Hume on infinity

Question

I know Hume argued against dividing finite space into infinitely many regions, but I can't seem to find anything regarding his thoughts on infinity itself. From his Enquiry you sort of get that he thinks it is not productive to talk about the infinite generally speaking, but he doesn't explicitly say he is against it as far as I can tell.

I am interested in this as I am trying to make sense of his concept of infinity. In sections 2-3-4 of the above mentioned work, he says that we are free to combine however many simpler ideas and form more complex ones. He also says that The most lively thought is still inferior to the dullest sensation. This would suggest to me that, in his view, you could add up infinitely many low-vivacity parts (he says that thoughts and ideas are less vivacious, therefore they must have some vivacity) and the result would still be duller than the dullest first hand sensation. In order for this weird airthmetic of vivacity to make sense, one must have a different understanding of infinity.

Answer 1

At least ever since Anaximander of Miletus, who introduced the notion of apeiron, man is well aware of the definite need to get to grips with boundlessness and its kin. One of the critical issues raised in this problematic is whether infinity is a complete and real object, that, for example arithmetically, we could denote by a numeral 'ω' just as we denote the number three by the numeral '3', or a fictive object; a disposition to point beyond itself each time we reach its limit. Conventionally, the former one is called actual infinity, and the latter one is called potential infinity.

In empiricist frameworks, actual infinity is generally regarded as epistemologically flawed and dispensable, a façon de parler at best. Understandably, Hume is no different. Though his theological view diverges from the baseline of his thought (recall that he was an 18th-century man of letters), the following passage from An Enquiry concerning Human Understanding (sect. II) offers a good insight about his approach to the notion of infinity:

The idea of God, as meaning an infinitely intelligent, wise, and good Being, arises from reflecting on the operations of our own mind, and augmenting, without limit, those qualities of goodness and wisdom. We may prosecute this enquiry to what length we please; where we shall always find, that every idea which we examine is copied from a similar impression.

For a comprehensive discussion of this topic, you may wish to consult Dale Jacquette's David Hume's Critique of Infinity (Brill's Studies in Intellectual History, 2000).