Number theory posits that natural numbers (i.e., "counting numbers") group, classify and order both empirical objects, such as apples, and abstract or imaginary objects, such as ideas. Through quantitative mathematical reasoning, humans conceptualize properties such as magnitude and amount. Numbers are the basic symbols of quantitative reasoning. People use number theory as a kind of symbolic shorthand by which to label, identify and think about certain properties of an empirical or abstract entity.
Proofs are mathematical models that represent rules internal to mathematics. Mathematical theoreticians develop proofs by assuming the truth of certain basic statements called axioms, then logically deduce conclusions. Mathematical proofs demonstrate relationships between mathematical properties. In geometry, for example, proofs demonstrate the relationship between an area of a circle and its diameter.
Investigation is the research-based process of applying mathematical reasoning to other disciplines. Mathematicians and theorists from the social sciences and natural sciences use investigation to determine the relationship between real-world entities and mathematical symbols. Through investigation, mathematicians have discovered unique mathematical properties governing natural properties like the shape of a snowflake and the flying patterns of bees.
Mathematical abstraction represents the opposite of the process of investigation. Through abstraction, mathematicians create ideal statements that represent idealized versions of the real world. For example, the area of a rectangle in real world is never precisely its base length multiplied by its height. Physical rectangles are irregular and not perfectly geometric. By contrast, mathematical equations represent the world in an abstract, perfected state.