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Example 1. Euler's proof that there are infinitely many primes.




Example 3. Variational methods.


Example 4. Erdős's lower bound for Ramsey numbers.











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64 Answers 64

Use of Lagrange theorem (group theory) to prove Fermat's small theorem?

Use of fixed point methods (and completeness) to prove existence of solutions to differential equations?

Use of Fields theory to prove the impossibility of the trisection of the angle?

Use of group theory to prove insolvability of 5th degree equation?

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Fermat´s small theorem is in its nature group-theoretic. I don´t see anything surprising in proving it via Lagrange. Application of fixed point methods to differential equations is applying analysis to analysis. It only asks one to realize that differential and integral operators are maps... :-) But I think the 3rd and 4th point of your answer definitely suit to the original question. –  efq Dec 9 '10 at 19:00

How about Bolzano's 1817 proof of the intermediate value theorem?

In English here: Russ, S. B. "A Translation of Bolzano's Paper on the Intermediate Value Theorem." Hist. Math. 7, 156-185, 1980.

Or in the original here: Bernard Bolzano (1817). Purely analytic proof of the theorem that between any two values which give results of opposite sign, there lies at least one real root of the equation. In Abhandlungen der koniglichen bohmischen GeseUschaft der Wissenschaften Vol. V, pp.225-48.

Not fully rigorous, according to today's standards, but perhaps his method of proof could be considered a breakthrough nonetheless.

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More specifically, Bolzano was first to recognize that a completeness property of the real numbers was needed, and he proposed the principle that any bounded set of real numbers has a least upper bound. –  John Stillwell Aug 29 '11 at 16:51

I am always impressed by proofs that reach outside the obvious tool-kit. For example, the proof that the dimensions of the irreducible representations of a finite group divide the order of the group relies on the fact that the character values are algebraic integers. In particular, given a finite group $|G|$ and an irreducible character $\chi$ of dimension $n,$ $$\frac{1}{n} \sum_{s \in G} \chi(s^{-1})\chi(s) = \frac{|G|}{n}.$$ However, since $\frac{|G|}{n}$ is an algebraic integer (it is the image of an algebra homomorphism) lying in $\mathbb{Q},$ it in fact lies in $\mathbb{Z}.$

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Novikov's proof of the topological invariance of rational Pontryangin classes, for which he was awarded the 1970 Fields Medal. Fundamentally new (complicating a fundamental group to simplify geometry), and also fundamentally important. Here is what Sir Michael Atiyah had to say (as cited in the introduction to Raniski's Higher Dimensional Knot Theory):

Undoubtedly the most important single result of Novikov, and one which combines in a remarkable degree both algebraic and geometric methods, is his famous proof of the topological invariance of (rational) Pontryagin classes of a differentiable manifold... As is well-known many topological problems are very much easier if one is dealing with simply-connected spaces. Topologists are very happy when they can get rid of the fundamental group and its algebraic complications. Not so Novikov! Although the theorem above involves only simply-connected spaces, a key step in its proof consists in preversely introducing a fundamental group, rather in the way that (on a much more elementary level) puncturing the plane makes it non-simply-connected. This bold move has the effect of simplifying the geometry at the expense of complicating the algebra, but the complication is just manageable and the trick works beautifully. It is a real master stroke and completely unprecedented.
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Some more proofs that startled me (in a random order):

Liouville theorem to prove that Weierstrass P-function satifies the differential equation you know.

Complex methods to establish the addition law on an elliptic curve.

Cauchy's formula (for P'/P) to prove that C is algebraically closed.

Pigeon hole principle to prove existence of solutions to Fermat-Pell's equation

Kronecker's solution to the same equation, using L-functions.

Minkowski's lemma (a convex compact, symmetric, of volume 2^n contains a non trivial integer point) and its use to prove Dirichlet's theorem on the structure of units in number fields.

Fourier transform to prove (versions of) the central limit theorem.

Multiplicativity of Ramanujan's tau function via Hecke operators.

Poisson formula and its use (for example, for the functional equation of Riemann's zeta function, or for computing the volume of SL_n(R)/SL_n(Z), or values of zeta at even positive integers).

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How about Rabinowitsch's proof of the Nullstellensatz?

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I was about to go post about the 3-5 switch in the proof of Fermat's Last Theorem when I read this answer. It made me realize both are tricks rather than "proofs which require a fundamentally new way of thinking." Indeed, I think a computer could easily come up with the idea of introducing a new variable to simplify what needs to be proven (Rabinowitsch) or of switching tactics to deal with one case separately (Wiles). I'd go so far as to say computers are much better than humans at this kind of equational reasoning. –  David White May 19 '11 at 20:01

I work in automated theorem proving. I certainly agree, in principle, that there are no proofs that are inherently beyond the ability of a computer to solve, but I also think that there are fundamental methodological problems in addressing the problem as posed.

The problem is to come up with a solution that would not be regarded as 'cheating', i.e., somehow building the solution into the automated prover to start with. New proof methods can be captured by what are called 'tactics', i.e., programs that guide a prover through a proof. Clearly, it would not be satisfactory to analyse the original proof, extract a tactic from it (even a generic one) that captures the novel proof structure and then demonstrate that the enhanced prover could now 'discover' the novel proof. Rather, we want the prover to invent the new tactic itself, perhaps by some analysis of the conjecture to be proved, and then apply it. So we need an automated prover that learns. But anticipating what kind of tactic we want to be learnt may well influence the design of the learning mechanism. We've now kicked the 'cheating' problem up a level.

Methodologically, what we want is a large class of problems of this form. Some we can use for development of the learning mechanism, and some we can use to test it. Success on a previously unseen test set would demonstrate the generality of the learning mechanism and, hence, the absence of cheating. Unfortunately, these challenges are usually posed as 'can a prover prove this theorem' rather than 'can it solve this wide range of theorems, each requiring a different form of novelty. Clearly, this latter form of the question is hugely challenging and we're unlikely to see if solved in the foreseeable future.

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The Lebesgue integral seems to have been a fundamentally new way of thinking about the integral. It's hard to prove the convergence theorems if you have the Riemann integral in mind. I suppose there are probably many instances where you can give a computer a very ineffective definition of something and ask that it prove theorems. Ask it to prove anything about the primes where you start with the converse of Wilson's theorem as the definition of a prime. Can the computer figure out that its definition is terrible? Can it figure out what a prime really "is"?

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Turing's solution of Hilbert's Entscheidungsproblem. The new idea was to invent the Turing machine and "virtualization" (the universal Turing machine).

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Would the "quantum method" fit the bill here ?

"Quantum Proofs for Classical Theorems" Andrew Drucker, Ronald de Wolf

"Erdös and the Quantum Method" Richard Lipton

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Lovasz's proof of cancellation in certain classes of finite structures still bewilders me; I can only imagine that he found the proof first and then came up with the theorem afterwards. The basic idea is to look at homomorphisms between a given structure and a sequence of other structures. A comparison of two such sequences involving structures of the form AxC and BxC can be taken to a comparison between A and B. The condition that there exists a one-element substructure is used to show a certain nontriviality of the comparison, and a few more details result in showing A is isomorphic to B if(f) AxC is isomorphic to BxC.

I should have asked Lovasz how he came up with the proof; I am confident that most people would not be able to come close to the method independently if they were only given the theorem statement. (Not to mention the analogous statement of unique nth roots in the same class.)

Gerhard "Ask Me About System Design" Paseman, 2010.12.09

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Hochster and Huneke's tight closure theory to prove various theorems in Commutative algebra (Cohen-Macaulayness of rings of invariants, existence of big Cohen Macaulay algebras)?

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Proving that subgroups of free groups are free requires the knowledge of topology, a completely different field which a priori does not have anything to do with groups.

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Though the topological proofs are beautiful and slick, you don't need topology to prove this fact (and the original proofs were completely algebraic; see Lyndon and Schupp's book on combinatorial group theory for nice accounts of Nielsen's original proof as well as the later Reidemeister-Schreier approach). –  Andy Putman Dec 18 '10 at 6:54

The Ax-Kochen theorem about zeros of forms over the $p$-adics which was proved using model theory.

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There are two ways to prove the compactness theorem for propositional logic - either using the completeness theorem and going from semantic entailment to syntactic proof, or by a topological argument in Stone spaces. The latter, I feel, is an unexpected way of doing it - but I don't know the history of the subject so I'm probably not qualified to comment whether it was fundamentally new or not. Certainly in light of Stone's representation theorem, it seems unsurprising that there could be a topological proof of a theorem in logic, and as I understand it this connection is further investigated in topos theory?

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The topology on Stone spaces is precisely the Zariski topology on the spectrum of a Boolean ring. It is one of three historical sources of the idea that commutative rings can be thought of as topological spaces, along with various work in algebraic geometry and Gelfand's work on C*-algebras. I have never been very clear on the historical relationship between the three. –  Qiaochu Yuan Jan 18 '11 at 16:27

Here are two more candidates for new ways of thinking in proofs but I am not sure about the historical picture. One is Brunn sieve which led to new results in number theory. The other is Kummer's method that have led to proofs of many cases of FLT. (Frey's new way of thinking regarding FLT was already mentioned in a Roy Smith's comment.)

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I am surprised that noone mentioned Hilbert's proof of Hilbert's Basis Theorem yet. It says that every ideal in $\mathbb{C}[x_1,\ldots,x_n]$ is finitely generated - the proof is nonconstructive in the sense that it does not give an explicit set of generators of an ideal. When P. Gordan (a leading algebraists at that time) first saw Hilbert's proof, he said, "This is not Mathematics, but theology!"

However, in 1899, Gordan published a simplified proof of Hilbert's theorem and commented with "I have convinced myself that theology also has its advantages."

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I have two favorite examples.

A. H. Weyl's 1916 proof of the equidistribution (in $[0,1]$) of the sequence $x_n=n\alpha \bmod \mathbb{Z}$, $\alpha$ irrational. He formulates an even more complicated question, namely to prove that

$$ \lim_{n\to\infty} \frac{1}{n}\sum_{k=1}^n f(x_k)=\int_0^1 f(x) dx, $$

for any Riemann integrable $f$. (The uniform distribution follows by setting $f=$ the characteristic function of an interval.) To prove the more complicated question he observes that the space $X$ of $f$'s satisfying the above equality is a vector space and it is closed with respect to a natural topology. He then observes that $X$ contains all the trigonometric polynomials (trivial computation) and thus $X$ must contain all the functions that can be approximated by trig polynomials. This implies that $X$ contains all the Riemann integrable functions. This a soft touch a proof, with no brute force computation, using ideas of functional analysis at a time when the ideas of functional analysis were not part of the mathematical arsenal.

B. Forty years later A. Grothendieck gave a beautiful proof to the (then) recently discovered Riemann-Roch-Hizebruch formula. He formulated a more complicated problem, observed that the more complicated problem has a rich structure encoded in the object he invented and now called the $K$-theory of coherent sheaves, and then used functoriality to show that to prove the most general case it suffices to prove it for two special classes of examples.

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"unexpected technique"

Sometimes the result itself is unexpected. Cantor's diagonal proof (and other counterexamples), Godel's incompleteness, Banach-Tarski and nonmeasurable sets, independence results generally.

I think you want cases where the result is anticipated, but the technique seems unrelated?

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I would count Cantor's proof of the existence of transcendental numbers. –  gowers Dec 9 '10 at 16:39
However, I think I have come up with a reasonably plausible account of how somebody could have thought of it: gowers.wordpress.com/2010/12/09/… –  gowers Dec 9 '10 at 23:00

Use of the Hardy–Littlewood circle method towards Waring's problem.

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Malliavin's proof of Hormander's theorem is very interesting in the sense that one of the basic ingredients in the language of the proof is a derivative operator with respect to a Gaussian process acting on a Hilbert space. The adjoint of the derivative operator is known as the divergence operator and with these two definitions one can establish the so called "Malliavin Calculus" which has been used to recover classical probabilistic results as well as give new insight into current research in stochastic processes such as developing a stochastic calculus with respect to fractional Brownian motion. What makes his proof more interesting is that Malliavin was trained in geometry and only used the language of probability in a somewhat marginal sense at times - alot of his ideas are very geometric in nature which can be seen for example in his very dense book: P. Malliavin: Stochastic Analysis. Grundlehren der Mathematischen Wissenschaften, 313. Springer-Verlag, Berlin, 1997.

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How about Goodwillie Calculus? I'm not an expert in this field, but it seems to capture a lot of very deep ideas in stable homotopy theory and in category theory more generally. Here is a stub which includes some of the traditional concepts you can get back from Goodwillie Calculus: http://ncatlab.org/nlab/show/Goodwillie+calculus

Here are some lecture notes which go over the Goodwillie calculus and use it to derive the James splitting $\Sigma^\infty \Omega \Sigma X$ and the Snaith splittings of $\Sigma^\infty \Omega^n \Sigma^n X$ in a new way (this is an example of the "proof" the question is asking for): http://noether.uoregon.edu/~sadofsky/gctt/goodwillie.pdf

Finally, I recently saw an amazing talk given by Mark Behrens which used the Goodwillie Calculus to lift differentials in a particular spectral sequence to differentials in the EHP Spectral Sequence, meaning this abstract machinery could also lead to powerful new computational tools. This is discussed in a recent paper: http://www-math.mit.edu/~mbehrens/papers/GoodEHPmem.pdf

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Barwise compactness and $\alpha$-recursion theory. The idea many properties of the following are captured by thinking of how to define analogs in $V_\omega$:

(1) Finite sets are elements of $V_{\omega}$.

(2) Computable sets can are $\Delta_1$ definable over $V_{\omega}$.

(3) Computable enumerable sets can are $\Sigma_1$ definable over $V_{\omega}$.

(4) First order logic is $L_{\infty, \omega} \cap V_\omega$.

Then, if we replace $V_\omega$ by a different countable admissible set $A$, many of the results relating these classes have analogs. E.g. Barwise compactness, completeness, the existence of an $A$-Turing jump, ...

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Nicolas Monod's genius two pages new counterexample to the von Neumann conjecture, inspired by Mary Shelley's novel Frankenstein: http://arxiv.org/abs/1209.5229

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I would like to propose the theorem of J.H.C. Whitehead that if $X$ is a path connected space, and $Y$ is formed from $X$ by attaching $2$-cells, i.e. $Y=X \cup_{f_i}e^2_i$ for a family of maps $f_i: S^1 \to X$, then the crossed module $\partial: \pi_2(Y,X,x) \to \pi_1(X,x) \;$ is the free crossed module on the characteristic maps of the $2$-cells.

The proof spreads over three of his papers.

  1. On adding relations to homotopy groups. Ann. of Math. (2) 42 (1941) 409--428.

  2. Note on a previous paper entitled ``On adding relations to homotopy groups.''. Ann. of Math. (2) 47 (1946) 806--810.

  3. Combinatorial homotopy. II. Bull. Amer. Math. Soc. 55 (1949) 453--496.

The essential geometric content of the proof uses transversality and knot theory, and was in his paper 1. The definition of crossed module was given in his paper 2. Finally the definition of free crossed module was given in his paper 3, together with an outline of the proof, referring back to paper 1. You can find my own exposition of the proof here. The referee wrote that: "The theorem is not new. The proof is not new. But the paper should be published since these papers of Whitehead are notoriously obscure." I explained the proof once to Terry Wall, and he said it was a good 1960's type proof! What my paper does is repackage Whitehead's proof for a modern audience, and with pictures and consistent notation.

It seems to me pretty good to give the essence of a proof years before you have the right definitions for the theorem!

The notion of crossed module has over recent years become more widespread, partly because of its relation to $2$-groupoids and double groupoids. This is discussed a little in a seminar I gave in Chicago last year. See also the Wikipedia entry and that from the nlab.

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I think the concepts of Archimedes which are at the birth of infinitesimal calculation, as the definition of length of a circle (hence the concept of $\pi$), and how to calculate the area of ​​a circle from $\pi$.

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Lovasz proof of Shannon Capacity of the Pentagon (the only proof known). Introduces Semidefinite optimization. Geometrizes and introduces analytic techniques to Graph Theory. Descartes introduced coordinate space approach to geometric problems. In the same spirit, Lovasz's proof coordinate space approach to graph theory problems.

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  1. Minkowski's geometric methods in algebraic number theory.
  2. Kolmogorov's application of Shannon's entropy to classify Bernoulli dynamic systems.
  3. Applications of low-dimensional geometric topology (the fundamental group) to the abstract group theory.
  4. Archimedes' applications of mechanics to geometry (he considered them to be but heuristics, and provided additional "purely geometric" proofs for his results, but it was, as we know now, profound mathematics, something like affine geometry, with linear coefficients of points, adding to $1$, more or less being normalized weights).
  5. Banach's method of applying Baire Theorem (Baire Property).

(I have one more recent too, which took specialists by surprize, but 5 is a nice number).

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The post,the answers generated, the comments...wow! They are so nice!


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