Well, the "epistemic hand wringing" has a very serious point, which is that it spells big trouble for philosophy of science, which is (among other things) concerned with the "problem of demarcation". Put simply: how do you tell what is and isn't "science"?

Hume's formulation of the problem of induction actually pointed to two things: one, the "logical" problem of induction, was simply the standard critique of inductive generalization as an unsupportable method of inference. The other, the "psychological" problem of induction, claimed that inductive generalization was nonetheless how human beings actually think, and so we're screwed. But in the late nineteenth century, and then again in the mid-twentieth century, you get two thinkers who challenge this.

Charles Saunders Peirce took a view of science and of human thought which was not based on induction: in Peirce's view, the "average" person simply believes something until it causes some sort of conflict (at which point, Peirce claimed, other methods of justifying belief would be developed in response, leading to a chain which eventually ends up at the scientific method). Peirce also didn't view science as being able to give ultimately true answers to questions (thus sidestepping the need to justify inductive generalization, even if it does end up as part of scientific method); rather, science can get better and better approximations to the truth over time (as more observational data becomes available and new theories are proposed to explain the data), but will never actually arrive at "the truth" (and we wouldn't be able to tell even if it did). In other words, Peirce's view of human knowledge and of science is based around fallibility.

Karl Popper, immersed in the world of German-speaking philosophy, came to very similar conclusions much later on, and proposed a solution to the problem of induction in the following form. First, he accepted in its entirety the logical problem of induction, but declared that it need not cause problems for science, because science need not be inductive in nature. Second, he proposed that the psychological problem of induction was a fiction: he asserted that the way people actually reason is far closer to fallibilism (just like Peirce), and framed it in common-sense terms as a process of trial and error.

Popper built a theory of demarcation around flipping the problem of induction on its head: it is true, he happily conceded, that no number of observed instances is sufficient to establish a generalization to all instances (including those as-yet-unobserved, or unobservable). But this turns out not to be such a big deal, because all it takes is one observed counterexample to demonstrate that a theory is false. Thus we can still proceed scientifically, but instead of speaking of theories which are "verified" by observation, we speak of theories which survive attempts at falsification.

Popper came to the same sort of conclusion as Peirce regarding the "truth" of scientific theories: he felt that there was no useful distinction between, say, a "hypothesis" or "conjecture", and a "theory", because none of them can be said to be true -- the best that can be said is that they have not yet been proven false. And so he developed a system in which "science" consists of those theories which can be subjected to falsification: a theory is scientific only if there is some test which, if it gives a negative result, will be taken as showing that the theory is false.

He talked occasionally of this system as applying a form of Darwinian selection to theories: there is never a final "best" or "true" theory, but there is a selection process at work which eliminates false theories through observation of counterexamples. The theories which stay with us and form the basis of everyday working science, then, are not those which are "true" but are merely those which, so far, have survived that selection process. And in judging between competing theories, Popper preferred the theory which was boldest in terms of possible falsification: theories which make assertions that are easy to test for falsity, he claimed, tend also to be those which -- if they survive such tests -- provide the broadest and most useful basis for further scientific work.

Of course, both Peirce and Popper are terribly unfashionable in philosophy of science these days. Peirce is reviled for having the gall to claim that science advances toward truth over time even if it never arrives at truth (a position which every good postmodern Kuhnian disciple will dogmatically reject). And Popper is often viewed as a sort of semantic charlatan whose attempt to shift from verification to falsification was merely a critique (albeit a devastating one) of communism, Freudian psychology and logical positivism.