# Proofs that require fundamentally new ways of thinking

I do not know exactly how to characterize the class of proofs that interests me, so let me give some examples and say why I would be interested in more. Perhaps what the examples have in common is that a powerful and unexpected technique is introduced that comes to seem very natural once you are used to it.

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

If you haven't seen anything like it before, the idea that you could use analysis to prove that there are infinitely many primes is completely unexpected. Once you've seen how it works, that's a different matter, and you are ready to contemplate trying to do all sorts of other things by developing the method.

Example 2. The use of complex analysis to establish the prime number theorem.

Even when you've seen Euler's argument, it still takes a leap to look at the complex numbers. (I'm not saying it can't be made to seem natural: with the help of Fourier analysis it can. Nevertheless, it is a good example of the introduction of a whole new way of thinking about certain questions.)

Example 3. Variational methods.

You can pick your favourite problem here: one good one is determining the shape of a heavy chain in equilibrium.

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

One of the very first results (Shannon's bound for the size of a separated subset of the discrete cube being another very early one) in probabilistic combinatorics.

Example 5. Roth's proof that a dense set of integers contains an arithmetic progression of length 3.

Historically this was by no means the first use of Fourier analysis in number theory. But it was the first application of Fourier analysis to number theory that I personally properly understood, and that completely changed my outlook on mathematics. So I count it as an example (because there exists a plausible fictional history of mathematics where it was the first use of Fourier analysis in number theory).

Example 6. Use of homotopy/homology to prove fixed-point theorems.

Once again, if you mount a direct attack on, say, the Brouwer fixed point theorem, you probably won't invent homology or homotopy (though you might do if you then spent a long time reflecting on your proof).

The reason these proofs interest me is that they are the kinds of arguments where it is tempting to say that human intelligence was necessary for them to have been discovered. It would probably be possible in principle, if technically difficult, to teach a computer how to apply standard techniques, the familiar argument goes, but it takes a human to invent those techniques in the first place.

Now I don't buy that argument. I think that it is possible in principle, though technically difficult, for a computer to come up with radically new techniques. Indeed, I think I can give reasonably good Just So Stories for some of the examples above. So I'm looking for more examples. The best examples would be ones where a technique just seems to spring from nowhere -- ones where you're tempted to say, "A computer could never have come up with that."

Edit: I agree with the first two comments below, and was slightly worried about that when I posted the question. Let me have a go at it though. The difficulty with, say, proving Fermat's last theorem was of course partly that a new insight was needed. But that wasn't the only difficulty at all. Indeed, in that case a succession of new insights was needed, and not just that but a knowledge of all the different already existing ingredients that had to be put together. So I suppose what I'm after is problems where essentially the only difficulty is the need for the clever and unexpected idea. I.e., I'm looking for problems that are very good challenge problems for working out how a computer might do mathematics. In particular, I want the main difficulty to be fundamental (coming up with a new idea) and not technical (having to know a lot, having to do difficult but not radically new calculations, etc.). Also, it's not quite fair to say that the solution of an arbitrary hard problem fits the bill. For example, my impression (which could be wrong, but that doesn't affect the general point I'm making) is that the recent breakthrough by Nets Katz and Larry Guth in which they solved the Erdős distinct distances problem was a very clever realization that techniques that were already out there could be combined to solve the problem. One could imagine a computer finding the proof by being patient enough to look at lots of different combinations of techniques until it found one that worked. Now their realization itself was amazing and probably opens up new possibilities, but there is a sense in which their breakthrough was not a good example of what I am asking for.

While I'm at it, here's another attempt to make the question more precise. Many many new proofs are variants of old proofs. These variants are often hard to come by, but at least one starts out with the feeling that there is something out there that's worth searching for. So that doesn't really constitute an entirely new way of thinking. (An example close to my heart: the Polymath proof of the density Hales-Jewett theorem was a bit like that. It was a new and surprising argument, but one could see exactly how it was found since it was modelled on a proof of a related theorem. So that is a counterexample to Kevin's assertion that any solution of a hard problem fits the bill.) I am looking for proofs that seem to come out of nowhere and seem not to be modelled on anything.

Further edit. I'm not so keen on random massive breakthroughs. So perhaps I should narrow it down further -- to proofs that are easy to understand and remember once seen, but seemingly hard to come up with in the first place.

• Of course, there was apparently a surprising and simple insight involved in the proof of FLT, namely Frey's idea that a solution triple would give rise to a rather exotic elliptic curve. It seems to have been this insight that brought a previously eccentric seeming problem at least potentially within the reach of the powerful and elaborate tradition referred to. So perhaps that was a new way of thinking at least about what ideas were involved in FLT. – roy smith Dec 9 '10 at 16:21
• Never mind the application of Fourier analysis to number theory -- how about the invention of Fourier analysis itself, to study the heat equation! More recently, if you count the application of complex analysis to prove the prime number theorem, then you might also count the application of model theory to prove results in arithmetic geometry (e.g. Hrushovski's proof of Mordell-Lang for function fields). – D. Savitt Dec 9 '10 at 16:42
• I agree that they are difficult, but in a sense what I am looking for is problems that isolate as well as possible whatever it is that humans are supposedly better at than computers. Those big problems are too large and multifaceted to serve that purpose. You could say that I am looking for "first non-trivial examples" rather than just massively hard examples. – gowers Dec 9 '10 at 18:04
• My feeling is that when someone says "X is fundamentally new" (for various values of X) in reference to some mathematics, IMO this usually demands as a prerequisite that one has a pretty narrow perspective on the kinds of thinking that came beforehand in order to believe the statement. This doesn't take anything away from novel mathematics, it's just that fundamentally new is almost always too hyperbolic expression for the mathematics it describes. I imagine the main reason mathematicians use such hyperbolic terminology is that hype draws people's attention, and that helps ideas propagate. – Ryan Budney Dec 11 '10 at 5:17
• It seems to me that this question has been around a long time and is unlikely garner new answers of high quality. It also seems unlikely most would even read new answers. Furthermore, nowadays I imagine a question like this would be closed as too broad, and if we close this then we'll discourage questions like it in the future. So I'm voting to close. – David White Oct 13 '13 at 18:52

Dvir's proof of the finite field Kakeya conjecture.

I know this is closely related to the Guth--Katz result which has already been dismissed, but I will justify why it answers the question. Dvir's proof dramatically changed the point of view in the restriction problem (a big subfield of Euclidean harmonic analysis) and allied problems in PDEs and geometric meaure theory by importing 'the polynomial method' into the field. In particular, it is highly unlikely that the Guth--Katz result would have came as and when it did without Dvir's insight. Prior to Dvir's insight a surprisingly large amount of effort was spent on working towards the finite field Kakeya conjecture (Dvir's theorem) including many works by a Field's medalist. While Dvir's original proof was a few pages, it was quickly distilled into a paragraph proof that is accesible to undergraduates who have just understood what a finite field is.

The obvious critique is that the polynomial method existed before this. Certainly it did and polynomials for a long time before that. I am not saying that Dvir invented the method. However, no one saw that this connection nor expected it. Similarly complex analysis and number theory were investigated in parallel before Riemann introduced his famous eponymous hypothesis. In analogy with the above example on the infinitude of primes, it would be like we started with Euler's proof, then discovered Hadamard--de la Vallee Poussin's proof with the twist that their proof was only as transparent and elegant as Euclid's proof.

I think Fürstenberg´s proof of the infinitude of primes, taken for itself, could be considered in the spirit of the original question, even though its mathematical value is questionable.

• A plea in advance - can we not have this discussion again? – HJRW Dec 9 '10 at 19:16
• And others think it's the usual proof in disguise :-) – Robin Chapman Dec 9 '10 at 19:44
• Well, ok, I was not aware of all these discussions (I have just found some more, lol). I agree though that its mathematical value is doubtful as it seems to be just a reformulation of Euclid´s proof, yet I find it amusing that such reformulation is possible... No more discussion about it. Period. :-) – M.G. Dec 9 '10 at 21:18
• @ Max Muller: In the spirit of accuracy (but without wanting to move towards opening up any cans of worms) I believe it is the case that Furstenburg's is not a repackaging of analytic ideas, but rather Euclid's original argument. – Nick Salter Dec 9 '10 at 21:43
• @all: Please consider Henry Wilton´s plea! Thanks! :-) – M.G. Dec 9 '10 at 22:17

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

Though I am not going to answer in exactly the way required, I believe including occasions were new insights helped to give support a great unsolved problem are worth-noting. For instance, we can consider a case in Random Matrix Theory: the statistical interplay between the distribution the zeros of the Riemann Zeta function and the eigenvalues of a random Hermitian matrix which has provided a basis for the Hilbert–Pólya conjecture. .

• empslocal.ex.ac.uk/people/staff/mrwatkin/zeta/random.htm – Unknown Dec 12 '10 at 9:10
• I think it's fair to say that this interplay has led to a lot of conjectures but no proofs; whether random matrix theory can actually be used to prove theorems about L-functions remains to be seen... – David Hansen Dec 12 '10 at 9:28
• I agree! The conjectural relationships and Odlyzko's computations thrived along this new avenue and this was what I wanted to emphasize. – Unknown Dec 12 '10 at 9:50

The first formal proofs using limits. (the oldest ones I know are in Newton's Principia)

• Newton didn't consider them formal--so much so that he switched back to the ancient Greek method. – Włodzimierz Holsztyński Jul 3 '13 at 5:39

I don’t believe that Hilbert’s 10th problem has been mentioned ... all of what went into that.

I also don’t believe that all of what went into making it possible to prove the four color theorem with the help of a computer has been mentioned.

And although much is mentioned about set theory, introducing the actual axioms of ZFC was revolutionary and I don’t think it has been mentioned ... apologies if I missed it in the long set of replies ... I guess this answer can always be edited (or deleted if it is a duplicate).

Hilbert's proof of Hilbert-Waring theorem.