In Berkeley's idealism God holds the world together, enabling us to avoid chaos and experience a shared, orderly reality. But Schopenhauer is an idealist and an atheist. How can we then explain the shared experience of reality? What holds everything together? This seems quite odd.
Berkeley populated the world with entities, ideas, which were in their essence perceptions. Berkeley's famous formula was "esse est percipi", to be is to be perceived. Those perceptions, the ideas, are things which are necessarily perceived by someone, by some perceiver. There are, on the other hand, no substantial things behind the perceptions, in Berkeley's world. When I perceive e.g. a tree, there is never a real, substantial tree behind the perception, according to Berkeley. That is because, there are in Berkeley's world no entities that could exist without being perceived. This is the peculiar reason why God is required to "hold together" Berkeley's world. My perceptions of e.g. a tree are not grounded in a substantial entity, such as a real tree. Therefore the perceptions must be grounded in something else. And Berkeley concluded (after some more reasoning) that that something is God.
The situation is different with Schopenhauer, which is, in the aspect under discussion, following in the footsteps of Kant. In Kant's world, unlike that of Berkeley's, there are substantial things behind our perceptions. These are what Kant called the things-in-themselves. It is not only that things-in-themselves can exist without being perceived. They cannot possibly be perceived, as they are. When you perceive a tree, in Kant's world, there is something substantial behind the perception. This something is not a real material tree (which is why Kant has been called an idealist) but it is a real something, a thing-in-itself.
Since there are, then, substantial things behind our perceptions, these things, together with the perceiving subjects, are the grounds of our perceptions. Therefore Kant's world, unlike Berkeley's, is held together by these things-in-themselves (together with the perceiving subjects). Therefore the need for God does not arise in Kant's system, in the way that it arises in Berkeley's. And Schopenhauer's system, in the aspect under consideration, is just a variation on Kant's system. Schopenhauer upheld Kant's version of idealism, including the doctrine of the thing-in-itself. (Schopenhauer further held that there was just one thing-in-itself, which he called the Will.) That's why the need for God did not arise in Schopenhauer's idealism, as it did in Berkeley's.
The World Will, primordial, blind, and irrational, "holds the world together" in Schopenhauer's philosophy. It doesn't do a very good job of it, but then he was not called the father of pessimism for nothing. Schopenhauer's ethics was influenced by Oriental philosophy, in particular the Buddhist idea that the world is full of suffering, and the cause of suffering are human desires, confused, misguided, and ultimately futile. All living beings, and even inanimate objects, are merely "individuations" (his term) of the Will, the sole source of action in the world, and the root of its pointless suffering. Schopenhauer's ethics shares with Buddhists, and Stoics, the ideas of ascetic self-limitation and perseverance, the only way to "beat" the Will, which manifests itself in us through its incessant compulsive urges.
Schopenhauer's metaphysics, on the other hand, closely follows the classical German idealism, albeit mixed with also Oriental idea from the Upanishads: of Atman (soul) being mystically identical with Brahman (world). The world as Will for Schopenhauer is the Kantian thing-in-itself (and Kantian appearances form the world as representation). Unlike Kant, however, Schopenhauer admits a direct "intuition" of it, which manifests however not in artistic exaltation, as for Schelling, or moral law, as for Fichte, but in the base urges of the Will to Live. The dynamic of the Will can be described as the polar inversion of Hegel's boundless rationalism. His Absolute Geist starts from abstract notions, that are delimited through negations, and then sublated into the multiplicity of concreteness through negation of the negations. This dialectic is seen as both self-creation and self-cognition, at the end of which "the rational is the real". In parallel, the World Will tears itself apart, but not through rational moves of Hegel's Logic, but through violent urges that splinter it into a myriad of individuations, controlled by "umbilical cord" of unconscious volition.
Now concerning the title question. Idealism is the position that the fundamental "stuff" of the world is more like what we make contact with through our mental faculties of reason, understanding and intuition, rather than through our senses, as in the case of matter. Materialism is the opposite position. There is no logical connection between either of them and the existence of God as in the Western Christian tradition, an omnipotent, omniscient, personal being. There is no reason why the world, either ideal or material, can not "hold itself together", or be held together by some other mechanism or principle. That holding belief in God correlates with philosophical idealism is more a matter of tradition and psychology. Plato and Aristotle gave versions of idealism without such a God, but most later philosophers belonged to a Christian tradition, and so strived to find a place for him in their metaphysics. However Spinoza, and especially Hegel, reduced "God" to little more than a mere principle, which Schopenhauer inverted and termed the World Will.
Schopenhauer offers a double epistemological aspect of the world, which is to say, existence, as being at once Will and Idea. His work is not explicitly a study of ontology, i.e. essence/being, beyond the essences of Ideas, which for Schopenhauer are simply the intelligible, a priori objectifications of Will, for the subject. Although this system appears self-contained, and there is a stated atheism in Schopenhauer's later works, the master work nevertheless - and quite mysteriously - qualifies the World, as Will "for us". This is probably only an acknowledgment that even great geniuses like Schopenhauer, despite tremendous powers of reason, are hopelessly constrained by the principle of sufficient reason and the principium individuationes, from intellectually transcending one's own mind in order to attain to the whole truth (as opposed to knowledge) of ultimate realities. As Schopenhauer himself says, this would be like the Baron Münchausen lifting himself up by his own bootstraps. It is thus, in passing, that Schopenhauer recognises that the world "as Will" must itself only be an Idea (of the nature of the world), "for us", and thus can never truly describe the ineffable totality of the noumenon. But who is this "us" that the world is "for"?
Schopenhauer's Idealism is distinguished from German Idealism in its refutation of the 'divinity' transcendental character or absolute independence of the Subject. Instead, the subject is one half of one face of the World Will. Schopenhauer dispenses with this issue early on as mere "speculative theology", whereas this "for us" remains a paradox at the heart of his system: does the Will "for us" not in fact presuppose some higher standpoint, a knower who best understands the world, who is presupposed by its being understood according to, and attenuated by, the degree of species-consciousness and who, when one of its manifestations attains to this very understanding of the world and subdues it, can be said to overcome it "like a god"? It is my view that Schopenhauer's system tantalisingly invites us to undertake precisely such "speculative theology", if not, at least, to doubt the rigorousness or necessity of atheism to his system.
To the specific claims about the nature of the Will and its knower, Schopenhauer would respond, firstly, that the "knower" is simply the a priori subject of knowledge and secondly, therefore, it is just this same Will, seconded with knowledge (of its terrible manifest nature), that overcomes itself. In other words, the denial of the Will is achieved when the a priori subject of knowledge, in pure, will-less contemplation of the world as Idea (a subject which is itself ultimately also, Will), realises that the outer manifestations of the inner nature ought continually to be denied.
The 'divinity' that German Idealists like Fichte and Schelling situated in the Subject, Schopenhauer criticised as a species of sophistry, demonstrating his account of the dependence of the Subject upon Will just as objects depend on it, in the aspect of the world as Idea. Yet he doesn't offer an explanation for this "us" that nevertheless looms over his the Will just when his system demands or even suggests one. If the subject itself is not "the Absolute" or "transcendental", being merely Will, what relation to it does this "us"-ness stand, if the subject is not, ontologically, a two-way conduit of knowledge between the world and "us"? Why else would the moral injunction, ought, be imposed on the object-self by the subject-self, if there wasn't some eternal and sovereign standpoint pertaining to the claims of self-knowledge over the more universal and vociferous claims of worldy knowledge, which belong to desire?
Schopenhauer thinks this sovereign standpoint is simply the subject's knowledge that the Will, broken up into representations or phenomena, necessarily strives against its own true inner unity in consequence of our mental mapping of things in time, space and causality. This knowledge, the quieter of the Will in our object-being, can be achieved by contemplation of the world only by the pure subject of knowledge free from the worldy lusts, and which leads to an ascetic denial of all will to life.
The problem that Schopenhauer ignores is that this romantic "contemplation" is nonetheless still willing, a fact he practically concedes when later describing the daily struggle and suffering that the ascetic ethos - the rather insufferable summum bonum of his system - requires, as symbolised by the futility of slaying of the Hydra. Well we may ask, is the Will really "overcoming itself," or is it simply employing novel means of willing, commingled with self-deception, in cunning mimicry of what we would recognise ethically as holiness or blessedness? He has elided what appears to be a far more parsimonious explanation for this self-overcoming behaviour and that is to recognise a Self, being or essence of the world that is not Will alone, but includes it, is included in it, and is therefore superior to it by virtue of being itself capable of moral wisdom leading to will-lessness, or better, will-lessness transfigured by joyful wisdom, to compassion.
Instead, by describing the Will as cruel and evil, Schopenhauer proffers a judgment on the world. Describing the world as having both an autonomous, unified nature and a knower of its multiplicity of Ideas who is that very same Will manifest a priori as subject of knowledge, he fails to see that the basis for the Brahmanic/Buddhistic ethic of denying the Will to life, i.e. this "cruelty" of the manifest Will, is itself only a representation to that subject, or "for us", thus only an interpretation, and an unsatisfactory or incomplete one at that, as Nietzsche thought. Since our rational knowledge remains subject to the principle of sufficient reason our judgment of the nature of the ultimate reality of the world must remain limited, fallen.
It is thus that Nietzsche asked, rather than deny the Will because it is strife, why wouldn't you embrace the strife, because it is Will? A perverse conclusion, and no doubt one that Schopenhauer himself did not and would not have supported, but it follows ineluctably upon sound Schopenhauerian axioms. As Chesterton remarked, "Nietzsche scales staggering mountains but he turns up ultimately in Tibet. He sits down beside Tolstoy in the land of nothing and Nirvana. They are both helpless - one because he must not grasp anything, the other because he must not let go of anything. The Tolstoyan will is frozen by a Buddhist instinct that all special actions are evil. But the Nietzscheite's will is quite equally frozen by his view that all special actions are good; for if all special actions are good, none of them are special. They stand at the crossroads and one hates all the roads and the other likes all the roads. The result is - well, some things are not hard to calculate - They stand at the cross-roads."
It is my view that Schopenhauer can only avoid the absurd paradox of the Will overcoming itself, on the one hand, and the equally absurd Nietzschean Will to Power, on the other, if he transcends the desiccated Buddhistic atheism that governs the Will and finds a higher significance to it, thereby liberating his ethics from the overwhelming Tolstoyan pessimism ("it would be better to have never been born"), that does his masterpiece little justice. I believe that, by some uncharacteristically complacent atavism of philosophical thought (I dare not attribute motive, to such a great genius), when speaking of the World as Will "for us", Schopenhauer includes the "us" in the Will because he has not undertaken a meaningful analysis of the distinction between the Logos of Heraclitus, which he tacitly follows, and the Christian Logos of John, which he never mentions. This was partially explained by his (and his contemporaries') contempt for scholasticism and Judaism, which led to common misunderstandings of the inner meaning of Scripture (and particularly the doctrine of Original Sin, which he thought was merely Will, manifest as sexual desire) and partly, despite his extraordinary genius, his inability to overcome the rationalistic and empiricist temperament of his epoch in order grasp the true significance of the Christ event, as an anthropological, as well as theological revelation; anthropological, because it revealed the true origin of Man - thus also the true origin and problematic of the principium individuationes, or "knowledge of Good and Evil" - founded on a collective murder (of a scapegoat in a social crisis) and a lie (that the victim was guilty, dying as a propitiatory sacrifice ordained by violent gods), and theological because in demystifying this violence, as totally human, at the foundation of the species, the true being of the world, the "asher ehyeh asher (I am who I am/I am the Being)" or "Tat twam asi (this Thou art)" reveals itself as an undivided and non-violent essence.
To condemn the world as Schopenhauer does, therefore, as "will against itself" or, as Heraclitus wrote, as "polemos (strife), the king and father of us all", is to fall for the very fallacy that is Original Sin, i.e. to see differences in the world and to fail to see that the ever-present strife, chaos and cruelty is merely apparent "for us", constrained as we are in the fallen state to think according to the principle of sufficient reason and the principium individuationes (our brilliant but still limited way of understanding as causality and seeing as multiplicity), blinding us to the true identity of world, which as the Logos of John characterises it, is Love.
Your question actually contains quite a lot.
- How can there be idealism without God?
- How can we explain the shared experience of reality?
- What holds everything together?
In any endeavor one should ask if there exist a causality linking these events together. Is the causality a necessary truth, or a contingent truth?
It is indeed a contingent truth. An ideology is basically a contingent truth, unique and relative to none but the beholder. Whether other people adopt it as their own, or choose their own ideology is the "sharing" in the experience.
Your last question seems to jump from everyday causality to a first cause such as in the idea of creation from nothing. The problem with this is causality itself is "held together" by time which is really a temporal expression of our imagination. Remove time from the equation and what is left is a beautiful world of objects and space, the two absolutes we can be certain about.
I don't see how there can be an idealism that has a basis without God, particularly anything in terms of an objective idealism.
The natural world doesn't know what oughtness is. Chemistry can't answer that. Physics can't answer that. Biology can only periferally answer that.
A scientific or rather scientistic approach would have us believe that we are "only chemistry, biology, and physics." The key word there is only. We are certainly chemistry, biology, and physics, but it would require a massive burden of proof to assert that that is all we are (eliminating other options systematically). Not the least of which is the actual human experiences of love, relationship, community, wonder, and awe. But beyond that the experience of freedom, deliberation, conscience, consciousness, imagination and other human experiences seriously call any reductive naturalist approach into serious question (and especially those that are only physicalist or materialist in nature).
A further problem is skepticism (ideological and philosophically as practiced), which in practice is undermining either all beliefs or all beliefs I disagree with. Both ultimately serve as the acids of belief. Without some basis of belief--its impossible to exist as a human being. Moreover the reductive approaches which are used tend to eliminate key aspects of reality (ie the big picture, history, and context). There is a certain populism or common sense involved in these approaches, but when you add specificity, nuance, history, context, or the big picture--these approaches usually falter. Not only that, their philosophical worldview as a whole is seriously brought into question by the above.
For instance, the illusion is that this kind of thinking will bring us freedom, but it actually leads to a very deterministic outlook--one in which human choice is entirely removed. Thats why Huxley undermined human dignity and human choice in his philosophy. That goes against the principles of freedom and human dignity that allow a platform like this to exist.
Moreover, such an approach runs aground of a kind of relativism. If we end in relativism, we end up with nothing. We have no notions of justice, accountability, or respect from systems of power. Governments, businesses, and the powerful become the rulers--because everything is relative. Its a Machiavellian outlook, rather than one in which freedom, dignity, justice, and everything we value most (for instance the US Constitution) is actually valued.