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Re: More Mathematica CAN'T do than CAN???
Mathematica is good at what it does - namely manipulating lists and equations, solving systems of equations and differential equations, and a myriad of other well integrated state-of-the art procedures algorithms. Personally I use it for almost everything, but that is related to both the material I work on and the fact that I am most familiar with Mathematica (a soft form of vendor lock-in). For some of my recent work, which involved large expressions with Grassmann fields, a program like GiNaC, FORM or maybe Cadabra would probably have been more appropriate - but would have had the overhead in learning how to use them properly. For other work where I've needed to analyse graph automorphism groups and Lie algebras, GAP (accessed via a Sage notebook) was a more sensible choice. Finally, some appropriate quotes from "The Magma Algebra System I: The User Language" http://www.math.ru.nl/~bosma/pubs/JSC1997Magma.pdf (It's from 1997, but still relevant since the core of Mathematica hasn't changed much) On Universality: "A system design that provides a satisfactory computational environment for all areas of mathematics has not yet appeared, and for good reason: it is probably impossible." On First class status for structures: "Since, operationally, algebraic structures and individual =93elements=94 of a structure are each mathematical entities possessing properties which we seek to investigate, structures need to have first class status in an algebraic language. Thus, it should be as easy to define a polynomial ring R and perform operations on R as it would be to define and perform arithmetic with a polynomial. This is in sharp contrast to the approach taken in systems such as Mathematica and other systems which have adopted an element-centred model of algebra and provide virtually no support for structural computation." Simon