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Re: Language vs. Library why it matters

• To: mathgroup at smc.vnet.net
• Subject: [mg61563] Re: Language vs. Library why it matters
• From: "Richard J. Fateman" <fateman at eecs.berkeley.edu>
• Date: Sat, 22 Oct 2005 00:36:32 -0400 (EDT)
• Organization: UC Berkeley
• References: <dja2ht$fll$1@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

Bill Rowe wrote:

> 
> 
> My intent was merely to make the point Godel's results must apply 
 >to Mathematica since they clearly apply to mathematics and Mathematica
 >by design is intended to encompass mathematics.

Not so.  Mathematica does not prove theorems, generally. Some mathsource
programs may construct some proofs in some limited context, but those 
don't contradict Godel's result.

Mathematica running on your computer is not, actually, as
powerful as the simplest Turing machine. The "Halting Problem"
for Mathematica on your computer is solvable.  You have a finite
machine with a finite number of states. It is possible in principle
to tell if a program is in a loop or if it will halt. The testing
procedure may take a very long time to run, but the testing procedure
is also finite.

Or perhaps you are trying to state something else about decidability.

One can state classes of problems for which no algorithm can exist in 
Mathematica. (or C or Fortran or executable by the finite collection of 
atoms that constitute a single human brain, or for that matter the
collective brains of all humans living and dead....)
For example, Hilbert's 10th problem
http://mathworld.wolfram.com/DiophantineEquation.html

But this does not prevent one from finding solutions to a subset of
that class of problems. Perhaps the subset of Diophantine Equations
that can be expressed in less than 2^40 ASCII character in Mathematica
syntax can be solved. That would not contradict the theorem, but
it would take the bite out of it.

If you want to explore decidability, Frege, Whitehead/Russell, Godel, 
Turing, ... that's fine, but don't confuse it with defects in
computer algebra system design.

As for whether Mathematica or anything else can "arithmetize" all of
mathematics, you might look at the QED project, archived at
http://www-unix.mcs.anl.gov/qed/

or at some of the more recent work.. google for MKM.




RJF