Hilbert's famous 24th problems handles the problem of the simplest possible proof of a mathematical statement, in a nutshell.

It is said that there are a few problems with this problem. First of all, which problems arise from this problem?

Besides that, I'm wondering, can't we just define the simplest proof of a statement as the least number of times axioms are used + the number of axioms used in the non axiom statements used in a proof?

Like in:

To prove statement A we need 3axiom 1 + 2statement B

And to proof statement B we need 1axiom 1 + 2axiom 2.

Hence we need 9 axioms to prove statement A? (Assuming this is the minimum. There may also be another proof using axiom 3, but this proof needs 12 axioms, this it's not the minimum).

It seems to be a very reasonable way of defining the simplicity of a proof, and I don't see any way how it could lead to contradictions or interpretations.

Nevertheless I'm always open for corrections, since I'm just an amateur, passionate about math.

  • 3
    $\begingroup$ Well, it's not "unsolvable" as much as it is "open-ended". The problems asks if it is possible to devise a coherent theory of "simplicity" and what that might look like if it existed. Trying to make that more precise certainly seems like an interesting and important project. this question provides some relevant discussion. $\endgroup$
    – lulu
    Apr 21, 2022 at 23:23
  • 2
    $\begingroup$ What I learnd from this place was to try to answer my questions first, then show where have not been able to progress any further and why then ask for insight here. If you have ideas try to see how far they can take you , then post them here for a very specific topic. Your question is too broad . It is more of a discussion and there is a discussion chat available for that. My advice is to make this question as specific as possible and show the work you have tried to answer it. $\endgroup$
    – jimjim
    Apr 21, 2022 at 23:24
  • $\begingroup$ The key issue is : What is "simple" ? A short proof need not be a simple proof. What we surely can find are all shortest proofs (if we know there is one). $\endgroup$
    – Peter
    Apr 29, 2022 at 7:32
  • $\begingroup$ This can be trivially shown to be uncomputable. Since you say you are an amateur but passionate about mathematics, simply pick up any proper introductory text on logic (such as those I recommended on my profile), and learn the basics first before looking at strange things like Hilbert's problems. $\endgroup$
    – user21820
    Apr 30, 2022 at 15:08


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