Chapter 2: Turtles All the Way Down?: Recursion and Infinity in the Human Sciences

This chapter began (and begins) as a survey, from one mathematician’s point of view, of how of the terms recursion, recursive, and the like are used in mathematics and in the human sciences. That survey, and an analysis of its results, led (and leads) to a similar survey and analysis of the various uses of infinity, infinite, and the like.

The first two sections of the chapter are mostly about mathematics as a social and cultural activity. I give some history of mathematicians’ use of the terms recursion and recursive. I discuss differences of form and function among ellipsis and aposiopesis in general discourse, in typical “scholarly”/”scientific” discourse, and in “paramathematical” discourse. I draw various connections among aposiopesis, infinity, well-foundedness, recursion, and computation in mathematicians’ discourse and (other) behavior. I describe the history and nature of the “horror of infinity” and its not always recognized twin, enthusiasm for (or at least pleased acceptance of) infinity. I conclude with speculation about the role that two axioms first introduced during the mathematical formalization of set theory between 1874 and 1925, the Axiom of Infinity and the Axiom of Foundation, might have played in essentially eliminating the “horror of infinity” among mathematicians.

The last two sections of the chapter are mostly about the human sciences, and human experience more generally. I describe and distinguish several families of uses (not too closely related to each other) of the terms recursion and the like in the human sciences and trace their lineages back to “base cases” (not all of them in mathematics), then do the same for infinity and the like. I observe that in the human sciences there is considerable enthusiasm for infinity, although “horror of infinity” persists here and there; I argue that the enthusiasm is misplaced and that the horror is unnecessary. The former argument—partly empirical, partly theoretical—arises out of a discussion of finities and infinities in human experience, framed in terms of some axioms of “evolutionary ontology,” including von Uexküll’s Axiom of Subjective Finiteness. I only sketch the latter argument; it applies Aczel’s nonstandard set-theoretical Anti-Foundation Axiom to derive a formalism for infinity-free modeling of several lineages of ‘recursion’ in the human sciences.