Math Proof and Axioms

Knowledge Question: If we treat mathematics as a kind of game with it’s own set of rules. Then to what extent can the rules of mathematics be changed, but can still be treated as the same kind of “game”?


The scope of this knowledge question (KQ) is mathematics and is the study of quantity, space, shape and change. By applying the important knowledge in mathematics regarding quantities, spaces, shapes and changes, we can learn more definite truths about the world around us. Some practical problems that maths helps to solve can be as simple as how much change you should receive after paying for something, to something as difficult as black holes. Currently some of the main questions that mathematicians are trying to find answers to are the remaining unanswered millennial prize problems.


The foundation of mathematics, mathematical axioms are the initially concepts that mathematicians use for their mathematical proofs. However in order for mathematicians to be able to solve new and harder problems, they have been forced to bend some of the fundamental axioms in order to solve these problems. However in doing so, have mathematicians stepped into a new “game” of mathematics; or are they simply playing the same “game” with revised rules?


Answering this question would require one to fully understand all the different mathematical axioms there are, and then look at all the different “games” of mathematics that are a result of changing the basic Euclidian axioms. Then one should see whether the kind of mathematics done in these new “games” are fundamentally the same as regular, traditional mathematics; or are something entirely different.

Historical Development:

Axioms in Euclidian geometry can be considered to be the fundamental “rules” for the game of mathematics. Much later on in time, Hyperbolic and Elliptic geometry modified these “rules” so that they can work in other situations. One can look into these relatively newer forms of geometry and compare the mathematics done in Hyperbolic, Elliptic and Euclidian geometry to see whether they are fundamentally the same; or completely different.

Links to Personal Knowledge:

As of right now, I am only versed in the axioms of Euclidian axioms (ie Things that coincide with one another equal one another) and the mathematics behind Euclidian geometry (ie the sum of the interior angles of a triangle is 180º).

Faith and Intuition

As WOK’s, Faith and Intuition are similar in some aspects and different in others. Knowledge that is derived from faith is often simply widely accepted without need for any prior knowledge or reasonable deduction, whereas knowledge derived from intuition is often based on past experience or a “sixth sense”. Both of these WOK’s have the capacity to be useful and problematic to people when it comes to knowledge.

Knowledge that is derived from faith can provide people with an answer to commonly unanswerable questions such as “Why are we here?”. The answers to these questions provide society with moral guidelines and provide people with a sense of purpose and clarity. However faith can also be problematic, because there is no real proof that the knowledge derived from faith is accurate. For example most religions believe that there is a higher being, whether it be god, jesus, allah or some other higher being. However there is no real proof that these beings exist, which does suggests that the existence of these higher beings may simply be something the human mind made to cope with the vast unknowns of the universe.

Knowledge that is derived from intuition can be justified because in some cases, it allows people to be somewhat of a harbinger in that it can allow people to foretell what is to come. This is useful because it allows people to be ready in a moments notice, without an immediate need to be. For example if you see a glass teetering at the edge of the table and you see someone nearby who is unaware of the glass and whose elbow is almost touching the glass, then you intuitively know that the person will knock the glass off the table and send it crashing down to the ground. The observer to then position himself in a way to catch the glass if it falls, or to move the glass to a safer position beforehand. However knowledge garnered by intuition is also a gamble in a way, because it is not always 100% accurate. Take the above example for instance, there is only a possibility that the person will knock over the glass. It is just that there is a high probability based on the observers past experiences and prior knowledge, that he has come to the conclusion that the person will knock over the glass.

Memory and Imagination

Claim: Despite the imperfections of imagination and memory as ways of knowing, the Areas of Knowledge have developed in such as way as to overcome them. Discuss this claim with reference to at least two AOKs.


Basing knowledge solely on imagination and memory is not societies ideal way of getting knowledge, because there are many problems that come with using imagination and memory to attain knowledge. For example when trying to build a case in court, sometimes lawyers utilise eye witness testimonies to make their points. However this is not the ideal since people’s memories are not always precise and there are bound to be limitations to what the human mind can accurately recollect, meaning that eye-witness testimonies are always subject to bias which affects the reliability of the knowledge derived from memory. Furthermore knowledge gained from imagination is, in a way, a kind of extrapolation of all your prior knowledge to create new knowledge. While this may be useful in some areas, it is still just an extrapolation and therefore it does not take into account other external factors that may affect the results.

However there are AOKs that have developed ways to overcome the limitations of imagination and memory, and these AOKs are history and the natural sciences. This is because even though new knowledge is initially conceptualised through memory and imagination, it is ultimately backed up by further evidence of the initial concept. In history we do not know for 100% what something from the past looks like, and therefore we must use our imagination or memory to visualise it. However we are also able to use external facts and evidence to corroborate the knowledge that we derive from our imagination and memory, in order to ensure that the knowledge we are attaining is more accurate. Similarly with the natural sciences, if a scientist has a theory about something that is based on his/her imagination of something, then the theory will require physical evidence in order to be generally accepted by the general public.

Language as a Source of Knowledge

Question: “The vagueness and ambiguity of language always limits the production of knowledge”.

Language is a method of communication between humans that is either written or spoken, and consists of the use of words in a generally accepted structure and convention. Humans constantly uses language to express themselves and their ideas to others, and thus spread their knowledge throughout the global/local community. However there are some drawbacks when one must be careful of when trying to transmit knowledge through language, because not only are some concepts hard to express in language, but the interpretation of language very fluid at times. What this means is that because of people’s different life experiences and what they already know, different people’s interpretation of language can differ from person to person making it hard to find a common base in language to communicate your specific and intended message clearly.

Religion is an Area of Knowledge (AOK) that shows how the ambiguity and vagueness of language can sometimes lead to people having different religious ideals. This can be seen through how christians, muslims and others treat people and situations differently. For example, a verse in the Bible states that “Vengeance is Mine”. Many people interpret this verse as saying that if someone wrongs you, than you have the right to exact vengeance upon the wrongdoer. However what this verse really means is that if someone does you wrong, god will be the one who will provide justice, kind of like karma. As a result many devote Christians may misinterpret this verse and use it as an excuse to exact revenge upon someone, and not allow god to pass judgement upon the individual himself. Therefore, the vagueness of this Bible verse may impede the progress of knowledge by creating individuals who do not think about why they were rejected or wronged, and simply think they are right and deserve vengeance on those who rejected/wronged them without thinking about the circumstances from an outside point of view. In addition, the vast majority of muslims believe that the word jihad symbolises their internal struggles to live a life that abides by the moral codes of the Koran. However some muslims believe that jihad represents an external struggle against those who would threaten the faith or the faithful, sometimes by using arms. As a result of the vagueness of the term jihad, some people have taken it up as an excuse to wage a global war of terror against western society. Therefore the misinterpretation of jihad impedes the religious knowledge of many muslims, because nowadays people relate jihad to islamic terrorists and not it’s true meaning of spiritual struggle.

However the ambiguity and vagueness of language does not always impede the production of knowledge, and this can be seen in the arts. In fact, I believe that the fact that the ambiguity and vagueness of language in the arts is what allows this AOK to be so diverse, impactful and eye-opening. For example, scripts done by playwrights may contain language that is really vague. However this allows for many different interpretations of certain scenes, which allows different directors to convey different meanings to the audience. Thus allowing the audience to gain more insight into different situations, and thus increase their understanding of the different scenarios and reasons for these scenarios. Therefore I believe that vagueness and ambiguity does not always limit the production of knowledge.

Further Questions: (The ones crossed out are answered in above)

  • Do vagueness and ambiguity always limit the production of knowledge
  • Does any kind of knowledge benefit from vagueness and ambiguity of language
  • Is all language vague and ambiguous
  • What is language
  • What limits language
  • What is role of language in producing knowledge
  • Does knowledge require language to be made?

Does Emotion Hinder Knowledge in some AOKs?

Claim: A good historian strives to be as unemotional as possible, this is the only way to write accurate history

“With great power comes great responsibility”, the famous quote from the wall-crawling hero Spider-Man is a lifelong mantra that drives the superhero forward through his superhero career. This quote can also be applied to historians, because their version of history is usually societies version of history. The history that they deem accurate is the basic framework with which we build modern society upon, and as a result they hold immense power to shape and mould society. Therefore their jobs require unbiased perspectives regarding historical events, in order to not lead themselves and society astray. However the question remains, is being unemotional the only way to write accurate history?

Emotional historians may over/under exaggerate certain events in order to make history conform to their ideas, as a result only half of the total story will be told. As a result these historians will superimpose their biased perspectives on society and will affect how society progresses. Overall, their version of society will become askew and will bend history to their ideas and imagination.

On the other hand, historians that are emotional may be able to convey certain ideas and events better than unemotional ones. In various literary pieces, it is often the ones that better express the author’s emotions that people tend to better imagine the scenery being described. The same can be said about historical recounts or descriptions, because recounts with more emotive words may be able to better describe certain events than unbiased, unemotional words. For example when describing a war, it would be better to use emotional words to describe how people felt about that war. As a result, a history without emotion would be a history that has lost an important element, emotion. History is riddled full of emotional conflicts, events and people, therefore recounting and describing history without emotion will only provide a part of the story.

In conclusion I believe that while it is important for historians to write accurate history, I do not believe that to do so emotion must be thrown out of the picture. This is because emotion plays an important part in historical events, therefore ridding history of emotion will take away an important part of the story which blinds society to the whole truth.

Sense Perception

Today in TOK, we were exposed to a mind-opening experience that informed us that our minds usually do not allow us to perceive the world as it is, but as how we are preconditioned/engineered to perceive the world. While I realise that there are problems with our perceptual systems, this doesn’t mean that knowledge gained from our senses is completely unreliable. This is because the knowledge that we gain from our perceptual systems is still knowledge, regardless of whether it is true or not. This is because regardless whether the knowledge we receive accurately reflects all there is to an object, we believe that the knowledge we receive is accurate. Furthermore in recent years as our understanding of ourselves and the universe grew, we have managed to develop technology in order to combat the illusions that our mind may cast on how the world really is. One example is how the Hubble telescope can provide scientists with images of distant nebulas, galaxy systems or other cosmic phenomena, but as seen through different parts of the electromagnetic spectrum.Image result for Hubble Telescope Image of Galaxies through different spectrums

Thus providing us with new information about how the universe works and looks, that is not based on our biased and inaccurate senses of perception.

A counter-argument is that if the knowledge we receive is flawed in some way, then other presumptions or knowledge derived from knowledge gathered by our sense of perception could also be equally flawed. I say that knowledge is knowledge, regardless of whether it is accurate or not. Furthermore the knowledge that we derived from our sense of perception is something that our civilisation has relied on for centuries, and that knowledge was all that we had for centuries. Another argument that can be made is that despite the possibility that the information we receive may be flawed, we have been able to capitalise on our knowledge and develop a greater understanding of what the universe actually looks like.

WOKs are Double-Edged Swords

In total there are 8 ways of knowing and 8 areas of knowledge, and through discussions held in class I believe that the Way of Knowing (WOK), reason is one of the more important ways of knowing. Reason is important in a few of the Areas of Knowledge (AOK), but I think that mathematics is one of the AOKs that relies more on this WOK. Unlike the arts, ethics or religion, math is an AOK that is destitute of emotion and requires logical reasoning to gain any ground. For example, it does not matter whether you feel happy or sad in order to solve a math problem, because the rules apply anywhere in the world. In any part of the world, no one would dispute that 1+1=2 or 100+500=600. Even though reason cannot always bestow knowledge upon someone through lack of reason or by being impaired by some external factor, I believe that mathematics is a perfect match for reason. This is because mathematics only works if all the factors are considered and used in a logical manner, which is determined through fair and impartial reasoning since there is nothing that appears more favourable as the only factors to consider are numerical. However this is not to say that external factors could affect one’s ability to rationally use reason to solve a math problem, therefore one does need to consider how other WOKs affect each other.

In mathematics, it is undisputed that 2+2=4 because when you raise 2 of your fingers alongside with 2 other fingers already upright, then you have 4 fingers in total. Don’t trust me? Count them. However a real good question was raised during one of our class discussions, “How would you prove that −2−2=−4?”. Of course we cannot prove it using the same method as above, since we cannot produce negative numbers using our hands. However if we use reasoning, then we can reason that “−2−2” should follow the same principles as “2+2”. This means that the numbers should add up to a 4, however since the numbers are going in the negative direction then the 4 should also be negative.

IB Gold Coast Retreat Reflection

From the 17th-18th of August, the entire Grade 11 cohort of CDNIS travelled to the Gold Coast (the one in Hong Kong) to familiarise ourselves with the Diploma Programme. I particularly liked how the retreat was an easy way for me to socialise and get to know other people in my grade other than the ones that I usually hang out with. I was able to hang out with the other people in my grade through participating in various activities like the poverty simulation at Crossroads, our free-time at the pool after the simulation and the “IB Core remix” on the second day of the retreat.

During the poverty simulation, we were put into ‘families’ and made to act like impoverished families and make paper bags for very little pay. All the while undergoing various trials at the same time, all the while trying to survive the day. I was able to bond with the various members of my ‘family’ through the pressure of trying to survive the day’s work (I also bonded with them through the loss of my phone, shoes and a kidney to keep the ‘family’ from falling into debt). While this activity was fun and provided insight into how impoverished families lived, I also gained more insight into what I might be able to do to help these impoverished family. This is because  the simulation leader’s question, “How could someone as insignificant and seemingly-powerless as me could do anything to help anyone?” really resonated with me. I have racked my brain trying to find solutions to no avail, which led me to prematurely accept that I could do nothing about the situation and that only the ‘big people’ could do something. However the simulation director has finally provided me with a solution, making micro-loans or backing a child’s education. I now realise that despite all the challenges that are seemingly posed against normal people are simply figments of our imagination, and if we are determined enough to help, we can.

Spending my free-time hanging around with my friends at the pool and for dinner are the fondest memories I have of the retreat, because recently I have been constantly stressed about my DP course choices and this retreat was a great way for me to relieve and forget some of my stress. Furthermore I did not hang out with many people my age throughout the summer, so this was a great opportunity for me to let loose and basically be me. Although things did get slightly out of hand when I a small get-together in my hotel room turned into a big get-together, which was annoyed me and reminded me of how tight-knit our grade can be.

During the “IB Core Remix”, I was brought back to the reality that I am now a DP student and will face many new challenges throughout the next 2 years of my academic life. I found that many of the activities in the remix helped me gain and idea of what path I would like to walk on, and this idea was created after going through the university talk, figuring out a personal statement for myself, the talk with the alumni and thinking about CAS opportunities. While the idea I have now is not that specific, it’s a start that was only possible through listening to Ms. Irvine talk about different universities, thinking of my accomplishments and shortcomings, interacting with the alumni and thinking of CAS opportunities for me to make a mark on society.

If you cannot explain something, you do not know it.

When two or more parties disagree with each other, there is an assumption that one of  parties involved is wrong and the other is right. However this is technically not the case, because there is another possibility where both parties could be correct in there statements. This is because while some topics may have definitive and irrefutable facts, other topics are more conceptual and do not have right or wrong answers. One example is in literature where a word could mean two completely different things to two different people, and this is due to how the person is raised and what experiences they have. Furthermore, the ideas or facts that people use to support their ideas are based on what the person knows, or at least what the person thinks they know. Therefore if one person argues that god is real and another person argues that god is simply a human construct based on faith in a higher power, both arguments could still technically be correct since the arguments are based on what the people think are irrefutable facts. Which in turn suggests that being ‘right’ does not matter what the larger community states is ‘right’, it only matters what the individual thinks is right to make an individual’s opinion right.

Inspired Summative Composition Reflection

Source of Inspiration:


I attempted to make my composition reflect the sense of tranquility that the picture above instills in me when I look at it, and the sense of longing that I sometimes have when I look at this photo to go back to this place where I could just relax and not have anything to worry about. My song has a relatively slow tempo of 92 bpm to represent the tranquility that the photograph represents. At the same time, the song was composed in a major key (A flat Major) and utilizes mostly eighth notes which gives the overall composition a sense of longing and fondness, which reflects my fondness that the memory evokes in me and the longing I have to return to this place and witness this spectacle once more. Even though my choice to use saxophone was mainly due to the fact that it is the only instrument I really know how to play, my choice for using the flute was because of it’s higher pitched sound. I wanted to use the flute to give the composition a more graceful tone to imitate the fondness that I have of this memory, and to create a lighter mood for the overall composition.

In terms of the structure of the song, I made a small intro to give the song a positive tone which reflects the initial positive feeling I get when I see this image. The intro is then followed immediately by a chorus, which uses it’s imperfect and plagal cadences and the overall sad quality to represent the feeling of longing that I have to return to this place. The chorus is then followed by a verse representing the sadness I feel that comes along with the longing. Afterwards the chorus is played again to represent the renewed sense of tranquility I have after realising that even after 2 years, the fact that I still remember this particular scene is a testament to show that the memory of being at this place will be with me forever.

Initially, I wanted to create a piece of music that was based on a scene from a movie because I thought it would have been easier to compose based on something that also had a sense of progression and was also accompanied by visuals. After attempting to create a score for a movie scene from Batman vs Superman, I realised that because of the limited instruments I have on finale and my own limited knowledge on how to play instruments other than saxophone, plus the fact that the main character was Batman who already had a theme song made me decide not to compose music for the movie scene. This was because the scene’s main character, Batman already has a theme song which will entice me to copy the theme song. In addition, the scene already has a score, which might affect my composition by making me subconsciously compose with ideas similar to the score in the movie. Afterwards I decided to create a pop song composition using other songs as sources of inspiration, but my attempts at making an upbeat song were not successful due to my tenacity to try and replicate segments of other songs and somehow incorporate these segments into my composition.

However after looking through one of my photo galleries, I found a picture I took and felt the happiness in the memory that the image evoked, the longing the sadness I had to return to this beautiful place, the sadness that accompanied with the longing and the eventual happiness I felt. Afterwards, I went straight into experimenting with different sounds in Finale and researching useful sets of chords to help me figure out what notes to use. After experimenting with notes and finally figuring out a chorus that I liked,

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I was able to easily complete the introduction, the verse and the ending by making them sound something similar to the chorus to give the overall composition a sense of unity. Throughout the chorus, I mainly use Neighbor Tones accompanied closely by the occasional Passing Tone to both mimic the waves in the image and to signify the gradual change in emotions that I have whenever I look at the image. However once this was done, I noticed that the transition between the chorus and the verse and vice verse was too obvious for my liking. As a result, I used transitional measures to transition between different segments of my piece.

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Once the first version of my composition was completed, I seeked out external feedback from Ms. Po who informed me that even though she liked my rhythmic ideas; the chords, the instrument choice and arrangements of my chosen instruments simply was not right. After providing me with feedback and helping me make adjustments, I was able to to make my composition sound better through more thought-through choices. For example, I set the composition’s key signature as A flat major instead of F minor to convey the happy yet sometimes sad emotions the inspiration photo evokes in me; Ms. Po also helped me to change the chords of my piece so that the chords did not clash with the melody of the composition; she helped me arrange the instruments I used (Flute, Alto Sax and Electric Guitar) so that I can actually hear what is being played by each section; and she also advised me to use the Electric Guitar instead of the Electric Bass to play my chords which helped greatly since I had no idea that the Electric Guitar played the bass chords and the Electric Bass did not. In the end, I was able to keep most of my melody the same as it was originally. Changing only the chords used, and one of the three instruments that I used in my original composition.

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