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Particles and Waves in Conceptual Time Systems

Particles and Waves in Conceptual Time Systems. Karl Erich Wolff Mathematics and Science Faculty University of Applied Sciences Darmstadt Ernst Schröder Center for Conceptual Knowledge Processing Research Group Concept Analysis at Darmstadt University of Technology. Outline.

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Particles and Waves in Conceptual Time Systems

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  1. Particles and Wavesin Conceptual Time Systems Karl Erich Wolff Mathematics and Science Faculty University of Applied Sciences Darmstadt Ernst Schröder Center for Conceptual Knowledge Processing Research Group Concept Analysis at Darmstadt University of Technology

  2. Outline • Introduction: Problems in Physics • Temporal Concept Analysis • Examples of Conceptual Time Systems • Time Dimension and Branching Life Tracks • Object Identification: Conceptual Representation of a Tale • Particles and Waves

  3. Problems in Physics • Reality – Observation - Theory • Formal Representations: Numbers, Scales, and Systems • Continuity and Discreteness • Granularity, States, and Situations • Transitions and Life Tracks • Objects, Particles, and Waves

  4. Theories of Space and Time • Aristoteles: continuum, point of time, duration, time as a category • Classical Physics: example: x“(t)=a; x(t)=x(0) + v t + ½at² • Special and General Theory of Relativity: Space-Time (curved) • Quantum Theory: Time-dependent Schrödinger function ψ(t) • Automata theory: States, Transitions (without an explicit time description) • Mathematical System Theory: State? System? • Quantum Gravity: Looking for a theory with an appropriate time description

  5. Einstein’s Granularity Remark Albert Einstein: „Zur Elektrodynamik bewegter Körper“ Annalen der Physik 17 (1905): 891-921 Footnote on page 893: „Die Ungenauigkeit, welche in dem Begriff der Gleichzeitigkeit zweier Ereignisse an (annähernd) demselben Orte steckt und gleichfalls durch eine Abstraktion überbrückt werden muß, soll hier nicht erörtert werden.“

  6. A General Theory for Objects in Space and Time ... • should cover discrete and continuous descriptions • needs a theory of granularity • needs a suitable notion of ‚system‘ and a notion of ‚states‘ depending on the granularity • transitions: time-dependent changes between states • objects as subsystems (and not as „atoms“) • should construct a temporal logic based on these temporal systems

  7. Conceptual Time Systems with Objects and a Time Relation (CTSOT) Time part T Event part C g v w Object 1 h i Object 2 j Time scales Event scales K(C) derived context K(T) | time states states object concepts: situations

  8. Harmonic Oscillator: State Space y x

  9. Harmonic Oscillator: Situation Space

  10. Stars, Morning Star, Evening Star, Venus Monday morning evening east west

  11. Stars, Morning Star, Evening Star, Venus Tuesday morning evening east west

  12. Morning Star Evening Star morning evening Monday Tuesday Tuesday Monday east west

  13. Venus morning evening Monday Tuesday Tuesday Monday east west

  14. Venus: Morning Star and Evening Star

  15. Life Tracks: Morning Star, Evening Star

  16. Example: A Journey of John and Mary

  17. Time Dimension = 1 ??? • Using the order dimension we define: • The time dimension is the order dimension of the concept lattice of the time part • The time scale dimension is the order dimension of the time scale • These dimensions may be greater than one! • Are life tracks 1-dimensional ?

  18. Branching Life Tracks? Unique-State-Lemma: A conceptual time system is at each time granule in exactly one situation, in exactly one state, and in exactly one time state. But branching of life tracks is possible! Example: We send an „abstract letter“ in two copies. A B C D

  19. Two Gymnasts: Roll and Jump

  20. The Waves of Rolls and Jumps

  21. Wave of a Population

  22. Rigid Body ? Translation Rotation Reflection

  23. Indistinguishable Objects ? An observer of a temporal system who describes that system by a CTSOT has to decide how to choose the objects of the CTSOT. Like tennis balls they may be indistinguishable with repect to some parts of the chosen description. In the following German tale Der Wettlauf zwischen dem Hasen und dem Igel (The Race Between the Hare and the Hedgehog) the hare cannot distinguish the hedgehog and the hedgehog‘s wife.

  24. Hare and Hedgehog: partial data

  25. Hare and Hedgehog: State Space hedgehog hh-wife hare

  26. Hare and Hedgehog: Situation Space

  27. The Hare‘s View: Situation Space

  28. Thank you! www.fbmn.fh-darmstadt.de/home/wolff/index.htm

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