I Love 23

The 23 Problems of Hilbert: A Guide Through the Mathematical Challenges of the 20th Century

Posted on Jul 22, 2023

In 1900, at the International Congress of Mathematicians in Paris, the famed mathematician David Hilbert laid out a list of 23 unsolved problems that he believed would set the course for the mathematical research of the 20th century. These problems covered a wide range of topics, from number theory and algebra to geometry and analysis, touching on some of the most profound questions of his time.

The number 23, in this context, represents the challenges and the enigmas that mathematicians set out to unravel. Each problem represented a milestone, a beacon guiding the journey of countless mathematicians.

Some of the most notable among these are:

  • The Second Problem: This problem dealt with proving that the arithmetic axioms are consistent - a challenge that led to major developments in logic and set theory, and also to Gödel’s incompleteness theorems.

  • The Eighth Problem: This problem, known as the Riemann Hypothesis, concerns the distribution of prime numbers and is still unsolved today. It is considered one of the most important unsolved problems in mathematics and carries a million-dollar prize for its solution.

  • The 23rd Problem: Fittingly, the last problem on Hilbert’s list is more of a challenge than a specific question. Hilbert asked for further development of the methods of calculus of variations - an area of mathematics with wide applications in physics and engineering.

Hilbert’s problems were so influential that they guided much of the mathematical research conducted during the 20th century. Even today, several of these problems remain unsolved, continuing to inspire and challenge mathematicians around the world.

In this sense, the number 23 not only symbolizes the ingenuity and ambition of David Hilbert but also embodies the spirit of mathematical inquiry and the eternal pursuit of knowledge. Through the lens of Hilbert’s problems, the number 23 takes on a profound and enduring significance in the field of mathematics.