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Re: Re: Language vs. Library why it matters

• To: mathgroup at smc.vnet.net
• Subject: [mg61503] Re: [mg61496] Re: Language vs. Library why it matters
• From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
• Date: Thu, 20 Oct 2005 00:56:22 -0400 (EDT)
• References: <200510200308.XAA12895@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

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:
>
>
>> I'm not sure Gödel's (anti-)proof is the core reason that
>> Mathematica is difficult to understand, though it certainly plays
>> into the whole picture.
>>
>
> I would agree Godel's theorem's do not explain the steep learning  
> curve for Mathematica. I believe the steep learning curve for  
> Mathematica is due to the richness of the tool set made available.
>
>
>> 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).

Of course nobody has ever found any "inconsistencies in mathematics",  
and is extremely unlikely to do so. In fact it is doubtful even if  
the phrase "inconsistency in mathematics" has any meaning at all.


Andrzej Kozlowski
Tokyo, Japan

• References:
  • Re: Language vs. Library why it matters
    • From: Bill Rowe <readnewsciv@earthlink.net>