What Are the Three Types of Mathematicians? Insights from Shing‑Tung Yau

Shing‑Tung Yau, a world‑renowned mathematician, proposed that mathematicians fall into three broad categories, reflecting their work patterns and thinking.

First type: Mathematicians who create theory

Yau says these mathematicians drive the development of mathematics. They can be subdivided as follows:

Discovering commonality and formulating theory : For example, S. Lie in the late 19th century observed symmetries in mathematics and physics and introduced the theory of continuous transformation groups, laying foundations for modern mathematics.

Extending and transplanting theory : For instance, extending calculus from finite‑dimensional to infinite‑dimensional spaces, or applying it to the study of surfaces.

Using comparative methods to seek common ground across disciplines and develop new results: Weil compared integer equations with algebraic geometry to develop arithmetic geometry; Langlands used group representation theory and automorphic forms to propose the Langlands program.

Developing theory for new phenomena : Gauss’s discovery of surface curvature led Riemann to create Riemannian geometry.

Solving important problems : Nash developed an implicit function theorem while solving the isometric embedding of general Riemannian manifolds into Euclidean space.

Deepening theory after new theorem proofs : e.g., further study after the Atiyah–Singer index theorem.

Introducing new structures : Kähler introduced the eponymous scale in complex manifolds; Thurston introduced the concept of “geometrization” in three‑dimensional manifolds.

Second type: Mathematicians who seek patterns from phenomena

These mathematicians typically discover problems through data experiments or natural and social phenomena and make meaningful conjectures . Examples include Gauss’s study of prime distribution, Pascal and Fermat’s work on gambling odds, and the Black‑Scholes equation for option pricing.

Third type: Mathematicians who solve hard problems

These mathematicians focus on solving major mathematical challenges , such as Yau’s own solution to the Calabi conjecture. They not only solve the problem but also use it as a springboard to further develop theory and advance mathematics.

Yau’s view highlights the crucial roles of mathematicians in theoretical innovation, problem solving, and pattern discovery . He emphasizes that mathematics is not only about truth but also the pursuit of beauty. Through his perspective we gain a deeper understanding of the mathematician’s world and the unique place of mathematics in human civilization.

Reference: Qiu Cheng Tong. (2023). Truth and Beauty: Qiu Cheng Tong’s View of Mathematics. Jiangsu Phoenix Literature Publishing House.