In France, most math/CS grad students hear the following story:
On the day of his defense, Jay, a PhD candidate in pure Maths, presents his results, a series of highly non-trivial theorems which brilliantly establish universal properties for objects of a set $\cal B$, defined as the subset of $\cal A$ satisfying an additional property $P$. Both the characterization of $\cal A$ and the definition of $P$ are quite non-trivial, and belong to different subfields of maths (e.g., topology and number-theory). After acknowledging the originality and technicity of the proofs, the jury moves on to the questions, and ask:
Could you give us an example of an object in $\cal B$?
Jay starts building the object on the board, fails, try again, fails again and the defense moves on to avoid any further embarassment (at that moment, Jay is silently weeping in a corner of the room).
Unfortunately, during the following deliberation, the jury manages to show a contradiction in the axioms defining $\cal A$ and $P$! All the theorems were legit, as they all started with universal quantifications ($\forall x\in {\cal B}$...). Jay does not get his PhD, and moves on to make buckets of money doing applied maths for the finance industry (happy end version).
Now, I suspect this story is a urban legend, as I already heard two distinct versions of it (please comment with your local variant if you have one), but I was wondering if there were instances of controversies in the history of Mathematics that may have resulted from a similar situation. The key ingredients in this story are:
- Set $\cal A$ and property $P$ such that ${\cal B} := \{x\in {\cal A}\mid P(x)\}=\varnothing$ non-trivially (think "Millenium prize non-trivial");
- A second property $P'$;
- Instances of ${\cal B}$ being difficult to construct or handle in a human lifetime (e.g. in number theory);
- A first result by author X, showing that $ P'(x), \forall x \in B$, using somewhat obscure techniques.
- A second result by author Y, showing that $\neg P'(x), \forall x \in B$, using even obscurer techniques.
- An undignified academic squabble involving X and Y (+ their respective proponents) pointing out typos and mistakes in the respective proofs.
- Once the war is over, both sides reluctantly accept the other's result, or simply lose interest (and forget to collect the 1M prize for settling the ${\cal B} =\varnothing$ question).
Is anyone aware of any instance of such a scenario in the history of Maths? If not, how did communities (note that several communities may be simultaneously involved in this game) manage to prevent such a situation to occur?
Edit: As funny as they may be, this question is not about "epic failures" of grad students (or about trivial/useless results in Maths), rather about the history/sociology of Mathematics.
The canonical example is probably the reluctance of the Pythagoreans to accept the existence of irrational numbers.
The story goes that when a student named Hippasus presented his defense of the irrationality of $\sqrt{2}$, his advisory committee listened carefully, conferred with the gods, and subsequently had him drowned at sea in an apparent show of disapproval.
By contrast, graduate school today is much more civil. They merely drown your soul and financial future.