The 20th century transformed mathematics from a discipline with ambitions to completeness into one that explicitly faced its own limits. Along the way it witnessed the solution of centuries-old problems, the birth of new fields, and the arrival of computing as a mathematical tool. More mathematics was produced in these hundred years than in all previous centuries combined.
Hilbert’s 1900 address
At the 1900 International Congress of Mathematicians in Paris, David Hilbert presented a list of 23 unsolved problems spanning the whole of mathematics. The list was a challenge and a prophecy: here are the questions that will shape our century.
Many of the problems were solved in the decades that followed. Others — the Riemann hypothesis (Problem 8), parts of the continuum problem (Problem 1) — remain open. Hilbert’s list gave 20th-century mathematics a coherent research agenda.
Russell and the foundations
In 1901, Bertrand Russell discovered a paradox in Cantor’s naive set theory: consider the set of all sets that do not contain themselves; does it contain itself? Either answer leads to contradiction. This and similar paradoxes (Burali-Forti, Richard) threatened the entire foundation of mathematics.
The reaction split into schools:
- Logicism (Frege, Russell, Whitehead): reduce all of mathematics to logic. Principia Mathematica (1910–1913) attempted this, taking 362 pages to prove that .
- Formalism (Hilbert): treat mathematics as formal symbol manipulation and prove its consistency within itself.
- Intuitionism (Brouwer): reject non-constructive proofs and actual infinity.
Gödel
In 1931, the Austrian logician Kurt Gödel (1906–1978) published his incompleteness theorems. He showed:
- Any consistent formal system containing arithmetic contains true statements that cannot be proved within the system.
- No such system can prove its own consistency.
These theorems destroyed Hilbert’s program in its original form. They showed that mathematical truth genuinely exceeds mathematical proof. The implications — philosophical, logical, and practical — are still being worked out.
Turing and the birth of computing
In 1936, the English mathematician Alan Turing (1912–1954) solved Hilbert’s Entscheidungsproblem — the question of whether there is an algorithm to decide the truth of mathematical statements — in the negative. To do so, he introduced an abstract model of computation now called the Turing machine. This is the theoretical foundation of all digital computers.
Turing’s wartime codebreaking work at Bletchley Park contributed decisively to the Allied victory in World War II. His 1950 paper “Computing Machinery and Intelligence” essentially founded the field of artificial intelligence. He was prosecuted for homosexuality in 1952 and died in 1954, officially by suicide.
The Bourbaki school
From the 1930s through the 1970s, a collective of French mathematicians writing under the pseudonym Nicolas Bourbaki systematically rewrote the foundations of modern mathematics in rigorously axiomatic terms. Éléments de mathématique ran to dozens of volumes. Bourbaki’s abstract, structuralist style dominated mid-century mathematics and shaped how mathematics is still taught.
Great solutions
The 20th century produced solutions to several very old open problems:
- Andrew Wiles proved Fermat’s Last Theorem in 1994, 358 years after Fermat’s margin note.
- Wolfgang Haken and Kenneth Appel proved the Four Color Theorem in 1976 — the first major proof to depend essentially on computer verification.
- Grigori Perelman proved the Poincaré conjecture in 2002–2003 using Hamilton’s Ricci flow — settling the first of the Millennium Prize Problems.
- Deligne proved the Weil conjectures in 1974, the key step in Grothendieck’s program of rewriting algebraic geometry.
Paul Erdős and collaborative mathematics
Paul Erdős (1913–1996) traveled the world living out of a suitcase, producing over 1500 papers with more than 500 co-authors. His way of doing mathematics — collaborative, problem-oriented, often brief — shaped combinatorics, graph theory, and number theory for decades.
The expansion
The 20th century saw the emergence or transformation of many fields:
- Topology becomes a major discipline (Poincaré, Brouwer, Alexander)
- Functional analysis (Banach, Hilbert, von Neumann) underlies quantum mechanics
- Algebraic geometry is rebuilt on commutative algebra (Zariski, Weil, Grothendieck)
- Category theory (Eilenberg, Mac Lane) gives structural languages for all of mathematics
- Probability becomes rigorous (Kolmogorov’s 1933 axiomatization)
- Computer-assisted proof emerges as a new kind of mathematics
At the century’s end
The 20th century ended with the Clay Mathematics Institute announcing seven Millennium Prize Problems in 2000, each with a one-million-dollar reward. Perelman solved Poincaré in 2003. The other six remain open.
It also ended with mathematics more diverse, more specialized, and more productive than ever before — and more aware of its own limits.
Frequently asked
What was the biggest idea of 20th-century mathematics?
Probably the realization — crystallized by Gödel in 1931 — that there are true mathematical statements that cannot be proven within any consistent formal system containing arithmetic. This shattered Hilbert's program and changed the foundations of the discipline permanently.
Is there more mathematics now than ever before?
Yes. More than half of all mathematics ever produced was produced after 1950. The number of working mathematicians has exploded, and subfields have proliferated. A 19th-century mathematician could still be conversant in the whole discipline; a 21st-century one cannot.