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Re: Re: Mathematica 7.0.1.0 and some General Comments

  • To: mathgroup at smc.vnet.net
  • Subject: [mg97413] Re: [mg97384] Re: Mathematica 7.0.1.0 and some General Comments
  • From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
  • Date: Thu, 12 Mar 2009 05:44:09 -0500 (EST)
  • References: <goqphr$lt2$1@smc.vnet.net> <gp5fou$9nr$1@smc.vnet.net> <200903120719.CAA23555@smc.vnet.net>

On 12 Mar 2009, at 08:19, Mariano Su=E1rez-Alvarez wrote:

> On Mar 11, 7:21 am, mike.honeychu... at gmail.com wrote:
>> On Mar 10, 5:35 am, Sebastian Meznaric <mezna... at gmail.com> wrote:
>>
>>> I don't think Mathematica should replace mathematics. It is an
>>> important tool, but very importantly, a commercial closed-source 
>>> tool.
>>> It costs a great deal of money and you do not know what it is doing
>>> (although most of the time it gives correct results). Only systems
>>> that can be considered to generally replace Mathematics have got 
>>> to be
>>> open source (although I admit I do not use any). If at least the 
>>> basis
>>> of Mathematica was made open-source with paid-for support from 
>>> Wolfram
>>> that would make things a lot better. As it stands, we should not 
>>> chain
>>> people to commercial software.
>>
>> Other than people who use pencils and paper, or blackboards and
>> chalk*, everyone is "chained" to commercial products in their
>> workplaces. We need to "free" our minds a bit from the idea that
>> software should somehow be an exception to everything else that 
>> occurs
>> in our workplaces. Or alternatively perhaps someone can explain to me
>> why software should be any different to scientific equipment, cars,
>> dishwashers... I cannot use an open source mass spectrometer, drive =

>> an
>> open source car [although GM and Ford are verging on open source :),
>> or at least maybe publicly owned soon] etc.
>
> Well, if you come up with a proof of a theorem
> which depends on non-trivial Mathematica code
> to do non-trivial computations, in what way can
> you possibly say that you know how the proof works,
> if *you* yourself, the author of the proof, do not
> know what Mathematica is really doing? Using
> closed-source code simply goes against the very
> spirit of open review which is essential to
> the scientific endeavor.
>
> There was a recent discussion in this subject
> on the AMS Notices, which you can get at
> <http://www.ams.org/notices/200710/tx071001279p.pdf>.
>
> -- m
>

The truth is that, on the one hand, a great many mathematicians 
(perhaps the majority) rely on results of other mathematicians whose 
proofs they have not fully (and sometimes not at all) understood, or 
even tried to understand. Moreover, many of such proofs have only been =

published in a sketchy form with various parts "left to the reader" 
and or accompanied by comments like "it can be shown" etc. Anyone who 
does not know that must have had no contact with real world research 
in mathematics.

On the other hand, all algorithms use by Mathematica are standard and 
can be found in books and papers. So the issue is only whether they 
are correctly implemented. While we cannot check this for sure, they 
are ways to check this with a very high degree of confidence, which is 
at least as high as that of the correctness of proofs checked by human 
mathematicians, many of which have only been read carefully by a 
handful of persons.

Whether you choose to believe in a purely "human" proof or a computer 
assisted one you are always relying on trust and can never have 100% 
assurance. There are cases of mathematical proofs that have been 
accepted as correct for many years before it was discovered that they 
contained gaps or mistakes. There are also many "theorems" that have 
been "fully proved" by humans and yet if you talk to the experts you 
will find out that they would not be very surprised if some day they 
turn out to be incorrect or incomplete.
There are lots of examples of this phenomenon (though usually it is 
better to avoid mentioning specific ones on public fora) but if you 
want one very well known example you might read the last three 
sections of
=
http://www.ams.org/bull/2001-38-03/S0273-0979-01-00909-0/S0273-0979-01-00909-0.pdf

Andrzej Kozlowski




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