It is one of the deep mathematical insights that foundational
systems
like first-order logic or set theory can be used to construct large
parts of existing mathematics and formal reasoning. Unfortunately
this insight has been used in the field of automated theorem proving
as an argument to disregard the need for a diverse variety of
representations. While design issues play a major rôle in the
formation of mathematical concepts, the theorem proving community has
largely neglected them. We argue that this leads not only to problems
at the human computer interaction end, but that it causes severe
problems at the core of the systems, namely at their representation
and reasoning capabilities. In order to improve applicability, theorem
proving systems need to take care about the representations used by
mathematicians.

Donald Norman gives a fascinating introduction into "The Design of
Everyday Things." His insights are of a very general nature and we
argue that the principles for good design hold in mathematics as well.
The design of concepts in mathematics takes a lot of the burden on
getting things right from the human user and puts it into an
appropriate representation. The different representations are used to
keep information together, hide unimportant details and allow to
concentrate on the important parts. Sometimes the right representation
is the key step in the process of problem solving. If one were to use
a foundational system directly, however, everything would have to be
expressed explicitly in a uniform representation, which offers no or
only little structural support.

o

d_{1}

...

d_{n}

d_{1}

c_{11}

...

c_{1n}

. . .

. . .

. . .

. . .

d_{n}

c_{n1}

...

c_{nn}

To exemplify this, we will take a closer look at multiplication tables.
The
information accessible from the table is that it is a binary operation, it
is
discrete and defined on a finite domain. Domain and range are directly
given.
The table has its own notion of well-formedness, that is, all d_{i} have to
occur and have to be different, the table must be fully filled. In the
design
we find natural and cultural constraints. Multiplication tables are
designed
in a way that their structure puts "information in the world" that makes
it
difficult to violate well-formedness. An under-specification would leave a
hole in the structure, it is impossible to enter more than one entry per
field. Furthermore, although the order of the d_{i} in the columns and rows
could in principle be different, cultural conventions prevent that. This
in
turn makes particular reasoning methods possible which are connected to the
representation. For instance, the commutativity of o is checked by
verifying that the table is symmetric with respect to the diagonal. For
more
details, see the full technical report of this contribution at