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The Relevance of Sequential Time

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Knowledge and Time

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Abstract

Chapter 12 is devoted to the relevance of sequential time, a parameter time measuring the temporal distance between facts with a clock. But the apparently simple question of how facts are generated already leads to severe challenges that to date have not been uncontroversially resolved. In any event, once there are facts, they can be causally ordered, and experiments can be causally described. Most issues related to determinism, causation, prediction and retrodiction in science are based on sequential time.

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Notes

  1. 1.

    This subsection and some more material of Sect. 12.1 are based on a planned but unfinished chapter on Disentangling and Entangling Systems of Primas and Müller-Herold (1978). The draft version of that chapter remained unpublished since is was still in a not quite correct preliminary form.

  2. 2.

    The dynamics of group-theoretically defined elementary systems is usually automorphic, but this does not imply that the dynamics of interacting systems is automorphic. The dynamics of interacting systems must be derived in an unbiased manner from the known interactions without presupposing automorphic actions. Although this has not been achieved in general, many physically reasonable C*-algebraic systems without an automorphic dynamics are known.

  3. 3.

    In classical thermostatics, the entropy of a system is well-defined only if the system is in thermal equilibrium. Therefore, entropy is not a generally appropriate tool to order past events in a temporal sequence: non-equilibrium entropy cannot be defined in a unique way, see Meixner (1968).

  4. 4.

    Editor’s note: The remainder of Sect. 12.1 derives how this can be formulated in detail. It connects directly to Sect. 8.4.

  5. 5.

    Operations in this sense were introduced by Haag and Kastler (1964). Compare also Davies and Lewis (1970) and Davies (1976).

  6. 6.

    Compare Cornfeld et al. (1982, Theorem 2, p. 457), Schröder (1984), and Narnhofer and Thirring (1990).

  7. 7.

    Editor’s note: A recent study of real-world systems along these lines is due to Antoniou et al. (2016). Much in the spirit of the author, Antoniou and colleagues utilize the theory of so-called innovation eigenspaces of time operators to define an internal time scale (“age”) for the growth of facts in the evolution of dynamical systems.

  8. 8.

    See Leibniz’s third letter to Clarke, quoted from Alexander (1956, pp. 25–26).

  9. 9.

    See Leibniz’s fifth letter to Clarke, quoted from Alexander (1956, p. 77).

  10. 10.

    Compare also Mach (1902, p. 555): “The concept of cause is replaced …by the concept of function; the determining of the dependence of phenomena on one another”, and (p. 483): “There is no cause nor effect in nature; nature has but an individual existence; nature simply is.”

  11. 11.

    It is even difficult to express purely logical implications in a non-temporal manner. For example, the statement “a implies b” is usually explained by the sentence “when a occurs, then b follows”, or by the sentence “if I can prove a, then I can show that b is valid”.

  12. 12.

    Leibniz’s definition of co-existence is: “Given the existence of a multiplicity of concrete circumstances which are not mutually exclusive, we designate them as contemporaneous or co-existing.” Quoted from Wiener (1951, p. 202).

  13. 13.

    Compare for example Reichenbach (1956), Grünbaum (1973), van Fraassen (1970).

  14. 14.

    For a careful analysis of the problems of prediction and retrodiction cf. Watanabe (1969).

  15. 15.

    Editor’s note: See Primas (1997) for more details about forward and backward concepts of causation and determinism. It should be noted, though, that there are subtle differences between causation and determinism that the author does not address here. See for instance Atmanspacher and Filk (2012).

  16. 16.

    A dynamical system is a system whose state changes in time. A dynamical system is bi-deterministic if fixing its state at some time \(t\) implies that every state before or after \(t\) is uniquely determined. A system is called closed if all variables that can influence it have been taken into account in the specification of its initial state at time \(t\) (compare Havas 1965, 1968). Any other system is called open.

  17. 17.

    According to Collingwood (1998, p. 296), the term “cause” in experimental science “expresses an idea relative to human conduct, because that which causes is something under human control and this control serves as means whereby human beings can control that which is caused. In this sense, the cause of an event in nature is the handle, so to speak, by which human beings can manipulate it.”

  18. 18.

    A remarkable consequence of local quantum field theory is the PCT theorem, implying that invariance under proper Lorentz transformations implies invariance under the product of time reversal (T), charge conjugation (C), and space reflection (P). For a review, compare Grawert et al. (1959). If PCT-symmetry is broken, then Lorentz symmetry is also broken. See also Sect. 7.3.3.3.

  19. 19.

    A direct consequence of time-reversal symmetry is the following famous theorem by Felix Bloch (around 1933, unpublished; compare Brillouin 1933, p. 360f): Every system in a state of lowest free energy and not subject to external fields has vanishing current density everywhere. Or, in a provocative statement ascribed to Wolfgang Pauli: All theories of superconductivity are wrong.

  20. 20.

    For example Herbert Feigl (1953, p. 408) claimed: “The clarified (purified) concept of causation is defined in terms of predictability according to a law (or, more adequately, according to a set of laws).”

  21. 21.

    According to Fetzer and Almeder (1993, pp. 100f.), “ontic questions concern what is the case” while epistemic questions refer to our beliefs. “An ontic answer to an epistemic question (or vice versa) normally commits a category mistake.” In the context discussed here, see also Atmanspacher (2002).

  22. 22.

    For example, Max Born (1955a, 1955b) claimed that classical point mechanics is not deterministic since there are unstable mechanical systems which are epistemically unpredictable. For a critique of Born’s view compare von Laue (1955). Similarly, Leon Brillouin claimed that “we should never discuss what happens while we are not making any observation”, and that “initial conditions are not ‘given’; they must be measured and observed” (Brillouin 1960, p. 92, 1962, p. 314).

  23. 23.

    The publication dates (von Neumann: communicated December 10, 1931, published 1932; Birkhoff: communicated December 1, 1931, published 1931) do not give the proper chronological order. As Birkhoff and Koopman (1932) explain, von Neumann communicated his results to them on October 22, 1931, and “raised at once the important question as to whether or not ordinary time means exist along the individual path-curves excepting for a possible set of Lebesgue measure zero”. Shortly thereafter Birkhoff proved his individual ergodic theorem.

  24. 24.

    Borel (1914, p. 98) discussed a purely classical mechanical model of an ideal gas and assumed that a relative change of the gravitation potential of \(10^{-100}\) is not under our control. Such a change corresponds to an external perturbation due to a translation by 1 cm of a particle of mass 1 gram located on Sirius, \(8.3\times 10^{16}\) meters distant from earth. Then the prediction from classical mechanics for the positions of the molecules in a macroscopic sample becomes completely wrong after \(10^{-6}\) seconds. This instability derives from the fact that a slight change in the direction of motion of a particle is amplified at each collision.

  25. 25.

    Hermann Wold (1938) called a stationary forward predictable process singular if knowledge of its past allows an error-free prediction. It is called regular if it is not singular and if the conditional expectation is the best forecast. Unfortunately, Doob (1944) renamed singular processes “deterministic” and regular processes “nondeterministic”. Since determinism has nothing to do with predictions, this now popular terminology is based on a notorious category mistake (see Sect. 12.3), which we will avoid in what follows.

  26. 26.

    Compare Wiener’s papers and the editorial comments by Masani (1981, pp. 164–370).

  27. 27.

    For an elementary introduction, compare for example Cramér and Leadbetter (1967, Sects. 5.7 and 7.9).

  28. 28.

    An example can be found in Ash and Gardner (1975, problem 3 on p. 68).

  29. 29.

    Compare for example Cramér and Leadbetter (1967, Sect. 7.4).

  30. 30.

    For details, compare the example discussed by Doob (1953, Chap. XI, §4); or Cramér and Leadbetter (1967, Chap. 7.5).

  31. 31.

    The crucial imprimitivity relation has been recognized by Hanson (1958, Eq. 2.3 on p. 163) and by Kallianpur and Mandrekar (1965, p. 560). The time operator itself has been introduced by Tjøstheim (1976), who also discussed the canonical commutation relation between the time and frequency operator. Independently, these relations have also been found by Gustafson and Misra (1976).

  32. 32.

    Wold’s decomposition theorem refers to the special case of discrete-time stationary processes, and was extended by Hanner (1950) to the case of continuous-time processes. The general decomposition theorem is due to Cramér (1961).

  33. 33.

    See Kreı̌n (1945a, 1945b) and Wiener (1949). Wiener’s original work appeared as a classified report in February 1942. Editor’s note: The notion of a Wiener-Kreĭn criterion, unusual in the standard literature, was adopted by the author to appreciate essential ideas by Kreĭn in the development of Paley-Wiener criteria.

  34. 34.

    An algorithm which calculates the future values of a trajectory \(t\mapsto x(t| \mathbf{\omega })\), \(\mathbf{\omega }\) fixed, of a singular stochastic process for almost all \(t>0\) provided the past \(\{x(t| \mathbf{\omega })| t\le 0\}\) is given has been discussed by Scarpellini (1978a, 1978b, 1978c, 1978d).

  35. 35.

    For details compare for example Doob (1953, Chap. XII), Yaglom (1962), Rozanov (1967), Cramér (1961, Sect. 7.9).

  36. 36.

    In the context of statistical prediction theory, Masani and Wiener (1959b) called a non-anticipative forward moving-average representation an innovation representation. For details compare the monograph by Rozanov (1967) and the review by Kailath (1974).

  37. 37.

    Compare the review of Furstenberg (1960) by Masani (1963, p. 196).

  38. 38.

    Wiener’s early work was classified, his book Extrapolation, Interpolation and Smoothing of Stationary Time Series appeared in 1942 as Classified Report to Section D2, National Defense Research Committee of the USA. This report had an extraordinary influence in engineering circles and was released for general use in 1949.

  39. 39.

    For a summary of Wiener’s work on anti-aircraft fire control, compare his autobiography (Wiener 1956) and the biography by Masani (1990, Chap. 14).

  40. 40.

    Such averages were first used by Harald August Bohr (1932), Abram Samoilovitch Besicovitch (1932) and Vyacheslav Vassilievich Stepanov (1925) in their investigations of almost periodic functions and their discrete spectra. See also Maak (1950).

  41. 41.

    For convenience, we shall consider scalar continuous-time processes. However, vector-valued processes and discrete-time problems can be addressed in the same framework.

  42. 42.

    This formulation is taken from Masani (1990, p. 139f).

  43. 43.

    Wiener and Akutowicz (1957) proved that such a process is weakly mixing, which implies ergodicity.

  44. 44.

    Of course, Wiener did realize the virtue of thinking of certain irregular individual functions as trajectories of stochastic processes. He also introduced Gibbs’ statistical viewpoint into communication engineering.

  45. 45.

    Compare Bass (1974) and Bass (1984, Chap. IX). In his valuable commentary, Masani (1979) stressed the importance of the role of generalized harmonic analysis for non-stochastic approaches to turbulence and for the quest for randomness.

  46. 46.

    This stochastic process was introduced by Uhlenbeck and Ornstein (1930) as a physical model of Brownian motion.

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  1. (deceased), Lab. für Physikalische Chemie, Eidgen. Technische Hochschule Zürich, Zürich, Switzerland

    Hans Primas

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  1. Collegium Helveticum, Eidgen. Technische Hochschule Zürich, Zürich, Switzerland

    Harald Atmanspacher

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Primas, H. (2017). The Relevance of Sequential Time. In: Atmanspacher, H. (eds) Knowledge and Time. Springer, Cham. https://doi.org/10.1007/978-3-319-47370-3_12

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