This mail is publicly posted to a distribution list as part of a process
of public discussion, any automatically generated statements to the
contrary non-withstanding. It is the opinion of the author, and does not
represent an official company view. (01)
The objection to ontologism from the mathematical point of view is that
alternative choices of axioms give alternative mathematical systems. The
non-Euclidean geometries are variations on the fifth postulate that all
non-parallel lines meet. The way I was taught, say, ring theory, was
that it was group theory with additional axioms. That is, the
mathematical system you get is dependent on the axioms you use.
Similarly, the logic you get depends on the axioms, such as the number
of truth values you choose. (02)
Given that systems can contradict each other if they have different
axioms, it is difficult to see that a single meta system could derive
all mathematical principles, unless the principles themselves are not
sufficient to detail all mathematical systems. This is rather like
saying all English words can be written with the 26 letters of the
alphabet - it is not something that allows us to generate all valid
English without generating a larger number of invalid ones. (03)
As a matter of engineering practice, for complex products, there is no
single product breakdown system that adequately represents a product.
For example, the assembly breakdown structure of an aircraft defines the
components that make up an assembly stage, and we can ship these round
or store them as we need. However, this breakdown is no use when we look
at the installed systems, which are proper subsets of multiple assembly
stages, and which must be functionally analysed as a whole. (04)
Sean Barker
BAE SYSTEMS - Advanced Technology Centre
Bristol, UK
+44(0) 117 302 8184 (05)
BAE Systems (Operations) Limited
Registered Office: Warwick House, PO Box 87, Farnborough Aerospace
Centre, Farnborough, Hants, GU14 6YU, UK
Registered in England & Wales No: 1996687 (06)
> -----Original Message-----
> From: ontolog-forum-bounces@xxxxxxxxxxxxxxxx
> [mailto:ontolog-forum-bounces@xxxxxxxxxxxxxxxx] On Behalf Of Azamat
> Sent: 02 March 2008 18:36
> To: rick@xxxxxxxxxxxxxx; [ontolog-forum]
> Subject: Re: [ontolog-forum] Search engine for the ontology
>
>
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> Saturday, March 01, 2008 1:12 AM, Rick wrote:
>
> > Could you please define *ontologism*, how it replaces
> logicism and how
> it *covers* category theory ? The references I'm finding
> don't seem relevant to the discussion.
>
>
> Ontologism is the theory that all of scientific and
> mathematical principles can be derived from a universal
> ontology, its fundamental classes, rules, axioms or
> constraints. The core of it is an intersection of fundamental
> mathematics and universal ontology, resulting in mathematical
> ontology or ontological mathematics; don't mix with formal
> logical ontology.
>
> In such extensive knowledge system, the axioms of the set
> theory and order theory, the category theory and mereology
> will appear as derived truths
> (theorems) of ontological axioms of entities and relations.
>
> For instance, the undefined notions of the above mathematical
> theories (the set membership relation, the set inclusion
> relation, the mapping relation, the part-whole relation) come
> as special kinds of general relationship.
>
> Considering the lack of a single foundation of mathematics
> and science, such a scenario is very and very possible, if
> unavoidable.
>
> Since the proposal is rather new and highly controversial, I
> better refer you to a more detailed account:
>
> Part III. THE WORLD CODE: Mathematical Ontology as the Real
> Road to Reality.
>
>
> > I don't see the A word here: Abduction. I don't go anywhere
> without it.
>
> I am aware of the Peirce's idea of scientific activity as
> involving induction, abduction, and deduction. As far as the
> range of connotations for abductive reasoning is fuzzy
> http://en.wikipedia.org/wiki/Abductive_reasoning, i try to
> avoid its use.
> It looks as a kind of inferential reasoning, where also
> included implication, corollary, analogy, extrapolation,
> derivation. But abduction is hardly an intuitive generalization.
>
> >As an aside, have folks been tracking what Mark Greaves is up to at
> >Vulcan ?
> > videolectures.net/iswc06_graves_img/
> > Seems like he found an ideal customer in Paul Allen.
>
> That might be so. But Allen is worth much respect as a smart
> even intelligent billionaire, funding such a mostly high risk
> but high impact speculative projects as Digital Aristotle.
> Look at our Russian brainless
> multi-billionaires: supermodels, football teams, castles,
> etc. Think, the same may apply to 99% of a thousand and more
> of this sort of people.
>
> azamat abdoullaev
>
> ----- Original Message -----
> From: <rick@xxxxxxxxxxxxxx>
> To: "[ontolog-forum]" <ontolog-forum@xxxxxxxxxxxxxxxx>
> Sent: Saturday, March 01, 2008 1:12 AM
> Subject: Re: [ontolog-forum] Search engine for the ontology
>
>
> > Ravi, Azmat & All
> >
> > >
> > Sharma, Ravi wrote:
> >> Azamat
> >>
> >> Yes
> >> As you say, observation, experiment and reasoning, induction and
> >> deduction I also concur that any connection with validation or
> >> verification, are pillars for accuracy of any scientific theory?
> >> Thanks.
> >
> > If you really mean to limit inference to induction and
> deduction, this
> > puts us in a pretty small box. Isn't this the problem with normal
> > science awaiting the next paradigm shift.
> >
> >> Ravi
> >>
> >>
> >>
> >>> 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.
> >>
> >> 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.
> >
> > This is a closed system and just leads to inconsistency issues.
> >
> >>> 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".
> >>
> >> It had, remember logicism, which can be replaced by
> ontologism, covering
> >> the
> >> axiomatic set theory and the category theory.
> >
> > Could you please define *ontologism*, how it replaces
> logicism and how
> > it *covers* category theory ? The references I'm finding don't seem
> > relevant to the discussion.
> >
> > http://www.newadvent.org/cathen/11257a.htm
> >
> >> 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 problem-specific axioms. The axioms always come at the *end*,
> >>> not the beginning of any mathematical research. And they are
> >>> *always* problem specific, not universal.
> >>
> >> The axiomatic system basing on basic concepts and axioms
> and deduction,
> >> in no way excludes induction. Induction, either as an intuitive
> >> generalization of your experiences or an inference from
> experimental
> >> data, is the
> >> initial source of axioms in empirical sciences and
> theoretical sciences
> >> as
> >> ontology.
> >>
> >> 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.
> >
> > But, not all *logical principles* are as you presume them
> to be. Along
> > with induction and deduction, we require abduction. I think
> that's what
> > Azmat's referring to above as *informal intuitions.*
> >
> >> Extend a bit this approach and you may come to a more
> ground-breaking
> >> statement, all of science, its basic principles, postulates and
> >> assumptions,
> >> can be derived from the ontological principles.
> >
> > So how does this square with Kuhn's Structure of Scientific
> Revolutions
> > in which he claims that paradigm shifts come from anomaly
> and crisis ?
> >
> >> construct N-relational ontology, natural language ontology
> model, and
> >> how we
> >> can build the Digital Aristotle, or the Virtual Aristotle
> Machine (VAM).
> >>
> >
> > As an aside, have folks been tracking what Mark Greaves is
> up to at Vulcan
> > ?
> >
> > videolectures.net/iswc06_graves_img/
> >
> > Seems like he found an ideal customer in Paul Allen.
> >
> > --
> > Thanks Rick,
> > blog http://spout.rickmurphy.org
> > web http://www.rickmurphy.org
> > cell 703-201-9129
> >
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