Mathematics and culture

William Byers' How Mathematicians Think (Princeton, 2007) states that the deepest motivation for mathematical practice rests on the fact that "There exists that which cannot be expressed yet we must express it." (p. 120; italics in original.) He amplifies this as "putting the ineffable into words, describing the indescribable, or expressing the inexpressible." Exploring the example of the "infinite," he finds that historically two clusters of characerizations are found: on the one hand the infinite is boundless, endless, immeasurable; and on the other hand it is complete, whole, perfect, and absolute. The first cluster belongs more to the context of expression, the second to the intuition of the wholeness that lies beyond expression. In my terms, the indescribable infinite belongs to World Two, and the attempt to say the infinite in terms such as "immeasurable" to World Three, the realm of culture and specifically mathematical culture. Godel's incompleteness theorem is a paradigmatic example of mathematical culture because it shows that "within the context of logical thought one can deduce limitations on that very thought." (Byers, p. 282). This is quite similar to the argument in the Samkhya Karika that leads the highest embodied intelligence (sattvic buddhi) to see that "I am not, I have nothing, there is no self in me" (naham na me nasmi, SK 64).

The self referentiality of the Samkhya text closely parallels that of Godel. This seems to be a general feature of (genuine) culture. There is always self-reflection in the service of self-transcendence. We are bound into the world of duhkha but that suffering life contains within it the possibility--indeed the necessity--of a beyond. Great ideas in mathematics parallel the great realizations of figures like the Buddha and Patanjali (of the Yoga Sutra) that show the limitations and thereby the potentialities for release in our World One of constriction and pain.

Buddhist and Hindu postmodernism

I have long thought that much of modern and postmodern thinking can be categorized as "Hindu" or "Buddhist," depending on whether or not it allows the possibility of transcendence. Of course taken historically or philosophically that is not an easy question, and I think it is quite clear that many forms of Buddhism do allow for transcendance. I think the location of the Buddha's enlightenment at the moment of seeing the morning star figures this "going beyond." So when I use the rubrics "Hindu" and "Buddhist" I do not intend the terms literally. Rather, I mean that modernism and after divides fairly well into tough-minded cultures of suspicion, avid to dissolve or deconstruct all experiences of a beyond into sexual, political, economic, or gender interests, and tender-minded cultures of mysticism and ecstasy that believe in the reality of those moments of flight into joy surpassing our worldly understanding. Psychoanalytic and Marxist culture criticism fall into the former category, most of the counterculture and New Age (e.g., Jeff Kripal's Esalen, the sixties, etc.) into the second.

Where does postmodernism lie? I think that "officially" postmodernism is "Buddhist;" it wishes to eliminate anything essential and allow only the webs of difference (and consequent deferral) that most of its cannonical texts describe. In my usage, postmodernism tries to dissolve World 3 culture, i.e., culture that points toward experiences of enlightenment, into World 1, the realm of suffering (duhkha). But (in my view inevitably) postmodernist thought finds itself confronting moments reached ostensibly through its own deconstructive methods where the beyond breaks through, or at least beckons. Several of these occur in the thought of Jacques Lacan, which I think is why Derrida attacked him so severely. The structure of postmodern or poststructural thinking rests on a grid of oppositions that fail to capture any essential "self" in our material or intellectual worlds. In (real) Buddhist terms, there is a "failure to find" (anupalabh-) any permanent or self-subsiding entity there. Lacan's concepts of "the real" and "jouissance" raise the posibility of a transcendence that cannot be named or found. Transgressive within the patriarchal repression that has arisen within postmodernism Lacan's ideas suggest to me the relationship between prakrti and purusa in the Samkhya Karika. The nay-saying of self reached by prakrti (nasmi na me naham) points with overwhelming force at the seeing and I-ness that prakrti knows with all her being she is not. Lacan's "real" seems to me to be like this, something we can never know, and that we know more and more fully that we cannot know as we become wiser. On the other hand, the more sharp becomes our knowledge the greater becomes the bliss that rises from this understanding. Thus the real and jouissance are inextricably connected.

Cultural embers

The purpose of culture is to remember and create pathways from World 1 (the realm of suffering, duhkha) to World 2 (fulfillment, joy, enlightenment). As such culture is a "world of its own" that I call World 3. Tantra makes it very clear that this culture (including its version of "culture of the self") must be recreated at every moment, and that otherwise it tends downhill towards decay and ossification. A movie I saw last night pictures old culture at a moment not exactly of decay but of dying down, like a wood fire fading to coals. The Hermitage Dwellers is a recent Dutch film in Russian about three old and one young employees of the great St. Petersburg museum who find there a world apart from the continuing darkness and deprivation of the city (still "Leningrad" to the old people who lived through the German siege in the 1940s). The art gives off an afterglow of its original esthetic power and exposure to it--standing close by as if next to a dying fire--is enough to warm the souls of the museum "dwellers," but only if they take regular and sustained doses. Standing next to Rembrandt's "Prodigal Son" the young man, a former soldier with a guilty conscience, irradiates himself with the blissful moment of forgiveness that the art work shows, and the painting seems to translate him into a state of redemtion that is deep and genuine if also partial and fleeting. This, the film seems to say, is the most that old culture can do, give a modicum of comfort and distant intimation of transcendence to men who did not experience the original World 2 moment on which the work rests and lack World 2 moments of their own that these old beauties might find ways to touch more vitally. One thinks of the women in T.S. Eliot's museum "talking of Michaelangelo" or evening-clothed bourgeoisie at the Vienna opera immersed in the "Magic Flute."

Where is new culture today? Do others feel as I do that the last upwelling of nutrients from a lasting World 2 happened in the 1960s? The last great classical composers worked fifty years ago or more, and in popular music where are the equals of the Beatles, Velvet Underground, and Bob Dylan (aside from Mr. Dylan himself, who does not decay musically though he's now post sixty-four). Literature may be an exception here, though surely not in the English novel. And painting, while subject to the same drifiting lack of center (predicted by the great Yeats) does have moments of fierce originality of vision in the work of men who are physically old but still young in art (Cy Twombley and Brice Marden are two). Overall, though, there seems little but afterglow at which to warm our spirits. The only hope I can see, the same hope that was always there, lies in breakthrough to new Worlds 2. At any rate the old culture tells us, admonishes us, that breaktrough is possible and refuses our drift to the couch and the grave.

Siddhis and World Two

The "psychic powers" (siddhis) of yoga have been problematic, or even embarassing to some. What is the point of these apparently ego-focused abilities when the person is on the brink of release from ego? Chandogya Upanisad chapter 8 makes clear that the ability to obtain all desires lies in the atman-brahman identity, even as it equally shows that fulfilled desires below the level of that atman-brahman are temporary and ultimately undesirable. What is clear here is that only at the creative source are desires (kama-s) fulfilled, and that being at that creative source (recognizing that one is that source, that atman = brahman) is bliss. There is a push back toward beginnings in the Upanisads as in the Yoga Sutra, and those beginnings are realms of bliss. This is what I call "World Two," the state of fulfillment which is not only an "experience" but a stance. [Parenthetically, this is what Heidegger is about, always shifting from the purely mental and interior (experience, concepts) to an orientation, a stance or comportment towards Being.] Standing in ananda there is a sort of explosion of joy, akin to the moment of kensho in Zen Buddhism, which is expressed in exclamations of achievement of desired things--whatever happens to be desired. "Shivoham!" in the Sankara song is similar.

The state of bliss (brahmaloka = anandaloka) is a world, as van Buitenen suggests in his article "Ananda or all desires fulfilled." It is a positive reality in its own right, not just a relief from suffering. Siddhis and the various desired things spoken of in Chandogya 8 are alike in representing the spontaneous outpouring of World Two into this world ("World One"). In this way they are moments of culture, which I call "World Three." The joys of life are real only when they flow from World Two, from a stand in World Two. Pleasures confined to World One (Chandogya 8 shows that this comprises the waking, dream, and deep sleep states) are temporary and not really "enjoyable" (bhogya). "Magic," then, becomes an essential part of culture, which opens up a perspective on much of popular tantra.

How many purusas? As many as moments?

An important question in early samkhya is how many purusas there are. Upanisadic atman = brahman thinking might well suggest only one. Nevertheless, in order to account for the fact that we don't all share the same karmic burden or own the same psychophysical personality, samkhya concluded that there are as many purusas as there are sentient beings ("from Brahma to a blade of grass"). These purusas own different endowments but in their essential nature all amount to nothing more than consciousness. In the terms of Peter Pesic (in his book Seeing Double, MIT Press) the purusas are similar to elementary particles like electrons that are the same in all essential respects, and share "identicality." Some Buddhist thinkers (see Matthew Kapstein, Reason's Traces, Wisdom publishers) claim that there is a givenness of self-reference moment by moment that amounts to nothing but the momentary event of consciousness. In a sense there is something like purusa in each moment but a purusa that does not carry over to subsequent moments. Nevertheless there appears to be an "identicality" of consciousness in these moments that is very similar to one that applies between purusas in samkhya. An identity of "nature" between the consciousnesses within different moments seems implied in Buddhist concepts such as the Zen talk of "seeing eye to eye with the Buddha and patriarchs." The essence of liberation in samkhya is to see that the consciousness in all moments of experience is this completely selfless essence, and not an ahamkara (hence the recognition of "I am not" (naham) at Samkhya Karika 64). This is similar to Buddhist realization which also sublates the ahamkara (ego). The apparent difference is that samkhya ends with a permanent purusa and Buddhism with an endlessly repeated experience of momentary consciousness. That repetition is what I call culture, and I have argued that samkhya, rightly interpreted, implies the same thing (paper at AAR/DANAM, 2006).