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]
Original Communication [353]
Response
index [355]
Communication from Kathy Blackmond Laskey, Virginia USA
Footnotes by Paul Prueitt
Not to be overly pedantic, but the body of statements one accepts initially is, in modern mathematics, not necessarily finite. For example, many if not most mathematicians regard Zermelo / Fraenkel set theory with the Axiom of Choice (ZFC) as the foundational axiom system for mathematics. ZFC has infinitely many axioms. CYC uses an alternative axiomatization for set theory, due to von Neumann, Godel and Bernays (NGB), who developed it because they wanted a finitely axiomatizable set theory. The price of finite axiomatization is the existence in NGB of entities, called "proper classes", that are not sets. Mathematicians have worked on the problem of the relative strength of ZFC and NGB, and have concluded they are essentially of equal strength. Many important mathematical and scientific theories are not finitely axiomatizable. In fact, mathematical theories may be based on axiom systems of uncountable cardinality.
In mathematics, an axiom is a statement that is accepted without proof. A formal system consists of a logic (logical axioms and rules of entailment) together with a set of proper axioms that capture the content of some domain of application. One attempts to study the consequences of the logical and proper axioms according to the rules of entailment, and to construct reasoning systems that are sound (i.e., don't allow the derivation of contradictions) and complete (i.e., allow one to deduce all consequences of the axioms). Godel proved that this is not possible in general.
In the 20th century, mathematics backed away from the requirement that axioms be "self evident." [1] With the advent of quantum theory and general relativity, much of modern physics is based on axioms very few people (if any) see as "self evident." Applied mathematicians attempt to find axioms that formalize the essential aspects of the domain they are modeling.
In this enterprise, there is
usually an interplay between intuitive naturalness and empirical adequacy --
but for 20th century science, empirical adequacy trumps "self
evidence". Pure mathematicians study "interesting" axiom
systems for their intrinsic beauty, and again, there is often nothing
"self evident" about the axiom systems they study. [2]
Kathy Blackmond Laskey
[1] Moderator comment: This is
historically true and is a subject with very high esthetic value. It is from this point that I feel the
discussion with Drs Werbos, Ballard and Abdoullaev (and others) in the
generative methodology thread should start.
See [127]
[2] Moderator comment: At issue with the Semantic Web and W3C use of language is whether the historical qualities of the term “axiom” are fully preserved in the common usage now found in the Protégé forum and in other forums, such as the ONTAC (US federal CIO council working group) where “axiom” does not always mean “self evident” to “any” reasonable persons. The axiom is often merely a structural constraint in the specification of an ontology, such as that a specific “class” has a property. This “axiom” is then “asserted along with sometime hundreds of thousands of others “axioms”, to produce “inferences”