Material Scaffolds in Numbers and Time

The present paper develops a framework for interpreting Upper Paleolithic artifacts from an analysis of material complexity, numeration systems, and timekeeping using cultural categorizations (Hayden & Villeneuve, 2011), insights on the emergence of number terms in language (Menninger, 1992), and the astronomy practices of 33 contemporary hunter–gatherer societies (Yale’s Electronic Human Relations Area Files World Cultures database). Key findings: (1) an absence of societies with minimal material complexity and later-stage numeration systems, suggesting that material scaffolding may be important to realizing explicit number concepts, (2) the consistency of material complexity with both early- and later-stage numeration systems, emphasizing that material complexity may precede and inform the development of complexity in numeration systems, (3) the compatibility of astronomical practices with the spectrum of complexity in material culture and numeration systems, suggesting that the awareness of time may precede both, and (4) the increasing quantification of time consistent with greater material and numeration complexity, suggesting the availability of numbers as a cognitive technology may enable the structuring of time. These findings suggest that astronomy originates in the ability to estimate and infer contextual relations among natural phenomena and transitions from these natural associations to material representations and cognitive technologies that mediate conceptual apprehensions of time as a substance that can be quantified. Given that artifacts may act as scaffolds for explicit concepts of numbers and numbers scaffold explicit concepts of time, prehistoric artifacts such as the Blombos Cave beads (ca. 75,000), Abri Blanchard and Cellier artifacts (ca. 28,000), and Taï plaque (ca. 14,000) may represent similar scaffolding and conceptual development in numbers and time. It is proposed that the prehistoric societies making these artifacts possessed, in addition to material complexity, the abilities to express quantities in language and to use material externalization and cognitive technologies. Further, the Abri Blanchard artifact is proposed to represent externalized working memory, a very modern interaction between mind and material culture.

On p/q-recognisable sets

Let p/q be a rational number. Numeration in base p/q is defined by a function that evaluates each finite word over A_p={0,1,...,p-1} to some rational number. We let N_p/q denote the image of this evaluation function. In particular, N_p/q contains all nonnegative integers and the literature on base p/q usually focuses on the set of words that are evaluated to nonnegative integers; it is a rather chaotic language which is not context-free. On the contrary, we study here the subsets of (N_p/q)^d that are p/q-recognisable, i.e. realised by finite automata over (A_p)^d. First, we give a characterisation of these sets as those definable in a first-order logic, similar to the one given by the B\"uchi-Bruy\`ere Theorem for integer bases numeration systems. Second, we show that the natural order relation and the modulo-q operator are not p/q-recognisable.

The Ethnomathematics Practices of Eskaya Tribe

This research sought to describe the Ethno-mathematical practices of the Eskaya tribe of Taytay, Duero, Bohol using the ethnographical research design to explore the ethnomathematics practices through the lived experiences of the informants. Employing purposive sampling, selected teachers, parents, and students from the tribe served as the key informants of the study. Data collection took almost a year of observation, lived experiences documentation, and interviews. The study was able to describe some of the ethnomathematical practices of the Eskaya tribe such as the skills and processes of the Eskaya tribe commonly use in their daily life in counting, measuring, ciphering, ordering, classifying, inferring, and modeling patterns. These skills and techniques were used in studying their Eskaya numeration systems such as the Eskaya numbers and numerals, the Eskaya name of the basic shapes and the four fundamental operations, and the use of Eskaya numbers in measuring time, days, and months.

Lattice Bounded Distance Equivalence for 1D Delone Sets with Finite Local Complexity

Spectra of suitably chosen Pisot-Vijayaraghavan numbers represent non-trivial examples of self-similar Delone point sets of finite local complexity, indispensable in quasicrystal modeling. For the case of quadratic Pisot units we characterize, dependingly on digits in the corresponding numeration systems, the spectra which are bounded distance to an average lattice. Our method stems in interpretation of the spectra in the frame of the cut-and-project method. Such structures are coded by an infinite word over a finite alphabet which enables us to exploit combinatorial notions such as balancedness, substitutions and the spectrum of associated incidence matrices.