Saturday, March 01, 2008 1:12 AM, Rick wrote: (01)
> 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. (02)
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. (03)
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. (04)
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. (05)
Considering the lack of a single foundation of mathematics and science, such
a scenario is very and very possible, if unavoidable. (06)
Since the proposal is rather new and highly controversial, I better refer
you to a more detailed account: (07)
Part III. THE WORLD CODE: Mathematical Ontology as the Real Road to Reality. (08)
> I don't see the A word here: Abduction. I don't go anywhere without it. (09)
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. (010)
>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. (011)
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. (012)
azamat abdoullaev (013)
----- 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 (014)
> 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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