Falsifiability is usually defined as "the extent to which a scientific theory can be proven wrong". Does this mean that falsifiability is basically the extent to which a scientific theory is testable?
While "yes" would be a valuable answer in its own right, I'm going to deliberately introduce a pedantic basis for distinguishing falsifiability from testability that will hopefully provide a modern motivation for both.
The relationship between falsifiability and testability is more complex than it first seems. Sec. 7.1 here summarises a relevant 1984 theorem. The gist is as follows: given $\delta,\,\varepsilon\in(0,\,1)$ and a class of predictive hypotheses (e.g. all the parameter choices for a physical theory), we seek a large enough dataset to achieve:
Probably approximately correct (PAC learning): With probability $\ge1-\delta$, all as yet unfalsified hypotheses in the class are right in a proportion $\ge1-\varepsilon$ of their future predictions.
"Large enough" is proportional to the VC dimension, the largest finite dataset size that cannot falsify the entire hypothesis class (or, if no finite dataset can falsify it all, said dimension is infinite and no finite dataset size achieves PAC learning). We can't do better: we can't delete "probably" (i.e. set $\delta=0$) or "approximately (i.e. set $\varepsilon=0$) without making the dataset requirement infinite, even for a finite VC dimension.
Roughly speaking, the dataset size is logarithmic in $\delta$ and inversely proportional to $\varepsilon$, and for practically interesting values of $\delta,\,\varepsilon$ is feasible as long as we use physical theories of small enough VC dimension (which, again, is doable). So, in short, science can work well.
But that's just the gist. I've skipped over some terms and conditions, and will now focus on just one of them: the original choice of hypothesis class. This is basically the class of hypotheses we're willing to entertain. Whenever "a theory has been falsified", this really means we realized that hypothesis class was too restrictive. So here's the falsifiability-testability relationship:
When a hypothesis class is falsifiable, the data we gather can test its individual hypotheses, thereby allowing parameter estimation, future predictions, statistical analysis etc. The merit of choosing a falsifiable hypothesis class is that we can perform such testing.
Physics addendum: the VC dimension of a theory depends in practice on the regime. For example, the energy scale of a particle accelerator (as well as what it can measure) determines the VC dimension of e.g. the Standard Model, which affects how much data needs to be gathered. Similarly, non-renormalizable theories can be useful in regimes where only finitely many of their parameters matter to prediction, so the VC dimension is effectively reduced to some finite value. Ongoing problems with quantizing gravity can be described in terms of us struggling to make the VC dimension not only finite, but low enough for practical purposes, all while ensuring at least one correction to classical gravity is measurably large.
Philosophy addendum: these ideas, albeit less quantified, are somewhat anticipated in pre-1984 philosophical works. For example, Popper argued (modulo terminology) for working with theories of not only finite but minimal VC-dimension, and that we must in practice respond to falsification by switching to a brand new falsifiable hypothesis class. It is also an old observation in the philosophy of science that, while occasionally we must try not to throw out the baby with the bathwater when awkward evidence comes along, we should as a rule of thumb try not to increase VC-dimension with too many ad hoc hypotheses.
Although they sound similar, I would not say that falsifiability is testability. The whole point of falsifiability, at least as it was originally introduced by Popper, was to avoid what he believed was a logical fallacy that arose from verificationism (where a theory is meaningful if it can verified). This is because given the relation $H\rightarrow O$ (where H is some hypothesis/theory and O is an observation that it predicts), observing O does not logically guarantee H, however, observing not O does logically guarantee not H.
The reason I bring this up is because a scientific theory could be called testable, even if it was not falsifiable. Pseudo-sciences of the past, like Marxism (not its political interpretation... people used to think this was a science) or psychoanalysis were testable in the sense that there were observations that aligned with their predictions, however, they could never be falsified because whenever there was an unexpected observation, proponents would come up with a new explanation that aligned with their theory.
A good scientific theory is falsifiable, such as what happened to Newton's theory of gravity being falsified by things like Mercury's orbit, and eventually being replaced by General Relativity. Ultimately, I guess it depends on how you interpret testability, but note that falsifiability literally means falsifiability.
Not just tested. But more clearly speaking, tested and negated. Something that cannot be negated isnt falsifiable. Many things can be tested to the point of affirmation and reinforcement. Something more is needed. If science isnt trying to debunk its own conclusions, its doing it wrong.
It happens frequently that this condition of falsifiability is violated, particularly in fringe science. Whether its particle physics or cosmology, you frequently find counter evidence that, instead of debunking the model, merely results in "scientists" declaring that the model is more complicated than previously thought.
I have observed that instead of falsifying the status quo, counter evidence is frequently disregarded as "exceptional" or "potentially false" or a "statistical outlier". Convolution is added to the theory to rationalize the evidence.
In general, the practice of disregarding statistical outliers that dont jive well with the theory is questionable all by itself. That happens in all sciences though.
Cosmology is an easy example of some of the more extreme practices. For example, the act of taking one or two hundred years worth of local stellar observations... most of the earlier observations were made with imprecise instruments, mind you... a finite quantity of recent data... and then extrapolating it 13 billion years into the past. Highly dubious. They are attempting to use statistical science, but no pure statistician would ever condone this because it violates some of the basic rules of the math. Interpolation is bad enough, but to extrapolate orders of magnitude beyond the range of observation is ridiculous from a mathematically rigorous perspective. The math doesnt justify the practice. Does anyone question this though? Mathematicians and statisticians like myself do - and we are more qualified in the math than the cosmologist is - yet we are called anti-science nutjobs (or even theists) for it. There is a certain religiosity to blindly following "the science", and letting "scientists" come to amuck conclusions based on poorly implemented mathematics whose foundations they lack a theoretical understanding of.
Here is another example. Cosmologists couldnt figure out why they observed more gravity in our galaxy than the matter they saw... so they invented the concept of dark matter, without substantive evidence of its existence. Not necessarily unreasonable by itself... but juxtapose that with the invention of dark energy, to justify why there appears to be less gravity in the universe than what the observed matter would justify. Is that a contradiction? Why not take the more simplistic approach of questioning the model of gravity, or the methods by which gravity and matter are measured, than to invent two opposing theories without evidence to justify two opposing observations?
How do we measure distances on these grand scales? Oh, I can explain the math and the science of it as much as the next guy, but here's the question: How have these results been verified? No one took out a tape measure and proved that parallax accurately predicts the 4.3 light year distance to alpha centauri, did they? Its just something we presumed, and ran with. After all, it works on small local scales; so why not on big remote ones? Other cosmologists and astronomers "verified" the same calculations and observations, sure, but not the same actual distance measurement (which no one has ever actually done). Is it irrational of me to point that out? Or is it irrational of the cosmologist not to?
The paradox of NGC-7603 hasnt been resolved yet, has it? Two parts of a single group of galaxies measure distances spanning the universe, according to modern measurement methods. But visual evidence proves them directly connected. Last I heard this was an unanswered question. Has it been answered yet? Or as I suspect is more likely, merely brushed under the rug. And what about that star whose age was measured older than the universe itself?
To some extent the particle physicists is just as guilty. Heisenberg uncertainty. Quantum physics is a fascinating intellectual exercise but does the science really support it? I havent studied it enough to say for sure, but it seems to me that arguing particles literally exist in superposition is absurd... not from a biased mindset of newtonian reasoning, but from the perspective of someone who's studied the concept of randomness and recognizes that ignorance of absolute truth doesnt justify a relativistic or superposition of multiple simultaneous truths. We cant precisely measure particles. A certain uncertainty is inherent. Probability can be used to represent that fact. Does that necessarily mean that a particle is literally here AND there at the same time? I dont think so. The number of marbles in a jar is predetermined. The probability distribution of peoples guesses only models human uncertainty, not physical reality. In my humble opinion Heisenberg uncertainty is simply bad reasoning, the presumption of the equivocation of different interpretations of probability. If you run the marble jar count experiment multiple times you might find variability in the actual number. But the marble is still newtonian, and the quantity in the jar is not in flux. I have no doubts that particles behave with Heisenberg uncertainty: pv>h implies a limitation on our ability measure. I do have doubts, however (as no one has proved to me otherwise), that uncertainty in the predictability of outcome somehow legitimizes superposition and other quantum gobbledygook theory.
Biological science is similar. If you supported "spontaneous generation" of life, youre a midlevel hack and theist. We have microscopes now. Okay, but where did the very first life ever come from? Why, abiogenesis, of course. Its just "spontaneous generation" rebranded. Old ideas become new again. As long as you can put an atheistic spin on it, it just as acceptable today as it ever was then. Evidence? You dont need that. I remember hearing about early archeologists resisting the idea of dinosaurs for close-minded reasons akin to what we see now among fringe science.
Ah, evidence... the thing that separates a theory from a mere hypothesis. We have evidence of genetic variation, of mutation, of natural selection. But not of evolution. We have some, limited (in time and in space) observations of galactic expansion... but no evidence of a big bang that started it all. The latter are just speculative, natural limiting extremes of observation that themselves arent qualified to have the label "theory". To tack "theory" onto the end of the NAMES "big bang" and "evolution" are poor attempts of branding these hypotheses with more credibility than they actually have. After all, how do you falsify these things? If you cant, it isnt science.
Of course, you cant disprove truth. But thats where we are horribly wrong. Science is not and has never been about "the truth". The push to believe otherwise is a rather recent and purely cultural and political one, to deify the academic and the scientist, to secularize politics. No, in fact science - good science - is in a constant process of refinement and proving its own prior theories false. No theory is ever "true". Its only better than the previous one. Good theories are supplanted by better ones. More accurate, more predictive. And I dont mean truer ones; I mean ones that are more predictive. Subtle distinction. All this for one purpose. Its been about finding predictable, fruitful theory that help improve decision making, giving rise to innovation, etc. The proof is in the pudding, as they say. Cosmology has no practical, innovative value and do not improve lives. Neither does evolution. Biology and electricity, however, are theories that do.
I bring all this up to point out that modern fringe science is like a religion. Methods and theories are never questioned. They are simply over-complicated. Scientists that question interpretation of the facts are typically laughed out of the professional community, careers ended because those in the in-group resent the threat of their preconceptions that a few pose.
Falsifiability is not happening. Only verifiability. So... what should science do and what scientists actually do are not necessarily on the level.
To prove my point, watch as the down votes and negativity and calls for censorship to roll in, while no one directly confronts my specific points.