In the post “What is math?”, we described mathematics as the art of creating and exploring mathematical structures. It is not unlikely, however, that the reader is slightly unfamiliar with the notion of a mathematical structure. If this is the case, then our definition of mathematics is rather unsatisfying. This post aims to rectify this.
Structures in general
When we think of a structure in the everyday sense, we might think of buildings, houses, and bridges. We may also think of a structure as a more abstract object involving some form of complex organization. The plot of a movie, a musical composition, and government bureaucracies all are structures in some sense. All of these are instances in which small sub-structures are organized in ways to create larger, more complicated patterns. A building is nothing but the complicated organization of smaller sub-structures such as bricks, cement, wood, and iron. A musical composition is a complicated organization of melodies and harmonies, which are in turn complicated organizations of notes and rhythms.
Math is no different. A mathematical structure is nothing but a (more or less) complicated organization of smaller, more fundamental mathematical substructures. Numbers are one kind of structure, and they can be used to build bigger structures like vectors and matrices (the definitions for which will be posted in the future).
There are plenty of other kinds of mathematical structures that exist in a rather fundamental way, and that can be used to build other remarkably beautiful structures. Sets and functions are both incredibly fundamental in mathematics, and they can be used to build crazy things like topological spaces (again, which haven’t been defined (yet) on this site, but will be soon). Sets and functions can also be tools for exploring different types of infinities.
Building your castle
Studying math is like building a castle in your head. When building a castle, you first must learn to build a brick, and once that is mastered, you can use it to build a wall. Once you can build a wall, you can build a tower. Stronger bricks allow for higher walls and bigger towers. Additionally, powerful tools allow you to build faster and more efficiently.
The beauty of a mathematical structure comes from its ability to have larger structures built from it. Certain mathematical concepts allow for faster building than others. For example, a mathematician will find a mathematical crane much more useful than a mathematical wheelbarrow.
While you were in high school, you likely only learned about one type of structure—those that could be built up from numbers. Although some of this is interesting, the real beauty of math lies in the flexibility and deep interconnectedness of various kinds of mathematical structure. We explore several of these structures throughout this site, and I believe that once you start on your castle, you’ll never want to stop.