Mathematics is sometimes seen to have a degree of certainty that is unmatched by other areas of knowledge or is seen to be founded on a set of more or less universally accepted definitions and basic assumptions. This makes mathematics an excellent source of material for TOK discussions.
One interesting focus for discussions could be the status of mathematics as an area of knowledge. Students could consider why disciplines in the human sciences are often keen to cast their conclusions in mathematical terms, or why mathematical treatments of a topic are often taken by many to be a sign of intellectual rigour. They could also consider why mathematics is often given a privileged position in many education systems. Another rich source of material for TOK discussions can be the role of creativity, imagination, beauty and elegance in mathematics. Despite, or perhaps because of, the strict confines of mathematical logic, mathematics can be an enormously creative subject, asking its practitioners to make great leaps of imagination. This could lead to discussion of whether, or why, elegance and beauty should be relevant to mathematical value.
Another interesting focus could be the relationship between mathematics and the world around us. Mathematics is often used to model real-world processes. Yet, in some ways, mathematics can also seem quite abstract and detached from the real world, strongly focused on the application of reason rather than relying on experience and observation of the world.
Students could also consider the role and significance of proof in mathematics, and how this relates to concepts such as truth. They could reflect on whether the term “proof” is used differently in mathematics compared to how it is used in our everyday lives or in other areas of knowledge.
Examples of knowledge questions arising from this area of knowledge are suggested below.
• Why is mathematics so important in other areas of knowledge, particularly the natural sciences?
• How have technological innovations, such as developments in computing, affected the scope and nature of mathematics as an area of knowledge?
• Is absolute certainty attainable in mathematics?
• Is there a distinction between truth and certainty in mathematics?
• Should mathematics be defined as a language?
• Is mathematics better defined by its subject matter or its method?
• Does mathematics only yield knowledge about the real world when it is combined with other areas of knowledge?
• Is there a hierarchy of areas of knowledge in terms of their usefulness in solving problems?
Core theme:
• Why do you think mathematics enjoys a privileged status in many education systems? (scope)
• What is it about mathematics that enables mathematical results to remain unchanged over time?
• How significant have notable individuals been in shaping the nature and development of mathematics as an area of knowledge?
• What is the role of the mathematical community in determining the validity of a mathematical proof?
• Is mathematical knowledge embedded in particular cultures or traditions?
• Does personal experience play any role in the formation of claims in mathematics?
• Is progress harder to make in mathematics than in other areas of knowledge?
• If mathematics is created by humans, is it still possible to accept mathematical truths as objective facts about the world?
• Are all of the areas of knowledge in the TOK course themselves embedded in a particular tradition or bound to a particular culture?
Core theme:
• Who judges the validity of a proof? (perspectives)
• Is mathematical reasoning different from scientific reasoning or reasoning in other areas of knowledge?
• What is meant by the term “proof” in mathematics, and how is this similar to, or different from what is meant by this term in other areas of knowledge?
• How do mathematicians reconcile the fact that some conclusions seem to conflict with our intuitions?
• What does it mean to say that mathematics is an axiomatic system?
• How is an axiomatic system of knowledge different from, or similar to, other systems of knowledge?
• Do mathematical symbols have meaning in the same way that words have meaning? Is personal experience more important or less important in mathematics compared to other areas of knowledge?
Core theme:
• What steps can we take to help ourselves avoid being misled by statistics used in unclear or disingenuous ways in the media? (methods and tools)
• If mathematical knowledge is highly valued, does this place special ethical responsibilities on mathematicians when they are making claims?
• On what criteria could we decide whether mathematicians should be held responsible for unethical applications of their work?
• How are unethical practices, such as “data dredging”, used by statisticians to deliberately manipulate and mislead people?
• Is it ethically justifiable for academic mathematicians to spend time doing research that does not have immediate useful applications?
• Do mathematical judgments and ethical judgments face similar challenges in terms of the evidence available to support them?
• Are mathematicians the people best placed to create codes of ethics for professional mathematicians?
Core theme:
• To what extent do you agree with the claim that mathematics “serves as a training that shapes thinking in an ethics-free and amoral way” (Paul Ernest)? (ethics)
As part of their theory of knowledge course, students are encouraged to explore tensions relating to knowledge in mathematics. As an area of knowledge, mathematics seems to supply a certainty perhaps impossible in other disciplines and in many instances provides us with tools to debate these certainties. This may be related to the “purity” of the subject, something that can sometimes make it seem divorced from reality. Yet mathematics has also provided important knowledge about the world and the use of mathematics in science and technology has been one of the driving forces for scientific advances. Despite all its undoubted power for understanding and change, mathematics is in the end a puzzling phenomenon. A fundamental question for all knowers is whether mathematical knowledge really exists independently of our thinking about it. Is it there, “waiting to be discovered”, or is it a human creation? Indeed, the philosophy of mathematics is an area of study in its own right. Students’ attention should be drawn to questions relating theory of knowledge (TOK) and mathematics, and they should also be encouraged to raise such questions themselves in both their mathematics and TOK classes. Examples of issues relating to TOK are given in the “Connections” sections of the syllabus.
International Baccalaureate Organisation. Theory of Knowledge First Assessment 2022. Wales, International Baccalaureate Organization, Feb. 2020.
CERTAINTY: As far as the laws of mathematics refer to reality, they are not certain, and as far as they are certain, they do not refer to reality. Albert Einstein
CULTURE: Mathematics knows no races or geographic boundaries; for mathematics, the cultural world is one country. David Hilbert
JUSTIFICATION: Thus, in a sense, mathematics has been most advanced by those who distinguished themselves by intuition rather than by rigorous proofs. Felix Klein
TRUE: Mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true. Bertrand Russell
POWER: Mathematics is a place where you can do things which you can't do in the real world. Marcus du Sautoy
Key thinkers
Daniel Tammett - ways of knowing Tammet’s talk reminds us that the mind has many different ways of seeing and understanding the world.
Hannah Fry - mathematics of love Fry discusses whether we can apply mathematical rules to understand the most human of feelings - love.
Jean-Baptiste Michel - mathematics of history Michel’s talk proves that we can draw on seemingly incongruous knowledge (mathematics) in order to make sense of another field (history).
Eugenia Cheng - an unexpected tool for understanding inequality Cheng discusses the way in which abstract mathematics can help us to understand why we get angry, and the why privilege exists.
A Wired article, looking at the flawed assumptions behind the algorithm that led to major problems with UK A-Level results.
This is one of the biggest science stories of the last few years: the images of the M87 Black Hole, created from mathematical data created by telescopes positioned around the world. See Time, The Guardian, Vox, and The Washington Post.
This Wired article looks at how a ‘predictive policing tool’ is being used by the police in the UK to calculate the risks posed by individuals, and determine the actions taken against them by the authorities.
A Vox article, showing you the mathematics behind Covid-19, and why this is essential to understanding the pandemic.
An Aeon video, looking at why Bayes’ Theorem has an increasing relevance in the world of ‘big data’, and how it can help us to “reprogramme our intuition”.
This short Aeon video by Cathy O’Neil ponders the nature of algorithms, and challenges the idea that they represent “indisputable mathematical truths”.
Kevin Slavin - algorithms Few of us realize the extent to which algorithms control our lives. This talk provides some impactful real life examples to demonstrate the implications of the situation.
An Al Jazeera article, considering whether we should trust algorithms. For suggestions on how to further unpack this question, follow this link.
A Wired article, looking at how algorithms underpin so many different aspects of modern life, and help determine ethical decisions. For suggestions on exploring this source, click here.
Dunn, Michael. “Theoryofknowledge.net.” Theoryofknowledge.net, theoryofknowledge.net/. Accessed 9 Feb. 2021.
International Baccalaureate Organisation. Mathematics: analysis and approaches guide . Wales, International Baccalaureate Organization, Nov. 2020.
International Baccalaureate Organisation. Mathematics: applications and interpretation guide First assessment 2021. Wales, International Baccalaureate Organization, Nov. 2020.
Image:
Towards data science. “Image Illustrating ‘Why Is Mathematics Vital to Thrive in Your AI Career?,’” Towards Data Science, 15 Jan. 2020, miro.medium.com/max/780/1*1CziRAeRnF4UsCrhryH9yw.png.