Monday, June 21, 2010
Mathematics in China and India
Having seen some fascinating Greek Mathematics, it is time to move on to study Mathematics done in other parts of the world. This week we will focus on China and India. We will start with the Chinese mathematics and then talk about mathematics in the Indian subcontinent. Share your thoughts on the Chinese math and Indian math here. Both China and India have a very long and rich history. So there is a lot to learn from these ancient cultures besides Mathematics. When you read these chapters, you should compare and contrast them with the Greek and pre-Greek civilizations. Take notes as you read the book and please feel free to discuss your thoughts on the blog and in class.
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Bravo...Bravo to Courant and Robbins. If the rest of their book is anything like their introduction I would suggest reading its entirety for any student (high school, undergrad, or grad) that may pursue any field related to mathematics. What a great balance between the need for rigor and at the same time recognizing the necessity of investigation. I still contend that the "Ancient" mathematicians were on equal terms as their present day contemporaries. Yes the Greeks had their setbacks and limitations as did the Babylonians, Chinese, etc., but what do you expect when they were working from nothing. To pursue and investigate a theory must have been discouraging to say the least when they realized they where chasing shadows. On the other hand, to realize that you have truthfully validated your thoughts must have been very emotionally rewarding.
ReplyDeleteWhat is Mathematics? is a book that is recommended for every prospective student of mathematics. It is a classic and contains a sparkling collection of mathematical gems, one of whose aims was to counter the idea that "mathematics is nothing but a system of conclusions drawn from definitions and postulates that must be consistent but otherwise may be created by the free will of the mathematician." In short, it is giving the true meaning of mathematics. As David pointed out it has a perfect balance between
ReplyDeleterigorous axiomatic approach and the necessity of investigation.
I love the last sentence in the introduction which says that for scholars and laymen alike, it is not philosophy but the actual experience of doing mathematics that can answer the profound question "What is Mathematics?"
That is really true. That is why it is said that the best way to learn Mathematics is to *do* mathematics. It is not a spectator sport.
I think it's interesting enough to mention the use of the counting board. The Chinese were very dependent on this tool and we can see many uses in solving equations, creating pascal's triangle, as well as assisting in the computation of the Euclidean Algorithm. When compared to the Greeks, nothing really comes to mind that resembles the counting board. The counting board was where most of the Chinese math started. The ideas were worked out on the board and helped solidify the answer to the equation. Perhaps the book Elements is the closest resemblance. When solving theories, the Greeks always turned to the Elements first in their constructions and analysis. This may not be the best comparison but I found it interesting that the Chinese regularly used this prop to support their findings.
ReplyDeleteGood comment Christine. The counting board was something that I had on mind but I forgot to mention today.
ReplyDeleteStudies have shown that kids who are exposed to the abacus (modern counting board) tend to grasp basic math at a much faster rate. Even for adults counting boards are good. It is more fun to do calculations on a counting board than on a calculator :)
I found section 8.2.1 interesting as it hit upon the decimal place value system, a topic that seems to come up no matter what part of the world we are talking about. The importance of such a system was recognized throughout, but unfortunately for us, some of the history in India regarding this was lost. The figure provided in the book on page 234 is very interesting as it shows the progression over the centuries, and how the numbers 1 through 9 came into the symbols we use today.
ReplyDeleteI thought that the Chinese number system was very well thought out, developed, and complex and simple at the same time. It is interesting to see how the Chinese number system was set-up compared to the Greek number system. The Chinese number system definitely seems easier. I thought it was intersting that the Chinese are the ones that first started using a system of 10.
ReplyDeleteWith the diagram to prove the Pythagorean Theorem I was surprised that it went all the way back the Chinese. In my geometry class this past spring we went over several different proofs of the Pythagorean Theorem and one used this diagram. I had no idea until reading this chapter that it was developed so long ago.
I think the Chinese and Indian styles of mathematics are very different from the Greeks. They both seem to be a lot more algebra centered whereas the Greeks were mostly focused on geometry. I think this may have been because the Indians and Greeks both had better numeral systems which made this type of math easier to calculate. I found it very interested that both Chinese and Indian mathematicians used similar similar methods of finding the height of different objects.
ReplyDeleteIt is really interesting to see how religion has affected the study of mathematics. A common phrase in America is “Church separated from state” but, that also effects the ideas within that country. We have great examples of how religion has sparked inspiration in mathematics. Some examples from class include the Pythagoreans, the rituals of India, or the astronomers of Greece. Unfortunately, earlier times were less understanding and some mathematicians were exiled, imprisoned, or killed for their discoveries. I’m glad we live in a time where ideas like irrationals numbers can roam freely through our minds.
ReplyDeleteNick Witt: In your comment above, you obviously meant to say "Indians and *Chinese* both had better numeral system... " That is absolutely right, and that is the reason why they were able to advance so much in the field of algebra. Also, their number system allowed them to represent astronomically big numbers and that is one reason they were also good at astronomy. Greeks, for instance, could not handle very large numbers which is a major drawback for their progress.
ReplyDeleteBugs: You have a very good point. This relationship between religion and mathematics or science has a very long and interesting history. We saw some examples as you already mentioned. We shall see more. There are cases where they were also conflicts between religion and science. Religion was a source of inspiration for mathematics in some cultures, but in other cultures it was a major obstacle. Ask yourself: why was mankind stuck with the erroneous geocentric model for almost 2000 years? Answer: Religion. I shall say more about this in class.
ReplyDeleteI was looking up some stuff about India and read some interesting things:
ReplyDeleteChess was invented in India.
Bhaskaracharya rightly calculated the time taken by the earth to orbit the Sun hundreds of years before the astronomer Smart. According to his calculation, the time taken by the Earth to orbit the Sun was 365.258756484 days.
The value of "pi" was first calculated by the Indian Mathematician Budhayana, and he explained the concept of what is known as the Pythagorean Theorem. He discovered this in the 6th century, long before the European mathematicians.
Jenny: These are very good points to be noted. These are not mentioned in the book. Thank you for sharing them here. As we understand the history, there was very little or no communication between the east and the west. Europeans were completely ignorant of the developments in the East (especially in China and India) for a very long time. Even the decimal number system came to Europe only after a 1000 years after its inception in India and China. This is only one example, Jenny provided few above, and there are lots of such examples in the history of Mathematics.
ReplyDeleteAfter working thru some of the Vedic math yesterday I was intrigued at the amount of time it took to realize such shortcuts? Did the shortcuts come first by changing the digits to variables or did patterns exist first and then tested with variables? In Mathematics of Ancient India (Dutta) another "shortcut" that could be very useful (when teaching upper-level high school math students in ab = ((a+b)/2)^2-((a-b)/2)^2 and na^2 = ((n+1)/2)^2a^2-((n-1)/2)^2a^2. I wish that I had the time to "play" around with numbers (Better yet - Geometry) like that.
ReplyDeleteA mathematician’s apology
ReplyDeleteI think Hardy had some really interesting things in his essay. First, I never really realized how young some of the mathematicians were when they were discovering these new formulas and equations. I thought it was interesting when he said as you get older you cannot carry on the job as well and how mathematics in general is a “young man’s game”. In high school, it was my older math teachers that people had more respect for and that would teach the higher level subjects in math. Secondly, I liked how he posed those two questions, is my work worth doing and why am I doing it. It was very interesting to read his follow up on those questions and how easily some people can answer those questions. Finally, I liked two things at the end: the saying that language dies but mathematical ideas don’t and his comparison to math as the maker of patterns. There two paragraphs stuck out to me and I found it interesting his explanation of the both
I agree with you Jenny. I as well thought the age comment was interesting. We have all probably had situations where older teachers were just as influential as younger teachers are or have the potential of being. I liked the three outlining points he described as the reasons people tackle research. Intellectual curiosity, professional pride, and ambition. In a math world, it made sense but I think these are great motivations to really explore anything in life. We can always learn more, it's great to be ambitious and go after something you desire or like, and it's great to feel a sense of accomplishment when you complete a task or perform well. I do think however that some people's driving force is to better humanity. Not everyone is motivated the same way... For instance, some teachers may get into the field to impact students' lives, I know that is a driving force for me but not my only reasons.
ReplyDelete