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Is it possible to provide a proof of some unsolved result using elementary methods? I get the feeling it would be looked down upon and/or not taken seriously. Why is this?

Is there no merit to proving some conjecture using elementary methods?

Mike Pierce
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user5826
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    Generally, if a hard problem can be solved using elementary methods, someone will have already solved it. – William Stagner Mar 18 '15 at 23:31
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    Any proof is welcomed, elementary or not. Mathematicians sometimes go out of their way to provide an elementary proof. – littleO Mar 18 '15 at 23:32
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    Is your question why you have a certain feeling? – Mirko Mar 18 '15 at 23:37
  • Ha. No. Generally, when an amateur has a proof, it's usually looked down on. – user5826 Mar 18 '15 at 23:40
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    There have been many cases where a result has been known and is proven with a much cleaner elementary approach later. Take Shelah's proof for bounds on van der Waerden Numbers. The community was ecstatic. – JMoravitz Mar 18 '15 at 23:41
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    Suppose this elementary proof was the first proof of the theorem. Would the community have been as open then? – user5826 Mar 18 '15 at 23:43
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    Why would you think not? Unless the author claims to be the first while its not the case, or unless the proof was incorrect, I see no reason why that would be the case. – JMoravitz Mar 18 '15 at 23:44

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This text is reproduced from the preface of Ramsey Theorey (2nd edition) by Ronald L. Graham et al

The romanticized view of mathematics is that it proceeds in sudden bursts of brilliant insight. Sometimes it happens just that way. Van der Waerden's theorem, the central result of Ramsey theory, was proven in 1926. As van der Waerden recalled:

After lunch we went into Artin's office in the Mathematics Department of the University of Hamburg, and tried to find a proof. We drew some diagrams on the blackboard. We had what the Germans call "Einf$\ddot{a}$l": sudden ideas that flash into one's mind. Several times such new ideas gave the discussion a new turn, and one of the ideas finally led to the solution.

<p>[van der Waerden 1971]</p>

Van der Waerden's proof used a subtle double induction and when expressed quantitatively led to an extremely fast growing function. Mathematicians --we three included-- searched for a different proof technique without these features. In 1987 Saharon Shelah was shown van der Waerden's theorem and within a day or two found a new proof. Whether Einf$\ddot{a}$lle or not, Shelah's proof avoids the double induction, involves only "reasonably" fast growing functions, and --best of all-- is totally elementary.

My combinatorics professor this semester was at a mathematics conference at the time when someone suddenly burst through the doors shouting "You all must see this!" holding aloft a copy of the newly published proof. The conference took a halt while they excitedly examined and praised the newly found proof.

There can be places for respect and admiration for elementary proofs of both new and old results, assuming that it sufficiently adds to our current understanding. If it is proving a result which already has an elementary proof, or if the proof is incorrect in some way, the community might not greet it so warmly, however I see no reason why that should be thought of as the norm.

JMoravitz
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