Notes

Chapter 4: Systems Based on Numbers

Section 3: Recursive Sequences


History [of recursive sequences]

The idea of sequences in which later terms are deduced from earlier ones existed in antiquity, notably in the method of induction and in various approximation schemes (compare page 918). The Fibonacci sequence also appears to have arisen in antiquity (see page 890). A fairly clear idea of integer recurrence relations has existed since about the 1600s, but until very recently mainstream mathematics has almost never investigated them. In the late 1800s and early 1900s issues about the foundations of mathematics (see note below) led to the formal definition of so-called recursive functions. But almost without exception the emphasis was on studying what such functions could in principle do, not on looking at the actual behavior of particular ones. And indeed, despite their simple forms, recursive sequences of the kind I discuss here do not for the most part ever appear to have been studied before—although sequence (c) was mentioned in lectures by John Conway around 1988, and the first 17 terms of sequence (e) were given by Douglas Hofstadter in 1979.


From Stephen Wolfram: A New Kind of Science [citation]