The Cosmological Argument and Free Will

[Mudhammam]

If naturalism is true, and thus determinism, then [within the context of this argument] the 'beginning of the universe' is in fact not the 'beginning' because the cause of the beginning of the universe would be an effect of a previous cause.

Correct. Material antecedents would mean that the big bang is not the absolute beginning of the Universe, though it may still be taken as a discontinuity so radical that we cannot explain it, because we can find no laws which we can extrapolate backwards through this discontinuity.

This would go on and on into infinite regress and the question would still remain, what is the first uncaused cause? This is a question unanswerable by determinism.

This is not even a logically sound question. It's like asking, "Can an omnipotent god create beings more powerful than her?" A "first" cause already presumes that you're talking about a finite starting point in time.