15. Structures & Relations

Structures & Relations

Often spotting a pattern is the first step to understand the underlying structure involved.
And here lies the strength of abstraction in mathematics: the same mathematical structures can appear in widely different settings. Revealing a structure in one context can help solving problems in another context, no matter how different. 

A mathematical structure consists of a collection of associated objects and relations,that must all satisfy certain requirements, called axioms.

Mathematical objects are what we talk and write about when we do maths: numbers, functions, triangles, matrices, graphs, sets, etc. Despite their abstract nature, we think and talk about them as if they physically existed. 

A mathematical relation is a relationship between sets of elements. Well-known relations in mathematics are functions, which associate to each element of a first set exactly one element of a second set.

In physics, many relationships are expressed by equations where quantities are represented by letters or symbols. Algebra is the branch of mathematics that studies these symbols and the rules for manipulating them.

Axioms are statements taken to be always true, to serve as a premise for further reasoning and arguments. They are fundamental building blocks of both mathematical logic and philosophy.

Combining those abstract concepts forms coherent systems, languages, and frameworks: structures allowing us to elaborate and communicate ideas, as well as to understand and model our world.