I. Introductory remarks In this paper we will discuss some basic issues which may lay the ground for an historical approach to the teaching of mathematics in a novel way. Our project relies primarily on developing the concept of ethnomathematics. Our subject lies on the borderline between the history of mathematics and cultural anthropology. We may conceptualize ethnoscience as the study of scientific and, by extension, technological phenomena in direct relation to their social, economic and cultural backgrounds [1]. There has been much research already on ethnoastronomy, ethnobotany, ethnochemistry, and so on. Not much has been done in ethnomathematics, perhaps because people believe in the universality of mathematics. This seems to be harder to sustain as recent research, mainly carried on by anthropologists, shows evidence of practices which are typically mathematical, such as counting, ordering, sorting, measuring and weighing, done in radically different ways than those which are commonly taught in the school system. This has encouraged a few studies on the evolution of the concepts of mathematics in a cultural and anthropological framework. But we consider this direction to have been pursued only to a very limited and we might say timid extent. A basic book by R.L. Wilder which takes this approach and a recent comment on Wilder's approach by C. Smorinski [2] seem to be the most important attempts by mathematicians. On the other hand, there is a reasonable amount of literature on this by anthropologists. Making a bridge between anthropologists and historians of culture and mathematicians is an important step towards recognizing that different modes of thoughts may lead to different forms of mathematics; this is the field which we may call ethnomathematics. Anton Dimitriu's extensive history of logic [3] briefly describes Indian and Chinese logics merely as background for his general historical study of the logics that originated from Greek thought. We know from other sources that, for example, the concept of "the number one" is a quite different concept in the Nyaya-Vaisesika epistemology: "the number one is eternal in eternal substances, whereas two, etc., are always non-eternal," and from this proceeds an arithmetic [4, p. 119]. Practically nothing is known about the logic underlying the Inca treatment of numbers, though what is known through the study of the "quipus" suggests that they used a mixed qualitative-quantitative language [5]. These remarks invite us to look at the history of mathematics in a broader context so as to incorporate in it other possible forms of mathematics. But we will go further than these considerations in saying that this is not a mere academic exercise, since its implications for the pedagogy of mathematics are clear. We refer to recent advances in theories of cognition which show how strongly culture and cognition are related. Although for a long time there have been indications of a close connection between cognitive mechanisms and cultural environment, a reductionist tendency, which goes back to Descartes and has to a certain extent grown in parallel with the development of mathematics, tended to dominate education until recently, implying a culture-free cognition. Recently a holistic recognition of the interpenetration of biology and culture has opened up a fertile ground of research on culture and mathematical cognition (see, for example, [6]). This has clear implications for mathematics education, as has been amply discussed in [7] and [8].

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Ethnomathematics and its Place in the History and Pedagogy of Mathematics.

Ritualized behavior, intuitively recognizable by its stereotypy, rigidity, repetition, and apparent lack of rational motivation, is found in a variety of life conditions, customs, and everyday practices: in cultural rituals, whether religious or non-religious; in many children's complicated routines; in the pathology of obsessive-compulsive disorders (OCD); in normal adults around certain stages of the life-cycle, birthing in particular. Combining evidence from evolutionary anthropology, neuropsychology and neuroimaging, we propose an explanation of ritualized behavior in terms of an evolved Precaution System geared to the detection of and reaction to inferred threats to fitness. This system, distinct from fear-systems geared to respond to manifest danger, includes a repertoire of clues for potential danger as well as a repertoire of species-typical precautions. In OCD pathology, this system does not supply a negative feedback to the appraisal of potential threats, resulting in doubts about the proper performance of precautions, and repetition of action. Also, anxiety levels focus the attention on low-level gestural units of behavior rather than on the goal-related higher-level units normally used in parsing the action-flow. Normally automatized actions are submitted to cognitive control. This "swamps" working memory, an effect of which is a temporary relief from intrusions but also their long-term strengthening. Normal activation of this Precaution System explains intrusions and ritual behaviors in normal adults. Gradual calibration of the system occurs through childhood rituals. Cultural mimicry of this system's normal input makes cultural rituals attention-grabbing and compelling. A number of empirical predictions follow from this synthetic model.

http://pascalboyer.net/articles/BoyerLienardBBS.pdf

Why ritualized behavior? Precaution systems and action parsing in developmental, pathological and cultural rituals.

1 Cognitive constraints on religious ritual form: a theory of participants' competence with religious ritual systems 2 Ritual and memory: frequency and flashbulbs 3 Two hypotheses concerning religious ritual and emotional stimulation 4 Assessing the two hypotheses 5 General profiles of religious ritual systems: the emerging cognitive science of religion

/pdf/bringing-ritual-to-mind-psychological-foundations-of-5curkanx0e.pdf

Bringing ritual to mind : psychological foundations of cultural forms

Vocal imitation in human infants and in some orders of birds relies on auditory-guided motor learning during a sensitive period of development. It proceeds from 'babbling' (in humans) and 'subsong' (in birds) through distinct phases towards the full-fledged communication system. Language development and birdsong learning have parallels at the behavioural, neural and genetic levels. Different orders of birds have evolved networks of brain regions for song learning and production that have a surprisingly similar gross anatomy, with analogies to human cortical regions and basal ganglia. Comparisons between different songbird species and humans point towards both general and species-specific principles of vocal learning and have identified common neural and molecular substrates, including the forkhead box P2 (FOXP2) gene.

Twitter evolution: converging mechanisms in birdsong and human speech.

This book develops a new view of logic as a theory of information-driven agency and intelligent interaction between many agents - with conversation, argumentation and games as guiding examples. It provides one uniform account of dynamic logics for acts of inference, observation, questions and communication, that can handle both update of knowledge and revision of beliefs. It then extends the dynamic style of analysis to include changing preferences and goals, temporal processes, group action and strategic interaction in games. Throughout, the book develops a mathematical theory unifying all these systems, and positioning them at the interface of logic, philosophy, computer science and game theory. A series of further chapters explores repercussions of the 'dynamic stance' for these areas, as well as cognitive science.

/pdf/logical-dynamics-of-information-and-interaction-3fy0u23au4.pdf

Logical Dynamics of Information and Interaction

The Agnicayana, a 3000-year-old Vedic ritual, was performed in 1975 in a village .in southwest India by Nambudiri. brahmins. This event, which lasted twelve days, was filmed, photographed, recorded and extensively documented. From twenty hours of rough footage, Robert Gardner and I produced a 45-minu~e film, \"Altar of Fire.\" Two records are planned with selections from th~ eighty hours of recorded recitation and chant. Photographs of the ceremonies were taken by Adelaide de Me nil. In collaboration with the chief N ambudiri ritualists and other scholars, I am preparing a definite account of the ceremonies, which will appear in two illustrated volumes entitled: \"Agni The Vedic Ritual of the Fire Altar.\" I shall here be concerned not with empirical deS<:tiption, hut with theoretical implications. Vedic ritual is not only the oldest surviving ritual of ·mankind; it also provides the best source material for a theory of ritual. This is not because it is close to any alleged ''original\" ritual. Vedic ritua~l is not primitive and not an Ur-ritual. It is sophis· . . ticated and already the product of a long development. But it is the largest, most elaborate and (on account of the Sanskrit manuals) best documented among the rituals of man. Hubert and Mauss, who noted these facts in 1909, used the Vedic animal sacrifice as source material . for the construction of a ritual paradigm (\"un scheme abstrait du sacrifice\") . 1 However, they did not know that these rituals are still performed, so that many data were inaccessibLe to them. I shall use data from the 1975 performance and textual ·material from Sans:l(rit manuals, in particular :the srauta siUras, a literature exclusively devoted to ritual which dates from clle eighth through fourth centuries B.C.

The Meaninglessness of Ritual

Rules Without Meaning: Ritual, Mantras and the Human Sciences

Word Order in Sanskrit and Universal Grammar

The achievements of Pānini and the Indian grammarians, beginning nearly 2500 years ago, have never been fully appreciated by Western scholars--partly because of the great technical difficulties presented by such an inquiry, and partly because relevant tutorial articles have been confined to obscure and inaccessible publications.This book makes available to linguists and Sanskritists a collection of the most important articles on the Sanskrit grammarians, and provides a connected historical outline of their activities. It covers studies and fragments ranging from early 7th-century accounts of the grammarians--recorded by Buddhist pilgrims from China and Tibet, by Muslim travelers from the Near East, and by Christian missionaries--to some of the best articles that have appeared during the last century and a half.Chapters in the book cover the foundation of Sanskrit studies in the West laid by British scholars working in India and including the detailed and accurate information provided by Henry Thomas Colebrooke; the linguistic evaluations of Pānini by von Schlegel and von Humboldt; the work of Bhandarkar and of Kielhorn; William Dwight Whitney's low evaluation of the "native" grammarians; and the philological work of modern Western, Indian, and Japanese scholars.The editor observes that materials in the "Reader" reveal problems tackled by the Sanskrit grammarians which closely parallel developments in contemporary linguistics. He has provided historical and linguistic commentary and bibliographic data in the introductions and notes that accompany each selection. Articles are in their original English, German, and French. Texts or passages in Chinese, Tibetan, Arabic, Sanskrit, Latin, and Greek have, for the most part, been translated into English, and all Sanskrit passages have been translated into the Latin alphabet.

A Reader on the Sanskrit Grammarians

The work records one of the most complicated vedic rituals - Fire Altar - in full length currently used among Nambudiri brahmins. A project of Monumental proportions, Agni provides a description and a study of one of the world's most ancient rituals.

Frits Staal

Papers

The Meaninglessness of Ritual

Rules Without Meaning: Ritual, Mantras and the Human Sciences

Word Order in Sanskrit and Universal Grammar

A Reader on the Sanskrit Grammarians

Stalking the Sacrifice@@@Agni: The Vedic Ritual of the Fire Altar.