 Original Message 
From: "John F. Sowa" <sowa@xxxxxxxxxxx>
To: "[ontologforum]" <ontologforum@xxxxxxxxxxxxxxxx>
Sent: Thursday, February 28, 2008 5:00 PM
Subject: Re: [ontologforum] Search engine for the ontology (01)
> AA> The universality of mathematics had been accepted since Euclid
> > and Nicomachus, who put quantity with its key species, multitude
> > and magnitude, as its subject matter.
>
> The *idea* of universality would go back farther, at least
> to Pythagoras. But he spent many years studying in Egypt and
> later in Babylon. There is no clear record of what any of
> those mathematicians believed, not even Pythagoras. But in any
> case, the idea of universality is a *goal* that has *never*
> been achieved in any closed, finished accomplishment. (02)
That's correct. Such nontrivial ideas, the world has a mathematical
structure, ''all is number'' and everything can be reduced to numerical
relationships, come from his school. The reason why i put forward Euclid is
rather simple: the axiomatic paradigm was first established by his geometry.
The axiomatic method suggests that a genuine scientific theory is a body of
original constructs: meaningful concepts and fundamental statements
(axioms, definitions, rules, laws). The meanings of other concepts are
defined from the primitive ones as well as the truths of subordinate
statements are deducted from a fundamental set of axiomatic truths. (03)
> AA> While Descartes, Whitehead, Russell extended the mathematical
> > universe by introducing order and relation. Its universality
> > implies a single axiomatic foundation regardless your practicing
> > mathematicians disregarding the mathematical foundation.
>
> That is the fundamental flaw in the argument. Mathematics does
> not have and never has had anything that could remotely resemble
> "a single axiomatic foundation". (04)
It had, remember logicism, which can be replaced by ontologism, covering the
axiomatic set theory and the category theory. (05)
That was a goal that had been
> proposed by Hilbert and pursued vigorously during the early
> 20th century. But it had been criticized by many professional
> mathematicians, even before Goedel. Afterwards, the goal seems
> hopeless  and *useless* even if it were possible.
>
> Practicing mathematicians  people who actually solve problems
> that other people pay somebody to solve  dismiss the study
> of foundations as *irrelevant*. For any given problem, they
> *never* start from axioms. Instead, they have a large toolkit
> of methods and techniques, which is constantly being enlarged
> by new methods all the time. For any particular problem, they
> start with informal intuitions, and only *after* they have found
> a solution do they state it in a closed form with a small set
> of problemspecific axioms. The axioms always come at the *end*,
> not the beginning of any mathematical research. And they are
> *always* problem specific, not universal. (06)
The axiomatic system basing on basic concepts and axioms and deduction, in
no way excludes induction. Induction, either as an intuitive generalization
ofyour experiences or an inference from experimental data, is the initial
source of axioms in empirical sciences and theoretical sciences as ontology. (07)
All great scientific minds followed Euclid' axiomatic approach trying to
establish a singel foundation in their field of knowledge. Leaving its
validity, take logicism, promoted by Frege, Russell and Whitehead, that all
of mathematics, its key principles, can be derived from the logical
principles. (08)
Extend a bit this approach and you may come to a more groundbreaking
statement, all of science, its basic principles, postulates and assumptions,
can be derived from the ontological principles.
>
> JFS>> The state of physics is much worse. See _The Road to Reality_
> >> by Roger Penrose, for a very good overview.
>
> AA> Here you need a great conceptual design, uniform ontological
> > design, single conceptual framing, a consistent and comprehensive
> > top ontology.
>
> Such an exercise would be worth writing a book, and I congratulate
> you on doing so. (09)
Thanks. It has citing of your works, and could be interesting reading for
you. Also for all who are interested in objective, i hope, analysis of the
SUO ontologies and the semantic web foundations. It also proposes how to
construct Nrelational ontology, natural language ontology model, and how we
can build the Digital Aristotle, or the Virtual Aristotle Machine (VAM). (010)
> But we had both come to the same conclusion: *all* understanding and
> problem solving is *always* task specific. (011)
Only at the initial stage, it is casespecific. At the final stage, you come
to generalizations. How the scientific method works: stating a problem,
collecting of data, formulating a hypothesis or law or rule, testing it by
deductions, proving a consistency with other hypotheses, establishing a
theory as a deductivehypothetical system. Scientific understanding is not
only accurate and exact but systematic also. Intelligence implies general
knowingness. (012)
Real science, not immature practical quasisciences, as politics, economics,
jurisprudence, is the sum of universal knowledge organized as the material
axiomatic system. Note the difference, not as the formal axiomatic syntactic
system with inherent inconsistency and incompleteness, but as the material
axiomatic semantic system. (013)
> 'I am just opposed to the idea that there must be exactly one or exactly
> two foundations or exactly N foundations. (014)
Right. Actually they are four. Any scientific theory (ST) involves several
foundations:
1. the factual laws and hypotheses (F);
2. the logical assumptions and principles (L);
3. he mathematical postulates and laws (M);
4. the semantic rules and formulas (S).
But there is always one and single background: ontological classes and
axioms (O). Or, the structure of science founds on a single ontological
foundation and four conceptual pillars, as in the formula:
ST = {O; F, L, M, S} .
The science means knowledge and certainty, not opinion and judgment. Its
principles are facts, definitions, axioms, hypotheses and unifying models
and theories. It's object is not the measurable and sensible but the
substantial and necessary, the abstract and universal, the universal laws of
reality. To reach this high aim, the science is using three techniques:
observation, experiment and reasoning, induction and deduction.
An enterprise not corresponding to the standard axiomatic structure of a
scientific theory, as practical sciences of ethics, politics, economics,
history, etc., is not strictly scientific. (015)
azamat abdoullaev
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