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In a math thesis, no matter it is in undergraduate or PhD, is it okay that the objective of a math thesis is to give a new proof of old theorem? Even though the new proof may be more complicated or lengthy than the original one. Is it valuable?

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    Sure! This happens all the time and it is extremely valuable, especially when it is done in a way that clarified what was originally essential. Forget theses, professional mathematicians write papers on this! – Andres Mejia Aug 16 '18 at 03:52
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    'll expand on my comment. Excellent work often involves reproving old results, in a way that clarifies their point. I'm no historian, but here are two examples:

    1. Grothendieck's cohomological Proof of Zariski's Main Theorem which was conjectured by Serre.

    2. Serre's proof of rieman-roch was considered very important

     – Andres Mejia Aug 16 '18 at 04:01
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    You can find these all over the place. I guess the catch is you need a Reason to publish a new proof. I believe Erdos +Selberg famously gave the first elementary proofs of the prime number theorem https://en.wikipedia.org/wiki/Prime_number_theorem – Andres Mejia Aug 16 '18 at 04:03
  • @AndresMejia Thanks ! –  Aug 16 '18 at 04:06
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    No problem. In the future, I suspect the downvote may have been because this belongs somewhere in between this website and academia.SE. I can't read minds but perhaps recasting the question in terms of math would be better: how much mathematical content is there in proving old theorems? Is it worthwhile to revisit proofs of old theorems? etc. etc. – Andres Mejia Aug 16 '18 at 04:07
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    A proof by a different method might be a way to find additional knowledge..... Euler's analytic proof that there are infinitely many primes is longer and more complicated than Euclid's. But it gives extra info: That the sum of the reciprocals of the primes is infinite. – DanielWainfleet Aug 16 '18 at 09:13
  • @AndresMejia. Erdos also published a new proof (when he was 18) of Bertrand's Postulate (first proved in 1857). – DanielWainfleet Aug 16 '18 at 16:03

1 Answers1

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Warning. My answer heavily relies on the comments to your question.

The critical point is to bring something new. If you give a new proof to an old theorem, this can happen either by improving the statement of the theorem or its proof. Let me consider these two cases separately.

Improving the statement of the theorem.

  1. You use a weaker hypothesis. For instance, you generalise a theorem on complete metric spaces to complete uniform spaces, or a result valid for fields of characteristic $0$ without any characteristic assumption.
  2. You prove a stronger result. For instance, you give a better upper [lower] bound. See for instance Timothy Gowers's upper bounds for Van der Waerden's numbers. See also DanielWainfleet's comment about Euler's analytic proof that there are infinitely many primes, a very good example.

Improving the proof of the theorem.

  1. You give a simpler proof with the same mathematical tools.

  2. You give an elementary proof of a theorem involving difficult mathematics, even at the price of a lengthy proof. See Andres Mejia's example on Erdos-Selberg famous elementary proof of the prime number theorem.

  3. You give a more sophisticated proof but it clarifies the argument (and often leads to a more general statement). See Andres Mejia's examples (Grothendieck's cohomological Proof of Zariski's main theorem and Serre's proof of Riemann-Roch theorem)

  4. You give a proof within a weaker logical system. Typically, you manage not to use the axiom of choice, or you prove that a result still holds in a weak axiomatic system (logicians are fond of such results).

EDIT. As I was posting this answer, Hans Stricker asked whether Fermat's last theorem is provable in Peano arithmetic?, a good example for (4).

J.-E. Pin
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