The first argument, however, is obscure and dealt with rather sketchily. It is that 'all the points of time and space, in which we can suppose any object to begin to exist, are in themselves equal; and unless there be some cause, which is peculiar to one time and place, and which by that means determines and fixes the existence, it must remain in eternal suspense.’ Hume replies ‘Is there any more difficulty in supposing the time and place to be fix'd without a cause, that to suppose the existence to be determin'd in that matter? The first question that occurs on this subject is always, whether the object shall exist or not: the next, when and where it shall begin to exist. If the removal of a cause be intuitively absurd in the one case, it must be so in the other: And if that absurdity be not clear without a proof in one case, it will equally require one in the other. The absurdity, then, of the one supposition can never be a proof of that of the other, since they are both upon the same footing, and must stand or fall by the same reasoning.’
Why does he say 'The first question is whether an object shall exist'? This seems to consider the matter from the point of view of a creator, which suggests that Hume is consciously in Leibniz country; for Leibniz argues for the identity of indiscernibles on the ground that God must have a reason for putting A here and now, and B there and then, which there could not be unless something distinguished A from B. Yet Hume doesn't consider this. The questions are not easy to reformulate and seem better left out.--Elizabeth Anscombe , 'Whatever Has a Beginning of Existence Must Have a Cause': Hume's Argument Exposed,” Analysis, Vol. 34, No. 5 (Apr., 1974), p. 147.
The obscure argument quoted from Hume by Anscombe is Treatise 188.8.131.52, which starts follows: "Accordingly we shall find upon examination, that every demonstration, which has been produc’d for the necessity of a cause, is fallacious and sophistical. All the points of time and place,17 say some philosophers, in which we can suppose any object to begin to exist, are in themselves equal" (etc). Treatise, T 184.108.40.206 Anscombe pays no attention to Hume's footnote, which mentions Hobbes, and suggests he is in "Leibniz country." Leibniz country is a nice place where one's passport allows one entry to all possibility.
Before I get to Leibniz country, the distinguished editors, the late David and Mary Norton, comment, "Hume appears to paraphrase Hobbes, who said that ‘a man cannot imagine anything to begin without a cause...but if he try [so to imagine], he shall find as much reason, if there be no cause of the thing, to conceive it should begin at one time as another, that he hath equal reason to think it should begin at all times, which is impossible, and therefore he must think there was some special cause why it began then, rather than sooner or later; or else that it began never, but was eternal" (Of Liberty and Necessity, 276 [see for example, here].)* If Hume is really paraphrasing Hobbes, he has also made non-trivial changes or improvements: in the Hobbes passage (ii) there is no symmetry between time and place (which goes unmentioned in Hobbes). In addition, (ii) in context, Hobbes is really interested in agency (which is why place is irrelevant), whereas Hume seems -- as Anscombe notes -- to be focused on creation (which is why place also matters). In addition, in larger context (iii) Hume is taking aim at the principle that whatever begins to exist, must have a cause of existence (220.127.116.11) -- a key premise in the cosmological argument(s) for God's existence --, while Hobbes is only interested in a subjective or psychological version of the principle: "a man cannot imagine anything to begin without a cause." And finally, (iv) Hume wants to challenge the epistemic status (certainty) of the (cosmological) principle in order to undermine the claim that causation is a necessary relation, whereas Hobbes is taking aim at the idea that every action must have a preceding cause (the will). Now, with Paul Russel, I believe that Hume was very influenced by Hobbes, but I think the note to Hobbes by Hume, if it is by Hume, was an error.
Anscombe's suggestion, that Hume is focused on creation, is not without merit. With Hume, but not Hobbes, we are in the realm of ex nihilo nihil fit, or a weak-ish version of the PSR (akin to the principle of causality). This is indeed Leibniz land. It's also Clarke's land, especially Clarke of the Demonstration of the Being and Attributes of God; more particularly in answer to Hobbes, Spinoza, and their followers. As it happens, in the very next paragraph, Hume footnotes Clarke.
Now Clarke relies on a very particular metaphysical principle (it's also in Newton [recall]), I quote from his letter to Butler explaining it, “Necessity absolute and antecedent in the order of nature to the existence of any subject has nothing to limit it; but if it operates at all (as must needs do), it must operate (if I may so speak) everywhere and at all times alike….” (Dec 10, 1713) I call this principle 'absolute metaphysical necessity.' The background context is Clarke's (neo-Spinozistic) version of the cosmological argument in which he explains God's existence as a peculiar kind of self-cause (in which such metaphysical necessity grounds existence of God as a kind of formal cause).
Hume's argument does justice to the principle which basically says there is no reason why God's existence should start at any given time or at any given place--no place or time has any distinctive quality/property which could single it out. This principle grounds God's omnipresence and eternality. (It also underwrites monism and the lack of limitation on God's power.)
But as Hume discerns (and Clarke exploited) this principle makes it had to see why the universal isn't a pure, homogeneous plenum (something that Clarke things Spinoza should be committed to). So, something special is required to ground any (non-Godlike) particular, non-necessary existence (which breaks the pure homogeneity of a universe governed by necessity alone). For Clarke it's obvious this means you need something like God's (completely free) will. Hume kind of agrees with Clarke, for Hume, however, it entails there is no necessary connection between creation at a given time and place and the object caused then and there.
There is, thus, a sense then in which a theory that emphasizes God's will is akin to a theory that sees us governed by brute facts. Leibniz would invoke the PSR and identity of indiscernibles in order to explain God's choice and so avoid the unwelcome result (as Anscombe discerns). But the nice thing from Hume's perspective is that he can show that from an embrace of (i) absolute metaphysical necessity, and (ii) the existence of particular variety, he can show that (iii) not all causal relations are necessary. And he does so in a way that does not rely on his separability principle.
What about Anscombe's Leibniz? I doubt Hume was thinking of Leibniz in context. And Hume has other reasons to challenge final causes and the PSR. Of course, Leibniz has resources to defend the PSR or the principle of causality (etc.), but on Hume's account one can't simply assume that causal relations are necessary in one's argument for these, and that's no small matter.**