Karl Erich Wolff

Mathematics and Science FacultyDarmstadt University of Applied Sciences

Ernst-Schröder-Center for Conceptual Knowledge ProcessingResearch Center Conceptual Knowledge Processing

A Conceptual Granularity Theory

for Objects in Space and Time

Contents:

• Philosophical and Physical Aspects of Space and Time

• Granularity Theories

• Temporal Concept Analysis

Aristotle: Space – Time – Object

Physics, book III – VI:

Aristotle

• motion, change

• continuum, infinity

• place, empty place

• rigid body, object

• time, moment

• (state)

Aristotle: Motion and Change

Physics, book III starts:

Aristotle

„Nature is a principle of motion and change,

and it is the subject of our inquiry.

We must therefore see what motion is;

for if it were unknown, nature too would be unknown.“

Immanuel Kant: Space and Time

Kritik der reinen Vernunft, Band 2, Transzendentale Ästetik:

Immanuel Kant

Space and Time are: • not empiric, necessary • pure perception a priori• infinite

Time: • „has only one dimension“• „a form of the inner sense, i.e. a perception of our self and our inner state.“

Immanuel Kant: Change and Motion

Kritik der reinen Vernunft, Band 2, Transzendentale Ästetik:

Transzendental discussion of the concept of time: „The concept of change and, together with it, the concept of motion (as a change of the place) is possible only by and within an idea of time: that, if that idea would not be an (internal) perception a priori, no concept whatsoever, could make comprehensible the possibility of a change, i.e. a connection of contradictory opposite predicates (e.g. being at a place and not being of just the same thing at the same place) in one and the same object.“

Planck’s Act of Despair: Energy Quanta

Max Planck: (1858 – 1947)

„Über das Gesetz der Energieverteilung im Normalspectrum“

Annalen der Physik 4 (1901): 553-563

„... the whole procedure was an act of despair because a theoretical interpretation had to be found at any price, no matter how high that might be.“

Einstein’s Granularity Remark

Albert Einstein:

„Zur Elektrodynamik bewegter Körper“

Annalen der Physik 17 (1905): 891-921

Footnote on page 893:

„The inaccuracy which lies in the concept of simultaneity of two events at (about) the same place and which has to be bridged also by an abstraction, shall not be discussed here.“

Feynman Diagrams

It shows an electron, which starts out at 1 and moves through time-space to point 5, where it emits a photon, altering its path and moving along to point 3 where it goes off of the graph.  Another electron starts out at point 2 where it travels through space-time to point 6 and absorbs the photon emitted by the first electron.  Its path changes and it travels to point 4 where it leaves the graph. 

http://www.scs-intl.com/online/

x

t

R.P.Feynman 1918 -1988

van Benthem’s Logic of Time

Institute for Logic Language and Computation (ILLC), University of Amsterdam

Center for the Study of Language and Information (CSLI), Stanford

Main publications  The Logic of Time (1983), Modal Logic and Classical Logic (1985),  Essays in Logical Semantics (1986), Language in Action (1991),  Exploring Logical Dynamics (1996), Logic in Games (2001). 

Johan van Benthem

... is mainly concerned with predicate logic over chains referring to Hans KAMP and Dov GABBAY.

Ernst Schröder Conference 2001Logic and Knowledge

Temporal Logic

Augustus De Morgan Professor of Logic King‘s College London

Temporal Logic: Mathematical Foundations and Computational Aspects, vol 1: Mathematical Foundations (with I. Hodkinson and M. Reynolds) Oxford University Press, 1994, 671 pp.

This monograph is the standard reference work in the area.

Dov Gabbay

„We should, therefore, pay special attention to discrete future-branching past-linear flows of time.“

Connections to Theoretical Physics

Joachim Lambek (McGill Univ. Montreal) told me his connections to the theoretical physicistChris Isham (Imperial College London)and the similarity of my transition diagramsto the Feynman diagrams.

Goro Kato (Mathematics Department, California PolytecState University, San Luis Obispo) visited me in the summerof 2001 to discuss with me the relations between Temporal Concept Analysis and his temporal descriptions using categories, topoi and presheaves. He also has

connectionsto Chris Isham working in quantum gravity theory.

Goro Kato

Joachim Lambek

The Emergence of Time in Quantum Gravity

• the 'problem of time' in quantum theory: time is not quantized: (t)

• „Inaccessibility“ of the Planck-scale: No measurements possible at:• log(Planck-length [cm]) -33, • log(diam(quark) [cm]) -16• log(diam(atom) [cm] -8• log(Planck-time [sec]) -42

• „The fact that general relativity treets matter classically, • and gravity as curvature, while our best theories of matter• are quantum theories using a flat metric,• is enough to show that some sort of reconciliation is needed.“

• Two approaches: • Quantized General Relativity• The Superstring Programme

J. Butterfield: All Souls College Oxford, C. Isham: Imperial College, London

The Emergence of Time in Quantum Gravity (continued)

• Should we assume a manifold?• Chain of successively richer structures • (set of spacetime-points topology differential structure Lorentzian metric)

• Butterfield and Isham cite Riemann (Habilitationsschrift 1854):• „Now it seems that the empirical notions on which the metrical determinations of • space are founded, the notion of a solid body and of a ray of light, cease to be valid • for the infinitely small. We are therefore quite at liberty to suppose that the metric • relations in the infinitely small do not conform to the hypotheses of geometry; • and we ought in fact to suppose it; if we can thereby obtain a simpler explanation • of phenomena.“ [Translated by Clifford]

J. Butterfield: All Souls College Oxford, C. Isham: Imperial College, London

The Emergence of Time in Quantum Gravity (again continued)

• Butterfield, Isham (after the Riemann citation):• „Here then is a more radical sort of sense in which time, or better space-time,• might emerge. ... The usual tools of mathematical physics depend so strongly • on the real-number continuum, and its generalizations (from elementary calculus• ‚upwards‘ to manifolds and beyond), that it is probably even harder to guess what• non-continuum structure is needed by such radical approaches, than to guess what• novel structures of dimension, metric etc. are needed by the more conservative • approaches that retain manifolds.“

• „Furthermore, current approaches to quantum gravity face severe conceptual,• as well as mathematical, difficulties, and we must be ready for a complex and • unfamiliar relation between the conceptual frameworks of our present theories –• using the standard notions of space and time – and those of the, as yet, unknown • theory of quantum gravity.“

J. Butterfield: All Souls College Oxford, C. Isham: Imperial College, London

Precision and Granularity

Aristotle (Physics, book VI, 239a, line 23):

During the time when a system is moving, not only moving in some of its parts,

it is impossible that the moving system is precisely at a certain place.

We need a theoretical treatment of granularity!

Granularity Theories

• Fuzzy Theory

• Rough Set Theory

• Formal Concept Analysis

Zdzislaw Pawlak

Rudolf Wille

‘States’ in Modern Time Theories

• Automata Theory: States, Transitions

generalized by

• Labeled Transition Systems with Attributes:

No explicit time representation

• Petri Nets: States, Transitions, Token, Time

States and Transitions are

not related to Time

• Mathematical System Theory

What is a state? (Zadeh 1964)

What is a system? (Lin 1999)

• ... is the theory of temporal phenomena described with tools of Formal Concept Analysis

• It is the first formal theory defining states based on a general time description.

• It is a general temporal theory in the sense that it covers the continuous as well as the discrete case, the data table description as well as the law description of systems.

Temporal Concept Analysis

Seven Main Ideas in Temporal Concept Analysis

• States are defined as formal concepts• Instants (moments) are introduced as formal objects

• Basic Transitions are defined as elements of a binary relation on the instants• Transitions in some space (state space, situation space) are introduced

• Objects are introduced as subsystems having actual objects as formal objects• Life tracks of objects are defined using transitions

• Phenomena are abstracted from actual objects

• Something which is stable for a moment?

• Stable: Constant as to some granularity

• Moment (instant, point of time): A time object described in some context

• For each moment the system should be in exactly one state!

What do we mean by a State of a System?

Conceptual Time Systems(with a Time Relation)

g

h(Time Relation)

Time part T Event part C

Time scales Event scales

v w

K(C)K(T)

Time states States

Situations

Object concepts in

Transitions and Life Tracks

Transition: ( (g,h), (f(g), f(h)) )

Life Track f := {(g,f(g)) | g G }

Objects, Actual Objects, Systems

• object p

• actual object (p,g) (e.g.father in his youth)

• objects life tracks in the situation space

• Hence: „object = system“

Conceptual Time Systems with Actual Objects

and a Time Relation (CTSOT)

g

h

Object 2

Object 1

i

j

Time part T Event part C

Time scales Event scales

v w

The state space of a family (in the language of the

therapeut)

The Map Reconstruction Theorem

A state-transition covering: Linearly ordered CTSOT:  time part event part

(p,0) a s0

(p,1) b s1

(p,2) / s2

(q,0) a s0

(q,1) c s1

(q,2) d s3

(q,3) / s2

scales S(S,M,I)|(S,S,=)

Particles and Waves (1)

jump

roll

John

Mary

Phenomena

1. Idea: Generalize objects and waves to phenomena !

2. Idea: Study interaction!

3. Generalize Power Context Families (temporal and many-valued)

Next talk by Wendsomde Yameogo!

Thank you!

karl.erich.wolff@t-online.de