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Dynamical Systems & the Technological Singularity (Part 0)

Being a more pragmatically-minded grinder, I try not to get hung up on predicting the future, or writing theoretical articles about how great the singularity is going to be. In this entry (which will probably end up being a series), I'm going to partially break that rule, and analyze the singularity from the perspective of dynamical systems theory; in particular, I'll attempt to show what the mathematics say and, more importantly, what they don't say. This is vitally important, as transhumanists and singularitarians frequently draw conclusions about the singularity that are unjustified by the actual mathematics. For those who don't know what the technological singularity is, you can read this Wikipedia entry for a basic understanding.
So, dynamical systems theory is, not surprisingly, the study of dynamical systems. The term "dynamical system" itself has a few equivalent definition, but the one I shall use is the following:

*Definition*: Suppose we have a map F: U ⊆ ℝ → ℝ, a time interval I ⊆ U, and an initial condition (t_{0},x_{0}) ∈ (I,ℝ). Then we have a *(1-D, homogenous, continuous) dynamical system* given by the solutions x to the differential equation

|x'(t) = F(x) |x( t_{0}) = x_{0}

Usually, for convenience, we set t_{0} = 0.
Why is this important? Because the Law of Accelerating Returns (LAR) is itself a dynamical system, and can be analyzed according to the rules of dynamical systems theory. It is given by the equation

|x'(t) = ax
|x(0) = x_{0}
where a < 0 is some constant representing the rate of decay, x(t) is the cost of information technology at time t, and x_{0} is the current cost. In ordinary language, it states that the cost of information technology falls at a rate proportional to the cost itself.

This is very important to note: *this is all the LAR says!* Kurzweil himself will say this. The LAR therefore cannot be used to derive any additional conclusions about the so-called march of technology, at least not without plugging in additional facts or assumptions about society, the economy, etc.
*EDIT: It looks like I'm going to have to make this a series after all. The next entry in the series is going to contain examples of how to solve the LAR under certain societal conditions.*
*~ian*
Ian,

ReplyDeleteI just ran across your blog. I'm actually starting some research on this very topic! Specifically, analyzing the probability of the technological singularity using dynamical systems and chaos theory. I'm starting by considering the mathematical models presented in Anders Sandberg's article "An overview of models of technological singularity". I'm still in research phase, any suggestions??

Thanks