Is it okay that the objective of a math thesis is to give a new proof of old theorem?

Question

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?

Answer 1

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).