Leibniz's argument posits that everything that exists has some definitive cause or reason for its existence. The foundation of his cosmological argument is built on the "Principle of Sufficient Reason" (PSR): the explanation as to why it is X and not otherwise. Whether discernible by human methods of inquiry or not, no matter how complex and adaptive, every event, fact, and occurrence — physical or metaphysical — has a cause, a means of determining said cause (even if humans can never discern it), and is linked to other occurrences in space and time via regress (X caused by Y | Y caused by Z | Z caused by …). The argument for God's existence is that no phenomenon occurs without cause or reason in the universe; therefore there must be an entity that exists to be the "first cause."
Leibniz posits that this "first cause" must be God, the entity responsible for creating the chain reactions that is the universe unfolding. God must be the creative entity beyond the universe's contingencies to create the causal chain of the universe. Leibniz goes as far as to say in PSR that of all infinite realities in God's mind there must be a perfect reason why God selected this one. The mathematical constant of a circle is ~3.14 for the perfect reason, Italy is shaped like a boot for the perfect reason, I am a student attending UChicago taking a Philosophy of Religion course for the perfect reason, etc. God knows and we need not discern it in order for it to be true — just because it lies beyond the scope of human intelligibility does not mean things are not true. Gravity was a concept long before Newton "discovered" it. From simple daily occurrences like misplacing your keys or choosing chocolate instead of vanilla ice cream, to highly coordinated systems like the formation of the Roman Empire, the United States Government, or Google, to complex universal phenomena like the Heisenberg Uncertainty Principle or the Big Bang: everything has a linkage of causal events that "sufficiently" explain their reason for being.
So where does God fit into this equation? The cosmological premises for God's existence as posited by Leibniz are as follows: (1) things in this world exist; (2) everything that exists has a causally dependent reason for being; (3) to make a causally dependent reason for being coherent, there need be a self-existing, independent "first cause"; (4) this self-existing, independent "first cause" must be an all-seeing, all-powerful being capable of universal creation; (5) therefore, God must exist. That is the ethos of Leibniz's first cause argument for the existence of God.
Can we definitively claim whether the First Cause argument for God is sound or not? Not without a few notable objections. The irony is that regardless of how I bring forth these arguments, Leibniz would suggest that sufficient reason exists for all of them, even lying beyond my understanding. There is a certain degree of intellectual humility we must evoke here. We may not be able to prove anything, but in the spirit of Leibniz, the answer is out there.
Objection I: Who caused God?
My first objection is that if everything needs a sufficient reason for being, then shouldn't God need one too? The idea that God is granted a free pass exemption to correct the infinite regress seems like an intellectual cop-out. We could ask: what caused God? We don't have to accept God as the root cause for everything. If we make the jump to say that there is a divine creator, can't we extend that assumption further? God could be flawed, God can make mistakes, or God reports to a higher God. While we have been evaluating God from an Abrahamic perspective, we must not lose sight of the fact that at substantial moments through history, polytheism was a dominant theology. Brilliant minds like Socrates and Plato subscribed to it.
This relates to a recurring pattern in religion where God is summoned as a placeholder for uncertainty. We reason things empirically using scientific inquiry, and throughout this process we keep moving the goalpost back to what we consider to be the root cause of God's creation. At some point in civilization, when the sun would disappear at the end of the day, people prayed to God to bring it back. Then humanity discovered that astronomical and planetary motions dictate how the Earth revolves around the sun. Science moves the proverbial goalpost of first cause further and further down the stack. The question becomes: will we, in perpetuity, ever completely explain God away with science? The day will come where we either have sound empirical justification to reject the existence of God, or we accept a reality where our inquiries lead us to an answer that this universe had to be created by one. Currently, our knowledge repository is too ambiguous — too many known unknowns and unknown unknowns — to say either with confidence.
Objection II: The problem of free will
My second objection concerns the implications of free will. There is a slippery slope in devoting your understanding of the universe to Leibniz, as the implications of "first cause" bias heavily toward determinism. If all events were caused by an infinite regression started by God, then all actions are preordained and we lack free will. If the world is inherently deterministic, then the significance of morality becomes meaningless: the terrible wars and evils in this world would be pre-determined by God's creation. If we accept Leibniz's first cause argument, then God must have causally created the universe and simultaneously endowed humans with free will, which seems logically incompatible. How could you create causal links as the creator of all while permitting humans to act freely? Maybe this lies beyond the scope of human reason and only God can explain.
Objection III: Quantum non-determinism
My third objection is that we have observed phenomena in the universe that occur non-deterministically, without cause, in explicit violation of Leibniz's axiom. The concept stems from quantum physics, where matter in a contained system can appear with no chain of causality supporting its existence. This could imply that things can exist with no explanation. Interestingly, this cuts both ways. The justification of self-existent beings could explain the creation of God and reconcile qualms about God simply coming into being. But those same principles can extend to the creation of the universe itself, meaning the universe could be self-existent, coming into being contingent on no first cause at all.
Conclusion
All of this distills down to one important consideration: it is difficult to reason beyond ourselves. Since we are caught in the chain of dependence, we may be blindsided by our own position within it. There is a fundamental flaw to self-referential analysis. Using the "stuff" to explain the "stuff" is bound to produce some incompleteness. Leibniz co-created calculus to mechanize human reasoning and develop a universal language — characteristica universalis — to understand God's ordered creation. But how can we evaluate whether our methods of inquiry are leading us any closer to the answer of God's existence? We cannot, and probably will not be able to, for the foreseeable future. Things happen to make sense only in hindsight. Going back to humanity's early knowledge of the solar system: how many things will future centuries look back on in our present moment and simply think, "you fools."