Hilbert’s Problems: Part 1

In 1900, during a lecture in front of the International Congress of Mathematics in Paris, David Hilbert, a German mathematician, described 10 problems from a list of 23, all of which were unsolved at the time. The full list of the 23 problems was published in a paper in the Proceedings of the conference. Although these problems range in topics, they were all designed to serve as examples of the kinds of problems whose solutions would lead to the furthering of mathematics, and several of them were highly influential to 20th Century mathematics.

The problems mentioned are merely samples of problems; yet they are sufficient to show how rich, how manifold and how extensive mathematical science is today, and the question is urged upon us whether mathematics is doomed to the fate of those other sciences that have split up into separate branches. … I do not believe this nor wish it.”

In the first part of this series I will describe problems 1 – 11.

1st

Description: The continuum hypothesis – there is no set whose cardinality (the number of elements of the set) is strictly between that of the integers and that of the real numbers.

Status: The answer to this problem was proven to be independent of Zermelo-Fraenkel set theory (including the axiom of choice) by Paul Cohen in 1963 (building on work by Kurt Gödel in 1940). This means that either the continuum hypothesis or its negation can be added as an axiom to ZFC set theory, with the resulting theory being consistent if and only if ZFC is consistent. However, there is no real consensus on whether this is a solution to the problem.

2nd

Description: Prove the consistency of the axioms in arithmetic.

Status: Gödel’s second incompleteness theory, proved in 1931, showed that no proof of its consistency can be carried out within arithmetic itself. In 1936, Gentzen showed that a consistency proof can be obtained in a system that is much weaker than set theory. It has not been decided whether either of these is an adequate solution to the problem.

3rd

Description: Prove the equality of two volumes of two tetrahedra of equal bases and equal altitudes.

Status: This problem is resolved in 1902. The result turned out to be no, proved using Dehn invariants.

4th

Description: This problem was to find geometries whose axioms are closest to those of Euclidean geometry if the ordering and incidence axioms are retained, the congruence axioms weakened, and the parallel postulate discarded.

Status: Although there have been some solutions to the problem, in particular one proposed by Rouben V. Ambartzumian in 1976, Hilbert’s original statement of the problem is too vague to say whether or not it has been resolved.

5th

Description: Are continuous groups automatically differential groups?

Status: This problem can be seen to be resolved by Andrew Gleason in 1954, however if it is seen to be equivalent to the Hilbert-Smith Conjecture, then it remains unsolved.

6th

Description: Hilbert’s 6th problem is concerned with the axiomatization of the branches of physics in which mathematics is widely used.

Status: Currently there are two foundational theories in physics: the Standard Model of particle physics and general relativity. Many parts of these have been put on an axiomatic bases, however, physics as a whole has not – the Standard Model is not consistent with general relativity, showing the need for quantum gravity. This means that this problem is still open.

7th

Description: This problem concerns the irrationality and transcendence of certain numbers. In particular, two questions were asked:

  1. In an isosceles triangle, if the ratio of the base angle to the angle at the vertex is algebraic but not rational, is the ratio between base and side always transcendental?
  2. If a \not\in \{0,1\} is algebraic, and b irrational, is a^b always transcendental?

Status: Hilbert’s 7th problem was shown to be true in 1934 by Aleksandr Gelfond, and refined by Theodor Schneider in 1935. The result is now known as the Gelfond-Schneider theorem.

8th

Description: Prime number problems including Golbach’s conjecture, the twin-prime conjecture and the Riemann hypothesis.

Status: These problems all remain unsolved. The Riemann Hypothesis, which I discussed in a previous post, is now a Millennium Prize Problem. I personally think that the Riemann hypothesis is one of the most fascinating, unresolved problems in mathematics today!

9th

Description: This problem asks to find the most general law of the reciprocity theorem in any number field.

Status: It has been partially resolved by Emil Artin, who established the Artin reciprocity law. Combining this with the work of Teiji Takagi and Helmut Hasse, class field theory was established, which was an abstract answer to the problem.

10th

Description: Does there exist a universal algorithm for solving Diophantine equations? A Diophantine equation is an equation in which only integer solutions are allowed.

Status: This problem has been resolved. The result was obtained combining the work of Martin Davis, Hilary Putnam, Julia Robinson and was completed by Yuri Matiyasevich in 1970. It showed that no such algorithm exists.

11th

Description: Solving quadratic forms with algebraic numerical coefficients. That is, creating a mode of classification so we can tell is one algebraic form is equivalent to another.

Status: In 1923, Helmut Hasse accomplished this in a proof using his local-global principle: “the idea that one can find an integer solution to an equation by using the Chinese remainder theorem to piece together solutions modulo powers of each different prime number”.

Sources: 1 | 2 | 3 | 4

Hope you enjoyed this post! Part 2 will be coming on Wednesday. M x

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7 comments

    1. Hi, thank you for the link! I’ll definitely watch it sometime this week.
      No, it’s not my first language but I’ve studied at British schools all my life.
      Yes, I definitely want to integrate myself more into this community so I’ve been starting to read and comment on more blog posts. 🙂

      Liked by 1 person

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