Why is history relevant in the abstract world of mathematics?
The key to answering this question is in a particular field of mathematics called 'Ethnomathematics'... What is it? Well... according to Delgado, P. (2008):
The origin of the term 'ethnomathematics' can be traced back to Brazilian educator Ubiratan D'Ambrosio, who pioneered educational programs for 'third world' countries in the 1970's (Gerdes, 2001). He coined the term 'ethnomathematics' at a presentation before the American Association for the Advancement of Science in 1977. Ever since his introduction of the term, the mathematical community has fiercely debated the true significance and relevance of the term itself. The following is a list of different definitions given by various scholars, varying from 1985 to 1998:
"The mathematics which is practiced among identifiable cultural groups such as national-tribe societies, labor groups, children of certain age brackets and professional classes" (D'Ambrosio, 1985).
"The mathematics implicit in each practice" (Gerdes, 1986).
"The study of mathematical ideas of a non-literate culture" (Ascher, 1986).
"The codification which allows a ... group to describe, manage and understand reality" (D'Ambrosio, 1987).
Despite the varying opinions, it is clear to see that there is a unifying element among all defintions of ethnomathematics. The basic premise is that there exists a relationship between mathematics and culture. How social values influence the development of mathematics within a culture and how mathematical thought within a culture influences the culture itself are among the primary relationships investigated within the field of ethnomathematics. These relationships refer to a broad cluster of ideas ranging from distinct numerical and mathematical systems, to anthropological and cross-cultural comparisons of unique mathematical concepts, methods and tools, to historiography of the development of mathematics, to multicultural mathematics education, etc. The goal of ethnomathematics is to contribute both to the understanding of culture and the understanding of mathematics, but mainly to appreciating and understanding the connections between the two.
However, there is an overlying philosophical implication of ethnomathematics. In order to accept ethnomathematics, one must accept Inventionism as the true philosophy of mathematics. That is, one must accept the idea that mathematics was created, not discovered. In doing so, one must also accept that equations, formulas, concepts, and even numbers themselves did not exist before humans thought about them, or wrote about them. This philosophy further implies that while mathematics may be abstract, this 'abstractness' exists only within the mind of those who think about them. The minds of the individuals within a specific culture are the true locations where mathematics exists. Moreover, other influences surrounding a specific culture may also have an influence on the mathematical byproducts of that culture.
While a platonic philosopher of mathematics would contend that mathematics is, always was, and always will be the same throughout history, empirical evidence contradicts this notion. Archaeological evidence suggests that not everyone throughout history has perceived mathematics in the same way. Unique and divergent mathematical conceptualizations, perceptions, and problems within a specific culture are as rich and diverse as the fields of modern 'academic' mathematics. While Platonic philosophy may only account for these unique mathematics as 'discoveries', it does not account for why certain cultures 'discovered' certain answers, while others did not. Ethnomathematicians contend that culture is the factor which leads to the diversity among mathematical thought throughout history, and in the modern world as well.
But how can culture influence a concept so abstract and removed from the realm in which humans live? Let us explore some of the possibilities...
Common Themes in Ethnomathematics
Aside from unwarranted etymological associations with racist doctrine, studies within the field of ethnomathematics tend to reveal controversially racist tendencies within mathematics itself. The current view is that in contemporary mathematics, there exists a sociological bias to the study of the western traditions of mathematics, specifically the European traditions and contributions to this discipline. The roots of such bias are not historically justified, as many significant contributions to mathematics have origins from around the world. However, ethnocentrism- and more specifically eurocentrism- is a dominant force influencing modern mathematics. The origins of such beliefs are the subject of great debate among mathematicians, ethnomathematicians and historians alike. Many believe that the European colonialism of latter half of the previous millennium, driven by a lust for land, wealth, prestige, and dominance, lead to an extension of various European countries' territories to various corners of the globe. However, once in contact with a culture which europeans did not understand, it was natural for europeans to consider their own culture superior to that of others. Consequently, many of cultures around the world faced subjugation and slavery once in contact with europeans at that time. Among the various forms of culture that were subjugated, indigenous knowledge was especially subjugated and ignored. Among the forms of knowledge subjugated, mathematics was included. Thus, much of modern ethnomathematics focuses on addressing the eurocentrism of the previous millennium by countering the "common notion that most worthwhile mathematics known and used today was developed in the western world". The area stresses that the history of mathematics has been oversimplified, and seeks to explore the emergence and mathematics from various ages and civilizations throughout human history.
The base 10 number system seems so obvious to ourselves that it is hard for us to believe that mathematics can be done in any other way. However, it is important to note that this is not the only mathematical base system used (past or present), but it is also not the "best" system either. Various cultures have used different bases, ranging from base 2, all the way to base 60. However, surprisingly, there are no inherent mathematical disadvantages to using systems with a base other than 10. So, another common theme in ethomathematics is to explore the sociocultural reasons why other civilizations (past and present) chose to develop a mathematical system with a base other than base 10.
It is also important to note that central to many cultures throughout is their world view; that is, the concepts ideas and explanations that a culture has developed in order for that culture to make sense of the world around them (and more specifically, the microenvironment around them). Among the components of world view are the concepts and ideas which answer questions such as:
Where did we come from? Why are we here? Why do we do the things that we do presently? Why did the past happen the way it did? What is bound to occur in the future? What forces govern the world around us? etc...
The collection of concepts and explanations of these and many other questions lead to what is referred to as cosmovision. Furthermore, the underlying guiding principals or ideas which a culture believes are important and inherent to their cosmovision are called 'socialcultural values'. These inherent sociocultural values are often subtle, but almost always detectable in almost all aspects of the culture. Thus, if we hypothesize that if there is a relationship between mathematics and culture, then it should be possible to detect the sociocultural values that influence the ways in which mathematics developed within a specific culture.
It is often the case that one of the primary manifestations of cosmovision in culture is religion. Obviously, in the past and present, religion plays a role in the formation of social values. These values should, as a consequence of sociocultural and ethnomathematical theory, have effects on the way mathematics is perceived and developed in a culture. In fact, for some cultures, religion is a driving factor which influences mathematics. In others, religion is an obstacle to mathematical development. Analysis of the ways in which religion and mathematics interact is both fascinating and essential to the complete understanding of the mathematics developed within the culture.
The Origins of Mathematics
Another important aspect of ethnomathematics is the research into the sociocultural origins of various mathematical concepts. For example, it is of great debate which of the great ancient civilizations was the first to invent the concept of zero. It is known that not every ancient civilization had developed a symbol for zero; in fact, most did not. Therefore, it is important to understand why one culture produced a symbol for zero, where other cultures did not. Were all cultures capable of conceiving of the concept of zero? It is likely so. But if so, then why did only a few cultures create a symbol for zero? Also, for those cultures that did develop a symbol for zero, did their culture influence its conceptualization? It is one of the goals of ethnomathematicians to understand the social values associated with a culture's conception of this and other mathematical concepts.
Another example of this is the great debate of the origins of algebra. Who invented algebra? It is often mistakenly credited to the ancient Arabic civilizations of 600-1200AD. However, various mathematical documents of algebraic problems and solutions have appeared all over the world; such as in ancient Egypt, Babylon, China, India, etc... With such widespread and independent development of algebra all over the world, it is easy to see algebra was not invented by any one person. However, modern ethnomathematicians believe not only that algebra was not invented by any single person, but that human beings are born with an inherent algebraic reasoning ability that is believed to be prehistoric in origin. In other words, algebra is as old as mankind's ability to sense quantity.
The ability to sense quantity (also referred to as the 'number sense') is a skill that is shared not only by mankind, but also by other animals as well. Somehow, the ability to visually distinguish between facing 5 tigers versus only one tiger made a difference to mankind's survival. So, it is assumed that the 'number sense' is an essential survival skill for many species of animals; not just humans. However, the ability to reason or deduce unknown quantity (also known as algebraic reasoning) appears to accompany the number sense in humans. For example, a prehistoric human being could have reasoned that if he had 3 pieces of meat in his cave this morning, but later only has 1 of the original pieces, he can deduce that two were either stolen or eaten by someone else. And it is this ability to reasonably deduce unknown quantity that gave mankind an adaptive advantage that helped him to survive, both in prehistoric times as well as today. This ability to reason an unknown quantity is precisely the subject of algebra. And, it appears that this ability is of prehistoric origin, possibly even genetically programmed in our DNA.
Quite ironically, most of the criticisms of ethnomathematics comes from mathematicians themselves. Other criticisms come from anthropologists and educators as well.
From an anthropological perspective, one of the challenges faced by ethnomathematicians is the fact that they are limited by their own mathematical and cultural mentalities. In other words, it is only possible to understand other cultures' mentalities in terms of one's own sociocultural framework. The same can be said of mathematics. Because most ethnomathematicians are educated in the western traditions of mathematics, their discussions of the mathematical ideas of other cultures tend to recast these indigenous ideas into the modern Western framework in order to identify and understand them. However, ethnomathematics strives to acknowledge that the mathematical ideas of another culture must be understood within the sociocultural context of that particular culture. This raises great criticism among the modern mathematical community in general, who often claim that mathematics itself is independent of culture (according to the platonic view of math) and thus can only be interpreted in accordance to the modern perceptions of mathematical identity. In other words, mathematicians tend to believe that mathematics, in its current form, can be understood by any one from any culture and from any time in history because it has been, and always will be, perceived in the same way. Despite the empirical archaeological and ethnographical evidence to the contrary, the platonic view of mathematics is held very strongly among many, if not most, mathematicians.
The main criticism from mathematicians of ethnomathematics is that of mathematicians objecting to the application of the word "mathematics" to subject matter that is not developed abstractly and logically, with proofs, as in the academic traditions of mathematics. Mathematicians argue that ethnomathematics is not a field of mathematics at all. They claim that understanding how other cultures have arrived at different ways of counting is not as insightful, on objective terms, as Sir Issac Newton's development of Calculus, or Cantor's work on infinity, for example. Moreover, some academic mathematicians feel that ethnomathematics is more properly a branch of anthropology than mathematics. An ethnomathematician might reply that ethnomathematics is not meant to be a branch of mathematics nor of anthropology, but combines elements of both and creates something uniquely different from either one. Moreover, to compare an indigenous culture's methods of counting with Cantor's set theory is not only unfair, but also unjustified. A method of counting is just as valuable to the people who use it as Calculus is to those mathematicians who use it. The difference in judgment is a reflection of a difference in cultural values.
Many have claimed that ethnomathematical techniques can be used to teach mathematical knowledge in many 3rd world communities. The idea of using a culture's common knowledge to recognize modern mathematical concepts seems undoubtable. However, from an educational point of view, critics claim that the use of ethnomathematics to teach modern mathematics is absurd, because it forces autonomous, isolated cultures to accept the socio-cultural values inherent in modern (western) mathematics.
An ethnomathematician answers such criticisms by saying that the devaluation of the importance of the role of culture in mathematics seems to reflect the sociocultural values among modern mathematicians; values that seem to reflect exclusivity and an authoritarian superiority not unlike that of ethnocentrism. Although the contributions of ethnomathematicians may not significantly contribute to the mainstream body of work to which modern mathematicians contribute, there exists a large body of unquestionably significant contributions to the overall understanding of mathematics as a human activity. This work not only legitimatizes the field of ethnomathematics, but also clearly contributes to the overall understanding of mathematics as a whole.
So, to answer your question, to understand mathematics is not only to understand the mathematical concepts themselves, but also the culture from which they were created. This involves history, anthropology, and mathematics as well.