Say you’ve spent years working on a doctoral thesis in a scientific discipline that can change significantly almost overnight: genetics, high energy physics, astronomy, etc. What happens if someone makes a breakthrough that completely nullifies your work? You’ve been doing research that outlines how x is possible, and someone demonstrates that x can’t be true because he/she has proven y to be true. Do you have to scrap years of work and start over again?
It’s extremely unlikely that a question in a controversial area of science will be decisively and unequivocally settled in the usual time span of thesis research.
In any case, thesis research ideally is supposed to be evaluated on its own merits. If you have solid evidence to demonstrate the case you want to make, the thesis will normally be accepted; if not, not. (However, a thesis that contradicts a well-established theory is likely to have to face a much higher standard of proof than one that upholds the conventional wisdom.)
Even groundbreaking discoveries don’t come out of nowhere. You keep up with the field, and what other folks are doing. This is not hard, in a highly specialized field such as the ones you describe: There are probably only a few thousand people in the world working in the field, less in a subfield that you’re working on, and most of them go to the same conferences. If someone else is working on a topic closely related to yours, you’ll at the very least talk with them about it at conferences, and you might even be collaborating with them. If it looks like they’re getting results relevant to yours, you either incorporate those results, or determine that they’re incorrect.
What’s more likely to happen is that someone will produce a paper or a thesis that closely mirrors the work that you are doing. Happens a lot. In which case, you have only a couple of choices.
a. Find a way to make your contribution distinct from the competing paper – find some subproblem or area that hasn’t been completely analyzed or understood.
b. Drop back and punt (e.g. find another thesis topic).
I’m wondering if the OP’s situation also includes being “scooped”, which does happen more often than you’d expect and the possibility keeps grad students terrified awake at night. In that situation, yeah, you’re pretty much up sh!t creek.
Directly on point, though, and in science rather than math (my field): if you’re showing how x is possible and someone else claims to show that x is impossible, it’s far from insurmountable. You still did the research and can get the Ph.D. on the strength of the fact that your work seems to show x is possible, unless the other person sees something you definitely overlooked. Basically, one of your defense board will probably bring up the other work and you can, well, defend your point of view the best you can. Who knows, you might turn out to be right in the end.
To elaborate on what Mathochist and Finagle said, I think the closest you’re going to come to the OP’s scenario is to get scooped.
My degree is in molecular microbiology, a very fast moving field. Unlike in the days of old, my dissertation was only red tape. My degree was really based on my publications. We were up against a competing lab and did get scooped, meaning they published the same (actually, very similar) results first. That meant that good journals were no longer interested in our work; it was old news. As Finagle says, one option is to try to make your work unique. That meant finding that weird tangent and exploring it.
A true disagreement of results (a situation I now find myslef in) is actually easier to publish. If your work is strong and you can adequately address why you think the previous work is wrong, the pub should go through.
Now, if you continually get scooped (which can happen in a really hot area), I think a dissertation committee would let you graduate without pubs. It has happened. However, even with a great dissertation, you are at a serious disadvantage with no pubs. It would be better in the long run to stay in school a little longer and get published.
I know of a case where the following happened: A student worked for a year or so on a mathematical proof that his thesis advisor and everybody else who had thought about the question agreed was a valid subject for research. He finally finished the proof and turned in a draft of the thesis to his advisor, who initially agreed that he had indeed proved the theorem. Two weeks later the advisor called him and told him that he had found a problem with the proof. For several weeks the two of them thought about the proof and convinced themselves that he hadn’t proved the theorem. Indeed, they discovered a counterexample which showed that the theorem wasn’t true. A few more weeks of thought convinced the student that there was no easy way to modify the theorem to make it true and it would take him another year to produce an acceptable thesis. At that point the student gave up on a math Ph.D. He took his math master’s and got a computer science master’s and then got a job in computers.