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System Surfaces: There Is Never Just Only One Structure

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Abstract

When making modes, it is a constitutive question, how a system can be described and where the borderlines between a system and its environment can be drawn. The epistemic position of system realism (there are systems as ontic entities) and system nominalism (systems are only descriptions) leads to the concept of natural surfaces on the one side and to the authorship of defined borders on the other side. A more mediated position, the system descriptionism, is unfolded here: it is shown that changing objects or processes, which are described as systems, require new system theoretical concepts. As one of them, the notion of non-classical system is introduced, and a classification thereof is proposed.

Ich weiß ein allgewaltiges Wort auf Meilen hört’s ein Tauber. Es wirkt geschäftig fort und fort mit unbegriffnem Zauber ist nirgends und ist überall bald lästig, bald bequem. Es passt auf ein und jeden Fall: das Wort - es heißt System.

Franz Grillparzer (1791–1872)

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Notes

  1. 1.

    Franz Grillparzer (1791–1872), Austrian Poet, cf. [32, GG VI 316, p. 463].

  2. 2.

    The following section has been adopted and modified from a former paper, cf. [68].

  3. 3.

    Here we do not use the term “ontology” as it is used in the realm of computer science like in artificial intelligence, data base technology, and others (universe of discourse). We refer to the original term ontology as it is used in philosophy. Ontology is the teaching about being (Sein), existing things (Seiendes), and modalities thereof.

  4. 4.

    Cf. for instance Heraklit: Über die Natur. Fragment 1 and 2 [15, p. 77] or Fragments 10–13 [15, p. 131f.]. For medieval thinkers take for an example Heinrich Gent [29, p. 564 ff.]. Cf. also Hübner’s article about order (Ordnung) [50, pp. 1254–1279].

  5. 5.

    Like the Aristotelian book (1965) with the headline de generatione et corruptione [4].

  6. 6.

    Cf. [3, Poetry, Book 18, 1456a11] or in [5, Nicomachian Ethics, Book 9, 1168b32].

  7. 7.

    Cf. a comprehensive summary by [112, p. 1463].

  8. 8.

    For a more extensive presentation of the history of system concept cf. [35, 102, 104].

  9. 9.

    Heidegger [40] pointed out that a system is encompassing all what can be deduced. It induces structures and changes all knowledge of science, it saves it (the knowledge; the author) and brings it to communication; it may be conceptualized as the totality of grounding relations, in which the matter of science (as objects, germ.: “Gegenstände”; the author) is represented with respect to its basis, and where they can be understood conceptually [40, p. 56] (transl. by the author). This has been also explicitly pronounced by Heidegger in his famous SPIEGEL-Gespräch (1976) with Augstein [42]: What will come after Philosophy? Cybernetics. But this will not be Philosophy anymore. (transl. by the author).

  10. 10.

    Like cybernetics or control theory [109], theory of automaton [88, 91], theory of syntax [20], game theory [90], theory of communication [57, 58, 103], and general system theory [9]. All these theories deal with state space approaches, at least homomorphism between syntax theory and theory of automaton can be shown, discrete cybernetics are automata, and continuous automata can be modeled as control circuits. Game theory can be modeled as automata, and the theory of communication can be formulated either as statistics of stochastic signal processes or even as noisy automata (i.e., communication channels, receivers, and emitters). These theories may considered to be classical theories (see Sect. 4).

  11. 11.

    In this rigidity the statements hold exactly only for linear type behavior of the elementary behaviors.

  12. 12.

    The structure, shown in Fig. 2, serves as an example.

  13. 13.

    Cf. for this distinction at [41, Sect. 15, p. 69 ff.] between a present object, found by man, i.e., Vorhandenes = presented-at-hand (as an objet trouvé), and a tool as Zuhandenes = the readiness-to-hand. For the translation of this concept by J.M. Macquarrie and E. Robinson cf. [41, p. 98 f.].

  14. 14.

    Cf. Democrit in [15] pp. 396–405 (fragments 68 A) and p. 418 (fragment 68 A 71). Notation according [22], Vol. 2, p. 41 ff. and p. 95 ff. respectively.

  15. 15.

    The set theoretic definition of a system is given according to [86, p. 254 ff.]. A general system is a relation on abstract sets S ∈ X{V i ; i ∈ I} with X as the Cartesian product. I is the index set. If I is finite, it is \(S \in V _{1} \times V _{2}\ldots V _{n}\); the components of the relation V i are called objects of systems. They can be divided into the input variables X ∈ { V i ; i ∈ I X } and the output variables Y ∈ { V i ; i ∈ I y } such that an input/output system can be written as S ⊂ { X ×Y }. The definitions in Chap. 1.2 can be regarded as specifications of this concept.

  16. 16.

    Let us presuppose that some modifications in the mathematical apparatus of system theory are possible. These modifications are discussed in [87, 108], and [61] with respect to numerous applications and disciplines.

  17. 17.

    Structurally seen, a method to found a system of rules, i.e., for moral philosophy, in a pragmatic-transcendental way (acc. to [2]) is coextensive with a method of foundation for rules of the use of a system. More in detail: any pragmatic interpretation of scientific law leads only to a rule of action, if the validity of the law and the acceptance for the necessity of existence thereof are unanimously accepted as an a priori in the realm of a dispute procedure. If, when designing a system model, validity and acceptance cannot be denied without contradictions, then one has a constitutive presupposition for the design itself.

  18. 18.

    For this section cf. also [63, Chap. I.4.].

  19. 19.

    Oral communication by Walter v. Lucadou, 1974.

  20. 20.

    This is not very original, but it used to be forgotten quite often. The relevant passage at Kant in Critique of Pure Reason: Philosophical cognition is the cognition of reason by means of conceptions; mathematical cognition is cognition by means of the construction of conceptions. The construction of a conception is the presentation a priori of the intuition which corresponds to the conception. For this purpose a nonempirical intuition is a requisite, which, as an intuition, is an individual object; while, as the construction of a conception (a general representation), it must be seen to be universally valid for all the possible intuitions which rank under that conception. [“Die philosophische Erkenntnis ist die Vernunfterkenntnis aus Begriffen, die mathematische aus der Konstruktion der Begriffe. Einen Begriff konstruieren heißt: die ihm korrespondierende Anschauung a priori darstellen. Zur Konstruktion eines Begriffs wird also eine nicht empirische Anschauung erfordert, ohne folglich, als Anschauung, ein einzelnes Objekt ist, aber nichtsdestoweniger, als die Konstruktion eines Begriffs (einer allgemeinen Vorstellung), Allgemeingültigkeit für alle möglichen Anschauungen, die unter denselben Begriff gehören, in der Vorstellung ausdrücken muss.”] (cf. [53, B 741], translated by J. M. D. Meiklejohn; cf. http://www.gutenberg.org/cache/epub/4280/pg4280.html).

  21. 21.

    Einstein has paraphrased this Kantian position in his famous lecture for the Prussian Academy of Science about “Geometry and Experience”: As long as mathematical propositions refer to reality they are not certain, and as long as they are certain [well proved], they don’t refer to reality. Cf. [23, p. 119 f.] (transl. by the author).

  22. 22.

    See textbooks for orbital calculations [11].

  23. 23.

    These conditions are not valid, if the concept of derivability and the concept of conclusion (i.e., the concept of truth and provability) don’t coincide anymore. This is the case in higher level calculi, proved by Gödel [31]. This leads to a lot of restrictions for system descriptions when dealing with complex systems that must be modeled by such higher level calculi. Cf. [63, Chap. III.2.].

  24. 24.

    See above; for [17, Part. Sect. 9–11], this was a pseudo-problem of philosophy.

  25. 25.

    Heisenberg [45, p. 92] reported about an entretien with Albert Einstein in the year 1927. According to that, Einstein said: From a heuristic point of view it may be useful to remember what we really observe. In principle it is completely wrong to ground a theory on only observable entities. The reality runs the other way around: It’s just the theory which decides what you can observe. (Transl. by the author). Later, Heisenberg adopted this point of view: We have to remember that what we observe is not nature [here] itself, but nature exposed to our method of questioning. Cf. [44, pp. 58, 41].

  26. 26.

    The concept of function has not yet made explicit. Nevertheless it is comprehensible in terms of a mathematical function as a mapping; cf. [60].

  27. 27.

    This is expressed by the so-called Ramsay Theorem; cf. [16, pp. 246–254].

  28. 28.

    For reductionism see also [82] and [63, Chap. III.4.].

  29. 29.

    This is due the fact that we are used to design concepts. Adopting the picture of Klix [56, pp. 536–549], it is clear that each observation based science presupposes the possibility of invariants. Invariant properties can be characterized by a structural difference to what one can observe. Each object seems to be higher structured as its environment—this is what an observer states if he speaks about an object. What can be observed potentially from the forms of observation are possible structures of objects. These and only these structures correspond with the possible forms of observation. This is the reason to use the term “coextensive.”

  30. 30.

    This is not a concept which could be interpreted as an operative measurement rule (Messvorschrift). To each observable there is a concept, but not vice versa, cf. [16, p. 59ff.].

  31. 31.

    Here structure has the meaning we ascribed to it in Sect. 1, Fig. 2.

  32. 32.

    The concept of preparation is discussed intensively with respect to quantum mechanical measurement. The preparation of the constraints is an ontic presupposition for the availability of a system and therefore for an experiment. On the level of description we have the definition of the constraints as initial and boundary conditions, together with the dynamics of a system in order to make predictions.

  33. 33.

    As proposed by Weizsäcker [107, p. 259].

  34. 34.

    This is no contradiction to the saying above according to which system theory could be a theory of possible structures. A measurement device presupposes an elementary structure, realized in an apparatus that is defined by the concept, which is the base of the observed variable. Thus an observed structure must have a relation to the elementary structure of the measurement device. This relation is contained within a theory about the field of objects in question.

  35. 35.

    Cf. [93, Chap. 4, particularly Chap. 4.4, pp. 233–271]. In quantum theory Hilbert spaces are applied. When controlling systems non-interaction-free observed, dual spaces like Hilbert spaces are applied, too, e.g., for linear time discrete systems. Even the problem of nondistinguishable state is discussed there.

  36. 36.

    Cf. Heisenberg’s original paper [43]; with respect to this see also the canonical textbook of Landau and Liefschitz [76, vol. 3, Sect. 16, p. 49ff.].

  37. 37.

    Nevertheless, an estimation of these properties with observers free from interaction is also restricted when dealing with nonlinear systems. For exact definitions see textbooks of control and mathematical system theory. For illustration here some verbal definitions: Controllabilty is given, if it is possible, to “move” the system through the state space from a defined start state into a finite state (or area if states) of interest in a finite time by applying an input. Stability is the property of the system to remain at or come back to a finite state or area of states asymptotically, invariant from a given input. Reachability: The system behavior has an area or points within the state space that can be reached by a given input from an arbitrary starting point in a finite time. Feed-forward (or feed-through): the system output can be controlled to a given degree directly by the input. To this degree the system can be considered as a channel for the input. Observability means that one can estimate the starting point of a system by measuring the input and the output dynamics. Cf. [80, 93]. A real theoretically satisfying formalization of the control problem has been given by [8, 51, 97]. For history cf. also [111].

  38. 38.

    The picture of incompressibility of an algorithmic description of complex processes has been developed independently by Kolmogorov and Chaitin [18, 19, 58].

  39. 39.

    A critique of the concept of emergence is discussed in [67] and [70].

  40. 40.

    Attractors are points or finite areas within the state space, in which the system moves and remains asymptotically. Therefore systems with attractors are stable, but not necessarily controllable, since stability and reachability is only a necessary but not sufficient condition for controllability. See also textbooks about control and system theory.

  41. 41.

    In contrast to the modus ponens ([ ∀x(A(x) → B(x)) ∧ A(e)] → B(e)), the abduction concludes from the particular to the general: ([ ∀x(A(x) → B(x)) ∧ B(e)] → A(e)). The proposition logical expression ([(a → b) ∧ b] → a) is falsifiable, i.e., not always true.

  42. 42.

    For badly understood phenomena one is not successful to reduce them to law-like principles within the realm of a theory.

  43. 43.

    I refer here to several oral communications in talks with F. H. Zucker in Cambridge, M.A. in 1981 and 1982. Cf. also some hints in [113] and presumably in [114]. I have not yet found an extensive exploration of this hypothesis within literature, but this could be an interesting research question.

  44. 44.

    A detailed discussion of such complementarily related observables has been given by Röhrle [99]. He tried to show the hypothesis that the appearance of complementarily related entities within a system description indicates sign that the field of interest is not yet satisfyingly understood.

  45. 45.

    This has been shown by Kornwachs et al. [63, 71, 72] using a measure of complexity which is defined by the degree of time dependence of system structure.

  46. 46.

    Cf. [46]. For modern discussion see, e.g., [7, 13].

  47. 47.

    Nevertheless, it is not yet clear how the interaction between variables dependent upon time and location and invariant variables can be explained in an ontologically satisfying manner. From a phenomenological point of view one can model this within a system description if one has an idea how this interaction could be hypothesized. Cf. [64, 101].

  48. 48.

    It is really astonishing that processes with changing structures and patterns of behavior like decay and growth cannot be modeled satisfyingly with usual system theoretical descriptions, based on elementary set theory and mappings. This would require designing sets mathematically, which can generate or annihilate their own elements without a choice process from another finite set, of already “stored” elements. There is a correspondence to this problem with the fact that there is no real calculable GOTO in the theory of algorithms.

  49. 49.

    To introduce a wave function ψ in quantum mechanics is an actual strategy to do so. As a state ψ is no observable, but it obeys a linear “equation of motion,” the well-known Schrödinger equation. Even mathematical operations on this wave function will allow a calculation of the possible values (eigenvalues) of observables.

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Acknowledgments

This paper is based on a presentation, held at Philosophisches Kolloquium, December 3rd, 2008 at the Institute for Philosophy, University of Jena, Frege Centre for Structural Sciences. The inspiring discussions with Bernd-Olaf Küppers and Stefan Artmann are gratefully acknowledged.

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  1. Brandenburgische Technische Universität Cottbus, Cottbus, Germany

    Klaus Kornwachs

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  1. , Frege Centre for Structural Sciences, Friedrich Schiller University Jena, Zwätzengasse 9, Jena, 07743, Germany

    Bernd-Olaf Küppers

  2. Jena University Language & Information E, Frege Centre for Structural Sciences, Friedrich Schiller University Jena, Fürstengraben 30, Jena, 07743, Germany

    Udo Hahn

  3. , Frege Centre for Structural Sciences, Friedrich Schiller University Jena, Zwätzengasse 4, Jena, 07743, Germany

    Stefan Artmann

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Kornwachs, K. (2013). System Surfaces: There Is Never Just Only One Structure. In: Küppers, BO., Hahn, U., Artmann, S. (eds) Evolution of Semantic Systems. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-34997-3_3

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