Hilbert's 24th problem

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Hilbert's 24th problem is a mathematical problem, the formulation of which was found in David Hilbert's estate and is considered to be a supplement to Hilbert's list of 23 mathematical problems . Hilbert asks the question of criteria or proofs for whether a proof is the simplest for a mathematical problem.

prehistory

David Hilbert gave a lecture on research topics at the 2nd International Congress of Mathematicians in Paris in August 1900 and published 23 problems in the autumn of the same year. In October 2000, the science historian Rüdiger Thiele from the University of Leipzig reported that he had found a 24th problem in Hilbert's estate.

text

According to Teun Koetsier (University of Amsterdam), the entry reads (German original text, partly in facsimile ):

“As the 24th problem in my Paris lecture, I wanted to ask the question: criteria for simplicity with respect to Provide proof of the greatest simplicity of certain proofs . Develop a theory of the methods of proof in mathematics in general. Given the given conditions, there can only be one simple proof. In general, if you have 2 proofs for a sentence, you do not have to rest until you have traced them back to each other or have recognized exactly which different assumptions (and aids) are used in the proofs: If you have 2 ways, so you don't just have to go these ways or look for new ones, but then explore the whole area lying between the two ways. Approaches to assess the simplicity of the evidence are offered by my investigations into syzygies and syzygies between syzygies. The use or knowledge of a syzygy makes it much easier to prove that a certain identity is correct. Since every process of adding is application of the commutative law of addition - this always corresponds to geometrical propositions or logical conclusions, these can be counted and z. B. when proving certain theorems in elementary geometry (Pythagoras or about strange points in the triangle) it is very easy to decide which is the simplest proof. "

Individual evidence

  1. ↑ News from the Royal Society of Science in Göttingen (1900)
  2. ^ Mathematical notebooks (three), Göttingen State and University Library, Manuscript Department
  3. ↑ Rüdiger Thiele: Hilbert's Twenty-Fourth Problem, American Mathematical Monthly, January 2003 (PDF; 201 kB)
  4. ↑ Teun Koetsier: Hilbert's 24th problem (PDF; 150 kB)
  5. ↑ Hilbert writes according to the facsimile in Koetsier's essay “largest”, “Ueberhaupt”, “Hülfsmittel”, “only” and “ueber”.
  6. ↑ Dominic Hughes: Towards Hilbert's 24th Problem: Combinatorial Proof Invariants, Stanford University, 2006 (PDF; 399 kB)