The (not-so) Exponential Growth of Knowledge

A couple of weeks ago I enjoyed a dinner in London with other alumni of University College Oxford. In speaking about the state of the college, the Master of Univ said that it was a constant struggle to decide what to include in undergraduate courses because,

"...our knowledge increases exponentially."

I know the Master was speaking figuratively, but being an engineer I started to wonder if knowledge actually does increase exponentially, and what conditions would be necessary to make this happen?

Where to start? Well, let's start with Einstein, or rather one of Einstein's collaborators, John Archibald Wheeler. In 1992 he said,

"We live on an island surrounded by a sea of ignorance. As our island of knowledge grows, so does the shore of our ignorance"

How fast does this island of knowledge grow? Let's start by making some assumptions about Wheeler's universe, namely:

  1. We expand the island by reclaiming 'knowledge land' from 'ignorance sea'.
  2. We always have enough resources on the island to reclaim land, but...
  3. We don't have any boats, so we have to stand on the island to do the island expansion work.

All of these assumptions add up to the realization that the rate of expansion of the island is proportional to the length of the shoreline. So if our island is round, or roughly round, and the area of the island corresponds to the amount of knowledge, K, then it will have a shoreline length, s, equal to the circumference of the circle. So we can write:

\frac{dK}{dt} = As = A\cdot2\sqrt{\pi K}

 

John Archibald Wheeler's Island of Knowledge surrounded by the Sea of Ignorance.

John Archibald Wheeler's Island of Knowledge surrounded by the Sea of Ignorance.

With an ignorance-reclaiming-speed of A m/s. This doesn't give us the exponential rate that the Master spoke of, it's merely quadratic:

 K(t) = A^2\pi t^2 .

Perhaps if the island were a more convenient shape we could achieve exponential growth? In fact a circle is the worst possible shape for the island because it has the lowest ratio of shoreline to area of any 2D shape. The best possible shape would be long and thin, infinitely thin in fact, then area and shoreline would be proportional, and so our island would grow exponentially, wouldn't it? Well, no, it wouldn't, at least not for very long. In order to maintain exponential growth the island must stay infinitely thin and so it can only grow at its ends, but this isn't really the land reclaim model we started off with. You can hardly say that shoreline length is the limiting factor in the growth of an island if you insist on reclaiming land only at the infinitely thin ends of the island! In fact we find an interesting result that if we grow uniformly from each part of the shoreline then no matter what clever shape we start off with we'll always end up with a circular island! Even if we start with exponential growth of the island (perhaps from some clever fractal geometry), we will soon settle to the growth rate given above, which increases with time, but is nonetheless far from exponential.

In fact there may be a paradox even in the Master's statement of the problem: if knowledge grows exponentially, then the growth rate must be proportional to the amount of knowledge. Growth in knowledge requires research, and researchers, but the Master's statement was itself concerned with the fact that each year we must select a smaller fraction of our ever growing knowledge to teach to the next generation of researchers. If this fraction is decreasing, then it looks like we're on a circular island of knowledge - it will grow ever faster, but at the same time, ever slower than exponential growth.

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