You want proof that you're a philosophical heavyweight? Inspire a group of the most serious, no-nonsense, mathematically- and scientifically-inclined philosophers, a group whose members' IQs are surpassed only by their level of disdain for the history of philosophy. That's what Hume did—150 years after his death, at that.
Who are the 20th-century sourpuss smarty-pants we're talking about? Why, the logical positivists, of course. They never did give Hume full credit for all he did for them, but they sure did buy Hume's general empiricist principle that all genuine knowledge is based on experience.
The logical positivists also specifically built on Hume's account of meaning to arrive at the notion of "verifiability," the idea with which they are most often associated. According to the logical positivists' verifiability theory of meaning, for a sentence to be meaningful, or for it to even make sense, there must be observable conditions under which it is true and observable conditions under which it is false. If there are no such empirically observable conditions, then that sentence is simply nonsense.
Sounds pretty impressive, huh? But the basic notion is already contained in Hume, in particular in his distinction between ideas and impressions. In his Enquiry Concerning Human Understanding, Hume says explicitly if a given idea isn't derived from an impression gained through experience, then that idea is meaningless. Sounds familiar, right?
Now, do the logical positivists admit that they derived their basic idea from Hume? No way. But then everyone wants to be original, don't they?
Carnap is the best known and most influential of the logical positivists, and he sure is into the idea of verificationism. He even wrote a book about it. That should be enough to express his eternal indebtedness to David Hume.
But Carnap dives still deeper still. His other key idea involves the distinction between "analytic" and "synthetic," which is a way of dividing up the two basic sorts of sentences in any language. Synthetic statements are claims that are true or false depending on the facts, or depending on the way the world is. They are, in other words, subject to empirical verification and falsification.
Analytic claims include the statements you find in logic and mathematics; they are true simply by virtue of the meaning of the terms involved. To know that 2+2=4, for example, you don't have to do experiments; the truth of this equation follows once you understand what each of the symbols means.
So, here is your final exam, Shmoopers. What does this remind you of? That's right—Hume. The analytic/synthetic distinction is just like Hume's distinction between relations of ideas and matters of fact. You make us so proud.
Another logical positivist, another closet Humean. Hempel applies the principle of verificationism to psychology and formulates the position of logical behaviorism. We're not saying Hume came up with that notion, too, but it is a natural application of his views.
Popper wasn't exactly a logical positivist, though he is often classified as one. His views are pretty similar to theirs, anyway: the positivists emphasize that the aim of science is to attempt to verify its theories; Popper, though, says that since you can never completely verify a theory, science actually is seeking to falsify its claims. Translation: Popper thinks you can't really prove a scientific theory; all you can do is prove theories false—like when an experiment fails and disproves the theory (source).
From the sidelines, though, we say verification, falsification—who cares? The basic idea is the same, and that basic idea is really just an extension of Hume's critique of causality. Hume showed that we could never establish once and for all that any given causal connection must hold. Popper then uses this point as a way of characterizing science in general. We can't definitively establish the truth of a scientific law, but we can establish its falsity.
Best of all, Popper had the graciousness to credit Hume with the inspiration for his insight: "I approached the problem of induction through Hume. Hume, I felt, was perfectly right in pointing out that induction cannot be logically justified" (source.)
For that, Popper receives an honorary lifetime membership in Positivistically Proud to Be a Humean Being.