Demonstrating that rigour is important

Question

Any pure mathematician will from time to time discuss, or think about, the question of why we care about proofs, or to put the question in a more precise form, why we seem to be so much happier with statements that have proofs than we are with statements that lack proofs but for which the evidence is so overwhelming that it is not reasonable to doubt them.

That is not the question I am asking here, though it is definitely relevant. What I am looking for is good examples where the difference between being pretty well certain that a result is true and actually having a proof turned out to be very important, and why. I am looking for reasons that go beyond replacing 99% certainty with 100% certainty. The reason I'm asking the question is that it occurred to me that I don't have a good stock of examples myself.

The best outcome I can think of for this question, though whether it will actually happen is another matter, is that in a few months' time if somebody suggests that proofs aren't all that important one can refer them to this page for lots of convincing examples that show that they are.

Added after 13 answers: Interestingly, the focus so far has been almost entirely on the "You can't be sure if you don't have a proof" justification of proofs. But what if a physicist were to say, "OK I can't be 100% sure, and, yes, we sometimes get it wrong. But by and large our arguments get the right answer and that's good enough for me." To counter that, we would want to use one of the other reasons, such as the "Having a proof gives more insight into the problem" justification. It would be great to see some good examples of that. (There are one or two below, but it would be good to see more.)

Further addition: It occurs to me that my question as phrased is open to misinterpretation, so I would like to have another go at asking it. I think almost all people here would agree that proofs are important: they provide a level of certainty that we value, they often (but not always) tell us not just that a theorem is true but why it is true, they often lead us towards generalizations and related results that we would not have otherwise discovered, and so on and so forth. Now imagine a situation in which somebody says, "I can't understand why you pure mathematicians are so hung up on rigour. Surely if a statement is obviously true, that's good enough." One way of countering such an argument would be to give justifications such as the ones that I've just briefly sketched. But those are a bit abstract and will not be convincing if you can't back them up with some examples. So I'm looking for some good examples.

What I hadn't spotted was that an example of a statement that was widely believed to be true but turned out to be false is, indirectly, an example of the importance of proof, and so a legitimate answer to the question as I phrased it. But I was, and am, more interested in good examples of cases where a proof of a statement that was widely believed to be true and was true gave us much more than just a certificate of truth. There are a few below. The more the merrier.

Answer 1

I once got a letter from someone who had overwhelming numerical evidence that the sum of the reciprocals of primes is slightly bigger than 3 (he may have conjectured the limit was π). The sum is in fact infinite, but diverges so slowly (like log log n) that one gets no hint of this by computation.

Answer 2

I would like to preface this long answer by a few philosophical remarks. As noted in the original posting, proofs play multiple roles in mathematics: for example, they assure that certain results are correct and give insight into the problem.

A related aspect is that in the course of proving an intuitively obvious statement, it is often necessary to create theoretical framework, i.e. definitions that formalize the situation and new tools that address the question, which may lead to vast generalizations in the course of the proof itself or in the subsequent development of the subject; often it is the proof, not the statement itself, that generalizes, hence it becomes valuable to know multiple proofs of the same theorem that are based on different ideas. The greatest insight is gained by the proofs that subtly modify the original statement that turned out to be wrong or incomplete. Sometimes, the whole subject may spring forth from a proof of a key result, which is especially true for proofs of impossibility statements.

Most examples below, chosen among different fields and featuring general interest results, illustrate this thesis.

1. Differential geometry

a. It had been known since the ancient times that it was impossible to create a perfect (i.e. undistorted) map of the Earth. The first proof was given by Gauss and relies on the notion of intrinsic curvature introduced by Gauss especially for this purpose. Although Gauss's proof of Theorema Egregium was complicated, the tools he used became standard in the differential geometry of surfaces.

b. Isoperimetric property of the circle has been known in some form for over two millennia. Part of the motivation for Euler's and Lagrange's work on variational calculus came from the isoperimetric problem. Jakob Steiner devised several different synthetic proofs that contributed technical tools (Steiner symmetrization, the role of convexity), even though they didn't settle the question because they relied on the existence of the absolutely minimizing shape. Steiner's assumption led Weierstrass to consider the general question of existence of solutions to variational problems (later taken up by Hilbert, as mentioned below) and to give the first rigorous proof. Further proofs gained new insight into the isoperimetric problem and its generalizations: for example, Hurwitz's two proofs using Fourier series exploited abelian symmetries of closed curves; the proof by Santaló using integral geometry established more general Bonnesen inequality; E.Schmidt's 1939 proof works in $n$ dimensions. Full solution of related lattice packing problems led to such important techniques as Dirichlet domains and Voronoi cells and the geometry of numbers.

2. Algebra

a. For more than two and a half centuries since Cardano's Ars Magna, no one was able to devise a formula expressing the roots of a general quintic equation in radicals. The Abel–Ruffini theorem and Galois theory not only proved the impossibility of such a formula and provided an explanation for the success and failure of earlier methods (cf Lagrange resolvents and casus irreducibilis), but, more significantly, put the notion of group on the mathematical map.

b. Systems of linear equations were considered already by Leibniz. Cramer's rule gave the formula for a solution in the $n\times n$ case and Gauss developed a method for obtaining the solutions, which yields the least square solution in the underdetermined case. But none of this work yielded a criterion for the existence of a solution. Euler, Laplace, Cauchy, and Jacobi all considered the problem of diagonalization of quadratic forms (the principal axis theorem). However, the work prior to 1850 was incomplete because it required genericity assumptions (in particular, the arguments of Jacobi et al didn't handle singular matrices or forms. Proofs that encompass all linear systems, matrices and bilinear/quadratic forms were devised by Sylvester, Kronecker, Frobenius, Weierstrass, Jordan, and Capelli as part of the program of classifying matrices and bilinear forms up to equivalence. Thus we got the notion of rank of a matrix, minimal polynomial, Jordan normal form, and the theory of elementary divisors that all became cornerstones of linear algebra.

3. Topology

a. Attempts to rigorously prove the Euler formula $V-E+F=2$ led to the discovery of non-orientable surfaces by Möbius and Listing.

b. Brouwer's proof of the Jordan curve theorem and of its generalization to higher dimensions was a major development in algebraic topology. Although the theorem is intuitively obvious, it is also very delicate, because various plausible sounding related statements are actually wrong, as demonstrated by the Lakes of Wada and the Alexander horned sphere.

4. Analysis The work on existence, uniqueness, and stability of solutions of ordinary differential equations and well-posedness of initial and boundary value problems for partial differential equations gave rise to tremendous insights into theoretical, numerical, and applied aspects. Instead of imagining a single transition from 99% ("obvious") to 100% ("rigorous") confidence level, it would be more helpful to think of a series of progressive sharpenings of statements that become natural or plausible after the last round of work.

a. Picard's proof of the existence and uniqueness theorem for a first order ODE with Lipschitz right hand side, Peano's proof of the existence for continuous right hand side (uniqueness may fail), and Lyapunov's proof of stability introduced key methods and technical assumptions (contractible mapping principle, compactness in function spaces, Lipschitz condition, Lyapunov functions and characteristic exponents).

b. Hilbert's proof of the Dirichlet principle for elliptic boundary value problems and his work on the eigenvalue problems and integral equations form the foundation for linear functional analysis.

c. The Cauchy problem for hyperbolic linear partial differential equations was investigated by a whole constellation of mathematicians, including Cauchy, Kowalevski, Hadamard, Petrovsky, L.Schwartz, Leray, Malgrange, Sobolev, Hörmander. The "easy" case of analytic coefficients is addressed by the Cauchy–Kowalevski theorem. The concepts and methods developed in the course of the proof in more general cases, such as the characteristic variety, well-posed problem, weak solution, Petrovsky lacuna, Sobolev space, hypoelliptic operator, pseudodifferential operator, span a large part of the theory of partial differential equations.

5. Dynamical systems

Universality for one-parameter families of unimodal continuous self-maps of an interval was experimentally discovered by Feigenbaum and, independently, by Coullet and Tresser in the late 1970s. It states that the ratio between the lengths of intervals in the parameter space between successive period-doubling bifurcations tends to a limiting value $\delta\approx 4.669201 $ that is independent of the family. This could be explained by the existence of a nonlinear renormalization operator $\mathcal{R}$ in the space of all maps with a unique fixed point $g$ and the property that all but one eigenvalues of its linearization at $g$ belong to the open unit disk and the exceptional eigenvalue is $\delta$ and corresponds to the period-doubling transformation. Later, computer-assisted proofs of this assertion were given, so while Feigebaum universality had initially appeared mysterious, by the late 1980s it moved into the "99% true" category.

The full proof of universality for quadratic-like maps by Lyubich (MR) followed this strategy, but it also required very elaborate ideas and techniques from complex dynamics due to a number of people (Douady–Hubbard, Sullivan, McMullen) and yielded hitherto unknown information about the combinatorics of non-chaotic quadratic maps of the interval and the local structure of the Mandelbrot set.

6. Number theory

Agrawal, Kayal, and Saxena proved that PRIMES is in P, i.e. primality testing can be done deterministically in polynomial time. While the result had been widely expected, their work was striking in at least two respects: it used very elementary tools, such as variations of Fermat's little theorem, and it was carried out by a computer science professor and two undergraduate students. The sociological effect of the proof may have been even greater than its numerous consequences for computational number theory.

Answer 3

When I teach our "Introduction to Mathematical Reasoning" course for undergraduates, I start out by describing a collection of mathematical "facts" that everybody "knew" to be true, but which, with increasing standards of rigor, were eventually proved false. Here they are:

1. Non-Euclidean geometry: The geometry described by Euclid is the only possible "true" geometry of the real world.
2. Zeno's paradox: It is impossible to add together infinitely many positive numbers and get a finite answer.
3. Cardinality vs. dimension: There are more points in the unit square than there are in the unit interval.
4. Space-filling curves: A continuous parametrized curve in the unit square must miss "most" points.
5. Nowhere-differentiable functions: A continuous real-valued function on the unit interval must be differentiable at "most" points.
6. The Cantor Function: A function that is continuous and satisfies f'(x)=0 almost everywhere must be constant.
7. The Banach-Tarski paradox: If a bounded solid in R^3 is decomposed into finitely many disjoint pieces, and those pieces are rearranged by rigid motions to form a new solid, then the new solid will have the same volume as the original one.

Answer 4

Claim

The trefoil knot is knotted.

Discussion

One could scarcely find a reasonable person who would doubt the veracity of the above claim. None of the 19^th century knot tabulators (Tait, Little, and Kirkman) could rigourously prove it, nor could anybody before them. It's not clear that anyone was bothered by this.

Yet mathematics requires proof, and proof was to come. In 1908 Tietze proved the beknottedness of the trefoil using Poincaré's new concept of a fundamental group. Generators and relations for fundamental groups of knot complements could be found using a procedure of Wirtinger, and the fundamental group of the trefoil complement could be shown to be non-commutative by representing it in $SL_2(\mathbb{Z})$, while the fundamental group of the unknot complement is $\mathbb{Z}$. In general, to distinguish even fairly simple knots, whose difference was blatantly obvious to everybody, it was necessary to distinguish non-abelian fundamental groups given in terms of Wirtinger presentations, via generators and relations. This is difficult, and the Reidemeister-Schreier method was developed to tackle this difficulty. Out of these investigations grew combinatorial group theory, not to mention classical knot theory.

All because beknottedness of a trefoil requires proof.

Claim

Kishino's virtual knot is knotted.

Discussion

We are now in the 21^st century, and virtual knot theory is all the rage. One could scarecely find a reasonable person who would argue that Kishino's knot is trivial. But the trefoil's lesson has been learnt well, and it was clear to everyone right away that proving beknottedness of Kishino's knot was to be a most fruitful endeavour. Indeed, that is how things have turned out, and proofs that Kishino's knot is knotted have led to substantial progress in understanding quandles and generalized bracket polynomials.

Summary

Above we have claims which were obvious to everybody, and were indeed correct, but whose proofs directly led to greater understanding and to palpable mathematical progress.

Answer 5

Maybe some of the answers to this question about "eventual counterexamples" - ie, which could plausibly be true for all $n$ but which fail for some large number - are relevant?

Some highlights from that question:

• $gcd(n^5−5,(n+1)^5−5)=1$ is true for n=1,2,…,1435389 but not for n=1435390; and many similar
• factors of $x^n−1$ over the rationals have no coefficient exceeding 1 in absolute value - until $n=105$
• The Strong Law of Small Numbers, a fun paper by Guy

Answer 6

Surely calculus is the ultimate treasure trove for such examples. In antiquity, the Egyptians, Greeks, Indians, Chinese, and many others could calculate integrals with a pretty good degree of certainty via the method of exhaustion and its variants. But it is not without reason that Newton and Leibniz are credited with the invention of calculus. Because once you had a formalism- a proof- of the product rule, chain rule, taylor expansions, calculation of an integral- in fact, once you had the formalism in hand to make such a proof possible- then with that came an understanding, and from that sprung the most powerful analytic machine known to man, that is calculus. Without a formalism, Zeno's paradox was just that- a paradox. With the concept of limits and of epsilon-delta proofs, it becomes a triviality.
Thus, in my opinion, proof is important in that it leads to mathematics. Mathematics is important in that it leads to understanding patterns, and patterns govern all of science and the universe. If you can prove something, you understand it, or at least "your concepts understand it". If you can't prove it, you're nothing more than a goat, knowing the sun will rise in the morning from experience or from experiment, but having not the slightest inkling of why.
The specific example, then, is "calculating integrals" and "solving differential equations".

With the reader's indulgence, an example of a mathematical proof saving lives. My friend's mum is an aeronautical engineer at a place which designs fighter jets. There was some wing design, whose wind resistance satisfied some PDE. They numerically simulated it by computer, and everything was perfect. My friend's mum, who had studied PDE's seriously in university and thought this one could be solved rigourously, set about finding an exact solution, and lo-and-behold, there was some unexpected singularity, and if wind were to blow at some speed from some direction then the wing would shear off. She pointed this out, was awarded a medal, and the wing design was changed. Lives saved by a proof. I'm sure there are a thousand examples like that.

Answer 7

Mumford in Rational equivalence of 0-cycles on surfaces gave an example where an intuitive result of Severi, who claimed the space of rational equivalence classes was finite dimensional, was just completely wrong: it is infinite dimensional for most surfaces. This is a typical example of why the informal non-rigorous style of algebraic geometry was abandoned: too many of the "obvious" but unproved results turned out to be incorrect.

Answer 8

I think the recent work on compressed sensing is a good example.

As I understand from listening to a talk by Emmanuel Candes - please correct me if I get anything wrong - the recent advances in compressed sensing began with an empirical observation that a certain image reconstruction algorithm seemed to perfectly reconstruct some classes of corrupted images. Candes, Romberg, and Tao collaborated to prove this as a mathematical theorem. Their proof captured the basic insight that explained the good performance of the algorithm: $l_1$ minimization finds a sparse solution to a system of equations for many classes of matrices. It was then realized this insight is portable to other problems and analogous tools could work in many other settings where sparsity is an issue (e.g., computational genetics).

If Candes, Romberg, and Tao had not published their proof, and if only the empirical observation that a certain image reconstruction works well was published, it is possible (likely?) that this insight would never have penetrated outside the image processing community.

Answer 9

The evidence for both quantum mechanics and for general relativity is overwhelming. However, one can prove that without serious modifications, these two theories are incompatible. Hence the (still incomplete) quest for quantum gravity.

Answer 10

I don't think that proofs are about replacing 99% certainty with 99.99% (or 100%, if the proof is simple enough). In one of his problems he studied early on, Fermat stated that it was important to find out whether a prime divides only numbers $a^n-1$, or also numbers of the form $a^n+1$. For $a = 2$ and $a = 3$ he saw that the answer seemed to depend on the residue class of $p$ modulo $4a$. He did not really come back to investigate this problem more closely; Euler did, but couldn't find the proof. Gauss's proofs did not remove the remaining 1 % uncertainty, it brought in structure and allowed to ask the next question. Just looking at patterns of prime divisors of $a^n \pm 1$ wouldn't have led to Artin reciprocity.

Answer 11

[Edited to correct the Galileo story] An old example of a plausible result that was overthrown by rigor is the 17th-century example of the hanging chain. Galileo once said (though he later said otherwise), and Girard claimed to have proved, that the shape was a parabola. But this was disproved by Huygens (then aged 17) by a more rigorous analysis. Some decades later, the exact equation for the catenary was found by the Bernoullis, Leibniz, and Huygens.

In the 20th century, some people thought it plausible that the shape of the cable of a suspension bridge is also a catenary. Indeed, I once saw this claim in a very popular engineering mathematics text. But a rigorous argument shows (with the sensible simplifying assumption that the weight of the cable is negligible compared with the weight of the road) that the shape is in fact a parabola.

Answer 12

Here's an example:
In the Mathscinet review of "Y-systems and generalized associahedra", by Sergey Fomin and Andrei Zelevinsky, you find:

Let $I$ be an $n$-element set and $A=(a_{ij})$, $i,j\in I$, an indecomposable Cartan matrix of finite type. Let $\Phi$ be the corresponding root system (of rank $n$), and $h$ the Coxeter number. Consider a family $(Y_i(t))_{i\in I,\,t\in\Bbb{Z}}$ of commuting variables satisfying the recurrence relations $$Y_i(t+1)Y_i(t-1)=\prod_{j\ne i}(Y_j(t)+1)^{-a_{ij}}.$$ Zamolodchikov's conjecture states that the family is periodic with period $2(h+2)$, i.e., $Y_i(t+2(h+2))=Y_i(t)$ for all $i$ and $t$.

That conjecture claims that an explicitly described algebraic map is periodic. The conjecture can be checked numerically by plugging in real numbers with 30 digits, and iterating the map the appropriate number of times. If you see that time after time, the numbers you get back agree with the initial values with a 29 digit accuracy, then you start to be pretty confident that the conjecture is true.

For the $E_8$ case, the proof presented in the above paper involves a massive amount of symbolic computations done by computer. Is it really much better than the numerical evidence?

Conclusion: I think that we only like proofs when we learn something from them. It's not the property of "being a proof" that is attractive to mathematicians.

Answer 13

1. Nonexistence theorems can not be demonstrated with numerical evidence. For example, the impossibility of classical geometric construction problems (trisecting the angle, doubling the cube) could only be shown with a proof that the efforts in the positive direction were futile. Or consider the equation $x^n + y^n = z^n$ with $n > 2$. [EDIT: Strictly speaking my first sentence is not true. For example, the primality of a number is a kind of nonexistence theorem -- this number has no nontrivial factorization -- and one could prove the primality of a specific number by just trying out all the finitely many numerical possibilities, whether by naive trial division or a more efficient rigorous primality test. Probabilistic primality tests, such as the Solovay--Strassen or Miller--Rabin tests, allow one to present a short amount of compelling numerical evidence, without a proof, that a number is quite likely to be prime. What I should have written is that nonexistence theorems are usually not (or at least some of them are not) demonstrable by numerical evidence, and the geometric impossibility theorems which I mentioned illustrate that. I don't see how one can give real evidence short for those theorems other than by a proof. Lack of success in making the constructions is not convincing: the Greeks couldn't construct a regular 17-gon by their rules, but Gauss showed much later that it can be done.]

2. You can't apply a theorem to all commutative rings unless you have a proof of the result which works that broadly. Otherwise math just becomes conjectures upon conjectures, or you have awkward hypotheses: "For a ring whose nonzero quotients all have maximal ideals, etc." Emmy Noether revolutionized abstract algebra by replacing her predecessor's tedious computational arguments in polynomial rings with short conceptual proofs valid in any Noetherian ring, which not only gave a better understanding of what was done before but revealed a much broader terrain where earlier work could be used. Or consider the true scope of harmonic analysis: it can be carried out not just in Euclidean space or Lie groups, but in any locally compact group. Why? Because, to get things started, Weil's proof of the existence of Haar measure works that broadly. How are you going to collect 99% numerical evidence that all locally compact groups have a Haar measure? (In number theory and representation theory one integrates over the adeles, which are in no sense like Lie groups, so the "topological group" concept, rather than just "Lie group", is really crucial.)

3. Proofs tell you why something works, and knowing that explanatory mechanism can give you the tools to generalize the result to new settings. For example, consider the classification of finitely generated torsion-free abelian groups, finitely generated torsion-free modules over any PID, and finitely generated torsion-free modules over a Dedekind domain. The last classification is very useful, but I think its statement is too involved to believe it is valid as generally as it is without having a proof.

4. Proofs can show in advance how certain unsolved problems are related to each other. For instance, there are tons of known consequences of the generalized Riemann hypothesis because the proofs show how GRH leads to those other results. (Along the same lines, Ribet showed how modularity of elliptic curves would imply FLT, which at the time were both open questions, and that work inspired Wiles.)

Answer 14

One can rigorously prove that pyramid schemes cannot run forever, and that no betting system with finite monetary reserves can guarantee a profit from a martingale or submartingale.

But there are countless examples of people who have suffered monetary loss due to their lack of awareness of the rigorous nature of these non-existence proofs. Here is a case in which having a non-rigorous 99% plausibility argument is not enough, because one can always rationalise that "this time is different", or that one has some special secret strategy that nobody else thought of before.

In a similar spirit: a rigorous demonstration of a barrier (e.g. one of the three known barriers to proving P != NP) can prevent a lot of time being wasted on pursuing a fruitless approach to a difficult problem. (In contrast, a non-rigorous plausibility argument that an approach is "probably futile" is significantly less effective at preventing an intrepid mathematician or amateur from trying his or her luck, especially if they have excessive confidence in their own abilities.)

[Admittedly, P!=NP is not a great example to use here as motivation, because this is itself a problem whose goal is to obtain a rigorous proof...]

Answer 15

Many examples that have been given are of statements that one could at least formulate, and conjecture, without rigorous proof. However, one of the most important benefits of rigorous proof is that it allows us to step surefootedly along long, intricate chains of reasoning into regions that we previously never suspected existed. For example, it is hard to imagine discovering the Monster group without the ability to reason rigorously.

In any other field besides mathematics, as soon as a line of abstract argument exceeds a certain (low) threshold of complexity, it becomes doubtful, and unless there is some way to corroborate the conclusion empirically, the argument lapses into controversy. If you are trying to search a very large space of possibilities, then it is indispensable to be able to close off certain avenues definitively so that the search can be focused effectively on the remaining possibilities. Only in mathematics are definitive impossibility proofs so routine that we can rely on them as a fundamental tool for discovering new phenomena.

The classification of finite simple groups is a particularly spectacular example, but I would argue that almost any unexpected mathematical object—the BBP formula for $\pi$, the Lie group $E_8$, the eversion of the sphere, etc.—is the product of a sustained search involving the systematic and rigorous elimination of dead end after dead end. Of course, once an object is discovered, you might try to argue that mathematical rigor was not really necessary and that someone could have stumbled across it with a combination of luck, persistence, and insight. However, I find such an argument disingenuous. Mathematical rigor allows us to distribute the workload across the entire community; each reasoner can contribute his or her piece without worrying that it will be torn to shreds by controversy. Searches can therefore be conducted on a massively greater scale than would be possible otherwise, and the productivity is correspondingly magnified.

Answer 16

Here is an example: 19 century geometers extended Euler's formula V-E+F=2 to higher dimensions: the alternating sum of the number of i-faces of a d-dimensional polytope is 2 in odd dimensions and 0 in even dimensions. The 19th centuries proofs were incomplete and the first rigorous proof came with Poincare and used homology groups. Here, what enabled a rigorous proof was arguably even more important than the theorem itself.

Answer 17

I have found that a strong indicator of research ability is a student wanting to know why something is true. There is also an interesting distinction between an explanation and a proof. (I gave up using the word "proof" for first year students of analysis, and changed it to "explanation", a word they could understand. This was after a student complained I gave too many proofs!)

A proof takes place in a certain conceptual landscape, and the clearer and better formed this is the easier it is to be sure the proof is right, rather than a complicated manipulation. So part of the work of a mathematician is to develop these landscapes: Grothendieck was a master of this!

Of course the more professional a person is in an area the nearer an explanation comes to a rigorous proof. But in fact we do not write down all the details. It is more like giving directions to the station and not listing the cracks in the pavement, though warning of dangerous holes.

The search for an explanation is also related to the search for a beautiful proof. So we should not neglect the aesthetic aspect.

Answer 18

Tim Gowers wrote:

But I was, and am, more interested in good examples of cases where a proof of a statement that was widely believed to be true and was true gave us much more than just a certificate of truth.

How about Stokes' Theorem ?

The two-dimensional version involving line and surface integrals is "proved" in most physics textbooks using a neat little picture dividing up the surface into little rectangles and shrinking them to zero.

Similarly, the Divergence Theorem related volume and surface integrals is demonstrated with very intuitive ideas about liquid flowing out of tiny cubes.

But to prove these rigorously requires developing the theory of differential forms whose consequences go way beyond the original theorems

Answer 19

Based on the recent update to the question, Fermat's Last Theorem seems like the top example of a proof being far more valuable than the truth of the statement. Personally it's a rare occurrence for me to use the nonexistence of a rational point on a Fermat curve but for instance it is quite common for me to use class numbers.

Answer 20

Circle division by chords, http://mathworld.wolfram.com/CircleDivisionbyChords.html, leads to a sequence whose first terms are 1, 2, 4, 8, 16, 31. It's simple and effective to draw the first five cases on a blackboard, count the regions, and ask the students what's the next number in the sequence.

Answer 21

This question is begging for someone to state the obvious, so here goes.

Take for example the existence and uniqueness of solutions to differential equations. Without these theorems, the mathematical models used in many branches of the physical sciences are incapable of making actual predictions. If potentially the DE has no solutions, or the model provides infinitely many solutions, your model has no predictive power. So the model isn't really science.

In that regard, the point of proof in mathematics is to create a foundation that allows for quantitative physical sciences to exist to have a firm philosophical foundation.

Moreover, the proofs of existence and uniqueness shed light on the behaviour of solutions, allowing one to make precise predictions about how good various approximations are to actual solutions -- giving a sense for how computationally expensive it is to make reliable predictions.

Answer 22

A rich source of examples may be found in the study of finite element methods for PDEs in mixed form. Proving that a given mixed finite element method provided a stable and consistent approximation strategy was usually done 'a posteriori': one had a method in mind, and then sought to establish well-posedness of the discretization. This meant a proliferation of methods and strategies tailored for very specific examples.

In the bid for a more comprehensive treatment and unifying proofs, the finite element exterior calculus was developed and refined (eg., the 2006 Acta Numerica paper by Arnold, Falk and Winther). The proofs revealed the importance of using discrete subspaces which form a subcomplex of the Hilbert complex, as well as bounded co-chain projections (we now call this the 'commuting diagram property). These ideas, in turn, provided an elegant design strategy for stable finite element discretizations.

Answer 23

Take the usual definition of Eisenstein functions $$G_k (\tau) = \sum\limits_{(c,d) \in \mathbb{Z}^2 - \{(0,0)\}} (c \tau + d)^{-k}$$ for integer $k \geq 2$. It is easy to prove that $$G_k \left( \frac{a \tau + b}{c \tau + d}\right) = (c \tau + d)^k G_k (\tau)$$ by manipulating the infinite sum, proving that $G_k$ is a modular form of weight $k$.

However the above formula is true only for $k \geq 4$ and not true for $k=2$, and the reason for that is because the sum is not absolutely convergent for $k=2$. One can prove that $$G_2 \left( \frac{a \tau + b}{c \tau + d}\right) = (c \tau + d)^2 \left[ G_2 (\tau) - \frac{2 \pi i c}{c \tau + d} \right]$$ The annoying additional term spoils modularity, and is of tremendous importance. In particular, there are no modular forms of weight 2, which is a good example of how crucial the "rigorous" step of checking whether a sum is absolutely convergent or not may be.

Answer 24

I think that the question itself is entirely misleading. It tacitly assumes as if mathematics could be separated into two parts: mathematical results and their proofs. Mathematics is nothing other than the proofs of mathematical results. Mathematical statements lacks any value, they are neither good nor bad. From the mathematical point of view, it is entirely immaterial whether the answer to a mathematical question like `Is there an even integer greater than two that is not the sum of two primes?' is yes or no. Mathematicians simply do not interested in the right answer. What they would like to do is to solve the problem. That is the main difference between natural sciences or engineering on the one hand, and mathematics on the other. A physicist would like to know the right answer to his question and he does not interested in the way it is obtained. An engineer needs a tool that he can use in the course of his work. He does not interested in the way a useful device works. Mathematics is nothing other than a specific set consisting of different solutions to different problems and, of course, some unsolved problems waiting to be solved. Proofs are not important for mathematics, they constitute the body of knowledge we call mathematics.

Answer 25

The answer to another MO question What did Ramanujan get wrong? cites Bruce Berndt (Ramanujan's Notebooks, Part IV) for a discussion of some cases where Ramanujan's legendary intuition went astray and led him to incorrect beliefs about the accuracy of certain approximations and asymptotic expansions. So this is a concrete example of a generally reliable but non-rigorous approach yielding false statements sometimes.

Answer 26

I tend to think that mathematics or---better---the activity we mathematicians do, is not so much defined by (let me use what's probably nowadays rather old fashioned language) its material object of study (whatever it may be: it is surely very difficult to try to pin down exactly what it is that we are talking about...) but by its formal object, by the way we know what we know. And, of course, proofs are the way we know what we know.

Now: rigour is important in that it allows us to tell apart what we can prove from what we cannot, what we know in the way that we want to know it.

(By the way, I don't think that it is fair to say that, for example, the Italians were not rigorous: they were simply wrong)

Answer 27

The way current computer algebra systems (that I know of) are designed is a compromise between ease of use and mathematical rigor. Although in practice, most of the answers given by CASes are correct, the lack of rigor is still a problem because the user cannot fully trust the results (even under the assumption that the software is bug-free). Now, it might sound like just another case of "99% certainty is enough," but in practice it means having to verify the results independently afterwards, which could be considered unnecessary extra work.

The root of the problem seems to be that a CAS manipulates expressions when it should output theorems instead. In many cases, the expressions simply don't have any rigorous interpretation. For example, variables are usually not introduced explicitly and thus not properly quantified; in the result of an indefinite integral they might even appear out of nowhere. Dealing with undefinedness is another problem.

All of this is inherent in the architecture of computer algebra systems, so it cannot be fixed properly without switching to a different design. The extra 1% of certainty may indeed not justify such a change. But if rigor had been emphasized more from the start, maybe we would have trustworthy CASes now.

I think this line of thought can be generalized. (As a non-mathematician) I can't help but wonder how mathematics would have progressed without the widespread introduction of rigor in the 19th century. I can't really imagine what things would be like if we still didn't have a proper definition of what a function is. So maybe rigor is indeed not strictly necessary in particular cases, but it has shaped mathematical practice in general.

Answer 28

"Sufficient unto the day is the rigor thereof."-E.H.Moore.

There's a lot of discussion over not only the role of rigor in mathematics,but whether or not this is a function of time and point in history.Clearly,what was a rigorous argument to Euler would not pass muster today in a number of cases.

Passing from generalities to specific cases,I think the prototype of statements which were almost universally accepted as true without proof was the early-19th century notion that globally continuous real valued functions had to have at most a finite number of nondifferentiable points.Intuitively,it's easy to see why in a world ruled by handwaving and graph drawing,this would be seen as true. Which is why the counterexamples by Bolzano and Weirstrauss were so conceptually devastating to this way of approaching mathematics.

Edit: I see Jack Lee already mentioned this example "below" in his excellent list of such cases. But to be honest,I don't think his first example is really about rigor so much as a related but more profound change in our understanding how mathematical systems are created. The main reason no one thought non-Euclidean geometries made any sense was because most scientists believed Euclidean geometry was an empirical statement about the fundamental geometry of the universe.Studies of mechanics supported this until the early 20th century;as long as one stays in the "common sense" realm of the world our 5 senses perceive and we do not approach relativistic velocities or distances,this is more or less true. Eddington's eclipse experiments finally vindicated not only Einstein's conceptions,but indirectly,non-Euclidean geometry-which until that point,was greeted with skepticism outside of pure mathematics.

Answer 29

I have the tendency to think that the need for absolute certainty is related to the arborescent structure of mathematic. The mathematics of today rest upon layers of more ancient theories and after piling up 50 layers of concepts, if you are only sure of the previous layers with a confidence of 99%, a disaster is bound to happen and a beautiful branch of the tree to disappear with all the mathematicians living on it. This is rather unique in natural sciences with the exception of extremely sophisticated computer programms but, in mathematics, you will have to fix by yourself an equivalent of 2K bug.

Of course, there are people who are willing to take the risk to see what they have achieved collapse in front of their eyes by working under the assumption that an unproven, but highly plausible, result is true (like Riemann hypothesis or Leopoldt conjecture). In some cases this is actually a good way to be on top of the competition (think of the work of Skinner and Urban on the main conjecture for elliptic curves which rests upon the existence of Galois representations that were not proven to exist before the completion of the proof of the Fundamental Lemma).

Answer 30

Isn't the point that human reason is generally frail altogether, especially when making conclusions by using long serial chains of arguments? So in mathematics where such extended arguments are routine, we want their soundness to be as close to ideal as possible. Of course, even generally accepted proofs are occasionally later seen to be lacking, but to give up proofs as the ideal changes the very nature of mathematics.

I heard that the example of parts of the Italian school of algebraic geometry of the 19th century was an important example of this overextension of intuition.

Furthermore, it is only in the attempt at proof that the real nature of the reasons why a statement is true are finally exposed. So the reformulation and refoundation of algebraic geometry in the 20th century is said to have exposed revolutionary new ways of seeing mathematics in general.

Finally, it is only by proof that the limits of applicability of a theorem are really understood. This comes into play many times in physics, say where some "no-go theorem" is elided because its assumptions are not valid in some new realm.

Answer 31

In response to the request for an example of a statement that was widely but erroneously believed to be true: does Gauss's conjecture that $\pi(n) < \operatorname{li}(n)$ for every integer $n \geq 2$, disproved by Littlewood in 1914, qualify?

Answer 32

I'm not sure it answers the question, but for me, providing (and understanding) a rigorous proof is a way to be sure you understood the deep thing behind what may seems "obvious", for good or wrong reasons: An example I like to give to non-mathematicians, is the Cantor's proof that the cardinality of $\mathbb [0,1]$ is strictly bigger than the one of $\mathbb N$.

The next metaphor is, I can imagine, well known, but that serves the purpose of the question: You start by asking what "two sets have the same cardinality" means. For that, I like to tell the story about the shepherd which can count until, say, 100, but has 1000 sheeps. Every year they go away to pasturelands, and come back. The clever shepherd uses small rocks and associate in his head "one rock = one sheep" to see if he lost, or won, sheeps the year after. Once this notion of bijection is somehow acquired, you ask about the bijection between $\mathbb N$ and $2\mathbb N$, and uses a piece of paper to illustrate. Finally, you go absurd and prove you can't have "one rock = one sheep" for $\mathbb [0,1]$ and $\mathbb N$. Surprisingly, even the more mathematophobics always feel their intuition is wrong and they seem to enjoy this feeling of admitting to themselves it was something deep; infinity is a complicated notion, and they were able to touch that. Maybe a mathematical proof is something which helps to transform a philosophical discussion into a statement nobody accepting elementary axioms can't argue against.

Answer 33

Richard Lipton recently blogged about this question in the context of why a potential proof of $P \neq NP$ would be important. I am probably bastardizing his words, but one of the reasons he gives is that a proof may give new insight and methods of attack to other problems. He cites Wiles' proof of Fermat's Last Theorem as an example of this phenomenom.

Answer 34

Gowers is particularly interested in cases in which a "proof of a statement that was widely believed to be true and was true gave us much more than just a certificate of truth." Reading through the answers, I am struck by a recurring theme. There are many cases where

• something seemed impossible;
• it was impossible;
• a lot of people didn't see the point of proving impossibility;
• the search for an impossibility proof led to the discovery of structures that arguably would not have been discovered otherwise.

Just to list a few examples explicitly:

• It seemed impossible to prove the parallel postulate from the other axioms. The proof of impossibility led directly to non-Euclidean geometries.
• It seemed impossible to "square the circle." The rigorous proof (Lindemann–Weierstrass) forms the foundation of modern transcendental number theory.
• It seemed impossible to solve all polynomial equations using radicals. Rigorous investigation of this impossibility led to what we now call Galois theory.
• (Daniel Moscovich's answer) It seemed impossible to unknot a trefoil. The search for a rigorous proof led to the discovery of all kinds of knot invariants. More generally, topology and geometry are rife with examples of "obviously inequivalent" structures, and the search for rigorous proofs has uncovered all kinds of important invariants.
• It seemed impossible to write down a procedure that would mechanically determine the truth or falsity of an arbitrary mathematical question. The rigorous proofs (incompleteness/undecidability) revealed fundamental limits to human knowledge.

In some cases, you could perhaps quibble with my claim that most if not all people thought these things were impossible, or that people didn't see the point of trying to prove impossibility. In this regard, let me mention the article Why was Wantzel overlooked for a century? The changing importance of an impossibility result by Jesper Lützen. Lützen persuasively argues that Wantzel's proofs of the impossibility of trisecting the angle and duplicating the cube were all but ignored because people just weren't that interested in an impossibility proof of something that everybody already believed, or suspected, was impossible. So even mathematicians are not immune to the tendency to undervalue negative results.

Answer 35

In my experience, the two greatest difficulties in mathematics are:

1. The obvious is not always true.

2. The truth is not always obvious.

Rigour is the essence of mathematics. A rigorous proof provides an explanation of why a particular mathematical statement is true, and, at the same time, takes care of all the "Yes, but what if"s.

Rigour and proof provide the guarantee of correctness and reliability.

Rigour and proof refine our mathematical insights and instincts so that the superficially "obvious" misleads us less frequently.

When I pose the problem "1, 2, 3, x Find x." the initial response is usually a derisory laugh, of disbelief that I am serious, because "the answer is obviously 4". It is easy to demonstrate using practical examples that this statement is, as it stands, nonsense. A rigorous analysis is required.

Answer 36

Michael Atiyah's discussion of the "proof" and it role seems to relevant to be posted here.

Taken from: "Advice to a Young Mathematician in the Princeton Companion to Mathematics." http://press.princeton.edu/chapters/gowers/gowers_VIII_6.pdf This link was provided by "mathphysicist" in answer on another MO question: A single paper everyone should read?

Quotation from M. Atiyah:

"In fact, I believe the search for an explanation, for understanding, is what we should really be aiming for. Proof is simply part of that process, and sometimes its consequence."

"... it is a mistake to identify research in mathematics with the process of producing proofs. In fact, one could say that all the really creative aspects of mathematical research precede the proof stage. To take the metaphor of the “stage” further, you have to start with the idea, develop the plot, write the dialogue, and provide the theatrical instructions. The actual production can be viewed as the “proof”: the implementation of an idea.

In mathematics, ideas and concepts come first, then come questions and problems. At this stage the search for solutions begins, one looks for a method or strategy. Once you have convinced yourself that the problem has been well-posed, and that you have the right tools for the job, you then begin to think hard about the technicalities of the proof."

Answer 37

Allow me to quote part of the introduction of chapter 9 of Lovász: Combinatorial Problems and Exercises.

The chromatic number is the most famous graphical invariant; its fame being mainly due to the Four Color Conjecture, which asserts that all planar graphs are 4-colorable. This has been the most challenging problem of combinatorics for over a century and has contributed more to the development of the field than any other single problem. A computer-assisted proof of this conjecture was finally found by Appel and Haken in 1977. Although today chromatic number attracts attention for several other reasons too, many of which arise from applied mathematical fields such as operations research, attempts to find a simpler proof of the Four Color Theorem is still an important motivation of its investigation.

So here it's not so much the proof but the search for a proof that has given something extra over just believing the theorem. Does that still count as an answer to this question?

Answer 38

This question already has an amazing amount of great answers. Being a physicist with very limited knowledge of mathematics, I certainly cannot expect to contribute something of equal value, however after reading through the answers I'm missing a certain aspect. The missing issue is: "What is a proof"? The "rigorization" process of Calculus with $\epsilon$/$\delta$ proofs was mentioned as a major progress. However I saw nothing doubting that the current foundations of mathematics might still be "improved", where it is of course an interesting question what that would mean. A while ago, when I was trying to find the answer to above question, I came across the following story:

Fields medalist Vladimir Voevodsky, when working on rather scary stuff that I do not begin to understand (motivic cohomology...), came across many cases where ground-breaking published papers with proofs contained errors, which would only be noticed years after. Sometimes this would render a lot of later work worthless. Errors were not noticed, although people were studying the papers in seminars. When it happened to him, it genuinely scared him and got him to start working on computer-assisted proof, as well as an axiomatization of mathematics that goes beyond ZFC, called "Univalent Foundations". It has several conceptual advantages but apparantly is very unknown (and not complete yet!). The aim is to produce a framework where computer verified proofs are a practical option (contraty to now, where such proofs are extremely cumbersome and impossible to use on a regular basis in publications).

Returning closer to the question at hand: A quote from his motivation (a lengthy article containing many references and statements relevant to this question) for the foundational work of his is:

A technical argument by a trusted author, which is hard to check and looks similar to arguments known to be correct, is hardly ever checked in detail.

Granted, the results I'm describing here are far removed from practical applications, such as would interest an engineer. However it isn't entirelly unreasonable that fields like theoretical physics would eventually come into contact with parts of mathematics which seem similar in terms of obscurity. The verdict is: Even in the 21st century, having a published result with a proof is not enough to be sure it's true, not even in the most renowned journals and by the most trusted authors. To achieve that, one would need to push the limits of rigour even further, thus acknowledging its importance (where of course it is up to debate whether and how we want to do this).

By the way, Voevodsky has a recorded talk of his where he considers the question "what if the current foundations of mathematics are inconsistent?", where he tries to imagine how one would work in a framework known to be inconsistent, rather than one where one hopes but can never prove that verything is fine

Answer 39

The fundamental lemma is an example that most believed and on whose truth several results depend. According to Wikipedia, Professor Langlands has said

... it is not the fundamental lemma as such that is critical for the analytic theory of automorphic forms and for the arithmetic of Shimura varieties; it is the stabilized (or stable) trace formula, the reduction of the trace formula itself to the stable trace formula for a group and its endoscopic groups, and the stabilization of the Grothendieck–Lefschetz formula. None of these are possible without the fundamental lemma and its absence rendered progress almost impossible for more than twenty years.

and Michael Harris has also commented that it was a "bottleneck limiting progress on a host of arithmetic questions."

Answer 40

Mathematics wasn't that rigorous before N. Bourbaki: in the Italian school of Algebraic Geometry of the beginning of the XXth century the standard procedure was Theorem, Proof, Counterexample. Also at the time of Cauchy some theorems in analysis began like "If the reader doesn't choose a specially bad function we have..."

The use of rigour in analysis, which Cauchy began, avoided that by being able to explain what was a "good function" in each case: analytic, $C^{\infty}$, being able to do term-by-term derivation in its expansions series...