Theorems with many distinct proofs

Question

I was told that whenever one learns a new technique, it is a good idea to see if one can prove a well-known theorem using the new technique as an exercise. I am hoping to build a list of such theorems to test my technique.

My Question. What are some theorems with a.) many different proofs, or b.) proofs using strikingly different techniques from mathematics?

Theorems with multiple proofs:

• Fundamental theorem of algebra
• Fundamental theorem of arithmetic
• Existence of Jordan normal form. (Standard minimal polynomial proof, PID proof, Terry Tao's proof, proof using complex analysis)
• (Related to prev.) Cayley-Hamilton theorem
• Spectral theorem
• Quadratic reciprocity law
• Pólya's recurrence theorem
• Basel problem
• Stirling's formula

(Arguably:)

• Uniform boundedness principle. (Pf 1. Baire category, Pf 2. Gliding hump)
• Brouwer fixed point theorem (although all the proofs I know of boil down to showing that the ball is not homeomorphic to the sphere)

I think the Pythagorean theorem also satisfies my description, but it is a bit too elementary.

Clarification 1: In order to prevent this question from being "what are some theorems that have higher level generalizations," a trivial specialization of a harder theorem (e.g. Hilbert space Pythagorean theorem implying Euclidean Pythagorean theorem) will not be considered a new proof, for the purposes of this question.

Clarification 2: This question differs from this similar stackexchange question in the following sense. The stackexchange post asked for a.) very elementary theorems (lower level undergrad), and b.) short proofs. My question asks for theorems from all levels of mathematics (up to say third year graduate level), which are (preferably) central to the theory.

Answer 1

Terry Tao has called Szemerédi's theorem the "Rosetta stone" of mathematics, and for a good reason: this result has gathered many different proofs using tools from many different areas of mathematics. Wikipedia has a brief overview, but for completeness here is a list (feel free to edit to add ones I missed):

• Szemerédi's original, purely combinatorial proof (based on his regularity lemma and prior proof of the 4AP case)
• Furstenberg's proof using ergodic theory,
• Gowers's proof using Fourier-analytic methods (greatly refining Roth's method for 3APs)
• Another proof by Gowers, using hypergraph regularity results,
• Polymath1 project has resulted in an elementary proof of density Hales-Jewett, of which Szemerédi's theorem is an easy consequence.

They also have countless variants, e.g. this simplified proof of the density Hales-Jewett by Dodos, Kanellopoulos, Tyros.

Answer 2

"what is worth proving is worth proving again" (Attributed to N. Katz in D. Ruelle's paper, The nature of properly human mathematics.)

You are asking for a very long list: most deep and important theorems have several proofs. I can give three examples, two of them classical (original proofs of the late 19th century), another one very recent.

The first example is Picard's theorem:

If an entire function omits 2 values then it is constant.

Some of its proofs, based on very different ideas are: The original proof of Picard; soon he gave another proof. The proof of Emile Borel, based on growth estimates and Wronskians. The proof of Wiman-Valiron using power series. Nevanlinna's proof based on the lemma on the logarithmic derivative. The differential-geometric proof by Raphael Robinson, using Ahlfors' comparison Lemma. Proofs based on isometimetric inequalities (specializations of Ahlfors' theory). Probabilistic proof by Burgess Davis. Larry Zalcman's elementary proof. My own proof using potential theory. Derivation of Picard's theorem from Harnack inequality by John Lewis,... And many others. Once I started writing a survey of different proofs, but abandoned this project: the survey would be too long and touch too many diverse areas. Most new proofs lead to new generalizations.

The second classical example is the Uniformization theorem, one of the deepest results of 19 century analysis. There is a book by H. P. Saint-Gervais with a survey of many proofs, not including the very recent one based on the Ricci flow. (BTW, Picard's theorem, is a simple corollary of the Uniformization theorem, which gives yet another proof of Picard's theorem).

My third example is more recent, it is a former B. and M. Shapiro conjecture.

Theorem. If $f$ is a rational function whose all critical points belong to a circle (on the Riemann sphere) then $f$ maps this circle into some circle.

This was proved for the first time by Gabrielov and myself. Then we found another proof. Next proof obtained by Mukhin, Tarasov and Varchenko used completely different ideas, coming from mathematical physics (they actually found two different proofs, leading to different generalizations). And later there were two other completely different proofs: one by Levinson and Purbhoo, and the most recent by E. Peltola and Yilin Wang, based on considerations from a different area of mathematical physics. Thus since 2004 we have at least 6 different proofs. (Several incorrect proofs have also been circulated).

Answer 3

The Riemann hypothesis for curves over finite fields has a number of distinct proofs.

Weil gave two proofs, one based on the Jacobian and one based on intersection theory on the product $C \times C$.

Deligne's proof of the Weil conjectures also implies the theorem, and is independent of Weil's proofs. This proof itself has many variants, though Katz eventually found a very simple version of the proof, restricted to the case of curves, which is almost certainly the "best" for this problem.

Stepanov and Bombieri gave a proof involving only Riemann-Roch.

Answer 4

The Nielsen-Schreier theorem from combinatorial group theory ought to qualify, as it is one of the most fundamental theorems of this area. It was originally proved in a restricted form by J. Nielsen in 1921, who showed that finitely generated subgroups of free groups are free. O. Schreier extended this to all subgroups a few years later in 1926. These proofs are entirely combinatorial. Baer & Levi published the first topological proof of the theorem, based on regarding a free group as the fundamental group of a bouquet of circles, in 1936.

Many years later, the theorem would be realised in geometric group theory via the key insight that a group is free if and only if it acts freely on a tree; the theorem then becomes obvious. J.-P. Serre gave a short proof using the action of groups on trees (i.e. Bass-Serre theory) in 1970, see also his 1977 Arbres, amalgames, $\operatorname{SL}_2$). It's important to note that this geometric idea had been around for quite some time: even M. Dehn (see Chandler & Magnus History of Combinatorial Group Theory) regarded the theorem as obvious in view of the fact that connected subgraphs of trees are trees. There are also many miscellaneous proofs scattered about; see e.g. this proof by B. Steinberg.

On the other hand, although free groups can be defined using universal properties, it seems like there is no direct category-theoretical argument for why the theorem should be true; indeed, the analogous theorem does not hold for e.g. subsemigroups of free semigroups.

Answer 5

Faltings's theorem, originally known as Mordell's conjecture, is a bit of a crown jewel of arithmetic geometry. It has been proven and reproven many times along with many generalizations. Here is a (surely incomplete) list of its proofs:

• Faltings's original proof, a tour de force which in one go has proven Mordell's conjecture, Shafarevich's finiteness conjecture and Tate's isogeny conjecture, using various deep tools in algebraic geometry, for instance Arakelov theory,
• Vojta's proof using ideas from Diophantine approximation theory,
• Bombieri's simplication of Vojta's method (which I believe is distinct enough to deserve its own spot),
• Lawrence-Venkatesh's recent proof using methods from $p$-adic Hodge theory.
• Hrushovski has proven Mordell-Lang conjecture, a much more general statement of which Mordell's conjecture is a consequence, using methods from model theory.
• Masser-Wüstholz give a proof of a finiteness result on isogenies of abelian varieties using some methods related to Baker's methods in transcendental number theory. Mordell's conjecture follows as in Faltings's proof.

Answer 6

The Brouwer fixed point theorem can be proven by quite different tools; this wiki article lists 9 proofs.

Answer 7

Here's a theorem of practical relevance:

A compactly supported smooth function in the plane is uniquely determined by its line integrals. That is, the X-ray transform (or the Radon transform) is injective.

This is what underlies computerized tomography, as measurements only give the line integrals of the unknown function (the attenuation coefficient, related to density). The 3D reconstruction is typically done slice by slice.

There are a number of different proofs, and I have collected many of them in my lecture notes:

• Radon's original proof relates certain circle averages of the data to circle averages of the unknown function, leading to an explicit inversion formula in terms of a Stieltjes-integral.

• Cormack's independent proof (from decades later) is based on writing the function as a Fourier series in polar coordinates, leading to a sequence of decoupled one-dimensional integral equations which can be solved explicitly.

• One can rephrase the problem in terms of the geodesic flow on the sphere bundle of a suitable domain containing the support of the function. The theorem turns out to be equivalent with unique solvability of a certain PDE on this bundle. The PDE is not of any standard type (elliptic, hyperbolic, parabolic), but there is an energy identity that saves the day. This proof works on many manifolds, too.

• Using the so-called Fourier slice theorem is perhaps the easiest approach, reducing the problem to injectivity of the Fourier transform in a space of one dimension less.

• The X-ray transform $I$ is not self-adjoint, so one is tempted to study its normal operator $I^*I$ instead. This operator turns out to be a Riesz potential, the injectivity of which is a result from the realm of potential theory.

• The somewhat ill-behaved operator on the plane can be transformed into a collection of nice self-adjoint operators on the torus. Then the theorem follows from how these operators intertwine with the Fourier series.

• …and there are variations on these ideas and completely new ones that have been designed to handle non-Euclidean geometry or partial data.

These proofs are all within analysis and geometry, but scattered pretty wide nevertheless. While there are some relations and shared tools between these approaches, the proofs are quite different. And this difference makes a practical difference, as it leads to different insights into the practical problem of various kinds of X-ray tomography.

Answer 8

The Atiyah-Singer index theorem has many different proofs: by cobordism theory and K-theory, by the heat equation method, and proofs coming from mathematical physics. I guess that there are further distinct proofs.

I am aware of two diffrent proofs of the theorem that Riemannian manifolds of constant curvature are locally isometric to spherical, euclidean or hyperbolic space: the first proof uses Jacobi fields, while the second uses Frobenius theorem.

Answer 9

The hook length formula for counting the number of standard Young tableaux of a given shape, has many different proofs.

Knuth have a nice probabilistic proof, and one can also derive it in the limit from R. Stanley's more general hook-content formula. There are also purely algebraic proofs, involving determinants.

Giving an alternative proof of the hook formula (or some generalization) is (or should be) on the bucket list of every fan of algebraic combinatorics (along with finding some new interpretation of the Catalan numbers).

Answer 10

There is an old MO question about Bott periodicity which lists 9 fairly different proofs, and it missed a few (heat equation, extensions of $C^*$-algebras, Toeplitz operators, coarse geometry...)

Answer 11

I'll give one of the most fundamentals results in maths as an example: the reals are uncountable. Similarly, see: Asking for Various proofs of uncountability of $[0,1]$

The first proof of Cantor used "Cantor's Lemma" about nested intervals.

His second proof uses his definition of perfect set, a set that is equal to its set of accumulation points, then he proved that no countable set is perfect

Cantor's third proof was the famous diagonalization method

Matthew proved that in 2007 using game theory (see Uncountable sets and an infinite real number game)

Another proof is a proof using measure theory, I think this is a proof by Harnack, but I don't have a source (if someone has please comment)

In 1969, B. R. Wenner. Classroom Notes: The Uncountability of the Reals proved that using Cauchy sequences

There is a new proof from 2009 by Levy: An unusual proof that the reals are uncountable, unfortunately I don't know the details of this proof.

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An interesting side story of the theorem, is Cantor's theorem about power sets.

Let's look at the following 3 theorems:

There is no surjective from $X$ to $\mathcal{P}(X)$

The classic Cantor theorem, secondly

There is no injection $\mathcal{P}(X)$ to $X$

And lastly

There is no bijection between $\mathcal{P}(X)$ and $X$

While all of those 3 looks almost the same, each of them carry a different reverse mathematics strength, so, under ZFC, there are at least 3 fundamentally different proof of "$|\mathcal{P}(X)|>|X|$"

Answer 12

The Grothendieck-Hirzebruch-Riemann-Roch Theorem has many proofs.

1. The proof due to Hirzebruch using Pontrjagin classes.

2. The "purely algebro-geometric" proof due to Grothendieck written up by Borel and Serre.

3. The proof using the Index theorem due to Atiyah-Singer and later Patodi.

4. The proof via Adams-Riemann-Roch using cyclic group actions due to Nori.

There are at least 3 proofs of the Weil Conjectures.

1. The first paper by Deligne "La conjecture de Weil I" gives a proof using the Lemma of Kazhdan-Margulis.

2. The second paper by Deligne "La conjecture de Weil II" gives a proof by Deligne's interpretation of the method of Hadamard and de la Vallée-Poussin.

3. There is also a proof due to Laumon which reduces the question to one about Weil sheaves on $\mathbb{A}^1$.

Answer 13

The prime number theorem has a bunch of different proofs. I would count most of the complex analysis proofs as being very similar in so far as they rely on there being no zeros of the zeta-function on the line ${\rm Re}(s)=1$, but there are a bunch of other proofs.

The elementary proofs of Selberg and Erdős are very similar.

Amitsur's proof, which uses the structure of arithmetic functions can be thought of as a distinct proof, although one can see it in some of the ideas in both the Erdős-Selberg type proofs and the analytic proofs.

Answer 14

Stan Wagon, Fourteen proofs of a result about tiling a rectangle, The American Mathematical Monthly, vol. 94, 1987, pp. 601-617. Summary at https://maa.org/programs/maa-awards/writing-awards/fourteen-proofs-of-a-result-about-tiling-a-rectangle

This paper recounts several proofs of the theorem, "Whenever a rectangle is tiled by rectangles each of which has at least one integer side, then the tiled rectangle also must have at least one integer side." The proof techniques range from double integrals to graph theory. Generalizations of the theorem are discussed, and the proof techniques are scrutinized for applicability to the more general problems.

There is a link to the full article at that site.

Answer 15

The spectral theorem is mentioned. There are two proofs I'm aware of:

• Via the fact that every matrix has an eigenvalue. It remains then to show that the eigenvalues are real, and that eigenvectors of distinct eigenvalues are orthogonal.

• A proof using Lagrange multipliers and the Extreme Value Theorem.

These proofs are fundamentally different. The first assumes something equivalent to the Fundamental Theorem of Algebra, which can fail in some toposes (like ironically over the sheaf topos over $\mathbb C$). The second one can be weakened to show that there is a maximum and minimum eigenvalue in a way that is valid in every topos. I think the full eigenspectrum can be recovered using something like the min-max theorem.

The failure of the FToA in toposes can be total. The statement that every complex number has a square root can be false. The claim that every monic cubic polynomial with real coefficients and no non-real roots has a real root can fail. The latter fact shows that merely arguing that the characteristic polynomial has no non-real roots is not enough to prove the existence of a real eigenvalue.

Vague question: Does this tell us anything deep about the Hermitian matrices?

Answer 16

Proofs that Jones polynomials are invariants of links:

1. The original one via von Neumann algebras by Jones.

2. An elementary proof by Kauffman.

3. Via Temperley-Leib algebras by Kauffman.

4. Quantum group representations by Reshetikhin & Turaev.

5. Several approaches in the HOMFLY paper.

6. Via Chern-Simons theory by Witten.

7. Khovanov homology.

Answer 17

Benson Farb said (in various talks, ages ago) that another proof of Helly's theorem is published every ten years. As I recall, he did not make clear how often a new proof is published. Anyhow, the OP inspired me to do a bit of digging. Here are a few bits and pieces.

1. Wikipedia says that the theorem dates to 1913 but the first three proofs in 1921, 1922, and 1923 are due to Radon, König, and Helly, respectively.

2. The MathSciNet review of Rabin's 1955 paper "A note on Helly's theorem" calls it

The much-proved theorem of Helly...

3. In 1963 Danzer, Grünbaum, and Klee published "Helly's theorem and its relatives"; the authors give an overview up to that time.

4. Farb's 2009 paper "Group actions and Helly’s theorem" is already alluded to above. In Section 3, Farb discusses the "topological Helly theorem" (proved by Debrunner in 1970).

Answer 18

Two different proofs of Torelli's Theorem: one due to Andreotti which can be found in Griffiths and Harris, and a second proof due to Martens.

Answer 19

The Weierstrass Approximation Theorem has many distinct proofs. One uses convolutions and approximate identities, one uses Taylor polynomials and the associated remainder estimate together with some ad hoc arguments, and one (or two?) use Bernstein polynomials.

If I recall correctly, a proof using approximate identities appears in Rudin's Principles of Mathematical Analysis, and a proof using Taylor polynomials appears in Abbott's Understanding Analysis. I'm sure there are many others; perhaps a list of all known proofs is available.

Answer 20

There are several different ways to prove Poincaré duality: given a closed, oriented, $n$-dimensional manifold $M$, there is an isomorphism $H^k(M)\overset\cong\to H_{n-k}(M)$. Here are a few that I know of:

• Poincaré's original proof giving a cell decomposition of the manifold, then taking the dual cell decomposition.
• A proof using singular homology (e.g. in Hatcher)
• A proof using de Rham theory/integration of differential forms (e.g. in Bott-Tu)
• The Morse-theoretic proof: given a Morse function $f\colon M\to\mathbb R$, one can obtain the dual chain complex via the Morse theory of $-f$.
• A proof obtaining the theorem from the existence of a Thom class given an oriented manifold.
• Using the Hodge star associated to a Riemannian metric on $M$

There are probably more proofs, too (e.g. coaxing it out of Serre duality?).

It's certainly not true that every technique in manifold topology can be used to prove Poincaré duality; the reason it has many different proofs is because (co)homology and orientations each can be defined in many equivalent ways.

Answer 21

The local monodromy theorem is a fundamental result on the topology of families of complex algebraic varieties. Let $X \rightarrow B$ be a smooth projective family of complex algebraic varieties over a smooth complex algebraic curve $B$, complement of a finite set $S$ of points in a compact curve $\overline{B}$. Then the local monodromy theorem states that for every point $s \in S$, the monodromy action around $s$ on the cohomology (with rational coefficients) of the fibers of $X \rightarrow B$ is quasi-unipotent (i.e. all its eigenvalues are roots of unity).

There are many proofs of this theorem, using very different ideas (a summary is given in section 3 of https://www.ams.org/journals/bull/1970-76-02/S0002-9904-1970-12444-2/S0002-9904-1970-12444-2.pdf ):

1. there is a "global geometric" proof due to Landman, using Lefschetz pencils and induction on the dimension.

2. there is a "local geometric" proof due to Grothendieck in the algebraic setting and Clemens in the Kähler setting, based on a local study of vanishing cycles.

3. there is a "complex hyperbolic geometric" proof due to Borel using the differential geometric fact that moduli spaces of polarized Hodge structures are negatively curved.

4. there is a proof by Brieskorn using the regularity of the Gauss-Manin connection and the Gelfond-Schneider theorem on transcendental numbers.

5. there is an "arithmetic" proof due to Grothendieck based on étale cohomology, reduction to positive characteristic, and general properties of l-adic Galois representations.

6. there is another "arithmetic" proof due to Katz, using the study of the Gauss-Manin connection in positive characteristic.

Answer 22

In 1940 Gödel proved the consistency of the continuum hypothesis with the Zermelo-Fraenkel axioms of set theory, by introducing the constructible universe $L$ and subsequently founding the subfield of set theory now known as inner model theory. In 1963, Paul Cohen introduced the method of forcing, which he used to show the continuum hypothesis is independent from the Zermelo-Fraenkel axioms, resolving Hilbert's first problem. However, forcing can also be used to give an alternative proof of the consistency of the continuum hypothesis, giving two proofs.

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The computable functions provably total in Peano arithmetic are exactly the $\varepsilon_0$-recursive functions. According to Buss's "The Witness Function Method and Provably Recursive Functions of Peano Arithmetic", this has been proven at least using Gentzen-style assignment of ordinals to proofs, Gödel's Dialectica interpretation, and model-theoretic methods.

Answer 23

Baker-Campbell-Hausdorff-Dynkin's formula has several proofs from distincts branches of mathematics: see for example the book Topics in Noncommutative Algebra from A. Bonfiglioli and A. Fulci

Answer 24

I think that the Amitsur-Levitzky Theorem is a good example as well.

Answer 25

Quadratic reciprocity had $6$ proofs by Gauß. Plus two were found posthumously. He proved it first in his Disquisitiones Arithmeticae. He called it "the gem of arithmetic". Euler and Legendre had conjectured it.

Now there are over $240$ known proofs.

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That $\Bbb Z_p^×$ is cyclic. Keith Conrad has amassed, at least, $10$ or $11$ different proofs of this fact

It was proven by Gauß in his $1801$ Disquisitiones Arithmeticae, an existence and a constructive proof. First Lambert, then Euler, then Gauß were playing with the notion of a primitive.

There's a constructive proof in Vinogradov's Elements of Number Theory.

Answer 26

There are at least 2 proofs of the existence of nonprincipal ultrafilters on $\mathbb{N}$. Terence Tao gives 2 in this informative article. The first is an application of Zorn's Lemma and the other uses the Stone–Čech compactification of $\mathbb{N}$. This might fall under the "arguably" category in your question.