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Re: More Mathematica CAN'T do than CAN???
*To*: mathgroup at smc.vnet.net
*Subject*: [mg113369] Re: More Mathematica CAN'T do than CAN???
*From*: Andrzej Kozlowski <akoz at mimuw.edu.pl>
*Date*: Tue, 26 Oct 2010 05:31:57 -0400 (EDT)
The statement of the "learned mathematician" is ridiculous, unless "less then half" is taken to mean "extremely few". The fact is (and I state it with complete confidence as another "learned mathematician" and a devoted Mathematica user) that Mathematica is incapable of "solving" even a tiny fraction 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://en.wikipedia.org/wiki/Millennium_Prize_Problems) (each worth a million dollars, by the way), but also most ordinary problems that ordinary mathematicians deal with in many branches of mathematics, such as, for example, topology, global analysis, probability theory, and so on. One obvious reason for that is that many of these problems are by nature non-computational and, at present, even the most sophisticated computer programs can do no more than, well, compu
t!
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 conjecture.
Of course Mathematica could, in principle, be useful in solving any of these problems, but before that happens a human mathematician would have to first 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 when this sort of thing has been done; some of them are described here:
http://en.wikipedia.org/wiki/Computer-assisted_proof#List_of_theorems_proved_with_the_help_of_computer_programs
In general, although computation is a central and essential aspect of mathematics, the driving force behind perhaps the majority of mathematical discoveries (along with geometric intuition), its role in actually proving difficult 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 consider proving theorems their "bread and butter". The current generation of computers and computer algebra programs, while able to do many marvellous computational 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 actually 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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