The Five Schools of Modern Mathematics
While most students of mathematics may never get to see them fully explored, there are five guiding branches in modern mathematics:
- Algebra - The study of the structure of sets and their properties (including related to groups, rings, fields, vector spaces, and lattices).
- Analysis - The study of limits and functions (including related to the topics of continuity, completeness, rates of change, and measure).
- Geometry - The study of shapes and space (including related to points, lines, surfaces, solids, and higher dimensional objects).
- Topology - The study of objects formed by the union and intersection of sets (including algebraic, geometric, and analytic objects).
- Discrete Mathematics & Number Theory - The study of countable sets and objects (including objects from all other fields).
Modern mathematics explores these fields and advanced topics in their intersection, and we can apply their core principles to questions beyond that of simple numbers or geometry to explore questions of philosophy, morality, and the unknown.
Algebra
Algebra is the study of structures which use algebraic properties to understand the relationship between sets and operations. While early Algebra primarily focused on the development of strategies to simplify expressions and solve equations, modern Algebra focuses more on the study of structures and the relationships which can arise from the abstract.
For those who have only experienced high school Algebra, think of Algebra as the study of mathematics when you abstract it beyond the numbers and variables and focus more on what lies beneath. The motions of a clock or a puzzle cube for instance may share the properties of closure, associativity, identity, and inverse, but only the clock has the property of commutativity. These structures and properties are what drives an algebraist.
Analysis
Analysis is the study of limits and functions, in particular as related to continuity, rates of change, and measure. It is a subject which begins with the idea of calculus but then extends it beyond the real numbers into the abstractness of arbitrary sets. It is an exploration into infinity and the infinitesimals, developing tools to explore all things great and small.
One core topic in Analysis is the idea of completeness; when we think of the Rational Numbers (which includes all fractions), it is missing Irrational Numbers (numbers such as pi). Analysis explores ways to extend the Rational Numbers to develop the Real Numbers, and once the Real Numbers are established ideas related to differentiation and integration can be explored, as well as a richer understanding of measure beyond that of Euclidean norms.
Geometry
Geometry is the study of shapes and space, in particular as related to points, lines, surfaces, solids, and higher dimensional objects. It is one of the earliest explored of the subjects, with basic principles being understood by even animals; what is a bird’s nest but a geometric object designed to protect an egg? Even artwork and the written language carry with it the earliest elements of geometry, the idea that we can gain an understanding of things through examining shapes and spaces.
While Euclidean Geometry has been the standard for over 2000 years, geometry can take many forms. Analytic Geometry explores graphing, Non-Euclidean Geometry explores geometry on the surface of a curved object, and Differential Geometry explores ways to examine motions and shapes in higher dimensions. It is all derived from the simplest structures that surround us all.
Topology
Topology is the study of objects formed by the union and intersection of sets. It is one of the newer major fields in the study of mathematics, and can be thought of as an exploration into the nature of Venn diagrams, atlases, and how sets are connected. It is geometry in the absence of shape. Though the subject is often not taught until students are far beyond calculus, the underlying principles of unions and intersections can be understood by all.
A common way to think about Topology is to consider an atlas which contains all the maps of the Earth. Each of those maps is but a representation of a small part of the Earth, but when combined we can see what the entire world looks like. What is not evident solely from the maps, though, is if the earth is spherical or pear shaped; this structure without shape is what drives Topology, and allows topologists to consider the structure of a doughnut and a coffee cup to be equivalent.
Discrete Mathematics & Number Theory
Discrete Mathematics is the study of countable sets and objects, and Number Theory is its largest subfield which explores discrete number sets. All things countable fall under the heading of Discrete Mathematics. Every piece of computer software, every counting principle, every information system; it’s all Discrete Mathematics. As soon as a child tries to count their fingers and toes, they are exploring the field of Discrete Mathematics.
While Discrete Mathematics and Number Theory are very broad fields, they are the topics that we use every day. Every computational algorithm from cross multiplication to long division was at some point a major breakthrough in these fields. Even the idea of counting beyond ten was at one point a major breakthrough in Number Theory; it is a field as old and fundamental as geometry itself. Still many questions elude number theorists; even the distribution of the primes remains just out of reach.
Modern Mathematics & Applied Mathematics
Modern Mathematics does not typically reside purely in any one of these schools, but instead focuses on the intersection of two or more such fields. For example, Differential Geometry focuses on applying differential structures from Analysis to geometric objects, and often incorporates topology to describe more abstract objects. As such, skilled mathematicians rely on years of dedicated study in each of these fields to become experts in the field.
While Mathematics focuses primarily on the study of certainty, Applied Mathematics is often used to analyze structures in other fields, such as Biology, Chemistry, Physics, Computer Science, and Economics. Even Statistics is a form of Applied Mathematics, using the structures of combinatorics and analysis to measure uncertainty. Where mathematics may be a form of philosophy, its applications are everywhere.