Mathematics Branches, Topics, and Sub-Topics

A structured visual guide to the major mathematical areas and their relationships.

Search by code, branch, topic, subtopic, or a keyword from the descriptions.

01Axx History of mathematics and mathematicians

This subtopic examines the historical development of mathematics and the contributions of mathematicians across periods and cultures, from ancient traditions through modern research mathematics. It is used to trace how concepts evolved, why methods became standard, and how institutions, notation, and pedagogy shaped the discipline. Applications include mathematics education, history of science research, and interdisciplinary scholarship where historical perspective sharpens conceptual understanding and communication.

Specific topics

01A05 General histories

Overview

This topic surveys broad histories of mathematics and is the best entry point for seeing the discipline as a long-running human practice rather than a sequence of isolated results. It helps connect ancient, early modern, and modern mathematics into a coherent narrative.

Related Wikipedia Page

History of mathematics (Wikipedia)

Useful Links

Key Ideas

  • Development of methods, notation, and institutions over time
  • Interaction between mathematics, astronomy, physics, and philosophy
  • Changing standards of proof, rigor, and abstraction

Typical Uses

Used for survey writing, education, and contextualising major theorems or schools of thought within their historical setting.

Applications

  • History of science and mathematics education
  • Biographical and cultural analysis of mathematical work
  • Comparative study of traditions across civilizations

References

Recommended Textbooks

01A16 Egyptian mathematics

Overview

This topic focuses on Egyptian mathematics, especially its practical arithmetic, geometry, and problem-solving methods as preserved in surviving papyri. It is a useful case study in how mathematical knowledge develops in administrative and construction contexts.

Related Wikipedia Page

Ancient Egyptian mathematics (Wikipedia)

Useful Links

Key Ideas

  • Unit fractions and practical computation
  • Geometric formulas used for surveying and land measurement
  • Problem texts that reveal working mathematical routines

Typical Uses

Used to study the origins of arithmetic technique, surveying, and early geometric reasoning in an administrative society.

Applications

  • History of mathematics courses and museum interpretation
  • Comparative study of ancient computation
  • Reconstruction of early algorithmic procedures

References

Recommended Textbooks

01A20 Greek, Roman mathematics

Overview

This topic covers Greek and Roman mathematics, especially the shift from practical calculation to deductive proof, geometric method, and systematic theory. It is central for understanding the emergence of rigorous mathematical argument in the classical tradition.

Related Wikipedia Page

History of mathematics (Wikipedia)

Useful Links

Key Ideas

  • Axiomatic geometry and proof by deduction
  • The role of Euclid, Archimedes, Apollonius, and later commentators
  • Transmission of Greek methods into Roman and later traditions

Typical Uses

Used to connect modern proof culture with its classical roots and to study the early development of geometry, number theory, and astronomy.

Applications

  • History of proof and geometry
  • Classical scholarship and translation studies
  • Teaching foundational mathematical ideas through historical examples

References

Recommended Textbooks

01A45 History of 17th century mathematics

Overview

This topic covers the 17th century, a formative period for symbolic algebra, analytic geometry, infinitesimal methods, and the mathematization of natural philosophy. It marks the transition toward modern mathematical research styles.

Related Wikipedia Page

History of mathematics (Wikipedia)

Useful Links

Key Ideas

  • Analytic geometry and algebraic symbolism
  • New methods for curves, motion, and infinitesimals
  • Interplay between mathematics, astronomy, and mechanics

Typical Uses

Used to study how modern mathematical notation and problem types emerged alongside early scientific revolution work.

Applications

  • Intellectual history of the scientific revolution
  • Historical reading of foundational texts by Descartes, Fermat, Newton, and Leibniz
  • Comparative analysis of algebraic and geometric methods

References

Recommended Textbooks

01A55 History of 19th century mathematics

Overview

This topic covers 19th-century mathematics, when rigor, abstraction, and institutional research culture expanded rapidly across algebra, analysis, geometry, and mathematical physics. It is a key century for the modern shape of mathematics.

Related Wikipedia Page

History of mathematics (Wikipedia)

Useful Links

Key Ideas

  • Rise of rigor in analysis and geometry
  • Emergence of modern algebra and group theory
  • Professionalization of mathematical research and publication

Typical Uses

Used to trace the emergence of modern abstraction, formalism, and the research university as a home for mathematics.

Applications

  • Historical analysis of Cauchy, Riemann, Weierstrass, and Klein
  • Study of the foundations of analysis and geometry
  • Contextual reading of the origins of modern mathematical language

References

Recommended Textbooks

01A60 History of 20th century mathematics

Overview

This topic covers 20th-century mathematics, including the diversification of pure and applied fields, the rise of abstract structures, and the increasing role of computation and formal foundations. It is the century in which mathematics becomes visibly global and highly specialized.

Related Wikipedia Page

History of mathematics (Wikipedia)

Useful Links

Key Ideas

  • Abstraction, axiomatization, and structural methods
  • Growth of topology, logic, algebra, and applied mathematics
  • Computers, computation, and the expansion of mathematical communities

Typical Uses

Used to study the modern professionalization of mathematics, the emergence of new subfields, and the interaction between mathematics and computing.

Applications

  • Intellectual history of modern mathematics
  • Biographical study of major 20th-century mathematicians
  • Context for the development of contemporary research programmes

References

Recommended Textbooks

01A70 Biographies, obituaries, personalia

Overview

This topic covers biographies, obituaries, and personal accounts that illuminate the lives of mathematicians and the social context of their work. Biographical study helps connect individual creativity to institutions, correspondences, and intellectual communities.

Related Wikipedia Page

List of mathematicians (Wikipedia)

Useful Links

Key Ideas

  • Lives, correspondences, and institutional roles
  • Biographies as a route into mathematical culture
  • Obituaries and memoirs as historical sources

Typical Uses

Used to humanize mathematical history, support archival research, and provide readable entry points into a mathematician's work.

Applications

  • Historical and cultural research
  • Teaching and outreach
  • Contextualizing major theorems and schools of thought

References

Recommended Textbooks