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Stillwell mentions in his book Mathematics and its History that:

Most of the really old unsolved problems in mathematics, in fact, are simple questions about the natural numbers...

What is it about Euclidean geometry that makes it have so few classical unsolved problems, relative to number theory?

* Inspired by this question, Why are so many of the oldest unsolved problems in mathematics about number theory? and motivated by this comment.

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Geometry is decidable, number theory is not. –  Asaf Karagila Feb 3 at 3:01
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@AsafKaragila Could you explain what you mean by "decidable" in a way that could make sense to me? –  user89 Feb 3 at 3:02
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@Jason: Yes, and it is in fact not directly relevant. But I think that decidable theories are inherently better understood than undecidable theories. Because it means exactly that a computer could easily (even if it would take a very long time) identify all the theorems from the non-theorems. twirlobite: The modern axiomatization of geometry is such that a computer software could take as an input a statement in the language of geometry and tell us if it is a theorem or not. The algorithm is very very slow, but it exists nonetheless. –  Asaf Karagila Feb 3 at 3:18
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You picked a time to ask that when Euclidean geometry was out of long-standing open problems. The classical construction problems and the problem of the fifth postulate lasted about two thousand years. Do you think the Riemann hypothesis or P = NP will hold up that well? –  bof Feb 3 at 3:50
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@RBarryYoung: I somehow feel that your two last comments, with a bit of fleshing out and a few more references, would be an awesome answer to this question. –  Willie Wong Feb 4 at 15:25
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1 Answer 1

up vote 3 down vote accepted

Because Euclidean geometry is currently not fashionable, most people do not study topics in it or discuss problems in it, and so you simply hear of fewer problems, solved or unsolved.

Any claims that all Euclidean geometry problems are decidable, as given in the comments to the question, will depend on some restricted definition regarding the form that a "problem" can take.

There are plenty of unsolved geometry problems. I recommend the book Unsolved Problems in Geometry by Croft, Falconer, & Guy (1991). In addition to hundreds of problems, the book even points to 17 other collections of unsolved geometry problems.

Many of these unsolved problems are fairly easy to state. For example: Among all configurations of $n$ points not all on one line, what is the minimum number of lines they might determine? Explainable to a small child, yet unsolved.

There are many, many beautiful unsolved geometry problems, including ones with Erdös rewards, and I cannot do them justice in this answer.

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Are you sure that you fully understand what the term "decidable" means? It's a technical term, rather than the natural meaning of the word which one can understand as "can be solved". –  Asaf Karagila Feb 26 at 20:44
    
@AsafKaragila Yes, I am sure, I have published papers on the topic. –  Matt Feb 26 at 20:53
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That's generally hard to tell from just "Matt". So excuse me for not knowing that. :-) –  Asaf Karagila Feb 26 at 21:01
    
No problem! :-) –  Matt Feb 26 at 21:02
    
@AsafKaragila To add a bit of content, you could imagine someone describing a Turing machine (or a proof formalism) in geometrical terms (like how Gödel numbering lets you describe such things in terms of arithmetic), so that a "geometry" problem is equivalent to the halting problem, or to a statement that the statement has no proof, etc. Of course this geometry problem will not be of the limited form which has been classified as decidable. –  Matt Feb 26 at 21:17
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