Before turning to our beliefs in causation and the continued and independent existence of objects, which together form the heart of Hume's argument in Book I of the Treatise, let me spend a few moments on two subordinate matters. The first is his treatment of the nature of abstract ideas, which shows in miniature the strategy Hume will adopt when he comes to the big questions. The second is his very odd, idiosyncratic treatment of mathematics, which reminds me of nothing so much as Nelson Goodman's theory of individuals in his first book, The Structure of Appearance. Hume takes his question about abstract ideas from Bishop Berkeley, who asked whether [in Hume's words] abstract or general ideas "be general or particular in the mind's conception of them." Berkeley said they are particular, and Hume agrees. His argument here has exactly the same structure as his arguments later on: First, a negative or sceptical argument, refuting what previous philosophers had said about the subject; Then a positive, constructive, psychological explanation for the fact that we believe what reason cannot demonstrate.
The negative argument here is quite simple: Every impression has some determinate degree or other of whatever properties it exhibits -- some particular shade of color, some intensity of sound, some spatial shape, some taste, some smell, some numerosity. Since every idea is a copy of a preceding impression, or a combination of copies of such impressions, every idea also must have some determinate degree or other of whatever properties it represents. We cannot possibly have an idea of Man in General that is not the idea of some particular man, nor can we have the idea of Color in General that is not the idea of some particular shade of color. There could be no impression of which such an idea is the copy, and, to repeat once again, all ideas are copies of preceding impressions. So much for the negative argument. You can see what a powerful weapon the Copy Theory of Ideas is.
But we most certainly think that we have the idea of Man in General. We are quite capable of forming sentences employing the idea, of engaging in reasonings that use the idea, of communicating our thoughts concerning the idea with others. What on earth are we doing? Hume's answer is, I think, quite imaginative. The mind has the ability to notice resemblances among disparate things. One Frenchman looks and sounds much like another [at least to a Scotsman], red looks much like orange, the sound of an oboe is not unlike that of an English Horn. It is, Hume claims, simply a fact of human nature, not more deeply explicable [at least by Hume] that we find resemblances among different objects of perception or experience. And it is also a fact of human nature that the mind tends to associate together resembling things. This fact of association is, like the noting of resemblances, just a feature of the mind for which no further explanation can be offered. It is certainly not necessary that the mind possess this tendency. So far as anything logic can tell us, we might be so constructed that when we were presented with instances of resembling things, we developed no tendency or habit at all of associating them -- which is to say, of thinking of the second when we encounter the first. But we do have this tendency, and from it arises what philosophers have mistakenly called abstract ideas.
The structure of a general or abstract idea is this: It is a word, such as the word "Horse," that through repeated use has become associated with a number of particular resembling ideas, for example of this horse, of that horse, of the other horse. When we use the word, we call to mind both a particular idea of some one horse, and also the habit, created by association, of thinking of all the associated ideas when we hear the word. So, as Hume says, while all ideas are "particular in their nature," some "are general in their representation." Indeed, it is a striking and convenient, but inexplicable, characteristic of the mind [of some minds more than others, by the bye] that if someone makes an assertion, with the aid of a general term, that is incompatible with the characteristics of some of the particular ideas associated with that term, the mind will call to mind just those other particular ideas whose particular characteristics contradict the assertion. For example, if I assert that all Texans are admirable, having in mind Jim Hightower, the mind will immediately recall the ideas of George W. Bush and Rick Perry, rather than the idea of Molly Ivens.
The important thing to see here is that abstract ideas have a complex structure that consists of a number of particular ideas, a set of resemblances, and a mental habit of association. On Hume's analysis, it would make no sense to ask whether abstract or general terms, such as "humanity", or "justice," or "color," or "government" can exist separately from our thinking them. The mental habit of association is an essential and constitutive part of the general idea.
Hume's treatment of space and of mathematics, particularly of Geometry, is truly quirky, and has had, so far as I know, no significant influence on subsequent discussions of the subject. But there is a certain mad consistency to his position. Briefly, Hume argues that our experience of space is of what later psychologists would call minimally discriminable regions. Space is clearly not infinitely divisible, he says, because as we divide each region of space into smaller and smaller subdivisions, there comes a point beyond which we cannot discern a sub-part of the region we are contemplating. Hence, our idea of space must be of a finitely divisible quantity. Furthermore, "[t]he infinite divisibility of space implies that of time, as is evident from the nature of motion. If the latter, therefore, be impossible, the former must be equally so." Book I, Part II, Section ii.] So, he concludes, the idea of space is "nothing but the idea of visible or tangible points distributed in a certain order."
I have never thought that Book I, Part II was Hume's finest hour. On to the really important matters of Parts II [causation] and IV [the continued and independent existence of objects.]