Wednesday, July 22, 2015
Infinite Ways of Knowing
This week, I invite you on a journey: to India, backward and forward through time, and along the number line. Along the way, you will consider multiple approaches to and embodiments of infinity, encounter a genius so profound we are only beginning to understand him and his work 95 years after his death, redefine the relationships of causal ideas, and examine the possibility of recreating the thoughts of others. Begin with this article about the life and ideas of the Indian mathematician Srinivasa Ramanujan. Note with special joy the author's reference to ways of knowing. Take the time to understand the mathematics explained. Harken back to our days of logic and discussions of attempts at proof of God's existence. And feel free to delve into the work of Vi Hart (in Links at the right) for clear, entertaining takes on several concepts touched upon in the article. For your post, please extract two KQs--specific and tied to AoKs and WoKs--from the article. Explore and answer them, and consider other moments to which they might be applied.
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Before I present any sort of knowledge question, I have to note that I spent an hour drawing Apollonian Gaskets and considering infinity in terms of elephants, because Vi Hart's youtube videos are fantastic.
ReplyDeleteKQ 1: How can the arts help to simplify mathematical concepts not easily explained by language?
As I just said, I spent an hour drawing Apollonian gaskets. The concept of infinity is pretty hard for me to understand. Reading through the article on Srinivasa Ramanujan was interesting, but the author's explanations of the concept of infinity were kind of difficult for me to understand. So, being an artist and procrastinator, I decided to watch a few Vi Hart videos on the subject. Vi Hart uses the idea of an endless line of elephants marching across a page to illustrate infinity. She says that as long as your elephants keep getting smaller in the same proportion (each elephant some fraction of the size of the last), the elephants will always approach 1, but never reach it. That was kind of cool, and I spent a while doodling infinite elephants on my Extended Essay notes instead of actually writing my Extended Essay, but I still didn't quite understand why a straight line of elephants could approach 1 but never reach it. Then, Vi Hart drew a circle with other circles in it. Simple as this sounds, it was what helped me get it. As long as there's still space to draw another circle inside your original circle, the big circle isn't full. It clicked. There's no way to fill the circle as long as you're always drawing the largest possible circle in the space you have remaining. Visual art helped me to understand a mathematical concept that I couldn't understand from language alone.
Another place where this may be applicable is in that infamous "two trains" problem. A train leaves Chicago at 3 PM, traveling 60 miles/hour, and another train departs Chicago at 4 PM, traveling 80 miles/hour. How long does it take for the second train to catch up with the first? As you might have guessed, I have absolutely no clue of how to solve this problem just by reading it. However, when I see one person moving 6 miles per hour, and another departing 10 minutes after them going 8 miles per hour, it clicks. In this case, movement explains what words could not. The same principle applies-- physical movement or visual stimuli, art, helps to ratify concepts that are difficult to grasp (at least for me) through words alone.
KQ 2: In what ways does reason hinder a knower's understanding of mathematics?
For me, "reason" is defined here as "common sense". For me, the notion of elephants infinitely approaching the edge of the page is confusing because at some point, you just can't draw another elephant. There's no way to do it. Likewise, there comes a point when you can't see any more space for a circle to fit inside a larger circle. My notions of what is and is not possible are not necessarily tied to the actual reality of what is and is not possible. I, personally, cannot keep drawing smaller and smaller elephants, but a computer program could. My attempt at reasoning through the notion of infinity actually hindered my understanding of infinity.
Another place where this is applicable is in the concept of space. Space: it's big, it's dark, and we really don't know much about it. Astrophysicists believe that the universe is spread out, generally, in a pattern. This pattern contains small voids of starless space, usually 1-2 billion lightyears across. Recently, a void has been found which is so much larger than the usual pattern that it is practically impossible for it to exist within the explanation of the universe that scientists are working with. Essentially, our current reasoning of how the universe works is not reconcilable with this new evidence of how the universe works. Scientists' understanding of the mathematics of space is hindered by their previous reasoning about it.
KQ1: How does faith exceed actual contents when people are making examination in mathematics?
ReplyDeleteSometimes people’s beliefs and confidence are more decisive in mathematical examination than actual contents, especially when the faith is relatively solid. In the article, when Hardy is confronted with the infinite “continued fraction” by Ramanujan, he initially does not have a deep mathematical analysis of such a formula; what he perceives are only the weirdness and elegance. However, he assumed Ramanujan’s formula to be correct, saying that “if they were not true, no one would have imagination to invent them.” He examines the formula using his trust of Ramanujan and beliefs in Ramanujan’s imagination and ability, which exceed his actual examination of the mathematical contents.
This KQ might also apply to my own study in math. In the last spring trimester, we had been studying to find derivatives of functions. For functions that contain more than one way to find the same derivative, when I use my textbook as the guidance, I always apply the method that the book gives in the first few examples rather than the ones given later in the chapter, because I always assume that the ones firstly given are more direct and simple to do than any other ones. Although all of the methods in the books would later be proved equal in their difficulty level, my examination of such mathematical methods is initially based on my beliefs in the first examples on the books rather than my thorough testing of those different methods.
KQ2: To what extent does an approach in math exclude mathematical reasonings?
To a very small extent, a mathematical approach can exclude reasonings, because math is a study mostly based on logics. In the article, Ramanujan had weird mathematics gushed out of him: in the beginning, he cannot explain the reasoning that leads to his formula, nor their significance. In this case, he demonstrates the seemingly lack of reasonings of his approach. However, while formulating his discovery, the process itself might include mathematical reasonings that comes from his internal mind and cannot be explicitly explained, because his formula contains strong logics in the form and structure. Hence, an approach in math could not be completely lack of reasonings.
This KQ might apply to circumstances when we are doing our own math. When we face something new, we might try to solve the problem using whatever methods that we know and think would work. When we are asked about how we solve it, it is always hard to explain the logics since the problem is something new that we do not know the exact reasoning. However, during the process of our attempt, we are likely to use mathematical reasonings that we have learned before, which totally contain reasonings. Even though we are not likely to say which logics we have used, partly because such reasonings are sticking to our mind for a long time and become a part of our conscious, the reasonings behind mathematical problem solving are always in existence.
KQ#1
ReplyDelete: Given the idea that what we know the world is simply what we perceive of it, what ways of knowing affect our ability to connect random things that seem to have no actual connection?
I think that to a large degree most humans desire some sort of order and logic that they can make sense of. We get from a sense of order it that all is well. Repetition of secure things reassure us that what we are doing Will keep us safe or at the very least will not endanger us more. When it comes to our ability to connect random events that have no obvious connection I think frequently we hinder ourselves. At first the idea of something completely different is scary because we're leaving that secure reassurance that all is safe behind and moving on to untested ground. We begin to use any way of knowing that we feel we can rely on to connect ourselves back to that safe place while still allowing ourselves to continue on. In my personal experience I found that I tend to rely a lot on reasoning. Today, I had a horse show. A few days ago, I had arrived early for a lesson and the person who my trainer was teaching for me is also having lesson is running a few minutes late. Not wanting to risk overtaxing my horse I simply gave him a very long walk warm up, but to keep him busy, I included a lot of turns and halt transitions make our walking more interesting. I then happen to have one of the best flight lessons I've had in a long long time. Today when I was getting prepared to show in dressage, I specifically plan extra time to be able to give my horse a long walk warm-up that he probably didn't need, but because it happened once and I really really was desperate to have a good test, I went back to my safe place and logically did the same thing. Reasoning told me that a long walk warm-up had worked up once and while my might have simply been feeling very good that day and since we didn't work very hard that day, everything happened to go well and the warm up might have had nothing to do with it. However, repeating that action gave me my sense of a safe place to start and I then used my reasoning to slowly work my way out of that place but still drawing my chains securing me to my safe place.
KQ number two: we define certain people such as Ramanujan or Einstein for example as geniuses because we cannot understand the work they created yet to what extent are the originality and ideas in works understandable to others?
ReplyDeleteTo ask this question I'll be going with the perspective of the author of the article. He did fully admit that he did not believe he could fully understand the works of Ramanujan but he took joy in his ability to learn and yet he still made his own discoveries that were entirely unique and that would only have been able to be created by him. However while he may have been able to explain those ideas to others very simply, is it not possible that their understanding of his ideas is simply related to their perspective and the creation of their own unique ideas that allowed them to say they understood what the author had thought? For each of us has a unique perspective and so for the author to be able to build another's perspective I think that would mean he would not only have to be able to understand the other's ideas in the sense that he uniquely comes up with the understanding through his own perspective but then build on it himself. So while he's not coming up with an idea that has never been seen by the world and the others cannot say they would ever be able to understand, he would still be genius because he is in a sense coming up with the same ideas to any other classified genius would have come up with that few others could understand. Therefore, I'm left to wonder if our language and strive to classify everything and everyone limits ourselves because we say we cannot come up with an idea because you cannot even understand someone else's but to be able to understand them not only would be you have to understand the basis of the idea but also have in our own perspective knowing that we could not possibly have is the same perspective as another. To give an example of my personal life, when I was little my parents decided to repaint the house. They picked out a color that they would call Greg and hire painters and the painters in the house. I came home from school one day, and I saw our house and the first thing I asked my mom when I walked in the door was "Mommy, why did you paint the house purple?" When we telling the story years later, my mom would recall how she was almost offended by my question and how I thought our house was purple. We would have to bathe her and her family were my mother father and brother would call me colorblind because they all thought the house was gray. However that was their perspective and mine was slightly different I saw a purple house and to this day I still cannot understand why they see it as gray. I do not think that this means one of us is more color blend then the others or that I should feel shamed of myself. I know I have a different perspective so what I perceive as purple is what they perceive as gray. I believe this is how the author felt when he was reading Ramanujan's work. He knew that he could not ever truly figure out exactly what reminded John understood and his perspective the world but he used his own perspective to understand new ideas in a different way making him in a sense just a genius as the man who originally thought of the ideas.
KQ1: How can our interpretation of advice be affected by our relationship with the speaker?
ReplyDeleteIn this article, the author writes "You have to find meaning in it, otherwise it was just an indulgence of the id. So, once again following my idols, I looked for the meaning". I find that relationships lie at the very base of interpretation, not actual language. For instance, my sister Alex and friend David have told me the exact same thing, "grades shouldn't be the most important part of your life". When my sister told me this over the phone on a particularly stressful night before winter exams, I just thought "how could she know, she's never been an academic person, not to mention an IB candidate". When my friend David told me the exact same thing yesterday while I was trying to write my extended essay and stay home from the beach, I actually listened to him and went diving. Not because he used different language, but because I know that he is an academic person, and an IB candidate. So, many of the times we take advice its not because we really agree with it, but because we agree with the person giving it.
KQ2: To what extent can instinctual knowledge be considered conscious knowledge?
"A thought appeared in my head, in words: This is not safe." The author describes his reaction to almost being hit by a car a thought, a written thought. However, the uneasy feeling of not being safe is derived solely from inherited instincts. One must also realize that experience can shape instinct over time, but at a much slower rate. Today, I have seen this in my ability to control and love my own fear. As a kid, doing dangerous things is a terrible, scary, and unthinkable, as it should be. However, as you grown up, especially in the inevitable teenage angst years, you learn to love the unexpected and the exciting. My instincts tell me not to ride on motorcycles, swim with sharks and walk up to full grown moose. However, I have either consciously rewritten my instincts or have chosen to simply ignore them. I find the strangest joy in ignoring my instincts, in doing the dangerous and in exploring unexpected.
KQ 1: What role does intuition play in the creation of "genius" works?
ReplyDeleteWhile reading the article, one thing that I was amazed by was Ramanujan's ability to figure out all of these theorems and mathematical concepts that trained mathematicians would spend years on and be unable to understand. Due to this, Ramanujan can be seen as a genius. Rather than reasoning, he used "leaps of pure intuition" in order to find new mathematical truths, ones that other ways of knowing, such as reason, were unable to initially achieve. He was even able to anticipate and solve future mathematical theories using his intuition and genius ability. This question can also be applied to the works of people such as Einstein or artists such as Vincent van Gogh and Beethoven. They are all considered geniuses in their fields and each had their scientific/artistic/musical intuition.
KQ 2: In what ways can language alter the knower's understanding of a mathematical concept?
One thing that I noticed in this article was the author's use of language and how it makes it easier to understand some of the concepts presented in it. For example, when describing the tau function, he writes "It’s a machine that takes some raw material and then stretches, compresses, reshapes, or transforms it into something else." In this case, the language really helps the reader understand the overall concept. This question was also inspired by my curiosity of what the language and organization in Ramanujan's notebooks must have been like that so many mathematician's did not recognize the genius in them and rather thought of the formulas as "crazy tricks" until giving them a closer look. This question can also be applied to to classroom settings where a teacher is explaining mathematical concepts to students. For example, the first time that I was taught the slope formula, I was unable to understand it because the teacher used complicated language, but I was able to understand it later after watching a youtube video which explained it in terms that were easier for me to grasp. Something that seemed impossible became easy to understand because of the language used.
To what extent does a lack of language create obstacles in understanding mathematical concepts?
ReplyDeleteWithin the article about Srinivasa Ramanujan, Robert Schneider stated repeatedly as to how different mathematicians tried and failed to understand Ramanujan’s genius work, although most of his work suggested answers and ideas related to mathematical theories of their time. Schneider stated that “A typical page in one of [Ramanujan’s] three “notebooks”...[contained] no words of explanation, just equations, symbols, and strings of digits.” This showed that Ramanujan did not use language, but rather examples to portray his knowledge. Schneider spoke about Ken Ono who, at first glance, thought little of Ramanujan’s formulas because he believed that they did not suggest anything deep. I believe that this could have occurred because there was a lack of language to help interpret Ramanujan’s goal or deeper meaning to his formula and therefore initially led Ono to interpret Ramanujan’s work incorrectly.
As a person who could not be farther from a mathematical genius, I found myself many a time frustrated over math homework that I did not understand although it had been explained thoroughly in class. My Dad, who is a lover of mathematics ( I fail to understand why I did not inherit that trait), would try and help explain the newest concept that I had learned to ease my frustration. Unfortunately, with his excitement for math, he would frequently go on mathematical tangents and thus confuse me more. In this scenario, a surplus of language created more of an obstacle in my understanding of mathematical concepts rather than a lack of it.
To what extent can imagination undermine sense perception in solving problems?
Robert Schneider desperately wanted to understand what Srinivasa Ramanujan had to say to the world, and hopefully what he would have to say to Schneider himself. To hopefully reach this understanding and an almost divine connection to Ramanujan, Schneider visited his home, and walked his way through the life that he lived by seeing the surroundings that Ramanujan saw such as the Sarangapani Temple and feel what Ramanujan felt, such as the tablet that he wrote on. However, by going through this sensory adventure of Ramanujan, Schneider did not understand what the great mathematician’s message was until his imagination helped fill in the gaps when he dreamed. Although his sensory adventure may have enhanced his ability to connect to Ramanujan’s message, it was his imagination that spearheaded his understanding.
In my final years of elementary school I managed to lose a lot of things. That included my new cell phone, jewelry, flash drives that I needed for school, and more important things. When I would lose something, I would walk my way throughout my day, literally, hoping that I would find whatever I had lost along the way both by looking at my surroundings more closely, and jogging my memory. Most times, I was unable to find what I needed. Within the same week that I would lose something, I would dream about finding it and, miraculously, I would find that item the next day in the spot that my dream suggested it would be in. In this situation, much like Robert Schneider, my sensory exploration to find my missing items proved to be a failure, whereas my imagination was able to guide me in solving my problem.
There are a few quote from this article that stuck out to me, and it’s those that I’m drawing my knowledge questions from.
ReplyDeleteFirst: “[the formulae] must be true because, if they were not true, no one would have the imagination to invent them.”
KQ: To what extent can we trust mathematical laws that we do not understand?
This stems from the idea that the quote is insinuating— it doesn’t matter if the formulae make sense or if they are easy to understand. They have to work, because why else would they exist in the first place, and be believed by so many people for so long a period of time? It’s the same thing that I might encounter in math class, if Ms. L-T teaches us something but doesn’t explain why it works (although that is rarely the case), we have to trust it because it exists, and it’s in a textbook and other people use it and say it works, so it has to work. Mathematical laws just are that way, but it’s hard to trust and believe in something we can’t wrap our minds around. Especially when these laws or formulae represent something intangible. We can’t see it or feel it. We have no real proof that’s it even exists and maybe it doesn’t. But it’s still important enough that we need a way to explain and define it.
Second: "If the formula was a tool for measuring the physical world, that implied that the physical world was the deeper reality, of which the equation provided only an approximation. If that was the case—if mathematics was subservient to reality—the equation would have failed, not the machine.”
KQ: In the sciences, what must take precedence: the logical explanation for something and how it should work, or the reality of what has in fact happened?
In scientific experiments, we have to make predictions for what is going to happen, based on data and logic and an understanding of the subject. However, as anyone who has ever taken a science class knows, experiments do not always go according to plan. Sometimes the truth of the experiment outweighs any statement of “this shouldn’t have happened because past trials have shown that it’s supposed to work this way.” We would like for our truths and our explanations to line up, but the fact is that a perfect harmony like that rarely happens. Things are going to simply go wrong and there is no way to explain why. They just happen. However, does that mean we should never trust the supposed “truth” ever again? Should an outlier data piece ruin perhaps years of study? No. But can we simply throw away that outlier and say it never occurred simply because it SHOULDN’T have occurred? Is a fluke that relevant? Those are the questions we have to consider when running tests like that.
KQ 1: How can emotional attachment to knowledge influence the use of reason in the situation?
ReplyDeleteAnswer: I saw this article as a manifestation of the emotion verses reason debate. Before opening the article I expected to read about the life and works of Ramanujan in a very informative matter. I understood that he had made many contributions to math so I saw him as a man of reason. However, the article played more on the emotional side instead of being written like an excerpt from an encyclopedia. I think that by writing about his experience discovering Ramanujan made it more emotional. I think that the author made many choices based on emotion over reason such as switching suddenly from music to math and traveling to India to learn more about Ramanujan. Often clear cut reason is overruled by emotion because we are naturally inclined to what is emotionally gratifying. In the article is comes up as the author who is interested in math chooses to go to India to learn about Ramanujan. The author’s emotional attachment to the story of Ramanujan leads to telling of his life works in an attempt to retrace his steps. Someone would expect a person studying Ramanujan and his theories to study him from a classroom using proofs and analyzing his equations. However, the former is more emotionally rewarding, especially for the author so it is chosen instead. The argument of emotion verses reason often presumes that one is a better but in the case of the article a path of emotion resulted in an informative and interesting telling of the story of Ranujaun in the same way a more reasoned or scholarly approach would.
KQ 2: In what ways are reason and imagination linked and how can they help each other?
Answer: According to the article Ramanujan had been creating mathematical formulae from a very young age and it is known that children have very active imaginations. This makes me think about the role his imagination may have had in helping his process. The use of reason is of course an inextricable part of math but I wonder how or if imagination could have been used to create or improve upon previous formulae or theories. In the case of Ramanujan it can be seen how imagination could have aided him in his autodidacticism. I think that a very good grasp of reason when applied to math allowed him to understand a wide range of very advanced theories however, by adding imagination to the mix he was able to create many of his own. Imagination and reason are used together in the process of creating writing. A story with a lot of imagination but no much reason can be unbelievable or unrelateable. On the other, a story with a lot of reason and a little imagination is often boring. Imagination and reason are examples of WOKs that blend together very well.
KQ 1: How can the faith one employs in indigenous knowledge systems be used to generate mathematical concepts?
ReplyDeleteIn Srinivasa Ramanujan's case, the faith involved in the understanding of his indigenous knowledge system, Hinduism, was not only directly responsible for but also inherent in guiding him to generate mathematical formulae and theorems. The author of this articles talks about how one of the essential ingredients to the making of any mathematician is a mentor; Srinivasa had no human mentor, but the window of his home had a view directly outwards to a temple honoring the infinite one, Vishnu, who in essence became Srinivasa's mentor. Of course, in order for Vishnu to become that mentor, Srinivasa first needed to have established a solid faith in him, and believe in his powers. In my understanding of the explanation given in this article, Hinduism teaches that Vishnu paradoxically embodies infinity and nothingness all at once; Srinivasa's faith in this God and subsequent faith in the concept this God embodies led him to explore infinite numbers and awed him in such a way that a Goddess transcended his conscious reality and "wrote formulas on his tongue" that flowed out of him without a reasonable explanation. Srinivasa put his faith into the divine, which delivered to him.
I feel that Srinivasa's circumstances are very unique; not many other records are supplied of people generating mathematical theorems directly from the mouth of God (however, there are myriad accounts in the Bible and many miracles about Saints that discuss an individual having visions and interpreting God's words). However, in clarification, I want to take a minute to address the phrasing of my KQ, which assumes that the faith employed in indigenous knowledge systems is different than types of faith employed in other areas. I think it is a unique type of faith, because it involves a certain amount of blindness (since we can't actually see God, we only see representations of him) and an understanding and acceptance of a world that transcends physical borders and exists in another reality: the spiritual reality. So how can I reapply this question? I think it can be applied to Ken Ono's mathematical journey in a less direct sense. His faith, which involved taking a blind leap and having a unexplainable assurance in a reality that transcends physical borders (the conceptual reality of numbers), brought him closer to advancing the mock theta function. In essence, the same components required of the type of faith involved in indigenous knowledge systems brought him closed to mathematical proof.
KQ 2: What role does language play in our interpretation of mathematics?
ReplyDeleteSrinivasa's documentation skills for his discoveries were not the most copious or effective, since he worked by slate and transcribed onto paper after several lines of proofs. He only wrote down numbers in his proofs and provided no explanation as to how he had gotten from one line to the next. His language was such that he could understand it although others might not, thus it was up to the next generations to be able to unlock his genius and determine how he would go down in history. Because of his loose language, multiple theories could be extracted from a single page of notes, hence his journals derived a plethora of new math concepts and theories depending on how people interpreted his numbers and lines of proofs. The ambiguity of his math language led to a series of breakthroughs because people were required to apply their creative perspective to make an understanding of his words in their functional societies.
This question can also be applied to my life, but instead of using pure numbers as a language, it requires using English to help explain the language of numbers. In math class, the way Ms. L-T explains concepts informs the way I interpret how the math should be performed. For example, when Ms. L-T explained vectors in 3-D, she first only explained how i, j, and k are aligned in space by telling us to visualize cubes. Thus, when I set out to perform my exercises for homework, I interpreted 3-D vectors as being in a cube shape, although they are actually single directional lines travelling through 3 planes.
KQ 1: How can imagination be the only reliable way of knowing when it comes to understanding foreign mathematical concepts?
ReplyDeleteI believe imagination is the least direct way of knowing. As stated in the article, the author had to resort to less direct ways of knowing when he discovered “there was nothing to grab onto.” There were no more than four pictures of Srinivasa Ramanujan and his known family members had all passed. Throughout the article the author talks much about how he imagined Ramanujan drew up his symbols and numbers. He tried to solely understand his work by putting himself in his shoes through this less direct way of knowing. In this case, with nothing to fall back on or grab onto to understand the creator of the ingenious theories, the only way to understand or acquire about Ramanujan was imagining what he would do. In another instance, when I am in math class and another student arrived at the same answer as I did with a different method I become curious. Sharing our homework in class then demonstrates the disparity and method and we work to understand each other’s thought processes. Now that I think about it, the only way for me to fully understand another’s method of arriving at an answer is to imagine every step and how the person may have arrived at that step in to solve the problem. The same process is visible in the author of the article in the sense that he used imagination to try and understand the thought process of Ramanujan in configuring his mathematical theories. This brings me to the idea that one way of knowing can be used to understand another. In this example, imagination is not only used to understand foreign mathematical concepts but the language in which these concepts were created.
KQ 2: How can language provide itself as a barrier when it comes to understanding mathematical theories?
With Ramanujan’s work, there was not a lot more than a few numbers and symbols to figure out widely expressed theories. His work was clearly in his own language and it’s evident that it is taking a long time for people to fully understand it. My theory behind this length of time it has taken and is still taking to understand his ingenious is simply the language in which is was written and the lack of. The other day my dad asked me to take a picture of the numbers on my car in order for the dealership to cut a new key. Yes, I locked my keys in my car. Anyway I wasn’t sure which numbers he was talking about because he wasn’t specific enough in his language. I sent him a picture of the inspection sticker at first thinking that’s what he needed. Not until later did I realize that there are number on the bottom right of the windshield of the car. Hence the reason why it took longer for him to receive the key from the car dealership. Here language and lack of explanation drew out the length of time it took for me to receive a key to open the car. As a sidenote, the dealership cut the key wrong and I still can’t unlock my car. Note to all: do not lock your keys in any car. Time is not to be wasted.
KQ 1: To what extend does Mathematics connect with creativity?
ReplyDeleteWhen Hardy saw Ramanujan’s formulas, which seemed to be weird and crazy, Hardy said they “must be true because, if they were not true, no one would have the imagination to invent them.” Therefore, this opinion evokes a question whether Math need other ways of knowing. As we all know, Mathematic proves can not be separated from logic. However, is logic the only thing the mathematicians need? I don’t think so. There is also a small portion of creativity interferes with logic and reasoning, which can be somehow be seen as the emotion or sense perception. Because something has inspired Ramanujan profoundly in the contribution of his interest in Math, otherwise a commonplace person would not just to come up with these formulas no one had seen before. And Ramanujan’s accomplishment in Math is somehow because of his creativity. Since he did not receive much education, it would be difficult to the come up with these complex theories. However, in the case of Ramanujan, it looks like he did not need much of learning in school, just training his logic and way of thinking, he created astonishment.
KQ 2: To what extent does extraordinary genius lead to one’s solitariness?
The story of this genius in Math makes me think of the story of Vincent Van Gogh. They both so talented in certain field, to the extent that no one can really understand them and their works. In the story of Ramanujan, before his death, “he produces the strangest work of his career: a series of mathematical formulae only recently understood.” When he was alive, his work is too complicate to understand in his time period, which someone may think he was crazy based on the formulas. The same thing happened with Van Gogh, his work was unappreciated while he was alive. Only after his death, the works became understandable and invaluable. Therefore, in a large extent, the extraordinary can cause other people’s bewilderment, which lead to the lack of understatement and language in communication.