Symbolic Mathematics
Much of the work of mathematicians involves symbolic transformation of mathematical expressions.
When confronted with an integral, a mathematicians first thought would be to obtain a symbolic expression for the value of the integral rather than to undertake numerical integration.
This has the advantage of better efficiency, accuracy and reusability.


Software Paradigm
Symbolic mathematics tools such as Mathematica provide an interactive environment with a special programming language oriented towards manipulation of symbolic expressions.
A rich library of numerical and symbolic functions is provided, which are used to evaluate or simplify expressions entered.
The user can extend the environment by defining new mathematical functions or new ways of transforming mathematical expressions.


Rigour Deficit
Symbolic mathematics tools provide good support for the application of mathematics, but the don't know about proof, and don't really know about truth either.
So they will blindly accept new transformation rules without regard to the soundness of the transformation.
As mathematical assistants, they are agile. and smarter than numeric software, but still, basically, dumb.
To get smart you need rigour, and to get rigour you need logic.

