Mathematics and Technology Before the Modern Era

The mapping of numbers onto space is fundamental to measurement and to mathematics. Is this mapping a cultural invention or a universal intuition shared by all humans regardless of culture and education? We probed number-space mappings in the Mundurucu, an Amazonian indigene group with a reduced numerical lexicon and little or no formal education. At all ages, the Mundurucu mapped symbolic and nonsymbolic numbers onto a logarithmic scale, whereas Western adults used linear mapping with small or symbolic numbers and logarithmic mapping when numbers were presented nonsymbolically under conditions that discouraged counting. This indicates that the mapping of numbers onto space is a universal intuition and that this initial intuition of number is logarithmic. The concept of a linear number line appears to be a cultural invention that fails to develop in the absence of formal education.

Is calculation possible without language? Or is the human ability for arithmetic dependent on the language faculty? To clarify the relation between language and arithmetic, we studied numerical cognition in speakers of Mundurukú, an Amazonian language with a very small lexicon of number words. Although the Mundurukú lack words for numbers beyond 5, they are able to compare and add large approximate numbers that are far beyond their naming range. However, they fail in exact arithmetic with numbers larger than 4 or 5. Our results imply a distinction between a nonverbal system of number approximation and a language-based counting system for exact number and arithmetic.

The transition from the Middle Stone Age (MSA) to the Later Stone Age (LSA) in South Africa was not associated with the appearance of anatomically modern humans and the extinction of Neandertals, as in the Middle to Upper Paleolithic transition in Western Europe. It has therefore attracted less attention, yet it provides insights into patterns of technological evolution not associated with a new hominin. Data from Border Cave (KwaZulu-Natal) show a strong pattern of technological change at approximately 44-42 ka cal BP, marked by adoption of techniques and materials that were present but scarcely used in the previous MSA, and some novelties. The agent of change was neither a revolution nor the advent of a new species of human. Although most evident in personal ornaments and symbolic markings, the change from one way of living to another was not restricted to aesthetics. Our analysis shows that: (i) at Border Cave two assemblages, dated to 45-49 and >49 ka, show a gradual abandonment of the technology and tool types of the post-Howiesons Poort period and can be considered transitional industries; (ii) the 44-42 ka cal BP assemblages are based on an expedient technology dominated by bipolar knapping, with microliths hafted with pitch from Podocarpus bark, worked suid tusks, ostrich eggshell beads, bone arrowheads, engraved bones, bored stones, and digging sticks; (iii) these assemblages mark the beginning of the LSA in South Africa; (iv) the LSA emerged by internal evolution; and (v) the process of change began sometime after 56 ka.