poetix (old content)

oh build your ship of death

A Note on Sets and Time

It is important to understand that a set can be uncountably infinite, and that a continuum can be a set just as much as an arrangement of discrete entities. A good example is the set of real numbers. It is uncountably infinite, which means that the real numbers cannot be put into a one-to-one correspondence with the natural numbers, and a topology can be defined on it, which means that it can be shown to satisfy a certain definition of continuousness*.

This has some consequences for the ways we can think about time and change in relation to a set-theoretic ontology. “Cinematic” time, a succession of discrete snapshots of the state of things, can be represented as a function from the set of natural numbers to a set of states: 0 maps to the first “frame”, 1 to the second, and so on. But what about a function from the set of real numbers to a set of states? This would guarantee that for any two states, at times T1 and T2, there would be an uncountably infinite number of “intermediate” states at times between T1 and T2. The complaint that such a function represented a “static” image of change would seem to lose some of its force in this case, since the resulting graph would have a sufficient “density” to capture the most smoothly graduated becoming.

For Deleuze as for Bergson, a becoming is not a series of instantaneous states (ways of being) but a transition taking place over a duration: a static “snapshot” of this transition cannot capture its essential property of being in motion. A flux is always subtracted from any “closed” set of images or crystallisations of the flux, flowing beneath and around them. But in what does this subtraction consist? If we consider an audio signal chopped up into samples, it is possible to state very precisely what characteristics of the signal will be lost in the process (depending on the sample rate, we will lose all frequencies above a certain threshold). Suppose we say that the signal contains an infinitely ascending harmonic series; then no sampling of the signal can possibly capture all of its “content”. The signal has a “density” greater than that of the sample-set: the two are not isomorphic, since there are differences in the signal that do not correspond to differences in the sample-set.

However, this is not so if the set we choose to represent the signal is the graph of a real-valued function. There is no difference that can manifest itself in the signal that such a set cannot register, and thus no sense in which the flux of its becoming can be said to exceed or overflow its overall state of being-over-time.

* This would be one of those moments where it becomes apparent that I’m not a proper mathematician. Any passing proper mathematician is welcome to tell me what I really mean.