1. 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

2. 1.Introduction: Problems in Physics 2.Temporal Concept Analysis 3.Examples of Conceptual Time Systems 4.Time Dimension and Branching Life Tracks 5.Object Identification: Conceptual Representation of a Tale 6.Particles and Waves Outline

3. 1.Reality – Observation - Theory 2.Formal Representations: Numbers, Scales, and Systems 3.Continuity and Discreteness 4.Granularity, States, and Situations 5.Transitions and Life Tracks 6.Objects, Particles, and Waves Problems in Physics

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. Einsteins 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) g h Object 2 Object 1 i j Time part TEvent part C Time scalesEvent scales vw K(C) time statesstates situations object concepts: derived contextK(T) |

8. Harmonic Oscillator: State Space y x

9. Harmonic Oscillator: Situation Space

10. Stars, Morning Star, Evening Star, Venus morningeveningMonday eastwest

11. Stars, Morning Star, Evening Star, Venus morningeveningTuesday eastwest

12. Morning Star Evening Star morningevening Monday eastwest Monday Tuesday

13. Venus morningevening Monday eastwest Monday Tuesday

14. Venus: Morning Star and Evening Star MoTumoeveawelu 1 2 3 4

15. Life Tracks: Morning Star, Evening Star

16. Example: A Journey of John and Mary

17. 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 ? Time Dimension = 1 ???

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. Time Relation DayTown (letter,0)0A (letter,1)1B (letter,2)2C (letter,3)3D 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 hedgehogs wife.

24. Hare and Hedgehog: partial data actual objects objecttime hare- object placerunning hedgehog0hedgehog0 houseno hh-wife0hh-wife0hedgehoghouseno hedgehog1hedgehog1 fieldno

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

26. Hare and Hedgehog: Situation Space

27. The Hares View: Situation Space

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