Animated Mathematics

A Mathematical formulation of TOK

Students and teacher when they first encounter TOK (Theory of Knowledge) are confused. This should not be seen as something negative, as confusion is often an indication that you have been mentally stimulated. There are many excellent articles that introduce TOK, and most of them are heavy reading and have probably been written by language/history teachers. Here I shall give a mathematical formulation of TOK.

Events that Create Knowledge

Human beings create some knowledge K during an event E. The event could be a reaction to a real-life situation (RLS). The person uses the ways of knowing (WOK) which are intuition (i), imagination (m), language (l), reason (r), emotion (e), faith (f), senses (s), and memory (m). This can be written as


The knowledge K can then be classified as falling into the one (or more) of the areas of knowledge (AOKs) which are the arts, ethics, history, the human sciences, indigenous knowledge systems, mathematics, the natural sciences, and religious knowledge systems. In reality, there are many events and every person on the planet contributes to human knowledge in some way. Such a complex system is extremely difficult to analyze, but it is useful to look at some simple examples.

Examples of Simple Events

Event Intuition Imagination Language Reason Emotion Faith Senses Memory AOKs
Ice denser that water 0% 0% 5% 40% 5% 0% 40% 5% Science
Find railway station 0% 0% 5% 5% 0% 0% 40% 50% Human Science
Is Painting beautiful 25% 25% 5% 5% 25% 5% 5% 5% Arts
wear good luck charm 25% 25% 5% 5% 25% 5% 5% 5% religious/indigenous Knowledge

The above table considers four relatively simple events, and the relative proportion of WOKs used to create knowledge; the proportions are only estimates and different people are likely to make different estimates. The first line considers is ice denser than water? In practice, one would have measure the volume and mass of ice and water and then calculate the density. The measurements would require using scientific instruments which will require the application of one’s senses. The density calculation would involve reason and memory. The conclusion that water is denser than ice would be true for everyone irrespective of how they feel, and this is an example of objective knowledge.

The second line considers a person who wishes to make their way to the railway station. A person who is familiar with the area will just use memory and then their senses to navigate during the journey, and there could also be some language used when reading street names. Persons who are unfamiliar with a city would most likely use google maps. The location of the railway station is the same for everyone so this is also an example of objective knowledge.

The third questions "is the painting beautiful?" is matter of personal opinion and the conclusion will vary; and it is reasonable to assume that different people will use a different proportion of AOKs to reach their conclusion. This is an example of subject knowledge, ince the answer is not the same for everyone i.e. "beauty is in the eye of the beholder".

The fourth question "should one wear a good luck charm?" is in the realm of magic and/or religion. Many indigenous cultures around the world have a tradition of good luck charms linked to varies deities, and often these charms have found their way into formal religion. The supposed/perceived value of good luck charms varies from individual to individual, and is another example of subjective knowledge.

Knowledge can be roughly divided into objective and subjective knowledge, the latter is usually associated with science and mathematics, and the former with the arts, religions, and other non-scientific diciplines.

The Presentation

A real-life situation (RLS) is the starting point of a TOK presentation. The student is free to choose a RLS and I would encourage you to choose one which you find personally interesting and to avoid clichés RLS that are temporarily popular. The third line in the table "is a painting beautiful" can be an interesting real-life situation. An appropriate knowledge question could be "when judging paintings is beauty in the eye of the beholder". A student can then use the above table to analyze how individuals judge the merits of a painting. The WOK of knowing is the key to obtaining a second order knowledge question which can often involve the interaction and interplay of the induvial WOKs. An appropriate second order knowledge question could be "which WOKs are relevant to judging beauty". A second order knowledge can be helpful in considering different perspectives of the knowledge question, relevant to the original RLS and other RLS. In practice, the student is required to create an argumentative presentation which has similar structure to an arguementative essay.

The Essay

The essay roughly has a reverse structure to the presentation. The IB provides essay prompts which are often second order knowledge questions. The student is then required to determine relevant knowledge questions. A WOK table analyzing the essay prompts can be one strategy to find relevant knowledge questions. The essay is recommended to have a structure involving claims and counterclaims which is in essence an arguementative essay.


TOK has been described by making an analogy with a mathematical function, such as the area A of a rectangle with width w and length l, where A(w,l)=wl. The width does not depend on the length since it can be any positive values of w and l. An event E that creates knowledge is a function of the eight WOKs. However, this does not mean the different WOKs are independent of each other, and according to the event there could be a relation between different WOKs. Moreover, perhaps the different WOKs somehow need each other to operate effectively, we consider some examples.

The dispute between the Pope and Galileo Galilei is well known. Galileo was imprisoned for heresy for suggesting that the Earth was not the center of the universe. This episode in history can be interpreted as the struggle between faith and reason. This off course does not mean that faith and reason are opposed to each other, it may be a social contruct of Christian civilization. In the Eastern traditions, there is no heresy and generally science and religion have not clashed historically.

The so-called clash between reason and faith is ongoing, and major front in this war has been opened by the author Richard Dawkins through his provocatively titled book "The god delusion". The tacit assumption by many is that science is closely aligned with reason, and reason is superior to the other WOKs and this often generates a certain amount of intellectual snobbery from scientists. The irony is that the greatest physicist in history Albert Einstein was not a proponent of reason only; to quote Einstein "The intuitive mind is a sacred gift and the rational mind is a faithful servant. We have created a society that honors the servant and has forgotten the gift". Einstein used imagination and intution as much as reason when he formulated the theory of special relativity which revolutionized the interpretation of space and time. He showed using mathematics not too beyond grade 10 that both space and time were not absolute but dependant on the observer's point of view. Art and music can stimulate a person's intuition and imagination, and it might be possible that both subjects can be just important as the sciences and mathematics in education.