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 Structures
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
One of several mathematical castles that you can build for yourself!
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.
The True Beauty of Math 
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I’m struggling to differentiate between, or inter connect, the definitions of patterns and structure.
Thanks for the question! Though I’m a little confused by the wording, can you elaborate on what you mean by “definitions of patterns”?
Thanks for your analogy between structures in everyday world and structures in mathematics. After reading the paragraph entitled “Structures in general” I get a sense of hierarchy i.e. there is something simple (in some sense) then it is used to build more complex object and so on. However, when I read the definition of a group I do not get any sense of hierarchy. For example the property of being closed, having identity or inverse do not invoke any sense of hierarchy in me. What is the starting point in case of groups? What do we do so that the starting point becomes a more complicated object? What is complex about the organization? How many levels of hierarchy are there?
I would be very thankful if you could provide some explainations with examples.
First of all sorry for my late reply! And secondly you’re entirely right: there does appear to be a sort of hierarchy of structure in mathematics, and indeed the same is true for groups. This hierarchy is not necessarily linear, but it does move from “more general” to “more structured” in any given direction. Namely, the “starting point” so to speak (though there are some issues here) could be seen as set theory. In particular, simply dealing with sets means that we’re dealing with the most general objects, and in turn these objects have almost no structure: they’re just collections of things.
We can go up to the next step in the hierarchy by endowing sets with the ability to have their elements combined in such a way that this combination process is closed, has an identity element, as is invertible. We’re then limiting ourselves to only certain kinds of sets, but these sets now have more structure (and we call these sets groups).
We can then (and will in much later lessons) go on to only explore those groups that have certain properties, like a commutative multiplication, or which or generated by certain elements, or what have you, and we would be continuing to go up in our hierarchy of structure by limiting ourselves to commutative groups or finitely generated groups.
We could also have taken another step up in the hierarchy by only considering the groups that can be given a different kind of multiplication as well (simultaneosuly as the above group multiplication), with a different identity, and in this case we would be studying rings. Rings are then just special kinds of groups which are (as we know) just special kinds of sets. We could then study special kinds of rings known as fields, and so on and on.
We could ALSO have started from sets and moved on to, say, more geometric considerations by endowing sets with a type of “distance function” and therefore studying what are called metric spaces, or endowed sets with a special subset of sets and begun our study of topology. In all of these cases we are considering sets with more and more structure, which are therefore less and less general, but which are also more and more interesting. This is what this “hierarchy” is all about, and groups play an important part in it all the same.
As a side note, a lot of this hierarchical perspective will change once we study category theory, but for now this is certainly a great perspective to have.
I hope that helps and don’t hesitate to ask about it more! 🙂
Thank you very much for your wondrful reply. I am often confused by these three terms – set, space, and structure. I know the examples but am not able to abstract from those examples those “essential features” that would help me differentiate between these three terms. What are these “essential features” of set, space and structure?
Sorry for my lateness! A set and a space are usually very concrete objects: a set is a collection of elements and a space is, in any given circumstance, a set with very particular added structure. For example, we usually use the word “space” as a shorthand for “vector space,” which is a set with a very particular kind of structure. A manifold might also be what is meant by “space” in some given circumstance, and again a manifold is just a set with (a lot of) very particular structure added to it. “Structure” itself is either just a general term for any kind of mathematical structure at all, or it is also used (as I have above in this answer) to refer to the added requirements that we place on an object. For example, when I say that a vector space is a set with added “structure,” I simply mean that a vector space is a set with such and such properties. A group is a set with the property of being able to associatively multiply elements, have an identity element, and invert elements. Thus, a group is a set with added structure. Does that make sense/answer your question?
A pattern is a subset of a structure. Similar bricks could build a variety of structures
Can using a table to reveal a pattern be considered a mathematical structure
The table itself isn’t the mathematical structure in that case, but rather the pattern is 🙂
I am much interested in dining out what tools are used in structure when working with set THEORY and functions. I would welcome one or two examples in plan english.many thanks.
Satisfiable explaination
Thanks! 🙂