The significant mathematical activity represented in this section is the reasoning involved in understanding mathematical objects in terms of their properties.

In any branch of mathematics, the mathematical object is the "thing" under consideration. The mathematical object is defined clearly and unambiguously, and it is straightforward to verify whether something is one of those objects or not. The characteristics that an object has are its properties. Mathmatical objects can be classified and sub-classified according to the properties that they do and don't possess, and this is how much knowledge about them is gained.