[Axiom-developer] RE: DistributedExpression (was: How can I get every term of an expression ?)

[Bill Page]

On September 22, 2005 5:24 AM Martin Rubey wrote:
> > ...
> > Axiom's approach to output as a coercion to the type 
> > OutputForm is radically different than any other computer
> > algebra system that I know. 
> 
> Yes.
> 
> > There is an attempt also in Axiom to do the same with 
> > input (InputForm and SExpressions etc.) 
> 
> I think that this attempt is doomed to fail. Still it may be 
> useful, at least for debugging.
>

Could you explain why you think this approach is "doomed
to fail"? To me it seems simply under developed in the
current version of Axiom. I don't see any conceptual problems
with extending the idea. So far I have found it quite
useful, for example in implementing the 'op()' operator
for manipulating parts of expressions. See

http://wiki.axiom-developer.org/ManipulatingExpressions
 
> 
> > Understanding how to use types and domains in Axiom is 
> > both "90% of the problem" and "90% of the reasons why
> > one might want to use Axiom in the first place". We need
> > to write more about this ...
> 
> No. There is enough written about it (in the Aldor User 
> Guide). A short introductory text is in the Axiom book.
>

?! I am rather shocked that you would make this claim! :(
I have read all of this material as well as all of the
papers and articles on this subject and still I think that
types and domains in Axiom are a big problem... and also
the main reason to use Axiom as I said above.

It seems to me that one of the reasons that Axiom is not
used as much and is not being developed as quick as Mathematica,
Maple and even Maxima is precisely because of the problem to
trying to understand and use strong typing in computer algebra
systems (the marketing and commercial reasons not withstanding).
In a sense, Axiom is/was an experiment in the application of
strongly typed programming languages in computer algebra and
to be quite honest and blunt, for the most part the experiment
seems to have failed. :(
 
> 
> > Yes, I do agree that "cosmetics" is useful. In fact I have 
> > been know to claim that "notation is (almost) everything"
> > in mathematics 
> > ...
> 
> NO, NO, NO. Good notation is important, and I'd agree if 
> you'd say that mathematics builds on good notation. But it's
> not nearly "almost everything".
> 
> What (good) notation provides is a means to make your ideas 
> and proofs clear and enable others to follow them. The idea
> comes before the notation. Often even the proof comes before
> the notation.
> 

My philosophical position is much more radical. I take the
strong "Whorfian" view

http://en.wikipedia.org/wiki/Benjamin_Whorf

that language (in this case mathematical notation) largely
determines what and how we think about problems.

I doubt that "proof comes before notation" because I doubt
that rigoursly expressing a proof is even possible without
the proper notation. So you may "think" that you have a
proof until you try to write it down ... :)

Regards,
Bill Page.