From the inception of GST the intention was to establish a formal approach to be used in various scientific disciplines, just as probability theory is a mathematical approach that can be used in many fields. As Bertalanffy announced, GST "is a logico-mathematical field, the subject matter of which is the formulation and deduction of those principles which are valid for 'systems' in general" (Bertalanffy, 1950, p. 139). Bertalanffy tried to illustrate the validity of GST for systems in general with differential equations, which are still a proper tool for many problems. Nevertheless, further formal attempts had to follow to tackle systems more generally.
In the analysed papers we found one attempt that strove to establish a new logic–the 'ternary description language'–a 'non-classical, deviant logic' which distinguishes itself from statement calculus and predicate calculus (Uyemov, 1999, 2002, 2003). An essential point is that besides thing, both properties and relations are basic categories. What is a thing in one context can become a property or a relation in another context. Uyemov (1999, p. 358) illustrates this by the following example: '[...] in the sentence "Love is a good affection", the word "Love" expresses a thing (=an object, an entity). In the sentence, "That affection is love", "love" is a property. In the sentence, "John loves Margaret", the word "love" denotes a relation'. Three further categories are definite, indefinite and arbitrary, which refer to the difference between 'the' thing, 'a' thing and 'any' thing. Uyemov formalizes the dual definition of a system into ordinary language: 'A system is an arbitrary thing in which a relation having a definite property is realized'; and 'A system is an arbitrary thing in which properties having a definite relation between them are realized' (Uyemov, 1999, p. 365). Only the future can tell whether this approach, which is currently not widely recognized, can lead to an advancement of GST.
The approach of Mesarovic and Takahara (1975, 1989) is the basis for several developments, such as the development towards the concepts of system transformation and adaptive systems by Saito (1999). Also, the language of general systems logical theory (GSLT) (Resconi et al., 1999) refers to Mesarovid and Takahara; GSLT is based on their input-output concept and is expressed by means of category theory concepts. The language of the more formal and general GSLT is said to be complementary to and has the same aim as the more problem-solving oriented general systems problem solver (GSPS) of Klir (1985). With the new language of GSLT, hidden structures of knowledge in various scientific areas can supposedly be investigated.
In combination with category theory and the mathematical conceptions of Mesarovid and Takahara, fuzzification was introduced in the area of system theory (Zlatos, 1996).
The work of Mesarovic and Takahara was also the basis for aiming at the development of a general physical systems theory. Such a unified theory within the realm of physics might also serve to inform GST with a 'general algorithm for recovering topological structure from unstructured data' (Bowden 1998).
Further related approaches, as mentioned by Takahara (2005), are based on the thread of mathematical general systems theory (MGST), as introduced by Mesarovid and Takahara: Forrester's system dynamics and Checkland's soft system methodology are examples of the GST methodology. Laszlo's natural system concept and Miller's living system concept are seen as typical grand theories. Zadeh's fuzzy system theory, Klir's formal general systems theory, Mesarovid's formal theory, Wymore's formal theory for systems engineering, Pichler's computer-aided system theory (CAST), Ziegler's formal theory and Wu's pansystem, which is said to be influential in China, are termed 'theories of general systems' (Takahara, 2005).
Another mathematical contribution – with roots in the work of Mesarovic and Takahara-was provided by Lin (1999), who mentioned open problems in the set-theoretic systems theory (Lin et al., 1997a,b). A system in this context denotes "the unification of 'isolated' objects, relations between the objects, and the structure of layers" (Lin et al., 1997a, p. 289). The assumed unsuccessfulness of GST was analysed by Lin et al. (1997a) and was to a certain extent attributed to a missing definition; Lin and Wang (1998, p. 24) gave a definition–first stated in 1987–that should have been a generalization of Mesarovid's definition: 'S is said to be a (general) system, if S is an ordered pair (M,R) of sets, where R is a set of some relations defined on the set M. Each 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 the system S, respectively'. Within Lin's approach, basic features of systems are interpreted in a formal way, such as centralized system, partial system, hierarchy, multilevels, feedback and input/ output. Epistemological and ontological questions on the relationship between objects and mathematical descriptions are addressed, but practical questions are also raised: 'Find topological structures and algebraic structures such that in systems analysis they appear to be the same' (Lin et al., 1997b, p. 595). Lin et al. (199719, p. 603) concluded that, among other things, GST should become more unified; that is it should be a 'blending [of] philosophy, mathematics, physical science, technology, etc. into one body of knowledge'. According to Lin and Wang (1998) a shortfall of the set theoretical approach to GST, which was initiated by Mesarovid and Takahara, lies in the difficulty 'to quantify any subject matter of research', but the former try to overcome this shortcoming. The new approach has also been applied to issues in physics, such as the rest mass of a photon.
A mathematically deduced structural model of general systems was developed by Lin and Cheng (1998). Behaviour, or the external action of a system, is a function of its internal state and input from the environment; the system itself consists of parts 'which react on each other', and the relationship equations of the parts can be governed by natural or social laws. Hierarchical structures and the motion of the structure can be described. With this mathematical framework the behaviour of storage cells in a computer is demonstrated, and the model can supposedly also be applied to the brain. Moreover, the model should serve not only for a 'better understanding and control of systems in nature and society' but also to solve 'particular scientific problems' (Lin and Cheng, 1999, p. 82).
Information systems theory (IST) is a new formal approach that builds on further developed GST and information theory to arrive at (computer) models with minimized uncertainty for various fields (physics, chemistry, biology, sociology, economics, technology, etc.). 'An information system (IS) is an interconnected set of interactions that exchange information, and which are capable of integrating themselves into a common information unit (subsystem, system)' (Lerner, 2004, p. 406). Random microlevel processes are modelled by stochastic differential equations. Regularities from the microlevel are transferred into the macrolevel, and described by macroprocess's equations. The main goal is to reveal IS regularities.
An approach towards a new axiomatic theory was made by Krivov et al. (2002). They referred to GST and complexity research (especially multiagent systems) when they developed a logic-based approach to deal with qualitative and quantitative parameters in a rule-based context. Formal treatment of organization and patterns was emphasized by modelling complex adaptive systems.
Synergetics, a transdisciplinary field of research for studying self-organization within complex systems, and GST are rarely mentioned at the same time. Nonetheless, it does occur in conjunction with discussions of quantitative sociodynamics by Helbing and Weidlich (1995), and there are certain connections.
Some approaches, for example Lloret-Climent (2002), Uso-Domenech et al. (2002), were based on mathematical approaches to systems, but these approaches focused on only one particular area of interest such as the cell or an ecosystem.
In summary, two major threads were evident in the developments of formalizing GST. One thread was based on the works of Mesarovic and Takahara, the other on the works of Klir.