Download Aristotle

A Conceptual Granularity Theory
for Objects in Space and Time
Karl Erich Wolff
Mathematics and Science Faculty
Darmstadt University of Applied Sciences
Ernst-Schröder-Center for Conceptual Knowledge Processing
Research Center Conceptual Knowledge Processing
Contents:
• Philosophical and Physical Aspects of Space and Time
• Granularity Theories
• Temporal Concept Analysis
Aristotle: Space – Time – Object
Physics, book III – VI:
• motion, change
• continuum, infinity
• place, empty place
• rigid body, object
Aristotle
• 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:
Space and Time are:
• not empiric, necessary
• pure perception a priori
• infinite
Immanuel Kant
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.“
t
Feynman Diagrams
R.P.Feynman
1918 -1988
x
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/
van Benthem’s Logic of Time
... is mainly concerned with
predicate logic over chains
referring to Hans KAMP and
Dov GABBAY.
Johan van Benthem
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).
Ernst Schröder Conference 2001
Logic and Knowledge
Temporal Logic
„We should, therefore, pay special attention to
discrete future-branching past-linear flows
of time.“
Dov Gabbay
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.
Connections to Theoretical Physics
Joachim Lambek (McGill Univ. Montreal)
told me his connections to the theoretical physicist
Chris Isham (Imperial College London)
and the similarity of my transition diagrams
to the Feynman diagrams.
Joachim Lambek
Goro Kato (Mathematics Department, California Polytec
State University, San Luis Obispo) visited me in the summer
of 2001 to discuss with me the relations between
Temporal Concept Analysis and his temporal descriptions
using categories, topoi and presheaves. He also has connections
to Chris Isham working in quantum gravity theory.
Goro Kato
The Emergence of Time in Quantum Gravity
J. Butterfield: All Souls College Oxford, C. Isham: Imperial College, London
•
the 'problem of time' in quantum theory: time is not quantized: (t)
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•
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„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.“
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•
•
Two approaches:
Quantized General Relativity
The Superstring Programme
The Emergence of Time in Quantum Gravity
(continued)
J. Butterfield: All Souls College Oxford, C. Isham: Imperial College, London
•
•
•
Should we assume a manifold?
Chain of successively richer structures
(set of spacetime-points  topology  differential structure  Lorentzian metric)
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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]
The Emergence of Time in Quantum Gravity
(again continued)
J. Butterfield: All Souls College Oxford, C. Isham: Imperial College, London
•
•
•
•
•
•
•
•
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.“
•
•
•
•
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„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.“
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)
Temporal Concept Analysis
•
... 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.
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
What do we mean by a State of a System?
•
•
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!
Conceptual Time Systems
(with a Time Relation)
g
Time part T
Event part C
v
w
Time scales
Event scales
(Time Relation)
h
Object concepts in
K(T)
K(C)
Time states
States
Situations
Transitions and Life Tracks
Transition: ( (g,h), (f(g), f(h)) )
Life Track f := {(g,f(g)) | g  G }
Objects, Actual Objects, Systems
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•
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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
Time part T
Event part C
v
w
Time scales
Event scales
Object 1
h
i
Object 2
j
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
John
Mary
roll
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!
[email protected]