List of incomplete proofs
This page lists notable examples of incomplete published mathematical proofs. Most of these were accepted as correct for several years but later discovered to contain gaps. There are both examples where a complete proof was later found and where the alleged result turned out to be false.
Lecat (1935) is a list over a hundred pages long of errors made by mathematicians.
This section lists examples of proofs that were published and accepted as complete before a gap or error was found in them. It does not include any of the many incomplete attempted solutions by amateurs of famous problems such as Fermat's last theorem or the squaring of the circle. It also does not include unpublished preprints that were withdrawn because an error was found before publication.
The examples are arranged roughly in order of the publication date of the incomplete proof. Several of the examples on the list were taken from answers to questions on the MathOverflow site, listed in the external links below. The examples use the following symbols:
- Result is correct and was later rigorously proved.
- Result is wrong as stated, but a modified version was later rigorously proved.
- Status of the result is unclear
- Result is wrong
- Euclid's Elements. Euclid's proofs are essentially correct, but strictly speaking sometimes contain gaps because he tacitly uses some unstated assumptions, such as the existence of intersection points. In 1899 Hilbert gave a complete set of (second order) axioms for Euclidean geometry, called Hilbert's axioms, and between 1926 and 1959 Tarski gave some complete sets of first order axioms, called Tarski's axioms.
- Infinitesimals. In the 18th century there was widespread use of infinitesimals in calculus, though these were not really well defined. Calculus was put on firm foundations in the 19th century, and Robinson put infinitesimals in a rigorous basis with the introduction of nonstandard analysis in the 20th century.
- Gian Francesco Malfatti claimed to prove that a certain arrangement of three circles would cover the maximum possible area inside a right triangle. However, to do so he made certain unwarranted assumptions about the configuration of the circles. It was shown in 1930 that circles in a different configuration could cover a greater area, and in 1967 that Malfatti's configuration was never optimal. See Malfatti circles. In 1803,
- André-Marie Ampère claimed to prove that a continuous function is differentiable at most points, but in 1872 Weierstrass gave an example of a continuous function that was not differentiable anywhere: The Weierstrass function. In 1806
- Uniform convergence. In his Cours d'analyse of 1821, Cauchy "proved" that if a sum of continuous functions converges pointwise, then its limit is also continuous. However, Abel observed three years later that this is not the case. For the conclusion to hold, "pointwise convergence" must be replaced with "uniform convergence". There are many counterexamples. For example, a Fourier series of sine and cosine functions, all continuous, may converge to a discontinuous function such as a step function.
- Intersection theory. In 1848 Steiner claimed that the number of conics tangent to 5 given conics is 7776 = 65, but later realized this was wrong. The correct number 3264 was found by Berner in 1865 and by de Jonquieres around 1859 and by Chasles in 1864 using his theory of characteristics. However these results, like many others in classical intersection theory, do not seem to have been given complete proofs until the work of Fulton and Macpherson in about 1978.
- Dirichlet's principle. This was used by Riemann in 1851, but Weierstrass found a counterexample to one version of this principle in 1870, and Hilbert stated and proved a correct version in 1900.
- Alfred Kempe published a purported proof of the four-color map theorem, whose validity as a proof was accepted for eleven years before it was refuted. The proof did, however, suffice to show the weaker five-color map theorem. The four-color theorem was eventually proved in 1976. In 1879,
- Jordan curve theorem. There has been some controversy about whether Jordan's original proof of this in 1887 contains gaps. Oswald Veblen in 1905 claimed that Jordan's proof is incomplete, but in 2007 Hales said that the gaps are minor and that Jordan's proof is essentially complete.
- Vahlen (1891) published a purported example of an algebraic curve in 3-dimensional projective space that could not be defined as the zeros of 3 polynomials, but in 1941 Perron found 3 equations defining Vahlen's curve. In 1961 Kneser showed that any algebraic curve in projective 3-space can be given as the zeros of 3 polynomials.
- Mathieu group M24 does not exist, though in 1900 he pointed out that his proof was wrong. In 1898 Miller published a paper incorrectly claiming to prove that the
- Lebesgue tried to prove the (correct) result that a function implicitly defined by a Baire function is Baire, but his proof incorrectly assumed that the projection of a Borel set is Borel. Suslin pointed out the error and was inspired by it to define analytic sets as continuous images of Borel sets. In 1905
- Dehn's lemma. Dehn published an attempted proof in 1910, but Kneser found a gap in 1929. It was finally proven in 1956 by Christos Papakyriakopoulos.
- Italian school of algebraic geometry. Most gaps in proofs are caused either by a subtle technical oversight, or before the 20th century by a lack of precise definitions. A major exception to this is the Italian school of algebraic geometry in the first half of the 20th century, where lower standards of rigor gradually became acceptable. The result was that there are many papers in this area where the proofs are incomplete, or the theorems are not stated precisely. This list contains a few representative examples, where the result was not just incompletely proved but also hopelessly wrong.
- Perko pair, a pair of knots listed as distinct in tables for many years until Perko discovered in 1974 that they were the same. This gives a counterexample to a theorem claimed by Little in 1900 that the writhe of a reduced knot diagram is an invariant.
- Hilbert's sixteenth problem. Henri Dulac published a partial solution to this problem in 1923, but in about 1980 Écalle and Ilyashenko independently found a serious gap, and fixed it in about 1991.
- Hilbert's twenty-first problem. In 1908 Plemelj claimed to have shown the existence a Fuchsian differential equations with any given monodromy group, but in 1989 Bolibruch discovered a counterexample.
- Kurt Gödel proved in 1932 that the truth of a certain class of sentences of first-order arithmetic, known in the literature as [∃*∀2∃*, all, (0)], was decidable. That is, there was a method for deciding correctly whether any statement of that form was true. In the final sentence of that paper, he asserted that the same proof would work for the decidability of the larger class [∃*∀2∃*, all, (0)]=, which also includes formulas that contain an equality predicate. However, in the mid-1960s, Stål Aanderaa showed that Gödel's proof would not go through for the larger class, and in 1982 Warren Goldfarb showed that validity of formulas from the larger class was in fact undecidable.
- Grunwald–Wang theorem. Wilhelm Grunwald published an incorrect proof in 1933 of an incorrect theorem, and Whaples later published another incorrect proof. Shianghao Wang found a counterexample in 1948 and published a corrected version of the theorem in 1950.
- Mumford (1968) showed that this is false for surfaces of positive geometric genus. In 1934 Severi claimed that the space of rational equivalence classes of cycles on an algebraic surface is finite-dimensional, but
- Littlewood–Richardson rule. Robinson published an incomplete proof in 1938, though the gaps were not noticed for many years. The first complete proofs were given by Schützenberger in 1977 and Thomas in 1974.
- Jacobian conjecture. Keller asked this as a question in 1939, and in the next few years there were several published incomplete proofs, including 3 by B. Segre, but Vitushkin found gaps in many of them. The Jacobian conjecture is (as of 2010) an open problem, and more incomplete proofs are regularly announced. Hyman Bass, Edwin H. Connell, and David Wright (1982) discuss the errors in some of these incomplete proofs.
- Severi (1946) claimed that that a degree-n surface in 3-dimensional projective space has at most (n+2
3)−4 nodes, B. Segre pointed out that this was wrong; for example, for degree 6 the maximum number of nodes is 65, achieved by the Barth sextic, which is more than the maximum of 52 claimed by Severi.
One of many examples from algebraic geometry in the first half of the 20th century:
- Rokhlin invariant. Rokhlin (1951) incorrectly claimed that the third stable stem of the homotopy groups of spheres is of order 12. In 1952 he discovered his error: it is in fact cyclic of order 24. The difference is crucial as it results in the existence of the Rokhlin invariant, a fundamental tool in the theory of 3- and 4-dimensional manifolds.
- Class numbers of imaginary quadratic fields. In 1952 Heegner published a solution to this problem. His paper was not accepted as a complete proof as it contained a gap, and the first complete proofs were given in about 1967 by Baker and Stark. In 1969 Stark showed how to fill the gap in Heegner's paper.
- Hilbert's sixteenth problem. In the 1950s, Evgenii Landis and Ivan Petrovsky published a purported solution, but it was shown wrong in the early 1960s.
- Nielsen realization problem. Kravetz claimed to solve this in 1959 by first showing that Teichmuller space is negatively curved, but in 1974 Masur showed that it is not negatively curved. The Nielsen realization problem was finally solved in 1980 by Kerskhoff.
- Yamabe problem. Yamabe claimed a solution in 1960, but Trudinger discovered a gap in 1968, and a complete proof was not given until 1984.
- derived functor of the inverse limit functor under certain general conditions. However, over forty years later, Amnon Neeman constructed a counterexample. In 1961, Jan-Erik Roos published an incorrect theorem about the vanishing of the first
- Mordell conjecture over function fields. Manin published a proof in 1963, but Coleman (1990) found and corrected a gap in the proof.
- Schur multiplier of the Mathieu group M22 is particularly notorious as it was miscalculated more than once: Burgoyne & Fong (1966) first claimed it had order 3, then in a 1968 correction claimed it had order 6; its order is in fact (currently believed to be) 12. This caused an error in the title of Janko's paper A new finite simple group of order 86,775,570,046,077,562,880 which possesses M24 and the full covering group of M22 as subgroup on J4: it does not have the full covering group as a subgroup, as the full covering group is larger than was realized at the time. The
- Complex structures on the 6-sphere. In 1969 Alfred Adler published a paper in the American Journal of Mathematics claiming that the 6-sphere has no complex structure. His argument was incomplete, and this is (as of 2011) still a major open problem.
- Burnside's problem. In his proof he assumed the existence of a set of parameters satisfying some inequalities, but Adian pointed out that these inequalities were inconsistent. Novikov and Adian had previously found a correct solution around 1968. In 1973 Britton published a 282 page attempted solution of
- In 1975, Leitzel, Madan, and Queen incorrectly claimed that there are only 7 function fields over finite fields with genus >0 and class number 1, but in 2013 Stirpe found another; there are in fact exactly 8.
- Wilhelm Klingenberg published a proof that smooth compact manifolds without boundary have infinitely many closed geodesics. His proof was controversial, and there is currently (as of 2011) no consensus on whether his proof is complete. Closed geodesics. In 1978
- Classification of finite simple groups. In 1983, Gorenstein announced that the proof of the classification had been completed, but he had been misinformed about the status of the proof of classification of quasithin groups, which had a serious gap in it. A complete proof for this case was published by Aschbacher and Smith in 2004.
- Kepler conjecture. Hsiang published an incomplete proof of this in 1993. Hales later published a proof (currently believed to be correct) depending on some very long computer calculations.
- Busemann–Petty problem. Zhang published two papers in the Annals of Mathematics in 1994 and 1999, in the first of which he proved that the Busemann–Petty problem in R 4 has a negative solution, and in the second of which he proved that it has a positive solution.
- Algebraic stacks. The book Laumon & Moret-Bailly (2000) on algebraic stacks mistakenly claimed that morphisms of algebraic stacks induce morphisms of lisse-étale topoi. The results depending on this were repaired by Olsson (2007).
- Matroid bundles. In 2003 Biss published a paper in the Annals of Mathematics claiming to show that matroid bundles are equivalent to real vector bundles, but in 2009 published a correction pointing out a serious gap in the proof.
- Porter, Roy (2003). The Cambridge History of Science. Cambridge University Press. p. 476. ISBN 0-521-57199-5.
- Thomas L. Saaty and Paul C. Kainen (1986). The Four-Color Problem: Assaults and Conquest. Dover Publications. ISBN 978-0-486-65092-0.
- Yulij Ilyashenko (2002). "Centennial History of Hilbert's 16th problem" (PDF). Bulletin of the AMS 39 (3): 301–354. doi:10.1090/s0273-0979-02-00946-1.
- Boerger, Egon; Grädel, Erich; Gurevich, Yuri (1997). The Classical Decision Problem. Springer. p. 188. ISBN 3-540-42324-9.
- Goldfarb, Warren (1986). Feferman, Solomon, ed. Kurt Gödel: Collected Works 1. Oxford University Press. pp. 229–231. ISBN 0-19-503964-5.
- Roos, Jan-Erik (1961). "Sur les foncteurs dérivés de lim. Applications.". C. R. Acad. Sci. Paris 252: 3702–3704. MR 0132091.
- Neeman, Amnon (2002). "A counterexample to a 1961 "theorem" in homological algebra (with an appendix by P. Deligne)". Inv. Math. 148 (2): 397–420. doi:10.1007/s002220100197. MR 1906154.
- Bass, Hyman; Connell, Edwin H.; Wright, David (1982), "The Jacobian conjecture: reduction of degree and formal expansion of the inverse", American Mathematical Society. Bulletin. New Series 7 (2): 287–330, doi:10.1090/S0273-0979-1982-15032-7, ISBN 978-1-982150-32-7, MR 663785
- Burgoyne, N.; Fong, Paul (1966), "The Schur multipliers of the Mathieu groups", Nagoya Mathematical Journal 27: 733–745, ISSN 0027-7630, MR 0197542
- Coleman, Robert F. (1990), "Manin's proof of the Mordell conjecture over function fields", L'Enseignement Mathématique. Revue Internationale. IIe Série 36 (3): 393–427, ISSN 0013-8584, MR 1096426
- Laumon, Gérard; Moret-Bailly, Laurent (2000), Champs algébriques, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics] 39, Berlin, New York: Springer-Verlag, ISBN 978-3-540-65761-3, MR 1771927
- Lecat, Maurice (1935), Erreurs de mathématiciens des origines à nos jours, Bruxelles - Louvain: Librairie Castaigne - Ém. Desbarax
- Mumford, David (1968), "Rational equivalence of 0-cycles on surfaces", Journal of Mathematics of Kyoto University 9: 195–204, ISSN 0023-608X, MR 0249428
- Olsson, Martin (2007), "Sheaves on Artin stacks", Journal für die reine und angewandte Mathematik 603: 55–112, doi:10.1515/CRELLE.2007.012, ISSN 0075-4102, MR 2312554
- Rohlin, V. A. (1951), "Classification of mappings of an (n+3)-dimensional sphere into an n-dimensional one", Doklady Akad. Nauk SSSR (N.S.) 81: 19–22, MR 0046043
- Severi, Francesco (1946), "Sul massimo numero di nodi di una superficie di dato ordine dello spazio ordinario o di una forma di un operspazio", Annali di Matematica Pura ed Applicata. Serie Quarta 25: 1–41, doi:10.1007/bf02418077, ISSN 0003-4622
- Vahlen, K. T. (1891), "Bemerkung zur vollställndigen Darstellung algebraischer Raumkurven", J. Reine Angew. Math. 108: 346–347
- Ilya Nikokoshev, Most interesting mathematics mistake?
- Kevin Buzzard what mistakes did the Italian algebraic geometers actually make?
- Will Jagy, Widely accepted mathematical results that were later shown wrong?
- John Stillwell, What are some correct results discovered with incorrect (or no) proofs?