> Anyway, I'm hopeful that someday "CAS for the masses" will consist of
> not a whole new simplified system but of defining in Axiom or something
> like it a set of assumptions and constraints that creates the
> simplified environment. To me that makes the most sense for both
> correctness and "expandibility" - as people need new ideas they can
> incrementally add them, examining the consequences of each (for
> example, expanding the domain of numbers under consideration from Reals
> to Reals+Imaginary). There's no particular reason I can see to abandon
> rigor for simplicity - if the mathematics is valid it should be just a
> question of defining the assumptions made to "simplify" the notation
> for new users.
Hi CY,
you got precisely to the point. And, getting back to my original
email, my first sentences were about mathematicians looking at the
mathematics completely differently.
I, personally, don't know, if it is possible to have a simple CAS for
masses and also being it able to build on it more rigorous and
abstract mathematics. As you said, what is really needed in physics is
a very limited subset of mathematics (basically just calculus,
integrals on k-manifolds in n-dimensions, differential equations,
matrices, noncommutative algebras like Pauli or Dirac, series, limits,
asymptotic expansion, complex numbers, but that's all). Ok, then some
stuff like recurence relations, tensors, etc., but all of this I just
call a simple calculus. I want the CAS for masses to be really good at
this simple calculus.
If someones finds a way how to do more advanced stuff, why not, but I
am not interested in that primarily, as it is not needed. And if the
more advanced stuff renders this basic calculus less usable, then I am
completely against it. On the other hand, you have exactly the
opposite opinion - if something is not exact, let's not talk about it.
If, as a side effect, the CAS could also be used for masses, then why
not, but primarily you are concerned with mathematical formalism.
So I think this is the core difference between me and most developers of Axiom.
I am not saying that you or me should be wrong - this is just the
basic difference between mathematicians and physicists.
Ondrej