On Tue, Apr 11, 2006 at 08:25:27AM -0700, Vishvesh Obla wrote:
> 'Computer programming bears much more relation to
> language than it does to maths'
> 'if year++ then age ++, the expression is _not_
> mathematical; it is language.'

I tend to think that this distinction between "mathematics" and
"language" is an artificial one; the two are in many regards both merely
abstractions of methods of communication - we talk, after all, of
"computer languages" and of "the language of mathematics".  The English
language has a grammar in some respects more forgiving than does, say,
the programming language perl, which itself has a grammar more forgiving
than does assembly language.  

Amongst computer languages there is indeed a wide variety of different
levels of strictness; some people prefer more strict languages, which
may often have the advantage that the behaviour of a compiler is more
predictable; others prefer looser constructs.  Similarly, there can be
in programming a great deal of flexibility in the approaches to solving
a problem; often, the approach that makes most instinctive sense to many
people will also be the most computationally intensive, and vice versa. 

So, then, as to whether mathematics is like language is like 
programming--I think it's a huge oversimplification to act as though 
any of these things is a simple "thing" that can be either understood or 
not understood.  I am sure that after a point a large part of the 
challenge of mathematics, language, programming, music, or whathaveyou, 
is an understanding of the fundamental syntax and concepts that underpin 
any of them.  Although Peter might hold that it's possible to have a 
conversation without an understanding of the fundamentals of language, 
I'd disagree; I'd similarly argue that although we can all slap paint 
onto a canvas, if we spend the time to understand colour theory, 
perspective, etc., we will produce more pleasing results.  In my efforts 
to be a better musician, an effort to better understand musical theory 
make it much easier for me to come up with a pleasing tune or sequence 
of chords.  In my work as a programmer, I endeavour to understand the 
tools that are available to me to help me represent and to manipulate 
data.  And if I were interested in becoming a mathematician, I would 
work to systematically build up my understanding of fundamental concepts 
of mathematics so that I could build upon them.

Now, do different people have different aptitudes?  No doubt.  But I 
don't think that this stems from a fundamental difference between these 
different areas; after all, even if "computer programming bears more 
relation to language than it does to maths", any high-level computer 
program must eventually be translated into low-level code that is 
ultimately just a sequence of electrical instructions: again, it's a 
matter of abstraction, and of which method of abstraction suits 
different people.  But I am absolutely against Will's assertion that 
"knowledge beyond the trivial is beyond the scope of mathematics", from 
which, I think, this discussion grew: I think it's actually 
intellectually and morally bankrupt to suggest that there are things 
that "cannot be known" or that we shouldn't try to expand our 
understanding in all of these areas.

I've probably mentioned it before, by the way, but I think that 
Hofstadter's work in this area is quite essential, especially his 
seminar "Godel, Escher, Bach: An Eternal Golden Braid".