Written at Grenoble airport and dispatched to silicon.com from my car via 3G later the same day at 2.1Mbps.
Exponential functions are very useful in describing many natural and unnatural phenomena such as the diffusion of a gas or the growth of a technology.
But the word ‘exponential’ is often used out of context and the real meaning bastardised by erroneous context and misunderstanding.
I was recently at a presentation given by an economist who proudly declared that he did not believe in exponentials. These are functions of the form ab and very commonly expressed in the natural form ex or EXP[x].
His thesis was that markets and other activities such as technology growth do not follow the exponential curve. They are governed entirely by the logistic or S curve.
He also postulated that nothing progresses much beyond the 50 per cent point of the S curve before deviating, flipping and gong into decline.
There was nothing new or novel in this revelation. He was correct in a way but reality is much less kind to the theory. Many activities follow exponential growth or decline or follow the S curve, in an approximate manner – much in the same way that data aligns with the normal or bell curve in probability and statistics.
To get a good fit to any mathematical or theoretical curve we often have to think in terms of a sufficiently large sample over a sufficiently long time.
Even detailed and accurate scientific results suffer from scatter due to measurement uncertainties and the stability of the environment and materials.
Little surprise then that we see the growth of computing power, memory, networks and mobile phones as being about the exponential line rather than following it with great precision.
In textbooks and academic papers the representation of the product birth-to-death process tends to be somewhat idealised and we have to remember to colour all these models with a degree of practical experience and judgement. Even getting accurate and timely sales information in the real world can be a challenge.
When a market becomes saturated, or a gas has completely diffused, all activity will grind to a halt, and in the ideal case it will be defined by the logistic curve. There really are no surprises here.
However, what this economist was not getting is that technology does something rather magical. It…
…stacks the S curves of successive generations as processes are continually refined, and designs are improved. And this phenomenon has been understood for a long time too.
But, to further confuse the issue, technological growth cannot be illustrated on the simple linear scale of a standard graph because it just won’t fit.
For example, fewer than 10 transistors on a chip in 1958-59 had grown to well over two billion by 2011, and it’s impractical to record such a scale on anything linear. So this is where a little mathematical juggling comes in handy.
By applying a logarithmic transformation, one can make the S curve conveniently linear, and nine orders of magnitude or more can be accommodated on the vertical axis.
This approach is exactly where a lot of confusion now lies. By and large the use of logarithms is no longer taught in schools or many college or university courses.
So, when technologists and scientists present their results, they are very often misunderstood by lay and otherwise educated people. Politicians for example seem to be the most prone to talk of “exponential growth” while not understanding what it really means.
Gordon Moore’s Law is perhaps the most common example of people not getting it, because the use of the exponential and logarithmic, often in the same sentence, presents a bit of a false picture to anyone not on first name terms with the mathematical basics.
This issue is illustrated in the growth curve below, where we see a scatter of results along an S curve, which is now a straight line because of the logarithmic scale on the y axis.
No doubt my economist friends would point out that eventually Moore’s Law too will come to the end of the road and exponential growth will cease as silicon becomes obsolete. But there is absolutely no sign of this decline happening yet.
In fact silicon has allowed us into the nano and bio worlds in a manner and on a scale thought impossible just 30 years ago.
As a result we have in front of us a raft of new technology options that offer a route to a world that will go even further and faster that silicon.
So we can look forward to a new set of S curves based on nano-materials and structures sitting atop the many past generations of silicon. This is the real picture, and not one of impending collapse around the 50 per cent point.