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Re: Mathematica 7.0.1.0 and some General Comments

  • To: mathgroup at smc.vnet.net
  • Subject: [mg97429] Re: Mathematica 7.0.1.0 and some General Comments
  • From: Mariano Suárez-Alvarez <mariano.suarezalvarez at gmail.com>
  • Date: Fri, 13 Mar 2009 04:48:51 -0500 (EST)
  • References: <goqphr$lt2$1@smc.vnet.net> <gp5fou$9nr$1@smc.vnet.net>

On Mar 12, 8:44 am, Andrzej Kozlowski <a... at mimuw.edu.pl> wrote:
> 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.

Indeed. History is full of such events. Mathematics
is a human activity, so to expect anything else would
simply be against all evidence, and only people who
have no real understanding of the way mathematics
is done are surprised by that.

But that is really unrelated to the fact that
black boxes that produce results essentially out of
the blue are quite not exactly desirable tools
for the scientific endeavor.

An example: a while ago in this list there was an interesting
discussion <http://groups.google.com/group/comp.soft-
sys.math.mathematica/browse_frm/thread/6529a588a3fb12c4/
bed4bcf7c952334e?lnk=gst&q=#bed4bcf7c952334e> on what Simplify really does.
What it does is, of course, amazingly complex, but
Daniel Lichblau observed that "FullSimplify can make
mistakes on measure zero sets. We do not generally regard this
phenomenon as a bug, though we reconsider on case by case basis".
The documentation certainly does not include enough information
for me to tell what "generally" means in this context,
and which cases were special-cased, and so on---I conjecture
that such information is not available to normal users.
Of course, in my work as a mathematician I have seen (too!)
many papers in which I could not tell why something
followed from something, or why something was equal to
something else.

But: the difference between "traditional" computation (I do not
have a better term...) and computation done using
closed applications is that in the first, at least in principle,
there is complete openness as to what was done, while
in the second there is a party *actively* withholding
possibly relevant information. (The motivation for that
withholding is probably not confounding mathematicians
all over the globe; it is not difficult, e.g., to come up
with a few possible reasons which make WRI not
keep an openly accessible list of bugs, all more
trite than world domination)

-- m




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