Once I came to understand Heidegger's account of "world" and "the world-ing of the world" in Sein und Zeit, I have thought it fundamentally correct. If one begins with Descartes' cogito, there simply is not a way to "build a bridge" to the world. (Descartes famously tried to justify the existence of, and the determinate shape of, the external world on the fact that God is not a deceiver, and we are justified in asserting God because we have an idea of perfection.) Far better to begin with Heidegger with the fact that our be-ing is always already be-ing-in-the-world, that consciousness is always consciousness of a world.
Husserl famously developed the method of phenomenological inquiry that putatively bracketed the metaphysical questions of materialism and naturalism and advocated an ad fontes return to the things themselves in introspection, grasping, as it were, through the eidetic reduction things in their essential thingness. The method was to choose an object, vary imaginatively the features of it, and ultimately grasp what it is that cannot be eliminated if the object is to be the object it is.
While Husserl's phenomenological reduction of bracketing judgments about the ultimate nature of the world in favor of describing carefully one's experience of the world was supposed to leave in abeyance the metaphysical question of materialism and idealism, it is pretty clear that an argument can be built plausibly claiming that Husserl is committed to a type of idealism. (The transcendental reduction abandons our natural attitude on the world in favor of a description of the intersubjective space of the transcendental ego.)
The question that concerns me is whether Husserl's student, Martin Heidegger is also finally committed to a type of idealism. After all, is not his world the sum of significances in which one pre-reflectively finds oneself, a world in which one finds one's way? Is not this world and its complex relationships of meaning present only for Da-sein (Being-there), a world which is itself a pole of Da-sein and thus forever within its arena of consciousness. (My apologies to Heidegger for using "consciousness," but I think that an argument can be made that being-in-the-world just is to be conscious.) We are pre-reflectively always coping with the world, a world that tends to disclose itself when our regular coping breaks down. (Heidegger famously points out that we don't really know what a hammer is -- what it means -- until we are without it in a relevant context.)
Heidegger's distinction between Zuhandenheit and Vorhandenheit is meant to get at the distinction between our everyday dwelling in the world of the "ready to hand" and our occasional examination of objects in this world with a critical distantiation, a distance that allows us to investigate the object as it is in itself. (We might translate the latter as "present at hand.") When our hammering no longer happens pre-reflectively, we might instead attend to the properties of the hammer and thus attempt to consider the hammer as it is in itself, as disconnected from the web of significances within our being-in-the-world. In so doing, we might try to correct the hammer so that it can again recede into the ready-to-hand significances of our primordial dwelling in the world.
But this distinction between Zuhandenheit and Vorhandenheit seems to be a distinction in and for Dasein in its own consciousness as it attempts to grasp objects in its world in different ways. The reading off of the objective properties of the hammer is a function of the attitude in which Dasein engages the world, and it is difficult to locate the grounds to claim that the adoption of this attitude of reading off succeeds in getting us to the thing as it might be out beyond the world of Dasein. If ready-to-hand is a dwelling of Dasein in its being-in-the-world, then is not present-at-hand also a type of comporting, a comporting that is ultimately found in a web of meaning in and for Dasein, and thus not a deworld-ing of the world in favor of the objectivity of the thing?
Hubert Dreyfus has famously claimed that Heidegger escapes idealism through the de-worlding move of "formal indication" (formale Anzeige). He points out that Heidegger was really quite interested in questions of what the world is in itself, and that Heidegger thus thought it possible to refer to objects as the objects they are without the nature of the objects being determined within the holism of the context of meaning in which they are ingredient. Comparing this move to Kripke's notion of rigid designation, Dreyfus argues that Heidegger too could have understood reference to objects apart from their descriptions and contexts.
Kripke talked about an "initial baptism" that connected name to thing, and allowed for increasing understanding of the thing and finally a grasp of the essence of that thing apart from the ways we might describe or pick out the thing. (The atomic number of gold is essential to gold, its necessity is, however, a posteriori. That which first allowed examination of gold, those properties by which we might unreflectively pick it out, turn out not to be essential to the thing. Analogously, water is identified by being H2O, not by the properties of colorless, odorless, and tasteless.) Dreyfus suggests that Heidegger's formal indication functions like Kripke's rigid designation, and that this move allows Heidegger, like Kripke, to escape the idealist net. If this is so, then Heidegger like Kripke is committed to the ontology of natural kinds, the notion that there are, as Putnam says, self-identifying objects that exist apart from human perception and conception.
There is quite a literature on the formale Anzeige in Heidegger, and clearly there is no consensus that such a move takes one to realism. However, I do like the attempt to connect Heidegger's excellent analysis of what it is to be-in-the-world with resources that would allow the world to be in some sense without our being in it. But the problem here does seem Kantian. If the formale Anzeige takes us beyond the fuer sich of the world to the an sich of things, then how exactly does the an sich connect to the fuer sich? In other words, how exactly is deworlding of the world possible? How are natural kinds possible beyond descriptions when they themselves are articulated in terms of descriptions? What could a natural kind be apart from the language that articulates the kind as the kind it is, a language that operates both at the deworlding and worlding levels? What kind of faith is necessary to assert theoretical entities as having self-identifying being apart from their ingrediency in theories? Can we find this primal place before language when, as Heidegger later says, language itself is the house of being? Ultimately, can we locate essences out and beyond the results of an eidetic reduction? If so, what would be the grounds of this conceivability?