Hall (1963) stated that the goal of applied mathematics is the mathematical mastery of reality.

Where does an applied mathematician begin? Often, when formulating a problem, the process starts with identifying the main or essential properties of the phenomenon. In certain cases, particularly for physical phenomena, this process of idealization and schematization plays a key role, since a real phenomenon involves a multitude of processes, all of which are practically impossible to account for. Once the essential factors have been identified, the next step is to translate these factors into the language of mathematical concepts and quantities, and to establish relationships between these quantities.

As a rule, this is the most difficult part of the modelling process. A model begins with something very simple, taking on more complex forms as a deeper understanding of the phenomenon is achieved.