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Re: Re: Re: Simplifying and Rearranging Expressions

  • To: mathgroup at smc.vnet.net
  • Subject: [mg96090] Re: [mg96051] Re: [mg96008] Re: Simplifying and Rearranging Expressions
  • From: "David Park" <djmpark at comcast.net>
  • Date: Wed, 4 Feb 2009 05:23:06 -0500 (EST)
  • References: <gls1u8$hjl$1@smc.vnet.net> <15441402.1233316177571.JavaMail.root@m02> <gm0q6p$rpr$1@smc.vnet.net> <200902010940.EAA22741@smc.vnet.net> <18977710.1233662192327.JavaMail.root@m02>

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.

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.

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

  




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