Azamat, (01)
I'm switching this thread to ontolog-forum, 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
ontolog-forum for a continuation. (02)
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. (03)
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. (04)
> 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. (05)
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. (06)
But if you study the history of physics, there are four kinds of
breakthroughs: (07)
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. (08)
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. (09)
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.) (010)
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. (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. (012)
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. (013)
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. (014)
As far as formalizing metaphysics, it is safe to say that there are
*no* major insights that have resulted from the formalization. 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.) (015)
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. (016)
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. (017)
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. (018)
John (019)
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