The role of modeling throughout mathematics

Problems in applied mathematics are generally connected to questions which are non-mathematical in their original formulation. Therefore, some kind of translation is required to model the relevant situation in terms of mathematical objects (numbers, functions, sets,...). Creating such a mathematical model consists of two crucial steps:

• Naming all occuring objects and concepts
• Expressing all relevant relations between them

Once a model is established, one can formulate the original question as a mathematical statement using the newly introduced objects and relations. Then, the task is to derive its answer using formal rules.

This modeling process comes into play whenever a concept has to be cast into a mathematically precise form. In particular, modeling does not only occur at the interface to science and technology but also within mathematics itself in the form of definitions of single notions up to whole theories.

However, since the definitions in standard lectures are already well established, the modeling process is hardly ever discussed in its own right. As a result, students may have problems with vaguely formulated exercises which require a modeling step to obtain a problem that can be approached using formal methods. These difficulties express themselves in the typical statement "I don't know what to do here".

In view of this, one objective of the $\mmath$-project is to raise awareness for the modeling step throughout mathematics. For this purpose a formalization of the term model was developed and implemented in the $\mmath$-language, where it serves as a fundamental language construct.