Re: Applying Mathematica to practical problems

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• Subject: [mg131015] Re: Applying Mathematica to practical problems
• From: Andrzej Kozlowski <akozlowski at gmail.com>
• Date: Tue, 4 Jun 2013 01:57:43 -0400 (EDT)
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```On 3 Jun 2013, at 09:34, Richard Fateman <fateman at cs.berkeley.edu> wrote:

> On 6/1/2013 9:23 PM, Andrzej Kozlowski wrote:
>> This is one of many examples of "cross purpose" arguing. I was not
>> discussing implementing non-standard analysis at all. My point was
>> that that there is nothing logically more dubious about a finite
>> "number" x such that x+1=1 than there is about a positive "number" x
>> such that nx< 1 for every positive integer n, or, alternatively,
>> finite "number" x such that x/n >1 for every positive integer n.
>> Mathematicians often use the word "number" when referring to objects
>> belonging to some "extension" of the real line.
>
> Oh where to start.  Here's one place.
>
> You say it is ok to have a number x such that x+1=1.  I agree.
> I say it is NOT ok to have a number x such that x+1=x.
>   You respond
> It is ok to have a number x such that x+1=1.
>   Can you see the difference?

matter. It is O.K. to extend the real line to an algebraic and
topological sturcture which contains objects such x such that x+1=x.
Of course such objects do not have inverses, so you can't conclude that
0=1.  Admittedly, in Mathematica the underlying logic is obscured by
the fact that it equality is not identity, so you do have

1`0 == 1

True

1`0 == 0

True

but

1`0 == 0

True

All this means that equality is a non-transitive relation on this
extended set of objects (but identity is). That is also logically
perfectly sound.

>
>
> Also is the Grobner basis program in Mathematica the fastest?
> I suspect it is not, though I have not compared it to Faugere's
> work, or other unnamed systems. Is it the only one using significance
> arithmetic?  I suspect it is.  What would that prove?  What
> would that prove about use as a default?

This is nonsense. Faugere does not work on numerical analysis and has
not implemented a numerical Groebner basis. Other people have worked on
approximate Grobener bases using fixed precision arithmetic but (as far
as I know) there are no working implementations available.

Groebner bases with exact coefficients are an entirely different
subject, unrelated to this discussion.

>
> Finally, I would remind AK (and others)
> that proving some number of correct results
> does not prove an algorithm is correct.  Proving even one incorrect
> result demonstrates a bug.
>

You have never demonstrated one incorrect result proved by Mathematica.

AK

```

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