How should we view the relationships between the major branches of mathematics?

Mathematics is a field with many branches, which may be categorized into subfields, or they may be partially organized into different categories, depending on context. <br><br>Some of the distinct subfields that mathematics can be divided into, depending on its applications in turn, may include geometry, arithmetic, analysis, combinatorics, topology, algebra, differential geometry, measure theory, number theory, category theory, differential equations, logic, set theory, mathematical physics, operator algebras, operator theory, graph theory, dynamical systems, algebraic topology, control theory, representation theory, game theory, ergodic theory, statistics, combinatorial game theory, harmonic analysis, category theory, information theory, knot theory, mathematical finance, coding theory, chaos theory, symplectic geometry, network theory, homotopy type theory, homological algebra, and measure theory.<br>Depending on what you are using them for, the actual number of branches of mathematics is much larger than the above list, and for example, probability theory, machine learning, dynamical systems theory, or computer science could be subfields or else different branches of mathematics, depending on context (for example, they're often considered something separate from the core branches of mathematics per se, but are often considered branches of mathematics nonetheless, perhaps subfields of core branches). <br><br>One way to organize these branches is to be based in terms of the underlying technology or ideas of each field. <br>But another way to organize these different branches is to separate them into algebraic, geometric, and analytic branches. <br><br>So for example, the following different branches of mathematics would fall under the category of algebraic mathematics: <br>* abstract algebra - includes group theory, ring theory, Galois theory; <br>* homological algebra; <br>* representation theory; <br>* combinatorics - including the study of graphs, matroids, partial orders, designs; <br>* coding theory; <br>* algebraic geometry; <br>* algebraic topology; <br>* algebraic number theory; <br>* category theory; <br>* information theory. <br><br>The following branches of mathematics would fall under the category of geometric mathematics: <br>* convex geometry; <br>* discrete geometry; <br>* differential geometry; <br>* algebraic geometry; <br>* symplectic geometry; <br>* Riemannian geometry; <br>* topology; <br>* graph theory; <br>* algebraic topology; <br>* geometric group theory; <br>* geometric measure theory; <br>* differential topology; <br>* control theory; <br>* machine learning; <br>* network theory. <br><br>And the following branches of mathematics would fall under the category of analytic mathematics: <br>* measure theory; <br>* functional analysis - including operator algebras and operator theory, and including harmonic analysis; <br>* real analysis; <br>* complex analysis; <br>* mathematical physics - including quantum mechanics; <br>* dynamical systems theory; <br>* ergodic theory; <br>* probability theory; <br>* stochastic process theory; <br>* chaos theory; <br>* partial differential equations; <br>* ordinary differential equations; <br>* integral equations; <br>* approximation theory - including interpolation, spline interpolation, and function factorization; <br>* Fourier analysis; <br>* machine learning; <br>* calculus of variations; <br>* optimal control theory; <br>* numerical analysis; <br>* game theory; <br>* order theory; <br>* set theory; <br>* logic; <br>* number theory; <br>* combinatorial game theory; <br>* graph theory; <br>* statistics; <br>* mathematical finance. <br><br>Some branches of mathematics may fall under multiple categories in this framework, or they might be considered a combination of different subfields. <br><br>An example of this is how category theory is both an algebraic and analytic subfield of mathematics. Similarly machine learning is both a geometric and analytic branch of mathematics, but is often considered separately as its own branch of mathematics. Algebraic geometry is both an algebraic and geometric subfield of mathematics, and number theory is both an algebraic and analytic branch of mathematics as well, depending on context. <br>Differential equations is both an analytic and geometric branch of mathematics, as are dynamical systems, ergodic theory, control theory, graph theory, game theory, combinatorial game theory, network theory, homotopy type theory, combinatorics, and topology, depending on context, and whether or not you're focusing on the theory or the applications. <br><br>So given the non-rigidity of the categorization, for the different branches of mathematics, how do we view how they are distinct from each other, and how should we view their inter-relationships, depending on different contexts? <br><br>In other words, how do we categorize the different branches of mathematics, and distinguish between them, depending on different contexts?