TOK Task #12 – Math Scope

  1. What is the difference between a conjecture and a theorem?

A conjecture is one step shy of a theorem – it is the foundation of a theorem as an idea of a rule or concept is present, but not with the full supporting evidence. A theorem, on the other hand, appears to be fully supported with evidence and at least valid to a certain extent.

  1. In THE VIDEO  Eduardo Saenz de Cabezon uses the example of people being surprised that folding a normal piece of paper 50 times, will reach a thickness as high as the sun. He challenges us to ‘do the math’ and see that he is correct. What do you think meant when he said that Maths dominates intuition and tames creativity? Do you agree with this?

A large component of mathematics is based on theorems, which are supported by seemingly empirical evidence that is deemed logical. The Ways of Knowing reason (logic) and intuition are closely intertwined, and consequently, it could be argued that one’s intuition is based on logic, meaning concepts that could be derived from (but are not limited to) ethics and morals etc. Creativity runs wild, and logic and intuition then offers a framework to the mind when thinking, for example about creative solutions when attempting to find supporting evidence for a theorem. I agree with what Saenz de Cabezon said – I agree that maths does dominate intuition and acts as an indicator of what might seem logical at a certain time, for example when answering a maths question. I also agree that maths tames creativity, as creativity appears to work collaboratively with logic to form reasonable but simultaneously, possibly outlandish ideas and evidence.

  1. Saenz de Cabezon claims that the truths in maths are eternal. Do you think this gives maths a privileged position in TOK?

I think that this statement would only give maths a privileged position in TOK if the framework of TOK revolved around believing that logic is the supreme way that humans can know and learn. This is because in TOK, we are encouraged to think about how all Ways of Knowing and Areas of Knowledge have their flaws and strengths. I think that the claim that the truths in maths are eternal is not entirely true. There are periods of time in between trying to find new evidence for a theorem if the previous evidence was discovered to be purely coincidental in its support to the statement.

  1. List any of the knowledge questions related to maths that came out of your discussion in class.

How objective can maths be?

Should maths be a compulsory subject at school? Who should be studying it?

What is maths?

What Ways of Knowing would be most applicable to maths?

How do we determine when maths is right or wrong?

TOK Task #11 – Art and Truth

All topics can portray a “truth”, even if they are not the same. The arts and sciences are inextricably linked, as science can sometimes explain the beauty of art, and art is often used to convey scientific “truths” to the common man. Truth is the conveyance of the honest tendencies and experience of the human condition – it is not simply what happened, but how events were perceived by humans, and what impact they had on those same individuals. Truth can still be depicted even if not through direct means, as ideas can be communicated through a variety of ways. Ultimately, the quality of a “truth” is measured by its ability to allow individuals to grasp and fully comprehend the extend of a historical occurrence, economic trend, scientific phenomena, mathematical reality, or emotional universality.

Artists carry a special responsibility to convey the truth, as the impact of their work gives them power. The truth expressed by the artist can defeat the lie, as ‘one word of truth outweighs the whole world’. Factually true statements can be found in a work of literature just as much as in science, but there is something unique to art. However, by trying to figure out the truth in art, humans may be diminishing it. There is a value of art which goes far beyond the passing on of ‘truth’.

Both the essays show that knowledge can be produced in different ways, but knowledge is most definitely produced in art. The arts and the sciences are reliant on each other as empirical information and comparatively subjective information must work hand in hand to produce knowledge. There may be previously established knowledge that exists, but knowledge established from art can build upon that. This means that there are multiple perspectives and approaches to each story, and there is not just one objective way of viewing something.

TOK Task #10 – “Group” Knowledge in the Arts

“Without the group to verify it, knowledge is not possible.”

The role of the ‘group’ in the production of knowledge is to confirm and verify the information. Verification in the production of knowledge is crucial if it is intended to be shared so it can be universally understood, at least to a certain extent.

This links to personal and shared knowledge as shared knowledge would potentially need to be confirmed or verified as true or empirical for universal understanding. Shared knowledge would need to be verified if it was something people had to understand to a certain level to serve a certain purpose, for example learning maths to become an engineer, or learning physics to become an architect. Personal knowledge does not necessarily have to be verified, as personal knowledge can be something such as emotions, and all emotions, while not all right, are all valid.

In the arts, there are many types of knowledge, such as conceptual knowledge, moral knowledge, and aesthetic knowledge. Conceptual knowledge allows humans to understand human-established concepts, such as our emotions and beliefs. Art can provide us with conceptual knowledge by eliciting emotions from humans and hence, making us aware of them. Moral knowledge is also an example of knowledge provided by the arts, as art can appeal to a certain cultural context and allow the viewer to become influenced or educated about an issue. However, moral knowledge generally builds upon our preconceived notions, and thus, moral knowledge is developed instead of created.

Claims in art are generally open to being shared and discuss. Usually, most people form opinions and debate over art, such as whether a created piece is art or not. Everyone is entitled to their own opinion as the confines of art are not restricting. There is a lot of room for subjectivity in the arts, and consequently, not many definitive answers. Subjectivity and ambiguity in arts is what allows claims to be shared and discussed, as there is less of a “black or white” scenario.

TOK Task #9 – What Is Art?

Knowledge within arts is not objective & therefore not meaningful.

If knowledge is subjective, it is not meaningful because there are too many approaches and thus, diverges from the general sum of what is known and understood.

In the arts, it is difficult for a consensus to be reached about what a piece is about as it holds different meaning for individuals. When something is capable of eliciting different emotions for different people, knowledge about the piece cannot be formed as the way the piece is being perceived and approached is so different – it would not be easy for everyone to reach the same understandings, particularly if emotions affect the learning of knowledge. For example, Onement VI by Barnett Newman, was a painting that had a turquoise stripe separating two dark blue rectangles. Firstly, it would be difficult for most people to understand what the piece is, which means that the thoughts produced could not be knowledge and would be potentially superficial in terms of understanding. Secondly, no knowledge that is substantial can be reached because all the knowledge is stemmed from individual emotions. Knowledge from individual emotions are merely thoughts that have been generated after perceiving something. This knowledge cannot be transferred because it is not a total representation of what something is supposed to mean, and consequently, no widely accepted sum of what is known can be reached.

However, on the other hand, regardless of the source of knowledge absorption, there is a plethora of ways of knowing, subjective and objective, that make knowledge valid, particularly moral knowledge. If information produced is understood from subjective ways of knowing, then subjective knowledge is meaningful regardless of how opinionated it is, as it still provides us with certain cultural contexts and understandings about morals. Learning subjective ideas can be considered knowledge as learning about individual emotions can be related to learning about human psychology with a real life source. For example, Jean-Michel Basquiat’s “Irony of a Negro Policeman” is a piece of artwork pertaining to racial issues. The way we perceive the artwork may tell us more about racial issues from a different perspective, perhaps through the way Basquiat depicts this policeman. The artwork can allow the audience to gain a deeper understanding about the cultural contexts involved, and how race issues are prevalent.

TOK Task #8 – Intro to Natural Sciences

What distinguishes Natural Science from other AOKs? Identify any potential issues or questions that may arise when you consider your definitions.

In my opinion, the fact that Natural Sciences are the first fallback for any statement or claim because of its’ factual property distinguishes the AOK from other AOKs. From Naomi Oreskes’ TED talk on the topic of ‘why we should trust scientists’, a statement regarding the reason the Natural Sciences are our number one fallback when investigating and learning is because of the scientific method, meaning that we were taught that scientists follow a method that guarantees the truth of their claims, meaning that these methods are credible and can be relied on. Natural Sciences are also automatically the number one fallback for support for claims because of the factual quality of the sciences – we believe that science has the power to prove or disprove something because there is substantial evidentiary support, at least to a certain degree, that determines something to be true.

The primary question that would arise towards this definition is how true the idea that the Natural Sciences are all proven with facts is, considering that it can be argued that all scientific theories and discoveries are all opinions that have certain support that align at a certain time. This means that the facts are not substantially true, which can deter its reliability in terms of why the Natural Sciences is a primary fallback for most claims. Another problem about the factuality of the Natural Sciences is that they are inductive rather than deductive, meaning that scientists start with observations, from which they form theories from. This means that what we know, which is derived from scientific knowledge, is all based on their sense perception and how they saw a certain idea. This does not necessarily mean that the theory is true, even if there is evidence that just so happens to support it. This diminishes the factual credibility of the Natural Sciences, which is what humans are heavily reliant on when trying to be more credible with their claims.

TOK Task #7 – Faith and Intuition


Faith Knowledge
  • something that someone believes in (e.g. religion)
  • something that stems from an opinion
  • something that is learnt from external information → if taught, taught by what is determined to be ‘true’ at the time
  • can distract from reality → belief overtakes what is actually true and prevents someone from understanding/learning the full truth of some form of information
  • influences perception of something → pre-existing bias towards a certain belief causes an obstruction in acquiring information (ignorance)
  • knowledge is not always correct and the absolute truth, so when shared, false knowledge prevents one from understanding information to its truest
  • when sharing/acquiring knowledge, the language used to explain it can be difficult to understand which can result in inaccuracies when acquiring knowledge
Justifications as WOK
  • works collaboratively  with other ways of knowing to allow us to learn (e.g. faith is stored in memories as it is what we believe)
  • allows us to understand things from a certain perspective (e.g. someone sharing information pertaining to their faith and/or beliefs)
  • what we know is derived from knowledge from the past (developments in knowledge allow us to understand more)
  • works collaboratively with other ways of knowing to further enhance our understanding (e.g. cohesive language should be used when sharing information so knowledge can be universally acquired and understood)

TOK Task #6 – Memory & Imagination


Memory Imagination
Role in pursuit of knowledge
  • allows one to utilise prior knowledge to further understand something
  • what we learn is the definition of concepts we have previously understand (refined)
  • allows one to understand something in more vivid imagery to understand it further
  • cognitive biases → affects how we understand and know something when we learn it
  • we fill in the blank spots in our memory with ideas that may not necessarily be true/reliable (may be imagination)
  • faults in memory, problems pertaining to memory (e.g. Alzheimer’s)
  • makes it difficult for someone to fully understand non-abstract concepts (understanding of straightforward concepts being deterred by created ideas)
  • distracts from reality
  • fabricates certain facts when understanding/acquiring knowledge
Link between others
  • sense perception: we do not always recall things exactly the way they were when we originally perceived them
  • emotion: can cause a bias when recalling things from memory (e.g. traumatic experiences having influences on understanding)
  • language: one might not completely remember what someone said word for word, which may cause an alteration in understanding
  • sense perception: what we know allows us to experience imagination more vividly
  • emotion: someone’s mood can affect the way they believe or see something, and could potentially cause them to imagine something instead of actually seeing it
  • reason: almost acts like the counter of imagination, as imagination requires more creativity and reasoning may be more factual and straightforward

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.

In the arts, imagination is best employed to allow one to create something representative of a bigger concept in a visual way that can communicate a deeper message. Furthermore, imagination is also used in the interpretation of arts, where everyone’s individual mind and perceptive abilities derive different meanings from art depending on their personalities, and exposure. Imagination allows people to interpret difficult concepts or abstract concepts easier as it creates a visual aid of sorts to communicate a deeper message. On the contrary, imagination would be much more difficult to apply in a subject such as the natural sciences. In the natural sciences, memory would be a more prominent way of knowing as inventions are usually derived from prior knowledge, which is dependent on one’s memory and understanding. Also, it is important in learning the natural sciences that one understands the root concepts in order to further their understanding so they can apply it in the future. Without this base understanding, one’s scientific knowledge would be quite superficial and not thorough. This indicates that the respective ways of knowledge can not intertwine in between areas of knowledge. However, sometimes, utilising both memory and imagination can be helpful. If the examples of arts and natural sciences are used again, memory allows someone to apply previously learned skills in their art to imagine and thus, create more concepts. In natural sciences, imagination can be helpful when attempting to think out of the box for inventions or thinking about new ideas.

TOK Task #5 – Reasoning

Pure logic is only concerned with the structure of arguments. The validity of an argument is independent of the truth or falsity of its premises.

If something is logical, is it valid? Furthermore, what does it mean to be valid or have information that is valid? In terms of Theory of Knowledge, something that is valid is something that follows two premises and forms a conclusion.

In the natural sciences, validity is interpreted to mean that the result is something that agrees with what the scientific population has proven thus far and is what they believe at that time. Even if logical reasoning and deduction is utilised to make conclusions, the premises do not necessarily have to be true. For example, just because two premises form a sound conclusion does not mean that it is valid.

For example:

Premise 1: Eukaryotes are a type of cell
Premise 2: Eukaryotic cells have a nucleus
Conclusion: All cells have a nucleus.

According to scientific evidence this statement is not true despite the fact that the argument is valid. Both premises are true, yet this does not make the final argument true. This shows that the validity of an argument is independent from the truth or falsity of its premises.

However, on the other hand, the validity of an argument can also be connected to the truth or falsity of its premises.

For example:

Premise 1: The United States has a president.
Premise 2: The president lives in the White House.
Conclusion: The president lives in the White House.

This would show that the validity of an argument is related to the truth or falsity of its premises as in this case, both premises are known to be true. Therefore, the way the statements are interpreted is all dependent on the structure of the language.

TOK Task #4 – Vagueness & Ambiguity of Language

The vagueness and ambiguity of language always limits the production of knowledge.

In pursuit of knowledge, subjectivity and objectivity must come into play. However, what they both have in common is that both perspectives can be vague and ambiguous.

This claim states that vagueness and ambiguity of language always limits the production of knowledge. The term ‘production’ is vague in itself – by production, does one mean to generate knowledge? To convey and distribute knowledge? To interpret knowledge? The statement can be interpreted in many different ways and is still correct.

Being vague or ambiguous with language does not always limit the production of knowledge. For example, it is not necessarily limiting being vague or ambiguous with language in arts. Being vague or ambiguous in the arts allow the audience to let the art influence them in individual ways, which can be argued to be the point of art. A piece of artwork with two stripes painted on it would be considered ambiguous but it could impact people in different ways and produce different opinions and interpretations of it. These productions of knowledge are still valid regardless of the fact that they differ from each other because the art itself was created to be interpreted with influence from every individual’s mind.

On the other hand, being vague or ambiguous with language could limit the production of knowledge as well, for example in the natural sciences. If someone writes the methodology for a lab report with vague terms, the results produced using that method would be invalid because there could have been an error in the execution of the experiment as the method wasn’t clear enough. This could severely impact what humans know to this day and whether something is scientifically possible or not and cause a misinformation spread with humans.

There are limits to being vague and ambiguous as there are advantages to being vague and ambiguous. In the end, both are valid when it comes to the production of knowledge dependent on what the topic is. If the subject is more subjective, then being vague and ambiguous could work in its favor. If the subject is more objective, then being vague and ambiguous could be more harmful and limiting.


TOK Task #3 – Emotion in Science

The sciences are all about using reason to understand the world, there is no place for emotion in science.

There are many different ways emotion can affect the sciences – during the execution of experiments, understanding science etc. To determine how emotion affects sciences, one must understand what emotion entails. Emotion could mean the feelings one experiences when studying science, which can be affected by beliefs, past events, physical environment, incentives, or the feelings in general about studying science.

There is space for emotion in science because one requires passion to study and thus, understand the sciences more effectively. This means that one needs to have the motivation to commit and accurately progress through the learning process of sciences. Without emotion and passion when one studies science, the information will not be completely absorbed and learned to the best of one’s ability, which could prove to be a hinderance during the sharing or teaching of information. Furthermore, in fields such as psychology, emotions and feeling are important when it comes to sympathising, understanding, analysing and ultimately helping someone. It is important to understand that emotions have an impact on actions and to understand this, sometimes exercising emotion will help.

However, it can also be argued that there is no place for emotion in the sciences. Considering the fact that emotions can be influenced by belief, past events, physical environment, and incentives, the information or the study in the sciences can be negatively impacted if one lets emotion affect their thinking. If someone has religious beliefs that affects their view on sciences, for example evolution, this could have a negative effect on the results as it may be deemed unreliable or less valid because of their bias. Since the sciences are about using reason to understand the world, emotions can cause a serious negative influence on the quality of the science whether it be learning it or teaching it.

TOK Task #2 – “Even though there are problems with our perceptual systems, this doesn’t mean that knowledge gained from our senses is completely unreliable”

“Even though there are problems with our perceptual systems, this doesn’t mean that knowledge gained from our senses is completely unreliable.”

Knowledge gained from our senses isn’t completely unreliable despite the fact that there are problems with our perceptual systems, and perhaps that different people have different opinions.

These ‘problems’ with our perceptual systems could range from being deaf, blind etc. in terms of the five senses, but even if so, the use of their other senses cannot be invalidated when perceiving objects or notions as everyone is entitled to their own opinion. Also, the word ‘unreliable’ suggests that something is untrustworthy, but how can something be deemed unreliable if it is someone’s perception? It depends on the knowledge in question. 

This statement works in favour of the arts. Suppose that there is a sculpture and two people are viewing it. One is blind, and the other has senses that are fully functional. The blind person’s perception of the sculpture is not unreliable despite the fact that he cannot see the composition of it, but he can still feel it physically. This provides someone with fully functional senses with an exclusive perspective of how the sculpture feels. Simultaneously, the person with fully functional senses will experience one more sense than the blind person, but there might be an overwhelming amount of perceptive tools and thus, provides a different experience from the blind person. This knowledge is not completely unreliable because it is still a perspective, and it is important for perspectives to be considered when trying to judge something as subjective as art. In this case, perspective could be considered knowledge. 

Furthermore, perspectives are still valued in history, but it should be facts that dictate how our society acts upon different morals and ethics. If a soldier from WW2 gives a testimony, there is no way to define how reliable this is as he could be suffering from PTSD and thus, dramatise some events. This still doesn’t invalidate his testimony even if his perceptual systems were influenced as there is a level of bias, and this bias can be useful when analysing the different sides of the story in history in order for us to establish our own perspective.

Whether knowledge is reliable or not is highly dependent on what the subject matter is. However, the perspective, despite the problems in perceptual systems, is still valid as perspectives allow us to gain insight about the people around us, and that is knowledge in itself. Knowledge can be divided into quantifiable (e.g scientific data, statistics) and unquantifiable (e.g analysis of art) – if it is quantifiable, then the level of unreliability should be more heavily considered, but if it is unquantifiable, then all knowledge is reliable. 

TOK Task #1 – “If you cannot explain something, you do not know it”

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

With reference to the class activity today about knowing and explaining, in what ways might it be reasonable to suggest that people who disagree can both be right?

In reference to the claim itself, it might be reasonable to suggest that people who disagree and agree can both be ‘right’. This is because to be ‘right’ doesn’t always mean to arrive at the same factual definition, and that it pertains to beliefs. For example, some people with Christian beliefs may believe that abortion is wrong and that may be the ‘right’ for people that share the same set of beliefs. However, for atheists, for example, they may believe that abortion is a fair and justified choice, and to other people that share the same belief, this is ‘right’.

However, what does it mean to be ‘right’? Does this mean that one is correct, and that their belief or statement is true? To be ‘right’ could mean to have the current, widely accepted belief about something. To be ‘right’ means that to be appropriate; to know something as morally justified. To be ‘right’ is to follow the common thought that is perceived to be ‘right’ until it is proven opposite, and thus, is not considered ‘right’ anymore. I believe that what classifies as ‘right’ is ultimately what you choose to believe and that you think it is ‘right’. If you believe something is ‘right’, there is a high chance that your belief will not be swayed, because the way you have chosen to interpret something is how you perceive something to be ‘right’.

Therefore, this statement can not be reduced to being ‘right’ or ‘wrong’. In the statement, the language itself provides many opportunities for people to believe if something is ‘right’ or ‘wrong’. To explain is not limited to explaining something correctly and ‘factually’. It means to articulate an idea, even if this idea is absurd. Furthermore, there are limitations in terms of language as to how you can explain – just because you cannot articulate your idea does not mean that you do not know what it is. For example, in North Korean dialect, there is no word for ‘love’ – this does not mean that no one feels love, which counters the claim.

On the other hand, if we look at something as logical as mathematics, someone explaining a maths question needs to know what the question asks, and thus, needs to be able to explain it in order to teach someone else about it. This supports the claim that if you cannot explain it, you do not know it, because mathematics is very logical and one-sided (e.g. follows a set of steps), and if you know how to do it, then you should be able to explain it.

In my opinion, this statement heavily relies on the context given. ‘Knowing’ does not limit to only comprehending how to do something, it extends to meaning, purpose, intention etc. ‘Explaining’ does not limit to describing the truth and the fact. To be ‘right’ is not limited to having one overall, shared, belief, and therefore, people who disagree and agree can both be ‘right’.