Introduction
Algebra is the bridge between arithmetic (working with concrete numbers) and higher, structural mathematics. It introduces symbolic language, rules for manipulating expressions and equations, and ways to generalize patterns. These skills will be used in geometry, calculus, and advanced fields like abstract algebra or linear algebra.
Mathematics is a very broad field, and just like biology has a taxonomy for organisms, math has a taxonomy for its subjects and subfields. The taxonomy can be arranged in levels, starting from the foundations and branching into specialized areas. The "taxonomy of math" is a branching hierarchy:
Foundations → Pure Math (Algebra, Geometry, Analysis, Number Theory, etc.) → Applied Math (Probability, Computation, Physics, etc.) → Interdisciplinary Fields.
1. Foundations of Mathematics
These provide the logical and conceptual bedrock:
- Logic (propositional, predicate, modal, intuitionistic, etc.)
- Set Theory
- Category Theory
- Foundations/Formal Systems (Peano arithmetic, ZFC, type theory, etc.)
- Philosophy of Mathematics (constructivism, formalism, platonism, etc.)
2. Pure Mathematics
The study of abstract structures, independent of applications.
a. Algebra
- Elementary Algebra (YOU ARE HERE)
- Abstract Algebra (groups, rings, fields, modules, lattices, etc.)
- Linear Algebra
- Universal Algebra
- Homological Algebra
- Representation Theory
b. Geometry
- Euclidean & Analytic Geometry
- Differential Geometry
- Algebraic Geometry
- Topology (point-set, algebraic, differential)
- Non-Euclidean Geometry
- Discrete Geometry
c. Analysis
- Real Analysis
- Complex Analysis
- Functional Analysis
- Harmonic Analysis
- Measure Theory
- Calculus of Variations
- Nonstandard Analysis
d. Number Theory
- Elementary Number Theory
- Analytic Number Theory
- Algebraic Number Theory
- Diophantine Equations
- Transcendental Number Theory
- Arithmetic Geometry
e. Topology
- Point-set topology
- Algebraic topology
- Differential topology
- Geometric topology
f. Combinatorics
- Enumerative Combinatorics
- Graph Theory
- Design Theory
- Extremal Combinatorics
- Ramsey Theory
3. Applied Mathematics
Where math interacts with sciences, engineering, and technology.
Probability & Statistics
- Probability Theory
- Mathematical Statistics
- Stochastic Processes
- Information Theory
Computational Mathematics
- Numerical Analysis
- Symbolic Computation
- Scientific Computing
- Computer Algebra Systems
Mathematical Physics
- Quantum Mechanics (mathematical formalism)
- Relativity
- Dynamical Systems
- Partial Differential Equations
Optimization & Operations Research
- Linear & Nonlinear Optimization
- Game Theory
- Decision Theory
- Control Theory
Biomathematics
Economics
Cryptography & Coding Theory
4. Interdisciplinary / Modern Directions
Modern classes that do not fit neatly into the classical “pure vs applied†categories.
- Data Science & Machine Learning (mathematical side)
- Chaos & Complexity Theory
- Network Theory
- Mathematics of Computation
- Category-Theoretic Approaches to Physics and CS
- Homotopy Type Theory
5. Meta-Levels
This is the historical, cultural, and philosophical aspects of math.
- History of Mathematics
- Education of Mathematics
- Philosophy of Mathematics