The reading “The Rise of
Theoretical Mathematics in Ancient Greece” summarized the development of
mathematics in ancient Greece. Many people
think of the ancient Greeks and think of their application of mathematics to
various fields. A lot of what the Greeks
discovered (e.g. golden ratio which has applications to architecture, music,
nature, etc.) and expanded upon appeals to various senses, and this reading (in addition to explaining the evolution of math) mentions that this was a large part of Greek culture.
And going along with the idea of
mathematics influencing various fields, the New York Times article discusses
prime numbers and the mystery behind them.
The infinite quantity of primes also has fueled various attempts to
better understand this “mathematical anomaly.”
And out of curiosity, I went to the Prime Island site (http://yoyo.cc.monash.edu.au/~bunyip/primes/primeIsland.htm)
and “listened to primes” (http://primes.utm.edu/programs/music/listen/). It’s interesting to see numbers interpreted this
way.
But I think the essence of
mathematics is explained by the last sentence of “The Euclidian Synthesis”: “the
attempt to create a deductive system, far from hindering imagination, actually
stimulates it to create new ideas.”
Euclid’s work is one example.
Another is people who have tried to explain primes but have not been
able to establish a predictive pattern.
Instead, they have established things such as Prime Island and musical
interpretations, imaginatively creating something organized out of a random
pattern. Math is often used to model
situations, and I think this statement about mathematical discoveries applies to
other fields as well. When someone says
they have reached a breakthrough, their work is scrutinized by many
parties. And while the initial findings
may or may not hold true, the inquiries by others result in tangential findings
and subsequent breakthroughs. Society as
a whole is all about expanding knowledge bases, which is exactly what happens
in this process. And while we can sit
back and amass information, it’s what we do with that knowledge, ethical or
unethical, that defines us.
“The Rise of Theoretical Mathematics in Ancient Greece” does a great job outlining mathematical advancements in Ancient Greece and the conditions that made such advancements possible. As Gloria mentioned, the Greeks expanded upon the various senses as well as ideas they acquired from foreigners, which advocated innovation. With a shift from mythos to logos, logic and reason further encouraged intellectual growth, especially math related. Such growth may not have been possible without the social and economic stability that provided a necessary amount of security and tension to promote change. Arguably it was the lifestyle of Ancient Greeks that made their progress in mathematics and other subject matters possible.
ReplyDeleteThis idea is applicable to other ancient societies and even the modern world. The New Your Times article “From Here to Infinity: Obsessing With the Magic of Primes” is evidence of this. Curiosity is often what drives humans to understand more but without time exploration of the unknown would not be possible. I also believe competitive nature can influence such curiosity. The desire to further understand or enhance an idea contributes to innovation, such as in mathematics, and allows us to build upon ideas of the past. The various ways in which people try to better understand something, such as the prime numbers, also contributes to a broader view on the topic matter. Even “The Euclidian Synthesis” illustrates a different approach to understanding the underlying patterns of a concept. Overall the various ways to interpret a mathematical concept is unique to mankind and is very beneficial in terms of mathematical advancement and understanding.
Gloria, you make a good point. The ancient Greeks were well known for their mathematical developments and their ability to apply these developments to various fields. The Greek culture emphasized the importance of education and encouraged people to think beyond what was currently known in order to seek truth. The Greeks were well aware that they did not completely understand the physical world and so they were not content with the knowledge they had, but instead they were constantly looking for new ways of thinking about the physical world. However, I think it is a key point that the Greeks were not necessarily searching only for knowledge for the sake of knowledge. They were not focused on formally constructing an arbitrary mathematical system. Instead, their focus was centered on solving problems with practical applications in the physical world. For example, the Greeks developed points, lines, and planes because they saw them as essential constituents of the physical world, not merely undefined terms. Many of the Greek advancements have practical applications even today. Euclidean geometry can be applied not only in the classroom, but has practical applications in various fields. However, it is questionable as to whether or not the mathematical questions we continue to investigate today will ever really have any practical applications in the real world. Say for example, we were able to develop an algorithm to generate every prime number imaginable. That would be a great mathematical accomplishment, and would receive substantial recognition from the mathematical community. However, would this discovery actually have any practical application? The answer is very likely, no. Although this is a problem that many people have spent much time obsessing over, it is really a question of seeking knowledge for the sake of knowledge, something that seems to differentiate us from the Greeks.
ReplyDeleteI really like Gloria's last point about ideas and breakthroughs. Even if someone claims to have made some sort of breakthrough and it turns out to be false, it still sparks a lot of debate and thinking that can ultimately lead to new ideas and other 'breakthrough' concepts. One of the main reasons that Greece had so many great ideas was because of their heavy exposure to other cultures through colonization, trade, and warfare. They compared, borrowed, and assimilated selectively different mathematical lore to help build their own understanding.
ReplyDeleteI really like the quote "Whatever the Greeks have acquired from foreigners they have in the end turned into something finer." (pg 2) It really illustrates how important these outside influences were in advancing the Greek's ideas in math.
I like your and Gloria's point of view; however, an aspect of the first reading that sort of bothered me was the almost holy reverance the author seems to have for the Greeks. Each paragraph is practically glowing with the Greeks acheivments. This to me smacks a little bit of bias and Eurocentric thought. I am probably reading too much in to it, but thats what I thought when I read it - we must be aware of the inovations and advancements that other cultures had as well.
DeleteBrandon, this brought up an interesting perspective on the discovery for me. The author mentions that, "The Greeks favored attributing inventions to a single person, even when there was no sound historical basis for doing so." It makes me ask why must inventions be attributed to a certain someone? I know in America, 21st century, this question seems silly and moot. If inventions are for the benefit of mankind, why can mankind not be the inventor?
DeleteOn a culture standpoint: there seems to be an internal need for fame, for recognition of an achievement. To have this need, one must start to understand themselves a in a broader reality of time and history. Why would have a desire for tangent forms of recognition (i.e. credited in written history) if they did not understand that there feat would hold famous past their lifetime. There is always the possibility that they did do it either for pure altruistic reasons or curiosity, but there has to be some value change to want to attribute inventions to a person rather than a society or civilization.
Gloria I like how you pick out the last sentence of "The Euclidean Synthesis" as a representative statement of the essence of mathematics. I also found the statement "the illustrative diagrams and systematic nature of demonstrative geometry were characteristic of a people who generally favored a natural visual approach to the physical world and stressed order, as prominently reflected in their geometric pottery and their carefully sited and symmetrically designed cities," in the first reading to be very interesting. This gives a lot of insight into the culture and it's people. From this we can see that a society's innovations can give many clues to their values and way of living. However, in your last sentence about a society using it's knowledge ethically or unethically, I believe that is tough for us to judge because of the great differences in various societies' beliefs and values.
ReplyDeleteSomething that I find very curious regarding mathematics and deductive reasoning is how.. illogical (or counter-intuitive is perhaps the better word) the ways in which we discover it are. Because math and deduction are veritable schools of logic [or rather schools disciplined primarily in logic] and, logically , one would think that in order to advance one's understanding of math/deduction, they would have to do it via continual use of their logical faculties. Right? I mean the only way to strengthen your muscles, sharpen your wit, or temper your patience is through constant practice; so it makes sense that to improve your logic and/or abstract deduction, you would have to hit the books hard non-stop, right?
ReplyDeleteBut of course we know that isn't true - after all, most mathematical/scientific/technological breakthroughs are seen when the individual "genius" taps into parts of their brain that may (or may not) be completely unrelated to hard logic and deduction. Schrodinger (I'm told) didn't figure out his time-space dependent wave equation until he vacationed for weeks in a cabin out in the country. Einstein discovered the relations of hard physics through menial (and seemingly unrelated) events at the patent office. And - my personal favorite - Friedrich Kekule (the guy who discovered that the molecular shape of benzene is actually a circle instead of a straight chain of carbons) well he discovered the molecular structure of benzene in a dream about a snake of carbons devouring its own tail; more curious still is that that particular snake is known as the Ouroboros and is one of many archetypes that appears universally in the subconsciouses of humans [though in this case it's probably because the Ouroboros appears to be of Egyptian/Greek origin and Kekule probably saw it at some point in his studies].
One final - and personally striking - example is in the last paragraphs of Johnson's article about Primes in the Times, when he talks about the two autistic children [twins, no less] and how ludicrously adept they were at numbers. These kids could tap into some part in their brains [and possibly in ours as well] that accessed that "Pythagorean sensibility".......... and yet, as soon as they were separated and fully transitioned into the "logic" of society, that connection was lost.
Curious how the greatest revelations in logic come from the deep and mysterious recesses of the abstract.
I really like your last sentence on what we do with our knowledge is what defines us. I definitely agree with that sentence.
ReplyDeleteOverall, the final reading is the one that I thought was really interesting. I just like how the bigger you go the less prime numbers there are, which makes sense because there is more probability that your number will be divisible (or have a factor) by some other number. I liked the part he talked about going through his credit card numbers, his cell number, and any other big number that he could find. i thought that was quite interesting aspect to this story and it really reminded me of the movie "The Number 23" which I don't know why because they are quite different.
In the movie, he reads a book and think it's really written about him and then he starts seeing the number 23 everywhere he goes. it's in his address, if you add his credit card or cell number together, it equals 23...etc. I just find it interesting where I didn't think before but that movie could have been using some kind of mathematic technique that I didn't know about until now. It shows me how often math is around without me even realizing.
There was another movie it reminded me of, but I don't remember the name of that movie. In the movie he gets a list of numbers that predict all the disasters that are going to happen in the world. It was quite interesting because it got me thinking that if we would be able to do that in the future using math. I mean, numbers and mathematics is so complex, who knows what might happen in the future?
Gloria, I like how you mention that mathematics influences other various fields. After all, a lot of mathematics that was originally discovered was done for its own sake. The patterns that Euclid saw when organizing his geometry, the finding of the golden ratio, the Pythagorean Theorem, were all originally found with mostly just mathematics in mind. It would be many years later when people realized that some areas of mathematics could be used to help out with other fields. Additionally, a lot of mathematical topics currently do not have any practical application, but they might one day.
ReplyDeleteHowever, while this may be true, I also think that in some cases, the other “various fields” influenced math. After all, Newton thought up calculus in his attempt to understand physics. In a way, this shows that mathematics is also an attempt to understand the world around us in addition to “creating a deductive system”.
I'd like to pick up on the idea that the system doesn't hinder creativity, but rather stimulates it. In a world where we're told constantly to "think outside the box" it's also important to realize that "the box" (here mathmatical basics) frees us from trying to create a new box, and allows us to focus on different questions. I feel like in ancient fields, like math, we really depend on what was created before because it frees us from having to reinvent it.
ReplyDeleteIt's been said previously said that a lot of math exists without a practical application, and I think it says something about humans in that we are willing to pursue knowledge, even whe the knowledge seems like it might be (at the time) purposeless.
I would like to point out several things that were mentioned about the sharing of knowledge within society. Where I would like to go with this, seeing as I fell behind on my postings this week, is to touch briefly on the philosophical aspect that Gloria brought up about the ethical aspects of what we do with the knowledge we acquire and who it is shared with. Today in our society and throughout much of the world information and knowledge are being shared at a mind-blowing pace. It seems that every time we turn around information is being propagated from person to person, place to place at even faster rates. We all know that this has not always been the case; sure information would still be seen migrating throughout the cultures of Greece, Rome, Mesopotamia, etc. but it would be considered at a snail’s crawl in comparison to what we are witnessing today. This of course was due to the fact that information back then was so scares, new, and valued that only those in the ‘loop’ were allowed, or even educated enough to be capable of understanding much of the information.
ReplyDeleteOn the contrary, what we are experiencing in today’s society is radically and fundamentally different; with widespread access, spanning all social constructs, to a wealth of information that is even more so now than ever before, realistically at our fingertips. Not only can we now find information but we are granted the ability to also share our own personal knowledge of information as well. This is simply astonishing to think about when looking at the constructs of human existence, especially when looking at our own span of existence and the groundbreaking strides that mankind has made in the past twenty to thirty years, by means of technology making publically accessed knowledge possible. The Greeks showed us that knowledge is something essential to growth because it makes life easier allowing people more time to interject thought into their daily lives, henceforth stimulating intellectualism. Today with an abundance of knowledge available to all people we are seeing firsthand what societal growth by means of acquiring knowledge is capable of when it is shared in a society without favoritism to all who seek it.