Sunyata ("Emptiness"). The Mahayana tradition has put a special emphasis on sunyata. This was necessary, in part, because of the tendency among certain early Buddhist schools to assert that there were aspects of reality that were not sunya, but which had inherent in them their "own-being". Several important Buddhist philosophers dismantled these theories by arguing for the pervasiveness of sunyata in every aspect of reality. (Nagarjuna was among the most important of these.) The specific arguments are too complicated for us to deal with here. But it is important to appreciate that understanding absolutely everything as sunya could imply that even those things most revered by Buddhists (such as the arhant ideal and the rules laid down in the vinaya) were empty. Mahayanists tended to argue that members of the Hinayana traditions were attached to their ideal forms as if they were not sunya.
To some extent, sunyata is an extension of the concepts made explicit in the 3 Flaws. All things being impermanant, nothing can be seen as having an independent, lasting form of existence. And this is, in essence, what sunyata is all about. Strictly speaking, sunyata can be defined as "not svabhava". The concept svabhava means "own being", and means something like "substance" or "essence" in Western philosophy. Svabhava has to do with the notion that there is a form of being which "is" and "exists" in a form that is not dependent on context, is not subject to variation, and has a form of permanent existence. As such, the "soul" as understood in Abrahamic religions would have svabhava. God would certainly have svabhava. The Platonic forms (such as those described in the allegory of the Cave) would have svabhava.. Certain abhidharma teachings conclude that the building blocks of reality have such svabhava. But Mahayana philosophers like Nagarjuna concluded that sunyata is the fundamental characteristic of reality, and that svabhava could be found absolutely nowhere.
One of the images used to illustrate the nature of reality as understood in Mahayana is The Jewel Net of Indra. According to this image, all reality is to be understood on analogy with Indra's Net. This net consists entirely of jewels. Each jewel reflects all of the other jewels, and the existence of each jewel is wholly dependent on its reflection in all of the other jewels. As such, all parts of reality are interdependent with each other, but even the most basic parts of existence have no independent existence themselves. As such, to the degree that reality takes form and appears to us, it is because the whole arises in an interdependent matrix of parts to whole and of subject to object. But in the end, there is nothing (literally no-thing) there to grasp.
Pratitya-samutpada ("Dependent Co-arising"). The flip side of sunyata is pratitya samutpada. They are two sides of the same coin. They mean the same thing, but from two different perspectives. To the extent that sunyata is a negative concept (i.e., not svabhava), pratitya-samutpada is the positive counterpart. Pratitya-samutpada is an attempt to conceptualize the nature of the world as it appears to us, not (as with sunyata) by saying what the world is not, but by characterizing what is. I would say that pratitya-samutpada is probably just about my favorite religious-philosophical concept from within the traditions of the world. It is wonderfully subtle, and Buddhist philosophers have developed it beautifully.
As mentioned above, this concept is understood in two quite different ways in Theravada and Mahayana thought. In Theravada dependent co-arising (usually designated by its form in Pali, paticca-samuppada) is understood as a logical-causal chain which illustrates in a linear fashion the preconditions of suffering that can be analyzed and eliminated according to a strictly codified pattern of behavior. In Mahayana, on the other hand, which emphasizes the emptiness of things, dependent co-arising as a concept is used to clarify the nature of sunyata by showing that all things that appear to have independent, permanent existence are really the product of many forces interacting. Thus, in Mahayana it is stressed that all things are dependently co-arisen, because their seemingly independent existence really depends on the coming together simultaneously (the co-arising) of the various parts and forces that go into making them up. As such, pratitya-samutpada is more a metaphysical concept in Mahayana, and it is nonlinear inasmuch as it attempts to picture a universe in which all things are inextricably linked in a cosmic wholeness that cannot be unwoven into independent threads or pieces.
One illustration of sunyata and pratitya-samutpada is the Jewel Net of Indra (see above). Another is a rainbow. We know that a rainbow is real in some sense, because we can see it, locate it, measure it, and so forth. However, it is also clear that a rainbow is no "thing", but rather the product of various forces interacting as sunlight shines through an atmosphere that has water droplets in suspension. Mahayana thinkers have asserted that all phenomena, including especially individual human beings, are like this, inasmuch as it is impossible to locate any basic particle or entity that is dependent in no way for its definition and existence on the relationship that it has to other things. All things are, therefore, "empty" and "dependently co-arisen".
Many great Buddhist philosophers have thought through with great care the nature of shunyata and pratitya-samutpada. This is but a simple illustration of much more complex reasoning, such as that found in the writings of Nagarjuna, Chandrakirti, and other subtle thinkers. (See Smith, 82-112. See also Paul Ingram. 1990. "Nature's Jeweled Net: Kukai's Ecological Buddhism" on Electronic Reserve. )
It may seem that the articulation of such ideas "tends not to edification", or that it resembles absurd philosophical speculation such as "how many angels can dance on the head of a pin?" However, the study of these (and other) philosophical concepts has typically been linked with practices that train Buddhists to release themselves from attachment to or striving after "things" that might seem to offer some lasting sort of satisfaction. One of the most basic forms of attachment is the mind's tendency to grasp after objects of thought and perception as real (i.e., as having svabhava), and this tendency is reinforced in ideas that we have about the world. The use of philosophical reasoning to deconstruct such misconceptions (as they are understood within Buddhism) is a powerful vehicle for eliminating seeds that can eventually grow into very serious obstacles in one's orientation to the world.
Among the most important applications of these ideas with Mahayana has been to expose the emptiness and the co-dependently arisen qualities of even Buddhism itself. Mahayana claims itself to be an important vehicle to liberation, but it also points to its own provisional character. Mahayana does not see itself as an end, but as means to an end. That end is liberation, enlightenment, and an end to suffering. However, as with all religions, there is a tendency for the religion to reinforce itself as real, as an end in itself, within the minds of its adherents. The philosophical traditions of emptiness and dependent co-origination are important correctives to this tendency. There is an important saying within Zen: "If you meet the Buddha on the road, kill him." When people come to see the Buddha as a being to be revered merely for the sake of piety itself, or when Buddhism itself becomes the chief focus of its practitioners, then it is time to "kill the Buddha", to point to the emptiness and provisional quality of Buddhism itself.