Maria Ascher's Elsewhere Analysis

Maria Ascher's *Mathematics Elsewhere,* identifies mathematical ideas that are present all over the world, and is "intended as another step toward a global and humanistic history of mathematics." (Ascher IV) This important volume clarifies how many universal mathematical concepts, both simple and complex, are used and understood by countless cultures worldwide, regardless of differences in geography, language, and era. By studying and widening the scope of the history and breadth of mathematical thought, Ascher argues that "we are supplying complexity and texture... [and] in short, enlarging our understanding of the variety of human expressions and human usages associated with the same basic ideas." (2)

To study underlying mathematical principles in various cultures, it is important to recognize that in many cultures, mathematics usually arises as a secondary technique in order to satisfy another problem. As Ascher's book focuses on "traditional, small-scale cultures," most of the mathematical techniques
discussed are interwoven deeply into the people's way of life and need for survival. (3) Such cultures discussed include the Borana of Ethiopia with around 15 million people, the Basque community of Sainte-Engrâce with a population of 375 people, and the Balinese with two million people. Furthermore, among many other contexts; divination, calendrics, and constructing societal hierarchies, are common areas where mathematical ideas are embedded in many of these communities. Finally, one area: divination, is especially useful for pinpointing these mathematical ideas, because, as this subject is so common and yet esoteric, it enlarges the commonly held definition of mathematics.

Divination, on a superficial level, is defined as a decision-making process involving a randomizing mechanism to procure a particular result. (5) On a deeper level, divination is a way of organizing the world; it turns the unpredictable, chaotic world into a more manageable decision. Instead of being an infinite number of possibilities, divination filters out the extraneous possible results, and leaves the diviner and the client a few possible options which are then interpreted and studied in an unique context. This process of sifting through possibilities is really a specific way of thinking, an underlying belief there exists some kinds of knowledge that is not acquired by one's own means, but revealed to one by following a precise method. Ascher describes this as,

> Divination... [is a] shared, systematic, and structure approach to knowledge. They depend not on the state of the diviner but on... his careful adherence to procedures and on his reservoir of wisdom. These latter divination systems are, in fact, considered by some scholars to be sciences.

> [Additionally] belief in divination does not imply believing that *all* occurrences are controlled by extranormal forces; it is only believing that the outcomes of *particular* procedures, carried out under *particular* circumstances, and usually with *particular* materials, are expressions of specific deities, witches, or other supernatural forces.
(5-6)

This view of divination, as a system of thinking and reasoning, is closely related to the rules that govern a mathematicians work. As author Douglas Hofstadter writes "proofs are demonstrations within fixed systems of propositions," similarly, divination sessions result in conclusions stemming from a series of codified techniques. (Hofstadter 26) Altering any of these standardized methods in either situation would cause different outcomes would occur. Hence, by drawing parallels to the underlying backbone of the what divination really is, in turn, the definition of mathematics is also enlarged and
clarified.

Moreover, Ascher describes how many common divination tactics use procedures found in the Western mathematical subject: *Discrete Math.* Discrete denotes separation and Discrete Math is simply the collection of mathematical techniques to study separate, countable objects. As divination is the process of filtering out the infinite possibilities into a handful of separate, distinct predications, the similar use of discrete math techniques is logical. Finally, some of these topics that both disciplines use are modular arithmetic, probability, ordered pairs, and boolean logic.

Nevertheless, there are holes to Ascher's claims that many mathematical ideas can be found in other cultures. Particularly because this book focuses on small, traditional cultures, many of the mathematical ideas are in the context of other subjects. While these mathematical ideas are useful, one argument is to question if these similar mathematical ideas have a finite end. Since many traditional cultures do not study math apart from other subjects, a mathematician might ask if these cultures would ever reach the level of abstraction found in the modern, Western Academia. Ascher recognizes the limited scope of the book by writing:

> What particularly attracts our attention, however is the *formal* training received by some specialists in some cultures. We see this mode of learning as distinctive and as having important ramifications for the mathematical ideas being learned. What we mean by formal learning is... the training is organized, separated from daily routines, and carried out by members of a professional group... Perhaps the most significant aspect of this formal training is its separation from the experiential... [and] we are led to suspect that this formal mode is particularly conducive to the creative and transmission of substantial *systems* of ideas. (194-195)

Despite this argument, I still find this book's thesis to be strongly convincing. Having been educated in the Western Mathematical school, this book introduced me to many interesting viewpoints about mathematics and to new cultures. Learning other cultures' uses and relationships with mathematical ideas expanded my definition of mathematics, and I am able to appreciate math now in a more global and humanistic light.