What is the mathematical value of reproving theorems?

Many times in journals, people prove one theorem several times. I am puzzled as to why this is the case. Once it is proved, we already know it is true. Can someone explain why mathematicians reprove theorems? I apologize if this question is ill-suited for math stack exchange.

Solution 1:

A proof provides you often with more than with just an answer to the question:

• Is statement $A$ true?

A proof of $A$ contains information about why $A$ is true and is therefore some kind of characterisation of the mathematical object or mathematical relationship we want to study.

• One nice aspect is: The more characterizations (different proofs) we have the better is our understanding of this statement $A$.

• Another aspect is: The object or the relationship we want to study is not isolated in the mathematical world. We might want to find related statements. Do we find in different characterizations of the statement $A$ information about related objects? Do we find useful information to generalize statement $A$ or find some interesting specialisation of $A$.

• One more aspect is: We might have different demands for a satisfying proof. Some might want to look for a simplified proof, or for a proof avoiding specific techniques, or for a proof which is more beautiful.

Each approach might help us to better understand the statement $A$ and its mathematical essence.

Here are three examples of condensed versions of characterisations (proofs):

• The Harmonic Series Diverges Again and Again by S.J. Kifowit and T.A. Stamps presenting some rather elementary proofs together with the follow-up paper More Proofs of Divergence of the Harmonic Series.

• Thirty-two Goldbach Variations by J.M. Borwein and D.M. Bradley presents $32$ different proofs of the Euler sum identity

\begin{align*} \zeta(2,1)=\zeta(3)=8\zeta(\overline{2},1) \end{align*}

• Catalan Addendum by R.P. Stanley provides us with a wealth of different representations of Catalan Numbers extending his $66$ combinatorial representations from the second volume of his classic Enumerative Combinatorics.

which are nice opportunities to get an impression about the wealth of information about mathematical objects thanks to many different approaches.

Addendum 2021-05-23: Today when reading the AMS review by Ben Green of Additive Combinatorics by T. Tao and V. H. Vu I've found a fine reasoning for more than one proof. Read and enjoy (bold-face emphasis mine):

• Another major theme in the subject was initiated by Klaus Roth in his 1953 paper On certain sets of integers, the title being somewhat of a masterpiece of understatement. In this paper Roth addressed a question of Erdős and Turán, proving that every large subset $A \subseteq \{1,\ldots,N\}$ contains three distinct integers in arithmetic progression.

  He showed that a suitable notion of large in this context is that $|A|\geq cN/\log\log N$; the important feature of this bound is that the denominator tends to infinity with $N$, so that one may assert in a certain sense that sets of positive density contain three term progressions.

• It is natural to ask what happens for progressions of length $k\geq 4$. This issue was not resolved until the landmark work of Szemerédi, who proved in 1969 that sets of positive density contain $4$-term progressions and then generalized this to $k$-term progressions in 1975. His proof of the latter assertion, now known as Szemerédi’s theorem, is legendarily difficult, but aside from its intrinsic importance the paper led to one of the most important ideas in graph theory, the Szemerédi regularity lemma.

• Remarkably there have been several subsequent proofs of Szemerédi’s theorem, and it would scarcely be an exaggeration to say that each of them has opened up an entirely new field of study.

  • In 1977 Furstenberg proved the result by an ergodic theoretic approach.

  • In 1998 Gowers obtained the first sensible bounds, similar in strength to Roth’s bound mentioned above, using a kind of higher Fourier analysis. Intruigingly, this used Freĭman’s theorem as an essential tool.

    • Around 2003 Nagle, Rödl, Skokan, and Schacht and independently Gowers gave a fourth proof by developing a hypergraph regularity lemma.

• Tao has remarked that the many proofs of Szemerédi’s theorem act as a kind of Rosetta Stone. There is much to be gained by studying the relations between the different arguments, and indeed in proving that the primes contain arbitrarily long arithmetic progressions, Tao and the reviewer studied aspects of all four of the proofs mentioned above.