# Examples of theorems with proofs that have dramatically improved over time

I am looking for examples of theorems that may have originally had a clunky, or rather technical, or in some way non-illuminating proof, but that eventually came to have a proof that people consider to be particularly nice. In other words, I'm looking for examples of theorems for which have some early proof for which you'd say "ok that works but I'm sure this could be improved", and then some later proof for which you'd say "YES! That is exactly how you should do it!"

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Seems related to the earlier MO question mathoverflow.net/questions/43820/extremely-messy-proofs. –  Tom De Medts May 3 '12 at 10:13
It would be also interesting to hear of theorems where people didn't think that the proof could be much improved, but then were proven wrong. –  David Corwin Jul 9 '12 at 6:02
@LouisDeaett: Good question. There is something, though, still nicer about having a proof follow as a simple consequence of something complicated. It shows there is some larger (unifying?) idea. –  Manya Dec 3 '13 at 6:50

[Edit: This answer seems to fit the title of the question, though not the actual question in the body.]

Resolution of singularities in algebraic geometry seems like a good example. Hironaka's original proof was over 200 pages and hard to understand:

"Even A. Grothendieck [in Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 1, 7--9, Gauthier-Villars, Paris, 1971; MR0414283 (54 #2386)] admitted openly that he did not completely understand Hironaka's proof."

That quote is from Dan Abramovich's Math Review of the book Lectures on resolution of singularities by Kollár; the review goes on to say

"One can [nowadays] devote a few weeks in a first course on algebraic geometry to give just a complete proof of resolution of singularities in characteristic 0 (Chapter 3 of the present book, which is largely self-contained)."

I know almost nothing about this topic, but some names I know associated to the various approaches to simplification of Hironaka's proof are Bierstone, Milman, Encinas, Villamayor, Hauser, Cutkosky, Włodarczyk, Kollár, Cossart, Piltant... Please tell me any I missed!

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The alternating sign matrix conjecture was first proved by Zeilberger. Zeilberger's proof was extremely computational. A much shorter conceptual proof was later given by Kuperberg.

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Jordan's proof of the Jordan Curve Theorem was complicated enough that people still argue about its correctness. These days, an undergrad can prove it after learning the Mayer–Vietoris sequence.

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Do you know a good online source where the proof is well explained (not just outlined etc.)? I am equally interested in good textbooks discussing this approach. –  GH from MO May 3 '12 at 16:56
In typical US universities, undergrads do not learn Mayer-Vietoris, but your point is of course still correct. –  Henry Cohn May 3 '12 at 17:19
According to Tom Hales, there Jordan's proof should never have been controversial. An objection arose that he assumed the polygonal case without proof --- but that's a trivial omission! See mizar.org/trybulec65/4.pdf As for the idea that a student can prove it using Mayer-Vietoris, I disagree. Yes, a good undergrad can learn Mayer-Vietoris, but in order to use it here, you also need that the circle (or in generality, the sphere) is an ENR, which is a separate and clearly nontrivial result. Remember, the hard case of the Jordan theorem is the fractal case. –  Greg Kuperberg May 9 '12 at 19:19
There is a proof in my book "Topology and Groupoids", linking it with the Phragmen-Brouwer Property. Actually the proof is derived from one by Munkres in his book, but I think is improved by the use of groupoids. It was published in J. Homotopy and Related Structures 1 (2006) 175-183. (arXiv:math/0607229 ) –  Ronnie Brown Oct 18 '13 at 20:43

I think that Gelfand's proof of Wiener's $1/f$ theorem qualifies.

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Boone-Novikov theorem of existence of groups with undecidable word problem which originally has very long and complicated proof now has several (self-contained) proofs of length $\le 10$ pages (see Cohen, Daniel E. Combinatorial group theory: a topological approach. London Mathematical Society Student Texts, 14. Cambridge University Press, Cambridge, 1989. x+310 pp.).

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If you are prepared to allow an example from mathematical physics, then Penrose's proof that a ball moving relativistically appears as a circle to an observer. This had been proved previously by brute strength calculations with Lorentz transformations. Penrose reformulated it in terms of actions of the action of the Lorentz group on the celestial sphere. Since these are just conformal transformations, which take circles to circles, the boosted sphere appears circular.

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The prime number theorem, Newman's short proof is only three pages long.

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As the title of Zagier's paper makes clear, this proof is due to Donald Newman: "Newman's short proof of the prime number theorem," Amer. Math. Monthly 104 (1997), 705-708. –  Timothy Chow May 8 '12 at 20:50

The isosceles triangle theorem (pons asinorum), that the angles opposite the equal sides of an isosceles triangle are equal, was originally proved by Euclid by constructing several auxiliary lines. Pappus' proof uses no auxiliary lines, but only side-angle-side by "flipping" the triangle over to its mirror image.

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I described an example, Hindman's theorem, at http://mathoverflow.net/questions/94546 . The short version is that Hindman's original proof was unpleasantly complicated, whereas a later proof by Galvin and Glazer is now accepted as the standard proof. On the intuitive level, it's a definite improvement. Formally, though, from the viewpoint of reverse mathematics, Hindman's original proof is "better" because it uses far weaker set-existence assumptions.

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I think Furstenburg's Stone-Czech proof of van der Waerden's theorem would also fit the bill. –  Benjamin Steinberg May 3 '12 at 16:11

Kurosh's original proof of the subgroup theorem for free products used messy Kurosh systems. This was improved by covering space proofs (or equivalently covering groupoid proofs). One might argue the Bass-Serre theory proof is now the right one.

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Aigner and Ziegler's "Proofs from the BOOK" contains many good examples.

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My favorite from that book is Sperner's proof of Brouwer's fixed point theorem. –  Liviu Nicolaescu May 4 '12 at 12:18
With no further specification, this post is not an answer to the question asked. –  Did Nov 10 '13 at 16:58

A favorite of mine is the chirality of the trefoil knot, which can be proved easily using the Jones polynomial or some of its relatives. Louis Kauffman's paper "New invariants in the theory of knots", http://homepages.math.uic.edu/~kauffman/Bracket.pdf explains this nicely.

I don't know how it was proved before the Jones polynomial, but quoting from p. 204 of Kauffman's paper, "In the old days (before 1984) this was something that required a lot of mathematical background."

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Is it the case that using the Jones polynomial sort of hides away all that mathematical background, or does it somehow clarify the main idea of the proof (give one a sense of why it is true?) –  Manya May 4 '12 at 8:17
The first chirality proof, by Max Dehn in 1914, was indeed a lot more involved than the Jones polynomial proof. It involved finding the automorphisms of the trefoil knot group. –  John Stillwell May 8 '12 at 23:51

The Riesz-Thorin interpolation theorem is an example. As I understand it, the original proof published by Marcel Riesz was rather messy. Thorin found a much simpler proof of the theorem using complex analysis about ten years later.

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There are several examples from Tauberian theory. Around 1930, Karamata surprised people by giving much simpler proofs of Littlewood's original Tauberian theorems for power series. Wiener's Tauberian theorems were later given much slicker and arguably more conceptual proofs using operator theory.

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Example of a bounded linear operator on a Banach space without non-trivial closed invariant subspace.

The first example was given bei Enfo in 1975. Enflo submitted the full article in 1981 and the article's complexity and length delayed its publication to 1987 (see http://en.wikipedia.org/wiki/Per_Enflo). Simpler examples were constucted for example by Beauzamy and Charles Read.

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Worth mentioning that Read was subsequently the first to construct such an operator on $\ell^1$ –  Yemon Choi May 3 '12 at 19:00
Related: Aronszajn and Smith's theorem that a compact linear operator on a Banach space must have a nontrivial invariant subspace was later given a dramatically simpler proof by Lomonosov. –  Timothy Chow May 8 '12 at 20:36

PP (the class of languages decidable by a probabalistic Turing machine in polynomial time) is closed under union and intersection. This was conjectured by Gill in 1972 and stayed an open problem for 18 years, til resolved by Beigel, Reingold, and Spielman (BGS) in 1995, with a complicated proof involving rational functions. The same result fell out as an almost-corollary of Scott Aaronson defining quantum postselection for unrelated reasons: the new proof is less than a page. See:

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I disagree that the BRS proof is complicated. Given the rational function approximating sgn, the proof is just a paragraph. And the rational functions approximating sgn were mostly constructed already by Newman. In any case, BRS give their self-contained construction/proof in a couple dozen sentences. –  Ryan O'Donnell Jun 23 '13 at 23:01

Gauss's first proof of the Quadratic Reciprocity Law relied on an intricate induction argument and was not particularly illuminating. Later, Gauss's third proof (based on the Gauss lemma) and especially his sixth proof (based on quadratic Gauss sums) gave more insight. Perhaps, the proof with the biggest "wow effect" was given by Zolotareff using his lemma expressing the Legendre symbol as the sign of a permutation.

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Faltings' theorem (aka Mordell conjecture) can be taken as such an example. Different methods have been used so far with various difficulties.

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Widom's formula for calculating determinants of banded Toeplitz matrices. The original paper is hard to understand and uses quite intricate techniques.

Now, a quite simple proof can be found in Böttchers "Spectral Properties of Banded Toeplitz Matrices". Actually, it also follows quite directly from the formula on Hall-Littlewood polynomials here: http://en.wikipedia.org/wiki/Hall%E2%80%93Littlewood_polynomials

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Szemeredi's theorem and its special case roth's theorem have been given quite conceptual proofs by Hilel Furstenberg using ergodic methods which I think is quite natural while the initial proofs were extremely complicated.

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The original ergodic proof of Szemeredi's theorem is arguably not a perfect example for this list, since both Austin's and Polymath proofs of the density Hales-Jewett theorem are significantly simpler. –  pavel Dec 20 '13 at 22:31

Tverberg Theorem (1965): Let $x_1,x_2,\dots, x_m$ be points in $R^d$, $m \ge (r-1)(d+1)+1$. Then there is a partition $S_1,S_2,\dots, S_r$ of $\{1,2,\dots,m\}$ such that $\cap _{j=1}^rconv (x_i: i \in S_j) \ne \emptyset$.

Tverberg's theorem was conjectured by Birch who also proved the planar case. The case $r=2$ is a 1920 theorem of Radon which follows easily from linear algebra consideration.

(The first thing to note is that Tverberg's theorem is sharp. If you have only $(r-1)(d+1)$ points in $R^d$ in a "generic" position then for every partition into $r$ parts even the affine spans of the points in the parts will not have a point in common.)

The first proof of this theorem appeared in 1965. It was rather complicated and was based on the idea to first prove the theorem for points in some special position and then show that when you continuously change the location of the points the theorem remains true. A common dream was to find an extension of the proof of Radon's theorem, a proof which is based on the two types of numbers - positive and negative. Somehow we need three, four, or $r$ types of numbers. In 1981 Helge Tverberg found yet another proof of his theorem. This proof was inspired by Barany's proof of the colored Caratheodory theorem (mentioned below) and it was still rather complicated. It once took me 6-7 hours in class to present it.

What could be the probability of hearing two new simple proofs of Tverberg'stheorem on the same day? While visiting the Mittag-Leffler Institute in 1992, I met Helge one day around lunch and asked him if he has found a new proof. To my surprise, he told me about a new proof that he found with Sinisa Vrecica. This is a proof that can be presented in class in 2 hours! It appeared (look here) along with a far-reaching conjecture (still unproved). Later in the afternoon I met Karanbir Sarkaria and he told me about a proof he found to Tverberg's theorem which was absolutely startling. This is a proof you can present in a one hour lecture; it also somehow goes along with the dream of having $r$ "types" of numbers replacing the role of positive and negative real numbers. Another very simple proof of Tverberg's theorem was found by Jean-Pierre Roudneff in 1999.

For further details see these blog posts (I,II).

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Dear @Gil Kalai: I fixed the hyperlinks in this answer. Also, it should be noted that all the links other than the blog posts require an AMS subscription. –  Ricardo Andrade Nov 10 '13 at 10:44

I think that Ax's proof of the Chevalley-Warning Theorem qualifies.

The Chevalley-Warning Theorem is an affirmative solution of a conjecture made by L.E. Dickson in 1909 and taken up more seriously by Artin [I don't seem to have onhand much information about when Artin first got involved with this; if you do, please let me know] in the 1930's. The conjecture is that every finite field is a C1 field: namely, a homogeneous polynomial in more variables than its degree always has a nontrivial zero.

Chevalley's Theorem is stronger than that: it says that if you have polynomials $P_1,\ldots,P_r$ in $n$ variables with coefficients in a finite field, then if the sum of the degrees is less than $n$, it is not possible for there to be exactly one simultaneous zero. Warning sharpened this to showing that the set of simultaneous zeros is divisible by $p$, but in fact every proof I've seen of Chevalley's Theorem -- so in particular, Chevalley's proof! -- easily adapts to prove Warning's generalization.

(Warning's real contribution was a second theorem giving a stronger lower bound on the number of common zeros, assuming that there is at least one. But that is not the result I am talking about.)

Let me be honest: there is nothing clunky or technical about Chevalley's proof. It is completely elementary, has a clear moral, and takes a bit less than two pages. In an undergraduate course, it would fill one lecture nicely.

So how much room for improvement can there be? Well, Ax's proof literally takes ten lines. See for yourself. The big idea is that $\sum_{x \in \mathbb{F}_q} x^i$ is $0$ when $0 \leq i < q-1$. You could safely assign the proof of this as an exercise in any undergraduate course in which you cover the cyclicity of the unit group $\mathbb{F}_q^{\times}$.

Can anyone think of another serious conjecture made but unresolved by mathematical luminaries which turned out to have a ten line proof?!? I can't.

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It occurs to me that Morse theory is a good example. At the time of Morse, algebraic topology (even the notion of CW complex or cell complex) is barely developed, which made his combinatorial arguments extremely difficult to read.

Well, nowadays people can simply learn these topics by referring to the definite account of Milnor or Bott.

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Witten's proof of the positive energy theorem using spinors drastically simplified the original proof by Schoen and Yau.

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Kottman proved that in any infinite-dimensional Banach space one can find a sequence $(x_n)_{n=1}^\infty$ of unit vectors with

$$\|x_n-x_m\|>1$$ whenever $n\neq m$. The original proof is quite messy, but there is a yet another proof, attributed to Starbird, which can be found in Diestel's book Sequences and series in Banach spaces. It uses essentially linear algebra and the Hahn-Banach theorem only.

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an example from number theory (where such simplifications are not uncommon), Bertrands postulate

for any integer $n > 3$, there always exists at least one prime number $p$ with $n < p < 2n − 2$

was first simplified by Ramanujan and then later by Erdos who also proved a more general case.

another interesting case study here may be Lindemanns proof of transcendence of PI which is subsumed by later more general results. as Wikipedia states "Weierstrass proved the above more general statement in 1885. The theorem, along with the Gelfond–Schneider theorem, is extended by Baker's theorem, and all of these are further generalized by Schanuel's conjecture."

another "possible/controversial" famous/legendary case study here is Fermats Last Theorem; Fermat scribbled in the margin of his book that he had a remarkable proof, but modern consensus is that he must have been mistaken based on the 2020-hindsight of Wiles complex proof. however, strictly speaking, it has not been proven impossible that there exists a short proof.

it seems that later simplifications of proofs is a natural process of the historical/evolutionary progress of mathematics so that results once thought more arcane/inscrutable/complex become more accessable with the polishing/systematization of ideas/techniques.

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Emanuel Lasker's original proof of the Lasker-Noether Theorem was 98 pages long, but the modern proof fits into a few paragraphs and is standard undergraduate fare.

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I suggest Gödel's second incompleteness theorem. The first theorem states that every consistent, sufficiently strong, effectively presented formal system contains an undecidable formula. The second theorem states that such a formal system does not prove any theorem that implies its own consistency. Gödel never published a proof of the second theorem, after logicians accepted that it could be proved by encoding a proof of the first theorem within the formal system in question. However, actually to perform this encoding would be technically very difficult. Modern treatments derive the second theorem very easily after establishing the Hilbert–Bernays provability conditions.

Proofs of the first theorem have also been greatly simplified. Gödel's treatment required extensive technicalities to establish that certain existential quantifiers were bounded by a specific positive integer. These proofs can be replaced at the cost of modest other efforts.

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The global (or homology) version of Cauchy’s theorem was given an elementary proof by John Dixon. I believe this is mentioned in Rudin's Real and Complex Analysis. A proof is available online at http://www.math.uiuc.edu/~r-ash/CV/CV3.pdf. This states "The elementary proof to be presented below is due to John Dixon, and appeared in Proc. Amer. Math. Soc. 29 (1971), pp. 625-626, but the theorem as stated is originally due to E.Artin."

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The Krylov–Bogolyubov theorem states that a continuous map on a compact metric space admits an invariant measure. The original article is 50 pages long, but nowadays this is a one-liner. This is because all the measure theory involved has been neatly repackaged in functional analytic terms.

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