Number Sense & how we unlearn the intuition of a log scale

My favorite science/wonder reporting is a WNYC program called RadioLab.

One episode called Innate Numbers discusses how children in our culture are trained out of an intuitive and innate understanding of the logarithmic scale and into integers and linear thinking.  This is based on research and speculation by Stanislas Dehaene who wrote The Number Sense and Susan Carey.

Mirror Image

The thing I love about the exponential function is that it always has a mirror image.  For example, exponential growth in human population, would have many images that mirrored it such as exponential loss in square feet per person.  Now perhaps this is one of those tautological redundancies that I’m prone to, but I’m not so sure.  It has more to do with layered causes and effects in a complex and dynamic system.  So for example, exponential growth in fossil fuel use relates to exponential depletion of fossil reserves storing carbon in the deep earth, which relates to the exponential increase in gaseous CO2 to the atmosphere.  Really, we could come up with all kinds of chains of events that integrate to tell something about our deeply interrelated world.  The point here, is the exponential has momentum and as that momentum builds, one can start looking for the causal impacts resulting from that momentum.

A graph of the exponential function is a map of growth and decay.  In general, an exponential pattern can go unnoticed for generations before the culminating impact of its sequential doublings becomes observable.  The exponential function is the summative outcome of ‘normal’ behavior – what our parents did, what our neighborhoods are doing, what our children will do (unless something changes this pattern of behavior).  Climate change, population growth, AIDS, and consumption are status quo problems of exponential nature.  The change seems imperceptible in day-to-day life until it reaches a critical level when the culminating force of the doublings becomes abundantly clear.  The exponential function is self-similar change (1, 2, 4, 8, 16 or the reverse).  As one element grows, implicitly another recedes; the exponential function has a mirror image.

A good way to illustrate the exponential function is with the story of the poor boy who returns a lost princess to the king.  When asked by the king for any gift in trade, the boy only asks for a grain of rice, to doubled across each square of a chessboard (64 doublings).  The king knowing he has eight million bundles of grain (with one trillion grains per bundle) exclaims: That is all you want for returning my daughter?

But by the end of the simple doubling (1, 2, 4, 8, 16) across 63 squares, the poor boy has the king’s wealth in rice.  For the first 55 squares, the king is beaming because this request has barely made a dent in his stores.  By the 61st square, the king still has ¾ of his grain but by the 63rd square, the king is completely broke and encourages marriage between his daughter and the boy since he can no longer fulfill his end of the deal nor feed his daughter.  By simple doubling, the king experienced exponential loss of his finite resources and the boy exponential gain.

The ‘fact’ of the exponential function is that there is relatively little time to respond when the curve becomes noticeable.  In the example above, the simple doubling pattern can go for more than 55 generations, before the exponential function begins to be observable in the graph of the king’s rice.  Just two doublings from being broke, the king still has 75% of his resources.  This doubling is exponential growth coupled to exponential depletion of available resources.  When the resources are no longer sufficient, the king gives way to the boy.

The exponential function makes for a very simple graph – it is easy to name when you see it.  So easy, that the pound-in-your-heart implication embedded in the graph is lost and it is simply named:  ‘Oh, that is exponential’.  To most, it is a factual word without cultural embodiment.  It is a word lacking a literalization to bring it forth from the abstract.  What is the implication of ‘giving’ cultural meaning to ‘made’ abstract facts?  How do we culturally qualify the quantified?  There are nearly 7 billion people, each standing on his/her changing two foot square of earth with his/her own perspective on what is going on.

Meanwhile, the contemporary economic system thrives on infinite growth.  Infinite growth would be possible if we lived in an infinite world, but we don’t.  In ecology, the term carrying capacity is used to define the maximum population of a given organism that a particular environment or habitat can sustain (this includes biological and technological limits).

Awareness of the consequences of exponential growth is not recent.  In fact, Darwin cites the exponential function in the context of natural selection in the Origin of Species: ‘There is no exception to the rule that every organic being naturally increases at so high a rate, that, if not destroyed, the earth would soon be covered by the progeny of a single pair’.    And yet, modern civilization has not come to terms with the social, cultural, and political significance of the exponential function:  that our normal behavior will be changed with or without our thinking consent, with or without our ability to use the information embodied in the multiple expressions of the exponential function.

The doubling of a population mirrors the doubling of resource depletion which mirrors a doubling of waste products or pollution.  As a species advances on the ecosystem creating a network of chemical synthesis, spreading out from the originator, it both exponentially grows in number and simultaneously depletes it resource base.  The mirror informs us too.

jlw29aug11 tho much of this text came from JVC.2008.7(3).309-334.