Mathematical Independence Vs. Dependence
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Mathematical Independence vs. Dependence Joel A. Herr
February 4, 2003
Mathematical Independence vs. Dependence Where do mathematical theories and discoveries originate? Some mathematicians feel that culture and society play a big role in mathematical discoveries. Others consider mathematics to be independent from culture and regard it as being isolated from all societal influences. This has become a controversial topic since both sides of the argument hold valid points. Therefore, I will explain and analyze both sides of the argument in this essay. According to Reuben Hersh, mathematics is based on culture and its experiences. He describes his view as “humanistic.” This means that a culture’s experiences and history are very influential in the development of a culture’s mathematics. It also means that a culture should only be able to discover theorems or mathematical properties that have a relation to the cultural experience. Furthermore, the mathematics from one culture should be different from other cultures. In that case, a culture’s mathematical advancements would reflect the advancements of the culture itself, so the most advanced cultures would have the most advanced mathematics. In other words, the mathematics is only as advanced as the culture. The humanistic mathematics viewpoint also takes application and education into account. Supporters of this argument feel that if mathematics was truly independent from society (i.e. completely symbolic with meanings irrelevant to society), then it should not be able to be taught or learned. Therefore, the fact that it is learnable is proof that mathematics is not independent from culture. Furthermore, applied mathematics should not exist since it is directly involved with society.
Joel Herr
Page 1
2/4/03
The other argument is based on points made by Plato. This argument states that mathematics is independent from culture and remains so until discovered. As a result, all mathematical theorems and properties already exist and only wait to be discovered. Culture has no influence on this view of mathematics and, in fact, could be described in the opposite: culture is influenced by mathematics; rather, the culture is only as advanced as its mathematics. While the previous statement sounds similar to one I made earlier (i.e. “mathematics is only as advanced as the culture”), they are not the same. To prove this I will use a truth table. First, I will break down the statement, “mathematics is only as advanced as the culture.” This statement has two parts: “mathematics is advanced” (M) and “culture is advanced” (C). With these two parts, the statement “mathematics is only as advanced as the culture” can be rephrased as “if the culture is advanced, then the mathematics is advanced.” Likewise, the statement “culture is only as advanced as its mathematics” can be rephrased as “if the mathematics is advanced, then the culture is advanced.” Below is the truth table for these statements. M
C
M
C
C
M
T
T
T
T
T
F
F
T
F
T
T
F
F
F
T
T
Since the two statements are not logically equivalent, they do not mean the same thing. However, this statement was not meant to prove or debunk either side of the argument, but only to distinguish them.
Joel Herr
Page 2
2/4/03
Both arguments bring up some interesting contradictions. For example, if mathematics was based on a culture’s experiences and not independent, then how could discoveries made by two different cultures with different cultural experiences and backgrounds mean the same general thing? Also, as I mentioned earlier, how can mathematics be taught or learned if it is independent and its meanings are irrelevant to society? First, let me state that I do not agree with either of these arguments, but they do raise good points. The fact that mathematicians of different cultures have discovered similar, if not the same, ideas demonstrates that the “social mathematics” viewpoint is flawed. As mentioned in class, Ramanujan was a self-taught mathematician from India. His discoveries, without having read any previous theories, mirrored those of prominent mathematicians from the “more advanced” cultures, so mathematics is not based on culture. Similarly, mathematics has application to culture and society. Its ideas have meaning, thus it cannot be independent. I view mathematics as something not independent of culture, but not dependent on it either. Instead, I see mathematics as being a part of the collective societal memory or consciousness. This view describes thought as being shared throughout and ingrained into humanity and is thought to be the origin of instinct. It explains instances such as Ramanujan’s discoveries to mirror other mathematicians by saying that Ramanujan was using the collective memory when the discoveries came to him in his dreams. Similar occurrences outside of mathematics have also occurred. For example, devices that were frighteningly similar to the telegraph were introduced around the same time Samuel Morse brought his invention to the forefront. Likewise, more recently, advancements in
Joel Herr
Page 3
2/4/03
computing and software mimic this phenomenon (file sharing and instant messaging come to mind). Mathematics, then, has a meaning to society and can be taught and learned because it is derived from the collective consciousness. Since the collective memory is thought to be the basis of instinct, all people can understand it. Mathematics is embedded into people so they have the innate ability to learn, comprehend, and use it as if it were a human creation instead of a force of the universe. Now culture, instead of influencing mathematics, dictates the perception of it. Society gives math its “importance” and determines how the people view it. For example, not to use stereotypes, but the traditional view of Japanese culture details that the Japanese populace is better at mathematics. I do not believe that is the case, but instead that the Japanese culture puts more emphasis on mathematics and education, so the people perform better at it. Another example is the Barbie doll “math is hard” scandal. Mattel, the company that created the Barbie doll, was forced to change the voice clip in a line of the toys because Barbie, a cultural icon that millions of young girls play and interact with, was adversely influencing these young girls. The adverse affect was that girls were performing worse at math supposedly because in that voice clip, Barbie said, “math is hard.” To conclude, I feel that mathematics is part of human nature, much like logical thinking. People have the most basic concepts ingrained into them so that they may learn the deeper truths and nuances of it later. In that aspect the same conclusions can be reached through logical thinking, regardless of the cultural background. Culture plays the
Joel Herr
Page 4
2/4/03
role of modeling society’s perception of mathematics and determining its application. Nonetheless, this is still an important role. So is mathematics independent or is it cultural? The answer is yes.
Joel Herr
Page 5
2/4/03
Works Consulted Hersh, Reuben. What is Mathematics, Really? Oxford University Press, 1997. 343 p. Gouvêa, Fernando. “Read This! Review of What is Mathematics, Really?” MAA Online. 1999. Available: http://www.maa.org/reviews/whatis.html
Joel Herr
Page 6
2/4/03
February 4, 2003
Mathematical Independence vs. Dependence Where do mathematical theories and discoveries originate? Some mathematicians feel that culture and society play a big role in mathematical discoveries. Others consider mathematics to be independent from culture and regard it as being isolated from all societal influences. This has become a controversial topic since both sides of the argument hold valid points. Therefore, I will explain and analyze both sides of the argument in this essay. According to Reuben Hersh, mathematics is based on culture and its experiences. He describes his view as “humanistic.” This means that a culture’s experiences and history are very influential in the development of a culture’s mathematics. It also means that a culture should only be able to discover theorems or mathematical properties that have a relation to the cultural experience. Furthermore, the mathematics from one culture should be different from other cultures. In that case, a culture’s mathematical advancements would reflect the advancements of the culture itself, so the most advanced cultures would have the most advanced mathematics. In other words, the mathematics is only as advanced as the culture. The humanistic mathematics viewpoint also takes application and education into account. Supporters of this argument feel that if mathematics was truly independent from society (i.e. completely symbolic with meanings irrelevant to society), then it should not be able to be taught or learned. Therefore, the fact that it is learnable is proof that mathematics is not independent from culture. Furthermore, applied mathematics should not exist since it is directly involved with society.
Joel Herr
Page 1
2/4/03
The other argument is based on points made by Plato. This argument states that mathematics is independent from culture and remains so until discovered. As a result, all mathematical theorems and properties already exist and only wait to be discovered. Culture has no influence on this view of mathematics and, in fact, could be described in the opposite: culture is influenced by mathematics; rather, the culture is only as advanced as its mathematics. While the previous statement sounds similar to one I made earlier (i.e. “mathematics is only as advanced as the culture”), they are not the same. To prove this I will use a truth table. First, I will break down the statement, “mathematics is only as advanced as the culture.” This statement has two parts: “mathematics is advanced” (M) and “culture is advanced” (C). With these two parts, the statement “mathematics is only as advanced as the culture” can be rephrased as “if the culture is advanced, then the mathematics is advanced.” Likewise, the statement “culture is only as advanced as its mathematics” can be rephrased as “if the mathematics is advanced, then the culture is advanced.” Below is the truth table for these statements. M
C
M
C
C
M
T
T
T
T
T
F
F
T
F
T
T
F
F
F
T
T
Since the two statements are not logically equivalent, they do not mean the same thing. However, this statement was not meant to prove or debunk either side of the argument, but only to distinguish them.
Joel Herr
Page 2
2/4/03
Both arguments bring up some interesting contradictions. For example, if mathematics was based on a culture’s experiences and not independent, then how could discoveries made by two different cultures with different cultural experiences and backgrounds mean the same general thing? Also, as I mentioned earlier, how can mathematics be taught or learned if it is independent and its meanings are irrelevant to society? First, let me state that I do not agree with either of these arguments, but they do raise good points. The fact that mathematicians of different cultures have discovered similar, if not the same, ideas demonstrates that the “social mathematics” viewpoint is flawed. As mentioned in class, Ramanujan was a self-taught mathematician from India. His discoveries, without having read any previous theories, mirrored those of prominent mathematicians from the “more advanced” cultures, so mathematics is not based on culture. Similarly, mathematics has application to culture and society. Its ideas have meaning, thus it cannot be independent. I view mathematics as something not independent of culture, but not dependent on it either. Instead, I see mathematics as being a part of the collective societal memory or consciousness. This view describes thought as being shared throughout and ingrained into humanity and is thought to be the origin of instinct. It explains instances such as Ramanujan’s discoveries to mirror other mathematicians by saying that Ramanujan was using the collective memory when the discoveries came to him in his dreams. Similar occurrences outside of mathematics have also occurred. For example, devices that were frighteningly similar to the telegraph were introduced around the same time Samuel Morse brought his invention to the forefront. Likewise, more recently, advancements in
Joel Herr
Page 3
2/4/03
computing and software mimic this phenomenon (file sharing and instant messaging come to mind). Mathematics, then, has a meaning to society and can be taught and learned because it is derived from the collective consciousness. Since the collective memory is thought to be the basis of instinct, all people can understand it. Mathematics is embedded into people so they have the innate ability to learn, comprehend, and use it as if it were a human creation instead of a force of the universe. Now culture, instead of influencing mathematics, dictates the perception of it. Society gives math its “importance” and determines how the people view it. For example, not to use stereotypes, but the traditional view of Japanese culture details that the Japanese populace is better at mathematics. I do not believe that is the case, but instead that the Japanese culture puts more emphasis on mathematics and education, so the people perform better at it. Another example is the Barbie doll “math is hard” scandal. Mattel, the company that created the Barbie doll, was forced to change the voice clip in a line of the toys because Barbie, a cultural icon that millions of young girls play and interact with, was adversely influencing these young girls. The adverse affect was that girls were performing worse at math supposedly because in that voice clip, Barbie said, “math is hard.” To conclude, I feel that mathematics is part of human nature, much like logical thinking. People have the most basic concepts ingrained into them so that they may learn the deeper truths and nuances of it later. In that aspect the same conclusions can be reached through logical thinking, regardless of the cultural background. Culture plays the
Joel Herr
Page 4
2/4/03
role of modeling society’s perception of mathematics and determining its application. Nonetheless, this is still an important role. So is mathematics independent or is it cultural? The answer is yes.
Joel Herr
Page 5
2/4/03
Works Consulted Hersh, Reuben. What is Mathematics, Really? Oxford University Press, 1997. 343 p. Gouvêa, Fernando. “Read This! Review of What is Mathematics, Really?” MAA Online. 1999. Available: http://www.maa.org/reviews/whatis.html
Joel Herr
Page 6
2/4/03
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