 |
| Both of Sodachi's OPs are excellent. |
For being the "primary thing I do", mathematics is perhaps enjoying a much undeserved absence in this blog. I would say that it is the only thing that I really dedicated myself to study across these years. I have been trying to pick up different topics outside of mathematics, such as physics (QM, statistical physics, relativity and beyond...), philosophy and even some literature, and when I get actual work I would abandon my pursuits in favour of sleep. But out of all I would come back to mathematics most often.
I will start with an anecdote: My first foray into "proper" mathematics, outside of calculus, was real analysis, when I attended an enrichment programme in CUHK. Needless to say, the completely different way of looking at math was a shock. The regular high school textbook could perhaps be sprinkled with proofs here and there, but an actually serious university level pure math text would not be able to deviate from the defintion-theorem-proof (and -example if generous!) approach. It was the first time that I genuinely felt difficulty in math, trying to keep track of so many definitions and logical trickery, as opposed to the plug-and-chug style of computation expected from high school, where once you do enough exercises to be able to memorize the algorithms behind each problem type you are basically guaranteed success.
As such, I have been relentlessly trying to get myself to conquer real analysis, but every time I would frustrate myself over some elementary exercises, or, in an attempt to replicate some proofs, get bogged down on even the seemingly obvious. Maybe I really don't "do math"...
As a subject, math really get a lot of bad press, moreso in the (relatively) olden days, thanks to the decidedly uncreative (yet frustratingly apparently necessary) way of education in the elementary levels. These days, STEM cannot be more of a buzzword, testament to the fact that even my parents, sworn to avoid anything science and technology related know of it, so math is now associated strongly with the other three players in the field. Rightfully so indeed: there is no science without math, and surely no engineering without math. However, this kind of math is "applied", and once we bring out the actual "math" that is being taught in an undergraduate course (analysis, algebra, number theory, geometry etc.), the same old tale of "what is this useful for?" returns.
I must add that I myself is very guilty of asking the aforementioned question, all of the time. Analysis is indeed the foundation of much of the more serious attempts at computational engineering, and codes the language of papers I have yet to read, but such kinds of motivation, while motivating, is insufficient to cause an interest in the subject in itself, just like money is a motivation to work but not a motivation to be interested in working.
I was promised of a sense of "beauty" in mathematics by my high school teacher, who dedicated an hour to talk enthusiastically about the construction of real numbers from natural numbers. As anyone who has a modicum of knowledge about analysis would know, this is the foundation of any attempts at analysis. He also encouraged us to think about what math really is, what numbers really are. Do they live on some imaginary plane in another world, like what Plato thought, or are they actual, tangible (perhaps not!) objects as real as an apple and a person? Thus began my endless desire to "learn some actual math", despite how bad I am at it, all the while pondering about where exactly does this beauty lie?
Some say that the rigour of math is where its beauty lies. The relentless pursuit of exactness, unforgiving of any errors in any step of a proof is where the sense of admiration comes from. One counterexample is enough to disprove an entire theory. At the same time, anyone, regardless of social status, can contribute to mathematics. It is a subject composed only of cold hard truths, logically deduced from axioms in an absolutely watertight manner.
Is this really the nature of mathematics though? The media loves to portray mathematicians as borderline autistic savants, who always thinks about a seasonal unsolved problem of choice and forget his/her basic human necessities such as socializing. Well, to begin with, I have begun to understand that research is not something one can do while shut in a cubicle. It must involve openness to new ideas, criticism and peer review. Moreover, the majority of science involves incremental improvements over current ideas, famously termed "normal science" by Kuhn. The actual lives of mathematicians is thus probably quite far from what is portrayed in movies. (as always...)
I find myself repeatedly returning to a speech made by Francis Su, former president of the Mathematical Association of America. In it, Su made comments on not only why he thinks math is beautiful, but also the firm link between humans and math. Mathematics is indeed a human subject, not something for computers. It is surprisingly close to the visual arts. In one sense, both are almost obsessive pursuits towards elegance, something most would relate to the faculty of emotion in our mind. As much as there are good art and bad art, there are good mathematics and bad mathematics, and good mathematics do much more than being correct, much like good art doing more than being realistic. For example, a "proof" that basically lists out all of the possibilities is less elegant than one which leverages subtleties in logic to reduce the problem (simple and direct). Both are valid proofs, but clearly one is better than the other. Mathematics is as much about the communication of this kind of elegance as the appreciation of such elegance. In addition, most mathematicians would attest to the fact that the beginning of a proof is intuition, to find the right door into the path of light. Many personal characteristics of a successful mathematician -- perseverance, sagacity, conviction -- all refer to emotional qualities, rather than having hard knowledge.
Su also makes a connection towards justice. In front of mathematics, there should be no discrimination, as I mentioned earlier. In a sense, this is inextricably linked towards the topic of social justice. Should we adopt the mathematician's stance in governing a state, analogous to the philosopher's state of Plato?
Inevitably, the mention of higher mathematics would link towards "competitions" when in a high school setting. Indeed, the olympiads are the first chances of a high schooler to dabble in mathematics that is more than computations [1]. I have gained a rather strange, ambivalent attitude towards competitions (it is the reason why I am where I am today), but at the end of the day, we should all recognize that mathematics is not a competitive sport, and solving the most problems in a limited time is rather meaningless.
Perhaps I should make a final, passing remark about applied mathematics. I might have given a false impression that pure mathematics is in some sense superior to applied mathematics, but that is not true. We are fortunate that the laws of nature is crafted in a language that is consistent and well observable, something which makes mathematics a good tool for analysis. Is this necessarily the case, or have we reversed the logic here -- mathematics is a systematic and elegant way (not the only way) of understanding nature? We could also interpret reality as a representation of the elegance of mathematics. The appearance of the golden spiral, the carefully constructed structures around us, the elegance of renaissance paintings etc. are all examples of such.
Mathematics is a pursuit of elegance and order, even from the chaotic (chaos theory, turbulence modeling...). It is demanding, powerful and sublime (in the philosophical sense...). This is perhaps why so many are enchanted by this subject.
Convenient plug: For anyone interested in me talking about actual mathematics, among other more technical babble, I have a new "professional" website...
[1]: I am giving an overly unfair criticism of high school mathematics... There is a certain focus on proofs and rigour in high school mathematics, but it is indeed nowhere as important as in higher mathematics, where all we do revolves around building provable systems.
Is this really the nature of mathematics though? The media loves to portray mathematicians as borderline autistic savants, who always thinks about a seasonal unsolved problem of choice and forget his/her basic human necessities such as socializing. Well, to begin with, I have begun to understand that research is not something one can do while shut in a cubicle. It must involve openness to new ideas, criticism and peer review. Moreover, the majority of science involves incremental improvements over current ideas, famously termed "normal science" by Kuhn. The actual lives of mathematicians is thus probably quite far from what is portrayed in movies. (as always...)
I find myself repeatedly returning to a speech made by Francis Su, former president of the Mathematical Association of America. In it, Su made comments on not only why he thinks math is beautiful, but also the firm link between humans and math. Mathematics is indeed a human subject, not something for computers. It is surprisingly close to the visual arts. In one sense, both are almost obsessive pursuits towards elegance, something most would relate to the faculty of emotion in our mind. As much as there are good art and bad art, there are good mathematics and bad mathematics, and good mathematics do much more than being correct, much like good art doing more than being realistic. For example, a "proof" that basically lists out all of the possibilities is less elegant than one which leverages subtleties in logic to reduce the problem (simple and direct). Both are valid proofs, but clearly one is better than the other. Mathematics is as much about the communication of this kind of elegance as the appreciation of such elegance. In addition, most mathematicians would attest to the fact that the beginning of a proof is intuition, to find the right door into the path of light. Many personal characteristics of a successful mathematician -- perseverance, sagacity, conviction -- all refer to emotional qualities, rather than having hard knowledge.
Su also makes a connection towards justice. In front of mathematics, there should be no discrimination, as I mentioned earlier. In a sense, this is inextricably linked towards the topic of social justice. Should we adopt the mathematician's stance in governing a state, analogous to the philosopher's state of Plato?
Inevitably, the mention of higher mathematics would link towards "competitions" when in a high school setting. Indeed, the olympiads are the first chances of a high schooler to dabble in mathematics that is more than computations [1]. I have gained a rather strange, ambivalent attitude towards competitions (it is the reason why I am where I am today), but at the end of the day, we should all recognize that mathematics is not a competitive sport, and solving the most problems in a limited time is rather meaningless.
Perhaps I should make a final, passing remark about applied mathematics. I might have given a false impression that pure mathematics is in some sense superior to applied mathematics, but that is not true. We are fortunate that the laws of nature is crafted in a language that is consistent and well observable, something which makes mathematics a good tool for analysis. Is this necessarily the case, or have we reversed the logic here -- mathematics is a systematic and elegant way (not the only way) of understanding nature? We could also interpret reality as a representation of the elegance of mathematics. The appearance of the golden spiral, the carefully constructed structures around us, the elegance of renaissance paintings etc. are all examples of such.
Mathematics is a pursuit of elegance and order, even from the chaotic (chaos theory, turbulence modeling...). It is demanding, powerful and sublime (in the philosophical sense...). This is perhaps why so many are enchanted by this subject.
Convenient plug: For anyone interested in me talking about actual mathematics, among other more technical babble, I have a new "professional" website...
[1]: I am giving an overly unfair criticism of high school mathematics... There is a certain focus on proofs and rigour in high school mathematics, but it is indeed nowhere as important as in higher mathematics, where all we do revolves around building provable systems.
ED Discussion 24: Kizumonogatari II ED, Étoile et Toi by Clémentine
"Everything I see,
Everything I feel,
You are my universe."
One thing I really did not understand from the Monogatari series was Araragi's devotion to Oshino Shinobu, or rather should I say Kiss-shot Acerola-orion Heart-under-blade [1]. Why is Araragi a vampire (or rather, half vampire?) ? What truly is the nature of the bond between these characters? I am glad this series did clear things up, in a way that did not require me to remember much about the previously aired arcs.
The main question I kept returning to throughout the movies is the reason why Araragi decided to help Kiss-shot. While we see that the hallmark of Araragi's personality is his tendency to self-sacrifice, he begins the series with a lukewarm-at-best response to Hanekawa's reach for friendship, declaring friendships as burdens, or "weaknesses" in human character.
One clear difference these movies have with the other arcs is the step-up in somberness of tone, evident in not only the clear change in colour palette, but also the frequent portrayal of violence and bloody scenes, rivaling the most extreme moments in the main series. Clearly, this highlights the supernatural powers of Kiss-shot, that she is a vampire, an iron-blooded, hot-blooded, cold-blooded vampire that will devour humans for satiation. This begs the question raised further: why would Araragi sacrifice his body to help something clearly an enemy of mankind?
Perhaps it is to do with Kiss-shot's human appearance. Despite being non-human, she looks human, acts (rather) human, speaks a human language (conveniently, Japanese), and --- as we find out --- has human feelings of loneliness, joy, remorse and empathy. This is a large thematic point in the last movie though. As Araragi discovers Kiss-shot's appetite for human flesh, he is immediately repulsed extremely, wondering whether his actions in fact doomed his kin, whether he has helped the devil achieve the extinction of humanity. Yet, even after learning this fact, he decided not to completely exterminate Kiss-shot to restore his humanity, even when she explicitly threatens Araragi to do so. I would argue that Kiss-shot's human demeanor helped the audience sympathize with her easier, but evidently not much to convince Araragi plotwise.
Was it Kiss-shot's helplessness (she exclaimed that she didn't want to die, with crying noises of infants being played in parallel with her screams) which evoked pity in Araragi? I feel like this is very logical to a point of obviousness: The reason why one helps others is really out of pity, a feeling that the receiver deserves help because of belief that the receiver is in a dire situation. The "converse" is usually not true: Even when someone is in a dire situation, it does not mean that he/she will receive help. In fact it is quite rarely true; we turn blind eyes to people in need most of the time, understandably and reasonably so. There was a bit of hesitation on Araragi's side; he did run away initially, only to return with his offer. Araragi was described to be in sort of a trance when he returned (even more stream-of-conscious like than other parts), questioning himself in the process: "why did it have to turn out like this?" The desire to help is built into Araragi, even when it seems very dangerous for him to do so. He innately puts his life in an inferior position, as he promised to live a better life next time around. Is that so for the general person? (or, is Araragi a realistic person?) Do we need more Araragi's in the real world?
It is at this point that I started to wonder whether it was Araragi's lack of self esteem which motivated his actions. He didn't want friends because he did not think he deserved them, only masquerading this self-loath as egoism in front of others. He is willing to sacrifice himself because his self is of no worth; hence nothing is lost by his sacrificing himself. However, why does he want to return to being a human, when he can realize his image by being the villain to humanity?
Hanekawa played a major role here. He realized how he loved humanity. He may not have self-worth, but he thinks of others highly, all the more contributing to trigger his actions to annihilate himself (worthless) to save others (worth much more).
This series also made me ponder what I would feel if I were suddenly not human one day. What if I discovered that I am a human-devouring vampire? What if I found myself to be a cat one day? Is there a strong drive for me to return to being a human? Will I throw up in disgust at the thought of eating a human? (the present me would, but that is because I am not a vampire!) Even understanding the limitations of being human (the strong regenerative powers of vampires could be very useful), is there a fundamental will for us to be human? We do not know about Araragi's past, but would his past have influenced his decision and desire to be human?
As an afterthought, his tendency to self-sacrifice conflicted with his desire to be human throughout, and it is this tension which drove the conflicts in the movie. The one thing about Monogatari is that it is very savvy of its own... arguments and its weaknesses (shall we say). In Owarimonogatari, his "heroism" is subverted, but that is a story for another day.
I also found out that I love the bossa nova jazz style.
One clear difference these movies have with the other arcs is the step-up in somberness of tone, evident in not only the clear change in colour palette, but also the frequent portrayal of violence and bloody scenes, rivaling the most extreme moments in the main series. Clearly, this highlights the supernatural powers of Kiss-shot, that she is a vampire, an iron-blooded, hot-blooded, cold-blooded vampire that will devour humans for satiation. This begs the question raised further: why would Araragi sacrifice his body to help something clearly an enemy of mankind?
Perhaps it is to do with Kiss-shot's human appearance. Despite being non-human, she looks human, acts (rather) human, speaks a human language (conveniently, Japanese), and --- as we find out --- has human feelings of loneliness, joy, remorse and empathy. This is a large thematic point in the last movie though. As Araragi discovers Kiss-shot's appetite for human flesh, he is immediately repulsed extremely, wondering whether his actions in fact doomed his kin, whether he has helped the devil achieve the extinction of humanity. Yet, even after learning this fact, he decided not to completely exterminate Kiss-shot to restore his humanity, even when she explicitly threatens Araragi to do so. I would argue that Kiss-shot's human demeanor helped the audience sympathize with her easier, but evidently not much to convince Araragi plotwise.
Was it Kiss-shot's helplessness (she exclaimed that she didn't want to die, with crying noises of infants being played in parallel with her screams) which evoked pity in Araragi? I feel like this is very logical to a point of obviousness: The reason why one helps others is really out of pity, a feeling that the receiver deserves help because of belief that the receiver is in a dire situation. The "converse" is usually not true: Even when someone is in a dire situation, it does not mean that he/she will receive help. In fact it is quite rarely true; we turn blind eyes to people in need most of the time, understandably and reasonably so. There was a bit of hesitation on Araragi's side; he did run away initially, only to return with his offer. Araragi was described to be in sort of a trance when he returned (even more stream-of-conscious like than other parts), questioning himself in the process: "why did it have to turn out like this?" The desire to help is built into Araragi, even when it seems very dangerous for him to do so. He innately puts his life in an inferior position, as he promised to live a better life next time around. Is that so for the general person? (or, is Araragi a realistic person?) Do we need more Araragi's in the real world?
It is at this point that I started to wonder whether it was Araragi's lack of self esteem which motivated his actions. He didn't want friends because he did not think he deserved them, only masquerading this self-loath as egoism in front of others. He is willing to sacrifice himself because his self is of no worth; hence nothing is lost by his sacrificing himself. However, why does he want to return to being a human, when he can realize his image by being the villain to humanity?
Hanekawa played a major role here. He realized how he loved humanity. He may not have self-worth, but he thinks of others highly, all the more contributing to trigger his actions to annihilate himself (worthless) to save others (worth much more).
This series also made me ponder what I would feel if I were suddenly not human one day. What if I discovered that I am a human-devouring vampire? What if I found myself to be a cat one day? Is there a strong drive for me to return to being a human? Will I throw up in disgust at the thought of eating a human? (the present me would, but that is because I am not a vampire!) Even understanding the limitations of being human (the strong regenerative powers of vampires could be very useful), is there a fundamental will for us to be human? We do not know about Araragi's past, but would his past have influenced his decision and desire to be human?
As an afterthought, his tendency to self-sacrifice conflicted with his desire to be human throughout, and it is this tension which drove the conflicts in the movie. The one thing about Monogatari is that it is very savvy of its own... arguments and its weaknesses (shall we say). In Owarimonogatari, his "heroism" is subverted, but that is a story for another day.
I also found out that I love the bossa nova jazz style.
[1]: Haato andaa bleedo
