Knowledge from Pebbles: What Can Be Counted, and What Cannot

A pagan arithmetic: unstable sets in indigenous Amazonia

“…Nirenberg and Nirenberg use Borges's powerful image to critique the set theory that founds mathematics. According to the authors (Nirenberg and Nirenberg 2018, 3), if we ask a mathematician what counting is, he or she will reply that ‘to count a finite set is to assign to its elements, in a one-to-one manner, the numbers 1, 2, 3, …, n, without missing any one of the latter.’ Everything is given previously, such that this type of problem begins with the phrase: ‘given a set A,’ or ‘given the elements of the set A,’ and ‘given the natural numbers 1, 2, 3, 4, and so on’ (Nirenberg and Nirenberg 2018). In the authors’ words: ‘We assumed that there is no question as to what elements belong to the set.…”

“…The basic characteristic of the elements of the set, the authors state in another work (Nirenberg and Nirenberg 2011, 606), is that they must be ‘ apathés ,’ inert, which means that nothing happens to elements x or y when placed together: there is ‘no change in identity’ (Nirenberg and Nirenberg 2011, 607; also see 2018, 3, 8). The examples that the authors provide to contradict apathy are, among others, the chemical elements, which combine and transform when in contact (2011, 607).…”

“…The basis of the apathy pointed by the Nirembergs regarding the components of mathematical set is the principle of identity, the same cited above by Clastres: for every x, x = x. All any counting system requires is a first number, which may be zero or 1, and that the next number can be stated, ‘such-and-such a number + 1.’ What is not taken into account is that, in contrast to the stones that serve as the base for various counting systems (hence Borges's critique being based precisely on stones), ‘[o]ther things in the world are more difficult to count, whether because they are more subject to “becoming,” as the philosophers would say, that is, to change in the act of counting, so that it becomes difficult to speak of the 1; or because they are subject to transformative interaction when brought together with another, so that we cannot speak of the 1 + 1’ (Nirenberg and Nirenberg 2018, 5). The authors also assert, in the other article mentioned, that the axioms of set theory imply that ‘[a]ny rigorous attempt to base an ontology upon them will entail such a drastic loss of life and experience that the result can never amount to an ontology in any humanly meaningful sense’ (2011, 586).…”

“…An article co-authored by a mathematician and a philosopher (Nirenberg and Nirenberg 2018), suggestively titled 'Knowledge from pebbles. What can be counted and what cannot,' explores the question of the instability of the constitutive elements of sets in a very interesting way.…”

“…In the authors' words: 'We assumed that there is no question as to what elements belong to the set. And we assumed that an element x is not changed by being counted in the set (is not affected, for example, by being placed in the set, or coming into contact with some other element within the set)' (Nirenberg and Nirenberg 2018).…”