Advice to the Humanities scholar:
Do some math!
WILLIAM J. URBAN
Given the nature of my presentation, I thought we’d start with something mathematical:
This schema is taken from a lecture given in 1979 by the French philosopher Alain Badiou which can be found in his book entitled Theory of the Subject. And yes I put this graph up there just to scare you: if it momentarily brought you back to the horrors of high school geometry, my job is pretty much done as you’ve no doubt already made some sense of this for that very reason. I actually tried to find something more intimidating, but this is the best I could manage which still claims to directly involve the human subject.
Obviously Badiou put some work into this. He’s certainly trying to capture something with it. But if he ultimately fails to clarify subjectivity with such a complicated graph, we should remember that this is nevertheless our failure. After all, we are nothing if not subject to our own experience and this is especially the case with mathematics, as mathematics has surprisingly everything to do with the subject. I’d like to give you a sense of this today, of how we mustn’t secretly count one of the benefits of being Humanities scholars as having ended our days of calculating values for unknown algebraic variables. Indeed, would this not be the very mathematical expression of textual interpretation as such, since interpretation provides additional value to textual terms? But far more than this, I’m proposing that mathematics – more than any other field – most clearly involves the direct inscription of subjectivity at its very foundation. And if this is the case, we may not be simply dodging a small discomfort by avoiding mathematics, but rather much more severely evading the most pressing question that confronts us today in the Humanities, namely: what exactly is the status of the modern subject? It’s increasingly become my belief that we are doing ourselves a great disservice by ignoring the work done by mathematicians since the turn of the last century.
This is not to say that if you asked mathematicians how they formalized the inscription of subjectivity at the very foundation of their discipline, that you would get anything but odd looks. As with the Humanities student who finds himself before a text to be interpreted, the mathematician likewise finds himself before a math problem to be solved and more often than not, both immediately set to work without reflecting on what first initiated that work. I recall how my own high school math teacher – Mr. Noce – did not seem to have this reflexive capacity, despite often telling us that he was, and I quote: ‘the Best damn math teacher this side of the Mississippi.’ Mr. Noce was a giant of a man with a fist the size of a basketball which he used quite effectively to punctuate such personal maxims by pounding that fist into the blackboard. This certainly grabbed our attention. Suffice it to say, he largely taught us through intimidation which perfectly complemented our own teenage fear of calculus. But there was the rare occasion in which his lesson plan became suspended as something mysteriously would draw him to the blackboard and set him to work with his back turned to us for many silent minutes.
To begin to understand what entangled Mr. Noce in those moments, we need to examine the 9 axioms of set theory which compose the very foundation of mathematics. These 9 axioms were completely formalized by 1925. Today, I’ll only introduce you to 3 of these. The first axiom is the Axiom of the Void or the empty set:
AXIOM OF THE VOID
There is an empty set ∅ which provides
the sole existential basis of set theory.
It is unique.
This axiom is exceptional in that upon it alone rests the entire derivation of mathematical presentation as such. This is because the void, precisely as void, is universal and thus always included as part of all other sets. In fact, the very existence of numbers depends on a ‘zero-level’ of creatio ex nihilo, of how a One emerges from the empty set by merely taking the set of the initial empty set. And because the empty set can never be adequately signified, subsequent sets of these sets can continue ad infinitum. Now, if I tell you that for Lacan, this void or empty set is precisely the subject itself, you can begin to understand that what entangled Mr. Noce had something to do with his very own subjectivity. But it is crucial to understand that this is not so because this subject is pre-existing; rather, because the axioms of set theory are entirely prescriptive, each axiom only exerts a certain material ‘weight’ inasmuch as it results from a choice made by the subject. In the case of deciding on the void, this means that the subject must paradoxically choose itself, and must do so from an empty place. We must thus read the subsequent working out of a mathematical solution as a pseudo-problem for Mr. Noce, only meant to relieve him of the traumatic confrontation with that gaze of mathematics which initially drew him to the blackboard, a gaze which indexes his own subjectivity as void.
We Humanities students also relate to ourselves in just this manner. This fact is most evident when we’re directly asked to defend our choice of texts. Of course, we’re always prepared to relate a story to justify our selection. In our tradition, this involves what is known as ‘the context.’ But it’s often the case that the more persuasive the context, the more we suspect that things could yet be otherwise. In the end, no matter the circumstances, ultimately it is we who are held to account for the choice of texts and the interpretation of them. As they say, there is no accounting for taste and by extension, no textual engagement without its proper aesthetics.
But to avoid a fatal misunderstanding, if the subject is the empty set and thus obviously a finite being, this paradoxically cannot be directly asserted. The only way to do so is via the second axiom:
AXIOM OF INFINITY
There is at least one infinite set
which ensures actual endless expansion
of any existence.
The Axiom of Infinity involves a choice regarding the infinity of the universe. Let’s say you directly define yourself as a finite being, existing in a universe among other beings. But doing so is the height of arrogance, since you already objectify the limit between yourself and the world; ie, you adopt an infinite position from which you can observe reality and locate yourself within it. Rather, the only true way to assert your finitude is to accept that your world is infinite. Only in an infinite universe is there a lack of exceptional points such that you cannot locate your limit within it.
This much more radically de-centers us from the text than is commonly thought. As we cannot directly posit ourselves as finite to the infinity of the text, we must place ourselves at the very center of the textual universe in order to avoid any false claims to neutral readings. Perhaps surprisingly, this also calls for a new twist on today’s familiar story of how can we can never fully account for the context of the text, as every context must itself already be contextualized. What the empty set ensures is that this cannot be an infinite regress. This is because the ex-sistence of the empty set necessitates that the hermeneutical circle of meaning be always linked to a point within the field disclosed by it. This means that the frame of our view when we interpret a text is always itself framed by a part of its own content. Here is a point which resists all attempts to incorporate it into a meaningful context, as this point embodies meaning-as-such. And it is this point from which the text looks back at us. This is a disturbing experience no doubt, but one that can serve us well as it teaches us the proper interpretive gesture of finding the one paradoxical element in the text that “is” the absence of the empty set itself. If this is not the case, then there is no way to account for why the hermeneutical framework does not collapse in on itself. The empty set performs this crucial function of keeping such minimal distances. The subject – precisely as the empty set – is in the center of it all.
Some of you have likely noticed that despite my talk of choice of the empty set and of infinity, I’ve actually presented these two axioms as existential facts, precisely as you find them in math books. You may thus be tempted to conclude that everything I’ve just said only works this way if you in fact make these particular choices. But you would be moving too quickly, for we are not in the universe of the Marxist where we are free to choose but not under conditions of our own choosing. On the contrary, as Lacan said, we are free to choose only so long as we make the correct choice. And oddly enough our third axiom The Axiom of Choice, says exactly the same thing:
AXIOM OF CHOICE
The function of choice exists.
This is the precise concept of the being
(not act) of subjective intervention.
This axiom amounts to making a choice regarding choice itself. A moment’s reflection will confirm to you that this is unavoidable: if you’re confronted with a choice, you have already chosen. That is, you have chosen the very choice itself. The only thing remaining is the formal gesture of properly acknowledging that choice.
The ethical dimension of this tautological gesture is lost on most mathematicians, including my former teacher Mr. Noce. They overlook how pure subjectivity forms the very foundation of their discipline and argue that mathematics is on firm ground. However, we Humanities scholars know mathematics as a discipline that is self-grounding and completely dependent on a subject who is paradoxically produced by this very grounding activity. The only remaining thing for us to do is to apply this insight to our own field. This would require that we suspend our hermeneutical strategies long enough to see how these strategies themselves only endlessly defer our encounter with the traumatic gesture of subjectivity.
So to the Humanities scholar seriously interested in the human subject, my advice to you is to do a little math. But I warn you! There you will encounter an uncomfortable fact: if mathematics does indeed inscribe subjective choice at its very foundation, this means we avoid mathematics only because we can’t decide for ourselves.
Fun puzzles
