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.
An Obvious Question: Why did Hilbert chose to focus on unsolved problems rather than the new methods or results of the time?
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).
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
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.
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
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.
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.
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.
The First Problem: Or, put differently: “Prove Cantors 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.htmlhttp://mathworld.wolfram.com/CardinalNumber.html
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.htmSignificant contributors: 1940: Kurt Gödel 1963: Paul Cohen
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.
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.
The Other Problems:9 problems have a generally accepted solution8 problems have a controversial solution3 problems are open3 problems are too vauge. (4 if the canceled 24th is included…)
Final Thoughts:Hilbert’s problems1. 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.