The Mathematics of Ideas

Since the start of the Industrial Revolution around 1760 in England, mankind has experienced an explosive growth in ideas and products which have had an enormous positive impact on human lives.  From the Agricultural Revolution, where mankind first learned that food could be cultivated, to the start of the Industrial Revolution—a span of about seven thousand years—human productivity is estimated to have been about constant, at somewhere between $400 to $550 in annual per capita GDP (in constant 1990 dollars).  That is, for seven millennia there was essentially no improvement in things like caloric intake, life expectancy, or child mortality, on average, worldwide.  That’s depressing.

But with the Industrial Revolution, things changed.  By around 1800, a loaf of bread that took a fourth century worker three hours to earn could be earned in two hours.  By 1900 that loaf of bread was earned with fifteen minutes of labor, and the loaf could be earned with only five minutes of work by the year 2000.  To put that in perspective, the fourth century equivalent cost of a loaf of bread in the year 2000 would have been about $36.  [Insert witty Whole Foods joke here.]  The changes in the human life experience as a result of the Industrial Revolution are undeniably positive.  And not only in the cost of food.  Consider that a baby born in 1800s France had a life expectancy of only thirty years.  (The experience of giving birth in the mid to late 1800s was a very different experience than it is today.  One physician historian likens it to the American Frontier before the railroad.  At that time in the US, some 15-20% of all infants in American cities did not live to see their first birthday.)  But a baby born in the Republic of Congo in 2000 could expect to live fifty-five years.  In other words, living conditions worldwide have improved more in the last few hundred years than they did in the entire seven thousand years prior to the Industrial Revolution.  [Economic numbers are from William Rosen’s book, The Most Powerful Idea In The World]

It was the ideas of the Industrial Revolution that enabled this change.  Certainly there were a great deal of things going on, and a variety of factors that came together to enable the success of the Revolution.  But the ideas were the key.  And they were successful in no small part because of the growth in real knowledge.  Not only was the industrial world changing in the mid-late 1700s, but so was our ability to reliably learn things about our surroundings.  Isaac Newton founded modern physics when he published the Principia Mathematica in 1687.  Quite a few crazy ideas (i.e., phlogiston) persisted for a while, and some even helped push along the Industrial Revolution in spite of their wrongness, but the advancements and refinements of knowledge that came with the adoption of the scientific method helped to make these revolutionary ideas form and then come to reality.

Ideas, then, can be powerful things.  Good ideas, perhaps even more so.  So it bears asking, where do good ideas come from?  Where do we find the ideas that change the world and help to make it a better place?  Assuming we want more of them around, this is an important question to ask.  It also was the subject of an excellent, and appropriately named book by Steven Johnson in 2010.  I found his answer to be quite surprising and insightful.  Johnson suggests that the nature of good ideas is combinatorial.

“Good ideas are not conjured out of thin air; they are built out of a collection of existing parts, the composition of which expands (and, occasionally contracts) over time.” —Steven Johnson, Where Good Ideas Come From

That is to say, good ideas are novel combinations of already existing ideas, generally adapted to new purposes or solving new problems.  Putting things that we already know together in ways not done before can enable both incremental changes, and in some cases it can lead to sea changes in capability.  This combination process is the heart of what is known as innovation.  Every patent cites the prior work it is derived from.  Every scientific publication references the work that it is built on.  Every artist has other artists who inspire their creations.

Johnson is not, to be fair, the first to make this observation.  Abbot Payson Usher remarked in his book on mechanical inventions, that revision and reinforcement of previous inventions was how new inventions were created.  Indeed, when steam engines were used to drive grain mills in the days of the Industrial Revolution, the novelty of the invention was in how it coordinated existing ideas to make a more efficient system—doing what was already being done, but doing it better, faster, cheaper.  When Einstein would ponder problems he would often let his mind wander to make new connections between ideas floating around in his mind, a creative technique now known as “combinatorial thinking.”  And when Isaac Newton famously remarked that had he seen farther than others, it was because he stood on the shoulders of giants, this was another way of saying that he combined existing ideas and extended them to make new ones.

This combinatorial concept goes deeper, perhaps even to the heart of pretty much everything.  In 2003, Biologist Stuart Kauffman published the book Investigations, a title he borrowed from Wittgenstein.  In it he introduced a new take on the combinatorial concept—a simple observation with remarkable consequences.  His interest is in biodiversity and its generation.  He offers a scenario in the prebiotic Earth, where, as you might imagine, chemistry was likely much simpler than today, with considerably fewer distinct molecules than are on the planet now.  Certainly, complex molecules like proteins and DNA were nonexistent the planet at some point in the past, so this is a reasonable assumption. What, then, might be the process that takes us from that simple set to now?  Assume, he says, a relatively small set of simple molecules which, given time, may react with each other to create new molecules.  Take this founder set and call it the “actual.”  Now consider all of the possible reaction products—the ones that are just a single reaction step away.  None of these exist yet except in the realm of possibility.  This set of all potential products that are one step away gets the name of the “adjacent possible.”

The remarkable observation about the adjacent possible concept is that it forces us to view the things that can happen as a function of the things we have now.  This should seem obvious, but the it generally isn’t.  In the prebiotic Earth, you wouldn’t be able to have a sunflower, because the things that make up sunflowers didn’t exist yet.  If we have only a few simple molecules to start with, complex structures like DNA or chloroplasts that convert sunlight to energy are out of the question.  But in time, the actual set grows in number, which also means in complexity.  Run the clock for a very long time and the adjacent possible grows to include everything we have today, like sunflowers.  In Kauffman’s words:

“Four billion years ago, the chemical diversity of the biosphere was presumably very limited, with a few hundred organic molecular species.  Today the biosphere swirls with trillions of organic molecular species.  Thus, in fact, sunlight shining on our globe, plus some fussing around by lots of critters, has persistently exploded the molecular diversity of the biosphere into its chemically adjacent possible.”

In this view, biodiversity is then a very real form of innovation.

This combinatorial picture of growth is all well and good, but is it true?  If it is, how might we know?

I explored this once before, but it is worth revisiting.  The adjacent possible, as we said, is the set of all first order combinations of the actual.  There is an area of math that is all about counting combinations that we can use here.  Let’s say there are n objects in the actual.  We want to take them k at a time and let k vary from zero all the way up to n, thus taking all possible combinations into consideration.  That is, take all of the individual items themselves, then take them in groups of two, then groups of three, and so on all the way up to the whole bunch at once.  Sum up all of these numbers and that is the size of the adjacent possible set.

For a simple example, take objects AB, and C and combine them in all possible ways.  First you can have each of them individually (giving us back our original three: AB, and C), this is k=1.  For k=2 we have three combinations, ABACBC.  For k=3, we have just one combination, ABC.  We haven’t thought about k=0, which is no objects, but that’s a single real option which needs to be included.  So the total number of combinations is then just 3+3+1+1 = 8.

Determining the number of combinations for any value of n objects taken k at a time is known in mathematics as “n choose k.”  This is what we calculated in each step above.  Mathematically it is represented as:


What we want to know is the total when you add up all the different combinations as you let k vary—our value of 8 in the ABC example.  When you work it out, what you get is:


You may recognize the result as an exponential.  That is, the size of the adjacent possible set is exponential with respect to the size of the actual set.  Every time n increases by one, the sum of all the combinations doubles.  The question then, is how does n grow in time.  This can be found by the amount of time it takes for the sum to double.  A constant doubling time is a signature of exponential growth.

Now that we know what we’re looking for, the question is where should we look.  It is probably not a bad assumption to equate scientific literature with ideas.  That is, when a research group learns something, it is generally a good assumption that they will publish it.  Because the first to publish something new generally gets the discovery credit, publishing is important in the scientific community.  So measuring the quantity of scientific literature over time probably provides a realistic measurement of scientific knowledge or scientific ideas.

In 1961, a survey of scientific literature was done by Derek Price.  In the less technologically advanced days of scientific publishing, abstracts were popular.  It was far easier for a scientist or engineer to look through these abstract publications for papers he or she might be interested in reading than to carry around or dig through entire journal volumes.  So to count the quantity of scientific publications, Price went to the abstracts.  And what he found was that the cumulative sum of abstracts doubled every fifteen years.  It didn’t matter the field, Chemical Abstracts, Biological Abstracts, and Physics Abstracts all ended up with a cumulative value doubling in about 15 years.  The growth, then, of the actual set of scientific knowledge, grows at an exponential rate.

If we were looking for something, I daresay we found it.  While it falls short of absolute proof, the exponential cumulative growth of scientific literature is precisely what we would expect to find if the nature of new scientific ideas was combinatorial.  And as I mentioned previously, a scientific manuscript cites the work it is based on—the ideas it is based on, which signifies the combinatorial nature of the work.  While the exponential growth of scientific literature is not new, understanding it as a combinatorial process appears to be.

This is important because it helps to codify the route to new good ideas.  Being able to explore a larger collection of the available opens up a larger adjacent possible.  In Johnson’s words:

“The trick to having good ideas is not to sit around in glorious isolation and try to think big thoughts.  The trick is to get more parts on the table.”

Note that just a single extra piece in the available doubles the size of the adjacent possible.  It comes as no surprise, then, that some of the most famous inventors in history had lots of hobbies, interests that offer a wide range of pieces to be fit together in new ways.

There is another interesting aspect to Price’s work that bears mention.  It makes sense to ask ourselves when this 15 year doubling period behavior began.  That is, let’s follow the line back until it hits 1.  The result:  the year 1690.  In other words, within a small margin of error away from Newton’s original publication date of the Principia Mathematica (1687).  My thoughts are that this is not just coincidence.  Newton’s work defined the onset of the modern era of science, which enabled reliable knowledge to seed ideas which became the basis for the Industrial Revolution.  The innovations which have gone on to make every measure of living conditions in the modern world so drastically better than ever before in the history of mankind may very well be traceable back to the ideas of one notoriously talented British scientist.


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