Graphing the Future...

Naturally when reviewing non-fiction one needs to pay attention to the substance of the author's argument rather than just the style.  Even though I found Ray Kurtzweil's style in his "The Singularity is Near" extremely annoying, his arguments are cogent enough to require careful examination.

The first section of the book is crammed with graphs, most of them log-linear plots of some measure of technological progress over time.  The trends are all basically lines, indicating exponential growth, which he uses as evidence for his "Law of Accelerating Returns."  This is a generalized version of Moore's Law which he uses as the basis for his predictions.

Some of the math is misleading.  The most pernicious graph in this section is the "Countdown to Singularity", which has been replicated in many places all over the Internet.  This page, for example, pretty much makes the same observation that I make here.  Although this graph of "paradigm shifts" appears like a laser-straight line through history, it's entirely illusory.

Consider first that this is a log-log plot, with "Years Ago" along the bottom ranging from decades to millennia, and "Time to Next Event" up the side.  Since the vertical axis is the difference between successive values on the horizontal axis, it's impossible for any event to appear in the upper half of the graph above the 45 degree line, simply because no event can have a time to next event greater than its own age.  Using a log axis for time also makes it impossible to use a graph like this for prediction because it cannot represent future times, only times infinitely close to the present.  To account for the future we'd have to adjust all the "Years Ago" values for all events, changing the shape of the graph in a non-linear manner.

(You may wonder how the last dot gets its time to next event.  I believe the most recent event is the widespread adoption of cell phones, which does not itself appear on the graph since there hasn't yet been a next paradigm shifting event.  How he could consider cell phones a paradigm shift is beyond me, unless somehow getting the power to talk to yourself in public restrooms - a power once only reserved for schizophrenic homeless people - advances us on the path to transcending biology.)

But the main problem with the graph is the choice of events themselves.  Take the third event for example.  Labeled simply "Cambrian explosion (body plans)," this event summarizes and encompasses a vast number of biologically-significant historical changes taking place over a span of 100 million years.  We could count the first appearance of small shell fossils 550 million years ago (MYA), the first instances of brachiopods 530 MYA, or the first arthropods 525 MYA.  These were all certainly world-changing events, but including them would completely destroy the graph.  The 5 MY difference between the last two events would appear as a giant discontinuity since the "Time to Next Event" is so much smaller than the "Years Ago" for that pair.  For the people who study this period it has no shortage of interesting details, and yet Kurtzweil has chosen to ignore them.

Instead he lumps them together in an ad-hoc manner as if they were somehow all the same "paradigm shift."  One could even argue that they were part of an exponential process of differentiation that eventually petered out, as all exponential processes do, being replaced instead by differentiation within existing body plans rather than variation of body plans.

This process of summarizing and merging could continue into the future as well.  In 50 years, the last three events in the graph - Telephone, electricity, radio (an event which is already a collapse of three separate inventions); Computer; Personal computer - could easily be combined into a single "Information technology" bullet.  In another 50 years the previously distinct "Industrial revolution" event could be merged into that amalgam as well.  Thus it becomes clear that the rate of change of "paradigm shifts" is relative to our remoteness from the events themselves.

History unfolds before us like a landscape.  Sitting on the front porch of today, we see our glass of lemonade, the dog at our feet, the yard, the neighborhood, the buildings of the city, the far off mountains, and finally the sky and the sun.  This is by no means a correct ranking of the importance of these objects - just a measure of our distance to them.  The details of what's happening in the city are much more significant than the fly on our lemonade glass, and yet we overemphasize the latter because it subsumes a larger angle of our field of view.  History is the same only worse.  Unlike the inverse square law that governs our vision, our myopia of history is logarithmic.  Even if we know a lot about 100 years ago, we know one tenth as much about 1000 years ago, and one tenth again as much about 10,000 years ago.  By the time we get to billions of years ago we know just one thing about human history: life started.

It should not be a surprise in light of these observations that the graph looks like it does.  Since the number of things we know (or even if we know more, the way we combine them) decreases logarithmically as we look back in time, the position of events and time between them must look as it does.  It would look just like this for any list of significant events taken from observers at any point in history.  By way of demonstration I created this graph by generating random events distributed in a logarithmic fashion.  The pattern is exactly as expected.

But is there any significance to the other exponential plots that fill the first part of the book?  Yes and no.  Certainly they do show technologies that are currently advancing at exponential rates, but it's not clear how long each will continue.  It may be a day or decade but eventually all exponentials will hit a limit of one kind or another and taper off (like yeast multiplying in bread dough) or crash (like a gambler doubling his bet on every loss), depending on the underlying process.

The author is no idiot, however, and has prepared his readers for this inevitable historical reality check.  He argues that although each individual technology undergoes what he calls an S-curve (exponential growth followed by a damping phase and tapering off to zero growth) the overall technology trend is exponential.  The first piece of evidence he has for this is the "rate of paradigm shift" graph we analyzed above, and there is nothing more to say about that one.

Kurtzweil warns us not to use our "linear" intuition to evaluate exponential growth, so let's consider that.  Exponentials do indeed exhibit explosive properties if viewed over a long enough span relative to their doubling time.  Viewed over a short time they do appear linear, which gives us reason to doubt our complacency.  On the other hand, viewed over the medium term, all exponentials look the same.  We could consider any 100 year period in history where the population, growing exponentially at a rate of 2-ish percent per year, would multiply by 5 to 10 times.   If there were 1000 humans at some point in history then a century later there would be 5 to 10 thousand humans.  If there were a billion humans, then a century later there would be 5 to 10 billions humans.   The exponential plot for each of those expansions would look the same, except for the fact that one would be scaled to show 1000 people per tick while other would show 1 billion people per tick.  Other than that the shape would be identical.  Kurtzweil would like to find the "knee" of the exponential where it changes from roughly linear to unbounded.  There is no such thing if trends are viewed in their historical context.

More importantly, for any generation the experience that a cohort has of exponential trends is roughly the same.  When I was a child my father took me to visit the computer room at SRI, where a huge IBM mainframe filled the space.  The laptop I'm using to compose this post 35 years later is probably about a million times more powerful and hundreds of times cheaper.  The computers my daughter will use in her adult life will similarly more powerful, but she will see only the change from her childhood experience, which is now.  That next million fold increase will be impressive, but less impressive if your starting point for comparison is a PS3.

This is the second piece of evidence that Kurtzweil provides to support his "Law" - the steady increase in computing power across several generations of technology.  He shows that while each successive technology - relays, tubes, transistors, ICs - has grown in power and then petered out to be replaced by another, the rate of improvement of computing power has continued to grow exponentially.  He predicts that this will continue, with new materials and processes replacing the current generation of large-scale integrated semiconductors.  In this he may well be right.  After all you can make a computer out of anything you can finagle to work like a switch, and there are tremendous financial incentives to build switches out of smaller and smaller parts.  Since our appetite for computing bandwidth seems limitless there's no reason the trend shouldn't continue for a while.

The real question is what we are going to do with those 2040's computers that will be a trillion times more powerful than that 1970's IBM mainframe.  Will we be doing much the same kind of stuff we do today, only a lot more of it faster and cheaper?  Or will we have crossed a threshold and be doing something radically different?  Will we still use computers, or will we have become computers?  I would bet on the former, Kurtzweil on the latter.  Whether he is right or not, the evidence to back up his conclusion is not to be found in his graphs.

- jack*

UPDATE: see final part three here.