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Re: More Mathematica CAN'T do than CAN???

  • To: mathgroup at smc.vnet.net
  • Subject: [mg113414] Re: More Mathematica CAN'T do than CAN???
  • From: "Hobbs, Sylvia (DPH)" <Sylvia.Hobbs at state.ma.us>
  • Date: Wed, 27 Oct 2010 05:18:23 -0400 (EDT)

And imagine that! Such a software as mathematical proofware, select your le
Mathematica and press RUN!

Sylvia D. Hobbs, MPH, Director of Research & Evaluation
Bureau of Health Care Safety and Quality
Massachusetts Department of Public Health
Office of Emergency Medical Services
99 Chauncy Street, 11th Floor
Boston MA 02111
Phone:617-753-7304
Fax: 617-753-7320
cell:781-530-6381
e-mail: sylvia.hobbs at state.ma.us<mailto:sylvia.hobbs at state.ma.us>
P Please consider the environment before printing this e-mail



________________________________________
From: mathgroup-adm at smc.vnet.net [mathgroup-adm at smc.vnet.net] On Behalf Of =
Andrzej Kozlowski [akoz at mimuw.edu.pl]
Sent: Tuesday, October 26, 2010 5:31 AM
To: mathgroup at smc.vnet.net
Subject: [mg113414] [mg113369] Re: More Mathematica CAN'T do than CAN???

The statement of the "learned mathematician" is ridiculous, unless "less th=
en half" is taken to mean "extremely few". The fact is (and I state it with=
 complete confidence as another "learned mathematician" and a devoted Mathe=
matica user) that Mathematica is incapable of "solving" even a tiny fractio=
n of the problems in mathematics today. Not only it can't prove the Poincae=
 conjecture, Fermat's theorem, the (still unsolved) Riemann hypothesis, or =
any of the remaining Clay institute's "millennium prize problems" (http://e=
n.wikipedia.org/wiki/Millennium_Prize_Problems) (each worth a million dolla=
rs, by the way), but also most ordinary problems that ordinary mathematicia=
ns deal with in many branches of mathematics, such as, for example, topolog=
y,  global analysis, probability theory, and so on.  One obvious reason for=
 that is that many of these problems are by nature non-computational and, a=
t present, even the most sophisticated computer programs can do no more tha=
n, well,  comput!
e.  To mention a pretty trivial case, no amount of computing can ever show =
that there exist infinitely many Goldbach primes, or solve the Hodge conjec=
ture.

Of course Mathematica could, in principle, be useful in solving any of thes=
e problems, but before that happens a human mathematician would have to fir=
st reduce it to a finite computational problem, as was indeed, done in the =
case of the For Colour Theorem. However, there are still very few cases whe=
n this sort of thing has been done; some of them are described here:
http://en.wikipedia.org/wiki/Computer-assisted_proof#List_of_theorems_prove=
d_with_the_help_of_computer_programs

In general, although computation is a central and essential aspect of mathe=
matics, the driving force behind perhaps the majority of mathematical disco=
veries (along with geometric intuition), its role in actually proving diffi=
cult mathematical results is still very minor. This naturally leads to the =
question of how important "proofs" are in modern mathematics - while there =
has been some controversy on this matter, most mathematicians still conside=
r proving theorems their "bread and butter". The current generation of comp=
uters and computer algebra programs, while able to do many marvellous compu=
tational things, is still of very limited use where proofs are concerned.

Andrzej Kozlowski

PS. As an example of some of the remarkable things that Mathematica is actu=
ally able to do I would like to quote this:

 Reduce[(1 + 1/n)^n*(n/(n - 1)) > E && n > 1, n, Reals]

n > 1

Proving this "by hand" is an interesting exercise!



On 24 Oct 2010, at 12:05, Nicholas Kormanik wrote:

>
> A few evenings ago I was speaking with a "learned mathematician" at
> the local university here.  In the course of our wide-ranging talk, he
> stated that Mathematica is only capable of doing less than half the
> problems in mathematics today.
>
> I was floored at his assertion.  I have only scratched the surface of
> all that Mathematica can do.
>
> There's tons more that it CAN'T do???
>
> Your comments are welcome.
>
> Nicholas Kormanik
>



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