John, (01)
Your arguments are very persuasive but not entirely conclusive. The
substance of the message is that the empirical sciences are advancing by the
interplay of observations, measurements and theorizing; interaction of data
and generalizations; specific facts and mathematical demonstration and
theoretical reasoning and speculative intuitions. What is generally correct
and widely acceptable as a wellestablished scientific method. (02)
At its dawn, mathematics started as a simple counting of wealth and
measuring of real estate property. Now it's the deductive formal axiomatic
science dealing with structure, order, and relationship, at the most
abstract level, without any specific reference and meaning; what makes it a
highly valuable science. (03)
Take your example of thermodynamics. It is based on few number of concepts:
systems (material objects, particles), states (equilibrium), properties
(energy, entropy), changes (processes, irreversible, reversible),
interactions (work, heat), and fundamental laws (the first, second and
third). (04)
Take biology. It is also founded on several key principles: unity (of
origin, substance and function of all living matter); diversity (of
species); continuity (of generations); evolution (progressive change);
relationships (of living things and environment); homeostasis (state of
equilibrium, from the cell to the biosphere) (05)
It looks any fundamental scientific theory is nothing but a body of concepts
or principles and set of fundamental statements (existential and universal,
as definitions, axioms, rules, and laws). Defining all other specific
concepts and deducing all other particular propositions, this form of
knowledge organization secures consistency of meaning and completeness of
analysis. (06)
The same works for building an axiomatic real ontology as the general
theoretical system establishing the relationships of all things to each
other, with its application, computing ontology. Ontology is a body of most
primitive categories and basic truths of reality (universal definitions,
axioms, and fundamental rules), represented in an axiomatic and formal
manner; what makes ontology as the most valuable science. Since the meanings
of all other concepts can be defined in terms of basic ontological classes,
as all other meaningful propositions can be deduced from the basal
ontological truths. (07)
Azamat (08)
 Original Message 
From: "John F. Sowa" <sowa@xxxxxxxxxxx>
To: "[ontologforum]" <ontologforum@xxxxxxxxxxxxxxxx>
Cc: <standardupperontology@xxxxxxxxxxxxxxxxx>
Sent: Tuesday, January 29, 2008 10:03 PM
Subject: Re: Axiomatic ontology (09)
> Azamat,
>
> I'm switching this thread to ontologforum, since the topic is
> closer to the themes on that list. I'm cc'ing this note to SUO
> list to indicate that anyone who may be interested should go to
> ontologforum for a continuation.
>
> As I said, I have the highest regard for Descartes's mathematical
> achievements, and I also believe that his ideas in philosophy were
> interesting. The ideas of any great thinker, even when they're
> wrong or misleading, are worth studying because there are usually
> very instructive reasons why they are wrong.
>
> But the most important point to learn from them is what went wrong,
> what aspects were useful for further study, and what aspects misled
> several centuries of philosophers away from more promising ideas.
>
>> What matters is the whole new idea of subjecting metaphysical
>> systems to axiomatization, the rigorous and systematic analysis
>> of a system from precise definitions, axioms and rules, what
>> Spinoza essayed in his philosophy.
>
> That is indeed an interesting idea. In mathematics, it has proved
> to be very valuable. But in the empirical sciences, even physics,
> it has had mixed results. Newton's achievements came from applying
> mathematical techniques to ideas that had been developed (with some
> use of mathematics) by Galileo and Kepler.
>
> But if you study the history of physics, there are four kinds of
> breakthroughs:
>
> 1. Developing new mathematical techniques for solving difficult
> problems stated in terms of known physical ideas. Examples
> include the work of mathematicians such as Laplace, Hamilton,
> Lagrange, Minkowski, etc.
>
> 2. Discovering new physical principles by intuitive insights into
> the nature of the phenomena and testing them by experiment
> (either actual physical experiments or, as in the case of
> Einstein and Bohr, Gedanken experiments). Galileo, Faraday,
> and Einstein are examples of physicists whose intuitions were
> far more advanced than any of their colleagues, but whose
> mathematical techniques were modest in comparison.
>
> 3. Applying advanced mathematics to the task of formalizing
> the intuitions of physicists who were not as competent in
> mathematics. Examples include Newton, Maxwell, and Minkowski.
> (In fact, Einstein's first wife was a better mathematician
> than he was, and he gave her little or no credit for her work
> in collaborating with him.)
>
> 4. A judicious combination of mathematics and physical intuition
> to discover new principles that could not be discovered by
> observation directly. The first and one of the most impressive
> examples was the work on statistical mechanics and thermodynamics
> by Ludwig Boltzmann. The starting point was the intuition that
> heat was the result of enormous numbers of molecules bouncing
> around, but the numbers were so enormous that it was impossible
> to calculate the effects of individual atoms or molecules.
> So Boltzmann invented statistical methods for dealing with them.
> That was a brilliant achievement, but it was the result of deep
> intuition into both physical phenomena and mathematical methods.
> In effect, Boltzmann had the combined power of Faraday and
> Maxwell, but he did *not* begin with a priori axioms. (010)
Although created statistical thermodynamics as a new branch of
thermodynamics, Boltzman failed to show that the laws of thermodynamics can
be derived from the laws of mechanics (classical or quantum); since heat as
a non work interaction, by its nature, could not be defined in mechanical
concepts. (011)
> This is the state of physics, the most mathematical of all the
> empirical sciences. For other branches of science, ranging from
> chemistry to biology to psychology to sociology, careful observation,
> intuition, and experimental testing become more and more important
> than mathematical formalization.
>
> In fact, one can safely say that the great advances in chemistry and
> biology during the past century have *not* resulted from any use of
> axioms in those subjects, but from the applications of ideas and
> technology that were first developed in physics.
>
> Even in physics, mathematics has certainly been of overwhelming value,
> but the very few attempts at axiomatization, such as von Neumann's
> axioms for quantum mechanics, were little more than interesting
> exercises. They did not lead to any useful breakthroughs that could
> not be made more effectively by people with a good intuitive feel
> for the subject.
>
> As far as formalizing metaphysics, it is safe to say that there are
> *no* major insights that have resulted from the formalization. (012)
There is a principal distinction between formalization and axiomatization,
see my message on this subject. (013)
It
> is true that formal axioms are important for computer applications,
> since computers, by themselves, have no intuitions whatever. But
> the real insights have come from people whose intuitions were at
> the level of Galileo, Faraday, and Einstein. There are no formal
> achievements that are remotely comparable to the work of Newton,
> Maxwell, or Minkowski. (But there are insights into metaphysics
> that result from mathematical studies in physics, but the insights
> originated in physics, not formal philosophy.)
>
> In summary, the attempts by Spinoza and Descartes to formalize
> metaphysics were interesting failures. Even as late as the 20th
> century, when modern logic became available, attempts such as
> Carnap's Logische Aufbau were also interesting failures. Carnap
> was a good logician with a strong background in physics and
> mathematics. But his attempt was a dead end. Nelson Goodman
> made another interesting attempt, which developed some useful
> mathematics, but no new insights into metaphysics.
>
> The largest of all attempts was the Cyc project, which many people
> in AI regard as a failure. Cyc has had some useful applications,
> but none of them have been sufficiently successful to pay for the
> many millions of dollars that were invested in the project. (014)
> Those of us who are working with logic and ontology hope that
> some kind of formalization will be useful for major applications.
> But so far, there have been *no* new insights into metaphysics that
> have come from the process of axiomatizing Cyc or any other formal
> system. On the contrary, the real insights have come from the same
> source as all the other insights since antiquity: dedicated study,
> observation, intuition, and discussion with teachers, students, and
> colleagues. The insights from formalization, if any, were modest
> at best. (015)
The crucial problem with this project consists in its artificial selection
of toplevel categories, what makes a shaky foundation for the whole
enterprise. (more detailed analysis in the Universal Ontology book). (016)
>
> John (017)
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