In Treatise 1.4.5, Hume presents an argument for the following claim:
N An object may exist but be nowhere
In support of this Hume claims that our sense of place or locality is rooted entirely in our senses of sight and touch; other senses convey a sense of place only by association with these. In light of this he argues (T. 126.96.36.199):
1. Whatever has a place must either be extended or be a mathematical point.
2. Whatever is extended has a figure or shape.
3. A desire has no figure or shape
4. Mathematical points can be combined and disposed so as to form a volume.
5. A desire is not so disposable.
6. Therefore a desire has no place.
So, despite the apparently counterintuitive nature of N (which was explicitly denied by Samuel Clarke in his correspondence with Leibniz), Hume says of it that "this is not only possible, but...the greatest part of beings do and must exist after this manner" (T 188.8.131.52). We can say that an object is nowhere when its parts are not so related to each other as to form a figure or volume, and the whole is not related to other things so as to be distant or contiguous. Hume puts all our perceptions in this category (with the exception of those of sight and touch, which he thinks are extended). Indeed, not only are most of our perceptions (and their objects) nowhere, they are such that they could not possibly be in a place. If this is so, however, most of our perceptions (and their objects) cannot be 'locally conjoined' to matter, i.e., they cannot be united to matter in a place, because any relation requires that both be similar enough to serve as the ground of relation.
But we do often try to attribute local conjunction to the things Hume says can't be locally conjoined to anything. Suppose, says Hume, that we have a fig at one end of a table and an olive at the other end. We are naturally inclined to say that the taste of the fig is at the fig's end of the table, and the taste of the olive is at the other end. Hume thinks that the reason we do this is mere prejudice: we associate the fig's taste with the fig, knowing that the fig-body can cause a fig-taste to follow it in time. Since the fig-body and the fig-taste are related by causation and temporal succession, we assume that they are also related by local conjunction. Indeed, Hume's view is that this is almost impossible to avoid: when we are faced with an incomplete union the imagination has a natural tendency to fill in whatever relation is necessary to make the union complete; only then can we be satisfied. On reflection, Hume thinks, we would recognize that the result we've arrived at is clearly unintelligible and incoherent. For where in the fig-body is the taste located? There is no extended thing or set of points within the fig that constitute the fig-taste; therefore the locality we give the fig-taste is that of the fig-body and every part of it. In other words, we assume that the fig-taste is located in the whole fig-body, and located wholly in every part of the fig-body (totum in toto et totum in qualibet parte, as the scholastics would say). Hume thinks that this is an obvious contradiction: it is equivalent to saying that the fig-taste is both in a place and not there at the same time.
So, Hume argues, we are faced with a trilemma. Either
(a) some beings exist without any place;
(b) all beings (including things like desires and passions) are extended and figured;
(c) some things are so incorporated with extended objects that they are wholly in the whole and wholly in every part.
Since Hume has argued against (b) and (c), the only option left is (a), which is equivalent to N.
Such is only one of the more interesting and unusual arguments in Treatise 1.4.5, which is universally recognized by Hume scholars as one of the most bizarre and difficult to interpret sections of Hume's entire corpus.