Philo is basically satisfied with Cleanthes' objections, but he has one of his own to add. Demea says that either there must be an infinite chain of causes or else there must be some self-causing being, but Philo can think of another alternative: there might be a principle of necessity in the material world, some law governing nature that acts as the final explanation. He likens this type of necessity to the necessity found in mathematics: to someone who did not know algebra, he points out, certain arithmetical patterns might seem very mysterious. Someone would did not know math might feel the need to posit chance or design to account for these patterns. But anyone who knows algebra understands that these patterns arise because of mathematical necessity. The same might be true of the universe. Philo concludes the chapter by remarking that no one who was not already convinced of God's existence was ever convinced by this version of the ontological argument.
The ontological argument has a long and illustrious history in philosophy. The first known version of the argument was presented by the medieval thinker St. Anselm. This version of the argument appealed to the fact that our idea of God is an idea of a perfect being. (1) God is that of which nothing greater can be conceived. (2) That which exists is greater than that which does not exist. (3) Therefore, if God does not exist then we can think of something more perfect than Him, in which case He would not be God. (4) Therefore, denying the existence of God involves a contradiction.
The ontological argument played a big role in early modern philosophy. The rationalists, such as René Descartes, Baruch Spinoza, and G.W. Leibniz all used some version of the ontological argument in order to support their philosophical systems. This is because they believed that there is a reason for everything that happens in the world and that all of these reasons can be discovered just by thinking really hard. In other words, they thought that if we started with certain innate concepts (i.e. concepts they believed we were born with, such as the idea of God, the idea of infinity, the idea of matter and so on) we could use our faculty of reason to understand why everything in the world is the way it is. But in order for this to be the case, there must be some final cause that is its own reason for existence. If there is no such final cause then either there is an infinite chain of causes (in which case we can never understand everything, because we could never get to the end of this chain) or else there is some arbitrary end to the chain and so there is not really a reason for everything (i.e. there is no reason for the first link in the causal chain).
Hume, as an early modern empiricist, would not have been very sympathetic to the needs of the early modern rationalists. He would, in fact, have been very keen on showing that this argument does not work. However, the version of the ontological argument that Demea presents is not Descartes's influential proof, but a much weaker formulation. Also, despite the fact that Cleanthes does end up leveling very convincing arguments against this ontological argument, his most basic rebuttal is not very solid.
Cleanthes's argument for the claim that matters of existence cannot be proved a priori is worth examination. Cleanthes begins with the premise that all demonstrable truths (those that we can prove a priori) have a special property: to deny them involves a logical contradiction. For instance, consider the demonstrable truth "all bachelors are unmarried". If we try to deny this truth ("not all bachelors are unmarried") then we land ourselves in a contradiction. What it means to be a bachelor is to be unmarried, so you cannot logically maintain that not all bachelors are unmarried. Now consider another truth: "all men have a digestive system". If we deny this statement, we end up with a falsity, but not with a logical contradiction. There is nothing incoherent about claiming that some men lack a digestive system. The statement "all men have a digestive system" is true, but not demonstrably true; the only way to prove that it is true is to go out and look at human anatomy. One could, without a contradiction, imagine a man without a digestive system who survives by a miracle.
The relevant difference between this last truth and the previous truth about bachelors, according to Hume, is that the claim about men's digestive systems is a truth about the way the world is (a matter of fact) whereas the first truth are really just stating facts about our ideas or words (what we mean when we say "bachelor" is "unmarried man"). It is only this latter kind of truth (relations of ideas) that can be proved a priori. Since claims about existence are matters of fact and not relations of ideas, Hume does not think that such claims can be settled with a priori arguments. In his judgment, there can never be a contradiction in asserting the non-existence of anything (for instance, there is no contradiction in saying, 'the sun does not exist').