Thursday, January 12, 2006
Additional reading:
Cory Casanave's paper on Data
Access
work on ontology for biological signal pathways
[127]
ß parallel discussion in
generative methodology bead thread
[127]
ß parallel discussion on Rosen
complexity
On the meaning of “axiom”
Earlier
part of this discussion
[342], [343], [344], [345], [346], [347], [348], [349], [350]
John (Sowa) and
Paul (Werbos)
Assertion: The semantic web community has
created a new meaning for the term "axiom". This new
meaning is related to the classical (and well understood)
notions about the small number of axioms and postulates (5 plus 5) of
Greek geometry. But the new meaning is not clear and not consistent
from one semantic web group to the next. I will draw a conclusion about
the mainstream use of the term "complexity" as being or related to
the computations of a computer program.
quote from http://library.thinkquest.org/22584/emh1200.htm
Certainly one of the greatest achievements of the
early Greek mathematicians was the creation of the postulational form of
thinking. In order to establish a statement in a deductive system,
one must show that the statement is a necessary logical consequence of some
previously established statements.
These, in their turn, must be established from some
still more previously established statements, and so on. Since the
chain cannot be continued backward indefinitely, one must, at the start, accept
some finite body of statements without proof or else commit the unpardonable
sin of circularity, by deducing statement A from statement B and then later B
from A. These initially assumed statements are called the postulates,
or axioms, of the discourse, and all other statements of the discourse
must be logically implied by them. Where the statements of a
discourse are so arranged, the discourse is said to be presented in
postulational form.
So great was the impression made by the formal aspect
of Euclid's Elements on following generations that the work became a
model for rigorous mathematical demonstration.
It is not certain precisely what statements Euclid
assumed for his postulates and axioms, nor, for that matter, exactly how many
he had, for changes and additions were made by subsequent
editors. There is fair evidence, however, that he adhered to the
second distinction and that he probably assumed the equivalents of the
following ten statements, five "axioms," or common notions, and five
geometric "postulates":
A1 Things that are equal to the same thing are also equal to one another.
A2 If equals be added to equals, the wholes are equal.
A3 If equals be subtracted from equals, the remainders are
equal
A4 Things that coincide with one another are equl to one
another.
A5 The whole is greater than the part.
P1 It is possible to draw a straight line from any point to
any other point.
P2 It is possible to produce a finite straight line
indefinitely in that straight line.
P3 It is possible to describe a circle with any point as
center and with a radius equal to any to finite straight line drawn from the
center.
P4 All right angles are equal to one another.
P5 If a straight line intersects two straight lines so as to
make the interior angles on one side of it together less than two right angles,
these straight lines will intersect, if indefinitely produced, on the side on
which are the angles which are together less than two right angles.
John (Sowa) {John’s reply à [354]}
I will ask you to describe what the term "axiom"
means within this large and influential community.
You know I have objected to the appearance of many meanings and the
utter inability of most in the OWL user groups (for example) to clearly specify
what "axiom" means TO him or her. You talk consistantly
about "the axioms" of an ontology, and I quite literally do not know
what you are talking about, because as a trained PhD in mathematics the axioms
of a branch of mathematics, geometry, number theory or real analysis is
not what is being talked about. Nor is there the clarity that one sees in
the axioms and postulates of geometry.
If we talk about the axioms of First Order Logic, then how many are
they? What is the list?
Is an axiom an assertion that a OWL class has an is a relationship to
an other OWL class?
People talk about an ontology as a set of concepts, relationships
between concepts, attributes and/or properties of these concepts, then there
are the "axioms" and inference engines. These axioms and
inference engines seem in infinite variety. Is there, any more
in the new meaning, the classical notions of self evidence of anything called
an "axiom"?
I made the argument, not accepted by the ONTAC group, that an ontology
should be free of inference, because inference is a logical entailment, and my
training in works by Rosen and others tells me that entailment is the key issue
that is not gotten right as yet. But we, our society, could
really use controlled vocabularies and concept ontologies.
see notes by Judith Rosen on this at:
http://www.ontologystream.com/beads/nationalDebate/347.htm
and
http://www.ontologystream.com/beads/nationalDebate/353.htm
As part of this discussion Paul Werbos is a bit critical of my criticism
of computer science in it's defining "complexity" as something which
is computational...
I wrote something on this in 1997
http://www.bcngroup.org/area3/pprueitt/kmbook/Chapter2.htm
but all of this is background to the issue of importance and that is the
control of language by computer science ... in the use (misuse) of terms like
formal semantics
artificial intelligence
logical expressiveness
computational complexity
and
axiom
Can we get a handle on this?
Dr Paul S Prueitt
The Taos Research Institute
Taos New Mexico