# Hilbert's 23 Problems

**Type:** Concept  
**Domain:** History & Philosophy  
**Presented:** 8 August 1900, International Congress of Mathematicians, Paris  
**Presenter:** David Hilbert  
**Codex URL:** /mathematics/history-philosophy/hilberts-23-problems/  
**Entry status:** Live — v1.0 (2026-05-27)

## Summary
23 problems (10 given in lecture, full 23 in published paper) shaping 20th-century mathematics. Roughly half resolved; several remain open including Problem 8 (Riemann Hypothesis).

## Selected problems and status

| # | Problem | Status |
|---|---------|--------|
| 1 | Continuum Hypothesis | Independent of ZFC (Gödel 1940, Cohen 1963) |
| 2 | Consistency of arithmetic | Gödel's 2nd Incompleteness Theorem complicates original goal |
| 3 | Polyhedra decomposition | Solved (Dehn, 1900) |
| 7 | Transcendence (e.g. 2^√2) | Solved (Gelfond-Schneider, 1934) |
| 8 | Riemann Hypothesis | Unsolved |
| 10 | Diophantine equation algorithm | Proven impossible (Matiyasevich 1970) |
| 18 | Kepler Conjecture | Solved (Hales 1998, computer-assisted) |

## Hilbert's motivation
"Wir müssen wissen, wir werden wissen" ("We must know, we will know") — later inscribed on his tombstone. Reflected optimism later complicated by Gödel (1931).

## Modern successor
Clay Millennium Prize Problems (2000) — explicitly modelled on Hilbert's precedent. Only Poincaré Conjecture solved so far (2003).

## Sources
### Tier 1
- Hilbert, D. (1902). Mathematical problems. *Bull. AMS*, 8(10), 437-479.
- Gray, J.J. (2000). *The Hilbert Challenge*. Oxford University Press.
### Tier 2
- Yandell, B.H. (2002). *The Honors Class*. A K Peters.
### Tier 3
- Derbyshire, J. (2003). *Prime Obsession*.

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*Mathematics Codex entry v1.0 — added 2026-05-27 — thecodex.expert/mathematics/*