Math, or rather our knowledge of it, is continually expanding in both volume and diversity of subfields. However, to better understand a general, high-level organization of it, let’s take a snapshot of current mathematical knowledge. If we roughly partitioned this corpus into its main branches, based on topical similarity and general amount of (current) content, what would they be?
I’d organize it like so:
- Geometry and Analysis: The study of geometric objects, limits, differential and integral calculus, continuity, topology, manifolds, and related concepts.
- Algebra: The study of operations, or functions of the form
(where the sets
correspond to operation arities), on both specific and abstract systems (such as integers and commutative rings.)
- Discrete Math: The study of “discrete” concepts, such as integers, rational numbers, counting methods, graphs, graph-like/combinatorial structures (such as block designs), and discrete processes (such as algorithms, if the theory of computation is included within math.)
- Special Topics: Anything in math not covered above, such as probability and statistics, mathematical logic, set theory, and category theory.
(“Special Topics” is basically the cool-sounding word for “miscellaneous” in academia.)
We now discuss some further clarifications and points about this list.
Combining Geometry and Analysis
Some authors seem to separate analysis and geometry in their organizations; for example, MIT’s categorization of its math courses at the time of writing contains one header for analysis (18.1xx) and another for geometry and topology (18.9xx.) Other authors seem to further separate geometry and topology, for example MathWorld. However, one of the main concepts of analysis, continuity, is defined generally within the setting of topological spaces, and many concepts in analysis, like derivatives and integrals, have their roots and motivations in geometric constructs. Furthermore, generalized geometric constructions like manifolds are typically defined most abstractly with topology. Thus, it makes more sense to me to combine all these fields if we want to define branches by topical similarity.
Classification of Probability and Statistics
Probability can be most generally defined in the framework of measure theory, which is a part of analysis, but the subject of probability includes the study of specific probability measures too, such as discrete probability, which falls more under discrete math, and continuous probability, which falls more under real analysis. Furthermore, statistics as a whole does not seem to be fully well-encompassed by any of the descriptions of geometry and analysis, algebra, or discrete math. It thus makes sense to place probability and statistics in a separate branch, which just has intersections with all of these.
Mathematical Logic vs Set Theory
Unlike many other authors today, I consider mathematical logic to be separate from set theory. However, since these would be part of the same “main branch” in my classification anyway, my thoughts on this are best left to another article.
Intersections Between These Branches
Despite our goal to partition the corpus of mathematical knowledge into branches, the four main ones we have identified do intersect to some degree with each other. In particular, discrete math is defined (informally) by a notion of “discrete-ness” of a concept, which is a theme that extends across many different concepts, sometimes within subjects. For example, “discrete geometry” and “finite geometry” are both sometimes considered subjects in geometry, as well as in discrete math. Or as another example, take mathematical logic; since a proof is a “discrete process,” proof theory would be in an intersection with discrete math. Because of that, I had actually originally included logic as a sub-branch of discrete math, as some authors still do. However, this doesn’t seem fully representative of all of logic, which includes topics like Boolean algebra for propositions and model theory, which I wouldn’t classify in general as “discrete.”
Informal Definitions of the Branches
It’s important to note that the definitions of these branches are fundamentally informal. For example, I’m not defining discrete math formally in such a way that it would be limited to countable sets. Then, it would exclude topics like infinitary combinatorics, which by all rights should be part of combinatorics, or abstract analytic number theory, which by all rights should be part of number theory. These topics are motivated by and related to the “discrete” concepts despite the fact that their scope may ultimately extend beyond such discrete objects, and thus they seem best classified within those subjects.
Topicality vs Amount of Content
This organization attempts to delineate main branches of math based on both topical similarity and general amount of (current) content. In general, it can be very possible that these two goals may not be fully aligned. However, in our case, I believe we have achieved a great maximization of both, with a delineation that is short and sweet (four main branches), and that seems the most reflective of current mathematical knowledge and activity.
—
Updated November 2023.
