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Re: More Mathematica CAN'T do than CAN???
*To*: mathgroup at smc.vnet.net
*Subject*: [mg113379] Re: More Mathematica CAN'T do than CAN???
*From*: Simon <simonjtyler at gmail.com>
*Date*: Tue, 26 Oct 2010 05:33:55 -0400 (EDT)
*References*: <ia10ci$op7$1@smc.vnet.net>
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
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