The Profound Nature Of Exponential Growth: A Powerful Experiment

When it comes to 'exponential growth' the acronym 'IT' should be split into 'information' and 'technology' - and both have different fundamental limits.


Take a moment to imagine what the tech landscape looked like 10 years ago...

Do you have a sense that the pace of change is accelerating?

As a professional tech analyst I certainly have that sense. But while we all fixate on the capabilities of the hardware building blocks - storage capacity, processor speed etc. - this is not where I'm sensing an acceleration.

The feeling I have is one that the volume of information I have to deal with is increasing exponentially - along with the need to extract meaning, knowledge and insight from that information.

Such is the volume of information coming our way that we now rely on tools to deal with the tsunami, and nowhere is this more true than in the field of 'Big Data' where most of us feel that we do not actually want any more data because we are still trying to make sense of the data we already have.

To most people it is obvious that the reason for the information tsunami is that the rate of progress is increasing exponentially, thanks to Moore's Law.

But do we really understand what 'exponential' means as it applies to information technology, or IT? Is there something more to it than Moore's Law? 

I thought I understood Moore's Law pretty well, until I sat down with a spreadsheet and developed a startling thought experiment which allowed me to see that the 'exponential growth of IT' can be resolved into two different types of exponential growth: the first, and most familiar, relates to the 'technology' or hardware element of IT while the second relates to the 'information' element of IT.

But the most important realization is that the hardware and information elements are not just accumulating at different rates, they have very different fundamental limits.

The power of exponential growth

To conduct this experiment you will need:

  • One ticket to a sports match at a national stadium (e.g. Wembley Stadium, if you’re in the UK).
  • One thimble
  • One paper cup
  • One bucket
  • One life jacket (the purpose of this will soon become clear...)

Let's imagine that kick-off is at 3pm and that you're in your seat with your thimble, paper cup and bucket neatly stacked inside each other on your lap.

We will assume that the match consists of 2 halves, each of 45 mins with a 15 break for half time. So, ignoring injury time, the match will finish at 4:45pm.

Just as you've settled into your seat, an attendant comes up and stands beside you with a hosepipe. Starting at kick-off, at 3pm, she is going to deposit one droplet of water in your thimble. At 3:01, she will deposit two droplets of water – making three in total. Next, at 3:03, she will deposit 4 more droplets of water into your thimble. You get the idea. So we'll have a sequence like this:

3:00pm:

 1 droplet

3:01pm:

 2 droplets

3:02pm:

 4 droplets

3:03pm:

 8 droplets

3:04pm:

 16 droplets

3:05pm:

 32 droplets

So far this doesn't sound very interesting. But let's see how this plays out:

3:06pm

After six minutes, or after six doublings, the thimble is full so the water will start spilling out and into the paper cup.

3:14pm

We've now had 14 doublings: the paper cup is full and the first drops of water start spilling into the bucket.

3:21pm

The bucket is full 21 minutes after kick-off, after 21 doublings. Because we do not have any larger receptacle we will just have to tip the water down the steps. 

And don't worry about trying to capture any more water in the bucket because things are about to get interesting...

3:36pm

36 minutes after kick-off, or after just 36 doublings - and when the match is still in the first half - the amount of water is large enough to fill an Olympic swimming pool (about 2,500 cubic metres).

Safety warning: please put your life jacket on NOW!

3:46pm

After another ten minutes, the water is lapping at the top of the stadium and the game is long over as everyone floats on the surface (just as well you had that life jacket!)

But we're not finished - things are about to get really crazy...

4:25pm

Ten minutes before the end of the match the volume of water is equivalent to all the world's oceans.

Remember that this is the result of just 85 doublings. For reference, the number of water droplets deposited over the first five minutes was 1, 2, 4, 8, 16 - which represents 4 doublings.

85 doublings might not seem very different to 4 doublings, but when the volume is accumulating exponentially the difference is profound. 

4:35pm

Just 10 minutes later, the volume of water has grown to reach the volume of the Earth. 

After 95 doublings a single droplet of water has grown to the size of our planet.

8:00pm

Just to take the thought experiment to its final stage by 8pm, just as you are sitting down to have dinner, the volume of water has now reached the volume of the known universe.

So it takes just 5 hours or 300 doublings, to go from one tiny water droplet to a volume of water that equals the volume of the known universe.

Implications

So what can we conclude from our thought experiment? I’d offer three conclusions:

  • Conclusion 1 (not controversial) : exponential growth can quickly get really crazy.
     
  • Conclusion 2 (controversial) : no physical process can continue accumulating exponentially forever. An analogy would be the data storage capacity or computational density of matter (e.g. Bytes per kg of matter, or FLOPS per kg of matter);
     
  • Conclusion 3 (very controversial) : a non-physical processes, for example the accumulation of information or even intelligence, can continue accumulating exponentially after the physical limits have been reached.

    For example, when it comes to data and, specifically, the valuable patterns that lie ‘hidden’ within that data, accumulation occurs faster and can keep going for longer.

In other words, if raw storage and computational capabilities increase at a rate of, say, R per year, then the value of the associated information will increase at a rate that is greater than R – because this is not subject to the same physical limits.

Here's an interesting analogy that lends some support to this statement: we know from mathematics that there are many different infinities and that some are far, far larger than others.

Hence, it is possible that the 'T' part of IT will reach a fundamental limit before the 'I' part.

In other words, while our thought experiment shows there must be a fundamental limit to the raw storage and computational density that can be achieved with 1 kg of matter (and I'd say that limit is far beyond the capabilities of the human brain), then this may not be a problem because by the time those limits are reached, the data processing capabilities will be effectively infinite, even though they are not really infinite.

Ultimately, the real value lies in being able to recognise patterns in the data and, returning to our 'water droplet' thought experiment, it might be that when it comes to extracting value from the data that pertains to our current reality then we are still in the first few minutes of the match.