# Proofs of theorems that proved more or deeper results than what was first supposed or stated as the corresponding theorem

Recently, I figured out that a colleague of mine has had published during recent years a proof of a theorem in which he was actually proving a deeper result which we both thought to be still open. After a closer look at his proof I found that, taking a bit more care and putting some additional emphasis in certain parts of his previous proof, he was actually proving the other still-thought-to-be-open problem: the construction was absolutely the same and therefore the proof of the previously published theorem was certainly a better argument than we first thought. I am curious now about this phenomenon happening more often. Do you know some other recent (let's say from 1700 to the current day) examples of this phenomenon of proofs being stronger than initially stated or proving more than thought at first?

• I would venture to say that nearly every proof ever written does this to some degree, though in some cases the strengthening may be tiny. Mar 6, 2021 at 17:09
• My advisor, John Rhodes, used to say that proof was a closure operator on the set of true statements. He said that after observing a proof in my thesis showed something stronger than I claimed I proved. Mar 6, 2021 at 17:26
• @Benjamin Steinberg: That is truly beautiful. Mar 6, 2021 at 19:48
• One can make a good case that Maxwell equations contained Special relativity which was not noticed by the community until Einstein:-) Mar 6, 2021 at 20:50
• I don't know details, and I don't know if the account was ever published before, but in this Numberphile video (around 10:36, see earlier for more context) Ken Ribet talks about how he has initially proven a special case of the "epsilon conjecture", but Barry Mazur pointed out to him his argument essentially goes through generally. I would be curious if anyone knows more details. Mar 6, 2021 at 21:15

The example given by Wojowu in the comments seems worth posting as an answer.

In the NOVA special The Proof, Ken Ribet says the following.

I saw Barry Mazur on the campus, and I said, "Let's go for a cup of coffee." And we sat down for cappuccinos at this cafe, and I looked at Barry and I said, "You know, I'm trying to generalize what I've done so that we can prove the full strength of Serre's epsilon conjecture." And Barry looked at me and said, "But you've done it already. All you have to do is add on some extra $$\Gamma_0(M)$$ structure and run through your argument, and it still works, and that gives everything you need." And this had never occurred to me, as simple as it sounds. I looked at Barry, I looked at my cappuccino, I looked back at Barry, and I said, "My God. You're absolutely right."

• I was not aware of the NOVA special, I was not aware of it. I have added the link to the Numberphile video I was talking about in the comment above. I would be still interested if someone would happen to have a more technical account of the story. Mar 7, 2021 at 18:36

In theoretical computer science, an extractor is an algorithm that takes a weak source of randomness (i.e. a distribution that may be far from the uniform distribution) and produces a much stronger source of randomness (i.e. a distribution that is close to uniform). A pseudorandom generator is an algorithm that takes a very small amount of "pure" randomness (i.e. a small number of bits sampled from the uniform distribution) and produces a much larger amount of "pseudorandomness" (i.e. a much longer sequence of bits that no polynomial time algorithm can distinguish from bits sampled from the uniform distribution).

In 1999, Luca Trevisan showed that all pseudorandom generators of a certain sort can actually be seen as extractors. This was a surprising result since extractors are based on an information-theoretic definition of "randomness" while pseudorandom generators use a computational definition. Also the two concepts had different origins and had been studied using somewhat different techniques and applied in different ways. Trevisan's result was a major breakthrough not only for showing that two seemingly different concepts were actually (more-or-less) the same, but also because it showed that existing techniques for constructing pseudorandom generators gave extractors that were much better than those that had been constructed previously.

A few years ago, I learned that Trevisan originally thought he'd proved something much weaker until Oded Goldreich pointed out to him the full consequences of what he'd done. As he wrote on his blog:

Three years later, while I was a postdoc at MIT and Oded was there on sabbatical, he played a key role in the series of events that led me to prove that one can get extractors from pseudorandom generators, and it was him who explained to me that this was, in fact, what I had proved. (Initially, I thought my argument was just proving a much less consequential result.) For the most part, it was this result that got me a good job and that is paying my mortgage.

Henri Poincaré provides an example in mathematical physics, as discussed by Thibault Damour and Howard Stein.

Poincaré said in June 1905:

The essential point established by Lorentz is that the electromagnetic field equations are not altered by a certain transformation (which I shall call after the name of Lorentz), which has the following form: \begin{align} x'&=kl(x+\epsilon t)\\ y'&=ly\\ z'&=lz\\ t'&=kl(t+\epsilon x) \end{align} where $$x,y,z$$ are the coordinates and $$t$$ the time before the transformation, and $$x',y',z'$$ and $$t'$$ after the transformation. Moreover, $$\epsilon$$ is a constant which defines the transformation $$k=1/\sqrt {1-\epsilon ^2}$$ and $$l$$ is an arbitrary function of $$\epsilon$$.

One can see that in this transformation the $$x$$-axis plays a particular role, but one can obviously construct a transformation in which this role would be played by any straight line through the origin. The sum of all these transformations, together with the set of all rotations of space, must form a group.

In a longer version of this paper from July 1905, Poincaré added that this does not change the quadratic form written in different units as $$x^2+y^2+z^2-t^2$$ and that we can regard $$x,y,z,t\sqrt{-1}$$ as the coordinates in a 4-dimensional space, with the Lorentz transformation as a rotation of that space around the origin. Poincaré regarded this as only completing Lorentz's work "in a few points of detail"; this led to Albert Einstein saying that "for all his acuteness, Poincaré showed little understanding of the situation."

When Einstein saw the same mathematical properties, also in June 1905, he created the theory of special relativity.

• I think this post is misleading. It's widely known and accepted that the theory of special relativity had several creators, and it's false to say that Einstein created the theory of special relativity in 1905. Obviously, the general theory of relativity is a different thing. Mar 8, 2021 at 22:14
• @HollisWilliams The distinction is usually that most physicists looked at Maxwell's equations, saw their invariants, and thought "their is a preferred frame, the frame of the aether, in which Maxwell's equations are formulated and valid". Einstein is usually credited as simply taking Maxwell's equations at face value, in particular its conclusion that the speed of light is constant in all reference frames, period. So Einstein usually gets the credit for his interpretation of what Lorentz invariance really was (it was an abstract tool to most before then), and abandoning the aether concept. Mar 9, 2021 at 1:58
• I think it is unlikely that the history of the theory was that simple and I don't think that Einstein abandoned the aether concept for definite in 1905 (I might be wrong). Theoretical physics does not proceed in such a uniform way: the history which you see in the textbooks is very much sanitised and cleaned up. Mar 9, 2021 at 4:51
• @HollisWilliams, I agree textbook histories are overly clean, but the sources I cited are pretty detailed about the messiness and based in the primary documents. Mar 9, 2021 at 5:58
• I'm pretty sure that Einstein was the first to realize that Lorentz symmetry should be treated as a symmetry of all physical laws, and that Galilean symmetry is nothing more than a low-speed approximation to it. Lorentz and Poincaré saw it as a symmetry of electromagnetism only; that's the crucial difference. On the other hand, I think Einstein didn't understand in 1905 that Lorentz transformations are rotations, since he resisted that interpretation when Minkowski suggested it in 1907, so he didn't see "the same mathematical properties" as Poincaré in 1905. Nov 4, 2021 at 20:15

In his 1955 paper "Invariant of finite groups generated by reflections" Chevalley gave a uniform proof the the Chevalley-Shephard-Todd theorem which says that for a finite group $$G$$ acting on a complex vector space $$V$$, the following are equivalent:

(i) The algebra $$S(V)^G$$ of invariant polynomial functions on $$V$$ is a polynomial ring;

(ii) $$G$$ is generated by pseudoreflections.

Actually Chevalley only proved (ii) $$\Rightarrow$$ (i); the other implication already had a uniform proof in Shephard and Todd's original work, and anyways is not hard once you know (ii) $$\Rightarrow$$ (i).

However, Chevalley only stated his result for reflections (i.e., pseudoreflections of order 2) because he was mostly only interested in Weyl groups, and in fact as far as I can tell he was not aware of the work of Shephard and Todd.

Serre observed, in his 1967 paper "Groupes finis d’automorphismes d’anneaux locaux réguliers", that Chevalley's proof goes through verbatim in the case of pseudoreflections, because he only ever uses the fact that the reflections fix a hyperplane, and not that they have order 2.

(In that paper Serre also studies pseudoreflection groups over fields of positive characertistic, where (i) and (ii) are no longer necessarily equivalent.)

For more discussion of the history of the Chevalley-Shephard-Todd theorem, see this previous MO question: Chevalley–Shephard–Todd theorem.

I hesitate to offer an example from two millennia earlier than you have requested, but perhaps it may qualify as surely having passed unnoticed by many thousands of students. Euclid's proof of the Pythagorean theorem (his 1.47) shows not only that the big square is the sum of the smaller squares, but also how it is divided into parts equal to those squares.

• Several millennia earlier or not, good classical example! The time limit is more an indication to be more or less contemporary than an exclusion from classical facts. Mar 6, 2021 at 17:08
• Looking at the proof (here) it seems to me like this proof quite explicitly talks about this decomposition and the two parts of the square which are equal in area to the smaller squares. I'm not sure what kind of student would read the proof and miss what is so explicitly written there. OP seems to be content with this answer although I don't see how it fits their criteria. Mar 6, 2021 at 21:29
• @Wojowu Yeah, you are right. My approval of this answer was more allowing some flexibility if good exceptional classical examples are found. But you are definitely right that the mentioned decomposition is not hidden anywhere but quite explicit. Mar 7, 2021 at 1:04
• I remember I did this in school with wooden material showing how the squares cut and fitted into the larger square. This is infact called Perigal cutting. cut-the-knot.org/pythagoras/PerigalII.shtml Mar 7, 2021 at 4:23
• At the Museum in Mathematics in Manhattan, they have (or at least had, before the COVID-19 pandemic) a physical puzzle where your task is to take certain tiles (made of hard plastic) and (1) tile the larger square with them, and (2) tile the two smaller squares with them. In fact I think they have two versions of the puzzle, with different sets of tiles, but unfortunately I can't remember the details. Mar 8, 2021 at 18:12

Here is an example of this happening in 2019.

We had nearly given up on getting the last piece and solving the riddle. We thought we had a minor result, one that was interesting, but in no way solved the problem. We guessed that there would be another five years of work, at best, before we would be able to solve the puzzle

While reading our research article, we suddenly realized that the solution was before our eyes. Our next reaction was 'oh no – we’ve shot ourselves in the foot and given away the solution

• That article doesn't really say anything about what the result actually was, other than involving planar graphs somehow. The actual paper is at arxiv.org/abs/1911.03449 (linked in the article, but I went and found it myself before noticing that). Maybe someone familiar with the problem could add some context. Mar 8, 2021 at 6:05
• As far as I can tell, here's what this is about. Suppose you have a graph which is being gradually updated: at any point in time edges may be added or removed from the graph (but the vertex set stays fixed). You want to know at each point in time whether the current graph is planar. This problem is apparently called "fully-dynamic planarity testing." Before 2019, the best known algorithm took O(sqrt(n)) time per edge insertion/deletion on average (where n is the number of vertices in the graph). (continued in next comment) Mar 9, 2021 at 19:11
• (continued from previous comment) But in 2019, Holm and Rotenberg gave an algorithm that takes $$O(\log^3(n))$$ time per insertion/deletion on average. So an exponential speed up over the best previously known algorithm. Apparently much of the idea for this result was already present in a paper that they had written earlier in 2019. I think this Quanta article has a better explanation of this result than the article linked in the answer above. Mar 9, 2021 at 19:14
• @lambda The paper you linked to appears to be the paper where they prove there is a $O(\log^3(n))$ amortized time algorithm for fully-dynamic planarity testing. The paper linked to in the other answer that talks about this appears to be the paper from earlier in 2019 which contained the key ideas that led to the later result. Mar 9, 2021 at 19:16

Euler apparently discovered the formula for what we now call Fourier series, and thereby could have initiated Fourier analysis, without recognizing its significance. I learned about this from Strichartz' book "The Way of Analysis", where he describes that Euler had an axe to grind due to a quarrel with Daniel Bernoulli over the best way to solve the wave equation, and he attacked Bernoulli's claim that you could decompose an arbitrary function as a sum of sines. I can't tell the story better than Strichartz:

Bernoulli was not able to answer Euler well, except to repeat his lame argument that the equation $$f(x) = \sum_{k=1}^\infty a_k \sin \frac{k\pi}{L} x$$ was like an algebraic system of an infinite number of linear equations in an infinite number of unknowns. One of the most significant weaknesses in his argument was that he could not produce a formula for the coefficients $$a_k$$ in terms of $$f$$. This formula was actually first discovered by Euler several years later in the course of an unrelated investigation (Euler had cosines instead of sines), but as Euler was predisposed to reject Bernoulli's claim he never pointed out the possible relevance. Thus, did the two of them botch the opportunity of developing Fourier series a full half century before Fourier.

Strichartz goes on to describe how history might have played out differently, and then concludes with this gem:

If there had been as many mathematicians in those days as there are today, no doubt some graduate student looking for a thesis topic would have observed this, which might have been enough to push Euler and Bernoulli onto the right track.

In the paper P. Erdős and A. Hajnal, On the structure of set-mappings, Acta Math. Acad. Sci. Hungar. 9 (1958), 111-131, the authors came close to solving Ulam's measure problem by proving that the first uncountable inaccessible cardinal is not measurable, as they pointed out in their subsequent paper P. Erdős and A. Hajnal, Some remarks concerning our paper "On the structure of set-mappings" — non-existence of a two-valued $$\sigma$$-measure for the first uncountable inaccessible cardinal, Acta Math. Hungar. 13 (1962), 223-226:

In accordance with the notations of [4] we say that a cardinal $$m$$ possesses property $$P_3$$ if every two-valued measure $$\mu(X)$$ defined on all subsets of a set $$S$$ of power $$m$$ vanishes identically, provided $$\mu(\{x\})=0$$ for every $$x\in S$$ and $$\mu(X)$$ is $$m$$-additive.

It was well known that $$\aleph_0$$ fails to possess property $$P_3$$ and that every cardinal $$m\lt t_1$$ possesses property $$P_3$$ where $$t_1$$ denotes the first uncountable inaccessible cardinal.

Recently A. Tarski has proved, using a result of P. Hanf, that a certain wide class of strongly inaccessible cardinals possesses property $$P_3$$ (called strongly incompact cardinals). H. J. Keisler gave a purely set-theoretical proof of this result. After having seen these papers we observed that the special case of this result that $$t_1$$ possesses property $$P_3$$ follows almost trivially from some of our theorems proved in [1]. We are going to give this simple proof in §2. Our method for the proof is of purely combinatorial character, and although it is certainly weaker than that of A. Tarski and H. J. Keisler, we think that it is of interest to formulate how far one can go with these methods at present.