Mathematicians failing to solve problems despite having all methods required [closed]

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On this wikipedia page, there is the following quote by Anil Nerode:

Being attached to a speculation is not a good guide to research planning. One should always try both directions of every problem. Prejudice has caused famous mathematicians to fail to solve famous problems whose solution was opposite to their expectations, even though they had developed all the methods required.

What are some good examples of this?

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If nothing else, this should have a big-list tag. I really don't know if this question is suited to MO. – David Roberts Oct 3 2011 at 4:37

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I think this question is very broad. I mean this happens all the time in mathematics. Perhaps the question should be made more specific. – Martin Brandenburg Oct 3 2011 at 7:20
It is interesting that Gerry Myerson's answer and John Stillwell's are morally related. Non Euclidean geometries were at one point an imputes to model theory. – Lunasaurus Rex Oct 3 2011 at 9:16

4 Answers

I suppose plenty of people could have developed non-Euclidean geometry, had they not been so intent on proving that no such thing existed.

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As Girolamo Saccheri, the author of Euclides ab omni nevo vindicatus'' – Giuseppe Oct 3 2011 at 6:26

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Gödel 's failure to discover unsolvability of the decision problems for predicate logic and Peano arithmetic may be an example. Gödel had all necessary tools: arithmetization, diagonalization, and an equation calculus for defining computable functions.

However, as he admitted later, he was misled by his incompleteness proof into thinking that there could not be an absolute definition of computable function -- he expected one could always form new computable functions by diagonalization. It was only when Turing came up with the definition via Turing machines that Gödel realized he was mistaken.

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Bott-Tu has a comment in the introduction about Poincaré not discovering the computability of de Rham cohomology through combinatorial data associated to a finite good cover. I might as well give you the original:

"To digress for a moment, it is difficult not to speculate what kept poincaré from discovering this argument forty years earlier. One has the feeling that he already knew every step along the way. After all, the homotopy invariance of the de Rham theory for $\mathbb{R}^n$ is known as the Poincaré lemma! Nevertheless, he veered sharply from this point of view, thinking predominantly in terms of triangulations, and so he in fact was never able to prove either the computability of de Rham or the invariance of the combinatorial definition. Quite possibly the explanation is that the whole $C^\infty$ point of view and, in particular, the partitions of unity were alien to him and his contemporaries, steeped as they were in real or complex analytic questions."

EDIT: I guess this is more of a failure to observe the fact, rather than it being opposite to expectation. Nevertheless, it's interesting and perhaps close enough.

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The history of the first version of Poincaré's essay submitted to the competition sponsored by King Oscar II of Sweden, could be representative of this situation but at the same time of the capacity of overcome previous errors.

The problem of the stability of a planetary system was central from the dawn of newtonian mechanics. A father of Analytical mechanics such as Dirichlet thought to have proved the stability for the n-body problem, but he died suddenly before to write it.

The prize of King Oscar was aimed to obtain such a proof of the stability and infact in the first version of his essay, Poincaré claimed the stability of the restricted 3-body problem. This essay won the prize, but just after the publication of the paper in the Acta he realized the presence of a serious error for the presence of homoclinic orbits. Consequently the published issues were recalled and the second version of the essay was printed.

The existence of a first version of Poincaré's essay has been discovered only in 1994 by June Green-Barrow. The second version is known as the starting point of the qualitative geometric methods in mechanics.

For more information a possible source is Diacu, F., The solution of the n-body problem, Math. Intelligencer 18 (3) 66-70, 1996.

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Were there any useful ideas in the first version that were omitted from the second? (It is an unfortunate habit of modern mathematicians to, upon notification of a mistake in a paper, remove the whole paper along with its correct and useful parts.) – darij grinberg Oct 3 2011 at 18:07
Dear Darij Grinberg I have edited the answer to complete the information, I hope to have been more satisfactory. – Giuseppe Oct 3 2011 at 20:43
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