A negative hypothesis is just a testable statement framed as a negative, along the lines of humans can't fly. How can we ever prove that? We could certainly disprove it by producing even one verified instance of a human that flies. But to positively prove that humans can't fly, other than specific instances of specific humans failing to fly on specific days, defies logic. The best you can do is to show evidence that supports it (humans hitting the ground in undeniably un-flight-like fashion), and then to conditionally accept the hypothesis, or regard it as indistinguishable from true. For the time being. Until we someday get better at distinguishing.
The perfect hypothesis in the perfect experiment would either be fully confirmed or fully disconfirmed by the experiment. But this rarely happens. Usually the best we can do is to either disconfirm a falsifiable hypothesis, or to confirm a confirmatory hypothesis.
To falsify a falsifiable hypothesis means that the hypothesis is definitely disproved. It is conclusively rejected, never to rise again. If we fail to disprove a falsifiable hypothesis, then it might be true, but it might still be false as well. It just means we were unable to reject it this time on the basis of this evidence from this experiment. In the light of lots of other experiments all of which likewise failed to falsify the hypothesis, we might be forced to accept the hypothesis as indistinguishable from true around the time we run out of ways of testing the hypothesis.
The Natural Sciences tend to use falsifiable hypotheses because it is the most reliable way of finding out objective facts. Only something that is virtually indistinguishable from true can withstand a determined, persistent onslaught of experiments seeking to disconfirm it. It is just too easy to find evidence that confirms (erroneously) what you want to be true, because this is a basic operational bias of the human brain.
Falsifying is also a very efficient way of clearing out the myriad possible explanations that are wrong in order to zero in on the one that is right. And it is generally easier to falsify or prove wrong a positive statement about what is, rather than a negative statement about what isn't. All humans can fly is readily disproved; whereas no humans can fly is a lot harder to disprove, and these are not the only two possibilities.
You can see how easy it is to frame a hypothesis as either positive or negative. Just be careful about false binaries. If one explanation fails, it isn't automatically the competing one. If all humans can fly fails, it does not automatically follow that no humans can fly, or that all humans can't fly. It just means you weren't able to do it that time. Perhaps humans can sometimes fly, with the right sort of suit and a good running start. NO capes, though. No capes!
Applied Science (e.g. engineering) tends to use confirmatory or provable hypotheses because it is the more efficient way to invent things. "Cameras are possible, because look at this working prototype I just made." Hypothesis confirmed in one experiment. Or you could frame the hypothesis as cameras are impossible and disconfirm that with the same single experiment. But most of the time x is impossible is not the preferred way to frame a hypothesis in Applied Science. The first ninety-nine camera prototypes that didn't work could be taken (erroneously) as evidence supporting this poorly framed hypothesis. The injunction you can't prove a negative is more a rule against framing hypotheses as negatives - a rule that is also frequently broken when appropriate to do so.
I still haven't answered the question - can you ever prove a negative hypothesis of the type XYZ does not exist? Is it ever possible to confirm this kind of hypothesis? Many top scientists will say, on principle, "No." But what if I gave you an example of Science positively proving the non-existence of something? And to make it harder, what if the thing already had scientific evidence for it and the beginnings of a consensus on its existence?
Our story starts in France in 1903. Respected and accomplished physicist Prosper-René Blondlot (1849 - 1930) needed an explanation for some weird shit his experiments were doing, and he also really needed to keep up with the other scientists of the age who were discovering new rays left and right. X-rays. Alpha Rays. Gamma Rays. Beta Rays. Is there no end to the vast variety of rays to be discovered in Nature? Well, actually, that was . . . um . . . that's pretty much all of them.
In any case, Blondlot wishfully assumed an experimental glitch he was seeing must be caused by one of these new Rays everyone was constantly discovering, which he named N-Rays. He set up a new cockamamie experiment that was guaranteed to not detect any other previously discovered kind of ray, and immediately found what he was looking for. Once he reported this amazing discovery, other scientists started reporting confirmations of his discovery in their own labs. Soon, N-Rays were being reported emanating from a wide range of materials with the oddly specific exceptions of green wood and rock salt.
N-Ray Mania swept France, and in just the first half of 1904 over 50 scientific papers were published on the subject (compared to just 3 about X-Rays). Eventually there would be over 300 papers on N-Rays. Over 120 other scientists confirmed the existence and properties of N-Rays. A serious dispute arose over who was the first to discover N-Rays emanating from the human body. The French Academy of Sciences awarded Blondlot a prize of Fr 50,000 (almost $600,000 in today's money). He had international fame, was a national hero in France, and he had his eye on a Nobel Prize.
Only problem was, N-Rays don't exist. They never did.
And Science proved it with one simple experiment. Well, a meta-experiment actually. A wickedly clever and devious experiment conducted on the way N-Ray Science was being done.
So how do you confirm the hypothesis N-Rays Don't Exist? Well, if N-Rays are a mere phantasm in the minds of scientists (and there was some reason to suspect this alternative explanation) then crucially disrupting the experiment without the knowledge of the scientist should have no effect on the "positive" results of experiment.
The fact that many eminent German and British scientists were entirely unable to replicate Blondlot's discoveries, and that the detection required the involved natural senses of the researchers, made hallucination one possible explanation for the phenomenon of N-Rays. This hypothesis is testable in an experiment conducted on N-Ray Science rather than on N-Rays themselves. If N-Ray Science is not affected by disabling the apparatus in a way unknown to and undetectable by the researcher, then this should only be possible if N-Ray Science is basically hallucinations (or outright fraud) on the part of the researchers. An objective phenomenon is supremely sensitive to technical faults in the apparatus; a subjective experience requires only a belief in the apparatus.
Unfair, you say? Unethical? Well they started it. Those researchers inserted themselves into the experiment, so I say it is fair game to conduct experiments upon them without their knowledge. Today most science utilizes electronic or otherwise mechanized detection, measurement, and data collection instrumentation in order to prevent the biases of the faulty human brain from contaminating the data. In cases where this is not possible, the gold standard is a double-blind study, in which neither the researchers nor the subjects (where relevant) know what they are doing or often what the experiment is even about. Only the designer of the study knows, and she is explicitly forbidden from participating in the data collection or analysis, and definitely forbidden from winking and gesturing in the researchers' direction.
In this case, American physicist Robert W. Wood traveled to Blondlot's laboratory in Nancy, France in September 1904. His perceived impartiality as neither British nor German was essential to this project. Whilst Blondlot happily demonstrated the "detection" of N-Rays, Wood removed a crucial part of the experimental apparatus without Blondlot's knowledge, which according to N-Ray theory should have rendered detection of N-Rays utterly impossible. But Blondlot continued counting and recording N-Rays on his phosphor screen, oblivious to the sabotage. Wood meanwhile was unable to observe any N-Rays, either before or after disabling the apparatus. "Not those big, obvious flashes of light," Blondlot explained, "Look for the much, much fainter ones." Yeah right - the ones that look just like what you see when you close your eyes.
Since N-Rays were detectable only by certain people and detectable whether the apparatus was operational or not, since N-Rays were never able to be recorded photographically in spite of numerous determined efforts, and since N-Rays had neither theoretical explanation nor theoretical reason to exist, there was only one conclusion to make, and that conclusion did not have to be conditional, tentative, or subject to qualification. Science had confirmed the hypothesis that N-Rays Do Not Exist.
So it is possible to prove a negative if you can do the following:
- Demonstrate the absence of any solid evidence for the thing. Solid evidence means evidence which bears no other explanation and is repeatable. Absence of evidence is very much evidence of absence, in cases where the hypothesis demands such evidence to be found precisely where it is not.
- Propose and test alternative explanations for all the evidence that does exist, such as it is. In our story this test consisted of showing that N-Ray detection was unaffected by disabling the apparatus, a state of affairs not concordant with objective phenomena, but perfectly concordant with subjective ones.
- Demonstrate that the non-existence of the thing is entirely concordant with all other available data.
Things that definitely do not exist can be proven not to exist if these three hurdles can be cleared. It is not necessary to be agnostic about every potentiality and every absurd claim on the basis that "you can't prove a negative." Because quite often, you can.
Klotz, I.M. "The N-Ray Affair," Scientific American, May 1980
Wood, R.W. "The N-Rays," Nature, Sept 1904, 70 (1822): 530–531