25 of 28 people found the following review helpful:
2.0 out of 5 stars
For scholars,OK. Educational guide: No, June 8, 1997
By A Customer
This review is from: Native American Mathematics (Paperback)
This book contains 13 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. They were 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 measure 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 "5 sleeps" depends on the weather or terrain covereed, rather than on invariant 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 measurement 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, measurement units which are all body-spans of one of us can work. If I must communicate this information to someone not physically present, objective or abstract units, independent (context-free) of 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, there may be 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. Whereas the Golden Section is well-supported by numerous measurements and analyses, as well as ancient Greek writings on irrational numbers.
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. Written records of the several large civilizations conquered by the Spaniards were systematically destroyed, and only a few mss. survive, mostly inaccessible in European holdings. This makes analysis a pastime of the very rich, or well grant-supported.
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 been developing 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 relationships 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 actual data, as opposed to speculative assumptions, it is not of much interest to them, either, I should think. On the evidence of this book, thre was little or no mathematical 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 nature of mathematics seem to me simply wrong. He says for example that "symbolic description of these concepts is presented in a universal mathmatical notation independent 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 exist, 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; or the California Pomo who also used a base 20 numeration.)
Mathematical system is discovered 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 convenient common notation.
Closs asserts that historians of mathematics "have paid only scant attention, if any, to mathematics in cultures not directly contributing to [modern mathematics]." The suggestion is if only historians would turn enough attention to mathematical 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.
Such contributions as may be now made by mathematically-inclined indigenous youth are not encouraged by decoying them into dead ends with false claims about a non-existent history.
Counting, simple arithmetics, and the pragmatically-applied geometries of builders are not mathematics, and it is an educationally false trail, a trap for the minds of Native youth, to suggest, as Closs and others do, that such study will make mathematicians of any of them.
--Reviewed by Paula Giese, Editor of Native Books website, http://www.fdl.cc.mn.us/~isk/books/bookme
Help other customers find the most helpful reviews
Was this review helpful to you? Yes
No
