Thesis The contingency argument fails because a collection of contingent things needs no explanation of its own
Its coordinates: 1.8 Natural Theology
Where it stands
The argument from contingency is one of the most elegant in philosophy, and because it comes apart cleanly into steps, I think I can show exactly where it breaks.
Roughly: everything is either necessary or contingent; contingent things depend on something else; the whole collection of contingent things is itself contingent; so it depends on something outside it, which must be necessary.
My objection is the one Hume puts in Cleanthes' mouth in Part IX of the Dialogues, and Russell pressed in his 1948 radio debate with Copleston. If I explain each of twenty particles of matter, I have explained the twenty. "The collection" is not a further thing waiting for its own explanation; it is an act of the mind grouping them. Russell's version: every human has a mother, but the human race doesn't.
If that is right, step three is the fallacy of composition, and the argument doesn't get off the ground. Each contingent thing is explained by earlier or other contingent things, possibly forever, and there's nothing left over.
I'm open to being shown that the argument doesn't need step three in this form, but I'll be surprised.