Ravi, Azmat & All (01)
I don't see the A word here: Abduction. I don't go anywhere without it. (02)
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. (03)
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. (04)
> 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. (05)
This is a closed system and just leads to inconsistency issues. (06)
>> 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. (07)
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. (08)
http://www.newadvent.org/cathen/11257a.htm (09)
> 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
> ofyour 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. (010)
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.* (011)
> 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. (012)
So how does this square with Kuhn's Structure of Scientific Revolutions
in which he claims that paradigm shifts come from anomaly and crisis ? (013)
> construct N-relational ontology, natural language ontology model, and
> how we
> can build the Digital Aristotle, or the Virtual Aristotle Machine (VAM).
> (014)
As an aside, have folks been tracking what Mark Greaves is up to at Vulcan ? (015)
videolectures.net/iswc06_graves_img/ (016)
Seems like he found an ideal customer in Paul Allen. (017)
--
Thanks Rick,
blog http://spout.rickmurphy.org
web http://www.rickmurphy.org
cell 703-201-9129 (018)
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