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Re: Re: Re: Simplifying and Rearranging Expressions

• To: mathgroup at smc.vnet.net
• Subject: [mg96068] Re: [mg96051] Re: [mg96008] Re: Simplifying and Rearranging Expressions
• From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
• Date: Wed, 4 Feb 2009 05:19:07 -0500 (EST)
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On 3 Feb 2009, at 15:13, David Park wrote:

> Demonstrations Project yes. It is the dynamic style one can easily  
> get in a
> Mathematica notebook and not in a static document. My objection to the
> Demonstrations Project is that the ones I've looked at are just a  
> single
> Manipulate statement with very little textual explanation or  
> development.
> Much better to have a notebook that contains development, textual  
> discussion
> and various kinds of presentations, dynamic and otherwise, and tools  
> to
> explain and work with some concept.

The fact that there is only one Manipulate statement per demonstration  
is no real restriction since one can make any number of demonstrations  
on a single topic (you can easily find examples of this on the  
Demonstrations site). The amount of textual information that can be  
included in the Details section isn't much less than in a typical  
review in Mathematical Reviews or Zentralblatt MATH. My vision of what  
a Demonstration is something rather like that of a review: the purpose  
is not to replace books, journal articles etc. but rather to provide a  
guide to what is available together (in favorable cases) with usable  
code. References to books and articles where the user can find more  
details are, in my opinion, just as essential in a Demonstration as in  
a review.

>
>
> For human beings it is just easier to understand an action than it  
> is to
> understand a static object. We evolved to respond to actions.
>
> I have a theory about great mathematicians.
> 1) They have the ability to visualize actions and interactions of  
> abstract
> objects in their mind, probably often geometrically.
> 2) They are very good at calculating without mistakes, perhaps a  
> little like
> some autistic people.
> 3) They can immerse themselves in many specific cases and calculate  
> them out
> rather fast and this way they gain a lot of experience.

Personally I would leave "greatness" out of this, if for no other  
reason than the one that it comes in an infinite variety while  
representing an infinitesimal market share. However, my own experience  
of contact with unquestionably great mathematicians makes me doubt at  
least 2 above. I still remember watching as an undergraduate a Fields  
medal winner (http://en.wikipedia.org/wiki/Stephen_Smale) struggling  
with a system of 2 linear equations with two unknowns. More typical is  
the following comment by another really great mathematician (http://en.wikipedia.org/wiki/Vladimir_Arnold 
)

"Every working mathematician knows that if one does not control  
oneself (best of all by examples), then after some ten pages half of  
all the signs in formulae will be wrong and twos will find their way  
from denominators into numerators."

Which may, perhaps, be a good reason why even great mathematicians  
might benefit from using programs like Mathematica.


Andrzej Kozlowski

>
>
> It is just these things that Mathematica, if used properly, helps  
> with. The
> newer dynamics helps with (1). And also remember that step-by-step
> derivations with the actual rules or definitions made explicit are a  
> kind of
> 'action'. The standard CAS facilities help with (2) and because of  
> this we
> can do a lot of (3). This opens up real mathematics to a much  
> broader class
> of people and that is a worthy goal in itself. This won't make  
> everybody
> great creative mathematicians, but it might boost a few who  
> otherwise were
> weak in one of the skills.
>
> But it takes an active, interactive, dynamic and discursive style of  
> writing
> Mathematica notebooks to get the full benefit. Static documents are  
> the old
> technology.
>
>
> David Park
> djmpark at comcast.net
> http://home.comcast.net/~djmpark/
>
>
>
> From: Andrzej Kozlowski [mailto:akoz at mimuw.edu.pl]
>
> I think you are missing the point of what WRI is doing. Nobody is
> trying to impose a single format on the world of mathematics, and
> nobody is trying to force mathematica users to abandon other programs.
> The best proof of that is the large number of export formats that
> Mathematica supports. The whole point of Mathematica "integrated
> approach" is entirely different. In my opinion it lies in the idea of
> "Mathematica Demonstration", as exemplified here:
>
> http://demonstrations.wolfram.com/
>
> I consider the Mathematica demonstration to be a truly remarkable and
> revolutionary idea. Why, I will try to explain below. First, I just
> want to note that this could not be achieved without a fully
> integrated system that Mathematica provides. That's why I don't expect
> that Mathematica will see any competition in this area for quite some
> time.
>
> Why I think these demonstrations are such a great idea? If you only
> glance superficially at the demonstration site you may think that they
> are merely cute animations and mathematical toys. Indeed, there are a
> few of this kind, but be not deceived. Many demonstrations contain
> fully functional code that can be downloaded by the user and after
> minor adjustment be used to solve serious real life problems. At the
> same time, the Mathematica demonstration provides a remarkably
> intuitive and lucid way of conceptualizing what otherwise would be
> more or less incomprehensible piece of computational code. Some of the
> demonstrations I have contributed are based on papers I reviewed for
> Mathematical Reviews. In my opinion, these demonstrations have far
> greater explanatory power than any number of words (certainly any
> number of words written by me). Some others attempt to elucidate
> concepts in Mathematical finance while at the same time providing code
> that can be actually useful in real world computations. In the case of
> mathematical finance, I think there is an almost universal agreement
> that in the past computational techniques were  emphasized too much
> while conceptual clarity was neglected. Mathematica now offers a
> unique way to combine conceptual description of a model with a
> mathematical solution through an analytical or numerical process. If
> more people contribute demonstrations adopting this approach the
> demonstrations site could become a valuable repository of reusable
> code accompanied by conceptual visualizations with a very wide area of
> applicability.
>
> In my opinion a Mathematica demonstration is much more than a "new
> format", it is a completely new form of expressing and communicating
> mathematical ideas. As such it justifies everything that WRI has done
> to make it possible - which is essentially everything that you are
> objecting to in Mathematica. But of course, you are always free to
> ignore these new features if they hold no interest for you. You are
> also free to use CalcCenter, which may well do everything that you
> really wish to use Mathematica for (I can't guarantee that as I do not
> really know either how you use Mathematica or what exactly CalcCenter
> can do).
>
> Andrzej Kozlowski
>
>
>
>

• References:
  • Re: Simplifying and Rearranging Expressions
    • From: AES <siegman@stanford.edu>