On Thursday, February 28, 2008 10:01 AM, Ravi wrote: (01)
Is it not right to state that axioms have roots in metaphysics or
philosophy and need not address reality as is in the many other branches
of mathematics? What is the a priori basis of universality? (02)
This is both right and correct. All key axioms of fundamental sciences,
formal or empirical, have roots in ontology. The foundations of any valuable
scientific theory includes apart the permanent trio, mathematical postulates
and axioms, factual formulas and law statements, semantic formulas and
rules, the ontological background in the first place.
It might be a surprise for working mathematicians and logicians, but all key
logical predicates have been ontological predicates, such as ''exist'', ''is
true'', '' is actual'', '' is false'', ''is identical with'', etc. (03)
>
> John
>
> Most of what you say about physics, I agree with. One has to keep in
> mind the accuracy to which one is trying to compute. For example Fast
> movement of matter near black holes is covered by general theory of
> relativity, while movement of matter in near earth or solar system is
> covered by special relativity. But ordinary accuracy of projectiles and
> planets is adequate for most purposes using Newton's laws.
>
> Physical Sciences admit imperfection to a degree. Bio sciences and human
> behavior are inherently less accurate in the prediction and quantum
> basis is reaching close to understand mutation and unexpected behavior
> of life elements.
>
> Social predictability and behavior gauging is beyond my comprehension.
> So is the admirable quest of mathematicians and axiom seekers necessity,
> as a prerequisite to understanding?
>
> But our readers do not want to acknowledge that mathematics is as old as
> the metaphysics (5000 years +) that gave us numbers and decimals and
> zero and infinity as I have translated before in this forum. A date of
> 600 BC is rather recent. We seem to not go before or parallel to Greeks
> just because these translations have not yet widely reached the West but
> Schopenhauer and Max Muller recognized the concepts originating then in
> Sanskrit.
>
> Is it not right to state that axioms have roots in metaphysics or
> philosophy and need not address reality as is in the many other branches
> of mathematics? What is the a priori basis of universality?
>
> Thanks.
>
> Ravi
>
> (Dr. Ravi Sharma) Senior Enterprise Architect
>
> Vangent, Inc. Technology Excellence Center (TEC)
>
> 8618 Westwood Center Drive, Suite 310, Vienna VA 22182
> (o) 703-827-0638, (c) 313-204-1740 www.vangent.com
>
>
> -----Original Message-----
> From: ontolog-forum-bounces@xxxxxxxxxxxxxxxx
> [mailto:ontolog-forum-bounces@xxxxxxxxxxxxxxxx] On Behalf Of John F.
> Sowa
> Sent: Thursday, February 28, 2008 10:01 AM
> To: [ontolog-forum]
> Subject: Re: [ontolog-forum] Search engine for the ontology
>
> Azamat,
>
> I'll start with a paragraph from the later part of your note because
> it gets to the heart of our disagreement:
>
> AA> The whole history of science consists in attempting to found more
> > general theoretical representations of objective laws and patterns,
> > guiding the world.
>
> I have no quarrel with the opening sentence, and I would emphasize
> the word 'attempting'. Science has always been a work in progress.
> Every new discovery in *every* discipline raises more questions
> than it answers.
>
> 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.
>
> 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". 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.
>
> JFS>> The state of physics is much worse. See _The Road to Reality_
> >> by Roger Penrose, for a very good overview.
>
> AA> His understanding of reality is very narrow and specific. imho,
> > this book hardly makes here a good argument.
>
> Penrose was not making an argument against foundations, he was just
> presenting a survey of the chaotic mess that constitutes modern
> physics. But that is at the most fundamental levels. At the levels
> of applications, every subdiscipline from mechanics, to astrophysics,
> to nuclear reactors, to antenna design, to particle accelerators,
> to aerodynamics, to solid-state physics, to cryogenics, etc., etc.,
> has a *totally* different methodology and basic toolkit of techniques
> and assumptions. A physicist who has been working in any one of the
> specialties I mentioned above (and many, many more) cannot move from
> that field to any of the others without *years* of study to come up
> to speed. (Actually, no employer would pay anybody to make such a
> transition, since it would be cheaper to hire a new PhD who had
> just written a dissertation in the specialty of interest.)
>
> AA> I have no doubts about the high scientific quality of these people
> > [at Cyc]... What I doubt, the approach, the way they attack such
> > unprecedented problems as creating a common sense knowledge base.
>
> I also have many serious doubts, but my recommended solution is to
> go in the *opposite* direction from your proposal: Cyc is already
> much, much more unified than it should be. I would recommend
> eliminating *all* axioms from the upper levels, and put them *all*
> in the lower levels (or in a toolkit of miscellaneous techniques).
>
> 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. But trying to use such a design as a foundation
> for an intelligent computer system is a recipe for *failure*. I
> know that I can't convince you, but I suggest that you spend a few
> years working with people like the Cyclers (or any other group of
> implementers) and try to implement your approach. Following is
> another epigram by Alan Perlis:
>
> A year spent in artificial intelligence is enough to make
> one believe in God.
>
> Alan Bundy at Edinburgh has been working on mechanical theorem
> proving and problem solving for about 40 years, and he is widely
> recognized for his accomplishments in the field. At an AI conference
> in 2006, we were both presenting invited talks. I was talking
> about problems in natural language understanding, and he started
> with automated problem solving. But we had both come to the same
> conclusion: *all* understanding and problem solving is *always*
> task specific. Perhaps God might have universal axioms, but
> no mortal has anything that could be considered universal.
>
> For Bundy's recent publications on these and related ideas, see
>
> http://homepages.inf.ed.ac.uk/bundy/
>
> AA> Try and look at the situation from the other side. The natural
> > and social sciences are disorganized, they are uniformly unordered,
> > lacking single conceptual order and ontological and methodological
> > arrangement.
>
> Precisely! That is how people think -- and people, with all their
> flaws, are still much more flexible and creative thinkers than any
> AI system ever conceived. For simple subjects like chess, computers
> can beat the world champion. But for complex subjects, every attempt
> to find widely applicable axioms has *failed*.
>
> Systems such as Mathematica are much better at manipulating formulas
> than any human, and professional mathematicians and engineers use it
> for manipulating the symbols. But mathematics and physics are very
> *simple* special cases that abstract as much of the detail as possible
> from the incredibly complex world. The social sciences and commonsense
> thinking have to deal with the complexities as they are.
>
> John
>
>
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