How I Mapped the Interconnections Between Major Math Disciplines

My Understanding of the Relationships Between Mathematical Disciplines

After three years of university study and completing all required mathematics courses, I asked myself: what is mathematics? To answer, I created a diagram to visualise how the major branches relate.

Below I explain the reasons for this categorisation, add the status of other branches, and comment on ancient Chinese mathematics.

I place analysis and algebra together, starting from number theory because natural numbers are studied first, followed by rational, real, and complex numbers, leading to real analysis, complex analysis, and eventually to analysis in general.

Algebra also originates from number theory, progressing from natural numbers to higher algebra (real and complex numbers) and then to abstract algebra, which expands algebraic structures.

Geometry begins with Euclidean geometry, which studies simple lines and planes; the axiomatic approach of Euclid’s Elements influences later work. Analytic geometry algebraises geometric problems, focusing on conic sections and eigenvalues. Abstract algebra and analytic geometry have no direct inheritance, only chronological ordering. Differential geometry builds on analytic geometry using tools from analysis and algebra.

The advent of computers greatly amplifies mathematics; computer knowledge is introduced as a tool and language.

Above the three main categories are cross‑disciplinary topics. For example, differential geometry uses analysis; differential equations belong mainly to analysis but are solved using vectors and matrices from algebra. Graph theory can be seen as a special case of higher algebra yet originates from geometry. Probability and statistics are generally placed outside analysis, algebra, and geometry, focusing on uncertainty, though modern probability adopts an axiomatic, analytical proof style.

I did not classify topology or functional analysis because I have not studied them deeply.

I also regard some subjects as studies of mathematics as a whole: the history of mathematics examines its development, mathematical logic inspects internal logical relations, and mathematical modelling connects mathematical knowledge with real‑world problems.

Regarding ancient Chinese mathematics, although it produced rich results, it did not develop a system as complete as the Western tradition, and modern Chinese mathematics largely absorbed Western achievements, so I have not organised it here.

In fact, mathematical classification is mainly based on research objects and methods. As tools increase, the same objects are studied with different methods, making a single classification insufficient. Future classifications built on contemporary mathematics should reflect recent advances and may become a topic for mathematics education.