Derivations

The process of derivation 'loads' real world activity into equations. This loading process allows equations to 'act out' the changes in the world, such as bodies moving in a straight line, rotating bodies, oscillating bodies, etc. The equation can thus be thought of as a symbol-based 'machine', which can mimic -- act out -- changes in the real world.

As the symbol-based system can act out changes in the world, it can be used to explore, and predict, the changes in the real world. This makes the symbol-based system very useful, as it allows building and exploring many 'what-if' scenarios, without making (costly) changes to the real world.