Deep+Thoughts+-+Mathematics

=DEEP THOUGHTS=

Instructions: Let's do an "answer and question" session. Note: Everyone is expected to post on this page TWICE: once for the first observation and once more to respond to another's.
 * 1) Read and consider the following questions we discussed in class.
 * 2) Write a response in which you offer your answer to ONE of the prompts below.
 * 3) End your answer with a question of your very own, related to your thoughts
 * 4) Respond to SOMEONE ELSE'S question in a sub-bullet point.

Use the following format:
 * Name -- Observation plus new question. (2-3 sentences aiming for complexity/comprehensiveness)
 * Other person's name -- answering your new question
 * Another person (?)

**QUESTIONS**

 * Do people’s beliefs about math become determined by their ability in the subject?
 * Is mathematical insight only able to be taught, or is it something inborn and either you have it or you don’t?
 * Does solving math problems mean you understand math?
 * Is math real?
 * Are numbers real? Is π real? Is infinity a number; is infinity real?
 * Is math true because we can use it, or can we use it because it is true?
 * How important is emotion/intuition in math?
 * How important is beauty/art in math?


 * Chae Young -- Are numbers real? Is π real? Is infinity a number; is infinity real?: My observation to this prompt is that numbers are not real. Numbers are something we created in order to make our lives comfortable and to keep our lives in a certain universal rule that can be accepted by people from all over the world. Also numbers could be interpreted as languages used to communicate. By creating numbers, we can trade or send messages via computers using 0 and 1. The origin of numbers can be traced back in the ancient time. In ancient time, people had create an universal law that could be used by all the people in order to trade goods. As a result, numbers were created in order to measure the weight of something and measure quantity of the goods that were traded. As the time passed, these numbers were started to be used for various reasons. These numbers were started to be used to measure ratios or constructing buildings of different shapes. As a result, the numbers like π was created. As more and more scientific discoveries were made at atomic levels, infinite numbers and imaginary numbers were created in order to make the concepts of those complicated science to be interpreted by us in certain numbers. However, the question that can be derived from this kind of observation is will we able to present everything that is happening in this world with numbers?'
 * Gerard Belmans -- (Response to Chae Young) Scientists have been proving scientific theories and then providing them as math equations for years. Einstein did it with his theory of relativity, and countless more have too. There are people all over the world searching for an elusive formula that explains everything, but for now it remains just out of reach for everybody. So yes, I think that eventually we will be able to explain everything using numbers. ( How do you sub-bullet point?)


 * Harry O'Sullivan -- I believe that mathematical insight is only able to be taught. Nothing is inborn; often times, we speak of "talent" as being something that cannot be trained or taught. Take your favourite ice hockey player for example: Many who still follow Wayne Gretzky believe that he was born with an uncanny ability to foresee plays; many said he had "eyes on the back of his head." It is no exaggeration to mention that he possessed an incredible talent that allowed him to shine on the ice. However, was he born with this talent? I believe not. The amount of time he devoted to hockey was greater than any who wished to be as good as him. It took hours of practice for him to be the best. In other words, he did not suddenly become the best. He worked hard because of his love for the sport. Similarly, take Leonhard Euler for example. He was not born with mathematical insight; his love for the field lead him to be, as many consider, one of the best mathematicians of all time. Indeed, he was probably very fast with calculations, but I don't believe he was born with that quality. Mathematics, like anything else, must be taught. Simple formulas and rules must be fathomed by a student. Once the fundamentals are in place, an individual can apply what he/she has learned to the "real" world. This is what separates us (students of maths) to professional mathematicians. Their ability to apply their knowledge to solve what is unknown is greater than ours. Their enthusiasm for discovering new and innovated ideas and techniques exceeds ours. That said, does everything have to be taught, or are some qualities inborn?
 * Sean Williams -- what you're saying about Wayne Gretzky clearly seems to me that he WAS born with the incredible talent of Ice Hockey. The fact that he possesed the ability in him, he KNEW that he was special and different from the rest - that lead him to "practice" and therefore achieve what he was born to do. If I started playing hockey today and practiced like crazy, does that mean I'll become the next Wayne Gretzky??
 * Quentin Perrot - Hockey is a sport that requires a certain knowledge, from the rules to the balance of iceskating etc; but, for example, you take a fast runner, is his rapidity taught? You've got some people who run slowly, and others who are just too fast. When professionals take part in competitions, they are simply refining their rapidity and not learning to run fast. Okay, now you take that same example and use it for mathematics; don't you think that some people are just born with the ability to do mathematics? Of course, without a teacher, you wouldn't know anything about the subject. That having been said, think of a classroom whose students are supposedly at the same level; why are some students better than others? Is it because their "mathematical insight is greater than others.. I'm not saying there's a maths-gene, but only that the genes that are used for mathematics clearly affect your performance. If a C-student studies, and devotes countless hours to mathematics, and gets a B, don't you think there's some inborn factor that plays a role in our so-called talents? Isn't it a combination of both practice and inborn qualities that makes the next Wayne Gretzky?
 * Chae Young Moon - Well, the universal thing in a sports world is that all the good athletes that plays sports that requires fast decisions all have innately athletic. However, even if someone is athletic, he or she needs experiences and need to practice in order to become a great athlete. As you practice and play more matches, you will gradually get used to the pattern of the matches and at some point will able to predict what is going to happen in a match. For an example, in order for a tennis player to return a 200km serve, that person just have to play a lot of matches against the other players who can hit 200km serve. Through continuous practices and experiences from the match, tennis players can develop their sports visions and predict where the serve is going to go at certain situation. Some famous players, like Roger Federer or Rafael Nadal, can predict where the serve is going to go by looking at the opponents' movements. When they see the opponents' movements, they will automatically react to it and return the ball. Same for Wayne Gretzky. I think that Wayne Gretzky's uncanny ability to foresee plays are developed through constant practices and experiences form the real match and his brain just reacts really fast, reacting to every situations that happens during the match. So I think, if someone with some fundamental athletic ability practices and gain innumerable experiences from real matches, that person will be able to be next Wayne Gretzky.
 * Shawn- People are definitely born with different qualities, and I think it is rather natural for someone to be born with an innate talent that others do not have. As Kant mentioned, there are A Priori statements, and A Posteriori statements; the same goes for us. Many Africans are born with very muscular body due to genetics, and it is evident from the fact that many Africans excel in sports, and it is often because they innately have more mass of muscle than other races do. They are also born with unique vocal chords that few non-African races can mimic their distinct "black" feeling, more known as groove. This trait affects the people so strongly that many black people have difficulty learning the orthodox, "Western" style of classical singing. And yet, many scientists researching genetics are approving that while innate features are often significant, environmental factors are nearly as important, or even more important as they. Not many people point out how Koreans raised outside Japan are terrible at math, as when an Irish student can beat the Korean Koreans in math class. Well, I'm referring to myself and this other guy, but...all right. I think the point Quentin mentioned above is not as always as it is. People who get Bs in math perhaps study more often than those who get Cs. These C-people wouldn't necessarily be happy with the fact that they "suck" at math (unless you are like * *** who thinks getting Cs is cool), and most of them "don't know" how to study because they haven't. If those people were raised in the exact same environment as those of the B-students, I think it will make a considerable difference. So, please don't moan that you are bad at math because you lack the Korean math gene, but be aware that many Koreans are under heavy pressures from the parents, and here the famous adage goes "A=Average, B=Below average, C=Crap, D=Death, F=F*cked"


 * Sean Williams -- Mathematical insight is something that you either have or you don't; something that you are born with. I remember writing about a famous mathematician back in middle school and his name was Carl Friedrich Gauss. This guy was a GENIUS.. because when he was only 10 years old, he was able to add up the sum of all numbers from 1 to 100 in a matter of seconds. This shows that he was some sort of child prodigy, not normal. He was born with the ability to do math, and that was his "reason" to be brought onto the planet (as John Locke in LOST would say..). Do we humans have to face the fact and hard reality that some things are just "not meant to be" ?
 * Aaron Olin -- This is exactly the way that I feel, no matter how hard I try and study I will never be able to be the same mathematical skill as some of the people at our school (cough* Koreans cough*). I agree that you say that people have a "reason" to be brought onto the planet. I also feel that everyone has their unique set of skills they are born with. For example, a person like Bill Gates and Lebron James are born on the same day. If you train both of them basketball in the morning and how to build computer systems at night, King James will obviously succeed in basketball and Mr. Gates will succeed in things related to computers. This is because people are born with their own set of natural skills. Im not saying that they don't need to practice but their high skill level and the ability to play well comes naturally to them. Tying back to your original quote, I agree that people are born with mathematical insight and it cannot be taught. (as a response to your comment on Harry's post, you are not the next Wayne Gretzky because I obviously am.)
 * Soo Hyung Jung -- I think mathematical insight is not about if people are smart nor normal. It is mainly a factor of how well an individual understands certain concepts and how applications can be put to those concepts. 'Inborn' (something that you are born with), let's say inborn as smart. Smart is a vague term. You can be smart and be good at other subjects such as science or history. Seeing as how the majority of the population is not considered mathematical geniuses, I don't think that "normal" people are all good at math. So my answer for your question is mathematical insight is not only for talent people, because not all smart people are good at math, and nor are "normal" people.


 * Edward Cannell – I believe that mathematical insight is not a “natural talent,” and is something that is taught. No one can do their first math question without being taught anything – to them, an equation would just represent a bunch of numbers and symbols (they may not even recognize the numbers as “numbers;” they may just see them as other symbols). I think that why some people think that it is a natural talent is because they seem to “get it” faster than others. This is probably because they are interested in math, and so they explore it further. However, this raises the question: if people don’t like math, does that mean that they are likely to have less mathematical insight than those who do?


 * Quentin Perrot -- Personally, I believe mathematics is "real," because of its wide application to all aspects of our lives. Whether it is waking up in the morning, knowing that your alarm works based on a set of equations, taking the bus to the school based on a set of schedules calculated thanks to simple mathematics, or simply when counting your money when you get a receipt back from the convenience store. But, what does "real" actually mean? It's actually extremely hard to find a definition that encompasses what "real" actually means; this is why I believe this question, "is math real," can be asked differently, as Harry said in class: "was mathematics discovered or invented?" If it were discovered, then it is real; on the other hand, if it was invented, it is only real to a limited number of people, and not to humanity, and the world in general. Let's think about it - over thousands of years, spread across continents, different civilizations "discovered" similar things involving mathematics - the Sumerians developed a system of metrology, the Chinese developed complex geometry and so on. Mathematics was an important part of every civilization, and was a means to develop into what they are now remembered for, with the Egyptian pyramids for example. So, in answering, "is math real," I think we must simply decide whether or not mathematics is an invention (like the invention of the lightbulb) or a discovery of something that was pre instilled in our cognition. T he Lebombo bone, discovered in the Lebombo mountains of Swaziland and dated to approximately 35,000 BC, is the oldest known mathematical object; it clearly shows that mathematics was real to even hunter-gatherer societies thousands of years ago. Hence, in your opinions, was mathematics invented or discovered?
 * Jangho -- I would not answer your question in simply saying, "mathematics was invented" or "mathematics was discovered", because it could be answered in both ways. Because I believe that mathematics is a concept that was created in order to help people understand and visualize the universal laws, mathematics could be something discovered and invented at the same time. If I say it in a simpler way, the law of mathematics was "discovered", but the concept and the theories of mathematics were "invented."


 * Albert Takagi-- Is Math Real? I still have a hard time defining the word "real." Does it suppose to mean to exist? or to not be false? In anyways, I consider math real. Although we are not able to feel math with our five senses, math does exist in our world as a concept or a process to obtain knowledge. In my opinion, the math we are talking here can be split into two parts; Process and Knowledge. For example, what can you observe from the number "5", or what ever number? There is nothing we can obtain just from the number. Numbers can not stand alone. However, these numbers could be interpreted as Knowledge by collaborating with the Process part, which is equation. If we say, 2+3=5, then now we have something that we can obtain from this set of numbers. We are now able say that "number five is the sum of two and three" or if the equation is 6-1=5, we can say " five is composed by subtracting one from six." By adding the Process(equation), we can now have the Knowledge(number). Therefore, math is a tool or a process to find the knowledge, which is real. But Why do we use math? And who might have invented? Or did we just discover it?
 * Harry O'Sullivan -- But why do we use mathematics? Well, I think that is a very good question. Perhaps a better one would be, why do we study mathematics at school? Over the years, we have heard countless remarks from students who believe mathematics will come of no use when we go off to "the real world." Personally, I've heard comments like "when are we ever going to need to use the cosine rule in real life?!" I believe this to be an unintellectual remark. Reason being, I think mathematics trains our brains to work and solve problems in different ways. Take trigonometry, for example. Angle B of a particular triangle can be found using different methods and ways. In real life, we will tackle hard and complicated problems. Mathematics teaches us to try and solve the problem in the most efficient, and the most success-prone way; if one method fails, we shouldn't give up, we should look for another alternative. Learning mathematics at school is essential to our problem solving abilities, and thus I believe that it is essential to our well-being that we learn this subject at school.


 * Gerard Belmans: People's belief in math is determined, in part at least, by their abilities in the subject. If, for instance, one is not good at the subject and is good at other things, you will be more likely to call math out in error over something, as opposed to people who are good in math, who will most likely attempt to defend it. This is because mathematicians find that they can explain the universe through math. Albert Einstein made great discoveries in this region of math, along with people like Planck. The confusing aspects of this, which are incomprehensible to most normal people, result in disbelief in those that are less able at mathematics, for the sole reason that they don't understand what the mathematicians are trying to say. Also, people less able in mathematics find their own answers to the questions they may have through methods completely different to maths, and their own steadfast beliefs sometimes prevent them from being able to believe in maths. In conclusion, I do believe that a higher ability in mathematics leads to a higher belief in the subject overall. Of course, this then begs the question, can math actually definitively prove anything?
 * Albert Takagi-- I don't think that math can prove anything "definitively", however, I think it can get a point which is really close to the truth. Math is a tool that we use to make complex questions much simpler to get the answer we human can understand with our mind. Simply, it is a process or a procedure that we follow in order to obtain the answer, which is knowledge. However, my point is that this world is too complex and many of the questions that come up in our mind can not be easily solved with our brain, neither if we use math.
 * Shawn- A mathematician claimed that NO theories can prove that our current mathematics is correct. In other words, most of our achievements on math is not as definitive as it used to be. If I say 1+1=2, nobody will probably ask me why, but i have a doubt there. How can you prove that it really is 2? Einstein wrote on his autobiography that while math is elegant, it is sometimes delusive. So if we agree that math is real, I think we can assume that math will prove something. Note that people in the past could not draw perfect circles and squares. Then how did Plato develop his postulates? Isn't it that these mathematical figures are innately input into ourselves that when we encounter a similar object, we subconsciously want to shift it to what we imagine>?


 * Jangho Seo -- Is math real? This question could be answered both yes or no. Math is not real if you consider the fact that it is just a concept that us humans created in order to make it easy to visualize the law of the universe. However, math is real in terms of being able to apply to sciences and technologies. If you think about it, we use math everyday in our lives. You often find it hard to get away from math, since even just the simplest things, such as reading your clock, involves mathematics. Although it is only a concept that people created long time ago, it actually works and could be applied everywhere. Since it is told that everything in nature could be defined with math, people tend to take math seriously and force others to learn it. According to those people, math can help us find the absolute truth. However, is the "absolute truth" something that could be defined by our concepts and theories of mathematics? What is your answer to the question: Does math help us find the "absolute truth"
 * Nari -- I don't think that math will help us find the "absolute" truth because nothing can be absolute in this world. But, I do think that math can help us find the truth because it is found all over the place inside many concepts. Although we still have the doubt that nothing in this world may be real or true, assuming it is real, if we didn't have math or numbers, our life will be different, although we would most likely have had something different to represent what we know as numbers. Math can derive things and lead things to conclusion, so therefore, I believe that math will help us find the truth.
 * Shawn- Math involves lots of theories that go against each other...The super-string theory, theory of relativism, and others have just complicated the matter for years. Yet, I think math will lead us to something. My question to you is, how do you know that we can define the absolute truth? I'm starting to doubt whether humans are really able to find it or not, but based on the presumption that it is, I'd say math may contribute to the finding of the truth. If you look at the traditional math that the Westerners have been studying, math provides a perfect logic for us to apply in the real world as you've mentioned. This isn't always the case though.
 * Kiyo-- From my perspective, i think math 99% real and 1% real OR other way around because like what Shawn said math is contradicting to each other and sometimes math can be vague thats because ,i believe, math is both real and imagination. After all, math is just a tool to make sense of the nature around us.

> I think that numbers aren't "real" because this is something we "believe" is real or true. Like we discussed during class, numbers may be something we either created or discovered. If this world started from a place where there was nothing, then numbers must have been created on the way up until now. And if this is the case, numbers aren't "real" but something that humans created or perhaps invented for use in this world and made people "believe" that it is real. This also leads to the point that all mathematics is just a connection that was made between each other, meaning that it is all fake and that it must be some kind of an illusion. > Using the ideas developed in the previous paragraph, π is also not "real" but a symbol that people were told to believe to relate concepts inside mathematics and to make numbers look "real." This would be close to impossible to discover out of this world which had nothing to start with. Also, infinity is not a number because all it contains is a symbol that looks like a 8 that is turned around. But still, infinity is not "real" because infinity is not specific or "absolute." Infinity can refer to any large number, making it vague and "unrealistic." > The questions that I came up with are the following: > Why would such a symbol like π or ∞ or ℮ be used to represent pi or infinity or Euler's constant? Why would people create π or ∞ or ℮ ? Is it to make math look more realistic?
 * Nari -- Are numbers real? Is π real? Is infinity a number; is infinity real?
 * Edward Cannell – I think that the symbols for pi, infinity, and Euler’s constant were invented because we needed a symbol to represent these hard mathematical concepts. For example, pi goes on forever (3.14…), but it is impossible to write an everlasting number in an equation involving pi. This is why π was created, and the same goes for infinity and Euler’s constant; they are impossible to represent using just numbers, so a symbol was created in their place.

I think of math as a logic based idea that can be used to solve many of lifes otherwise hard-to-answer questions. And since most/almost all mathematical concepts do not incorporate abstract ideas and vague concepts, Unless you are a mathematician looking for a new formula, there are no "but maybe"s and "what if"s when you look at math. I think it is the most linear and direct "science" in the world. Therefore, asking how important emotion and intuition are to math is illogical. Subjects/sciences like theology, psychology, ans philosophy are the complete opposite to math when you compare the levels of logic and emotion. The 3 sciences I mentioned above all heavily rely on human emotion and all three cannot be explained completely using logic. This is why there are constant debates on concepts within these three subjects and the subject matter must change accordingly to how we think as a socirty. Math, on the otherhand is foolproof and has no "loopholes" where emotion can seep in. So is math different from psychology, philosophy, and theology because mathematical concepts that have been brought up in the past are still "alive" and reliable?
 * Matthieu--How important is emotion/intuition in math?
 * nari--When you talk about Euler's rule, it answers my questions as to wether mathematical concepts that have been brought up in the past are still reliable and useful today. Other mathematical concepts like the Pythagorean theorem and Pascal's triangle were discovered by people hundreds of years ago, but are still reliable and useful today. Psychology, for instance, is messy in a sense that respected psychologists like Freud, who's ideas seemed fit for that period of time have ideas that are really not concrete enough to be proven and might be extremely offensive today. What I am trying to say is that math is a solid, growing science because it lacks human emotion, while theology, philosophy and psychology are a volatile and fluctuating subject because it incorporates human emotion.


 * Taku -- Does solving math problems mean you understand math? I don't think solving math equals understand it, especially at our level. In my opinion, I think that math that we are studying is just plugging in numbers into the equation. The higher level students in the math class are able to solve some difficult questions, which most of the others cannot solve, because they have superior ability to break the question apart and put them together. Although those students seem to understand math, if you ask them how do they work, why do they work, and how do you know that works, they will not be able to answer. However, I am not actually sure to what extent are you considered as "you understand math"? Is it the concept, the physics, and/or history?


 * Taku -- Is mathematical insight only able to be taught, or is it something inborn and either you have it or you don’t? I think that math itself is something only able to be taught. Unlike the other subjects, math requires practices to be able to solve any kind of problems. Most people say that Korean people are good at math because they are born with it, but I don't think that is the reason. The only difference between the advanced students and not so advanced students that I think is the ability to break the questions apart. The way to solve any kind of questions come from amount of practices and getting used to use any equations and mathematical techniques. The advanced students are able to use those techniques because they are used to any kind of situations and they know the simplest ways to solve them. However, why can't some people, including me, improve, even if they practice more than the others do?

Kiyohiro-- I think math is real since we use math to find out what is real. And since the process of solving a problem is real, the answer should be real. Math is a tool for us, human being, to make sense of what human has encounter. We can use math to support our concept of reality because math is real. At the same time math help us support our concept of reality, it also organize aspects in our mind, therefore, if math is not real, what we think or believe is real will be no longer real due to the fact, everything is based on math. When we solve mathematical questions and we got the right answer, it means we understand a part of math, but not whole. Because you have to know the concept of the question to find the accurate answer, which means you have to understand math.

Shawn- Mathematics is not the "Korean" specialty, but it is more of a universal order that structures the world. Mathematical orders all exist to formulate matters in this world, and I believe that this invisible world of numbers is actually how the world works. Pythagoras defined that the numbers are the basest unit of the matter. Nonetheless, I think that numbers show HOW a matter is created, not what it is made of. My favorite chocolate cake is made in certain ratio, certain number of ingredients combined, but it can't be that the cake is made of 120g of 7s and 1s....I also think that math is not always rational because the existence of numbers like radicals and "irrational" numbers complicate the orderliness of this subject. Advocates of "real" math like Pythagoras denied those numbers, but they just exist like we exist.

Aaron Olin -- Does solving math problems mean you understand math? I feel that solving math problems does not mean that you understand math. I feel this way because I do have some troubles in math. When my teacher gave us a packet of formulas I did not need to study. I walked into the test and I had no idea how to do any of the problems. Using the formulas given I just plugged it into my calculator to get the answer. I did always get the right answer but if I were asked to solve the problems without using the formula packet, I would have no idea to start. Also recieving the correct answer using the formulas and a calculator might mean that I know how to use a calculator and I understand how to read formulas of a piece of paper but it does not mean I understand math. I do feel that there are different levels of understanding math but then again, can anyone fully and truly understand math?

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 * Soo Hyung Jung -- Is math real? I personally think that math is a very abstract field. Once we have made some basic assumptions and added some definitions, world that emerges is not random. We can discover its properties but statements that are true are already true and statements that are false are already false, even though we don't know them yet. So when we work on some branches of math; we are not creating or devising any kind of mathematical process. We are just disclosing hidden process (mathematical) that was there from the very beginning. It was there even before we have created or defined it. I assume that math can be both real and abstract based on situations. If we use math in physics such as applyig mathematical formulas in order to find out intensity of light, it seems like math is real. However, it's hard for us to see math as a real in our daily lives. Therefore, I think it's kind of vague to define math either Hmm,,, maybe, I'm just confused between the concept and the actual math, itself.