Peter Cochrane's Blog: Tech growth and decline - why people just can't do the maths

Too many stumble over the basics...

...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.

Graph showing how stacked logistic curves create the illusion of a continual exponential

This graph shows how stacked logistic curves create the illusion of a continual exponentialImage: Peter Cochrane/

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.

Graph showing Gordon Moore's famous law of transistor packing density growth

Gordon Moore's famous law of transistor packing density growthImage: Peter Cochrane/

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.

About Peter Cochrane

Peter Cochrane is an engineer, scientist, entrepreneur, futurist and consultant. He is the former CTO and head of research at BT, with a career in telecoms and IT spanning more than 40 years.

Editor's Picks

Free Newsletters, In your Inbox