Hilbert, reductionism and sciences

[Sonnet Nguyen <Sonnet.Nguyen@fuw.edu.pl>]

Hi Friends,
I often have a pleasure to discuss with some people those are good
specjalists of formal logics and AI.  
Usually after disccusions with them I feel a headache...
Why?  
For their naive faith into reductionism!!!  

-----------------------SMALL TALK TO EVERYBODY----------------------

David Hilbert, great german mathematician, after 30 years of high
achivement on the frontier of math, walked into the blind alley of
reductionism.  In his later years he aimed to reduce the whole of math to
a collection of statements using a finite alphabetical symbols and a
finite set of axioms and rules of inference. 
This was a reductionism in the most literal sense!  Morely, Hilbert
proposed to solve the mathematical problems by finding a general decision
process that could decide, when given any statement composed of
mathematical symbols, whether that statement was true or false.
He called his program of finding this decision process by
ENTSCHEIDUNGSPROBLEM and dreamed of solving this problem.  (Of course, if
Hilbert solved this problem, then he solved all unsolved problems of
math!!!).
Obviously, the essence of Hilbert's program  was to find a decision
process that would operate on symbols in a purely mechanical fashion,
without requiring any understanding of their meaning!
So, in spite of prolonged efforts of Hilbert and his diciples, the
ENTSCHEIDUNGSPROBLEM was never solved!!!

KURT GODEL, one of the best mathematicians XX-th century, proved (when 
Hilbert was 70) that Hilbert's problem can not be solved!  Godel proved 
(in his great doctorate thesis in 1940 (?)) that in any formalization of
math which includes the rules of arithmetic a formal process for
seperating statements into true and false can not exist!!! 
(Morely, he proved that there are arithmetical statements that can not be
proved true or false!!!  Let see Godel theorem in any Logics Book for more
infor.  His proof is really great great work of ART!!!)  One of many
consequences of famous Godel theorem is that in pure
mathematics reductionism does not work!!! (i.e. to decide whether a
mathematical statement is true, it is not sufficient to reduce the
statement to marks on paper or in Computer.) 

Hilbert was a first example of a paradox that several greatest and
most creative spirits in Science after achieving great discoveries by
following their unfettered imaginations, were in their later years obssed
with reductionist phylosophy.  Eistein was a second example of this
curious paradox. Einstein spent almost last 20 years for search a set of
equations that would unify the whole of physics.   Like Hilbert,
Einstein (lonely at his time) wanted to reduce physics to a finite set
of marks on paper!!! (Einstein's attempt failed as Hilbert's attempt to do
the same with math!)

I want to remind here that in the history of Science it happens not
infrequently that a reductionist approach leads to spectacular success.
The Schrodinger (1926) and Dirac equation (1927) were triumphs of
reductionism!  These equations brought a miraculous order into processes
of atomic physics and complexities of physics and chemistry (and biology
(?)) were reduced to only 2 lines of algebraic symbols!!!!!
It is a typical phylosophy  of physicists which says that the
understanding of a complicated system as a whole is possible by an
understanding of its components. (through superposition principle, etc...) 
But it happens often in history of science that the understanding of the
components parts of a composite system is impossible without an
understanding of behaviour of the system as a whole! 

After knowing Godel theorem, one can say loosely that in the mathematics,
the whole is always greater than the sum of the parts!!! (Paradox or NOT?)
Every formalization of mathematics raises questions that reach beyond the
limits of the formalism into unexplored world.  It's same with other
sciences.

Hope some day I can discuss more about many interesting things in AI: 
automatic reasonings, automatic proving, etc.
and about "ending of physics" and "ending of sciences".   

-----------------------END OF SMALL TALK---------------------------

enjoys,
SN
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|  Sonnet Nguyen,    Polish Acad. of Sciences           |
|  E-mail:           sonnet@fuw.edu.pl;                 |
|  Internet:     http://theta2.ifpan.edu.pl/~sonnet;    |
|  Tel. (Office) (48-22) 43-70-01 ext. 1313             |
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