# Examples of falsified (or currently open) longstanding conjectures leading to large bodies of incorrect results.

In general, the way that modern mathematical research is conducted isn't the way that many would assume is the ideal method of research. That is, mathematics is not the linear progression of foundational results leading to new results which one might expect (although the actual framework is very close). Instead, one often makes generally accepted assumptions which one expects will be proven at a later date as a technical lemma. So, in some cases, mathematicians may base large bodies of results on an essentially unproven conjecture (a so-called conditional proof).

This process is fine and is often done to 'get to the bigger results' without having to wade around in a pool of technical proofs which everyone generally accepts as folklore. It is also worth noting that such work based on false conjectures is not entirely useless and might be easily specialised to cases where the conjecture is true, or easily fixed by working around the false conjecture. The methods used in such a proof may also shed insight on the area, even if the conclusions prove false.

However, this approach has its flaws in that, until a conjecture is made theorem, there is always the possibility that it may be proven false and so essentially make any 'theorems' proven based on the conjecture redundant. I'm particularly interested in such cases that may have occurred in history, or which have the possibility of occurring in the future and having a large effect on existing literature. I would like to place the emphasis of this question on the specific effect that such falsifications of conjecture can have, as there are already many questions on unexpected counterexamples on the site.

Some example:

• I imagine there were several pieces of work assuming the Pólya conjecture before it was eventually proved false.

• There are certainly whole areas of number theory based on the assumption that the Riemann hypothesis will eventually be proven true.

• When the triangulation conjecture was proven false in dimension 4, there were many proofs based on existence of triangulations that could then only be stated for 'triangulable simplicial complexes'.

• Similar to the Riemman hypothesis, there are countless papers based on the assumption that $P=NP$ (and indeed its converse).

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I like that someone asked this question, so +1. My feeling is that the answer will be that there are no really big examples of wrong conjectures leading to lots of useless conditional results. The reason is that lots of math is really an empirical science, and important false conjectures are found out pretty fast. –  S123 Mar 31 '13 at 17:51
Yes, there's something to be said for the accuracy of the intuition of the mathematical community. I imagine a conjecture is often only used as a basis for further results if the heuristics of the result suggest the conjecture is true and if the unlikelihood of the conjecture being false is seen as too great. Human intuition is definitely fallible though, so I expect there to be some nice examples. –  Daniel Rust Mar 31 '13 at 17:55
Well, I am not referring just to intuition, but also to lots of calculations, which are often not published. –  S123 Mar 31 '13 at 18:00
I don't really see "large bodies of results based on unproven conjectures"...Yes, here and there are some results based on this and assumption, but the overwhelmingly huge bulk of mathematics done, as far as I am aware, is based on solid, well proven results. Of course, this all may depend on one's understanding of "large bodies of results"... –  DonAntonio Mar 31 '13 at 23:08

Weierstrass' famous function disproved a lot of results.

Dunham1 reports that:

The renowned Andre-Marie Ampere had presented a proof that continuous functions are differentiable in general, and calculus textbooks throughout the first half of the nineteenth century endorsed this position... [The Weierstrass function] not only refuted Ampere's "theorem" but drove the last nail into the coffin of geometric intuition as a trustworthy foundation for the calculus.

Dunham lists several other examples (Cauchy disproving Lagrange's formulation of calculus in terms of Taylor series, Dirichlet finding a function which can't be written as a Fourier series) but I'm not enough of a historian to say if these had a "large effect" on results.

Edit: you might find this post interesting. E.g. this excerpt taken from an article by Erica Klarreich July 20, 2009 on the Simons Foundation website.

In the 1970s and 1980s, mathematicians discovered that framed manifolds with Arf-Kervaire invariant equal to 1 - oddball manifolds not surgically related to a sphere - do in fact exist in the first five dimensions on the list: 2, 6, 14, 30 and 62. A clear pattern seemed to be established, and many mathematicians felt confident that this pattern would continue in higher dimensions…Researchers developed what Ravenel calls an entire cosmology of conjectures based on the assumption that manifolds with Arf-Kervaire invariant equal to 1 exist in all dimensions of the form 2n−2. Many called the notion that these manifolds might not exist the Doomsday Hypothesis, as it would wipe out a large body of research. Earlier this year, Victor Snaith of the University of Sheffield in England published a book about this research, warning in the preface, …this might turn out to be a book about things which do not exist. Just weeks after Snaith’s book appeared, Hopkins announced on April 21 that Snaith’s worst fears were justified: that Hopkins, Hill and Ravenel had proved that no manifolds of Arf-Kervaire invariant equal to 1 exist in dimensions 254 and higher. Dimension 126, the only one not covered by their analysis, remains a mystery. The new finding is convincing, even though it overturns many mathematicians’ expectations, Hovey said.

1. Dunham, William. The calculus gallery: Masterpieces from Newton to Lebesgue. Princeton University Press, 2005.
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This second example is truly amazing. A great example of what this question was made for. –  Daniel Rust Oct 6 '13 at 12:54