The Derridean thing to do with a title like the above is to work the double genitive: mathematics is “of the trace”, arising out of the movement of the trace, rather than grounding the trace in some fixed structure. In one sense, this is fair enough: there is no mathematisation without a stock of marks, without iterability and so on. Mathematical writing displays nakedly what Derrida was calling, in Structure, Sign and Play, the “structurality of the structure”: mathematisation is structuration, by means of the mark; it’s a tracing, the delineation of a non-presence, and so on.

However, rather than talking about the way mathematical writing is embedded in Derridean écriture, what I want to do here is give a very basic account, using mathematical language, of an infinite referential structure. The basic idea is this: A refers to B, which refers to C, and so on; the referential chain never ends, and no entity ever appears except insofar as it is positioned within such a chain. What’s more, every entity may appear in multiple chains, as referent of many of other entities and referring in its turn to many more*.

Let’s start with a single referential chain: A refers to B, which refers to C, and so on. We have a set of “posts” in this chain: S={A, B, C…}, and a function f which has S as both its domain and codomain: f sends every member of S to some other member of S. In the case that f sends A to itself, we say that A is a “fixed point” of f.

If f sends A to B, and B to C, and so on, then we have a referential chain such that repeated applications of f will take you from A to any other “post” in the chain: f(f(A)) = C, f(f(f(A))) = D, and so on. Abusing notation a bit, we’ll write f ⋅ f (”f following f”) as f², and f ⋅ f ⋅ f as f³.

It’s possible for a chain to form a loop, e.g. A refers to B, B refers to C, C refers to A. In this case, A is a fixed point of f³, because f(f(f(A))) = A. Furthermore, every other post in the loop is also a fixed point of f³. If the set of posts is finite, then every chain must eventually run into a post that it’s run into before: every chain will terminate with a loop. There is therefore some fⁿ which will take every element in a finite S into an element that belongs to some loop. If we call the collection of elements belonging to loops S’, then there is some other fⁿ for which every element of S’ is a fixed point. We say that f is “permutative” for S’: after a given number of repeated applications, it will always return you back to where you started.

Only if S is infinite can we say that it is possible for some referential chain within S not to end in a loop; for example, S might be the set of natural numbers, and f a function which sends 1 to 2, 2 to 3, and so on. If S is infinite, then it’s possible to have an infinite number of “parallel” infinite chains that never join; for example, the natural numbers can be split into a chain of odd numbers and a chain of even numbers.

The question that arises here is: can a “signifying chain” ever be infinite, if the universe of signifiers is not itself infinite? Derrida’s answer here is something like this: as we pass along the signifying chain, we introduce an “alterity” into each signifier with each referential step. This is what “iterability” means: each time a mark is re-inscribed, each time a “post” is re-encountered in a referential chain, there is something novel about it: you can’t step in the same stream twice.

We can model this by introducing a temporal dimension into our universe of posts. Let S be a finite collection of posts, and T be an infinite set of temporal indices (or “moments in time”). The Cartesian join, S ⊗ T, of these two sets is a set of pairs, (A, t_n), or “A at time t_n”. We now redefine our function f as having S ⊗ T as its domain and codomain, so it takes A at time t₁ to B at time t₂. If it then returns to A at time t₃, this does not form a loop in the universe S ⊗ T, because (A, t₁) and (A, t₃) are not the same elements within this universe.

As is well known, Derrida defines différance in terms of both “spacing” and “temporalisation”: the referential movement of the trace crosses space as well as time, so that any finite set of referential posts is always crossed with an infinite set of times and places. This is how Derrida resolves the structuralist problem of how a finite symbolic domain can be transformed in any way that isn’t ultimately permutative, that doesn’t ultimately consist of going round in greater and lesser circles. Any such finite domain (e.g. Foucault’s episteme) is, according to Derrida, just a projection from the spatially and temporally infinite domain within which marks and symbols are actually inscribed; and this is true of any “present” in which the marks within a signifying chain can be gathered together to form a finite set (e.g. that of “the book”).

Seen in these terms, Derrida’s attempt to think past the impasses of structuralism has the twin merits of being both ingenious and fairly straightforward. If you pay attention to where, in Derrida’s arguments with structuralism, it becomes especially important that something should be “infinite”, then it should become apparent that the problem under consideration has a mathematical character: it hinges on the relationship between finitude and permutativity. The means of escape is to “reinscribe” the finite contexts examined by structuralist “science” in a “larger and more powerful” - i.e. infinite - context. What Derrida seems not to have considered was that this infinite context might also be presentable as a consistent multiplicity - or, to put it another way, that the infinite could be mathematically treated.

* To cover the many-to-many referential situation, we need to extend the picture given here of a set S with a single endofunction f to one of a set with multiple endofunctions - which can also be seen as a monoid. Each endofunction prescribes a single chain passing through each post in S, and the total referential situation of each post is given by all of the endofunctions that pass through it. Of course, part of the point of Derrida’s discussion of the trace is that there is no “total referential situation”; but this is because he assumes the non-totalisability of an infinite situation.