...+x,ifn>O; c. nx = - x - x -... - x, if n < 0. Any a-type hierarchyS of systemsover T now is called a time system 1281. Let to E T be a fixed element.The right transformation, denotedby S-to, of the time system S is definedby S-to(t) = S(t - to), for each t E T; and the left transformation,denotedby Sto, of the time system S is definedby S+to = S(t +t,,),for each t E T. The time system is said to be periodic,if there exists to E T such that to # 0 ahd St0 = S; the elementto is called a period of S. w 3.7. ([281).Supposethat the time system S is periodicwith a period to E T, then for each integern, Snto = S where Snta = Snto, if n -> 0; and Snto = S-[(-njtol,if n < 0. For more detaileddiscussionsand results,the reader is advisedto consultthe references118, 19, 21 - 25, 27 - 301. AFEWPfJIIDSOPHICXL?HOUGHTS E.P.Wigner[381 and R.W.Hansning studiedwhy mathematicswas so unreasonablyeffectivein the study of natural science,and what the nature of mathematicaltruth was. Many studieshave been done along this line. See, for example, [5, 8, 10, 151. Lin [261 shows how the conceptof systems can be appliedto the study of these questi0I-S.
h-cpositicm 4.1. (1261). Ey a formal language, we mean a languagewhich does not contain sentences with granunar mistakes,then any formal language can be describedas a system. Let L be a formal languageand (M,K)be the systems representation of the formal languageL, where M is the set of all words in the language L and K is the collectionof all sentencesin a fixed granmerbook. Let T be the collectionof all ZFC axions. Then, the orderedpair (M,KUT) can be seen as a systemsrepresentation of mathematics. Fran the definitionof systemsgiven by G.Klir 1171 that a system is what is distinguishedas a system and Proposition4.1 it followsthat if a system S = (M,R) is given, each relationr in R can be understoodas an S-truth.I.E., the relation is true among the objects in the set M. Therefore,each mathematicaltruth is a (M,TUK)truth, i.e., it is derivablefrom ZFC axioms,the principlesof formal logic and definitionsof scarenon-primitiveterms. Question4.1. ([261). If there were two mathematical statementsderivablefrom ZFC axioms with contradictarymeanings,would they still be (M,
Generalizingthe discussionabove about mathematical truths,the meaningof "truth"can be discussed as follows: Generallyspeaking,"truth"is a fuzzy concept. For example,the laws of nature are said to be the truth aboutthe~rld. Amathematicaltheorem is a truth in mathematics.Then, what isthe generalmeaningof truth ? The definitionof systemsgiven by Klir [171 impliesthat for any system S = (M,R),where M and R are the oh'ect set and the relationset of the system S? respectively,and any relationr in R, the relation r is a true relationin the system S. Therefore, the relationr can be said to be an S-truth.At the same time, Theorem 3.1 means that we cannot considera relationwhich is true in a system that containsall systemsas its objects.In other words, when the environmentalconditions are changed,a given truth may not be a true statement any more. As a conclusion,we have that for any given statement,if it is said to be a truth, it means that there must be an (environmental) system in which the given statementdescribes a relation. IS QNEBAL SYSIDlS'IHEORY A LANGJJAGE OF SCIENCEAND TEXXNOIDSY1 Scientificinvestigation has alternatedbetween particularproblemsand generalsimilaritiesand theoriessince the very beginning.Historically, theorieshave been developedwithin the boundaries of a particularfield, set up accordingto the subjectsstudied,such as biology,physics and chemistry.Rapid developmentof modem techniques of collectingand analysingdata, as well as the advancesof classicaltheoriesof special fields,have createda new problemon a higher level. That is a need to developa general theory which will serve as a unified foundationfor other specifictheories,crossingthe boundaries of disciplinesand ultimatelyresultingin a deeper understandingof the world. Can systemstheory be the generaltheory crossing the whole scienceand resultinqa deeper understandingof the world ? An extensivereview of applicationsof systemstheory shows thatthe
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theory is still too young to ulav such an imoortant role, but some currentr&e&h indicates that a well-developedsystems theorywill play such a role in the developmentof scienceand technolcgy.In the followingwe list a few examples to supportthis statement. Let Si = (MirRi),i = I, 2, be two Systems,and h: Mi + M2 be a mapping.Define two classesby tranfiniteinductionas follows: *i =Unsord(Mi)n,
i = 1, 2,
where Ord is the class of all ordinal numbers, (Mi)n is the Cartesianproduct of n copies of the set Mi, for any n E Grd, and (Mi)' = /a.Let *h be a class mapping from the class *Mi into the class '+I2defined in the following: *h(x) = (h(x,),h(x,),...,h(xe),...), for any x = (x~,xl,...,xe,...) E *Ml. 'Ihemapping h is said to be a S-continuousmapping from the system Si into the system S2 1311, if for any relationr E R2, *h-'(r) = (*h-i(x):x s r} E R1. Without confusion,the same symbol h will be applied to indicatethe mapping *h from *M 1 into *M,, and the mapping h will be denoted by h: S1+ SZ. Fact 5.1. Suppose that h: S1 + Sz is a mapping, where the systemsSi = (Mi,Ri)ri = 1, 2, are two topologicalspaces, then h is a topologically continuousfran the space S1 into the space .S*if and only if h is S-continuous. The proof follows from Example 3.lB and the definitionof S-continuousmappings. Fad 5.2. Suppose that h: S1 + S* is a mapping, where Sir i = 1, 2, are groups. Then the mapping h is S-continuousif and only if the mapping his homon-orphism fran the group Si into the group S2. The proof follows from Example 3.1C and the definitionof S-continuousmappingsand is canitted. The facts 5.1 and 5.2 show that the concept of S-continuityof mappings is a unified concept which containscompletelydifferentconceptsfran differentfields as specialcases. A mapping h: S1 + S2 is said to be a morphism fran a system Si = (M~,R~)into a system & = (M2,Rz) [321,providedthat h(R1)c Rz. The set of all morphismsfran S, into S, is denotedby H~an(Sr,s~). Suppose that {Si = (MirRi):i s I} is a family of systems,where I is an index set. A system S-is said to be a product of the family {Si: i s I} of systems,if there is a family of morphismsIoi E HOm(S,Si):i E I} such that for any system S* and a family [$i E HOm(S*,Si):i E I), there exists exactly one XsHcm(S*,S) sothat for all i E I. A result in tii = @ioX r [321 says that the product in the categoryof general systemsexists. The unique product system S = (M,R)of the family ISi: i E 11 of systems can be constructedas follows:M =n( Mi: i E I), *R = (*r = (r.)isxE IIic Ri: 27 !; ‘lr j E 11 and R = tr: r c *R$ wh;;z;o; (ri)isIE *R, let n(*r) = n(ri), for each i E I, r = {x E Mn(*r): there exists (Xi)iEI E Iii II?i such that for any CYsn(*r), x(u) = (xi(C4)FisI)* Let pi: IlisIMi+Mi be the projection,for i E I. 'Ihenpi E.HOm(S,Si). Fad 5.3. a. If each system in the family ISi: i
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EI } is a topologicalspace, the product systems is the box product topologicalspace of the spaces Si [391.b. Any ring (X,+,0)can be described as a system (X,(+,*})asfollows:+,. c X3, for any (x,y,z)E X3, (x,y,z)E + if and only if x + y = Z; (x,y,z)E . if and only if x.y = z. If each system in the family {Si: i c I} is a ring, the product system S is the direct sum of the rings Si. This fact shows that the conceptof box topology and the conceptof direct sum in algebra are unifiedby the conceptof product systems. AFEWFIiUAL~RDS Canpare the systemsmethod -- everythingunder considerationis a whole with basic objects and relationsconnectingthe objects -- with the classicalmethod of scientificresearch-- divide the problemunder considerationinto as small parts as possible,and study each of the isolated parts and simplifythe complicatedphenomenon into basic parts and processes.It can be seen that a completelynew understandingabout the world will be obtained.That is why each application of the general systgns theory requiresa revolutionto implement.It is not a process which often recamaendsmall incrementalchanges but one which more usually results in a ccmplete reassessmentof structures,roles and behaviours [391. REFERENCES l.D.Berlinski,%SystemsAnalysis.M.I.T., Press, Cambridge,Massachusetts,1976. 2.L.vonBertalanffy, Modern Theoriesof Development. Translatedby J.H.Wcodger,Gxf,,:d UniversityPress, 1934. 3.L.vonBartalanffy, General SystemsTheory. George Braziller,New York, 1968. 4.L.vonBertalanffy,Ihe History and Status of General SystemsTheory. In Trends in General SystemsTheory,edited by G.J.Klir,New York, (1972)21-41. S.S.~ochner,lbe Role of Mathematicsin the Rise of Science.Princeton,New Jersey:Princeton UniversityPress, 1949. 6. M.Bunge,Treatiseon Basic Philosophy,~01.4: A World of Systems.D.ReidelPublishing Company,Holland, 1979. 7. R.H.Cameronand W.T.Martin,'IheOrthogonal Developmentof Non-linearFunctionalsin Series of Fourier-Hermite Functionals. Annuals of Mathematics,48(1947)385-392. 8. P.J.Davisand R.Hersh,The MathematicalExperience.Boston: Birkhauser,1981. 9. R.Engelking,GeneralTopology.Polish Scientific Publishers,Warszawa,1977. lO.D.A.T.Gasking, Mathematicsand the World. AustralianJournalof Philosophy,8(1940)97116. 11. G.Gratzer,UniversalAlgebra.Springer-Verlag, New York, Heidelberg,Berlin, 1968. 12. A.D.Halland R.E.Fagen,Definitionof System. General System, 1(1956) 18-28. 13. R.W.Hanm&ng,The UnreasonableEffectiveness of Mathematics.AmericanMathematicalMonthly, 87(1980)81-90. 14. L.O.Kattsoff,A Philosophyof Mathematics. Ames, Iowa: Iowa State College Press, 1948. 15. M.Kline,Mathematicsand the PhysicalWorld. New York: Crowell,1959. 16. G.J.Klir,An Approachto GeneralSystems Theory. Van Nostrand,Princeton,New Jersey,
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on Mathematical
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17. G. J. Klir, Architectureof SystemsProblem Solving.Plenun Press, New York and Iondon, 1985. 18. K. Kunen, Set Theory:An Introductionto IndependenceProofs.North-Holland, Amsterdam, 1980. 19. K. Kuratowskiand A. Mostowski,Set 'Theory. North-Holland, Warszawa,1976. 20. D. Lilienfeld,The Rise of SystemsTheory. Wiley, New York, 1978. 21. Y. Lin, Can The World Be StudiedIn The Viewpoint Of Systems ?? Math. Coqut. Elodelling, ll(1988)738-742. 22. Y. Lin, On L-Fuzzy Systems.AppliedMathematics Letters,to appear. 23. Y. Lin, A Model of GeneralSystems.Math. Mcdelling,9(1987)95-104. 24. Y. Lin, CentralizedSystemsand the Existence. Sutmittedfor publication. 25. Y. Lin, On Multi-levelSystems.Int. J. SystemsScience,to appear. 26. Y. Lin, A Few Systems-Colored Views of the World. In "Mathematicsand Science", edited by R.E.Mickens,to be published by ScientificWorld Press. 27. Y. Lin, order Structuresof Familiesof C&nera1 Systems.Cyberneticsand Systems, to appear. 28. Y. Lin, On PeriodicLinked Time Systems.Int. J. SystemsSciences,to appear. 29. Y. Lin and Y.-H. Ma, Rarks on Analogybatween Systems.Int. J. GeneralSystems, 13(1987)135-141. 30. Y. Lin and Y.-H. Ma, A Note On the Models of General Systems.AppliedMath. Letters, l(1988)157-160. 31. Y. Lin and Y.-H. Ma, About the Conceptof ContinuityBetweenGeneral Systems.Int. J. General Systems,to appear. 32. Y.-H. Ma and Y. Lin, Limit Propertiesof Hierarchiesof General Systems.Applied Math. Letters,l(1988)181-184. 33. M. D. Mesarovic,Views on GeneralSystems Theory. Proceedingsof the Second SystesrsSymposiumat Case Instituteof Technology,New York, 1964. 34. M. D. Mesarovicand Y. Takahara,Generalsystems Theory:MathematicalFoundations. AcademicPress, New York, 1975. 35. M. Schetzen,The Volterraand Wiener Theories of Non-linearSystems.John Wiley & Sons, New York, 1980. 36. J. Schmidt,AlgebraicOperationsand Algebraic Independencein Algebraswith infinitaryOperations.Math. Japon., 6(19 61/1962)77-112. 37. A. Tarski,Contributionsto the Theory of Models I, II, IndagationesMathematicae, 16(1954)572-588. 38. E. P. Wigner, The UnreasonableEffectiveness of Mathematicsin the Natural Sciences. Comnunications on Pure and Applied Mathematics,13(1960)1-14. 39. A. T. Wood-Harperand G. Fitzgerald,A Taxonomy of CurrentApproachesto Systems Analysis.'IheComputerJ., 25(1982)1216.
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0895-7177/90 $3.00+0.00 Pergamon Press plc
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Methodology
Yi Lin', YonghaoMa* and Rick Port3 3
Departmentof Mathematics, Departmentof Psychology, lipperyRock University,SlipperyRock, PA 16057, USA ; Departmentof Mathematics,Universityof Utah, Salt Lake City, Utah 84112, USA. Abstract. A brief historicalbackgroundfor the general systemstheory to appear, and the followingproblemsare discussed.1. How do we know which definitionof general systemswill canfortus with the followingideal outccme: the general systemstheory,based on the definition,will offer a solid and unified theoreticalfoundationfor all systemsanalysismethods developed in differentfields ? 2. What is the differenceof the concept "set" in theories and in applications? 3. How can the general systemstheory be built on the axicaaticset theory ? 4. Why is mathematicsso unreasonablyeffectivein the study of natural science ? 5. What is the meaning of truth ? 6. IS the general systemstheory a languageof science and technology? 7. How can some results and conceptsin systemstheory be applied to unify differentconcepts in differentfields ? Keywords.General system, set, collection,nth-levelobject system,centralized system, a-type hierarchyof systems. systelns theory.
1RTR0DUc!r1oN
Cne of the basic problemson the study of general systemstheory is how to define the conceptof systemsrigorously,because the definitionwill affect the developmentof the general systrstheory greatly.Many effortshave been made ,,lthis problem.
Roughly speaking,the word "system"means a structure which consistsof a set of objects together with relationshipsbetween the objects and between their attributes1121.This idea of systems appeared in the early germinanttime of human society.For example,the ancient Chinese back many thousandsof years ago began to study huskanbody as an organicwhole, but not a pile of organs; and the Aristotle'sstatement"the whole is more than the sum of its parts" is a definitionof a basic systemsproblem . Along with the developmentof human kncNedge about the world and thatofmcdemtechnology, for details see [31, the concept of systemshas been describedby many greatestthinkersin the history in the languageof their times [41. However, the concept had never been introducedformally until the 1920s when L. von Bertalanffy noticed the importanceof the followingfact.The basic characterof the world is its organizational structure,the custcmaryinvestigationof single parts and processescannot provide a ccmplete explanationof natural phenomena.This investigationgives us no informationabout thecoordinationof parts and processes. Apollo dream, landinghuman foot on the moon, made the whole world realize that today'sscience and technologyhad been in such a historicalsituationthat we could design and producemiraculous things we never imaginedbefore if new systems of structurewere discovered. Since von Bertalanffyintroducedthe conceptof systems formallyin the 192Os,more and more systemsproblemshave been found in all scientific fields,and the word "system"has been given differentmeanings in differentdisciplines.Von Bertalanffyin 1968 [31 suggestedthe study of a general systemstheory to build a unified theoretical foundationfor all differentapproachesof
Historically,NorbertWiener in 1949 formulated a method, based on the work done by R.H.Camaron and W.T.Martin[71, for the analysisand synthesis of non-linearinput-outputfunctionalsystem 1351. 'Iheresearchalong this line was conducted by Y.W.Lee and the the researchgroup under his directionduring the period 1950-1965.Most of the resultsobtainedwere reportedin the R.I.E. QuarterlyProgressReports of that period. A.Tarski in 1954 [37] introduceda concept,termed as a system with relations,as a non-emptyset, called the danain of the system with relations, and a finite sequenceof familiesof relations defined on the set. A Ball and R.Fagen in 1956 [121 discussedthe definitionof systems in ordinary language (whichis given in the first paragraph above). J.Schmidtin 1961 [361 studieda concept in universalalgebra,called a general algebra,as follows:an ordered pair (A,f) = (A, {fi},sI) is said to be a general algebraof type A = iK+ 1 lc~ if and only if for any ie1, f. is a a fundamentaloperationfrcm AKi into A, where I is an index set, for any i E I, Ki is a set and AKi is the set of all functionsfrom Ki into A. M.D.Mesarovicin 1964 gave a definitionfor systems in the languageof set theory [331.His definitionwas that a general system was a proper subset of a finite Cartesianproduct: XC Xl x X2 X ... X Xnr where Xi, for i = 1, .... n, ware given sets. G. Gratzer in 1968 [ill introduceda mathematical definitionof systemsas follows:A relational system U is a pair (A;R),where A is a non-empty
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Proc. 7th Int. Conf. on Mathematical and Computer Modelling set and R is a family of (finitary) relations on A. In 1975, Masarovic and Y.Takahara [341 n-cdified Mesarovic's definition in 1964 as follows: A (general) system is a relation on non-empty sets: SC
II{Vi: i
E I},
II denotes the cartesian product and I is an index set. In 1979, M.Bunge [61 consideredthe inter-relationship between the systems under consideration and some environments of the systems. He gave the following definition: Let T be anonempty set. 'l&n the ordered triple a = (C,E,S) is a system over T if and only if C and E are mutually disjoint subsets of T and S is a non-empty set of relations on the union of C and E; and the sets C and E are called the composition and an environment of the system, respectively. G.J. Klir in 1985 [I71 surveyed the most important aspects of systems theory; and a philosophical definition of systems was given, which says that a system is what is distinguished as a system. In 1987, Lin and Ma 1291 applied the axicanatic set theory to the study of systems theory. Ccmbining the ideas listed above, their definition reads: S is said to be a system if and only if S is equal to an ordered pair (M,R) of sets, where R is a set of some relations defined on the set M. lhe elements in M are called the objects of the system S, and the sets M and R are called the object set and the relation set of the system S, respectively. Here, for any relation r in the relation set R, the relation r is defined as follows: there is an ordinal number n = n(r) such that r is a subset of the cartesian product Mn. Lin in [22] generalized the definition of relations in this concept of systems and introduced a concept, called L-fuzzy systems. In 1988, Lin and Ma gave a more general definition of systems as follows [301: Let I be an index set. S is said to be a general system of type K = {Ki}isI over I, where Ki is a partially ordered set, for each i E I, if S is an ordered triple (M,R,K) such that M is a set of all objects in S, for each i E I there is a subset ri of MKi as an element of R; Ki is called the type of the relation ri and K is called the type of the system S.
where
Fran the long list of definitions of systems, a question arises naturally: how do we know which definition will comfort us with the following ideal outcome: the general systems theory, based on the definition, will offer a solid and unified theoretical foundation for all systems analysis methods developed in different fields. The following discussion will show that each definition has its own advantages over all others. CNEERIMARYPROBLFMIN SYSTEMS THECRY General systems theory deriving frcm the work of von Bertalanffy [31 has had much influnce, but, has not been really concerned as a practical systems analysis method by scme systemstheorists 1391. Since the identification and propounding of the general systems theory many people have been trying to apply the theory to solving practical problems [161. This attempt has been notably unsuccessful [20, 11. The reason for the lack of success has been understood in many different ways. A.T.Wcod-Harper and G.Fitzgerald [391 believed that it was because the generality of the general systems theory made it difficult to use and to develop a methodological solution. Lin [21] believed that there were two sources
53
which caused the unsuccessful attempt in the applications of general systems theory. A well-known fact is that the basic concepts of the general systems theory have been introduced with a primary notation "set". The concept of sets can be defined in two different ways: 1) A set is a collection of some objects; 2) A set is a mathematical object satisfying ZFC axian system (see details in [21]). The first definition seems very intuitive and trustworthy. But,unfortunetely, the following famous Russell's paradox shms that this definition of sets cancause some inconsistency in any theory develom on the definition. Russell's puadx 1191. Let X be the totalityof all sets. Then, fran the definition of sets,the totality X is a set. The set X can be written ‘2 a union of t+zo sets A and B, where A contains all sets such that none of the sets containsitself as an element; equivalently, a set xbelongs to A if and only if x does not belong to x; and B contains all the sets such that each set in B contains itself as an element; equivalently, a set x belongs to B if and only if x belongs box. Now, consider the set A. If A belongs to A, frcm the definition of A and B it follows that A does not belong to A, a contradiction. If A does not belong to A, it follows frcgnthe definition of the set A that A belongs to A, a contradiction. Therefore, A does not belong to the totality X of all sets, a contradiction. It is now natural for us to accept the seconddefinition of sets. As discussed in [211, this definition can also cause problems in applica?ions of the general systems theory, built on tl:xdefinition of sets. For example, in set theray we have the following result : lhearem 2.1. (2X). If A is a collection of objects and there is a l-lontomapping h: A + X, for scme set X, then A is a set. In applications, situations vary greatly franthe theorem; and we generally do not know how to prove whether or not a collection of objects can be studied as a set. For example, A. let N bathe collection of all integers and define the membership relation "E" as follows: for any two integers n and m in N, n E m if and only if n < m. In this case, the collection N and the membership relation "El'do not satisfy ZFC axiom system, whence, the collection N is not a set. B. Let M be 10 people standing in a circle. Define the membership relation “E” as follows: for any two people m and n in M, m E n if and only if the person m is standing on the left of the personn. Now, the membership relation "E" does not satisfy the ZFC axiom system, either. The fact is that in applications, it has never been checked whether or not the collection ofobjects and the membership relation between the objects satisfy all the ZFC axioms. Gnly when a membership relation IIs"makes all ZFC aXimS!Xcane true propositions, the systems theory can be applied to the situation under consideration. Question 2.1. Develop a method to check whether or not a collection of objects and a membership relation defined on the collection satisfy ZEC axiom system. SCMElHEDREl'ICAL RESULTS
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In this section, a few current results will begiven to show what a general systems theory can be develo& on the definition of systems given by Lin and Ma in [291 and how the axiomatic set theory plays an important role in the study of the structures of systems. All unexplained symbolsappearing in the following are from 1181.
'Ihecaen 3.3. ([241). If A is a system with anuncountable object set, each object in A is a system with a finite object set, and there is anobject belonging to at least uncountable many objects in A, then there is a partial system B of A with an uncountable object set, and B forms a centralized system.
S is said to be a system, in the rest of this section, if S is an ordered pair (M,R) of sets,where R is a set of some relations on the set M. Any element in M is called an object of the system S; and the sets M and R are called the object set and the relation set of S, respectively. See [29] for details.
Let Si = (MirRi), i = 1, 2, be systems. The SYStern S1 is said to be l-level similar to the system S a if all objects in the bth systems are still systems and there is a similar mapping h: S1 -t S2 such that for each object m E Ml, the systems m and h(m) are similar [251. A system S = (M,R) is said to be centralizable [241 if there is a centralized system C = (i$.,Rc)such that S and C are l-level similar. Any center of the system C is also called a center of the system S.
Ezmmple 3.1. A. Any system described by equations can be redescribed in the form above. B. Each topological space (X,T) is a system with the object set X and the relation set T. C. Each group (G,+) is a system with the object set G and the relation set {+I, where the relation + c G3 is defined by: (x,y,z) E G3belongs to + if and only if x + y = z. Suppose that Si= (MirRi), i = 1, 2, are two systems. The two systems are said to be identical, denoted by SI = S2, If Ml = Mz and RI = RP. The system S1 is similar to the system S2, if there exists a mapping h:Ml + M2, called a similar mapping from S1 into S2, such that h(Rl) = {h(r): r E RI} = R2, where for any relation r E RI, h(r)= I(h(x,),h(xl), ....h(x.), ...). (xO,xl,...,xCl,.. .) E r). It can be shown that if the system S1 is similar to the system Sp, the system S2 is also similar to the system Si. Thus, we can just say that they are similar systems. The system S1 is a partial system of S2 1231, if Ml c M2 and there is a subset R*c R2 such that RI = {r/Ml: r E R*j. The following result can be proved in a way similar to Russell's paradox.
l't~eorem3.4. ([241). Let K be an infinite cardinality and 0 > K be a regular cardinality and satisfy that for any c1 < 0, IUIb: h < Kl I < 0. Assume A is a system with an object set of cardinality 2 0 and any object in A is a system with an object set of cardinality < K. Then there is a partial system A* of A such that the object set of A* has cardinality _> 0 and A* forms a centralizable system. Let T be a partially ordered set with ordering5 and the order type of the ordered set (T,li be a. An a-type hierarchy S of systems 1271 over the ordered set (T,c) is a function defined on the set T such that for any t E T, St = S(t) = (Mt, Rt) is a system, which is called the state of the a-type hierarchy S of systems at the mcanent t. Without confusion, we omit the words "over a partially ordered set (T,z)". For any t, r ; T wiz r 2 t, let Qtr: Mr -fMt be a mappinq .~ch
Qts = QtrOQrs meOren 3.1. ([231). There is not such a system that the object set consists of all systems. A system S, = (M,,R,) is said to be an nth-level object system of a system So = (M,,Ro) [291, if there are systems Si = (Mi,Ri), for i = 1, .... n-l, such that Si E Mi_1, i = 1, . . . . n,where n is a natural number. -rem 3.2. ([291). Assume ZFC. For any system S, there does not exist any &h-level object system Sn of S such that S is a partial system ofSn, for any natural number n > 0. A sequence {Si: i E ni of systems, where n is an ordinal number, is said to be a chain of objcet systems of a system S [291, provided that there is a natural number i, such that S, is an i,thlevel object system of S, and for any i, j E n with i < j, there is a natural number ij such that the system Sj is an ijth-level objectsystem Of Sj_. 'IY'EO~ 3.2. ([291). Assume ZFC. For any system S, each chain of object systems of S must be finite. A system S = (M,R) is referred to as a centralized system I241 if each object in M is a system and there is a system C = (M,_,G) with M, z fl such that for any distinct elements x and y in M, say x = (Mx,Rx) and y = (My,Ry), then Mc = MxfiMy and RcC (RxlMcc)fi(Ry/Mc). The system C is called a center of the system S.
and
Qtt = idMt,
for any r, s, t E T with s > r > t, where idMt, is the identity mapping on the set Mt. The faml1Y 1&j: s, t E T with s 2 t] is termed as a family of linkage mappings of the a-type hierarchy S, and each Qts is termed as a linkage mapping from the state Sr into the state St. An a-type hierarchy S of systems is referred to as a linked &type hierarchy of systems, if a family IQts : t, s E T with s 2 t} of linkage mappings of S is given; now, the linked cl-typehierarchy S of systems is denoted by iS,Qts,T1 or {S(t), RtsrTj. 'Theorem 3.5. ([27]). For anyn-type hierarchy S of systems over T, if for each t E T, the object set M * 0, there is a family {Qts: s, t ET with s > t! of linkage mappings of the u-type hierarchy of systems. meOrem 3.6. ([27]). Suppose that K is a cardinality > 0, and S is ana-type hierarchy ofgrstems over a partially ordered set (T,c) such that i. for each t E T, St = (Mt,Rt) and /Mtl 2 K; ii. there is a cofinal and coinitial subset DC T such that(Qts: s, tE Dwiths>t}isafamily of linkage mppings for the hierarchy {St: t E D) such that for any s, t E D with s > t, lQt,(M,)j ( K; then there is a subset PC D such that ,r* is COfinal and coinitial in T, and a family {fits: s, t E T with s 2 t} of linkage rrappinqs for S such
Proc. 7th Inr. Conf. on Mathematical and Computer Modelling
that for any s, t E Tk with s 2 t, &s= Qs.
(TUK)-truth ?
In the rest of the section,let the partially ordered set (T,z)be linearlyorderedby the ordering5 and there is an operation+ on the set T such that CT,+) is an abeliangroup satisfying that for any t, t', s and s' E T, t < s and t' < s' imply t + t' 5 s + s'. Let 0 be the identityin the group CT,+);and for any x E T, the symb01 1x1 indicatesthe absolutevalue of x defined by 1x1 = max{x, -x1, where t-x) is the additive inverseof x; and for any integern,
As a consequenceof the previousdiscussionin this section,the whole mathematicalstructure might be said to be true by virtue of mere definitions (namely,of the non-primitive mathematical terms) providedthat ZFC axioms are true. However,strictlyspeaking,we cannot,at this juncture,refer to ZFC axioms as propositions which are either true or false, for they contain free primitiveterms, "set" and the membership relation"E", which have not been assignedany specificmeanings.All we can assert so far is that any specificinterpretation of the primitives which satisfiesthe axioms -- i.e., turns them into true statements-- will also satisfy all the theoremsdeduced from them. For detailed discussions,see [18, 191. Examplescan be given to show that ZEC axiom systempermits of many differentinterpretations, in everydaylife as well as in the investigation of laws of nature. Frcaneach differentinterpretation we understand somethingm3re about nature.Becauseof this fact, we feel that mathematicsis so unreasonably effectivewhen appliedto the analysisof practicalproblems.
a. nx = 0, if n = 0; b.nx=x+x+...+x,ifn>O; c. nx = - x - x -... - x, if n < 0. Any a-type hierarchyS of systemsover T now is called a time system 1281. Let to E T be a fixed element.The right transformation, denotedby S-to, of the time system S is definedby S-to(t) = S(t - to), for each t E T; and the left transformation,denotedby Sto, of the time system S is definedby S+to = S(t +t,,),for each t E T. The time system is said to be periodic,if there exists to E T such that to # 0 ahd St0 = S; the elementto is called a period of S. w 3.7. ([281).Supposethat the time system S is periodicwith a period to E T, then for each integern, Snto = S where Snta = Snto, if n -> 0; and Snto = S-[(-njtol,if n < 0. For more detaileddiscussionsand results,the reader is advisedto consultthe references118, 19, 21 - 25, 27 - 301. AFEWPfJIIDSOPHICXL?HOUGHTS E.P.Wigner[381 and R.W.Hansning studiedwhy mathematicswas so unreasonablyeffectivein the study of natural science,and what the nature of mathematicaltruth was. Many studieshave been done along this line. See, for example, [5, 8, 10, 151. Lin [261 shows how the conceptof systems can be appliedto the study of these questi0I-S.
h-cpositicm 4.1. (1261). Ey a formal language, we mean a languagewhich does not contain sentences with granunar mistakes,then any formal language can be describedas a system. Let L be a formal languageand (M,K)be the systems representation of the formal languageL, where M is the set of all words in the language L and K is the collectionof all sentencesin a fixed granmerbook. Let T be the collectionof all ZFC axions. Then, the orderedpair (M,KUT) can be seen as a systemsrepresentation of mathematics. Fran the definitionof systemsgiven by G.Klir 1171 that a system is what is distinguishedas a system and Proposition4.1 it followsthat if a system S = (M,R) is given, each relationr in R can be understoodas an S-truth.I.E., the relation is true among the objects in the set M. Therefore,each mathematicaltruth is a (M,TUK)truth, i.e., it is derivablefrom ZFC axioms,the principlesof formal logic and definitionsof scarenon-primitiveterms. Question4.1. ([261). If there were two mathematical statementsderivablefrom ZFC axioms with contradictarymeanings,would they still be (M,
Generalizingthe discussionabove about mathematical truths,the meaningof "truth"can be discussed as follows: Generallyspeaking,"truth"is a fuzzy concept. For example,the laws of nature are said to be the truth aboutthe~rld. Amathematicaltheorem is a truth in mathematics.Then, what isthe generalmeaningof truth ? The definitionof systemsgiven by Klir [171 impliesthat for any system S = (M,R),where M and R are the oh'ect set and the relationset of the system S? respectively,and any relationr in R, the relation r is a true relationin the system S. Therefore, the relationr can be said to be an S-truth.At the same time, Theorem 3.1 means that we cannot considera relationwhich is true in a system that containsall systemsas its objects.In other words, when the environmentalconditions are changed,a given truth may not be a true statement any more. As a conclusion,we have that for any given statement,if it is said to be a truth, it means that there must be an (environmental) system in which the given statementdescribes a relation. IS QNEBAL SYSIDlS'IHEORY A LANGJJAGE OF SCIENCEAND TEXXNOIDSY1 Scientificinvestigation has alternatedbetween particularproblemsand generalsimilaritiesand theoriessince the very beginning.Historically, theorieshave been developedwithin the boundaries of a particularfield, set up accordingto the subjectsstudied,such as biology,physics and chemistry.Rapid developmentof modem techniques of collectingand analysingdata, as well as the advancesof classicaltheoriesof special fields,have createda new problemon a higher level. That is a need to developa general theory which will serve as a unified foundationfor other specifictheories,crossingthe boundaries of disciplinesand ultimatelyresultingin a deeper understandingof the world. Can systemstheory be the generaltheory crossing the whole scienceand resultinqa deeper understandingof the world ? An extensivereview of applicationsof systemstheory shows thatthe
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theory is still too young to ulav such an imoortant role, but some currentr&e&h indicates that a well-developedsystems theorywill play such a role in the developmentof scienceand technolcgy.In the followingwe list a few examples to supportthis statement. Let Si = (MirRi),i = I, 2, be two Systems,and h: Mi + M2 be a mapping.Define two classesby tranfiniteinductionas follows: *i =Unsord(Mi)n,
i = 1, 2,
where Ord is the class of all ordinal numbers, (Mi)n is the Cartesianproduct of n copies of the set Mi, for any n E Grd, and (Mi)' = /a.Let *h be a class mapping from the class *Mi into the class '+I2defined in the following: *h(x) = (h(x,),h(x,),...,h(xe),...), for any x = (x~,xl,...,xe,...) E *Ml. 'Ihemapping h is said to be a S-continuousmapping from the system Si into the system S2 1311, if for any relationr E R2, *h-'(r) = (*h-i(x):x s r} E R1. Without confusion,the same symbol h will be applied to indicatethe mapping *h from *M 1 into *M,, and the mapping h will be denoted by h: S1+ SZ. Fact 5.1. Suppose that h: S1 + Sz is a mapping, where the systemsSi = (Mi,Ri)ri = 1, 2, are two topologicalspaces, then h is a topologically continuousfran the space S1 into the space .S*if and only if h is S-continuous. The proof follows from Example 3.lB and the definitionof S-continuousmappings. Fad 5.2. Suppose that h: S1 + S* is a mapping, where Sir i = 1, 2, are groups. Then the mapping h is S-continuousif and only if the mapping his homon-orphism fran the group Si into the group S2. The proof follows from Example 3.1C and the definitionof S-continuousmappingsand is canitted. The facts 5.1 and 5.2 show that the concept of S-continuityof mappings is a unified concept which containscompletelydifferentconceptsfran differentfields as specialcases. A mapping h: S1 + S2 is said to be a morphism fran a system Si = (M~,R~)into a system & = (M2,Rz) [321,providedthat h(R1)c Rz. The set of all morphismsfran S, into S, is denotedby H~an(Sr,s~). Suppose that {Si = (MirRi):i s I} is a family of systems,where I is an index set. A system S-is said to be a product of the family {Si: i s I} of systems,if there is a family of morphismsIoi E HOm(S,Si):i E I} such that for any system S* and a family [$i E HOm(S*,Si):i E I), there exists exactly one XsHcm(S*,S) sothat for all i E I. A result in tii = @ioX r [321 says that the product in the categoryof general systemsexists. The unique product system S = (M,R)of the family ISi: i E 11 of systems can be constructedas follows:M =n( Mi: i E I), *R = (*r = (r.)isxE IIic Ri: 27 !; ‘lr j E 11 and R = tr: r c *R$ wh;;z;o; (ri)isIE *R, let n(*r) = n(ri), for each i E I, r = {x E Mn(*r): there exists (Xi)iEI E Iii II?i such that for any CYsn(*r), x(u) = (xi(C4)FisI)* Let pi: IlisIMi+Mi be the projection,for i E I. 'Ihenpi E.HOm(S,Si). Fad 5.3. a. If each system in the family ISi: i
and Computer
Modelring
EI } is a topologicalspace, the product systems is the box product topologicalspace of the spaces Si [391.b. Any ring (X,+,0)can be described as a system (X,(+,*})asfollows:+,. c X3, for any (x,y,z)E X3, (x,y,z)E + if and only if x + y = Z; (x,y,z)E . if and only if x.y = z. If each system in the family {Si: i c I} is a ring, the product system S is the direct sum of the rings Si. This fact shows that the conceptof box topology and the conceptof direct sum in algebra are unifiedby the conceptof product systems. AFEWFIiUAL~RDS Canpare the systemsmethod -- everythingunder considerationis a whole with basic objects and relationsconnectingthe objects -- with the classicalmethod of scientificresearch-- divide the problemunder considerationinto as small parts as possible,and study each of the isolated parts and simplifythe complicatedphenomenon into basic parts and processes.It can be seen that a completelynew understandingabout the world will be obtained.That is why each application of the general systgns theory requiresa revolutionto implement.It is not a process which often recamaendsmall incrementalchanges but one which more usually results in a ccmplete reassessmentof structures,roles and behaviours [391. REFERENCES l.D.Berlinski,%SystemsAnalysis.M.I.T., Press, Cambridge,Massachusetts,1976. 2.L.vonBertalanffy, Modern Theoriesof Development. Translatedby J.H.Wcodger,Gxf,,:d UniversityPress, 1934. 3.L.vonBartalanffy, General SystemsTheory. George Braziller,New York, 1968. 4.L.vonBertalanffy,Ihe History and Status of General SystemsTheory. In Trends in General SystemsTheory,edited by G.J.Klir,New York, (1972)21-41. S.S.~ochner,lbe Role of Mathematicsin the Rise of Science.Princeton,New Jersey:Princeton UniversityPress, 1949. 6. M.Bunge,Treatiseon Basic Philosophy,~01.4: A World of Systems.D.ReidelPublishing Company,Holland, 1979. 7. R.H.Cameronand W.T.Martin,'IheOrthogonal Developmentof Non-linearFunctionalsin Series of Fourier-Hermite Functionals. Annuals of Mathematics,48(1947)385-392. 8. P.J.Davisand R.Hersh,The MathematicalExperience.Boston: Birkhauser,1981. 9. R.Engelking,GeneralTopology.Polish Scientific Publishers,Warszawa,1977. lO.D.A.T.Gasking, Mathematicsand the World. AustralianJournalof Philosophy,8(1940)97116. 11. G.Gratzer,UniversalAlgebra.Springer-Verlag, New York, Heidelberg,Berlin, 1968. 12. A.D.Halland R.E.Fagen,Definitionof System. General System, 1(1956) 18-28. 13. R.W.Hanm&ng,The UnreasonableEffectiveness of Mathematics.AmericanMathematicalMonthly, 87(1980)81-90. 14. L.O.Kattsoff,A Philosophyof Mathematics. Ames, Iowa: Iowa State College Press, 1948. 15. M.Kline,Mathematicsand the PhysicalWorld. New York: Crowell,1959. 16. G.J.Klir,An Approachto GeneralSystems Theory. Van Nostrand,Princeton,New Jersey,
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