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| A painting of the Santa Cruz church mission From Wikipedia |
Doing a history of mathematics for American Indians is a major challenge as traditional methods of historical inquiry are of limited use. In its most familiar form, the history of mathematics focuses on the narrative history of how intellectual ideas were developed by individuals working within a western intellectual environment. Consider the titles of two award-winning books: Euler: The Master of Us All and Leonhard Euler: Mathematical Genius in the Enlightenment. As the titles suggest, they are biographies of the eighteenth century mathematician Leonard Euler, and they focus on presenting Euler's mathematical work to a modern audience with the goal of impressing upon readers the transformative impact of Euler's scholarly work (he was a genius who "mastered us all"). While Euler lived more than two hundred years ago, he lived a life that was not that different from the life of a typical modern-day math professor. Following an education in Switzerland that culminated in a dissertation accepted at the University of Basel, Euler worked at European scientific academies. He published prolifically in academic journals, so a modern researcher has a wealth of writings to draw on. Important personal papers were also preserved, so the authors are able to humanize Euler's mathematical work by placing it within a broader narrative of his life (the best known story is perhaps that Euler lost his eyesight yet continued to remain remarkably prolific, publishing an average of one mathematical paper each week.)
The type of history represented by Euler: The Master of Us All and Leonhard Euler: Mathematical Genius in the Enlightenment is basically impossible for the indigenous peoples of California. The type of written records needed to produce a detailed historical biography of any individual, mathematician or otherwise, simply do not exist. Indigenous people were also largely excluded from academia until relatively recently, so we can't trace their mathematical activities in scholarly journals.
We can, however, make progress in answering important questions if we change our focus. In this post, I will focus on how to extract historical information from linguistic data. While we can't use this to craft narratives of individuals, much less make a case for their "mathematical genius," we can answer important and interesting questions.
Perhaps the most interesting question is, "Could native peoples count?" That is, was the practice a European introduction or did native people develop their own counting methods? This question has been debated for over a century. Influential early ideas were set out in a 1863 paper, "On the Numerals as Evidence of the Progress of Civilization," by John Crawfurd. Crawfurd, a British colonial administrator and a member of the Ethnological Society of London, argued in that linguistic information about numbers could be used to assess how "civilized" a given people were.
Crawfurd asserts that the ability to express and calculate with numbers is a difficult, abstract skill. Following ideas surrounding Darwinism, Crawford takes for granted that the "different races of the human family" are progressing towards "civilization" A measure of how far a given "race" has "progressed," he argues, is how their language can be used to express numbers. More "civilized" people speak languages that allow for the expression of larger numbers. Explaining their languages, Crawfurd finds that the languages spoken by aboriginal Australians only include words for numbers up to four, while the Chinese language allows for the expression of numbers up to the range of a thousand million. Consequently, he concludes that the Chinese are more civilized than aboriginal Australians. This belief was widely by European scholars, so it would have been seen at the time as validation of Crawfurd's argument.
Anyone who has taken an introductory class in anthropology in the last one hundred years should be able to dismantle Crawfurd's paper. The theory that all cultures are progressing towards the same way of organizing society, towards "civilization," is regarded as a misapplication of the ideas of Darwin. The rejection has been so complete that I almost feel like I am arguing with a straw man in discussing it. Around the late nineteenth century, scholars largely associated with Franz Boas collected evidence to support the idea that a more useful (and less racist) conceptual framework is to replace the biologically based notion of different "races" with a concept of "culture" that is socially determined. The Boas school agreed that a culture changes by an evolutionary process, but this process consisted of responses to circumstances specific to the culture rather than a unified "progression" towards "civilization."
Nevertheless, Crawfurd's idea of deriving information about mathematical practices through a comparative study is an attractive one. Crawfurd's paper includes an appendix of number words for over fifty different languages, and a study of it clearly demonstrates intriguing and suggestive similarities and differences. (Many languages appear to express numbers using "base 10" as in English, but others appear to use a different base, and some seem to use a whole different way of reckoning.)
One can modify Crawfurd's underlying theoretical ideas so that they work within the modern framework of cultural evolution. For many cultures where the number system is well-document, the system appears to originate in the bureaucratic needs of the governing class of an agricultural society: the king wants to collect a portion of each farmer's crops as a tax, so his administrators need a methods to record things he wants to collect. In response, the administrators develop words for expressing the large numbers they need to record, and these words disseminate broadly through the society over time as different people interact with the administrators.
Despite the passage of over one hundred years, the validity of this idea continues to be debated. Is counting an abstract skill that had to be invented? Or is it a hardwired skill, similar to or even a part of Chomsky's universal grammar in linguistics? Chomsky himself is in the first camp. He argues that numerical facility is an invented/developed skill: it is separate from linguistic facility and could not have developed by natural selection. An obvious challenge to this school of thought is the empirical fact that people of all cultures seem to not only possess some facility with numbers but there are also striking commonalities in how numbers are expressed. The word for "ten" plays often plays a distinguished role, for example. Those who argue that counting is an invented skill explain this as a by-product of how things like human physiology impact the manner in which people count. The number "ten" is significant because people commonly count using their ten fingers.
Work in neurology and psychology provides a stronger challenge to the idea that counting is not a hardwired skill. Other animals have been shown to have some facility with numbers. A study of lionesses showed that they can distinguish between small numbers such as five and three. Other studies have shown that human infants can similarly distinguish between small quantities.
A productive framing of the issue is, what aspects of numerical facility are hardwired? Nobody would argue that humans are hardwired to understand complex numbers. Many of my calculus students, even very mathematically talented ones, are unable to grasp the concept. Much less clear is whether humans are hardwired to understand that the successor principle, that an infinite sequence of numbers can be generated by starting with small numbers like one, two, and three and then repeatedly "adding one."
The social processes driving the development of number systems has also been questioned. While, in cultures like ancient Mesopotamia, the early development was driven by the needs of a bureaucratic state is also challenged, but one can debate whether this is a widespread phenomenon. We can only document the early development of numbers for a handful of cultures, so perhaps the apparent significance of the bureaucratic state is an illusion created by the nature of the evidence that's available to us. One can point to other cultural practices that create a need to develop a number system. Large numbers hold a special significance in some religions (think about calculating the age of the Earth using information from the Bible) and are a natural object of intellectual interest (many young children get excited about naming "the biggest number").
Certainly, I am not going to resolve these issues, but they held direct attention to important aspects when trying to use linguistic evidence to say something about the history of mathematics. In any case, the first matter is the bread-and-butter issue of simply documenting in a careful way how a given language expresses numbers. For many native languages, this is no simple matter.
Consider the native people who lived in the area around the modern city of Santa Cruz. There are no records of their languages until the late 1700s when they encountered the Spanish. The first Spanish the native people interacted with were Catholic priests who had come to convert people to Catholicism. Many Indians were moved to church missions, sometimes forcibly, and were encouraged to learn Spanish and adopt other European cultural practices. Moreover, relations were often tense as some native people were forcibly moved to the mission, and many lived under harsh conditions that included performing difficult manual labor for the mission. In the mid-1800s, the Catholic missions were secularized, and most of the native people began living and working on large sheep and cattle ranches ("Ranchos") owned by Spanish immigrants.
Over the course of the 1800s, many native languages went "extinct" (although there are important ongoing language revitalization efforts). The information that we have about these languages is often information recorded by people with no linguistic training and little sensitivity to native culture (the Catholic priests who ran the missions, for example). Moreover, they were collecting information from speakers who had been in contact with Spanish and English speakers for decades.
One valuable source I have found for information about the language spoken by Indians living on the Santa Cruz mission is vocabularies that were collected by the French scholar Alphonse Pinart in 1878. The available records are lists of words collected from four different people: a woman who grew up on the mission in Soledad, a woman living on the Santa Cruz mission, a man living in the town of Aptos, and a man living in Carmel.
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| Location of the interviews with speakers of Ohlone speakers From Google Maps |
The numerals recorded by Pinart are displayed below. The first column is an English word, and the second is the corresponding word in Spanish. The remaining four columns record the words used by the people Pinart interviewed. ("Costanoan" is the term the Spanish used to referred to the people living around modern Santa Cruz. The modern term is "Ohlone.") You'll see that no information from second speaker was recorded, and there are significant differences in the words the other speaker used. Taking into account differences in pronunciations, the numbers for 1, 2, 3, and 4 appear to be similar, but others words seem largely unrelated. This could reflect differences in regional dialect. While speakers are separated by less than seventy miles, at least two of the speakers grew up on different church missions. In a number of accounts, residents remarked on the diversity of native languages, with people on each ranch and mission speaking their own dialect.
Despite the limitations of the data, we can make some significant deductions. The first speaker appears to count using a base ten system. The word for twenty, "utcihk matumn" is formed by modifying the word for ten, "matussu," to "matumn" and then prepending the word "utchihk" which appears to be a modification of the word for two, "utci."
The words used by the fourth speaker are even more intriguing. The word for "six" is created by joining the word "xali" to "sakken," and then the next two numbers are created by replacing "xali" with ucumai (perhaps a modification of the word for two, "uthis"?) and then with kapxami (a modification of "kappes" or "three"?) This suggests that the fourth speaker is counting numbers using base 5.
The use of base 5 is more than a curiosity. In both Spanish and English, speakers count in base 10, so the fourth speaker's manner of counting can't be an adoption from Europe. Rather, it suggests that the native people had developed ways of counting prior to contact with Europeans.
The fourth speaker also told his interviewer how to count small quantities of coins (reales or Spanish silver coins and pesos) in his language. It appears that the words for numbers undergo slight modifications when used in references to coins. For example, the word for two, "uthis," is modified to "utis" when used in references to reales. Of course, in reaching this conclusion, we're dependent on accuracy of the speaker's pronunciation and the researcher's transcription. The differences could, perhaps, be simple transcription errors.
We could try to investigate the language in greater depth by looking for additional records of the language or by using comparative methods. Before we get carried away, we should try to develop a more robust framework for describing and analyzing the manner in which a given language expresses numericals. I will do that in a subsequent blogpost.
