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extracts various patterns from the mass of empirical data and tabulates them in
natural laws. It is here that it comes in contact with mathematics, which provides
a rich supply of abstract structures for clothing these laws. In this process,
physicists have often stimulated deep mathematical research for establishing the
concepts asked for (e.g. distributions for modeling point sources), sometimes
they have also found ready-made material in an unexpected corner of “pure
mathematics” (e.g. Rogers-Ramanujan identities for kinetic gas theory); most of
the time, however, we see a parallel movement of mutual fertilization.
As a branch of technical engineering , physics is utilized for constructing
the machines that we encounter in the world of technology (item number 3
in the three-step rhythm). Engineers are nowadays equipped with a huge body
of powerful applied mathematics - often hidden in the special-purpose software
at their disposal - for controlling some processes of nature precisely in the way
desired at a certain site (e.g. the temperature profile of a chemical reactor). If
we are inclined to look down to this “down-to-earth math”, we should not for-
get that it is not only the most prominent source of our money but also the
immediate reason for our present-day prosperity.
Of course, the above assignment 1
natural sciences (e.g. theoretical phy-
sics), 2
technical sciences (e.g. engi-
neering physics) must be understood cum grano salis : Abstract “mathematical”
reasoning steps are also employed in the natural and technical sciences, and a
mathematician will certainly benefit from understanding the physical context of
various mathematical structures. Besides this, the construction of models is also
performed within mathematics when powerful concepts are invented for solving
math-internal problems in the same three-step rhythm (e.g. 1
formal science (i.e. mathematics), 3
extension fields,
2
Galois groups, 3
solvability criterion).
2
Algebraization: The Commitment to Computing
The above view of mathematics prefers its dynamical side (problem solving)
over its static one (knowledge acquisition), but actually the two sides are inti-
mately connected (knowledge is needed for solving problems, and problems are
the best filter for building up relevant knowledge bases). The dynamic view of
mathematics is also the natural starting point for symbolic computation, as I will
explicate in the next section. In fact, one can see symbolic computation as its
strictest realization, deeply embedded in the overall organism of less constructive
or “structural” mathematics.
Within symbolic computation, I will focus on computer algebra in this pre-
sentation. Strictly speaking, this means that we restrict our interest to algebraic
structures (domains with functional signatures and equational axioms like rings).
But this is not a dogmatic distinction, rather a point of emphasis; e.g. fields are
also counted among the algebraic structures despite their non-equational axiom
on reciprocals. In some sense computer algebra is the most traditional branch of
symbolic computation since rewriting functions along equational chains is maybe
the most natural form of “computation with symbols”. What is more important,
though, is the axiomatic description of the algebraic structures in use.
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