Anybody heard of the idea of The Singularity? Roughly, it goes like this: technological progress builds on itself, and this self-reinforcing feedback loop is eventually going to come to a head where humanity makes a quantum leap into an unknowable and godlike transhuman technological future – possibly as early as the middle of this century. The name of the idea comes from mathematics, where approximately speaking a singularity is a place where a function rockets off to infinity. Alternately, some adherents of this type of thinking believe that progress is exponential; formally this doesn’t result in a true singularity but in this type of scenario it makes little difference.

I’m (probably) not going to take sides today. Instead we’ll just look at some math that’s relevant. Say you have a function P(t), that measures technological Progress as a function of time. Understandably this is hard to quantify, but as a thought experiment you get the drift. We might say that the rate of increase of progress is proportional to the progress we’ve already made. This is itself just a statement of a differential equation:

The solution to that equation, assuming suitable initial conditions, is just the exponential function:

Ok, nice enough. But *is* the rate of progress really just a function of previous progress? It can’t always be. Take a heat engine, for instance. Throughout history people have turned heat difference into useful work in various ways, and today many heat engines are pretty efficient. They’re getting more efficient all the time. But we do know as a matter of basic physical law that there is a limit – no engine can beat the Carnot limit for efficiency any more than a vehicle can beat the limit for the speed of light. Progress there can’t rocket off to a singularity, physics causes the path of progress to lead into a deepening swamp terminating in an impenetrable brick wall.

So let’s try a different model. What if progress is proportional to previous progress multiplied by a factor that shrinks as progress approaches the physical limit? For convenience, scale things such that the limit is 1:

When there are physical limits, this is a much more realistic equation than unlimited exponential growth. It’s solvable, and the resulting P(t) is called the logistic function, and it’s our Sunday Function:

Graphed, it looks like this:

Now here’s a key thing to realize. If you’re looking at this function well to the left of the origin, it looks pretty darn exponential. This is because if you let t become large and negative, e^-t is huge (because negative times negative is positive). This means the 1 + e^-t is essentially just e^-t, because that term is so much larger than 1. And that means that we have effectively 1/e^-t, which is just e^t. So if you’re in the past relative to the origin, you might well think you’re looking at true exponential growth which will explode into infinite progress in the future. But you’d be wrong – logistic doesn’t stay exponential as time keeps marching on.

And since physics does set limits, be they light speed or Carnot efficiency or Heisenberg uncertainty or whatever, I suspect that technological progress is logistic. The remaining question is the height of the limit. If it’s much much higher than progress as of 2009, then maybe we will get what amounts to a singularity. I rather doubt it though, many types of technological improvement have already come near their limits – CPU clock speed, for instance, has been stalled around 3 GHz* for a while now. On the other hand, some things like our understanding of the human genome are not even within a tiny fraction of their probable maximum.

But maximums they have, and most of them don’t look like they’re completely out of sight. So count me out of the singularity boosting. The future will be amazing, but probably not impossibly so.

Did I say I wasn’t going to take sides? Sorry.

*For clarity, clock speed is not a measurement of performance as such. A modern processor can do much more per clock cycle than previous generations. But it is a good example of one branch of computational advance that’s not terribly likely to continue increasing indefinitely.