Hume seems unhappy with the idea that there are "perceptions" that are "dependent, interrupted, and different" and also "objects" that are "continued, identical [through time], and [mind-]independent." (A Treatise of Human Nature, 1.4.2, 52)
To solve the problem of "double existence," we can have recourse to my understanding of a "thing-in-itself." From my book:
When I say, (a) "This desk is brown," is it not the case that "brownness" is not "really" "in" the desk? It is just a perception, conditioned by our sensory organs and spiritual machinery. Brownness is a subjective experience. What has it to do with the actual desk?
In epistemology, we would have a truth-bearer (that thing which has a truth value), i.e., an (ideal) proposition; and a truth-maker (that thing which makes the truth-bearer true), such as the (real) state of affairs of the desk's being brown, to which the proposition corresponds. If (a) is to be true (and since in epistemology we do not indulge in justifications, I will certainly not lower myself to prove that to my reader), then there had better be a real brown desk out there. The brown desk is the thing-in-itself, and that is the end of it.
In philosophy of empirical science, on the other hand, all we have are perceptions. All we see and hear and so on are signs -- of something, perhaps, but who knows and who cares of what? We use these signs in our lives to pursue happiness. We manipulate them in order to cause them to conform to our desires. Simply put, we entertain ourselves. Mises would seem to agree: "We may define the external world as the totality of all those things and events that determine the feasibility or unfeasibility, the success or failure, of human action."
There are no longer truth-bearer and makers, only usefulness-bearers and makers.
Notice how the situation is reversed: the usefulness-bearer (that item which is useful) is (really) "out there," as represented by signs. The usefulness-maker is our (ideal) notions on how exactly the usefulness-bearer is useful.
Whether there is anything "behind" the perceptions is now irrelevant. The material world can be cut off with Occam's razor. "Solid reality," according to science, is a useful illusion.
Let me now put this another way. Consider a syllogism:
(Major) A dresser is used to store clothing.
(Minor) There is a brown wooden dresser to the left of me.
(Conclusion) I have a place to store clothing.
There are two aspects to this modus ponens. One is validity; the other, soundness.
Regarding the former, assuming the premises are true, does the conclusion follow? In this case, yes, it does. But what does it mean to say that "there is a brown wooden dresser to the left of me" is true? Only that the dresser is a really existing object -- not a perception! -- indeed, continued, somewhat identical to itself, and independent, of a certain kind in a certain location.
Regarding the latter, is the minor premise true? Well, it at first glance seems so, but I won't vouch for it. For example, physics teaches that what appears solid to the senses, on the atomic level is in fact mostly empty space. The sign that whatever-it-is emits that is delivered to us may well fail to "resemble" (Hume's term) the "real object." What is the "real" dresser apart from how to appears? I have no idea and don't care to speculate.
So, we can see that the double existence is in fact a useful device and a reasonable supposition. If Hume denies that there are objects, straightforwardly described by their appearances, then he will lose the capacity to judge the validity of inferences. If it is a hopeless case even to assume a premise such as our minor to be true, then we can't arrive to a conclusion, and that would be pretty sad, because I'd be forced to throw my clothes on the floor.