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MathGroup Archive 2005

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Re: Language vs. Library why it matters

  • To: mathgroup at smc.vnet.net
  • Subject: [mg61529] Re: Language vs. Library why it matters
  • From: Bill Rowe <readnewsciv at earthlink.net>
  • Date: Fri, 21 Oct 2005 00:38:17 -0400 (EDT)
  • Sender: owner-wri-mathgroup at wolfram.com

On 10/20/05 at 12:56 AM, akoz at mimuw.edu.pl (Andrzej Kozlowski)
wrote:

>
>On 20 Oct 2005, at 12:08, Bill Rowe wrote:
>
>>On 10/19/05 at 2:16 AM, hattons at globalsymmetry.com (Steven T.
>>Hatton) wrote:

>>>In particular, Gödel's results tells us any sufficiently powerful
>>>language necessarily leads to the possibility of formulating
>>>selfcontradictory statements.  I'm not aware of Mathematica
>>>producing any such selfcontradictions.

>>But Godel's theorems were statements about mathematics. And
>>Mathematica is clearly intended to encompass mathematics, to allow
>>the user to do any valid mathematics. Consequently, Mathematica
>>must either inherit any inconsistencies existing in mathematics or
>>restrict the user from performing certain mathematics.

>I am not sure I understood these remarks correctly, but in case
>anybody thinks Godel showed that there are "inconsistencies in
>mathematics", necessary or not, I hurry to point out that this is
>absolutely not so. Godel actually showed two things (in different
>papers). First, that in any sufficiently rich axiomatic system one
>can formulate unprovable statements (unprovable being entirely
>different form inconsistent) and secondly, that no proof of
>consistency of such a system can be given (if such a proof was
>found them the system would be inconsistent).

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. The usage of the word "inconsistencies" was a poor choice on my part since it appears to have detracted from the point I was trying to make.
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