NATIVE AMERICAN MATHEMATICS, Edited by Michael P. Closs; University of Texas Press, P.O. Box 7819, Austin, TX 78713; 512-471-7233; 1986, first paperback printing, 1996, 431 pages, reference bibliography, contributor notes, 0-292-71185-9
This book contains 13 rather technical articles, whose publication here was supported by the Canadian Society for the History and Philosophy of Mathematics. Most of the articles contain number systems -- really, mostly number words, with occasional notes on what might have been early linguistic forms suggesting simple tribal arithmetics. The strongest suggestion, however, is that arithmetical developments occurred only after contact with fur traders, and involvement in trade, made some such reckonings necessary; they were wholly unnecessary to hunting-gathering cultures in small bands. Similarly, measurement, with objective units never developed, as measurements with context-dependent relative units such as handspans, armlengths, or number of sleeps to determine distances, sufficed for constructions made entirely by the individual, who built what he or his family needed, and not communicated among construction or tool-making specialists. People who traveled well-known territories but did not claim land as property had no need to measured nor map and did not develop abstract (context-free) units nor techniques for doing so.
These articles may be of some use to those who want to develop cultural curriculum supplementary units on mathematics. This has already been done for Mayan numeration and some kinds of calculations done in their base-20 system. It will be much harder for the developer to make use of information in the other articles, because these really consist more of linguistics (numeration words) than anything else.
"Cultural Ecology of Mathematics: Ojibway and Inuit Hunters" shows a world view in which there existed many different forms of measurement that are context-dependent. For example, a distance measure of &q uot;5 sleeps" depends on the weather or terrain covereed, rather than on linear distance as we conceptualize it now. (Mathematics, numbers, measurement units and the logic of western mathematics are context-free, not context-dependent.)
The hunter-gatherer system of thought in J. Peter Denny's exposition is fascinating, and may suggest some exercises and teaching materials that could be developed, though considerable effort will be required, because teaching development also has to use such exercises to lead into non-relative measuremnt units. They key concept for context-free units, number, and measurement development appears to be communication. If I am building a canoe, or teaching someone who is physically present, measuremnt units which are all body-spans can work. If I must communicate this information to someone not physically present, objective or abstract units, independent (context-free) 0of a particular person's particular bodily measurements, must be agreed upon by both parties and methods that work, whoever carries them out can then develop.
"In search of MesoAmerican Geometry" by Francine Vinette is suggestive that on some Mayan sculptures, an aesthetic principle of organization that might have been to that culture as the Greek Golden Section organization is to western architectural and sculptural organization. She uses very few examples, and publishs only a couple of diagrams and no measurements or photographs. Mathematical ratios, if any, are not discoverable, and in any event have no significance if found in only a handful of objects.
This book sets ,forth the beginnings of a history of mathematicsd for the western hemisphere, but it seems most likely that although much could be done by analyzing buildings and objects of Meso-America and Peru, the absence of written records makes this kind of study unlikely to produce much. Mathematics is not a disconnected assortment of assertions and pragmatic techniques, but a vast logical structure, contributed to by relatively small numbers of men over thousands of years. Both at an individual level and as a discipline, it is inconceivable that mathematics can exist without a written and symbolic language of logic that can be used by the mathematician, and that can be shared by those rare scholars of diverse cultures, who build on one another's work although separated by centuries of time and thousands of miles of space. Written and symbolic logical language is essential to the abstraction -- context-free character -- of mathematics. Thus what may have developed in the southern parts of the western hemisphere before it was destroyed by Spanish invaders is mostly irrecoverable (if it once existed) except in the limited form of finding a few relationshiups among stone fragments.
On the whole, this book seems of interest only to specialists in the history of mathematics. And because there is so little data, it is not of much interest to them, either. That is, on the evidence of this book, thre was little or no mathmatical development in the western hemisphere, but what there was might be to some limited extent recoverable from what remains of the larger constructions in the southwest, meso- and south America. There may have been individual discoveries of mathematical import, but what distinguishes mathematics as a discipline is its logical construction over milennia by thousands of rare individuals, who can comprehend one another's work and contribute to the structure. This requires a system of writing and abstract symbology that can be comprehended by mathematicians of different cultures and societies and eras even where languages differ. This did not exist in the western hemisphere.
Some of Closs's assumptions (stated in his short introduction) about the m=natur of mathematics seem to me simply wrong. He says for example that "symbolic description of these concepts is presented in a universal mathmaticla notation indpendent of language. For example, as part of this notation, numbers are expressed in a decimal system, using Hindu-Arabic nmumerals." That example is simply wrong. The fact is that where mathematics of any kind does exidst, it can be extracted from various notations (for example the Mayan, where it is discoverable they had a zero and their system was base-20, not base-10 decimal) by its structure, which is the universal, because in fact it is entirely abstract: a network of relations -- not by its use of a convenient common language, or a convnient common notation. What is common to all mathematics is a universal and wholly abstract logical structure, expressed across culturs and historical eras in a variety of different notations, but perciptible (to those with the talent) because the structure is the same.
Closs asserts that historians of mathematics "have paid only scant attention, if any, to mathmatics in cultures not directly contributing to i[modern mathematics]." The suggestion is if only historians would turn enough attention to mathmatical matters in the western hemisphere, a similar structure might be articulated. I do not believe so. This is a "politically correct" attempt to claim the existence of something -- an abstract, logical structure that is independent of cultures and specific languages or notations -- that there is no evidence for in the western hemisphere. It is not a question of "indirect" contributions, either. There were no contributions, indirect as well asnone direct, in past eras, to the structure of mathematics.
Such contributions as may be now made by mathematically-inclined indigenous youth will not be fostered by false claims about a non-existent history. We are currently in an era of revolution in mathematics. The concepts of fractal gemoetries that use the complex plane of numbers and computers as investigative tools have opened an era in which new kinds of infinties (fractional) have appeared and new applications to natural phenomena are being discovered, along with the gorgeous computer-generated fractal patterns that depict iterative processes which use their own outputs to form inputs for successive operations. This is an area in which talented Native youth can make significant contributions to a vast and ancient structure, if they learn about it. This will not happen if Native youth are decoyed into what amounts to false histories. The pragmatics of counting, simple arithmetics, and the pragmatically-applied geometries of builders are not mathematics, and it is an educationally false trail to Native youth to suggest that such study will make mathematicians of any of them.
Closs ends his introduction by saying "in my opinion, native Americna mathematics can best b described as a composite of separate developments in many individual cupltures." But I suppose this is a way of also saying since there never was developed the abstract structure which not only could be but was shared among these separate cultures, as happened around the Meditrranean 6,000 years ago, there is no matheematics of the kind he would like to posit, because what he describes is just that situation: no common structure communicated among many cultures. Which (to me) means there is no mathematics, no common structure of abstract logical relations. He says a western-hemisphere discipline of history of indigenous math must "draw on disciplines outside its traditional structure: anthropology, linguistics, archaeology." Unfortunately, the reason for that is that in the western hemisphere an indigenous mathematics does not exist, did not develop, and so scraps of builders' pragmatics, bits of language that denote numbers, counting, and very simple arithmetic -- which have no history as contributions direct or indirect to the actual structure of mathematics -- are substituted.
I would rather see Native youth study actual mathematics and its real history. Reviewed by Paula Giese
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Last Updated: Friday, May 17, 1996 - 5:14:30 AM