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Due to the high degree of abstraction present in mathematics, the sharing of knowledge has a barrier of entry even among those of similar skill level. This is due to the fact that when we try to translate the abstraction into mathematical notation, we create an entirely new subset of the universally accepted mathematical language, and thus the sharing of knowledge among the expert knowers is also a challenge in some sense.

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Many perspectives are taken by the expert knowers in the Area of Mathematics, some believe it to be tangible while others believe it to be based on the foundations of formalizations of abstraction based on Cantor's set theory. However this is also pointed out to be somewhat false as it is believed that most people believe in both perspectives, and change their views when its convenient. This is a method of firstly grounding the abstraction in reality to find meaning, but also allowing the abstraction and formalization to underlie the philosophical backing of Mathematics.

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In this extract, Mathematics is manifested beyond the crunching of numbers and calculating of theorems. It writes about the role mathematics plays in mass culture, being popularized to the point of becoming commercialized and used to advertise completely unrelated products such as fragrances and movies. The ubiquitous nature of mathematics has caused it to be universally understood and relatable. Therefore, many different companies and brands use it as a device to connect with a larger crowd, due to how undisputed and absolute mathematics can be, even towards those who are not math enthusiasts. I think mathematics is a language that is mostly absolute and constant. It is not susceptible to the varied nuances of judgement that other languages and subjects may be vulnerable to. It is a concept that is enshrouded in large amount of truth, since regardless of race, language or religion mathematics is interpreted and perceived the same by all. Therefore, it becomes a very relatable and universally understood topic.

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Mathematics in the most basic form is the analysis and observation of patterns that surround us. It is described as a "thing of beauty" as it is a concept that allows us to make sense of the World around us, enabling humans to predict, understand and explain phenomena that occur in our world. It is a truth that exceeds the boundaries of time and perception. Mathematics has been a concept immune to objectivity and has existed since the dawns of time. It was only a matter of time before it was discovered and incorporated into our daily lives. However, as describe in the article, mathematics is familiar yet so foreign to us because it is a set of rules that are based on abstract assumptions that our pioneers had formed. It is so concrete yet so fluid. For instance, the foundation of mathematics, a set of 10 that resets to a repetition of two 1s, is a rule that was never something destined or set in stone. It was the sole decision of the people who had preceded the understanding of mathematics, "expert knowers", as well as those that had adopted this ideology that caused our entire World to perceive math in this light. Therefore, this striking contrast intertwined in mathematics makes it truly a marvel of magnificence.

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Mathematics, in this extract, is explored though it’s role in the relationship between order and chaos. The two are polarising yet both are crucial in our daily lives.

Mathematics is deriving order from chaos. Chaos is not necessarily disorder, but rather a natural state of being that w may simply fail to understand. Numbers on its own may lack a system, but we humans have decided to come up with a way to organise them, giving each of them a role, making them interact with each other. Through doing so, we are thus able to come up with formulas and solutions that are rational and equate. This, coupled with the satisfaction of discovery, enables math to become universally applicable.

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We often regard mathematics as something uncool or generally unappealing; I feel that this could be due to the high amounts of skill required to successfully engage oneself in this rational and systematic subject. However, this article explores the extent to which mathematics is involved in the workings of our society. Questions such as the existence of zero are not necessary for us to function, but employ our desire for gaining of knowledge and discovery to heighten our curiosity. We can entertain these questions, though never coming to a conclusion, allowing us to realise that math is not as straightforward as it seems.

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This extract describes how Mathematics has played an integral part in our society in terms of mass culture as it has been popularised to the point where it is used commercially to promote items that are irrelevant to Mathematics, such as perfumes, movies and even jeans. It shows that Mathematics is more than just proving theories and solving equations. Because Mathematics is known as the universal language, where people all around the world can understand it without having to face the problem of a language barrier or having to deal with the subjectivity of languages. Due to this, many brands use Mathematics to promote its products since it is an easy way to connect and reach out to a larger audience without having to worry is the audience understands the advertisement. Since Mathematics is backed up by facts and theorems that have been proven time and time again, it goes to show that Mathematics is generally recognises it the same way, making it a reliable and easy mode of communication.

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Expert knowers in Mathematics have different views on the topic itself that differ quite greatly from each other. Some expert knowers believe in Platonism, where all mathematical objects are real while some believe in formalism, where there are no mathematical objects and they just do not exist while constructivists believe that only what can be obtained by finite construction should be considered as genuine Mathematics. This shows that Mathematics may not be as universal and straight-forward as it seems considering people all around the world have greatly differing views on the same topic. It can also be seen that people can believe in more that one theory and are not restricted to just one, sometimes changing their view when it suits their needs.

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Mathematics is all about finding patterns and a reason why something is the way it is. Out of the four possibilities mentioned, the word order was in three of them. This goes to show how no matter how complicated something seems to be, there would be a higher change of finding patterns and order out of it rather than classifying it as 'chaos out of chaos'. Mathematics is something known by almost everyone. Mathematics has a certain order and law to how everything works unlike other aspects of knowledge such as languages and sciences where things can vary. Math follows a strict structure, which is what allows it to have patterns and thus be more predictable than other areas of knowledge

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Our lifes have been run by mathematics since the start of time. Whether we want it or not, since the beginning of time, our ancestors have been using math for the simplest of things for the sake of keeping track of the number of items they had. in every single century and every level of human advancement, humans have had to use mathematics for their needs. even though these needs differed greatly, one thing they all had in common was using mathematics to cater to their needs. mathemaitcs helps us to make sense of the world around us. by adding a pattern and rules as to why things happen, we can comprehend things going around us better.

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Mathematics in this article is described as the most arcane of academic disciplines that has implanted itself firmly in today's world. Mathematics is a universal language that is interpreted the same all over the world, no matter the difference between individuals. The article talks about the history of mathematics, including many different concepts such as the concept of the number zero. The mathematical knowledge that we now know and learn about comes from the expert knowers such as Archimedes. Mathematics can be found almost everywhere in the world, and this also leads to the study of physics. Therefore, mathematics being used universally, even in other areas of study can help knowers like us to better understand topics and concepts in the world

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Mathematics is defined as the science of total intellectual order. I understood this as a pursuit to understand relationships between arbitrary variables through identification of patterns. A random strand of numbers (aka. deriving chaos from chaos) is not math, but finding a pattern, described by numbers and symbols and diagrams, from chaos through the use of logic is.

The article discussed about how Chaos originally meant a void of nothingness and emptiness, hence it is meaningless. And later the definition of chaos is humanized to appear as mess and disorder, where no interconnectedness among events and phenomena. The definition is very interesting as it shows that  information that is not understood or processed(maybe through math) is equivalent to a having no knowledge at all.

In a nutshell, Math, which is finding order within chaos,  is one of the many ways of acquiring meaningful knowledge from the world full of meaningless information.

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Mathematics is a language. In order to converse in it, one has to study it and have much background knowledge to support. It makes it easier to discuss abstract and niche topics, so that knowledge can be constructed and produced. Mathematics allows construction of knowledge on top of other already available knowledge.

In Mathematics, there is a very distinct difference in correct proof and incorrect proof, since math is governed by absolute laws and definitions, yet there is much uncertainty in knowing what is complete and incomplete proof.

Though mathematics is objective, it is done with using critical thinking, not algorithmic thinking. This is reinforced by the article which quotes that "no computing machine ever built will accept would accept his definitions as inputs."

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Mathematics is a science, and has contributed to various sciences greatly. Mathematics have been applied from the beginning of human history, for various purposes. Our earliest ancestors used mathematics in their survival and tied mathematics to tangible objects. The Ancient Greeks mainly studied mathematics to satisfy their curiosity about the subject, and along the way they managed to unearth groundbreaking discoveries that are still in use to this day, or have been disproven by other philosophers/scholars along the way.

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In this extract, it states that mathematics is not only about numbers in business, trade, probability, projective geometry or even quantum theory, but it is also a system of logical reasoning and is "a thing of beauty" rather than only "a technological tool". The main purpose of mathematics is to help us to quantify things in this world such as money or time. However, it must be acknowledged that some of the mathematical concepts might not be true and based on new discoveries in the future, our understanding of mathematics might change with time.

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Mathematics can be accessible to people of all intelligence levels as well as a realm of knowledge only privy to a group of superior knowers. For the ideal mathematician, mathematics is a language that only he and his fellow elite knowers can converse in. Many scholars and philosophers have attempted to define Mathematics, eg. Aristotle naming it the study of quantity while Henri Poincare believes Mathematics to be the relations between objects. Mathematics is hence defined by the mathematician in relation to his work and discoveries.

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Mathematics is usually perceived as a subject that is objective, factual and with only one correct answer. However, as I read this article I realised that Mathematics can be subjective. In this article, the topic of zero and how it took Western mathematicians so long to accept that nothing could be something was intriguing to me. Here Mathematics is being shown as something that is not so straightforward and how it caused paradigm shifts that were quite difficult to accept. Mathematics can lead to deep thinking beyond the maths itself but to fundamental questions such as the fact that nothing is still something.

It can also be seen how Mathematics is applied in daily life not just through the actual use of Mathematics but through media and popular culture. Mathematics truly goes past the perceptions of being factual and objective, to create links between people, knowledge and culture

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In this article, Mathematics is described as more than a subject and more of a language and something that brings people together. As seen in the article, the author talks about how mathematics entered the popular culture after Andrew Wiles proved Fermat's Last Theorem, a problem that has not been solved for more than 350 years. This publicity made people more interested in math as more people went to find out the theorem and how it could have been proved. Next, mathematics is a form of language as it is able to convey messages effectively. For example, the person who proved Fermat's Last Theorem was offered modelling jobs, fragrance and more. This shows that through the use of mathematics, be it its fame or its usefulness, a company is able to make use of mathematics to sell something as well. Hence, the article mentions that mathematics is a language that can help convey messages and bring people together to work towards a common goal of proving other theorems.

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Mathematics is a language, and much like a language, its origin can also be lost in translation. This reading investigates a hypothetical "ideal mathematician", or "most mathematician-like mathematician". In this example, the mathematician's life work, the study of "non-Riemannian hypersquares", can only truly be understood by a select few who are the intended audience of the mathematician's writings. They are able to see his underlying intentions, purpose, and bias in his seemingly "mechanical" works. However, if this group of people that understand his work dies out, his writings will forever remain lost in translation, or "less translatable than those of the Maya". In this way, the work of a mathematician can be very unsatisfying, as the study of such an abstract language that tries to make sense of the world often does not yield much. Similar to spoken language, abstract mathematical concepts are of no use to the public, and their study eventually dies out when unapplicable to the majority. As such, the mathematics most people will come into contact with in their lives ends up being a very simplified, basic form of itself so as to become relevant and digestible. One singular person will never fully know mathematics, just like a singular person we will never fully know a language, as its rich course of study will always have some aspects enshrouded in mystery or having been lost in translation.

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Most people who encounter mathematics tend to disregard the philosophical side of it, hence the majority of the mathematical world (65%) is made up of Platonists. Platonism makes the most sense in execution, as it is easier to study information people believe is already there, and make sense out of the concrete and objective rather than the abstract. This stays true to other aspects of life; people may choose to believe that everything is a simulation (much like the Matrix) or say that nothing matters because "we all live on a floating rock" (a popular thing to say online nowadays). However, most of these philosophies exist as fleeting thoughts or lighthearted fantasies to ponder, but in execution, one will find it extremely rare that people exhibit reckless behaviour because they truly believe that nothing matters. In reality, regardless of one's inward beliefs, the philosophy everyone ends up subscribing to is the one that is most relevant to an individual's life and goals. Bringing this back to the topic of Mathematics, it seems like we need to base our exploration on things we must believe to exist, or else we wander aimlessly trying to navigate the unknown. Besides, most people will never reach the boundaries of what has been discovered in Mathematics such that they "become aware of the difficulties" in it. So does Platonism aid in Mathematical advancement or does it limit it?

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In this reading, it mentions how mathematics is a simplified problem-solving tool that can be used in terms of pattern, order and chaos. In terms of patterns, mathematical equations are able to help the people relate to the drawn diagrams to find out if there are any relation between each pattern. In terms of order, mathematics can help one arrange people or spaces systematically so that there would be a certain arrangement of people or spaces. Lastly, mathematics is in a way chaotic. Based on the equation formed for the "order out of chaos" situation, a mathematical equation can be chaotic when there are too many perimeters. This further shows that mathematics can be used to see patterns despite the equation having many perimeters which goes to show that mathematics is versatile and offers a simplified way to view things.

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Mathematics may seem like a blur, a chaotic mess, but actually is all about pattern and order. It can easily get messy but if you slowly follow the algorithm, it is easy to understand the rationale behind formulas and equations. It is up to the person to find the structure in mathematics and since mathematics is universal, it is one of the most versatile

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Despite Mathematics being seen as something straightforward which only consists of definite answers, it is interesting to see that expert knowers of Mathematics have very differing frameworks. Although the frameworks of platonism and formalism are very different, where they question the reality of Mathematics, they still share the same principles of reasoning and logic behind Mathematics. Being an expert knower of Mathematics, they also seem to go through a conflict of choosing between platonism and formalism, or of choosing both which shapes how they view this area of knowledge.

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Mathematics is a language that knowers can communicate with. It is a very selective language not very widely known other than a select group of people and without the necessary mathematical skills, a knower is unable to process and understand the knowledge that mathematicians produce and converse about. For example, ideal mathematicians have difficulty in talking about non-Riemannian hyper-squares to other knower without the expert knowledge or even with other expert knowers who may not be interested in that particular branch of the area of knowledge as mentioned that the college prefers ‘non- Eulerian’ semirings, showing how broad the area of knowledge is. Math is such a broad area of knowledge that many expert knowers have had trouble giving it a fixed definition, ranging from Aristotle ‘the study of quantity’ to Carl Hempel ‘axiomated deductive system’

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This article describes mathematics as something more than just numbers. It is something that connects everyone. Regardless of where people are or what language they speak, they will understand mathematics as it is universal. This allows knowers to share their knowledge about the same concept to the world and become the expert knowers to shape the knowledge of personal knowers

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Mathematics has been integrated into our lives ever since we were born and centuries before it. It is not just a mere collection of facts or technological tools, but rather, a “thing of beauty” that shines greatly in our everyday lives. Mathematics however, though is considered a science that relies heavily on formulas, deduction and analytical thinking, can still be deemed inaccurate, as the theories, experiments and investigations carried out to derive a formula may be inaccurate, which can be changed with technological advancements and a new perspective.

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The area of knowledge of mathematics in this article shows that mathematics can be linked and infused into entertainment and popular culture. It also shows that mathematics can be subjective and debatable in the examples of zeros and how whether the millennium began in 2000 or 2001 due to the absence of the year ‘0’ and how mathematics can be ambiguous as 0 can represent different things and applied differently in other areas of knowledge such as physics where 0 Kelvin, or Absolute zero represents 273.16 degrees celcius.

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Mathematics is a simplifying tool that organises a field of chaotic numbers into patterns. The term chaos dictates a sense of disorder that is difficult to interpret. Mathematics is how humans explore relationships between numbers to construct definitive patterns and equations that make it easier to comprehend. Its systematic treatment of relationships between quantities expressed symbolically allows humans better to understand a ‘chaotic’ string of numbers.

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Mathematics is a fundamental aspect of science, and it can be manipulated into complex theories, or simply be conveyed in mass media ways such as magazines, television shows, biographies as stated in the article. It is a binding tool that enables people of different cultures, languages, and backgrounds to assimilate information without the use of words, as it is a universal language of numbers. It also breaks the stereotype of mathematics being factual, algebraic, as it can be interpreted by knowers around the world.

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Mathematics is not only hard to grasp for knowers, but that is also the case for expert knowers. Expert knowers have difficulty understanding the reasoning behind the findings of another fellow expert despite having around the same achievements. This is seen in the article when Professor Nameless, the founder of the theory of non-Riemannian stated a problem and gave a partial solution, yet when displayed, no one except for himself was able to understand. In a much better scenario when a small group of them were able to understand his proofs, as no computing machine could accept his definitions as inputs, his discovery is threatened to be forgotten as he is unable to share his findings to other knowers due to language barrier.

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We have come a long way in embracing mathematics as in the past, it is believed that only the chosen/gifted few were able to take on the subject, but in modern times, it is more welcomed since even high schoolers like us are able to grasp ideas of mathematics. However, mathematics is more of appreciating than understanding. Since very long ago, mathematics has already benefitted many as shown in helping the ancient greeks with their own interest and Archimedes for his ingenious engines of war. There is so much more to unravel about mathematics yet very little has been discovered. Overall, mathematics is not something that can be comprehensible and claimed to be reliable. As such, we can only perceive and not come up with an explanation to the wonders of mathematics.

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Mathematics is a language which is used to express and communicate concepts which are very much a part of the structure of the world, containing truths which are valid forever. However, the some of the concepts expressed by mathematics exist only in the study of mathematics itself. It is similar to a word in one language (A) with no equivalent in another language (B), where mathematics is A and B is layperson terminology. Without familiarizing oneself with mathematics to a great extent, it becomes impossible to understand mathematics fully as, in a number of cases, there is no means by which certain mathematical concepts can be "translated" into a form that is already understood. Because of this "missing vocabulary", knowledge acquisition from mathematical texts is limited as there are only segments of texts which can be "translated" accurately, while they are nested in a sea of illegible writing. This leads to omission and "mistranslation" in many areas, causing knowledge to be "lost in translation". It is thus that the ideal mathematician's work is only intelligible to a few dozen, or at most a few hundred, people as only so many are well versed enough in mathematics to understand the work in its entirety, accurately.

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Our universe exists as a place of primeval chaos, and some even go so far as to say that entropy (an inclination towards disorder/the degree of disorder in a system) is a fundamental part of our universe that is constantly increasing within the closed system that is our universe. Meanwhile, mathematics attempts to explain phenomena in our universe using concepts that hold true regardless of the passage of time. Hence, it is said that, to some extent, "the whole object of mathematics is to create order where previously chaos seemed to reign and to extract structure and invariance from the midst of disarray and turmoil". Mathematics, therefore, deals with finding patterns across the phenomena of our universe because a pattern implies an adherence to some form of structure from which scientists can derive their laws.

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I realise that I often take Mathematics for granted, and I fail to remember that the mathematics I am studying today is based upon centuries of reasoning, proving, and the calculations and brains of the many genius mathematicians who came before us. This also shows how Mathematics have, to a certain extent, led to the evolution and development of human brains, whereby we are able to solve problems easily thanks to the past work and discoveries by mathematicians; this is seen in the article where “many of the ideas that are easily grasped by any high school student were inconceivable to the foremost thinkers only a few hundred years ago.” Unlike biology and chemistry whereby it higher chance of being proved wrong through the course of time, mathematics’ fixed rules, principles and equations will remain the same and true; it is “eternal and unchanging”.

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As a person who studies mathematics in school and is often drowning in eternal mathematical confusion, I never had the time to stop and ponder about the mere existence of math. From this article I got to know more about the expert knowers who debate about the foundations of mathematics and are split into Platonism, formalism and constructivism. Platonists believe that there is a universal truth underlying all of mathematics, whereby mathematics cannot be created; it only can be discovered. Formalists believe that there are no mathematical objects and that all of mathematics can be defined by a set of predefined rules. Constructivists regard mathematics as only what can be obtained by a finite construction. I found this very interesting as it made me question my thoughts and beliefs about math, as well as the learning attitude that I have for mathematics and whether or not it is affected by the biases and inaccurate preconceived thoughts I have about mathematics.

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This extract discusses the role of mathematics in society beyond a functional purpose.
The paradigm shifts in the mathematics area of knowledge meant that the knower’s knowledge was challenged which shifts their perception. This makes them “vulnerable” to corporations who sensationalise these ground-breaking discoveries and hook the viewers through the linking of their product to the paradigm shift.
As mathematics is an area of knowledge w a large range of deep concepts, the way we see values in different social contexts is very important. For example, how temperature is measured through Celsius and Fareheit.

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Mathematics was constructed through the social need for counting. In our world, different societies have different functions and purposes for mathematics. Hence, some societies like tribes in the Amazon may not be as developed in mathematics due to the absence of a need for counting. Instead, vague denominations and groupings are used.
This extract also discusses how knowers today can grasp concepts previously elusive to the most esteemed thinkers of yesterday. It made me question what factors could have influenced this shared knowledge. How could the advancement of human communication and the development of technology contribute to a faster advancement in discovering and deconstructing mathematical issues we encounter?