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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.

user107952
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    Often different proofs highlight different techniques and shed light on several approaches worth understanding or applying in other venues. – Ted Shifrin May 05 '21 at 21:24
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    Another aspect is that it is helpful to find the least amount of assumptions needed to prove a statement. This often requires finding a different way to go about the proof. If assumptions can be relaxed, then the statement applies to more things, which is super cool. – morrowmh May 05 '21 at 21:25
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    It is because "new" proofs often use "new" techniques, which can simplify the theorem significantly. Breakthroughs in certain branches of mathematics can transform a hard problem, which was open for decades into a homework question for students over night. Also can be very long, and in a lecture you do not have time for that. That also means to develop the necessary theory to do so. So other proofs can avoid this too. There are many other reasons why different proofs are interesting. – Cornman May 05 '21 at 21:25
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    Well for starters, there is strength in numbers. The more ways you can prove something the less likely it is that there’s a flaw in all of them. Also some proofs are better than others for demonstrations – Stephen Donovan May 05 '21 at 21:25
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    Also there are often easy theorems, which have an inductive proof. For example the fact that $\sum_{k=1}^n k=\frac{n(n+1)}{2}$. But then there is the proof of Gauss, which is much more insightful. So you also learn more, or understand a problem better, when using different approaches. Especially when you only have an inductive proof, as they tent to come "out of nowhere", and not really motivate the problem. – Cornman May 05 '21 at 21:28
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    @cornman agreed, and similarly for proofs via contradiction. Constructive methods are usually more desirable. – morrowmh May 05 '21 at 21:29
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    Also, the fact that a theorem is true is often way less interesting than the reasons why it is true. The more different proofs of a theorem we have, the better our understanding of the theorem. Also, there are enlightening proofs and less enlightening proofs (proofs by guessing and then confirming the guess, for instance). It's always good to be on the lookout for the former. – Vercassivelaunos May 05 '21 at 21:33
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    Maybe an example, I'm not an expert, but as far as I know, in the field of differential geometry, the reprove of Gauss-Bonnet theorem by Chern reveals the concept of characteristics class in geometry category. There are also many other examples in the history which shows the reprove work is valuable. – GK1202 May 05 '21 at 21:38
  • Also sometimes theorems are first proven using 'external' methods. Some theorems in combinatoric geometry can be easily proven using methods of toric topology and homological algebra and combinatoric people spent a lot of effort to try to make the proof use as little machinery outside of combinatorics as possible. – Sergey Guminov May 05 '21 at 22:07
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    The only purpose of a mathematical proof is as a certificate of correctness, is that right? One wonders why journals and textbooks publish proofs. We could save a lot of space in the libraries if they would just publish statements by referees or editors attesting that they have examined the proofs and found them to be correct. – bof May 05 '21 at 23:04
  • As Bill Thurston has put it in this post: "The product of mathematics is clarity and understanding. Not theorems, by themselves.". Given that point of view, it should be clear that a theorem is not the end goal. And having more proofs often equals more understanding. – M. Winter May 11 '21 at 22:40
  • I think there is a large component of trying to make the area more accessible for later generations. I remember thinking that The Gauss' Theorema Egregium (https://en.wikipedia.org/wiki/Theorema_Egregium) didn't seem to have a demonstration complicated enough to be worth the title. But this is because the quantities we use in the modern proof were not even defined back them. – Kernel May 12 '21 at 07:42
  • Another more recent example: Following Martin Hairer's "A theory of Regularity Structures", which got him the Fields medal, the area of stochastic partial differential equations got a lot more attention. With it, other mathematicians that have read his work made different connections between the pieces. In order to teach themselves and their students they rewrote the proofs via either simplified versions or with completely different approaches. Almost ten years later from the original paper, it is much easier to guide yourself through his theory thanks to the many researchers who did this. – Kernel May 12 '21 at 07:47

1 Answers1

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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):

\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.

Markus Scheuer
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