Henri Poincaré

Overview

Poincaré worked across nearly every field of mathematics of his day and is often described as the last person who could. His 1895 paper Analysis Situs and its supplements founded algebraic topology, introducing the fundamental group and the invariants used to distinguish shapes that cannot be told apart by inspection. Working on the three-body problem in the 1880s he found that orbits can depend so sensitively on initial conditions that long-term prediction becomes impossible in practice, which is the origin of what is now called chaos. In 1904 he asked a question about which three-dimensional shapes are equivalent to a sphere; it stayed open for a century until Grigori Perelman proved it in work released in 2002 and 2003. He also wrote for general readers, arguing that the axioms of geometry are conventions rather than facts, and he objected to non-constructive reasoning in a way that makes him a predecessor of the intuitionists. Start with Science and Hypothesis.

Where to start reading

• Analysis Situs 1895

  The founding paper of algebraic topology, introducing the fundamental group and homology as tools for telling spaces apart.

• Science and Hypothesis 1902

  Essays for a general reader on number, space, force, and the status of scientific theories, including his argument that geometric axioms are conventions.

  Read it on Project Gutenberg

• Les Méthodes Nouvelles de la Mécanique Céleste 1892-1899

  Three volumes on the three-body problem, containing the qualitative methods that led to the discovery of sensitive dependence on initial conditions.

Part of

• Topology
• Intuitionism

Sources

1. Henri Poincaré MacTutor History of Mathematics
2. Henri Poincaré Wikipedia
3. Topology history MacTutor History of Mathematics