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The Mathematics and the Liberal Arts pages are intended to be a resource for student research projects and for teachers interested in using the history of mathematics in their courses. Many pages focus on ethnomathematics and in the connections between mathematics and other disciplines. The notes in these pages are intended as much to evoke ideas as to indicate what the books and articles are about. They are not intended as reviews. However, some items have been reviewed in Mathematical Reviews, published by The American Mathematical Society. When the mathematical review (MR) number and reviewer are known to the author of these pages, they are given as part of the bibliographic citation. Subscribing institutions can access the more recent MR reviews online through MathSciNet.
Ascher, Marcia and Ascher, Robert. Ethnomathematics. Hist. of Sci. 24 (1986), no. 64, part 2, 125--144. (Reviewer: Jens Høyrup.) SC: 01A10 (92A20), MR: 88a:01005.
Discusses the danger of identifying non-literate mathematics with "primitive" mathematics. Warns against assuming that because a group has two sets of number words (as in the Blackfoot Indians, who are said to use different sets of numbers for the living and the dead), the group therefore doesn't understand the underlying identity between the different words. Regarding logic, when asked the question "All Kpelle men are rice farmers. Mr Smith is not a rice farmer. Is he a Kpelle man?", one Kpelle respondent answered "If you know a person, if a question comes up about him you are able to answer. But if you do not know the person, if a question comes up about him, its hard for you to answer." The authors emphasize that a response like this doesn't show a lack of ability in logical reasoning, but just differences in views in talking about people you don't know and about 'playing along' with a questioner. The authors discuss how the Sioux viewed the circle as a more natural shape than the (western) line. Kinship systems of the Aranda of Australia, and in Ambrym in the New Hebrides. How elders in Ambrym used diagrams to elucidate the kinship systems, and explicitly explained the patricycles of degree 2 and the matricycles of degree 3. An interesting question for a student might be to investigate if the Aranda system (with six groups) is optimal in ruling out certain types of marriages that are too close. Closely related topics: Ethnomathematics General, Number Words, Logic, Kinship Systems, The Aranda, Ambrym, New Hebrides, The Blackfoot Indians, The Sioux, and The Kpelle of Guinea.
Cordrey, William A. Ancient Mathematics and the Development of Primitive Culture. Mathematics Teacher 32 (1939), 51--60.
Discusses number words and systems of time reckoning for a wide variety of groups. Although many readers may be familiar with the Egyptian and Babylonian number systems, there are many interesting examples from the indigenous peoples of North and South America. The reader may want to ignore statements regarding the relative levels of different cultures. Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: Number Systems, Number Words, The Reckoning of Time, and Indigenous American Mathematics.
Court, Nathan Altshiller. Mathematics in the History of Civilization. The Mathematics Teacher 41 (1948), 104--11.
How different concerns of society influenced mathematics. How the development of the concept of number is reflected in language. How the concept of how many led to arithmetic. How the concept of how much led to geometry. (Taxation and agriculture also contributed to both.) Efforts to keep time led to trigonometry. Navigation and associated astronomical problems led to logarithms [and more trigonometry]. Problems in artillery led to graphs. Both required an understanding of motion. Analytic geometry and calculus were invented in part to better understand motion. Statistics developed to understand problems in the social sciences. Also discusses the nature of mathematics: mathematics for its own sake and the axiomatic method. Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: Why Study History Of Math, Number Systems, Arithmetic, Geometry, Taxation, Agriculture, Astronomy, The Reckoning of Time, Trigonometry, Artillery, Graphing, Navigation, Dynamics, Force, and Motion, Analytic Geometry, Calculus, Statistics, Social Science, and Proof.
Deakin, Michael A. B. The origins of our number-words. Austral. Math. Soc. Gaz. 23 (1996), no. 2, 50--66. SC: 01A07 (00A99), MR: 97c:01003.
Discusses how our number words evolved from Proto-Indo-European, and how English also has other traces of Proto-Indo-European number concepts. In the process, discusses a number of other languages, some Indo-European and some not. The author begins with the examples of the words eight and four and their derivation from the Proto-Indo-European *oktou and *kwetwores. He then discusses how languages with a one-two-many system also often have three forms of a noun, and how the form for 2 (the dual form) left remnants in English. Discusses how 1, 2, 3, 4 were adjectives in Proto-Indo-European, and 5, 6, 7, ... were nouns; other Indo-European languages have signs of this. In the case of 2, even English does. The author advances the interesting theory that the Proto-Indo-European word for five, *penkwe derives from a word for 4. As he notes, there is some evidence of an ancient base 4 number system in the Proto-Indo-European word for 9, *newn or *newm. See, for example, the Latin word novem, meaning new. Old Japanese may have some evidence of an ancient base 4 system as well; this is particularly interesting if, as is sometimes now believed, Japanese and Proto-Indo-European derive from a common source (Nostratic). The author gives examples of languages with coexisting bases. These lend plausibility to his argument about 5. Closely related topic: Number Words.
Dwornik, Henryk. A $2\sp{n}$-number system in the arithmetic of prehistoric cultures. Organon No. 16-17 (1980/81), 199--222 (1983). (Reviewer: Garry J. Tee.) SC: 01A10, MR: 85f:01006.
The author attempts to explain use of base 12 or base 60 in otherwise primarily base 10 cultures as an attempt to reconcile a base 10 and a base 2n system. As evidence of such a base 2n system, the author discusses the use of "base" 2 worldwide in systems for measuring distance, area, volume, and weight. He also discusses how Indo-European languages show evidence of an ancient base 4 or 8 system in the words for nine, such as in the well-known example of the Latin novem for both new and nine. The numbers 4+1, 4-1, 16+1, and 16-1 are all represented neatly in base 60. The author discusses some advantages of a number system where numbers are represented by bn...b2b1b0 as in base two, except where bi=1, 0, or -1. As the author admits, all of this is highly speculative. The author also makes the interesting observation that some of the numbers used in Mayan cosmology become very symmetric when expressed in base 2 on a 3x3 board. The suggestion seems to be that base 2 computation may have been a motivating force for the Mayans. As we still have little knowledge about Mayan arithmetic, it may be awhile before we have a definitive answer. Closely related topics: Number Systems, The Measurement of Distance, The Measurement of Area and Volume, The Balance and the Measurement of Weight, Number Words, and The Maya.
Gerdes, Paulus. On mathematics in the history of sub-Saharan Africa. Historia Math. 21 (1994), no. 3, 345--376. SC: 01A13, MR: 95f:01003.
This paper broadly surveys the recent research in sub-Saharan mathematics (and some related areas as well). Areas discussed include prehistoric mathematics (e.g., the Ishango and Border Cave bones), number systems and symbolism (including algorithms and education), games and puzzles (for example, a leopard-goat-cassava leaf river crossing problem and a "topological" puzzle), symmetry in African art, graphs or networks (e.g. Tschokwe sand drawings), architecture (one case involving magic squares; also a brief reference to fractals). Gerdes mentions string figures as a possibly productive future research area; he gives some starting points. He also discusses related areas, such as technology, and studies on language and mathematical concepts. A goal of the studies mentioned is apparently to better understand mathematics learning in Africa. Some studies focus on logic. Questions on interaction with ancient Egypt are still largely open. A better understanding of Islamic mathematics in sub-Saharan Africa is desirable as well. The author also touches on factors connected with the slave trade; e.g., the remarkable but not perhaps entirely atypical abilities of Thomas Fuller. Includes an extensive bibliography. Closely related topics: Sub-Saharan Africa, TallySystems, Games, Puzzles, Topology, Symmetry, Continuous Tracing Problems, Architecture, Magic Squares, Fractals in Art, String Figures, Ancient Egypt, The Reckoning of Time, Education, Logic, The Islamic World, and Thomas Fuller (1710-1790).
Harvey, H. R. and Williams, B. J. Aztec arithmetic: positional notation and area calculation. Science 210 (1980), no. 4469, 499--505. (Reviewer: M. P. Closs.) SC: 01A10, MR: 81k:01002.
It has long been thought that the Mayans were the only Mesoamerican people to have developed a positional number system. However, as the authors have noted, the Aztecs also had such a system (using lines and dots). The treatment of zero may be less consistent than it was with the Mayans. The authors discuss Aztec calculations of area as well. The Aztecs clearly used some sort of algorithm to compute these areas. (It's difficult to assess the calculations perfectly since areas of quadrilaterals are only determined by the lengths of the sides in the special case of triangles.) The authors discuss why the mathematics discussed in this article was unlikely to have come from the Spanish. The authors also discuss an interesting feature of the Nahua language which was spoken by the Aztecs, where a system of classifiers was used; the language included classifiers for round objects, for objects where length is a primary factor, and for objects that can be stacked. Closely related topics: The Aztec, Number Systems, The Measurement of Area and Volume, Number Words, and The Maya.
Hughes, Barnabas. Hawaiian Number Systems. Mathematics Teacher 75 (1982), 253--56.
Discusses the original mixed base (4 and 10) Hawaiian system and the introduction of a strict base 10 system after the arrival of missionaries. Gives many examples of both types of number words. (One theory, due to W. D. Alexander, 1864, is that groupings by 4 became popular from the the custom of counting fish and such by taking a couple in each hand or by tying them in bundles of four.) The transition between the two number systems was apparently not entirely smooth; younger Hawaiians understood only the decimal system had difficulty with older Hawaiians, who for example used different words for forty when speaking of forty canoes than speaking of forty fish. The author also discusses the introduction of some other words into Hawaiian. Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: The Hawaiians, Number Words, and Number Systems.
Hughes, Barnabas B. The earliest known record of California Indian numbers. Historia Math. 1 (1974), no. 1, 79--82. SC: 01A15, MR: 57 #15836.
The author discusses a document from 1775 which is now thought to be the earliest written record of the number system of a California Indian tribe. The document includes numbers for one through 14. One interesting feature is that some of the words suggest a base 4 number base. Also interesting is the fact that some of the number words that were recorded are different from the ones recorded by Dixon and Kroeber for related Costanoan Indians in 1907. These differences between this these number words (from Angel Island) and the others (from Mission Santa Clara) may indicate the influence of other tribes. The author notes that since Angel Island is nearly a centerpoint of various waterways, this influence is not surprising. The document was written by Fr. Vincente Maria, the chaplain of a Spanish expedition, and the author of the article seems to suggest that the confessional practices of the time may have encouraged Indians to use a decimal system for numbers larger than ten. This was because sins had to be identified at confession both by kind and by number, and because the Indians were not likely to be otherwise understood. Closely related topics: Number Words, Number Systems, California Indians, and Religion.
Seidenberg, A. km, a widespread root for ten. Arch. History Exact Sci. 16 (1976/77), no. 1, 1--16. (Reviewer: Richard L. Francis.) SC: 01A10, MR: 58 #4778.
Seidenberg studies number words in a wide variety of languages and finds some surprising similarities. He argues from these similarities that these number words, and therefore the corresponding number concepts, arose one place and spread throughout the world by a diffusion process. Here, and also in his article Seidenberg, A., The ritual origin of counting, he notes several similarities in the construction of number words in three languages that are built on the number words for one and two (Gumulgul in Australia, Bakairi in South America, and of the Bushmen in South Africa). These include the fact that in building odd numbers, the word one comes at the end, and also the fact that there is on connective. Similarities like these seem particularly natural under the diffusion hypothesis. However, the strongest evidence would come from number words themselves; for as Seidenberg notes, "If the number-vocabularies of the Gumulgul, the Bakairi, and the Bushman had been the same, and not merely the same in structure, probably everybody, or nearly everybody, would concede that the words derive from a single source." In fact, Seidenberg does find that one root, km, seems to appear in many number words world-wide. After looking at a wide variety of languages, Seidenberg concludes (p. 11) that the original word km meant "one", and thereafter began to be used for "one" larger unit, and particularly for the unit ten. He also finds at least one example of the root km meaning four; one wonders if perhaps this use might even have predated its use for ten. There have been attempts to explain the proto Indo-European root for ten in a way that conflicts with Seidenberg's theory; the notion of ten as "two hands" seems popular. Seidenberg discusses these attempts, but feels that they are rather ad hoc. Certainly from the point of a mathematician, Seidenberg's theory is very appealing. Closely related topics: Number Words, Number Systems, and Abraham Seidenberg.
Seidenberg, A. The ritual origin of counting. Arch. Hist. Exact Sci. 2 (1962b), 1-40.
It is common to argue that counting and other elementary mathematics arose spontaneously throughout the world in response to a practical, or perhaps psychological, need. Abraham Seidenberg argues instead for a diffusion theory, that counting arose only once, and then spread throughout the world. In fact, many common associations with numbers suggest such a common origin. One such association that Seidenberg is the idea that odd numbers are male and even numbers are female; this is certainly well known from the Pythagoreans, but turns out to be nearly universal. Seidenberg proposes that counting in fact originally arose in a ritual context. Seidenberg draws from a wide variety of anthropological sources for rituals and myths that hint at what this common origin might have been. He finds that counting "was frequently the central feature of a rite, and that participants in ritual were numbered." He focuses more specifically on creation rituals. He suggests that in the enaction of creation myths, men and women may have come onto the scene alternately, easily explaining the odd/male even/female association. He finds that his ideas clarify "pure 2-counting, which is the oldest stratum of counting we can detect." In pure-2 counting, there are separate words for one and two and these are used to form all other number words. He illustrates this with number words from diverse languages such as the Gumulgal of Australia, the Bakairi of South America, and the Bushmen of South Africa. He sheds additional light on his hypothesis with discussions of the possible origin of counting taboos (and connections with ritual sacrifice), of ancient one-one-correspondence "tally" systems (e.g., counting people with stones), of taxation systems, of money, and of gematria. Seidenberg also gives us some fascinating examples of counting in world religions. These include the analogy The Lord : His people = the shepherd : his sheep, the analogy The shepherd : his sheep = the moon : the stars. These two lead one to expect the moon to count the stars; and Seidenberg in fact finds evidence of this in ancient Babylonia. He argues from the equation The Lord's people = the stars of the heaven to The Lord's people = the sand upon the seashore that one would expect to find a ritual counting of sand. In fact, he finds the notion of Counter of the Sands both in Buddhism and among the Ancient Greeks. The equation The Lord = The Counter seems to be confirmed in two of the ninety-nine beautiful names of Allah, namely The Counter and the Reckoner; and there is further confirmation in Chapter's XV and XIX of the Qu'ran. This is a fascinating article, connecting mathematics with a wide variety of disciplines. Closely related topics: Myth and Ritual, Storytelling Traditions, Anthropology, General, Counting, TallySystems, Taxation, Number Words, The Pythagoreans, Gematria, Religion, The Islamic World, and Abraham Seidenberg.
Wren, R. L. and Rossmann, Ruby. Mathematics Used by American Indians North of Mexico. School Science and Mathematics 33 (1933), 363--72.
Surveys the use of numbers and geometric shapes in various North American indigenous peoples. Includes sacred numbers, number words, including an unusual instance of subtractive number words in the Bellacoola of British Columbia, number systems, reckoning of time and seasons. Also includes geometric characteristics of dwellings and (briefly) textiles, basketry, pottery, and tattooing. Often pottery designs were borrowed from textile art. A common principle in weaving is that no line, curved or otherwise could intersect itself. (Is this principle partly responsible for the popularity of spirals?) Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: Indigenous Mathematics of North America, Numerology, Number Words, The Bellacoola, The Reckoning of Time, Pattern, Weaving, Basket Making, Pottery, and Tattoos.
Zaslavsky, Claudia. Africa counts. Number and pattern in African culture. Prindle, Weber & Schmidt, Inc., Boston, Mass., 1973. x+328 pp. SC: 01A10, MR: 58 #20993.
This book is an excellent introduction to the mathematics of (primarily sub-Saharan) Africa. The best tribute to its importance may be in Gerdes, Paulus, On mathematics in the history of sub-Saharan Africa. Gerdes writes "In her classical study Africa Counts: Number and Pattern in African Culture ..., Claudia Zaslavsky presented an overview of the available literature on mathematics in the history of sub-Saharan Africa. She discussed written, spoken, and gesture counting, number symbolism, concepts of time, numbers and money, weights and measures, record-keeping (sticks and strings), mathematical games, magic squares, graphs, and geometric forms, while Donald Crowe contributed a chapter on geometric symmetries in African art." Regarding geometric symmetries, it is primarily the frieze patterns and plane patterns that are discussed; there is surely more work to be done on the bichromatic frieze and plane patterns. Many readers will wish to explore further. Gerdes' paper should be invaluable for this, not least for its extensive bibliography. Another useful resource is the newsletter distributed by the African Mathematical Union's Commission on the History of Mathematics in Africa (AMUCHMA). Closely related topics: Sub-Saharan Africa, TallySystems, Finger Numerals, Counting, Numerology, The Reckoning of Time, Money, Measurement, Games, Continuous Tracing Problems, Architecture, Magic Squares, Frieze Patterns, Plane Patterns, The Islamic World, and Anthropology, General.