Can everything simply exist for no reason?

The question is ambiguous between - does anything that exists have a cause, and does anything that exists have a reason or a purpose? In Platonism, mechanical causation is not possible without teleological causation and ‘reason’ is a broader category than causation. For example, for Leibniz, an infinite regress of cause and effects is possible, but the entire series must have a reason for existing, and this reason must be outside them. The infamous PSR is not a theory of mechanical causation, but a principle of harmony that says God orders everything according to the best and the existence of something is its compossibility in relation to the whole. This is to say that a monad is compossible or exists not because of its logical possibility but its ability to add to the overall purpose of things. In this worldview, the purpose is the ‘why’ of things, and this teleology is immanent within the thing itself and not externally imposed. To exist is to exist as a thing that acts to fulfil its purpose. If, however, we talk only of mechanical causation, we have to say that anything that begins to exist must have a cause, or else it is either non-existent because a thing cannot exist if it cannot be brought into existence or its non-existence is not negatable, or else it is eternal and does not require something other than itself to explain or cause its existence. The security of this principle has been sought in the Principle of Non-Contradiction, which holds that it is contradictory to say that something begins to exist without a cause. Hume famously questioned this, but his challenge depends to a great extent on his psychology, which is questionable. This is Hume’s challenge, given that we must understand something through its cause, and an uncaused existent would be unintelligible to us, why must we assume that what we think reflects the structure of reality? I would say this is to ask, given that the Pythagoras theorem is true, how do I know it is true for everyone? If I have validly determined what is real, then the result cannot be avoided by saying that the thing-in-itself must be different from the object of valid thought. If I am searching for all the rules of grammar that determine what it is for something to be counted as grammatical, if faced with a challenge, I can modify these rules, remove a couple of them or add new ones. But if you say that something can be grammatical without adhering to any rules of grammar whatsoever, then I would say you don’t really know what you are talking about.