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 >