Hilbert presented 23 problems at the International Congress of Mathematicians in 1900 to help guide and inspire future mathematical research. Some key problems included Cantor's continuum hypothesis as the first problem and developing criteria for determining the simplest proof for a theorem as the proposed but unpublished 24th problem. Over a century later, most problems now have solutions but some remain open or too vague, showing both the progress and ongoing challenges of mathematics. Hilbert's problems helped shape mathematics and provide benchmarks to measure advances in the field over time.

1. Hilbert’s 23 Problems
Math 350
Allison Simmons

2. Goals:
 Give a little background on Hilbert.
 Explain the creation and purpose of
his 23 problems
 Investigate a couple of the problems
 Check in on the problems’ current
status.

3. David Hilbert (1862-1943)

4. The Problems:
 Hilbert presented 10 of the
problems in 1900 at the
“International Congress of
Mathematicians” in Paris. This
talk was entitled “The Problems of
Mathematics”.
 The complete list of 23 problems
appeared in a later talk, which
was translated into English in
1902.

5. An Obvious Question:
 Why did Hilbert chose to focus on
unsolved problems rather than
the new methods or results of
the time?

6. His Intention:
“If we would obtain an idea of the probable
development of mathematical knowledge in
the immediate future, we must let the
unsettled questions pass before our minds
and look over the problems which the
science of today sets and whose solution
we expect from the future. To such a
review of problems the present day, lying at
the meeting of the centuries, seems to me
well adapted. For the close of a great epoch
not only invites us to look back into the past
but also directs our thoughts to the unknown
future.” – Hilbert (1900, The Problems of
Mathematics speech).

7. His Intention:
Hilbert’s problems were designed to be
 Simple
 Easily understood
 Inspiring
 A way of gauging mathematical
progress
 The first step in simplifying
mathematics as a system of axioms

8. The 24th Problem:
In 2000, it was discovered that
Hilbert had originally planned for
a 24th problem.
 This problem was not in his
lecture notes or any published
texts.
 His close friends and
proofreaders were also not aware
of this problem.

9. The 24th Problem:
“The 24th problem in my Paris
lecture was to be: Criteria of
simplicity, or proof of the
greatest simplicity of certain
proofs. Develop a theory of the
method of proof in mathematics
in general. Under a given set of
conditions there can be but one
simplest proof.”
-from Hilbert’s notes

10. Questions On The 24th:
 How should the 24th problem be
approached in relation to the
complete collection of 23?
 Why did Hilbert remove the 24th
problem? Did this affect his
research somehow?
These questions are still being
debated.

11. Critique Of The 24th:
 What defines “simplest”?
 In order to compare proofs, we
must be able to “measure” them.
How would this be done?
 Are all proofs finite? Consider the
role of computers in proofs.

12. The First Problem:
 Simple Interpretation:
“There is no set whose cardinality is
strictly between that of the integers and
that of the real numbers.”
 It may be useful to recall that the set
of all integers is countably infinite
whereas the real numbers are
uncountably infinite.

13. The First Problem:
 Or, put differently:
“Prove Cantor's problem of the cardinal
number of the continuum.”
 This is also known as the Cantor’s
Continuum hypothesis: “…there is no
infinite set with a cardinal number
between that of the "small" infinite set
of integers and the "large" infinite set of
real numbers (the "continuum").”
http://mathworld.wolfram.com/ContinuumHypothesis.html
http://mathworld.wolfram.com/CardinalNumber.html

14. Status The First Problem:
 There is no consensus as to if the
problem has been “solved”.
 “Instead of working out a solution of the
continuum problem, we falsified one of the
premises upon which Hilbert based it, proving
and verifying the proposition that there exist no
infinities beyond the infinity of the natural
numbers.”
http://bado-shanai.net/Platonic%20Dream/pdHilbertsFirst.htm
Significant contributors:
 1940: Kurt Gödel
 1963: Paul Cohen

15. First Problem’s Importance:
 Hilbert was very aware of the
mathematical community during his
lifetime. The first problem reflects this.
Hilbert was a friend of the problem’s
formulator, Cantor.
Cantor was discredited during his life for his
radical views on infinity and his later mental
illness.
At the time of the problem’s proposal
(1900), the tools to “solve” it did not exist.

16. The Other Problems:
• The handout attempts to group
Hilbert’s problems based on their
current status.
• Some information on notable
contributions is also given.
 Note: I accidentally put problem 16 as
both “open” and “too vague”. …..Oops.

17. The Other Problems:
9 problems have a generally
accepted solution
8 problems have a controversial
solution
3 problems are open
3 problems are too vauge. (4 if
the canceled 24th is included…)

18. Final Thoughts:
Hilbert’s problems
1. continue to fascinate
mathematicians from all over
the world.
2. allow us to easily track
mathematical progress over
time.
3. reveal the difficulty in achieving
consensus on a proof or result.

19. End.
Thanks for listening.