Number as a Second-Order Concept

Abstract

My contribution will focus on a central issue of Yehuda Elkana's anthropology of knowledge — namely, the role of reflectivity in the development of knowledge. Let me therefore start with a quotation from Yehuda's paper “Experiment as a Second-Order Concept.”

Numerical Abstraction in Prehistory. A View from Cognitive Archeology = Abstracción numérica en la Prehistoria. Una visión desde la arqueología cognitiva

“…In the creation of numbers, the previous existence of measurement marks, concrete numbers and their later evolution with the invention of writing to abstract or second order numbers has been accepted (Damerow 1996). Recently the abstractconcrete distinction of Ancient Near East numbers has been questioned, presenting a vision where numbers can be abstract from the beginning.…”

“…This is why various authors define the Babylonian mathematical tradition as sub-scientific (Høyrup 1994). This confusing situation would disappear with the study of the nature of number as a concept and subject of independent study, which Greek philosophers developed in the first millennium (Høyrup 1994;Damerow 1996;Overmann 2018b). Effectively, even among the ancient Greeks numerical abstraction acquired new connotations such as the mystical character they acquired with the Pythagoreans, which gives us an idea of the complexity and importance that numerical abstractions acquire in our culture.…”

Updating the Abstract-Concrete Distinction in Ancient Near Eastern Numbers

“…Early token-based accounting has long been understood as involving a "concrete" concept of number, with the invention of writing enabling the development of an "abstract" or "second-order" number concept (e.g., Damerow, 1996a). This distinction between "abstract" and "concrete" numbers is thought to have been facilitated by the invention of writing, which enabled the separate representation of quantity (by means of numerical signs) from commodity (by means of graphic labels), information that had previously been conjoined in the shapes, sizes, and quantities of clay tokens (Schmandt-Besserat, 1992a).…”

“…The abstract-concrete distinction has become fairly entrenched in the literature on Ancient Near Eastern numbers and thinking: "[T]he litany is often repeated that the Mesopotamians were incapable of abstract thought, that their languages lacked terms to express concepts" like numbers (Glassner, 2000, p. 55). It is particularly associated with the extensive research and publication on Neolithic tokens by Archaeologist Denise Schmandt-Besserat (e.g., 1977, 1982, 1992a, 1992b, as well as the cognitive analyses by developmental psychologist Peter Damerow (1988Damerow ( , 1996aDamerow ( , 1996bDamerow ( , 2007Damerow ( , 2010Damerow ( , 2012, who explicitly rooted the concrete-to-abstract notion in the work of psychologist Jean Piaget. Drawing in particular on work by sociologist Lucien Lévy-Bruhl (e.g., 1910, 1927, Piaget applied his ideas on cognitive development in children (e.g., how the thinking of children differs from that of adults; how the latter progressively develops from the former) to entire societies.…”

“…By the 2 nd millennium, the Old Babylonian mathematicians had developed a socalled pure mathematics involving complex algorithms for calculating answers to artificial as well as practical problems (Friberg, 2007;Høyrup, 2002aHøyrup, , 2002bRobson, 2007Robson, , 2008. However, they do not appear to have speculated much about the nature of number as a concept, an inquiry that would be taken up by the Greek philosophers in the 1 st millennium (Damerow 1996a(Damerow , 1996bHøyrup, 1994;Klein, 1968).…”