What are the most fantastic conjectures that were ALMOST true?

[andytoh]

When I say almost true, I mean that many mathematicians believed the conjecture was true, and for the longest time they all struggled really hard to prove it and the (long) proofs almost made it through, except for one line that wasn't quite right. They all tried to fix that one line, and realized they couldn't. Then someone eventually came up with a really sophisticated counterexample to prove that the conjecture was, despite all the excitement and effort, in fact false.

 

[Santa1]

Only one I can think of is the continuum hypothesis and Cantor, I think it partially drove him mad. Then after his death it was proved independent of ZFC (tell me if I'm really wrong here), not false though, just independent.

edit: then of course some types of equations that Euler and others stated had no solution etc. When we today with modern computers have found huge solutions to these type of equations.

 

[andytoh]

That example qualifies I suppose. It was the #1 unsolved problem back 1900. And it Cantor did try to prove it till his death.

Many topologists tried to prove that a countably infinite product of R is normal in the box topology. It's been quite a few decades of effort by many mathematicans, and Mary Ellen Rudin proved in the early 70's that it is true if the continuum hypothesis is assumed. If a counterexample comes up without the condition of the continuum hypothesis, all that effort of trying to prove it will have been a real frustration. But if the conjecture goes busted I guess it is not a huge loss, because the conjecture is not all that important I suppose.

 

[maze]

Everyone thought the parallel postulate was a theorem for thousands of years until the 1800's when other geometries were found. There have been a lot of false "proofs" of it over the ages.

 

[morphism]

One relatively recent example that comes to mind is related to the (still open) invariant subspace problem, which once asked if every operator on a Banach space has a nontrivial invariant subspace.

The answer is "yes" in the finite-dimensional setting (for complex spaces of dimension greater than 1 and real spaces of dimension greater than 2) thanks to existence of eigenvectors. In infinite dimensions things get a little more difficult. So let's stick close to home and consider compact operators only; these are operators which 'behave' very much like finite-dimensional ones. The spectral theory for compact operators tells us that if a compact operator has a nonzero element in its spectrum, then it has an eigenvector. Thus we need only consider compact operators with zero spectrum, i.e. quasinilpotent compact operators.

That these have nontrivial invariant subspaces was first proved by von Neumann in the 1930s for Hilbert spaces. Later, in the 1950s, Aronszajn proved it for reflexive Banach spaces, and a few years after that he and Smith proved it for general Banach spaces. Following this, a succession of mathematicians (Arveson, Feldman, Robinson, Bernstein, Lomonosov, ...) proved many generalizations; one of them (the Arveson-Feldman theorem) stated that a general (i.e. not necessarily compact) quasinilpotent operator (on a Hilbert space) has a nontrivial invariant subspace if the uniformly closed algebra generated by it and the identity operator contains a nonzero compact operator. This is a very neat and surprising result. Around this time a lot of people were starting to believe that general quasinilpotent operators have nontrivial invariant subspaces. This belief was strengthened when Hilden gave his very elementary proof for the case of quasinilpotent compact operators (on general Banach spaces). A lot of people were enthusiastically trying to adapt his methods to prove the general case, but all this came crashing down when Read gave (in 1997) an example of a quasinilpotent operator (on a Banach space) with no nontrivial invariant subspaces.

 

[andytoh]

Interesting. I suppose for every 10 mathematicians trying to prove a very believable conjecture, there should be 1 trying to find a counterexample. Without these counterexample experts, we would be forever trying to prove something that is not true!

 

[HallsofIvy]

Actually, a pretty standard method of starting on a proof of a difficult conjecture is to look for counterexamples. If the conjecture is true, there won't be any, of course, but looking helps to see why there are no counterexamples which itself helps to see why it is true.

 

[n_bourbaki]

A famous assumption, made by implicitly by Euler, was that rings of integers were UFDs. Gauss showed that this was not true, though people still kept assuming it: Lame's proof was based on the (incorrect) assumption that some cyclotomic extensions were UFDs.

 

[gel]

Another example is the http://en.wikipedia.org/wiki/Jordan-Schönflies_theorem" ). So it isn't true that embedding a sphere into 3d space splits it into two regions homeomorphic to the inside and outside of a sphere.