“Sideways” (2004): 5/5
Derrida closes the first section of “Of Grammatology as a Positive Science” with a meditation on the three most prominent neologisms in the text — or more specifically, on “this unnamable movement of difference itself, that I have strategically nicknamed trace, reserve or differance,” (93, emphasis in the text). It is the last of many definitions or descriptions of these words that occur throughout this first part. By this time we have already learned, from page 75, that “The trace is nothing…[it] exceeds the question What is?” These relatively straightforward definitions are sprinkled among a number of more circular allusions, particularly those concerning the ‘thought of the trace’; for example, from page 62, that “a thought of the trace can no more break with a transcendental phenomenology than be reduced to it,” and from that same passage on page 93, “a thought of the trace, of difference or of reserve…must also point beyond the field of the epistémê.”
These definitions are clearly paradoxical, both when compared to one another and, sometimes, internally. Derrida himself admits as much on page 62: “It is in fact contradictory and not acceptable within the logic of identity.” This is what many philosophers tend to use as grounds to attack Derrida (at least as I understand it) when they criticize his theory as practicing metaphysics while claiming to escape metaphysics, as Foucault seems to when he accuses Derrida of locating ‘the meaning of being’ in, precisely, the ‘reserve’ (p. lxii).
But for me, these paradoxes did not so much invalidate what Derrida was saying as remind me of the (once) paradoxical foundations of the concept of mathematical continuity. It struck me, while reading Of Grammatology, that Derrida might have been trying to usher in a new realm of philosophy in a similar manner in which Leibniz and Newton ushered in a new field for math. And for the purposes of this essay, I will argue just that — and moreover, that the conceptual ground that Derrida is trying to break open is homologous to the mathematical ground opened by calculus.
One of the key concepts in the development of calculus was the idea of the infinitesimal. It was taken in the 16th century to represent the idea of an infinitely small quantity greater than zero. The paradox inherent in such a definition is trivially easy to show. The infinitesimal was defined as above zero, but a quick proof would show that it had to be zero in order to be infinitely small. It had to be zero — “nothing”, you might say — but it was also not.
Despite these paradoxes, though, the infinitesimal was necessary in the proofs for calculus, and calculus worked. The infinitesimal was at once ‘necessary’ and wrong, and mathematicians used it in full knowledge of its wrongness — under erasure, you might say — because they needed it. Because what the infinitesimal theoretically allowed was the formulation of a continuous space out of our otherwise discrete number system. In such spaces, such as the so-termed “real line” between 0 and 1, an infinite quantity of numbers reside: 0.1, 0.01, 0.001, etc. And while the infinitesimal came in the shape of an algebraically discrete number, the thought of the infinitesimal, its use in theory, allowed mathematicians to go beyond discrete number space.
While it is almost certainly a gross simplification, it seems to me that Derrida’s trace — especially as it relates to the practice of deconstruction — is attempting a similar achievement. Derrida is attempting to open a space for infinite meanings within any ‘interval’ so defined by a text. Which is not to say that he is opening a space of unbounded meanings — just as there are an infinite quantity of number between 0 and 1, but 2 is not one of them, there are an infinite number of meanings to Hitler’s “Mein Kampf”, but it would probably be fair to argue that the idea that Hitler was indifferent to Jews is not one of them. And moreover, the role of the infinitesimal and the role of the trace are strikingly similar: just as the trace is the ever-shrinking space of association between any two things, the infinitesimal is the ever-shrinking proximity between one number and the next. If we think of mathematical distance as difference, both terms (infinitesimal and trace) fundamentally represent the phenomenon of an ever-disappearing difference that yet never entirely disappears.
In modern calculus, however, the infinitesimal has latterly been replaced by a more recent concept, that both stands up under scrutiny and plays the necessary role of the infinitesimal in the formulation of continuous space: the concept of the limit. The limit of a function is what a function approaches but never reaches. Importantly, the function doesn’t necessarily need to be defined at the limit point for the notion of the limit to make sense. This is another concept of ever-disappearing difference, but taken from another perspective: not a description of the difference itself, but of the (always deferred) process of differing — differance. It is the infinite approach towards a definite point, towards zero difference, that by definition never reaches its destination; the singular meaning of the word that is the infinite approach-towards-singular meaning. To use an example from Of Grammatology, if writing and speech are defined by the opposition between them, we might think of the two signifiers as functions approaching the same limit point/signified (arche-writing) from opposite sides, which neither ever reaches; the difference between them vanishes to the extent that neither is rigorously distinguishable from the other, as Derrida painstakingly shows in his book, while at the same time they are still defined by that never-quite-zero trace of difference between them. Both the infinitesimal and the limit (the trace and differance), in this manner, could be said to describe ‘the motion of difference’.
I’m tempted to take the analogy one step further, to bring in concepts from probability and start talking about a continuous distribution of meanings, none of which is ‘the meaning’ of a text (or of a word) but some of which are far more likely than others, and none of which exclude the possibility of infinitely distant meanings — but I imagine that I’m already well overstepping my bounds. I imagine that I’ve brushed a great deal of subtlety under the rug without realizing, that I’ve reduced my own understanding of Derrida’s unstable terms of instability to a stable set of concepts that I’m more comfortable with. But all the same, I find it useful to think of his approach in this manner, particularly as a defense of those who critique deconstruction as an ‘anything goes’ approach, allowing any interpretation that the reader wishes onto the text: to say that infinite interpretations are valid is not to say that any interpretation is valid. An infinity of meanings need not be unbounded. And just because a concept contains paradox doesn’t mean that it does not also contain truth.
