Alliance for Mystical Pragmatics

Alliance for Mystical Pragmatics

Harmonizing Evolutionary Convergence

Glossary Menus

exponential growth

Perhaps the easiest way to understand exponential growth is to note that exponentiation is repeated multiplication, just as multiplication is repeated addition. As an example of the latter 2 × 3 = 2 + 2 + 2, where 3 is the multiplicand. Similarly, 23 = 2 × 2 × 2, where superscript 3 is the exponent.

Here is a diagram of unfettered exponential growth. This is what makes exponentiation so difficult to relate to in human experience. For instance, it is estimated that there are between 1078 to 1082 atoms in the observable, physical universe. But that is just a tiny number compared to a googol (10100) or even a googolplex (10googol), where Googolplex is home to Alphabet Inc., the holding company for Google. And there is no mathematical limit to the expansion of these gigantic finite numbers.

Now while mathematicians have great fun exploring these vast finite numbers and the even vaster infinity of infinities, we need other mathematical tools to understand the exponential growth of evolutionary processes in practical terms. Pierre François Verhulst made the great breakthrough in this regard in 1844, when studying the potential for population growth in the newly formed Kingdom of Belgium.

Verhulst introduced a function, called ‘logistic’, which represents the evolutionary growth of structures under constraint, illustrated in the S-shape of the growth curve, which we experience as the learning curve when studying a new subject. D’Arcy Wentworth Thompson made much use of this function in his book On Growth and Form. In an extensive chapter on the rate of growth in biological processes, he pointed out that this one curve recurs in endless shapes and circumstances, for mathematics generalizes and “is fond of giving the same name to different things”.

This curve does not only apply to evolutionary processes. Positive feedback loops also follow a similar pattern. Most significantly, dozens of self-reinforcing feedback loops are causing abrupt, irreversible climate change to accelerate exponentially. The most critical of these is the potentially rapid release of methane gas in the Arctic into the atmosphere, for this is a far more potent greenhouse gas than carbon dioxide.

Etymology

Exponential, 1704, ‘of or pertaining to an exponent or exponents, involving variable exponents’, from French exponentiel, from present participle of Latin expōnere ‘put out’, from ex- ‘out’, from PIE base *eghs- ‘out’, and pōnere ‘put’, from PIE base *apo- ‘off, away’.

See also growth.

Common ancestor(s):