Is marriage a mathematical operation?

    One autumn Sunday in 1934 the director of the prestigious Parisian school École Normale Supérieure (ENS) called up a young philosopher called Claude Lévi-Strauss and asked him if he might be interested in applying for the position of Professor of Sociology at the University of Sao Paolo in Brazil. Lévi-Strauss, who had not yet reached his thirties, accepted the offer to leave for this faraway place mainly because he wished to distance himself mentally from the European intellectual scene. He thought it to be too involved in dealing with abstract problems and was more interested in immersing himself in anthropological field work.

    A World on the Wane

    Lévi-Strauss later remembered that the director of the ENS was, at the time he offered him the job in Sao Paolo, certain that natives of the country’s interior were already to be found in the suburbs of the Brazilian metropolis, and that the philosopher, interested in the relatively new discipline of gathering information about peoples of different cultures and their customs in the field, could study their culture during weekends.

    Naturally, there were no natives near the university where Lévi-Strauss spent the following couple of years lecturing. On several occasions, however, he did set off for more remote places in the rainforest where he could learn about the customs and way of life of the Indian tribes. Even though he spent a lot of time in the field and is today considered one of the principal names in twentieth-century anthropology and philosophy, he was neither a classical field anthropologist nor a classical philosopher who almost never leaves his desk.

    In fact, he was interested in solid data describing the customs of specific cultures and wanted to use the information he gathered to reveal the universal structure, characteristic not only of a certain tribe, but of all human societies. He later described his expeditions to the Amazon region in detail in his book entitled A World on the Wane, which combines his autobiographical notes with an analysis of the lives of Indian tribes.

    Are you the guy who makes jeans?

    On his return from Brazil he quickly realized that the situation in Europe was even worse than when he had left for South America. The air was thick with the anticipation and anxiety as the Second World War drew near, and because he was of Jewish origins, he only thought it wise to leave France once again. He decided to accept an invitation from the renowned private institution, The New School for Social Research in New York, which offered him tenure.

    Immediately after his arrival to the US, his acquaintances kindly suggested that he change his name as soon as possible. If he refused to do so, anyone who met him would think that he sold jeans. Even though it all seemed to be a joke at first, he quickly came to realize that the similarity of his name to that of the famous jeans manufacturer was more of a nuisance than an entertaining coincidence. In order to avoid unwanted misunderstandings he decided to sign his name Claude L. Strauss during his stay in the US. However, this change was not effective enough to save him from receiving orders for the legendary blue jeans to his address every once in a while.

    The French in Manhattan

    In New York, Lévi-Strauss made the acquaintance of the Russian linguist Roman Jakobson who also lectured at The New School. They quickly discovered they had very similar approaches to science. Lévi-Strauss tried to introduce the method that Jakobson had been developing in the field of linguistics to anthropology. In order to become familiar with each other’s past research they also frequented each other’s lectures.

    Many other French intellectuals spent the wartime years in New York. During the week, Lévi-Strauss spent his time with Jakobson, immersed in problems concerning anthropology and linguistics, while at weekends he joined the painter Max Ernst and the writer André Breton to look for Native American art in the local markets.

    The Elementary Structures of Kinship

    Jakobson was very impressed with Lévi-Strauss’s doctoral thesis so he encouraged him to publish it in book form as well. In his thesis, Lévi-Strauss examined the structures of family relations in different cultures around the world. Each culture has its own specific rules determining who can marry whom, and determines which newly created family bonds are strictly forbidden.

    The Elementary Structures of Kinship, as the book was called when it was finally released after the Second World War, is still considered to be one of the most important works in the field of anthropology. It also had a great influence on other fields of science.

    When Lévi-Strauss was preparing his manuscript for publishing in New York, however, he realized that he was still missing the essential part. He had plenty of data which he had gathered in the field, but he wanted to arrange it into a coherent whole, and in this he hit an obstacle. While using the same approaches that Jakobson had been applying to linguistics, he failed to find the inner logic within his field data on the acceptable and prohibited ways of creating new family relations.

    As he knew that finding a structure within data was also a mathematical problem, he turned to his mathematician colleagues, but they were not of much help. One of the representatives of the older generation even advised him to stop looking for principles because he would not find any: “Mathematics only knows four operations and marriage isn’t one of them.”

    A mathematician comes to the rescue

    After a while, he finally found some good fortune. In New York, he met a young French mathematician, André Weil, who also gave lectures at one of the American universities. Weil was a mathematician of a new breed who did not have much in common with his elderly colleagues. He was one of the founders of a group of young French mathematicians who published their scientific discussions and university textbooks under the collective pseudonym Nicolas Bourbaki.

    Their basic principle was to set mathematics on a new footing, mostly with the aid of set theory. In fact, The Bourbaki group was responsible for the famous new mathematics reform, moving the focus from arithmetic to sets and operations involving sets.

    In his research, Lévi-Strauss gathered a large quantity of data on family relations in different cultures of the world. He found out, for example, that similarities exist even between family structures of groups living as far apart as the Indian tribes of Brazil and the Aborigines of Australia. However, he could not succeed in revealing the general structure or system of all the acceptable newly formed family bonds.

    “When in doubt, look for groups!”

    After Lévi-Strauss explained his problem to him, André Weil immediately suspected that a structure that could organize the mass of anthropological data into a coherent whole could be a mathematical group. This completely abstract algebraic structure, which was not as well known in the mathematical community then as it is today, turned out to be one that can also be found governing the most unlikely situations, such as marriages between Australian natives.

    Weil was well known for his motto: “When in doubt, look for the group!” And this abstract algebraic approach was indeed useful when applied to anthropology. The essential principle that Weil applied to Lévi-Strauss’s data was that he ignored the very elements among which he was supposed to seek out regularities. He did not focus on the types of marriages, but on the relations between weddings. The structure was not hidden in the marriages as such, but in the differences between types of marriages. This approach was in complete accordance with the ideas of the Bourbaki group that relations and structures are the central elements of mathematics.

    Weil found out that the structure of relations between the marriages of the members of different generations and tribes was defined by what he called a permutation group. He described his findings in a discussion which was published as an appendix to Lévi-Strauss’s influential work. More than a concrete analysis of family relations, this book is important because of its influence, as it introduced the notion of structure as a system of differences, causing a veritable intellectual revolution.

    The interdisciplinary cooperation of Lévi-Strauss, Jakobson and Weil in New York in 1943 became a thing of legend. Three leading experts on anthropology, linguistics and mathematics, each brilliant in their own field, so different at first glance that it is hard to imagine how they could talk about anything other than the weather, came together to achieve an amazing scientific breakthrough. According to science historians, this was when the structuralist movement, which influenced all social sciences, mathematics and philosophy in the second half of the 20th century, came to life.