List of published false theorems
From Wikipedia, the free encyclopedia
The historical record of mathematics and mathematicians is excellent overall. Few published theorems have later been discovered to be false. However, a number of well-known exceptions are listed below.
- In 1803, Gian Francesco Malfatti proved 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 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, "continuous" must be replaced with "uniformly continuous".[1] 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.
- 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.[2][3]
Additionally, there have been many cases in which a published proof was incorrect, although the theorem in question eventually turned out to be true anyway. Some of the most notable are listed below.
- In 1879, Alfred Kempe published a purported proof of the four-color map theorem that stood 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.[4]
[edit] References
- ^ Porter, Roy (2003). The Cambridge History of Science. Cambridge University Press, 476. ISBN 0521571995.
- ^ Boerger, Egon; Erich Grädel, Yuri Gurevich (1997). The Classical Decision Problem. Springer, 188. ISBN 3540423249.
- ^ Goldfarb, Warren (1986). in Solomon Feferman (ed.): Kurt Gödel: Collected Works, vol I 1. Oxford University Press, 229–231. ISBN 0195039645.
- ^ Thomas L. Saaty and Paul C. Kainen (1986). The Four-Color Problem: Assaults and Conquest. Dover Publications. ISBN 9780486650920.
